EQUATIONS of STATE for FLUIDS and FLUID MIXTURES

E X P E R I M E N T A L T H E R M O D Y N A M I C S V O L U M E V

EXPERIMENTAL THERMODYNAMICS SERIES:

Calorimetry of Non-reacting Systems EXPERIMENTAL THERMODYNAMICS VOLUME I Edited by J.P. McCullough and D.W. Scott Butterworths, London, 1968

Experimental Thermodynamics of Non-reacting Fluids EXPERIMENTAL THERMODYNAMICS VOLUME II Edited by B. Le Neindre and B. Vodar Butterworths, London, 1975

Measurement of the Transport Properties of Fluids EXPERIMENTAL THERMODYNAMICS VOLUME III Edited by W.A. Wakeham, A. Nagashima and J.V. Sengers Blackwell Scientific Publications, Oxford, 1991

Solution Calorimetry EXPERIMENTAL THERMODYNAMICS VOLUME IV Edited by K.N. Marsh and P.A.G. O'Hare Blackwell Scientific Publications, Oxford, 1994

Equations of State for Fluids and Fluid Mixtures EXPERIMENTAL THERMODYNAMICS VOLUME V Edited by J.V. Sengers, R.F. Kayser, C.J. Peters, and H.J. White, Jr. Elsevier, Amsterdam, 2000

Measurement of Thermodynamic Properties of Single Phases EXPERIMENTAL THERMODYNAMICS VOLUME VI Edited by A.R.H. Goodwin, K.N. Marsh, and W.A. Wakeham Elsevier, Amsterdam, in press

Measurement of Thermodynamic Properties of Multiple Phases EXPERIMENTAL THERMODYNAMICS VOLUME VII Edited by R.D. Weir and T.W. de Loos Elsevier, Amsterdam, in press

International Union of Pure and Applied Chemistry Physical Chemistry Division Commission on Thermodynamics

EQUATIONS of STATE for FLUIDS and FLUID MIXTURES

P a r t I

E X P E R I M E N T A L T H E R M O D Y N A M I C S

V O L U M E V

E D I T E D B Y

J.V. S E N G E R S

Institute for Physical Science and Technology and Department of Chemical Engineering, University of Maryland, College Park, MD 20742, U.S.A.

and

Physical and Chemical Properties Division National Institute of Standards and Technology, Gaithersburg, AID 20899, U.S.A.

R.F. K A Y S E R Technology Services National Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A.

C.J. P E T E R S Laboratory of Applied Thermodynamics and Phase Equilibria, Faculty of Applied Sciences, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands

H.J. W H I T E , Jr. Institute for Physical Science and Technology and Department of Chemical Engineering, University of Maryland, College Park, MD 20742, U.S.A.

2 0 0 0

E L S E V I E R

Amsterdam - Lausanne - New Y o r k - Oxford - Shannon - S i n g a p o r e - Tokyo

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CONTENTS

List of Contributors xvii Preface xix

PART I

Introduction J. V. Sengers, R.F. Kayser, C.J. Peters, and H.J. White, Jr.

Fundamental Considerations

M.B. Ewing and C.J. Peters

2.1 2.2

2.3

2.4

2.5 2.6 2.7 2.8 2.9 2.10

2.11

Introduction 6 Basic Thermodynamics 6 2.2.1 Homogeneous Functions 9 2.2.2 Thermodynamic Properties from Differentiation of Fundamental Equations 10 Deviation Functions 11 2.3.1 Residual Functions 14 2.3.2 Evaluation of Residual Functions 15 Mixing and Departure Functions 15 2.4.1 Departure Functions with Temperature, Molar Volume and Composition 15

as the Independent Variables 2.4.2 Departure Functions with Temperature, Pressure and Composition as the

Independent Variables 19 Mixing and Excess Functions 20 Partial Molar Properties 22 Fugacity and Fugacity Coefficients 23 Activity Coefficients 25 The Phase Rule 26 Equilibrium Conditions 27 2.10.1 Phase Equilibria 27 2.10.2 Chemical Equilibria 29 Stability and the Critical State 30 2.11.1 Densities and Fields 30 2.11.2 Stability 30 2.11.3 Critical State 31

The Virial Equation of State 3 5 J.P.M. Trusler

3.1 Introduction 3.1.1 The Volumetric Behaviour of Real Fluids 3.1.2 The Virial Equation of State 3.1.3 Temperature Dependence of the Virial Coefficients 3.1.4 Composition Dependence of the Virial Coefficients 3.1.5 The Pressure Series 3.1.6 Convergence of the Virial Series

36 36 37 38 38 40 40

vi

3.2

3.3

3.4

3.5

3.2.1 3.2.2 3.2.3 3.2.4 Theory 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5

Thermodynamic Properties of Gases Perfect-gas and Residual Properties Helmholtz Energy and Gibbs Energy Perfect-Gas Properties Residual Properties

Properties of the Grand Canonical Partition Function The Virial Equation of State from the Grand Partition Function The Virial Coefficients in Classical Mechanics Quantum Corrections The Virial Coefficients in Quantum Mechanics: Helium and Hydrogen

Calculation and Estimation of Virial Coefficients 3.4.1 The Virial Coefficients of Model Systems 3.4.2 Estimation of Virial Coefficients Summary

41 41 42 43 47 48 48 49 51 55 60 67 67 70 72

Cubic and Generalized van der Waals Equations A. Anderko

4.1

4.2 4.3 4.4

4.5 4.6

4.7

Cubic Equations of State for Pure Components 4.1.1 Historical Perspective 4.1.2 Temperature Dependence of Parameters 4.1.3 Functional Form of the Pressure- Volume Relationship Equations Based on the Generalized van der Waals Theory Simple Equations Inspired by the Generalized van der Waals Theory Methods for Extending Equations of State to Mixtures 4.4.1 Classical Quadratic Mixing Rules 4.4.2 Composition-Dependent Combining Rules 4.4.3 Density-Dependent Mixing Rules 4.4.4 Combining Equations of State with Excess-Gibbs-Energy Models Explicit Treatment of Association in Empirical Equations of State Equations of State as Fully Predictive Models 4.6.1 Correlations for Binary Parameters 4.6.2 Group-Contribution Equations of State 4.6.3 Utilization of Predictive Excess-Gibbs-Energy Models Closing Remarks

75

76 76 79 82 86 92 94 94 98

100 102 107 116 116 117 119 119

Perturbation Theory T. Boublik

5.1 5.2 5.3

Introduction Basic Concepts of Perturbation Theory Perturbation Theories of Pure Simple Fluids 5.3.1 Van der Waals Equation of State 5.3.2 Equations of State and Radial Distribution Functions of Hard Spheres 5.3.3 Perturbation Expansion for Fluids with Infinitely Steep Repulsions 5.3.4 Second-Order Perturbation Term 5.3.5 Perturbation Expansion for Fluids with Soft Repulsions 5.3.6 Quantum Effects

127

128 131 133 133 136 140 142 144 148

5.4

5.5

5.6

5.7

vii

Mixtures of Simple Fluids 148 5.4.1 Mixtures of Hard Spheres 148 5.4.2 Perturbation Expansion for Mixtures with Infinitely Steep Repulsions 150 5.4.3 Perturbation Expansion for Mixtures with Soft Repulsions 150 Perturbation Theories of Molecular Fluids 151 5.5.1 Pair Potential of Molecular Fluids 151 5.5.2 Equations of State and Distribution Function of Hard Nonspherical 154

Molecules 5.5.3 Perturbation Expansion for Pure Molecular Fluids 157

5.5.3.1 Fluids of KiharaMolecules 157 5.5.3.2 Fluids of Molecules with Multicenter Pair Potential 159 5.5.3.3 Fluids of Molecules with Electrostatic Interactions 160

5.5.4 Water 163 Mixtures of Molecular Fluids 163 5.6.1 Mixtures of Nonelectrolytes 163 5.6.2 Mixtures of Electrolytes and Aqueous Solutions 165 Conclusions 166

Equations of State from Analytically Solvable Integral-Equation Approximations Yu. V. Kalyuzhnyi and P. T. Cummings

6.1

6.2

6.3

169

6.3.2

6.3.3

Introduction 170 6.1.1 Ornstein-Zernike Equation and Closure Relations 174 6.1.2 Short Review of Recent Progress in Integral-Equation Theory 176 Baxter/Wertheim Factorization of the Ornstein-Zernike Equation 179 6.2.1 One-Component Monatomic Fluids 180 6.2.2 Multi-Component Monatomic Fluids 182 6.2.3 Remarks 183 Equation of State for Analytically Solvable Models of Simple Fluids 183 6.3.1 Hard-Sphere Fluid 183

6.3.1.1 Solution of the PYA for the Hard-Sphere Fluid and Fluid Mixtures 184 6.3.1.2 Equation of State for Hard-Sphere Fluid and Fluid Mixtures 186 6.3.1.3 Semiempirical Correction to the PYA: The Generalized MSA 188 Adhesive Hard-Sphere Fluid 189 6.3.2.1 Solution of the PYA for the Adhesive Hard-Sphere Fluid and

Fluid Mixtures 190

6.3.2.2 Equation of State for the Adhesive Hard-Sphere Fluid and Fluid Mixtures 192

6.3.2.3 Percus-Yevick Theory for the Hard-Sphere Chain Fluid 195 Hard-Core Yukawa Fluids 196 6.3.3.1 Solution of the MSA for the Hard-Core Yukawa Fluid and Fluid

Mixtures 196

6.3.3.2 Equation of State for the Hard-Core Yukawa Fluid and Fluid Mixtures 201

6.3.4 Charged Hard-Sphere Fluids 204 6.3.4.1 Solution of the MSA for the Primitive Model of Electrolyte 204

Solutions 6.3.4.2 Equation of State for the Primitive Model of Ionic Fluids and

Fluid Mixtures 207

6.3.4.3 Corrections to the MSA: the Generalized MSA and the Associative MSA 209

viii

6.4

6.5

6.6

Equation of State for Analytically Solvable Models of Molecular Fluids 212 6.4.1 Interaction-Site Model Fluids 213

6.4.1.1 Site-Site Ornstein-Zernike Equation and Closure Conditions 214 6.4.1.2 SSOZ-PYA for the Fluid of Fused Hard-Sphere Diatomic

216 Molecules

6.4.1.3 SSOZ-MSA for the Fluid of Fused Charged Hard-Sphere Diatomic Molecules 218

6.4.1.4 Corrections to the SSOZ Formalism: Chandler-Silbey-Ladanyi 219 Equation

6.4.2 Hard Spheres with Dipolar Interactions 222 6.4.2.1 Analytic Solution of MSA for Dipolar Hard-Sphere Fluid 223

Equation of State for Analytically Solvable Models of Associating Fluids 228 6.5.1 Two-Density OZ Equation and Closure Conditions: Relation to the CLS

229 Equation

6.5.2 Two-Density PYA for Dimerizing Hard-Sphere Models 233 6.5.3 Two-Density PYA for the Smith-Nezbeda Primitive Model of Associating 237

Fluids 6.5.4 Two-Density MSA: RPM of Electrolytes and Dimerizing HCY Fluid 239 Conclusion 241

Quasilattice Equations of State for Molecular Fluids N.A. Smirnova and A. V. Victorov

7.1 7.2 7.3 7.4 7.5 7.6

255

Introduction 256 Basic Features of Lattice Theories; Structure of a Quasilattice EOS 257 Influence of Molecular Size and Shape 259 Contact-Site Models for Fluids with Strong Directional Attractive Interactions 264 Results of Thermodynamic Modeling by the Quasilattice EOS 270 Conclusions 282

The Corresponding-States Principle J.F. Ely and I.M.F. Marrucho

8.1 8.2 8.3

8.4

8.5 8.6

Introduction Theoretical Considerations Determination of Shape Factors 8.3.1 Other Reference Fluids 8.3.2 Exact Shape Factors 8.3.3 Shape Factors from Generalized Equations of State Mixtures 8.4.1 Van der Waals One-Fluid Theory 8.4.2 Mixture Corresponding-States Equations Applications of the Extended Corresponding-States Theory Conclusions

289

290 292 295 296 298 302 306 308 309 310 316

Mixing and Combining Rules S.L Sandler and H. Orbey

9.1

9.2

9.3 9.4

Mixing Rules for Cubic and Related Equations of State 9.1.1 The Van der Waals One-Fluid Model 9.1.2 Non-Quadratic Combining Rules for Van der Waals One-Fluid Model 9.1.3 Mixing Rules that Combine an Equation of State with an

Activity-Coefficient Model 9.1.3.1 Huron-Vidal Model 9.1.3.2 Wong-Sandler Model 9.1.3.3 Combination of Excess-Energy Models and Equations of State

at Low or Zero Pressure 9.1.4 Commentary on Mixing Rules for Cubic Equations of State Mixing Rules for the Virial Family of Equations of State 9.2.1 The Van der Waals One-Fluid Model 9.2.2 Generalized Extended Virial Equations of State and Their Mixing Rules Mixing Rules Based Upon Theory and Computer Simulation Conclusions

ix

321

323 323 327

330

330 332

340

346 346 346 348 352 354

10 Mixtures of Dissimilar Molecules E. Matteoli, E.Z Hamad, and G.A. Mansoori

10.1 10.2 10.3 10.4

10.5 10.6

10.7 10.8 10.9

Introduction Background Grand Canonical Ensemble Analytic Theory of Dissimilar Mixtures 10.4.1 Analytic Solutions for Low to Moderate Pressures Binary Mixtures Test of C 0 Closure 10.6.1 Hard-Sphere Mixtures 10.6.2 Lennard-Jones Mixtures 10.6.3 Real Mixtures Vapor-Liquid Equilibria Liquid-Liquid Equilibria Summary

359

360 360 361 363 364 366 367 367 368 368 372 376 378

11 Critical Region M.A. Anisimov and J. V. Sengers

ll.1 11.2 11.3 11.4 11.5 ll.6

11.7 11.8 11.9 11.10

Introduction Critical Region in the Van der Waals Theory Asymptotic Scaled Equation of State Corrections to the Asymptotic Scaled Equation of State Crossover between Scaling and Classical Asymptotic Critical Behavior Crossover between Scaling and Classical Nonasymptotic Critical Behavior in the Extended Critical Region Global Crossover Equation of State for One-Component Fluids Isomorphic Scaled Equation of State for Near-Critical Binary Fluids Renormalization of Critical Exponents Isomorphic Description of Fluid-Fluid Phase Equilibria

381

382 383 386 396 401

406

412 414 419 421

11.11 Crossover between Vapor-Liquid and Liquid-Liquid Critical Phenomena 11.12 Summary and Outlook

424 427

12

PART II

Associating Fluids and Fluid Mixtures E.A. Mfiller and K.E. Gubbins

12.1

12.2

12.3 12.4

12.5 12.6

Introduction 12.1.1 Associating Fluids 12.1.2 Experimental Evidence of Association Chemical Theories 12.2.1 Chemical Theory 12.2.2 Coupling of Equations of State with Chemical Approaches 12.2.3 Quasi-Chemical Theory and Group-Contribution Methods Statistical Mechanical Theories Wertheim's Theory 12.4.1 Brief Account of the Theory

12.4.1.1 Molecules with One Site 12.4.1.2 Mixtures of Molecules with Multiple Associating Sites

12.4.2 Tests of the Theory against Molecular Simulations 12.4.2.1 Hard Spheres with Two Association Sites 12.4.2.2 Mixtures of Associating Lennard-Jones Spheres

12.4.3 Limitations ofthe TPT1 Theory 12.4.4 Extensions and Applications to Complex Fluids

12.4.4.1 TPT2 Chains and Polymers 12.4.4.2 Inter and Intramolecular Bonding 12.4.4.3 Water and Electrolytes 12.4.4.4 Closed-loop Liquid-liquid Immiscibility 12.4.4.5 Amphiphilic Systems 12.4.4.6 Associating Liquid Crystals 12.4.4.7 Inhomogeneous Fluids

The Saft EOS and Applications Conclusions

435

436 436 438 441 441 441 443 444 445 445 445 447 447 448 449 450 451 451 453 455 456 457 459 460 460 466

13 Polydisperse Fluids D. Browarzik and H. Kehlen

13.1

13.2

Continuous Thermodynamics 13.1.1 Influence of Polydispersity on Phase Equilibrium 13.1.2 Approaches to Polydispersity 13.1.3 Fundamentals of Continuous Thermodynamics 13.1.4 Generalized Formalism of Continuous Thermodynamics 13.1.5 Analytical Distributions 13.1.6 Advantages of Using Continuous Thermodynamics Application of Continuous Thermodynamics to Equations of State 13.2.1 Fugacity Coefficients 13.2.2 Phase Equilibrium 13.2.3 Flash Problem

479

480 480 482 485 490 493 494 497 497 499 502

13.3 Critical Review of Past Work and Future Challenges 13.3.1 Description of Polydispersity and Numerical Procedures 13.3.2 Phase Equilibria of Real Systems 13.3.3 Stability of Polydisperse Fluids 13.3.4 Conclusions

xi

504 505 508 515 518

14 Equations of State for Polymer Systems S.M. Lambert, Y. Song, and J.M. Prausnitz

14.1

14.2

14.3

14.4

Introduction 14.1.1 Overview 14.1.2 Organization Equations of State for Pure Polymer Liquids 14.2.1 Cell Models

14.2.1.1 Prigogine's Cell Model (PCM) 14.2.1.2 Flory-Orwoll-Vrij (FOV) 14.2.1.3 Perturbed-Hard-Chain (PHC) 14.2.1.4 Further Modifications and Discussion

14.2.2 Lattice-Fluid (LF) Models 14.2.2.1 Sanchez-Lacombe (SL) 14.2.2.2 Costas and Sanctuary (CS) 14.2.2.3 Panayiotou-Vera (PV) 14.2.2.4 Mean-Field Lattice-Gas (MFLG) 14.2.2.5 Further Modifications and Discussion

14.2.3 Hole Models 14.2.3.1 Simha-Somcynsky (SS) 14.2.3.2 Further Modifications and Discussion

14.2.4 Tangent-Sphere Models 14.2.4.1 Generalized Flory Theories 14.2.4.2 Statistical Associated-Fluid Theory (SAFT) 14.2.4.3 Perturbed Hard-Sphere-Chain (PHSC)

Extension to Mixtures 14.3.1 Flory-Orwoll-Vrij (FOV) 14.3.2 Perturbed-Hard-Chain (PHC) 14.3.3 Lattice-Fluid (LF) 14.3.4 Mean-Field Lattice-Gas (MFLG) 14.3.5 Simha-Somcynsky Hole Model 14.3.6 Statistical Associated-Fluid Theory (SAFT) 14.3.7 Perturbed Hard-Sphere-Chain (PHSC) 14.3.8 Further Developments of Sphere-Chain Models 14.3.9 Cubic Equations of State for Polymer Mixtures 14.3.10 Specific Interactions Conclusion

523

524 524 524 524 525 525 527 528 529 530 530 531 533 534 535 536 536 538 539 54O 543 544 547 547 551 553 559 56O 563 567 574 575 577 582

15 Self-Assembled Systems R. Nagarajan and E. Ruckenstein

15.1 15.2

Introduction Thermodynamic Principles of Self-Assembly 15.2.1 Multicomponent Solution Model

589

592 595 595

xii

15.3 15.4

15.5

15.6

15.2.2 15.2.3 15.2.4 15.2.5 15.2.6

15.2.7 15.2.8 15.2.9

Modeling Self-Assembly from a Kinetic Perspective Pseudophase Model of Aggregation Estimation of Critical Micelle Concentration and Micelle Size Bounds on CMC and Micelle Size Size Dispersion of Micelles and Concentration Dependence of Micelle Size Micelle Charge Concentration of Singly Dispersed Surfactant Beyond the CMC Sphere-to-Rod Transition

Molecular Packing in Self-Assembled Structures Self-Assembly in Aqueous Solutions 15.4.1 Model of the Standard Gibbs-Energy Difference for Aggregates

15.4.1.1 Transfer Gibbs Energy of the Surfactant Tail 15.4.1.2 Deformation Gibbs Energy of the Surfactant Tail 15.4.1.3 Aggregate Core-Water Interfacial Gibbs Energy 15.4.1.4 Head-Group Steric Interactions 15.4.1.5 Head-Group Dipole Interactions 15.4.1.6 Head-Group Ionic Interactions

15.4.2 Computational Approach 15.4.2.1 Maximum-Term Method 15.4.2.2 Calculations for Rodlike Micelles

15.4.3 Molecular Constants for Surfactants 15.4.4 Influence of Gibbs-Energy Contributions on Aggregation Behavior 15.4.5 Influence of Tail and Head Groups on Aggregation Behavior 15.4.6 Influence of Ionic Strength on Aggregation Behavior 15.4.7 Influence of Temperature on Aggregation Behavior 15.4.8 Transition from Spherical to Rodlike Micelles 15.4.9 Formation of Bilayer Vesicles Micelles with Poly(ethylene oxide) Head Groups 15.5.1 Approach to Modeling Head-Group Interactions 15.5.2 Uniform Concentration Model

15.5.2.1 Head-Group Mixing Gibbs Energy 15.5.2.2 Head-Group Deformation Gibbs Energy 15.5.2.3 Aggregate Core-Water Interfacial Gibbs Energy

15.5.3 Non-uniform Concentration Model 15.5.3.1 Head-Group Mixing Gibbs Energy 15.5.3.2 Head-Group Deformation Gibbs Energy

15.5.4 Predicted Aggregation Behavior Self-Assembly of Surfactant Mixtures 15.6.1 Size and Composition Distribution of Aggregates 15.6.2 Gibbs Energy of Formation of Mixed Micelles

15.6.2.1 Transfer Gibbs Energy of Surfactant Tail 15.6.2.2 Deformation Gibbs Energy of the Surfactant Tail 15.6.2.3 Aggregate Core-Water Interfacial Gibbs Energy 15.6.2.4 Head-Group Steric Interactions 15.6.2.5 Head-Group Dipole Interactions 15.6.2.6 Head-Group Ionic Interactions 15.6.2.7 Gibbs Energy of Mixing of Surfactant Tails

15.6.3 Predictions of Molecular Theory 15.6.3.1 Nonionic Hydrocarbon-Nonionic Hydrocarbon Mixtures 15.6.3.2 Ionic Hydrocarbon-Ionic Hydrocarbon Mixtures 15.6.3.3 Ionic Hydrocarbon-Nonionic Hydrocarbon Mixtures

596 597 598 598

600

603 604 604 606 610 610 610 611 611 612 613 614 615 615 616 617 618 620 625 626 628 630 631 632 632 633 634 635 636 636 637 637 641 642 644 644 645 645 646 646 647 648 648 649 650 652

xiii

15.6.3.4 Anionic Hydrocarbon-Cationic Hydrocarbon Mixtures 659 15.6.3.5 Anionic Fluorocarbon-Nonionic Hydrocarbon Mixtures 661 15.6.3.6 Anionic Hydrocarbon-Anionic Fluorocarbon Mixtures 666

15.7 Surfactant Self-Assembly in Non-Polar Media 667 15.7.1 Gibbs Energy of Aggregation 668

15.7.1.1 Interaction between Head Groups 670 15.7.1.2 Quasi-Chemical Bonding between Head Groups 670 15.7.1.3 Steric Interactions between Surfactant Tails 671 15.7.1.4 Tail-Solvent Interactions 672

15.7.2 Predictions from the Model 672 15.7.2.1 Gibbs-Energy Contributions 673 15.7.2.2 Size Distribution of Aggregates 673 15.7.2.3 Influence of Surfactant Molecular Structure 677 15.7.2.4 Influence of Temperature and Solvent Polarity 681

15.8 Self-Assembly of Surfactants in Polar Non-Aqueous Solvents 683 15.8.1 Gibbs Energy of Aggregation 683 15.8.2 Prediction of Solution Behavior 684

15.8.2.1 Aggregate Size Distribution 685 15.8.2.2 Critical Micelle Concentration 686 15.8.2.3 Aggregate Polydispersity and Concentration-Dependent 686

Aggregate Size 15.8.2.4 Gibbs-Energy Contributions 688 15.8.2.5 Aggregation Behavior ofAlkyl Pyridinium Bromides 691 15.8.2.6 Aggregation Behavior ofAlkyl Trimethyl Ammonium Bromides 693 15.8.2.7 Comparison with Experimental Observations in the Literature 694

15.8.3 Micellization in Mixed Solvent System of Water-Ethylene Glycol 695 15.8.3.1 Aggregation Behavior of Cetyl Pyridinium Bromide 697

15.9 Solubilization in Surfactant Aggregates 698 15.9.1 Size Distribution of Micelles Containing Solubilizates 701 15.9.2 Gibbs Energy of Solubilization 702

15.9.2.1 Micelle Core-Water Interfacial Gibbs Energy 702 15.9.2.2 Head-Group Interactions for Poly(ethylene oxide) Surfactants 704 15.9.2.3 Surfactant Tail-Solubilizate Mixing Gibbs Energy 705

15.9.3 Predictions of Solubilization Behavior 705 15.9.3.1 Computational Approach 706 15.9.3.2 Solubilization in Ionic Surfactant Solutions 706 15.9.3.3 Solubilization in Micelles of Poly(ethylene oxide) Surfactants 708 15.9.3.4 Solubilization-Induced Rod-to-Sphere Transition 710

15.9.4 Solubilization of Binary Hydrocarbon Mixtures 716 15.9.4.1 Gibbs Energy of Solubilization of Binary Mixtures 718 15.9.4.2 Aggregate-Water Interfacial Energy 718 15.9.4.3 Surfactant Tail-Solubilizate Mixing Gibbs Energy 718 15.9.4.4 Predictions of the Binary Solubilization Model 719

15.10 Microemulsions 719 15.10.1 Geometrical Characteristics of Microemulsions 721 15.10.2 Size Distribution of Droplet Microemulsions 723 15.10.3 Size and Composition Dispersion of Droplets 726 15.10.4 Calculation of Interfacial Tension 727 15.10.5 Bicontinuous Microemulsions 727 15.10.6 Gibbs Energy of Formation of the Film Region 730

15.10.6.1 Transfer Gibbs Energy of Surfactant and Alcohol Tails 730 15.10.6.2 Deformation Gibbs Energy of Surfactant and Alcohol Tails 731

xiv

15.10.6.3 Interfacial Gibbs Energy of Film-Water Contact 15.10.6.4 Head-Group Steric Interactions 15.10.6.5 Head-Group Dipole Interactions 15.10.6.6 Head-Group Ionic Interactions 15.10.6.7 Gibbs Energy of Mixing inside the Film Region

15.10.7 Predictions for Droplet Microemulsions 15.10.8 Prediction of a Bicontinuous Microemulsion

731 732 732 732 733 733 737

16 Ionic Fluids H. Krienke and J. Barthel

16.1 16.2 16.3 16.4

16.5 16.6 16.7 16.8

16.9

16.10 16.11 16.12 16.13 16.14

Introduction Low Concentration Chemical Model Integral-Equation Methods at the MM Level Classical Systems with Long-Range Forces 16.4.1 Correlation Functions and Screening 16.4.2 Thermodynamic Functions Effective Ionic Interactions Solvation of Ions in Apolar Solvents Molecular Correlation Functions- Invariant Expansions MSA for Charged and Dipolar Hard Shperes 16.8.1 MSA of Charged Hard Spheres 16.8.2 MSA of Mixtures of Ions and Dipoles Molecular Pair Correlations 16.9.1 Interaction Site Model (ISM) 16.9.2 Molecular Ornstein-Zernike (MOZ) Theory 16.9.3 Site-Site-Ornstein-Zernike (SSOZ) Theory 16.9.4 Monte Carlo Simulation Solvent Properties Ion Solvation in Molecular Solvents Ion - Ion Potential of Mean Force Ion Association Concluding Remarks

751

752 755 758 763 763 766 767 768 771 772 773 776 781 781 782 784 785 785 787 792 792 798

17 Ionic Fluids Near Critical Points and at High Temperatures J.M.H. Levelt Sengers, A.H. Harvey, and S. Wiegand

17.1 17.2

17.3

Introduction Criticality and Ionic Fluids 17.2.1 Criticality and Critical Exponents 17.2.2 Criticality and Range of Forces 17.2.3 Phase Separation due to Coulombic Interactions 17.2.4 The Restricted Primitive Model, RPM 17.2.5 Criticality of the RPM 17.2.6 Tricriticality and Charge-Density Waves in Ionic Fluids Critical Behavior in One-Component Charged Systems 17.3.1 Water 17.3.2 Liquid Metals near the Vapor-Liquid Critical Point 17.3.3 Molten Salts near the Vapor-Liquid Critical Point

805

806 806 806 807 809 810 811 812 812 812 812 813

17.4

17.5

17.6 17.7

17.8

Experiments in Partially Miscible Ionic Liquids 17.4.1 The RPM as a Guide 17.4.2 Character of Criticality in Nonaqueous Ionic Solutions 17.4.3 Character of Criticality in Aqueous Ionic Solutions 17.4.4 Viscosity 17.4.5 Summary 17.4.6 Crossover: a Perspective Solution Thermodynami-c-s l~ar Critical Points 17.5.1 Principal Issues 17.5.2 Thermodynamic Properties of Interest 17.5.3 Criticality of Dilute Solutions 17.5.4 Ionic Effects on Critical Behavior Gibbs Energy Models for High-Temperature Aqueous Electrolyte Systems Helmholtz Energy Models; Equations of State 17.7.1 Models Assuming Complete Ionic Dissociation 17.7.2 Models Assuming no Ionic Dissociation 17.7.3 Models with Partial Ionic Dissociation 17.7.4 Solid Solubility Calculations Conclusions

XV

814 814 817 823 824 824 825 827 827 828 828 832 833 836 836 838 84O 840 841

18 Multiparameter Equations of State R. T Jacobsen, S.G. Penoncello, E.W. Lemmon and R. Span

18.1 18.2 18.3

18.4

18.5 18.6

18.7 18.8 18.9 18.10

Introduction The Development of a Thermodynamic Property Formulation The Use of Least-Squares Fitting in Developing an Equation of State 18.3.1 The Least-Squares Method 18.3.2 Multiproperty Fitting 18.3.3 Constraints 18.3.4 Optimizing the Functional Form of Equations of State Pressure-Explicit Equations of State 18.4.1 The Benedict-Webb-Rubin Equation of State 18.4.2 The Martin-Hou Equation of State 18.4.3 The Bender Equation of State 18.4.4 The Jacobsen-Stewart Equation of State Thermodynamic Properties from Pressure-Explicit Equations of State Fundamental Equations 18.6.1 The Equation of Keenan, Keyes, Hill, and Moore 18.6.2 The Equation of Pollak 18.6.3 The Equations of Haar and Gallagher and Haar, Gallagher, and Kell 18.6.4 The Equation of Schmidt and Wagner 18.6.5 The Equation of Jacobsen, Stewart, Jahangiri, and Penoncello 18.6.6 Recent Equations of State 18.6.7 Transition (or Unified) Equations Thermodynamic Properties from Helmholtz-Energy Equations of State Comparisons of Property Formulations Recommended Multiparameter Equations of State Equations of State for Mixtures 18.10.1 The Virial Equation of State for Mixtures 18.10.2 Extended Corresponding-States Methods 18.10.3 Mixture Properties Using Helmholtz-Energy Equations of State

849

85O 851 852 852 854 854 854 856 856 856 857 857 858 859 860 860 861 862 863 864 867 867 868 873 874 876 876 877

Subject Index 883

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LIST OF CONTRIBUTORS

xvii

• M.A. Anisimov (U.S.A.) • A. Anderko (U.S.A.)

• J. Barthel (Germany) • T. Boublik (Czech Republic)

• D. Browarzik (Germany)

• P.T. Cummings (U.S.A.)

• J.F. Ely (U.S.A.)

• M.B. Ewing (U.K.)

• K.E. Gubbins (U.S.A.) • E.Z. Hamad (Saudi Arabia)

• A.H. Harvey (U.S.A.)

• R.T Jacobsen (U.S.A.) • Yu.V. Kalyuzhnyi (U.S.A.) • R.F. Kayser (U.S.A.) • H. Kehlen (Germany)

• H. Krienke (Germany)

• S.M. Lambert (U.S.A.)

• E.W. Lemmon (U.S.A.) • J.M.H. Levelt Sengers (U.S.A.)

• G.A. Mansoori (U.S.A.) • I.M.F. Marrucho (U.S.A.)

• E. Matteoli (Italy)

• E.A. Mtiller (Venezuela)

• R. Nagarajan (U.S.A.)

• H. Orbey (Turkey)

• S.G. Penoncello (U.S.A.)

• C.J. Peters (The Netherlands)

• J.M. Prausnitz (U.S.A.) • E. Ruckenstein (U.S.A.)

• S.I. Sandler (U.S.A.)

• J.V. Sengers (U.S.A.)

• N.A. Smirnova (Russia) • Y. Song (U.S.A.) • R. Span (Germany)

• J.P.M. Trulser (U.K.)

• A.I. Victorov (Russia)

• S. Wiegand (Germany) • H.J. White, Jr. (U.S.A.)

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xix

P R E F A C E

The present volume on Equations of State for Fluids and Fluid Mixtures has been prepared under the auspices of Commission 1.2 on Thermodynamics of the International Union of Pure and Applied Chemistry (IUPAC). When the Commission requested me to pursue first a feasibility study and subsequently the production of a volume that would assess the range of approaches to develop equations for calculating thermodynamic properties of fluids and fluid mixtures, I was fortunate in finding Richard F. Kayser and Cor J. Peters and, subsequently, also Howard J. White Jr. as co-editors willing to assist me with this arduous task. The resulting volume deals with a broad range of equations of state for fluids and fluid mixtures including complex fluids like polydisperse fluids, polymer systems, self- assembled systems and ionic fluids.

This volume was originally scheduled to appear in 1998. However, after the volume had been in the production state for several months, the publisher assigned by IUPAC requested permission not to proceed with publication. Fortunately, after a second favorable independent review of the material in the book, Elsevier agreed to publish this volume. Because of the resulting delay of the publication schedule, the authors were provided with an opportunity to make (minor) revisions in their manuscripts which were submitted in the fall of 1999. It is hoped that, in spite of this delay, the reader will find this volume to be a valuable reference concerning the status of equations of state for fluids and fluid mixtures.

On behalf of all the editors, I thank the authors for their patience in dealing with the unexpected delay of the publication of their chapters. The Physical and Chemical Properties Division of the National Institute of Standards and Technology (NIST) provided the necessary office support for the editorial processing of this volume. We are especially indebted to Laurell R. Phillips, Jr. and Carol A. Thomas for their outstanding and dedicated service in preparing the volume for the publisher. The pleasant interaction with Huub Manten of Elsevier is also gratefully acknowledged.

Jan V. Sengers

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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 Intemational Union of Pure and Applied Chemistry. All rights reserved

1 INTRODUCTION

J. V. Sengers a'b, R. F. Kayser c, C. J. Peters d, and H. J. White, Jr. a

alnstitute for Physical Science and Technology and Department of Chemical Engineering University of Maryland College Park, MD 20742, U.S.A.

bphysical and Chemical Properties Division National Institute of Standards and Technology Gaithersburg, MD 20899, U.S.A.

C Technology Services National Institute of Standards and Technology Gaithersburg, MD 20899, U.S.A.

dLaboratory of Applied Thermodynamics and Phase Equilibria Faculty of Applied Sciences Delft University of Technology Julianalaan 136, 2628 BL Delft, The Netherlands

The voluminous literature on equations of state for fluids and their mixtures and the closely related equations for the Helmholtz energy or Gibbs energy over a range of variables, and their wide use in science, engineering and industry attest to their importance.

Commission 1.2 on Thermodynamics of the International Union of Pure and Applied Chemistry (IUPAC) considered the desirability of preparing a volume on equations of state for fluids and fluid mixtures under IUPAC sponsorship. The volume in question would cover the entire range of approaches to developing equations of state, including their theoretical bases and practical uses and their strengths and limitations. Furthermore, the volume would not be limited to equations of state for simple fluids and fluid mixtures, but would also include the more important classes of complex fluids and mixtures.

The Commission requested assistance in obtaining a broader view of the desirability and feasibility of such a volume. As a result, a committee, consisting of three of the editors of this volume, sent out a questionnaire to a wide range of experts and potentially interested parties. In addition to desirability, the questionnaire asked for suggestions for topics to be included and

the names of appropriate potential authors. The results of the questionnaire were strongly positive with respect to desirability of the

volume. The suggestions with regard to content, specific topics and potential authors were many and varied, as would be expected. Summarizing the suggestions made and integrating them with the original IIYPAC concept resulted in the following features: 1. All principal approaches to developing equations of state should be covered. 2. The theoretical basis and practical use of the various types of equations of state would be

covered. 3. The strengths and limitations of each type would be addressed. 4. In each case, what is available and what additional work is needed should be pointed out. 5. In addition to simple fluids and fluid mixtures, more complex systems would be covered.

These would include associating fluids, ionic fluids, polydisperse systems, polymers, and micelle forming and other self organizing systems. The editors have selected authors who are knowledgeable in the fields covered and asked

them to pay particular attention to the desired information listed in items 2, 3 and 4 above. There are some other areas, not mentioned specifically above, which deserve attention: The critical region is unique and has received a great deal of theoretical attention in recent

years. In addition, mixtures in the critical region have proved to have interesting properties with practical engineering potential.

Another area, namely that of chemically reacting fluids and mixtures, has not been covered in a general way, although it is treated in some of the chapters on more complex systems insofar as it pertains to these specific topics. Some limited aspects of this area have been studied for some time. For example, fluids that readily form dimers and mixtures containing weakly ionizing molecules, but the more general subject appears not to have received the study that the importance of the topic deserves. The editors hope that this area will receive additional theoretical and experimental study in the near future.

An increasingly more important area of research is the use of computer simulations for the calculation of thermodynamic properties of simple and complex fluids. This subject would deserve an in-depth coverage of its own and, because of size limitations, it was decided not to include the topic in the present volume. The structure of the volume closely follows the requirements suggested by the IUPAC commission and amplified and reinforced by the resulting questionnaire.

Chapter 2 provides the basic thermodynamic foundation that underlies all of the remaining chapters.

The next six chapters deal with the various approaches that have produced enlightening and useful equations of state with the emphasis on single fluids. They are: Chapter 3, the virial equation; Chapter 4, the van der Waals and related cubic equations; Chapter 5, perturbation theory; Chapter 6, the use of analytically solvable integral equations; Chapter 7, quasi-lattice equations; and Chapter 8, the corresponding-states principle.

Mixing and combining rules are dealt with in Chapter 9 with emphasis primarily on mixtures of simple fluids. Mixtures of dissimilar fluids are considered in Chapter 10.

The basic approach to the critical region with emphasis on the simpler fluids and their mixtures is given in Chapter 11.

The remaining chapters deal with more complex fluids and their mixtures. Associating fluids and their mixtures are the subject of Chapter 12. Polydisperse fluids are

the topic of Chapter 13. These are fluids of so many components that the composition of the fluid has to be considered in terms of a continuous distribution function, crude oil being an example. Polymer systems are considered in Chapter 14, and self-assembling systems are covered in Chapter 15. Self-assembling mixtures contain components that spontaneously form micelles and related structures.

The next two chapters deal with ionic fluid mixtures. Ionic fluids in general form the subject of Chapter 16 and ionic fluids and chemically reacting fluids in the critical region are the subject of Chapter 17.

Finally, multiparameter equations of state are covered in Chapter 18. Since these provide equations of state for individual fluids, they might have been grouped with the earlier chapters involved primarily with individual fluids. However, these equations are used for substances, such as water or ethylene, for which accurate data for a number of thermodynamic properties are available over wide ranges of temperature and pressure. Since the thermodynamic properties are interrelated, thermodynamic consistency is needed over the entire range. The resulting complexities put these equations in a special category.

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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved

2 F U N D A M E N T A L C O N S I D E R A T I O N S

M.B. Ewing a and C.J. Peters b

a University College London, Chemistry Department, Christopher Ingold Laboratories 20 Gordon street, London WC1H OAJ, United Kingdom bDelft University of Technology, Faculty of Chemical Technology and Materials Science Laboratory of Applied Thermodynamics and Phase Equilibria Julianalaan 136, 2628 BL Delft, The Netherlands

2.1 Introduction 2.2 Basic Thermodynamics

2.2.1 Homogeneous Functions 2.2.2 Thermodynamic Properties from Differentiation of Fundamental Equations

2.3 Deviation Functions 2.3.1 Residual Functions 2.3.2 Evaluation of Residual Functions

2.4 Mixing and Departure Functions 2.4.1 Departure Functions with Temperature, Molar Volume and Composition as the

Independent Variables 2.4.2 Departure Functions with Temperature, Pressure and Composition as the

Independent Variables 2.5 Mixing and Excess Functions 2.6 Partial Molar Properties 2.7 Fugacity and Fugacity Coefficients 2.8 Activity Coefficients 2.9 The Phase Rule 2.10 Equilibrium Conditions

2.10.1 Phase Equilibria 2.10.2 Chemical Equilibria

2.11 Stability and the Critical State 2.11.1 Densities and Fields

2.11.2 Stability 2.11.3 Critical State

References

2.1 INTRODUCTION

This chapter provides a thermodynamic toolbox and contains most of the important basic relations that are used in other chapters. The scope is restricted almost exclusively to the second law of thermodynamics and its consequence, but the treatment is still intended to be exemplary rather than definitive. New results are not presented as befits a discussion of fundamentals which are necessarily invariant with time.

2.2 BASIC THERMODYNAMICS

The state of a system may be described in terms of a small number of variables. For a phase in the absence of any external field, the second law of thermodynamics may be written as

d U = T d S - p d V + ~ ~tidn i (2.1) i

This equation shows that the change dU in the energy U may be described in terms of simultaneous changes dS in the entropy S, dV in the volume V, and dn~ in the amounts n~ of the C components. It is often convenient to eliminate the size of the phase by writing Equation (2.1) in terms of intensive variables. For example, division by the total amount

N = ~ n i i

gives

(2.2)

alUm = T d g m - pdVm + 2 ~idxi (2.3) i

where the subscript m denotes a molar quantity and the mole fraction of component i is defined by

X i = ?l /N (2.4)

Equation (2.1), or Equation (2.3) for intensive variables, is the most fundamental expression of the second law of thermodynamics. However, entropy, in particular, is not a very convenient experimental variable and, consequently, altemative forms have been derived from the fundamental equation (2.1). Introduction of the following characteristic functions:

Enthalpy H = U + p V (2.5) Hemholtz energy A = U - TS (2.6)

Gibbs energy G = U + p V - TS = H - TS = A + p V (2.7)

and use of Legendre transformations (1,2)with the fundamental equation (2.1) gives the following alternative forms of the second law

d H = TdS + Vdp + ~ ~idni i

dA = - Sd T - p d V + ~ g idn i i

dG = - S d T + Vdp + ~ gidni (2.10) i

Another modification of Equation (2.1), frequently used in statistical mechanics, is

(2.8)

(2.9)

d ( p V ) = S d V + p d V + E n i d ~ i (2.11) i

or, as the Gibbs-Duhem equation, in the study of phase equilibria

0 = S d T - Vdp + ~ rtid~t i (2.12) i

From Equation (2.1) and Equations (2.8) - (2.10) it can be seen that

T~p,I~ i

~t i =

S, V,I~ i S,p,fi i T~ V,I~ i

(2.13)

where the subscripted fig means that the amounts nj of all the components are constant except for component i. The quantity g~ is the chemical potential of species i. In terms of intensive variables these equations are

dHm = TdS m + VmdP + ~ ~t idx i (2.14) i

d A m - - S m d Z - pdVm + ~ ~l'idxi (2.15) i

dGm = - SmdT + VmdP + ~ ~idx i (2.16) i

d(o Vm) = SmdT + p d V m + Z nid~ t i (2.17) i

0 = S m d T - VmdP + E xid~ti (2.18) i

For a system of constant composition, Equation (2.1) and Equations (2.8) - (2.10) reduce to

d U = TdS - p d V

d H = TdS + Vdp

dA = - S d T - p d V

d G = - Sd T + Vdp

(2.19) (2.20) (2.21) (2.22)

A large number of thermodynamic relations may be derived from the above equations by conventional manipulations. Table 2.1 summarizes the most frequently used equations.

Table 2.1 Frequently used thermodynamic relationships with general validity.

d U = TdS - p d V

dA = - S d T - p d V

d H = TdS + Vdp

dG : - S d T + Vdp

v T p T

av) = Op) - ~ T= -~)

Maxwell:

U/,,

Helmholtz:

OU

- ~ , r

Gibbs-Helmholtz:

OT v T2 [ OT p T 2

2.2.1 Homogeneous Functions

A homogeneous function F of the first order in any number of the variables x, y, z . . . . is defined by:

F(X~, Zy, ~ , ...) - ZY(x, y, z, ...) (2.21)

where ;~ is an arbitrary number. If each independent variable is made X times larger, the function F increases ~. times. For large enough systems, all extensive thermodynamic properties are homogeneous and of the first order in amounts n, at fixed temperature and pressure. For homogeneous functions of the first order, Euler's theorem applies:

y,z,. , x,z,...

+ z + "'" (2.22) x,y,...

10

Equation (2.22) relates the value of the function to the values of its derivatives. For ~ = 1, Equation (2.22) reduces to:

• = + z + "'" (2.23) y,z,.., x,z .... x,y,...

By applying Euler's theorem to the various characteristic functions U = U(S, V, n l, n2,... ), H = H(S, p, n l, n2,... ), A = A(V, T, nl, n 2...) and G = G(p, T, nl, nz,...), respectively, the following expressions result

U = TS - p V + Z ni~i ( 2 . 2 4 ) i

n : TS + E n i~ti ( 2 . 2 5 ) i

A : - p V + ~ Rig i (2.26) i

G = ~_~ ni~i (2.27) i

For further details on Euler's theorem see references 1-3.

2.2.2 Thermodynamic Properties from Differentiation of Fundamental Equations

The quantities U(S, V, n 1, n 2, ...), H(S, p, n 1, n 2, ...), A(T, V, hi, n z . . . . ), and G(T, p , nl, n z . . . . ) are examples of thermodynamic potentials from which all properties of a system can be obtained without the need for integration. For example, Equation (2.10) gives directly the heat capacity at constant pressure

p,n -D-7-7),,,n

and the isothermal compressibility

KT = - -~P T,n Op2) T,n "~P T,n

The avoidance of integration is often an advantage in theoretical applications of thermodynamics because there are no troublesome constants of integration. On the other hand, very often the derivatives needed involve variables that are difficult to measure experimentally. Even with the Gibbs energy surface, which is closely linked to the convenient experimental variables of temperature, pressure and composition, differences in the Gibbs energy can be studied only at equilibrium. At constant composition the characteristic functions U = U(S, V, n l, n 2 .... ),

11

H = H(S, p, n 1, n2,. . . ), A = A(V, T, n 1 ,n2,. . . ) and G = G(p, T, nl, t/z,. . . ) reduce to U = U(S, V), H = H(S, p), A = A(V, T) and G = G(p, T). By differentiation of these functions and use of the fundamental equations (2.5) and (2.18) - (2.20), the value of any thermodynamic property can be expressed in terms of the derivatives of each characteristic function. Table 2.2 summarizes the most relevant results. Further details can be found elsewhere (3).

2.3 Deviation Functions

Almost all definitions of molar properties for mixtures lack an unambiguous definition in the sense that they can be related directly to measurable properties. Therefore, it is common practice to compare an actual mixture property with its corresponding value obtained from an arbitrary model, for instance, an equation of state. This approach leads to the introduction of deviation functions. For a general mixture molar property M m the deviation function is defined by (4):

M S = M'a_£ tual _ M m°del (2.28)

An important aspect in this definition is the choice of the independent variables. Almost all analytical equations of state are expressions explicit in pressure: that is, temperature, molar volume (or density) and composition x = x~, x 2, ... are the natural independent variables. Therefore, Equation (2.28) can be rewritten into:

MD(T, V, n) = Mactual(r, V, n)- Mm°del(T, V, n) (2.29)

where n denotes the amounts n 1, t/z, """ The value of M obtained from the model is evaluated at the same values of the independent variables as used for the actual mixture property. Alternatively, although not of direct interest for equation-of-state models, temperature, pressure and composition may be a suitable choice as independent variables for deviation functions, i.e.:

MD(T, p, n) = Mactual(r~ p, n)- Mm°a~l(T, p, n) (2.30)

In this case the value of M obtained from the model is evaluated at the same values of T, p, and n as used for the actual mixture property. Both approaches are interrelated as follows (4):

~ 0 M model MD(T'V'n) = MD(T'p'n) + ( Op )V,n dp (2.31)

Pr

In this equation Pr is the reference pressure at which the molar volume of the mixture obtained from the model is equal to molar volume of the actual mixture at the same temperature and composition as the mixture. A necessary feature of the model is that such a Pr exists.

12

= 0 o~..~

o~,-~

c~

0

o~._~

r~

o~,~

o c~

0

E-

~1 ~ I

cb

I

E~

~L

J

rb c~

I

f ~

~1 ~

I

E~

~ J

I

f ~

I

I

I

Cb

I I

~1~ ~1 ~

~L

J ~L

J

I

~1 ~ I

E~

~ J

I I

fir ~

I

~1 ~ c~

I

As pointed out in the previous section, the calculation of deviation functions requires a choice

13

~1 ~ cO

I I

r ~

~1 ~ cO

I

fr _

~

~1 ~

I

f ~

I I

fr ~

cO

I

f ~

c~b

I

r-o

f ~

r-o c~

C

As pointed out in the previous section, the calculation of deviation functions requires a choice

14

2.3.1 Residual Functions

As pointed out in the previous section, the calculation of deviation functions requires a choice of an appropriate model. If the model system is chosen to be an ideal gas mixture, which is an obvious choice for fluid mixtures, then the deviation functions are called residual functions. With temperature, volume, and composition as independent variables Equation (2.29) becomes

MD(T, V, n) = MR(T, V, n) = Mactual(T~ V, n)-/k/ag(T, V, n) (2.32)

or with T and p as independent variables

MD(T, p, n) = MR(T, p, n) = Mactual(T~ p , n)- Mig(T, p, n) (2.33)

which is a particular form of Equation (2.30). The two sets of residual functions are related by Equation (2.31) in the form:

MR(T,V,n)=MR(T,p,n)+P_f(OMig) dp (2.34)

Pr Op T,n

where the reference pressure Pr = RT/Vm is sufficiently low for the thermodynamic property M of the real fluid to have the ideal-gas value.

Some thermodynamic properties (U, H, Cv and Cp) of an ideal gas are independent of pressure, while, others like S, A and G are not. Consequently, from Equation (2.33), it can be seen easily that the following equations hold:

UR(T, V,n) = UR(T,p,n) (2.35)

I-1R( T, V,n) = H~( T,p,n) (2.36)

CRv( T, V, n ) = CRv( T, V, n )

CRp(T,V,n)=CR(T,p,n)

S" (T,V,n) = SR(T,p,n) = nlnZ

(2.37)

(2.38)

(2.39)

AR(T, V,n) = Ag(T,p,n) + RTInZ (2.40)

GR(T, V,n) = Gg(T,p,n) + RTInZ (2.41)

with Z =p V/nRT as the compression factor.

15

2.3.2 Evaluation of Residual Functions

With temperature, molar volume and composition as the independent variables, general expressions for the residual functions of thermodynamic properties are readily obtained from Equation (2.32). Abbott and Nass (4) have given a definitive list of expressions for various properties as residual functions and these are summarised in Table 2.3.

For temperature, pressure and compositions as the independent variables, Abbott and Nass (4) also evaluated general expressions for the residual functions. Table 2.4 summarizes the results. For further details see references 2 and 4.

2.4 Mixing and Departure Functions

In terms of the independent variables, temperature, pressure and composition, departure functions compare the value of a general thermodynamic property M(T,p,n) with the corresponding property in the ideal-gas state and at a reference pressure Pr, that is/h0g(T, Pr, n). According to the ideal gas law, the reference pressure Pr is related to the reference volume V r = nRT/pr. Similarly, as was the case for the residual functions, the independent variables, temperature, molar volume and composition can also be used to define departure functions. Based on these independent variables, the general thermodynamic property M(T, V, n) is compared with the corresponding ideal-gas property _aa0g(T, V, n).

2.4.1 Departure Functions with Temperature, Molar Volume and Composition as the Independent Variables

The following equality can be derived for the departure function of a general thermodynamic property M(T, V, n) "

T,n

- o M i g d V + dV O V r,, O V r,n

r

(2.42)

Applying conventional thermodynamic manipulations on Equation (2.42), the following relations can be obtained:

U - u i g = ! { / ( - ~ T )v,n -p}dV (2.43)

V,n - p} dV + nRT(Z- 1) (2.44)

16

Table 2.3 Residual functions with volume or density as an independent variable. (p = n/V and Z= pV/nRT). M R (T, V, n) Residual Function

R =

H R - -

- ~"~o~ ~~/~,~ do

nRT 2 OZ d 9 - ~ + n R T ( Z - 1) O " p, n [3

R _ _

A R .--

a R ._~

C n m

v

- nR - - ~ p,n P

n R T f ( Z - 1 ) dp o P

P

n R T f ( Z - 1) do + nRT(Z - 1) o P

(oz I }o R - ~ ) o , n + 2 --~ p,n P

C R _.~ P R t ~ O~TI l!t I ~~1 t -1 C v - R + R Z + Z + p

p ,n T,n

pR = pRT(Z - 1)

17

Tab le 2.4 Residual functions with pressure as the independent variable. (9 = n/Vand Z =pV/nRT). M ~ (T, p, n) Residual Function

R

R .__

A R - -

G R - -

R

v

C P

_ nRT 2 OZ dp _ nRT(Z - 1)

0 " p ,n P

_ nRT2 OZ dp

0 " p ,n P

- nR - ~ p,n

P

nRT f (Z - 1) dp - nRT(Z - 1) P

0

9

n R T f ( Z - 1) dp o P

+ 2 I } --~)p , , , p,n P

C v - R + R Z + Z + p p ,n T,n

p R __ R T ~ ( Z - 1) P

18

(2.45)

Note, that if the reference volume V r is replaced by the actual volume V of the system, then the corresponding residual functions are recovered (Table 2.3).

A - Aig = p - d V - nRTI V (2.46)

G - G ig = - t) - d V - nRT1 V + n R T ( Z - 1) (2.47)

The thermodynamic properties for the reference state are defined by the following relations:

g i g = E nig] g= Z nig] g - n R T ~ n i (2.48) i i i

sig = E niS] g - R ~ nilnn i (2.50) i i

n i g = E niH] g (2.49) i

A ig = ~ n c 4 / g + R T ~ n i l n n i = ~ n , ' t l ~ g - T.~n,S/g - R T ~ n i + R T ~ n i l n n i (2.51) i i i i i i

G i g = E niG] g+ R T ~ nihlni = E ntH~ g - T ~ niS]g + R T ~ nihln i (2.52) i i i i i

From Equations (2.48) through (2.52), the following mixture properties for ideal gases are obtained:

mmixU = uig - E niU] g = 0 (2.53) i

Amixn = n i g - Z n f l ] g - 0 (2.54) i

AmixS = sig - E niS] g = - R E nilnni (2.55) i i

AmixA = A ig _ Z n~] g = R T ~ nilnn i (2.56) i i

Amixa = Gig - Z nia] g = R T E nilnni (2.57) i i

Departure functions are conveniently evaluated from Equation (2.46) as the generating

19

function. The following calculation procedure is the appropriate route to follow:

1. Equation (2.46) gives (A - A ig)

2. ( S - sig) = _ ( c 3 ( A - Aig)/ OT V,n

3. (U- U ig) : (A- A ig) + T(S- S ig)

(2.58)

(2.59)

4. (H- /T g) : (A- A ig) + I"(8- S ig) + nRT(Z- 1) (2.60)

5. (G- Gig) = (A- A ig) + nRT(Z- 1) (2.61)

2.4.2 Departure Functions with temperature, Pressure and Composition as the Independent Variables

In this case, the general thermodynamic property is M(T, p, x) and the following equality for the departure function can be derived:

m(T,p,n)-mig(T,Pr,n) = ~ T,n Op T,n Op T,n Pr

The following relations can be obtained from Equation (2.62):

g - gig _ V - --~ p,n d p - n R T ( Z - 1) (2.63)

n - nig = i { V - ~( ~TIp,n}dP (2.64)

7 ;r (2.65)

A _Aig _ V _ n T dp - (2.66)

G - G ig = V - n T dp + nRTI P

Note, if the reference pressure Pr is replaced by the actual pressure p of the system, then the corresponding residual functions are recovered (Table 2.4).

20

Again, the thermodynamic properties for the reference state can be obtained from the Equations (2.48) - (2.52). In order to obtain the departure functions for this set of independent variables, the following procedure is recommended:

1. Equation (2.67) gives (G - G ig )

2. ( S - sig) : - { O(G-Gig),} OT ,p,n

3. (U- ~g) = (G- G 0g) + T(S- s~g) - nRT(Z- 1)

(2.68)

(2.69)

4. (H- /T g) = (G- G ig) + T(S- S ~g) (2.70)

5. (A- A ig) = (G - G'g) - n R T ( Z - 1) (2.71)

Further details on departure functions can be found elsewhere (5).

2.5 Mixing and Excess Functions

Deviation functions for mixtures are concemed mainly with variation in composition rather than pressure or density. Consequently, it is convenient to use molar quantities. Molar excess functions are defined by

M E = M g = Mm actual - Mim d (2.72)

where M~ is the molar value for the ideal mixture at the same temperature and pressure. If, for

example,/W d = pd then the molar volume of the ideal mixture (id) is given by

Vim d = E i x i V i * (2.73)

where Vi* is the molar volume of pure component i.

Other thermodynamic properties for ideal mixtures are defined by:

gim d = E x i g i * i

= , g* H~ Ex, i i

S~ : E x,Si* - RE xilnxi i i

id .4* + R T ~ xi lnx i Am = E Xt i i i

Gim d = E x i G i * + R T E x i l n x i i i

From Equations (2.73) through (2.78), it readily can be seen that for the mixing properties

(2.74)

(2.75)

(2.76)

(2.77)

(2.78)

21

the following expressions hold:

Amixgm = Vim d - £ x i V i * = 0

Amixgm = g i d - ~ x i g i * : 0

Amix/-/m : g ~ - Z x i g i * : 0

AmixSm : Sim d _ Z x i S i * : -RZxi lnx i i

id • AmixAm = Am - Z x t A i = R T ~ xilnx i

i

Amixam = Gim d - ~ x i G i * = R T Z x i l n x i i

(2.79)

(2.80)

(2.81)

(2.82)

(2.83)

(2.84)

Again an important aspect in this definition is the choice of the independent variables. From Equation (2.72) two different definitions of excess functions can be obtained:

ME(T,V,n) = M(T,V,n) - Mid(T,V,n) (2.85)

ME(T,p,n) - M(T,p,n) - Mid(T, pV, n) (2.86)

The two approaches are related by

MZm(T,V,n) E ~ OMimd) dp (2.87) = Mm(T'p'n) + Op r,~

Pr

Abbott and Nass (4) pointed out that in this case p is the pressure for which the molar volume of the ideal solution is the same as that of the real solution at given temperature and composition. This pressure is obtained by solving

Vm(T,p,x ) = ~ xiVi*(T~Pr) (2.88) i

A required feature of the model is that Equation (2.88) can be solved for the pressure. For Equation (2.87), Abbott and Nass (4) proposed an approximate relation suitable for practical purposes:

= Mm(T'p'n) + (2.89) Op r,~ Z x i K i V i *

i

In Equation (2.89), K i X is the isothermal compressibility of pure i:

22

, 1 ~:i : - , (2.90)

V i O P ) T

Based on Equation (2.89), Abbott and Nass (4) summarize expressions for the various thermodynamic properties.

Excess functions and residual functions are related. From Equations (2.33) and (2.86), it can be seen that the following equality holds:

ME(T, p, n) = M~(T, p, n)- {)14¢a(T, p, n)- Mag(T, p, n)} (2.91)

Considering Equations (2.48) to (2.52) and (2.74) to (2.78), one can see readily that for an arbitrary extensive thermodynamic property the following relation holds:

ig (2.92)

2.6 Partial Molar Properties

A partial molar property M~ of an arbitrary extensive thermodynamic property M = M(T, p,

n 1, n2.... ) is defined by the equation: Partial molar properties give information about the change of the total property due to addition of

(2.93) T,P,~i

an infinitely small amount of species i to the mixture. From Equation (2.17) it becomes apparent that, by definition, the chemical potential is the partial molar Gibbs energy, i.e., gi = Gr For this arbitrary thermodynamic property M the following expressions can be derived (2):

d M = d p + d Z + E Mi dn i T,n p ,n i

where M = ~2 n tM i, or equivalently, i

(2.94)

M m - ~ x M i (2.95) i

From Equations (2.94) and (2.95) the generalized Gibbs-Duhem equation is readily obtained:

dp + d T - ~ n = T,n p ,n i

(2.96)

23

In the case, when M = G, Equation (2.12) is recovered.

2.7 Fugacity and Fugacity Coefficients

The fugacity of a real fluid mixture with constant composition and at constant temperature is defined by the equation:

d G = R T dlnf (2.97)

Combination of this equation with its ideal gas equivalent leads to:

d ( G - 6 ¢g) = dG R = R T d l n ( f ) = RTdln~0 P

(2.98)

In this equation G R is the residual Gibbs energy and q~ is the fugacity coefficient. Integration of Equation (2.98) yields:

G R = R T lntp (2.99)

Comparison of Equation (2.97) and its equivalent in Table 2.4 leads to: P

ln~0 : f(z- 1)d --f (2.100) 0 P

This expression shows that an equation of state can be used to evaluate the fugacity coefficient. Similar expressions hold for an arbitrary pure component i:

dG~ = R T dlnf

and: R

G i : R T ln(,0i

In a real solution for species i the defining equation reads:

d G i : R T dlnj~.

From the definition of the residual Gibbs energy G R = G- G ~g, it follows that: R _ G~g

G i = G i

and for an ideal gas at constant temperature it holds that:

~i -- G~ g + R T lny i,

From Equations (2.103) through (2.105) the following expression results:

(2.101)

(2.102)

(2.103)

(2.104)

(2.105)

t ~ ) dln~. d ( a i - a ] g) = d a ? = R r dl ~i ~ = R r (2.106)

24

Note that from Equation (2.106) the relationship between the fugacity f. and the

fugacity-coefficient ~i in the mixture for component i follows:

~ . - f" (2.107) YiP

Integration of Equation (2.106) leads to a similar expression as represented by Equation (2.102):

Gi ~ = RT lndi (2.108)

For M= G, Equation (2.93) can be rewritten into:

(2.109)

Substitution of Equations (2.102) and (2.108) into Equation (2.109) gives:

ln~i = ( O(ln~)] On i r ,p ,n , i

(2.110)

Since ln~i is related to ln(p i as a partial molar property, from Equation (2.95) the following relation holds:

ln~ = ~ x iln(i (2.111) i

At constant temperature and pressure, the following Gibbs-Duhem equation can be obtained from Equation (2.96):

Sxid(]n~i) = 0 (2.112) i

Fugacities and/or fugacity-coefficients in mixtures are easily evaluated from all kinds of equation-of-state models. In case the equation of state has pressure and temperature as the independent variables, the following, generally valid, relationship can be applied:

RT ln~/ - RT 1 t ~ / i = (Vi - pRT)dp (2.113)

In this equation the partial :molar volume V~ is to be evaluated from the equation of state and Equation (2.93). For a pure substance the partial molar volume V~ becomes equivalent to the molar volume Vi* and Equation (2.113) simplifies to Equation (2.100). Most equation-of-state models have temperature and volume as the independent variables. In that case the following, generally valid, expression can be used for evaluation of the fugacity and/or fugacity-coefficient in mixtures at constant temperature and composition:

25

RT. ~ ] d V - R T lnZ (2.114)

V

In this equation Z = p V / n R T i s the compressibility factor of the mixture; the partial derivative in Equation (2.114) can be obtained from the equation of state under consideration. For a pure substance Equation (2.114) reduces to:

oo

_~~ RT_ RTln~o i = R T l n --f = f [ - ~ ] d V - R T l n Z + R T ( Z - 1) (2.115)

P v "

For an extensive treatment of the fugacity concept, one is referred elsewhere (1,2,6).

2.8 Activity Coefficients

Although activity coefficients, in general, are most conveniently evaluated from models which are specifically designed for the condensed phase only, this section demonstrates how the concept of activity is related to a similar formalism introduced for the fugacity concept. Equation (2.102) in Section 2.7 relates the residual Gibbs energy and the fugacity. In a similar way the excess Gibbs energy (Section 2.5) is related to the activity-coefficient formalism, which may be useful in describing the non-ideality of a condensed phase.

From Equation (2.86) the following equation is readily obtained by applying Equation (2.93):

E i d G i = G i - G i (2.116)

Integration of Equation (2.102) at constant temperature and pressure from the pure state of ,

component i, where G i -- G i and f. = f. , to an arbitrary composition in the solution

yields:

Gi - Gi * = R 7 1 ~ ~ i ) (2.117)

Application of Equation (2.93) to Equation (2.78) gives:

i d • G i = G i + R T lnx i (2.118)

Consequently, from the latter two equations the following result is obtained:

G i - G i = R T 1 = R T In ~i (2.119)

where, by definition, 3',. -- f , . / (x~) is called the activity coefficient of component i in the solution.

In a similar way, as fugacity coefficients are related to the residual Gibbs energy, identical expressions for the activity coefficient can be related to the excess Gibbs energy. For instance, the partial molar excess Gibbs energy of species i is related to the activity coefficient by:

26

G~ = RT lnYi (2.120)

Since G ~ is the partial molar property of G E , consequently, lnyi is also a partial molar property of G, the following two relations of general validity can be derived:

G E = R T ~_, xi ln~ ~ (2.121) i

RTlny, =(O(nGE)) O n i T,P,~ i

(2.122)

Additionally, at constant temperature and pressure, from Equation (2.96) it follows that:

E xtd(ln'Yi) = 0 ( 2 . 1 2 3 ) i

For further details one is referred elsewhere (1-3,5,6).

2.9 The Phase Rule

For a system consisting of C components and P phases in equilibrium the number of intensive variables required to specify the state of the system (that is the number of degrees of freedom F) is given by the Gibbs phase rule:

F - C - P + 2 - R (2.124)

In this equation R represents the number of restrictions imposed on the system. While the value for isothermal, isobaric or isochoric changes is obvious, the restrictions imposed by chemical reactions are often more subtle. For example, liquid water will exist as a mixture of H20, H +, H3 O+, and OH- but, if C is taken as 4, then the requirements of electroneutrality and the ionic equilibria lead to R = 3 and the system still behaves, quite correctly, as a pure component. Two restrictions are particularly important when studying phases in equilibrium. IfPa~ phases have the same composition, then there are (Paz - 1) phase boundaries across which the (C-1) compositions must be the same and the additional restriction is:

R = (Paz - 1)(C - 1) (2.125)

and the number of degrees of freedom is: F - C(2 - Paz) +1 (2.126)

This makes it clear that 3 phases can have the same composition only in the special case of a pure component, i.e., at its triple point. If Paz = 2, then F =1, and an azeotropic line always results irrespective of the number of components.

The second special case applies to the critical state. Here the Pc phases that become identical at the critical state are considered separately from the Pnc phases that behave normally. In this case, the additional restriction is R = 2P c - 1, and the number of degrees of freedom becomes:

27

F = C - Pnc + 2 - (2P c - 1) (2.127)

Consequently, a critical point (F = 0) is unique for a pure component while, for a binary mixture, a critical line (F = 1) is the simplest case and critical endpoints (P,~ = 1, Pn~ = 2) are unique points. We note also that since F cannot be negative:

C ~ 2P c - 3 (2.128)

As a consequence, a tricritical point cannot exist in a binary mixture. The combination of the restrictions for an azeotrope and a critical state shows that it is not possible for two azeotropic phases to become identical in a critical state, i.e., the critical and the azeotropic composition must be different. The restrictions that follow from the phase rule are simple consequences of geometry and are useful because they reduce the number of variables that must be used to describe the state of a system: small values o f f should result in simple equations of state. However, the phase rule gives no guidance as to which variables should be chosen.

2.10 Equilibrium Conditions

For a closed system with an arbitrary number of components and phases in which the temperature and pressure are uniform, the following combined statement of the first and second law is applicable (2,6):

dU t + p d V t - T d S t <_ 0 (2.129)

In this equation t refers to the total value of the various properties. The inequality symbol applies for infinite small changes between non-equilibrium states and the equality symbol holds for infinite small changes between equilibrium states, i .e., reversible processes.

For practical purposes, from Equation (2.129), the following alternative equation in terms of the Gibbs energy can be obtained:

dG t + S t d T - Vtdp <_ 0 (2.130)

Since in this equation temperature and pressure, which are easily controlled in experiments, are the independent variables, Equation (2.130) is the most useful representation of Equation (2.129). At constant temperature and pressure, Equation (2.130) reduces to:

(dat)p,T < 0 (2.131)

This equation states that at constant temperature and pressure any irreversible process proceeds in such a direction that the total Gibbs energy of a closed system will decrease. At equilibrium the Gibbs energy has reached a minimum value for the given temperature and pressure.

2.10.1 Phase Equilibria

For a closed multi-component system, Equation (2.131) can be used to derive the equilibrium conditions between two or more phases in a system at constant temperature and pressure. If we indicate the various phases by ~, [3, T ..... n, and the various species by 1,2,3,....,C, the following

28

equilibrium conditions in terms of the chemical potential result:

a ~ 7 _ n ~t i = ~t = ~t i - . . . = ~t i, i = 1,2,3,. .... ,C (2.132)

Alternatively, it easily can be shown (2,6) that phase equilibrium can also be defined in terms of the fugacity:

f~/ = f f = f7 = = ~ 1,2,3 ..,C (2.133) • ° ° / ' , o , ,

For a simple vapor-liquid equilibrium Equation (2.133) reads:

fv =~!, i = 1,2,3,. .... ,C (2.134)

Substitution of Equations (2.107) and (2.119) into this equilibrium condition gives:

" v 1 Yi~i P - xiT;f, i = 1,2,3,. .... ,C (2.135)

This formalism is the gamma-phi approach for calculating vapor-liquid equilibria. The fugacity coefficient (/9 i of each component that accotmts for the non-ideality of the vapor phase can be evaluated from an equation of state model, while the activity coefficient Yi to describethe non-ideal behavior of the liquid phase can be obtained from an excess Gibbs energy model.

The fugacity~ of pure species i can be obtained from the relation (2,6,7):

RT 1 = f V.dp (2. 1 36) f/)

In case the temperature is appreciably below critical and for not too high pressures, Equation (2.136) can be approximated by:

inf. _ V](p - Pi ) (2.137) f" Rr

Substitution of f,. - ~iPi gives:

ft . l = ~oi*P;exp[ V~(p - Pi )] (2.138) RT

In the latter expression the exponential is the Poynting factor. The contribution of this term

becomes significant only at higher pressures. For an ideal vapor phase (q~i= 1) , the liquid phase

is an ideal solution (Yi = 1) and if the Poynting factor does not contribute (low pressures), Equation (2.135) reduces to Raoult's law.

29

Equations of state, in principle, are able to describe the vapor and liquid phase ~V ~1

simultaneously, i.e., both the (j9 i and (/9 i c a n be evaluated from an equation of state model. Substitution of Equation (2.107) for both the vapor and liquid phase into Equation (2.134) leads to the phase equilibrium conditions:

"v " l yiqgi = xiq~ i , i = 1,2,3,. .... ,C (2.139)

Details of the various approaches to model vapor-liquid equilibria from equations of state can be found elsewhere (2,3,5-9).

2.10.2 Chemical Equilibria

Application of Equation (2.131), the general criterion for chemical equilibrium, is conveniently expressed in terms of the chemical potential of each species present in the equilibrium mixture:

E Vi~i -- 0 , i = 1,2,3 ...... ,C (2.140) i

In this equation v~ are the stoichiometric coefficients, which for products are taken with a positive sign and for reactants are taken with a negative sign. At constant temperature and composition, Equation (2.103) reads:

d ~ i - - d a i -- R T d l R j ~ i (2.141)

Integration of this equation from a standard state of pure species i to its actual state in solution gives:

0 gi g~ + R T In ~ = g i + R T In d i (2.142)

In this equation d i is the activity of component i in the mixture. Substitution of this expression

into the condition for chemical equilibrium gives the important relation:

/•. 0 - Vi~ i

I-l(ai) v' - exp (2.143) i R T

Since the right-hand side of Equation (2.143) is a function of temperature only, this term can be written as:

0 Ar GO - E vi~ti - - R T In K(T) (2.144)

i

Equation (2.144) defines the thermodynamic equilibrium constant, which is a function of temperature only. From Equations (2.143) and (2.144) it can be seen how the equilibrium constant is related to the activities of the various species in the mixture. Since the activities of the reacting

30

species are related to their fugacities, equations of state can be used for their evaluation. Further details on chemical equilibria can be found elsewhere (1,3,7).

2.11 STABILITY AND THE CRITICAL STATE

2.11.1 Densities and Fields

Griffiths and Wheeler (11) divided thermodynamic properties into two classes: fields f(for example T, p, and ~t) that must be uniform throughout a system at equilibrium; and densities 9 (for example S, V, and n) which, in general, are discontinuous across a phase boundary although they are uniform throughout each phase. With this nomenclature, the fundamental equation (2.1) for the second law of thermodynamics can be written, very compactly, in the form:

dU : ~ f j d p j (2.145) J

where the density U(S, V, n 1, -.., nc) is the thermodynamic surface and it should be noted that the hydrostatic field is -p rather than p. Equation (2.145) shows that conjugate densities and fields are related by

fj = (OU/Ogj)Oj o (2.146)

where the subscript bj indicates that all the densities except 9j are held constant. Griffiths and

Wheeler used an equivalent definition

9j : - (0fo/0fj)fj (2.147)

where f0 is a thermodynamic potential.

2.11.2 Stability

Just as the fields, which are the first derivatives (OU/09), characterise equilibrium, the curvature of the thermodynamics surface, which depends on the second derivatives (0 zU/09Jogk), determines the stability of the system. The stability determinant for a system with C components may be written as

D(S,V,n 1, ..., riG_l) =

" "

Uvs Uvv Urn, ""

U.1S U.1V Unlll 1 ""

i i i "..

(2.148)

31

where the elements of the determinant are given by

/ Pj, Ok n l , "", nc

'x oZu with S, V, (2.149)

In a stable system with C components, the thermodynamic surface U(S, V, nl,..., nc_l) lies above its tangent plane and has positive curvature and, consequently, all the (C + 1) determinants D(S), D(S, V), D(S, V, nO,..., D(S, V, n 1, ..., nc_ 0 are positive. Furthermore, since the variables may be chosen in any order, many more determinants may be formed and they are all positive. However, with C components there are only (C + 1) independent variables and a set of (C + 1) determinants is sufficient to establish the conditions for stability. For example, the system is stable provided

- _> 0 V, n CV (2.150)

D(S,V) D(S'IOD(S): D(S)(Op) D(S) 1 - - = > 0

D(S) -~ T, n VKT (2.151)

D(S,V,n,)DS (O~t) = >_ 0 (2.152) D(S,V, nl) = --~-(~-~ ( , V ) : D(S,V) ~ r,p, nl

and so on. The ratios of the determinants are obtained from

D(pl,-",pj ) _- [ Ofj.] (2.153)

D(p 1 , ..., lgj_ 1) Opj ) fi<j, Ok>g

which was derived by Gibbs (12).

2.11.3 Critical State

The critical state is the limit of stability at which all the determinants that were positive in Section 2.11.2 become zero. However, in the usual case where a transition between two phases is terminated, the critical state imposes only 3 additional restrictions, irrespective of the number of components. Similarly, although all the discontinuities in the densities vanish because the phases become identical, it is sufficient to consider the behaviour of the system with respect to a single density and to formulate the restrictions in terms of higher-order derivatives

2 2 3 4 (0 g/ooj )O j = 0"~ (03 g/opj )~j = O; (o4 g/opj )Oj > 0; (2.154)

The conditions are often defined in other thermodynamic surfaces where the variables more closely match an equation of state or the experimental conditions. For example, a gas-liquid critical point in a pure fluid is usually defined by

32

((~p/O V)T,n = O; ((]2p/(~ V2)T,n : 0"~ (03p/O V3)T,n < 0 (2.155)

which may be written in terms of the A(T, V, n) surface as where the notation introduced in Equation (2.149) has been extended in an obvious way.

/ Azv = ~ - ~ ) = ~ - ~ ) = 0; A4v = - ~ ] V , n > 0 (2.156)

Temperature is assumed to be uniform and constant for both Equations (2.155) and (2.156). However, if the H(S, p, n) surface is used, then an entirely equivalent set of conditions is obtained

H2S : - - ~ ) - ' ~ ) : O; n4s = - ' ~ ) p , n > 0 (2.157)

but now pressure is assumed to be uniform and constant. Equation (2.153) shows the relation between these surfaces and U(S, V,n), since A2v = D(S, V)/D(S) and Hzs = D( V,S)/D( V). Conditions equivalent to Equations (2.156) or (2.157) are obtained from the U(S,V,n) surface and Equation (2.154) with pj = V or S. While Equation (2.155) is more familiar through its use of (p,V,T) variables and association with gas-liquid critical points in pure fluids, each set of conditions can be used to describe the same critical state. For example Equation (2.157) or pj = S in Equation (2.154) might be very appropriate for a calorimetric study of a gas-liquid critical point.

The experimental conditions of a critical state in a binary mixture are closely matched by the Gibbs energy G(T, p, n~, n2) and the relation with the U(S, V, n 1, n2) surface is established with Equation (2.153) in the form

D(S,V, nl)/D(S,V) = (O~]l/Onl)T,p,n2 (2.158)

which leads to the following conditions for the critical state: 2 2 3

(O~tl/Onl)T,p, n2 = O; (0 ~tl/Onl)T,p, n2 = O; (03~l/Onl)T,p, n2 >_ 0 (2.159)

The Gibbs-Duhem Equation (2.18) allows these conditions to be expressed (1,10) in terms of the molar Gibbs energy G m and a mole fraction x

Gzx = (c~2G/(]xz)T,p = 0"~ G3x = (03G/Ox3)v,p = 0; G4x = (04G/Ox4)T,p ~ 0 (2.160)

Since most equations of state have temperature, molar volume, and composition as independent variables, while the Gibbs energy is explicit in temperature, pressure, and composition; a formulation of the critical conditions in terms of the Helmholtz energy is required. The following equations allow a transformation between G(T, p, x) and A(T, V, x) (1,10):

Gzx = Azx - (Avx)Z/Azv (2.161)

G3x = A3x - 3Avzx(Avx/Azv) + 3Azvx(Avx/Azv) 2 - A3v(Avx/A2v ) (2.162)

The Helmholtz energy A(T, V, x) and the derivatives required for Equations (2.161) and (2.162) may be obtained from any Equation of state that gives the pressure through

33

, o A(V,T,x) : (1 - x)A((V°,T) + xA 2 (V ,T) +

V RT{(1 - x)ln(1 - x) + xlnx} - fpdV

V °

(2.163)

In this equation AI*(V°,T) and Az(V°,T) are the molar Helmholtz energies of the pure

components and V ° is a reference volume. Multicomponent systems are handled in a similar way. For example, a ternary mixture can

be described in terms of four variables and Equation (2.153) gives

D(S, V, nl,n2)/D(S, V, nl) = (O].t2/On2)T,p,gl,n 3 (2.164)

and the conditions for a critical state are therefore 2 2

(C~tZ/(~nZ)T,p,~,,n 3 = O; (0 ~l,2/OnZ)T,p,p,,n 3 = O; (03].t2/On23)T,p,~tl,n3 > 0 (2.165)

The defining equations for higher-order critical points are straightforward in terms of the Gibbs energy and the composition variables. For instance, for a tricritical point in a (pseudo) binary mixture the following 2P c - 1 = 5 conditions have to be satisfied (1,10):

G2x : O; G3x : O; Gax : O; Gsx : O; G6x > 0 (2.166)

For higher-order critical points, the transformation equations from the Gibbs into the Helmholtz energy become extremely complex.

Acknowledgements

The authors are indebted to J.M.H. Levelt Sengers for numerous suggestions to improve the manuscript. The support of M.M. Abbott is also gratefully acknowledged.

REFERENCES

1. M. Modell and R.C. Reid, Thermodynamics and its Applications, 2nd ed., Prentice Hall, NewYork (1983).

2. H.C. Van Ness and M.M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions, McGraw-Hill, New York (1982).

3. S.M. Walas, Phase Equilibrium in Chemical Engineering, Butterworth (1985). 4. M.M. Abbott and K.K. Nass, Equations of State and Classical Solution Thermodynamics: Survey of

the Connection, in: K.C. Chao and R.L. Robinson, eds., Equations of State: Theories and Applications, ACS Symposium Series 300, American Chemical Society, Washington, DC (1986).

5. R.C. Reid, J.M. Prausnitz and B.E. Polin, The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York (1987).

34

6. J.M. Prausnitz, R.N. Liehtenthaler, and E. Gomes de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed., Prentice Hall, Englewood Cliffs (1986).

7. J.M. Smith, H.C. Van Ness and M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York (1996).

8. S. Malanowski and A. Anderko, Modelling Phase Equilibria, Wiley, New York (1992). 9. A. Anderko, Fluid Phase Equilib. 61, 145 (1990).

10. J.S. Rowlinson and F.L. Swinton, Liquids and Liquid Mixtures, 3rd ed., Butterworth, London (1982).

11. R.B. Griffiths and J.C. Wheeler, Phys. Rev. A 2, 1047 (1970). 12. J.W. Gibbs, Collected Works, Vol. 1, Yale University Press, New Haven (1928,1948)

Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 35

3 THE VIRIAL EQUATION OF STATE

J. P. M. Trusler

Department of Chemical Engineering and Chemical Technology Imperial College of Science, Technology and Medicine London SW7 2BY, United Kingdom

3.1 Introduction 3.1.1 The Volumetric Behaviour of Real Fluids 3.1.2 The Virial Equation of State 3.1.3 Temperature Dependence of the Virial Coefficients 3.1.4 Composition Dependence of the Virial Coefficients 3.1.5 The Pressure Series 3.1.6 Convergence of the Virial Series

3.2 Thermodynamic Properties of Gases 3.2.1 Perfect-gas and Residual Properties 3.2.2 Helmholtz Energy and Gibbs Energy 3.2.3 Perfect-Gas Properties 3.2.4 Residual Properties

3.3 Theory 3.3.1 Properties of the Grand Canonical Partition Function. 3.3.2 The Virial Equation of State from the Grand Partition Function. 3.3.3 The Virial Coefficients in Classical Mechanics 3.3.4 Quantum Corrections 3.3.5 The Virial Coefficients in Quantum Mechanics: Helium and Hydrogen

3.4 Calculation and Estimation of Virial Coefficients 3.4.1 The Virial Coefficients of Model Systems 3.4.2 Estimation of Virial Coefficients

3.5 Summary References

36

3.1 I N T R O D U C T I O N

3.1.1 The Volumetric Behaviour of Real Fluids

The virial equation of state is a relation between pressure p, temperature T and amount-of- substance density Pn which may be applied to real gases. For a perfect gas, these quantities are related by the well known equation

p = p,,RT (3.1)

or simply

Z = 1 (3.2)

where R is the universal gas constant and Z =p/PnRT is called the compression factor. For real fluids, the compression factor is found to behave in a rather more complicated

way than Equation (3.1) predicts. Such behaviour is illustrated in Figure 3.1 where experimentally determined values of Z are plotted as a function of Pn along a number of isotherms for fluid methane (1,2). At temperatures above the critical, the isotherms are smooth continuous functions of density but, at subcritical temperatures, they are split into two branches which each terminate on the vapour-liquid coexistence curve. At low densities, the isotherms converge in accordance with the experimentally proven result that

1.2

1.0

0.8

N 0 . 6

0.4

0.2

0.0

, , , / /

_

- ~ ~

[ I I E

0 5 10 15 20 25

Pn (m01"dm3)

Figure 3.1 Compression factors of fluid methane as a function of density (1,2). - . . . . . . , saturation curve; isotherms calculated from the equation of state of Setzmann and Wagner (3); c, critical point. O, 160 K; O, 190.555 K; I-I, 200 K; m, 220 K; ~, , 273.15 K; A,, 373.15 K; 'IF, 573.15 K.

37

Z ( T , p n --ff 0) ---- 1 (3 .3)

for all non-reacting gases. This behaviour will be referred to as the perfect-gas limit. At the other extreme of high densities, the isotherms adopt positive slopes and Z starts to increase rapidly as the fluid becomes nearly incompressible. It is also instructive to examine the same experimental isotherms as a function of pressure as in Figure 3.2. Here, one can see that both branches of a subcritical isotherm terminate on the coexistence curve at the saturated vapour pressure. The high compressibility of the fluid in the near-critical state is also apparent.

1.0

0.8

0.6

0.4

0.2

0.0

' ' ' ' ' ' ' ' I

1 2 5 10 20

p (MPa)

Figure 3.2 Compression factors of fluid methane as a function of pressure (1-3). Symbols and curves as for figure 1 e x c e p t - ---, tie line.

3.1.2 The Virial Equation of State

The accurate representation of p(T,Pn ) over wide ranges of temperature and density is a

problem of great practical importance to which there is no simple solution. The virial equation of state

p = p .RT {1 + Bp. + Cp2. + Dp3 +...} (3.4)

is one of the many hundreds of equations which have been proposed in an attempt to obtain a more accurate description of the behaviour of real fluids than is given by Equation (3.1). This equation may be viewed as a MacLaurin series in which (p/RT) is expanded in powers of Pn about the perfect-gas limit with density-independent coefficients B, C, D ... known as virial

38

coefficients. Conventionally, B is called the second viral coefficient, C the third virial coefficient, D the fourth and so on. Since it is an experimentally proven fact that Z is a function of both T and Pn, one must expect B, C, D, ... to be functions of the temperature and, as Z(T , Pn ) is not generally the same function for all gases, the virial coefficients for mixtures must also be functions of the composition.

If the virial equation of state had no better foundation than the empirical arguments introduced above, it is doubtful that it would have received the attention that it has. Indeed, some modem empirical equations of state (see, e.g., reference 3) are of much wider applicability and their greater complexity is not usually a real disadvantage. However, the importance of the virial equation is that it has a rigorous theoretical foundation in statistical thermodynamics which provides exact analytic relations between the virial coefficients and the interactions between molecules in isolated clusters. One finds that B depends upon interactions between pairs of molecules, C upon interactions in a cluster of three molecules, D upon interactions in a cluster of four molecules, and so on.

3.1.3 Temperature Dependence of the Virial Coefficients

One of the significant advantages of the virial equation of state is the existence of a very large database of experimentally determined virial coefficients (4). The second virial coefficient is by far the most well studied but C is known with some accuracy for a number of fluids and a few values of D have been reported. In view of this situation, the temperature dependence of the virial coefficients will be discussed first in the light of the experimental results. Figure 3.3 shows values of the second and third virial coefficients of methane over a wide range of temperature. The critical temperature T c and the Boyle temperature 7 9 (at which B = 0) are also indicated in this figure. At high temperatures, B is positive and varies only slowly with temperature while, at lower temperatures, it becomes large and negative. Like B, C is a slowly varying positive quantity at high temperatures and a rapidly varying negative one at low temperatures; in between these extremes it passes through a maximum. The curves shown in Figure 3.3 were calculated from intermolecular pair and triplet potential- energy functions determined for methane from speed-of-sound measurements (5). The behaviour shown in Figure 3.3 is typical of almost all gases although, of course, the magnitude of the virial coefficients and the characteristic temperatures such as 7 9 vary from substance to substance.

The temperature dependence of the higher virial coefficients is not well established experimentally but theoretical evaluations for model systems (see Section 3.3.6) indicate that these virial coefficients too are positive at high temperatures with a rapid divergence towards large negative values at temperatures below the critical (6).

3.1.4 Composition Dependence of the Virial Coefficients

For a pure gas, the phase rule requires that the pressure be a function of the two variables T and Pn only. Similarly, in the case of a mixture composed of v components, with amounts of substance n 1, n 2, n 3 ... n v present in volume V, the pressure must be a function of the (v+ 1)

variables T, nl/V, n2/V, n3/V.., nv /V and an expansion ofp, at constant temperature, in powers o f n l / V , n2/V , n3/V.., nv /Vshould exist. Carrying out this expansion, one again arrives at the

39

-50 O

E co

E 100 O -

-150

T c T B

| | | |

-200 0 600

I I I I

200 300 400 500

T (K)

5000

4000

3000

2000

1000

O

E to

E O

v

O

Figure 3.3 Second and third virial coefficients of methane (1,2). O, II B ; O, r-! c. Curves are calculated from intermolecular pair and triplet potentials determined from speed of sound measurements (5).

virial equation of state with Pn =(2iVlni)/V but with virial coefficients that depend upon

composition as follows:

i=1 j - - 1

Cmix =£££XiXjXkC~ik i=1 j = l k = l

Omix "-~£~£XiXjXkXlOijkl i = l j = l k = l /=1

etc.

(3.5)

Here, x, = n,/(E[=lni) denotes the mole fraction of substance s in the mixture and Bg, Co.k,

D/jkl, .-. denote a set of virial coefficients which depend only on T. Clearly, Bss, Csss, Dssss ... are the virial coefficients of pure s. The remaining coefficients are known as interaction virial coefficients and are related by the theory to interactions within clusters of dissimilar molecules• It is notable however that the composition dependence of the mixture virial coefficients (quadratic, cubic, quartic, etc. in the mole fractions) was arrived at without

4O

recourse to molecular theory. Interaction second virial coefficients B/j have been measured for many systems and the results collated by Dymond and Smith (4) but there are few experimental results for the higher-order terms.

3.1.5 The Pressure Series

In both experimental work and practical applications it is sometimes convenient to use (T, p) as the independent variables in place of (T, Pn)" For this purpose, an expansion of Z in p is used which may be written as:

Z = 1 + B ' p + C'p 2 + D ' p 3 +. . . (3.6)

The coefficients of this series are uniquely related to the virial coefficients as may be shown by eliminating p from the fight hand side of Equation (3.6) using Equation (3.4) and collecting terms. The first few such relations are given in Table 3.1. The composition dependence of the coefficients B', C', D' .-. in a mixture may be determined by combining these relations with Equations (3.5).

Table 3.1 Relations between coefficients in the density and pressure-explicit expansions of Z.

Pressure Series Density Series

B ' = B / R T B = R T B '

c ' = ( c - , ~ ) / ( R r ) ~

D' = ( D - 3BC + 2B 3) / (RT) 3

c = (RT)~(C' +/~ '~ )

D : ( R T ) 3 ( D ' +3B 'C ' + B '3)

3.1.6 Convergence of the Virial Series

It is important to recognize that the virial series might not converge for all experimentally realisable densities. The true radius of convergence is unknown in general and the experimental evidence is somewhat ambiguous. It is certainly possible to fit experimental compression-factor isotherms that extend over a wide range of densities to a truncated form of Equation (3.4). At supercritical temperatures, such representations appear to be satisfactory at densities of up to two or three times the critical (see, e.g., reference 7) while, at subcritical temperatures, good results may be obtained up to the density of the saturated vapour. However, it is often argued that the coefficients obtained in this way are not the true virial coefficients (8); indeed, the value of B determined in a fit to high-density data may differ noticeably from that obtained by analysis of precise low-density data (9). Whether or not such differences should be attributed solely to experimental uncertainties, or to the limitations of Equation (3.4) itself, is really not clear. On the other hand, it is known that the equation of

41

state is non-analytic at the critical point and it is therefore certain that some fluid states exist in the neighbourhood of the critical point at which the virial series fails to converge. Some theoretical results pertaining to the issue of convergence will be mentioned later in connection with model systems (see Section 3.3.6).

Whatever the ultimate radius of convergence might be, the question of real practical importance is to establish the region in which the series is rapidly convergent. Often, the virial series is truncated after the term in C; we are then interested in the associated truncation error. Provided that sufficiently accurate values of B and C are available, compression factors may be obtained with an accuracy of order 10 -4 at supercritical temperatures for densities up to roughly one sixth of the critical and the truncation error may be expected to rise to 10 -2 at about three quarters of the critical density. These estimates of truncation error are based largely on theoretical expectations of the magnitude of D and higher virial coefficients (6). The predicted divergence of these coefficients at low temperatures suggests that there might be a range of temperatures around the critical in which somewhat larger truncation errors occur. At lower temperatures still, the rapid decline in the saturated vapour density is such that the series converges rapidly for all accessible states of the gas.

3.2 THERMODYNAMIC PROPERTIES OF GASES

In this section, a general prescription is presented by means of which the thermodynamic properties of a fluid may be expressed in terms of an equation of state. Explicit results are presented for the virial equation of state.

3.2.1 Perfect-gas and Residual Properties

It is often convenient to express the thermodynamic properties of a fluid as the sum of a perfect-gas term and a residual term (10). The utility of this separation lies first in the availability of widely-applicable theory for the prediction, estimation and correlation of perfect-gas properties (11); and second in the dearth of such theory to account exactly for the effects of molecular interactions with the consequent need for some measure of empiricism.

For generality, consider a mixture of v components characterised by amounts of substance nl, n2, n 3 ... n v (denoted by the vector nV), temperature T, pressure p and volume V. If (T,V,n") are taken as the independent variables, a thermodynamic property X(T,V,n") may be written in the form

X(T,V,n v)= xpg(T,V,n v)+ xres(T,V,n v) (3.7)

where xPg(T,V,n v) denotes the property of a hypothetical perfect gas with the specified values of (T,V,n v) and where xres(T,V,n v) is the residual term. A similar decomposition may be applied with (T,p,n v) as the independent variables

X(T,p,n")= xPg(T,p,n")+ xres(T,p,n") (3.8)

but one should note that, although X(T,V,n")= X(T,p,n") , the perfect-gas terms and the residual terms in Equations (3.7) and (3.8) generally differ. This is a consequence of the fact that, while (T,V,p,n") characterise the state of the real fluid, the state of the hypothetical

42

perfect gas depends upon whether (T,V,n ~) or (T,p,n ~) are specified. The difference between xPg(T,p,n") and xPg(T,V,n ") may be obtained by noting that the volume of the

v hypothetical perfect gas is nRT/p in the one case, where n = ~i=l ni, but V in the other so that

XPg(T,V, nV)= xPg(T,p, nV)+ (¢~X pg/c:~V)T,nvdV RT/p

(3.9)

and

~n~" ( O X p g / 0 V)T,,,~ dV XreS(T'V'nV)= xres(T'p'nV)- R~'/p (3.10)

In the case of properties for which the perfect-gas term depends only on (T,n"), one then has XPg(T,V, nV)= xPg(T,p, nV).

3.2.2 Helmholtz Energy and Gibbs Energy

All of the thermodynamic properties of a homogeneous phase may be obtained from the Helmholtz energy or from the Gibbs energy, as discussed in Chapter 2. When (T,V,n v) are the independent variables, the Helmholtz energy A(T,V,n v) is the appropriate choice and the fundamental thermodynamic equation for the phase is

dA = -SdT - pdV + £~Aidn i (3.11) i=1

where S is the entropy and Pi is the chemical potential of component i. The partial derivatives of A with respect to T, V or n i then give -S, -p or/A i respectively. Once these quantities are obtained, the other state functions such as enthalpy H, energy U and Gibbs energy G follow from the appropriate combinations of A, TS and pV. Quantities such as heat capacity and compressibility may be obtained from second derivatives of the Helmholtz energy combined, where necessary, with the Maxwell relations (12).

When (T,p,n v) are the independent variables, the properties of a homogeneous phase are best obtained from the Gibbs energy G(T, p, n v) and the fundamental equation in the form

dG =-SdT + Vdp + £]Aidn i (3.12) i=1

S, V and ~i are then obtained from the first-order partial derivatives, other state functions from combinations of G, TS and p V, and the remaining properties from second-order partial derivatives of G.

Equations (3.11) and (3.12) are, of course, quite general and apply to any homogeneous phase. In the following sections, these relations are applied to obtain, first, expressions for the properties of the perfect gas and, second, expressions for the residual properties of a real gas in terms of either the virial coefficients or the coefficients of Equation (3.6).

3.2.3 Perfect-Gas Properties

43

The molar Helmholtz energy A~ g = APg/n of a pure perfect gas is a function only of (T, Pn ) and may conveniently be written

T Ipn(c~Apng/~;~Pn)TdPn A~g(T,p,) = A~ + ITO ( ~ A ~ g / g T ) o d r + o: (3.13)

where T ° and p O are reference values of the temperature and density at which A~ g takes the value Am °. Am ° is often specified in terms of reference values of molar energy and molar entropy: Am ° =Um ° - T°S°m . The integration with respect to temperature is best carried out after noting that

o (~A~ g / ~ T)pn "= - S Pm g = - o ( CPv ,gm / T ) d T - S m (3.14)

where C ~ is the isochoric perfect-gas heat capacity and is a function only of temperature. V,rn For the integration over density, it is also convenient to note that

(~Am pg / ~Pn ) T -- P / P ~ -- R T / P n (3.15)

Then, carrying out the integrations in Equation (3.13), one obtains the molar Helmholtz energy of the perfect gas as

o o

A~ g ' - C'~,, g d T - T C~,, g d ln T + R T l n ( p . / p. ) + Um - TS ~ o V,m o V,m (3.16)

Provided that C ~ (T) is known and that T ° o o V,m , pn, U m and S m a re defined, all of the thermodynamic properties of the pure perfect gas may then be obtained through Equations (3.11) and (3.16). Typically in the construction of an equation of state, the perfect-gas heat capacity is correlated by means of a suitable function which may then be integrated to obtain an analytic approximation to A~ g ( r ,Pn) .

For a mixture of v components, the molar Helmholtz energy is given by (11)

v v Affr~. ( T , P n , X V ) = Z x i A P g + R T Z x i l n x i (3.17)

i=1 i=1

where A/°g is the molar Helmholtz energy of pure i and x v denotes the set of mole fractions Xl, X 2, X 3 ... X v.

The principal perfect-gas thermodynamic properties of pure substances and multi- component mixtures determined from Equations (3.11), (3.16)and (3.17)with (T, Pn,X") as the independent variables are summarised in Table 3.2.

An equivalent set of equations to describe the properties of the perfect gas may also be obtained in terms of the Gibbs energy. Following arguments similar to those used to derive Equation (3.16), one may show that

44

GPm g : o Cl~-gp,m d T - T o C~pg.md lnT + R T l n ( p / P°) + Hm - (3.18)

Here, Cp,~m = C~V,m q- R is the isobaric perfect-gas heat capacity and T °, p °, H m° and S m° are reference values of temperature, pressure, molar enthalpy and molar entropy. In order to make Equations (3.16) and (3.18) consistent, one must choose the same reference states; this requires that p° = RT°Pn ° and H°m = U°m + RT. Mixture properties may be obtained by means of the equation

v

G ~ ( T , p , x ~ ) = xi Gpg -]- e Z 2 x i l n x i (3.19) i=1 i=1

The perfect-gas thermodynamic properties determined from Equations (3.12), (3.18) and (3.19) with ( T , p , x v) as the independent variables are summarised in Table 3.3.

It should be emphasised that the relations in Tables 3.2 and 3.3 give identical values of the thermodynamic properties when the reference states are defined as above and when Pn and p are related by p = pnRT. However, as indicated in Section 3.2.2, when these relations are applied to obtain the perfect-gas component of the thermodynamic properties of a real fluid, the results generally depend upon the choice of the independent variables. This completes the discussion of perfect-gas properties.

Table 3.2 Thermodynamic properties of pure gases and gaseous mixtures with (T ,p , ,xu) as the independent variables.

* Here B, = T ( d B / d T ) , B2 = T 2 ( d 2 ~ l d T 2 ) etc. for a pure gas, and B = B,,,, , B, = T(dB,, I d T ) , etc. for a mixture.

P V1

Residual Properties

Pure gas or mixture*

= R T { B P , + f c p : + : D ~ : + . . . )

S , ' = - R { ( B + B l ) p n + f ( ~ + ~ l ) p ~ + ~ ( ~ + ~ l ) p ~ + )

U,"" = -RT{BIp,, + iC ,p i + f DIP:+...)

CTm = -R{(2B,+B2)p, + i (2Cl+C2)pf + j( 2Dl+D2)pi+...}

H;; = R T { ( B - ~ , ) p , , + (c-:c,)pf + ( D - ~ D , ) ~ ; + . . . J

G,"" = RT{2Bp, + t C p : ++DP;+...)

paS = R T { ~ ( ~ X ~ B ~ , > P , + + ( C ~ X ~ ~ ~ C ~ ~ ) P S , I I j

+ ~ ( ~ ~ C X ~ X , X ~ D ~ ) P ~ )+...I i j k

Perfect-Gas

Pure gas

Xg = IT: C&dT - T IT: C;gmd ln T

+RT l r ~ ( ~ , / ~ ~ ) +u:- TS:

S ; g = ~ : + & a g m d l n ~ - ~ l n ( p n / d )

U:g = U. + IT: G a d T

C,":m = CyPgm ( T )

H:g = U. + RT + IT: G t d T

GF = AEg + RT

&PP p 3 * ~ g = G;

Properties

Mixture " U

&fX = ~ x i ~ ~ % R T ~ x , ~ n x , i=l !=I

" U

S : f x = ~ ~ i ~ ~ g - ~ ~ ~ , l n x i ,=I i=l

U

U::x = x x i v g i=l

U

C,"f;nix = C xiqg i (T) i=l

" H:~ = 1 xi H ~ P

,=I

" U

Gzx = C X ~ G P ~ + R T C X , l n x , ,=I ,=I

,us = ps*pg + RT In X ,

Table 3.3 Thermodynamic properties of pure gases and gaseous mixtures with ( T , p , x u ) as the independent variables. P m

* Here B: = T ( d B r / d T ) , Bl = T2(d2B' /d T 2 ) etc. for a pure gas, and B' = Bk, , B; = T(dBA,/d T ) , etc. for a mixture. p: is given in terms of

Residual Properties

Pure gas or mixture*

Perfect-Gas Properties

U u

4, = z x i ~ y g + ~ ~ z x i l n x i i=l i=I

M = p3*pg + RT lnx,

the interaction virial coefficients defined in Equation (3.5) with B = B-, etc.

Pure gas

= - R T { ~ C p 2 + 5 D f p 3 + . . . )

psres = R T [ { ~ ~ x ~ B , , - B)(pIRT) i

+ { + z x x i x j c i i , - ~ B ~ x , B , + + B 2 - C ) ( P / R T ) ~ i j i

+ - 3 ~ ~ ~ x ~ x ~ ~ ~ ~ i j k i j

Mixture

47

3.2.4 Residual Properties

All residual thermodynamic properties may be expressed in terms of an equation of state of the form p = p(T, Pn) or On = [Dn(T~P); Equation (3.4) is an example of the former and Equation (3.6) of the latter. The residual part of the Helmholtz energy for a phase of constant composition may be obtained by combining the identity

A~e=(T,p,,) = Io °" {(OAm/OP,)r-(OAff/OP=)rIdp= (3.20)

in which (OAm/OP,)r =p/p~, with Equation (3.4) for the pressure. All other residual properties may then be derived by manipulation of the result. Table 3.2 gives expressions for A~ es and for the other five common residual thermodynamic functions S res, Ur~ es, C~m, H re=

and G re~ in terms of the virial coefficients and with (T,p,,x ~) as the independent variables.

Also given in Table 3.2 is the expansion of the residual part of the chemical potential/a s of component s in a multi-component mixture. This is useful in a vapour-liquid equilibrium calculation where the virial equation of state is used for the vapour phase.* Such calculations are usually carried out for conditions of constant temperature and pressure and, to evaluate the chemical potential for specified values of (T,p,x ~), one must first solve Equation (3.4) for Pn" The partial fugacity fs of component s in a mixture is often used in place of the chemical potential to determine the equilibrium condition between phases. This quantity may be defined by the relation

ln(f~ / p) = (/a= -/a~ *pg) / RT (3.21)

where /a~*Pg is the perfect-gas chemical potential of pure s at the temperature and pressure in

question. In turn, f~ may be conveniently expressed in terms of the dimensionless partial fugacity coefficient d?= =(f=/x=p) which, in view of Equations (3.19) and (3.21), is most closely related to the residual chemical potential at specified (T,p,x v):

Rrln~=(T, Pn,XV)= /are=(T,p,x") = r e s " T , X v , X v /a= t ,P~ ) - RTlnZ(T, Pn )

(3.22)

The second part of Equation (3.22) may be obtained by using Equation (3.10). The residual part of the Gibbs energy may be obtained in a manner analogous to that used to obtain A~ es.

Table 3.3 gives the expansions of G res, S re=, H res, C~m, U re=, A re= and /a ~e= in terms of the

coefficients of Equation (3.6) with (T,p,x") as the independent variables. Other thermodynamic properties may be obtained through standard manipulations of the relations given in Tables 3.2 and 3.3.

* The chemical potential of components in the coexisting liquid cannot be obtained from the Virial equation but may be deduced from, for example, an activity coefficient model.

48

3.3 THEORY

In this section, the theoretical basis of the virial equation of state will be outlined and the relationships between the virial coefficients B, C, D, ... and intermolecular potential energies will be derived. Although the virial equation derives its name from the virial theorem of Clausius, only the more powerful statistical-mechanical development based on the grand canonical ensemble will be considered here (8,13,14). The development is carried out in both classical and quantum mechanics and the results are compared with experiment.

3.3.1 Properties of the Grand Canonical Partition Function

The ensemble methods of statistical mechanics have been discussed by many authors (11,13,15) and the well known results which form the starting point in the derivation of the virial equation are therefore reviewed here only briefly. The grand canonical ensemble is a hypothetical isolated assembly of some large number of systems of volume V each separated from its neighbours by walls which are permeable to both matter and energy. Since the entire ensemble is isolated it must be isothermal and each of its systems is therefore a replica on a macroscopic scale of a single open isothermal system with volume V, temperature T and absolute activity '~i for molecules of type i (~t, i "~" e~/Rr).

The number N i of molecules of type i in each system of the ensemble is not fixed but may fluctuate about mean values N i which, according to the postulates of statistical mechanics, are taken to be identical with the observable value of this property in the real system. The value of N i and of all the other thermodynamic properties of the open isothermal system may be obtained from a single function ~ called the grand canonical partition function.

For the general case of a mixture containing v components, the grand canonical partition function is given by

oo

Nl=O N2=O Nv=O c = l

(3.23)

The quantity 6)U,N~...Uv which appears as a coefficient in Equation (3.23) is the canonical-

ensemble partition function for a closed system containing the specified numbers of molecules in volume V at temperature T. This quantity is given by

ON, N~...N" = ~_~ exp(-Es/kT ) (3.24) J

where Ej is the energy of quantum state J and the summation is performed over all possible states of the system with the given volume and numbers of molecules.

The grand partition function is related to thermodynamic properties of the system by the equation

3= exp(pV/kT) (3.25)

This may be combined with the fundamental thermodynamic equation for the system with (T, V,~ v) as the independent variables (11)

49

d ( p V ) = S d T + p d V + R T ~ n~d ln3, i (3.26) i

to obtain all of the other thermodynamic properties. In particular, the amount of substance n i

of component i in the system is given by

L n i = N i = (OlnE/c31n3, i)T,V,~.j, i (3.27)

where L = R / k is Avogadro's constant. In the following sections, these relations are applied to derive the virial equation of state.

3.3.2 The Virial Equation of State from the Grand Partition Function

In principle, the thermodynamic properties of the system may be obtained directly from the grand partition function through Equations (3.23) and (3.25). Unfortunately, this requires computation of [~NlU2...U v for all possible values of N 1, N 2, --.; this is an impossible task. A

practical alternative which leads to the virial equation is to develop ~ as a power series in some suitably chosen set of variables which are then eliminated in favour of the number densities N i / V . Several authors have given derivations of the virial series for one-component (8,11,13,15), two-component (16) and multi-component systems (17,18). Here, the general case of the multi-component system is considered in outline.

It is first convenient to make two definitions. An 'active number density' z i is defined for each component in the mixture by the equation

z i = ( O } ° / V ) ~ , i (3.28)

This quantity has the property that z i ~ N g / V as X / ~ 0. A configurational partition function

Q~S...) is also defined for the system when it contains a total of N molecules of types (id', "") by means of the relation

(3.29)

In these equations, the partition functions O and Q are labelled in the same notation; thus

O~ ° ) - O u , N2...Uv, N = Ec \ lNc and (ij, ...) lists the type (species) of each molecule in turn. In

terms of these new variables, the grand partition function is given by

3 = 1+ ~Q(i)zi+ ~-'(Q~O)/2,)zizj + "~-'~~(Q}iJk)/3,)zizjzk+... i=l i=1 j=l i=1 j=l k=l

(3.30)

and takes the form of a power series in the z i in which the Nth group of terms pertains to a system of N-1 molecules. It is then a s s u m e d that an analogous expansion ofp in powers of the z i exists:

50

p / k r -- £ b(i)zi .-[- £ £ b~O')zizj -[- £ £ £ b~ijk)zizjzk -.[ -. . . i=1 i=1 j=l i=1 j=l k=l

(3.31)

The coefficients b~ j ) of this series may be related to the Q~J) by means of a simultaneous expansion of e x p ( p V / k T ) in powers of the z i and term-by-term comparison with Equation (3.30). The results of that operation are:

b(i) __ Q(i) /(I!V)= 1

b~U) : (Q~U)_ V 2)/(2!V)

b~iJ~) (Q~ij~) vQ~ij) VQ~ik) VQ~jk)+2V3 . . . . )/(3!V)

etc.

(3.32)

Incidentally, the quantities bff '), known as cluster integrals, which appear here play an important role in some formulations of statistical thermodynamics (15). They have the property that b~ j)--~ 0 when any one or more members of the group of N molecules is remote from the others.

The final step in the derivation is to eliminate the z i from Equation (3.31) in favour of the number densities N i / V . This may be done by noting that N i / V is given according to Equations (3.27) and (3.28)by

N~IV = (zi lV)(aln~lOZi)r .v,zm

= (zilkT)(c9 p 10zi)r.v.=j.,, (3.33)

from which it follows that,

( v / N i / V = z i - b(i) + 2~-'~ b~°)zj + 3~"~ b}O'k)zjzk +..

j=l j=l k=l

(3.34)

Equation (3.34) may be inverted to obtain a power series for z i in the number densities N i / V ,

Zi _ (~i/V)(a~i) q_ £a~iJ)(~i/V)_t_ ££a~ijk) (gjgk/V- - 2 )-k-.. "1 j=l j=l k=l

(3.35)

with coefficients:

a~ i) =1

a~O') : - 2b~0) ._ ( ik ) ( ik ) b~ iJ) a~ Uk, - 3bJ Uk, + 2b~U'(b~ °, + b~ik')+ 2b~ (b~ + )

etc.

(3.36)

51

Finally, combining Equation (3.31) with Equations (3.36) and collecting coefficients of (Ni /V) a, one obtains the virial equation of state with p, =(Y~iNi/LV) and with virial coefficients which are related to the cluster integrals as follows:

B U =-Lb~ °) (ik) (jk) Co~ = - L 2 {2b~ 'jk) - 41"~'(u)h(i~)3 ku2 '12 + "2~(~i)]'l(Jk)'2 + b~ 6 2 )}

etc.

(3.37)

These equations apply to any combination of like and unlike molecules and, by combination with Equations (3.32), the virial coefficients may be calculated from configurational partition functions. In the case of a one-component gas, the results simplify to B = - L b 2 and

C - - L 2 (2b3-4b 2). General relationships have been derived for the one-component gas by which any virial coefficient may be expressed in terms of a set of cluster integrals and by which each cluster integral may be expressed in terms of configurational partition functions (14,19).

In the following sections, the evaluation of Q~J) will be considered in both classical and

quantum mechanics for N = 1, 2, 3, .... However, one can see already that, in making the virial expansion, the intractable N-body problem posed by Equations (3.23) and (3.25) has been converted into a series of one-body, two-body, three-body .,. problems the first few of which one may expect to solve with relative ease.

3.3.3 The Virial Coefficients in Classical Mechanics

For almost all substances at accessible temperatures a classical treatment, augmented at low temperatures by small quantum 'corrections', may be used to obtain the configurational partition function and thus the virial coefficients with great accuracy. Only at low temperatures for the isotopes of hydrogen and helium is a fully quantum-mechanical treatment required. A fully classical treatment of Q~J) will therefore be considered first.

We begin by returning to Equation (3.24) for the canonical partition function. The energy levels over which the summation in that equation extends are those of the entire system and it is usual to make the assumption that they may be separated into two entirely independent sets. The first is associated with the internal degrees of freedom of the molecules and the second with the position, orientation and momentum of the molecules. With this assumption, which is amply justified for small rigid molecules, the partition function factorises into the product of two terms, the one (FN ~'j)) for the internal degrees of freedom and the other (0~u ~°) ) for the motion of the centres of mass:

@u(~ ) =.r(,Y)as(u)N =N (3.38)

By essentially the same assumption, ~7...) for N > 1 factorises into the product of N one- molecule partition functions with one term for each molecule. The configurational partition function is then entirely independent of the internal degrees of freedom and is given by

52

= (I ( V l )'<. ! c=l

(3.39)

The centre-of-mass term is given in quantum mechanics by the sum

q~") = ~ e x p ( - H j / k T ) (3.40) J

over all allowed (and distinguishable) translational quantum states of the N-molecule system. The quantity Hj which appears in this expression is the total (kinetic + potential) energy of the N centres of mass when the system is in the quantum state labeled J. The classical limit of q ~ ) is obtained by replacing the discrete spectrum of translational states by a continuum and the summation by integrals over the position, orientation and momentum of the molecules (20):

( )iS 3 N v • h FI .c = ~c Arc! e x p ( - H / kT) dP Ndr Ndo c = l

(3.41)

Here, H = 2~1(P i • P/2mi)+ U~ j''') is the classical equivalent of Hj, Pi is the momentum and

m i the mass of molecule i, U~ s ) is the total potential energy stored in the intermolecular

forces, and r N and o N denote the set of position and orientation vectors of the N molecules. In

addition, f2 c is a normalisation constant equal to the integral J'dco over all possible orientations in space of one molecule of component c.* The integration in Equation (3.41) is carried out over all possible values of linear momentum, over all positions of the molecules within the volume V and over all possible orientations of the molecules. The integration over momentum is easily carried out, yielding a factor (2rtmikT) 3/2 for each molecule, so that

q)(j...) = ( U v c=lA c 3Nc ,C2jNc N~,)-l lexp(_U~j...)/kT)drUdoU (3.42)

where

A i = h/(2xmikT) 1/2 (3.43)

is a de Broglie wavelength. Since, for a system containing only one molecule, there is no

intermolecular potential energy, q~(c) is given by

* The appearance of the remaining terms in the prefactor of Equation (3.41) is explained as follows. Since molecules of the same species are indistinguishable, configurations which differ only by the assignments of labels to the individual molecules are actually identical and should not be counted separately. It is therefore necessary to divide by the Arc! possible assignments of labels amongst the N molecules for each component c in the system, h 3u is a normalisation factor for the

c

integrations over position and momentum chosen such that the classical and quantum treatments of the partition function converge in the limit of high temperatures.

53

cI91(c) = ( V / A 3c) (3.44)

and, finally, the configurational partition function is

(n t ls Q~J) = ~¢'~c NC exp(-U~J)/kT)dr N d o N c=l

(3.45)

Note that, while 05u~/J'") is dimensionless, Q~J) has the dimensions of V u.

In order to evaluate the virial coefficients, knowledge is required of the potential energy of clusters of N molecules with N = 2, 3, .... For N = 2, the situation is comparatively simple and U 2 = u(r12,o12 ) is just the intermolecular potential-energy function for a pair of molecules with separation r12 =lr 2 -r l l and relative orientation given by a set of (body-fixed) coordinates

denoted by co12. The precise quantitative determination of u(r~2,o12 ) for a range of interesting molecules remains one of the most resilient problems in chemical physics and our present knowledge is quite restricted. The pair potential is however known with considerable accuracy for several monatomic substances, especially He, Ne, Ar, Kr, Xe and their mixtures, and for a number of simple systems containing molecular hydrogen (21,22). Considerable progress has been made for other diatomic gases, for mixtures of diatomic and monatomic components, and for a few other systems such as water (22). For other polyatomic systems, it has so far proved impossible to establish the multi-dimensional potential energy surface in any precise way. In these cases only crude models are available, many of which neglect completely the orientational dependence of the pair potential. The origin and determination of the intermolecular pair potential has been reviewed by several authors (21-23).

For virial coefficients above the second, U N is required for N > 3 and the question therefore arises as to whether or not this may be expressed simply as a sum of pair interaction energies. It is now well established that the approximation of pair-wise additivity is in fact fairly poor (24) and that one should write

N-1 N

U N -- Z Z Uij "+" AuN (3.46) i=1 j= i+ l

where u o. = u(r0.,o U) and Au N represents the deviations from pair-wise additivity in a cluster

of N molecules (8). Some information is available on the behaviour of Au 3 and, in particular, the asymptotic long-range form of the potential for atoms and spherical-top molecules is known from the theory of dispersion forces to be (25)

Au 3 = v3(1 + 3cosOicosOjcosOk) / ( r i j r i~ r j~ ) 3 (3.47)

Here, 0 3 is a dispersion coefficient and 0 i is the angle subtended at molecule i by moleculesj

and k. 03 is related approximately to the coefficient C 6 of r "6 in the asymptotic long-range expansion of the two-body potential by 03 = J a C 6, where a is the polarizability. The corresponding result for linear molecules is also known (26).

The cluster integrals b~ U) and b~ °k) may now be evaluated from Equations (3.32) and (3.45) and the results are

54

/52( U)= (2!V"Q2)-I ffo dridrjdc°idoj

= ( 3 ! V " Q 3 ) -1 fL f i j f i k f # + fijf ik + f i j f jk + fikf jkldridrydrkdc°idc° yd¢°k

+ (3 ! Vff23) -1 f(l+fj)(l+f/k)(l+fjk)f: k dridrjdrkd¢o idfh jdo3 k.

(3.48)

Here use has been made of the Mayer function f/j, defined by

f0 = exp(-uu/kT) - 1 (3.49)

and of an analogous function of the three-body potential:

f/jk = exp(-Au3 / k T ) - 1 (3.50)

For neutral molecules, bothf/j andfj k approach zero rapidly as the separation of the molecules is increased thus ensuring that the integrals converge. Notice that, if Au 3 is zero, fijk vanishes.

Since the intermolecular potential energy depends only on the relative position and orientation of the molecules, it is convenient to transform the integration variables from the space-fixed to the body-fixed frame. In the case of the position vectors, the transformations are

= 4 nri j dr, j V dridry 2 -1 8n r r r dr dr dr V dridrflrk= 2 i j i k i k ij ik jk

(3.51)

and the integrations for b~ ~/k) then extend over all values of (rij,r/k,rjk) which form a triangle. It

is also convenient to extend formally the range of integrations over molecular separations to infinity, rather than just to the walls of the container, and this is justified by the rapid convergence of the integrals. For the orientation vectors, one may adopt any convenient set of angular coordinates provided that the appropriate normalisation constant is used. Equations (3.37), (3.48) and (3.51) are now combined to determine B and C.

The second virial coefficient corresponding to the interaction of molecules 1 and 2 is given in body-fixed coordinates as

B12 =-(2nL/£212) ~ I,~,2)f2r~22dr~2d°~12 (3.52)

where ~r~12 is a new normalisation factor equal to the integral ~ dCOl2 over all possible relative

orientations. In the case of the third virial coefficient corresponding to the interaction of molecules 1, 2

and 3, the effects of additive and non-additive intermolecular potential energies separate so that

C123 = C ~ + AC123 (3.53)

55

The additive and non-additive contributions are given by

cla2d3 d = -(8~;2L 2/3~~23) If12f3f23 r12r13r23dr12dr13dr23dO~ldCO2do~3 (3.54)

and

AC123 = -(871:2L 2/3~"~23 ) I(l+f12)(l+f3)(l+fE3)f23 r~2r13r23dr~2dr13dr23&o ldf9 2df93 (3.55)

where £2 123 = I d~ldo,)2d~ 3 . Explicit expressions for the additive and non-additive parts of D have been given by Mason and Spurling (8); for higher virial coefficients it is preferable to use graphical notation to simplify the otherwise complicated expressions (15).

As a practical consideration, evaluation of B is easy by quadrature even for non-linear molecules (a six-dimensional integral is required). Similarly, evaluation of C, D or E for monatomic molecules is quite easy but, as soon as orientational dependence of the intermolecular potential is included, one is faced with a significant computational task. For example, to evaluate C for three linear molecules, one must integrate over nine coordinates and it is then probably necessary to resort to Monte-Carlo methods at least for the angular variables.

In the few cases where the intermolecular pair potential is accurately known, and quantum effects are unimportant, values of B calculated from Equation (3.52) are in close agreement with experiment. This situation is illustrated in Figure 3.4 by the example of krypton. The experimental third virial coefficients of krypton (28-30) are also shown in Figure 3.4 together with values calculated by means of Equations (3.54) and (3.55) on the assumption that Au 3 is given by Equation (3.49) with the theoretical value of o 3 (23). In view of the considerable scatter amongst the experimental results, the agreement is satisfactory and certainly much better than when AC is neglected.

3.3.4 Quantum Corrections

Except for hydrogen and helium at low temperatures, quantum effects may be accounted for rather accurately by the addition to the results of the previous section of small quantum corrections. It turns out that these terms may be obtained by expansion of the quantum- mechanical partition function about the classical limit in powers of h 2 (31). At the low temperatures where these corrections are significant, it is safe to treat the molecules as rigid rotators. Only translational and rotational terms will then be present in the expansion.

The treatment is most complete for monatomic systems where B is known correct to 0(]/6) (32) and C is known, including the effects of non-additive intermo_lecular forces, correct to O(1/4) (33,34). In the case ofpolyatomic molecules, the expansion of B is less well developed and there appears to have been no calculations for C. The translational and rotational quantum corrections to B were worked out for linear molecules to O(h 2) by

Kirkwood (35) and later to O(h 4) by Wang Chang (36). However, Pack (37) has pointed out that, in the derivation presented by Wang Chang, the transformation from space-fixed coordinates to body-fixed coordinates was made incorrectly with the consequence that a

56

-50

.--, -100 '7,

0 E -150

e0

E -200 0

m -250

-300

-350

4 0 0 0

3 0 0 0 o E

2 0 0 0 o,2.

o 1 0 0 0

o

t .0 t 200 300 400 500 600

T ( K ) I I I , I , I , I

100 200 300 400 500 600 700

T (K)

Figure 3.4 Second virial coefficients of krypton. O, recommended (experimental) values from reference 4; ~ , calculated from the interatomic potential of Aziz and Slaman (27). Insert: third

• ~ C add -I--A C ; C add . virial coefficients. O, (28); l , (29); O, (30); , . . . . . . ,

coriolis-type rotation-translation coupling term was neglected. It has also been observed that some terms of O(h 4) are missing from Wang Chang's formulae (38). A revised semi-classical

expression for B,

B = Bc~as s + ( h Z / 2 # ) B t l + (h2/24)B)(~ ) + (h2/212/~"rl]R(2) "3t- ( ]~2 /21 -~ )Bc l (3.56)

correct to O(h 2) has been derived and may be applied to all rigid molecules (both like and unlike) except asymmetric tops. In Equation (3.56), Bclas s is the classical second virial coefficient, # is the reduced mass and 11, I 2 are the moments of inertia of the two molecules.

#(°and Be1 determine respectively the leading translational, rotational The coefficients Btl, --rl and coriolis contributions to the quantum corrections. Pack (37) gives a general formula in space-fixed coordinates that relates the leading quantum corrections to the translation-rotation Hamiltonian. Specific expressions were also given in body-frame coordinates fo r linear molecules and for the interaction between an atom and either a linear molecule or a spherical top molecule. Here, only the results for a system of linear molecules are given:*

* Altemative expressions for B r and B e are given in reference 37 for the case in which the intermolecular potential is expanded in spherical harmonics.

2zrL ]" ~{exp ( -u l kT) 1} ,2 _ _ r122dr12do3 Bclass = ~('~2 0 (~12)

57

(3.57)

oo

zcL [ Sexp(_u/kV)(t3u/c3r)2 r122dr12d~12 gt' = 6(kT)3L'-~2 ;(~,2)

(3.58)

B r i) - - ,

oo

nL I [.exp(-ulkT){(c~ulOOi )2 +(~ula~i) ~ cs#O/} ~l~d~,~dco,~ (3.59)

rtL Be' = 6(kr)3D~2 S ~exp(-u/kr)F(r,O,,O2,~2 ) dr~2dco,2

0 (co12)

(3.60)

where (39)

F(r,O~,02,~2 ) = (0u/00~) 2 + (0u/002) 2 + {csc20~ + csc202 - 2}(0u/0~2) 2

- 2(c9u/0442) sin ~2 {(0u/00~ ) cot 02 + (0u/002) cot 01}

+ 2 cos~2 {(Ou/OO~)(Ou/OO2)-cotO ~ cotO2(Ou/O~2) 2 }

(3.61)

Here, 0 i and ~b i are body-fixed polar and azimuthal angles, $12 = ($2-$1), and

1 1 2

=-

(o211) -1 -1 0

(3.62)

Equations (3.57) - (3.60) are easily specialised to atom-linear molecule systems, where u no longer depends upon 02 and ~b12. Btl further reduces to the corresponding monatomic result when u is a function only of r. These formulae may also be applied to the interaction second virial coefficient B12.

An important theoretical result is that Bclas s always gives a lower bound for B itself so that the quantum effects are always positive (37). It appears that the semi-classical expansion of B in h 2 converges satisfactorily for neon and for heavier monatomic systems at all temperatures above the normal boiling temperature. This is illustrated in Figure 3.5 where B and, on an expanded scale, the first two translational quantum corrections are plotted for neon. For helium, the series converges satisfactorily only at temperatures above about 40 K. Except for the isotopes of hydrogen, molecular gases are generally sufficiently heavy that translational quantum effects are given with sufficient accuracy at temperatures of interest by the correction of 0 ( ] ; / 2 ) . Whether or not this is also true of the rotational term depends upon the moment of inertia and the degree of anisotropy of the intermolecular potential. For non-polar molecules, the rotational and coriolis terms together are generally smaller than the translational term but for polar molecules, especially polar hydrides, the reverse is true. It appears that B c is always significantly smaller than B r. These points are illustrated in Figures 3.6 and 3.7, where B, Btl

58

-30

-40

"7, _== 0

E

E 0 v

-10 0 E O3 E 20 0 "

i !

~ i ~ ' ~ " " 3 Btl .... '- t

Y --t / 1 40 60 80 100

I I I T (I K) I

50 75 100 125 150

T (K)

Figure 3.5. Second virial coefficients of neon. 0 , recommended (experimental) values from reference 4; , calculated correct to the second quantum correction from the interatomic potential of Aziz and Slaman (40). Insert: Btl and Bt2.

15

10

±

6

o t~~___. Brl o k - - - ~ 2 7 ""--B,~

1 0 0 200 300 T (K)

-5 I , I. I i 100 200 300 400

T (K)

Figure 3.6. Second virial coefficients of hydrogen. O, recommended (experimental) values from reference 4; ~ , calculated correct to the first quantum corrections from an anisotropic intermolecular potential model. Insert: Btl, Brl and Be.

59

-100

"T o -200 E

03 E o -300 133

-400

-500

4¢ ~ 2o

-"-" 15 ,_.... o E

10

rn < 5

0

I I I

200 250 300

J Brl

200 250 300 350 400

T (K) I I I

3 5 0 4 0 0 4 5 0

T K

Figure 3.7 Second virial coefficients of hydrogen chloride. O, experimental results of Schramm and Leuchs (42); , calculated correct to the first quantum correction from the Stockmayer potential (43) with s/k = 413 K and cr - 0.2591 nm, where s and cr are the scaling parameters for energy and length in the isotropic part of the potential. The dipole moment was taken as 3.53 × 10 -30

C.m (22). Insert: Btl and Brl.

and Brl are p lo t t ed for H 2 and for HC1; in these cases, Bcl is negl ig ib le . Fo r h y d r o g e n *, the second t rans la t iona l q u a n t u m cor rec t ion is also shown.

* The potential model used here is given by:

U(r, 0,, 02, ~b~2 ) = A exp(-Br)[1 + C{P2 (c~) + P2 (c2)} ] - F(r)IC6r-6 f -3(02 / 4rcs0)r-Sf~ + 9(02a/4ZCSo)f3]

where

2 2 2 f~ = 1 - ~¢ - 3~c(1 - ~) (c2q-c 2) --~1~.32(s1s2c - 2CLC2) 2, f2 = 1 - 5(c~ + c2) - 1 5c, c 2 + 2(s~s2c - 4qc2) 2,

f 3 = S4"JrS2 "Jr4 4C4+4C4, ~C = (ali-a±)/3~-, ~ ' - 2 ~ _ 1 _ + 1 ~ 1 ] , C i = COS0/ , S i = s i n 0 i , C = COS~)12 ,

and

F(r) = exp[-{(ro/r ) - 1} 2 ] r < r o

=1 r > r o

A and B were fitted to the high-temperature second virial coefficients while C and C6 were taken from (41): A/k = 3.545x107 K, B = 38.27 nm -1, C = 0.14, C6/k = 0.0899 K.nm 6, r0 = 0.3 nm.

Polarisabilities a I and a2_ and the quadrupole moment ® where taken from reference 24.

60

3.3.5 The Virial Coefficients in Quantum Mechanics: Helium and Hydrogen

For the isotopes of helium and hydrogen, the semi-classical expansion of the virial coefficients does not converge at low temperatures and, to obtain reliable results, a fully quantum-mechanical evaluation is then required. Although the relations given in Section 3.3.2 between the virial coefficients and the configurational partition function remain formally valid, the evaluation and treatment of the energy levels from which QN may be calculated is different in the full quantum treatment. The origin of these differences lies primarily in quantisation of the angular momentum.

It is important to recognise that we should observe quantum statistics as well as quantum mechanics in a correct treatment. It turns out that the statistical restrictions on the allowed states of a system become important at very low temperatures, and this aspect of the problem will be examined first. The essential limitations imposed by quantum statistics are well known and may be stated as follows. When a system is composed of two or more indistinguishable molecules, the only allowed states of the system are those which are either symmetrical or anti-symmetrical with respect to the interchange of two particles. Only symmetrical states are found for systems composed of indistinguishable atoms or molecules which contain an even number of elementary particles; such systems are said to observe Bose- Einstein quantum statistics. For systems composed of indistinguishable atoms or molecules which contain an odd number of elementary particles, Fermi-Dirac quantum statistics apply and only anti-symmetrical states are allowed. The adoption of correct quantum statistics restricts the number of quantum states from which the canonical partition function of a system of indistinguishable molecules is constructed. In the case of distinguishable molecules, no statistical restrictions apply.

The symmetry of a quantum state is determined by the parity of the wavefunction h TM

which describes it.* For indistinguishable atoms, the parity of h TM is determined by the nuclear- spin and translational states of the system and, in the case of non-zero nuclear spin, a number of degenerate nuclear-spin states is available. For molecules, the parity of 7* is determined by nuclear-spin, translational, rotational and vibrational states. However, the isotopes of hydrogen are the only molecular systems where statistical quantum restrictions have a significant influence on the virial coefficients and, even then, the effects are restricted to temperatures below about 10 K. Under those conditions, only the lowest accessible vibrational and rotational states will be populated in the gas and the treatment of internal modes is thereby greatly simplified. It is however necessary to recognise the existence of (distinguishable) ortho and para H 2 and D 2 molecules.

In view of the forgoing remarks, only a single (but possibly degenerate) internal energy level need be considered for an isolated atom or molecule at temperatures where statistical effects are of significance. The internal partition function F~ is therefore given by

F1 = go exp(-~o / kT) (3.63)

* A symmetric (or even-parity) state is characterised by a wavefunction which is invariant under the operation of exhanging the co-ordinates of any two molecules. Conversely, an antisymmetric (or odd-parity) state has a wavefunction which, under the same operation, changes sign.

61

where e0 and go are the energy and degeneracy. Since the zero of energy may be defined at will, one might as well measure all energies relative to that of the isolated molecule in the lowest allowed energy level. In that case, e0 = 0 and

F~ = go (3.64)

In the case of an atom with nuclear-spin quantum number s, go = (2s + 1) while, for a diatomic molecule, go = ( 2 s + l ) ( 2 j + l ) , where j is the rotational quantum number of the lowest allowed energy level. Since, for a system containing just a single molecule, the question of parity does not arise, the overall partition function O 1 is simply the product F~ q~l, where q~l is given by Equation (3.40) with its unrestricted sum over translational states. For all cases of practical importance, the sum over the translational quantum states may be approximated by an integral giving q~l = ( V / A 3) and thence

6) 1 = g o ( V / A 3) (3.65)

When the system contains two or more molecules, the calculation of the partition function is more involved. In the following treatment, attention will be restricted to the second virial coefficient and at first to the case of a one-component gas. The discussion starts with spherically-symmetric systems and proceeds to the case of linear molecules. The corresponding results for unlike atoms or molecules are also obtained rather easily by dropping the symmetry constraints.

In the case where the intermolecular potential is (or is assumed to be) spherically symmetric, it is still convenient to treat the translational and internal degrees of freedom of the molecules separately. However, in order to satisfy the symmetry constraints, both the internal and translational parts of the partition function should be separated into terms arising from quantum states of given parity. ~2 is therefore split into two terms, 052 and q~2, which are obtained by summing only over translational states of the specified parity (+ or -). Similarly, the (g0)2 degenerate internal states are separated into symmetric and anti-symmetric terms and the overall partition function 0 2 is then formed from products of internal and translational terms which have the required parity:

+ (3.66)

In this equation, W2 + and W 2- are statistical weights which are determined as follows. For

indistinguishable atoms or molecules which obey Bose-Einstein statistics, (go)2W2 + is the

number of symmetric internal states and (g0)2W2 - is the number of anti-symmetric internal states of the two molecules each in its lowest allowed internal-energy level. In the case of Fermi-Dirac statistics, the situation is reversed while, for distinguishable molecules, Boltzmann statistics apply and W2 + = W 2- = 1/2!. The statistical weights W2 ÷ and W 2- for the isotopes of helium and molecular hydrogen are given in Table 3.4. Consistent with the assumption of a spherical intermolecular potential, hydrogen molecules are treated as spherical particles with spin equal to the vector sum of the spins of the constituent nuclei.

62

Table 3.4. Statistical weights for the isotopes of helium and hydrogen (44,45).

System j s go W2 + W2- System j s

1 2 1/4 3/4 o-D 2 0 0,2 3He 2

4He 0 1 1 0 p-D 2 1 1 1 3 p-H 2 0 0 1 1 0 HD 0 2,2

o-H 2 1 1 9 5/9 4/9

g0 W2 + W 2-

6 7/12 5/12

9 5/9 4/9

6 5/12 7/12

In view of Equation (3.66), the configurational partition function of a pure gas is given by

Q2 = 2 ( V / ~ 1 ) 2 ~ 2 = 2A6(Wz+(I)~ + W2(I)2) (3.67)

and the second virial coefficient becomes

B = W:+B + + W2-B- (3.68)

Here, B + and B- involve only translational partition functions and are given by

B + = - ( L A 6 / V ) ( ~ - ½ ~ ) (3.69)

An important theoretical consequence of the adoption of quantum statistics is that B does not vanish in the absence of intermolecular forces. Indeed one can show that, for non-interacting indistinguishable atoms or molecules (14),

• ~ - ½ ~ =+_2-s/2(V/A 3) (3.70)

so that the second virial coefficient, n(°) is given by " " e x c h ,

Be °) = W2 + B(°) + + W 2- B(O) - xch

= _ 2-~LA 3(W2+ _ W2- )

with

(3.71)

B (°)+ = T-2 -~ L A 3 (3.72)

In view of the existence of an exchange term even for non-interacting molecules, it is convenient for real systems to deal with that part of B + which arises from the intermolecular forces:

63

B + - B {°)± = - ( L A 6 / V ) ( ~ 2 - ~o~±) (3.73)

The partition functions 052 and 05~ °)± which appear here are given by

052 = ~ exp(-H,~/kT) t ~{2°} ± = ~ exp( -H( f f / kT)

(y

(3.74)

where a represents the full set of six translational quantum numbers for the two-molecule system (subject to the implied symmetry restrictions) and H a and H2 °} are the translational

(kinetic + potential) energy of the system with and without intermolecular forces. Both Ho

and H(~ °) may be separated into the sum of two terms each associated with three quantum

numbers, the one term for the translational kinetic energy of the centre of mass, and the other for the (kinetic + potential) energy of the relative motion. Integration over the former yields a common factor of 2 3/2 (V/A 3) so that

B±-B(°)± =-23/2LA 31~. e x p ( - H , ~ . / k T ) - ~ exp(-H(~°)/kT)) (3.75)

where a' indicates that the summations now exclude the centre-of-mass terms. In the present case, with an isotropic intermolecular potential, the possible states of

relative motion are characterised by the quantum numbers (l, ml, n ). Here, l is the quantum number for the orbital angular momentum 1, m t gives the orientation of 1 in space, and n determines the energy. It is convenient to divide these states into bound levels, with discrete energy levels C.nl , and states with a continuum of positive energy levels H~I; each such state is (2l+1) fold degenerate corresponding to the (2l+ 1) allowed values of m l. The summation over n for positive energies may be replaced by an integral over the continuous variable ~c,

exp(-Hn~/ kT) = ~ exp(-J6rc2 )( dn/drc ) drc n

(3.76)

where rc is defined such that Hn~t = h2rcz/21a, ~ is the reduced mass, [3=h2/21akT, and

(dn/dtc) is the density of quantum states in the continuum. A similar equation holds for the

summation over the energy levels 14 (°) but with a different density of states: (dn/d~c) (°} The * " n l

quantity (dn/d~c)-(dn/d~c) (°) is known as the excess density of states and may be related rather easily to the quantum-mechanical phase shifts rh(rc ) of atom-atom scattering theory (46):

(1/ ~ )( drlt / d~ ) = ( dn/ d~c ) - ( dn/ d~ ) (°) (3.77)

Equation (3.75) then becomes

64

B + - B ( ° ) + --- - 2 3 / 2 LA3lZeven(2l + 1)B l t

B- - B (°)- = -23/2 LA 3y' (21 + 1)B 1 l odd

(3.78)

where*

B t = ~ . , [ e x p ( - e . n , l k T ) - 1] + (2fl/n) ~r/~(~:) exp(-/7~c2)~c d~: n

(3.79)

One can therefore write the complete second virial coefficient as

B = Bdirect + Bexch (3.80)

where

Bdirect -- -2 v2 LA 3 ~ (2l + 1)B t (3.81) l

and

(3.82)

In the case of unlike atoms of types 1 and 2, Bexch = 0 and it is easy to show that

B12 = -(LA 3/2)~-~(2l + 1)B~ (3.83) l

where A 12= ( h2/2n#kT) 1/2. In order to evaluate the second virial coefficient, one must calculate from Equation (3.79)

the quantity B t for 1 = 0, 1, 2, . . . . 6nl and r/t are required for this purpose and both may be obtained by numerical solution of the radial SchrOdinger equation for the two atom system. Although once considered to be a computationally demanding procedure, such calculations are today routine and may be performed for helium over a wide range of temperatures. Numerical methods for the precise evaluation of the phase shifts have been described by Boyd et aL (47), while Cooley (48) and Cashion (49) have described methods of calculating the bound-state energies. In practise, numerical calculation of the sums over l are usually truncated at some value l = lma x and the remaining contribution of terms with l > lma x are estimated using the Born approximation. Similarly, the integrals over ~¢ may be truncated at some upper limit and appropriate choices, together with some highly accurate calculations, have been reported by Aziz and Slaman for 3He and 4He (50). It appears that the intermolecular potential of helium

* The form of this equation is arrived at after integration by parts and noting that, according to Levinson's theorm, rll(O)/n is equal to the number of bound states in the well for the given l.

65

supports only one bound state for 4He (l = n = 0) with an energy given by Eoo/k = -0.0016 K (50). In the case of 3He, there are no bound states.

The results of full quantum-mechanical calculations of B for the isotopes of helium, based on an accurate intermolecular potential (50), are compared with experiment in Figure 3.8. Remarkably, the agreement is within a few tenths of a cm3/mol over the entire temperature range. For comparison, the semi-classical results for 4He are also shown. It is interesting to note that, while the contribution of #(°) appears to be significant up to temperatures of order *"exch 100 K, the total I Be~ch [ given by Equation (3.82) drops off very rapidly with increasing temperature and is negligible (< 0.01 cm3.mo1-1) for 4He above 5 K and for 3He above 6 K (47). At higher temperatures, quantum effects are confined to the difference between Bdirect

and the classical second virial coefficient; the origin of such differences lies is the quantisation of angular momentum. At temperatures where the semi-classical treatment of B is valid, these differences are accounted for correctly by the quantum corrections and it would therefore be incorrect to then add n (°) to the result. ~'exch

25

-25

"S -50 E

-75 E O

v -100 rn

-125

-150

-175

. . . . . . . . i . . . . . . . . i . . . . .

3He ~ ~ 7 ........ ':"- . . . . . . . . . . . . . . . . . . . .

/ / \ V ,.o / "4 e ,0: ; ; 4

. . . . . . . I . . . . . . . . ~ T (K ) . . . . .

10 100 700

T (K)

Figure 3.8 Second virial coefficients of 3He and 4He. ~, (51); O, (52); l , (53); El, (54); ,ik, (55); IF, (56); 0 , (57) . ~ , quantum-mechanical calculation of B (50); .............. , semi-classical B for 4He to O(h 2 ); , semi-classical B for 4He to O(h 4 ).

Insert: ~ , Bexch" -, . . . . . . , ~'exch#(0) (47).

Apart from 3He and 4He, a fully-quantum mechanical treatment is required only for the isotopes of hydrogen. For H 2 and D 2, but not for HD, the molecules exist in distinguishable ortho (o) and para (p) forms and the gas is best considered as a mixture with second virial coefficient

66

B z 2 X o X p B o p at - 2 -- Xo Boo -b xpBpp (3.84)

Since calculations indicate that Bexch is completely negligible in all cases at temperatures above 10 K, only B~ectwill be considered further here.

Molecular systems are characterised by the presence of angular-dependent intermolecular forces. These have the effect of coupling the angular momentum of the individual molecules to the intermolecular axis and thereby lifting the degeneracy of the states of relative motion. Strictly, only the total angular momentum J is a rigorous constant of motion but, in systems containing H 2, HD or D 2, the anisotropy is weak and the angular momenta J1 and J2 of the individual molecules are very nearly conserved. Each state with orbital angular momentum 1 is associated with sub-states characterised by the quantum numbers J and K. Here, J is the quantum number for J and K is the quantum number for the internal rotation angular momentum K = J l + J2.

In view of the coupling of rotational and translational states, one must return to Equation (3.67) and proceed with the full partition function. In the case where only the lowest allowed rotational energy level is populated, the development is rather simple and parallels closely that given above. The bound-state energies of the two-molecule system are now characterised by (n, l, K, J) and written c,~Ks. Considering for generality the case of unlike molecules with

rotational quantum numbers jl andj2, the result is

_LA 3 12 Bl'2 direct = 2(2jl + 1)(2j2 + 1)

0; ) x ~ )-"~(2J + 1) exp(-eSntz~/kT)+ [(dn/d~)-(dn/drz)(°)]exp(-[3rcZ)d~: \K=Ij~-j2I l=0 J=ll-KI

The excess density of states in the continuum is given by (41,58)

( dn/ drc) - ( dn/ d~¢) (°) = (i/2r0 Tr (S dSt/dr¢) (3.86)

where S = S(~c) is the S-matrix of molecule-molecule scattering theory, St is the adjoint of S, and Tr denotes the trace operation over the full set of quantum numbers (l, K, J). The result of

J this operation may be expressed in terms of a set of generalised phase shifts r/tK(~:) and, for

each l, (2 Jl + 1)(2 J2 + 1) such terms appear. These last two equations may be applied to obtain Baireet for any combination of like or

unlike molecules or atoms. In the case of an atom-molecule interaction, the sum over K drops out while, for atom-atom systems, J = l and the sums over both K and J drop out. In the latter case, only the diagonal elements of the S matrix are non-zero and they are related to the scattering phase shifts by S~/(r¢)= exp[2ir/t(~)]. Equation (3.86) then reduces to Equation (3.77) and Equation (3.85) reduces to Equation (3.81).

J and S(~:) again involves solution of SchrOdinger's equation for the Determination of ~'nlK system and such calculations are tractable for the low temperatures at which the quantum treatment is required. The S-matrix may be determined to high accuracy in the close-coupled approximation while the bound-state energies may be obtained by the same general method

67

(59) or by the more efficient secular equation method (60). It is also possible to rely on spectroscopically determined values of ~'nlK J and it is worth remarking that, for the systems of interest, only n = 0 states have been observed and there is little evidence of splitting of the l states due to anisotropy of the intermolecular potential (61). The sums over K and J in Equation (3.85) could therefore reasonably be collapsed leaving a bound-state contribution to Bdirect identical in form with that of a monatomic system.

Although Equation (3.85) is restricted to molecules in a single rotational level, it is clearly possible to derive a more general expression in which the full set of molecular rotational energy levels is retained. Alternatively, the gas may be treated as a mixture of species having different values of j. Unfortunately, close-coupling calculations become very time consuming once more than a few rotational levels are populated. It is notable however that, for p-H 2, thej = 2 level only reaches 1 per cent occupancy at about 75 K while, for o-H 2, the temperature must reach 145 K before the j = 3 level reaches the same occupancy. For purposes of comparison, the semi-classical formula, including the translational corrections to O(h4), should be of sufficient accuracy at temperatures above about 100 K (see Figure 3.6).

3.4 CALCULATION AND ESTIMATION OF VIRIAL COEFFICIENTS

Although an extensive database of experimental virial coefficients is available (4), methods for the calculation and estimation of virial coefficients are important for a number of reasons. In the first place, there are still many systems (especially mixtures) for which B and C are not known at the temperatures of interest and, in these cases, estimation methods can be very useful. Even when experimental data are available, in may be desirable to establish a soundly-based correlation and, for that purpose, a model intermolecular potential with appropriately chosen parameters can be very powerful. Finally, it is sometimes useful (for example, in thermodynamic perturbation theory) to have essentially exact virial coefficients for some simple model potential.

3.4.1 The Virial Coefficients of Model Systems

Virial coefficients have been evaluated for several isotropic model intermolecular potential-energy functions including the square-well, Lennard-Jones (n,6), exponential-six and Maitland-Smith functions.* A number of tabulations of B, C , D and E are available (6,21,62,63) and, as mentioned above, the evaluation of B and C by quadrature is quite routine. The parameters of such a model may be optimised fairly easily in a fit to virial coefficients or related quantities such as the Joule-Thomson coefficient or acoustic virial coefficients. The model may then permit extrapolation or the estimation of other properties, often with high accuracy. It appears that simple isotropic potential models can offer a very good account of the second and third virial coefficients of non-polar molecules provided that non-additive three-body forces are included in the calculation of C (5,64). Equation (3.47), with u3 treated as an empirical parameter, appears to be quite adequate (64).

Although the importance of analytical and tabulated results has been greatly diminished by the speed of modem computers, there is one class of model systems, namely the convex hard bodies, which are of special importance as reference systems in statistical

* Descriptions of these models may be found in, for example, reference 21.

68

thermodynamics and for which the existence of tabulated results is important. The virial coefficients of such systems are independent of temperature and some of the results are mentioned here.

The simplest of the convex hard bodies is the hard sphere. Analytical expressions for B, C and D, and precise numerical results up to the eighth virial coefficient, are available for a gas composed of such molecules (13,65). Estimates of the ninth and tenth virial coefficients have also been made (65) and the results are given in Table 3.5. A considerable amount of work has also been done on non-spherical convex hard cores for which the second virial coefficient is given by B = L v ( 1 + ?'), where v is the volume of one molecule and ?' is a shape factor which may be determined from the mean radius of curvature, the surface area and v (66). For spherical hard cores, ?' = 3 while, for other shapes, ?' > 3 and increases with the eccentricity of the molecule. Virial coefficients up to the fifth have been computed numerically for hard ellipsoids of revolution (67), spherocylinders (68) and chains of fused hard spheres (69). Except for highly eccentric molecules, the results are found to correlate rather well with the shape factor ), (6 7).

Table 3.5 Virial coefficients of hard spheres of diameter d (13,65).

B = 2toLd3~3 = b o

c = (5/8) bo 2

D = 0.2869495 b 3

E - (0.110252 + 0.000001) b 4

F = (0.038808 + 0.000055) bo 5

G = (0.013071 + 0.000070) b 6

H = (0.00432 + 0.00010) bo 7

I~, 0.00142 bo 8

J ~ 0.00047 bo 9

It appears that the virial series for hard spheres is convergent up to the density at which computer simulations indicate that solidification occurs (70). For non-spherical hard cores, Pad6 approximants to the compression factor formed from the available virial coefficients also agree closely with the results of computer simulations up to very high densities (71). These observations suggest that the virial equation is valid for all fluid states of this class of system.

Hard bodies do not exhibit a vapour-liquid phase transition but the addition to the intermolecular potential of an attractive field outside the hard core does give rise to this phenomenon. A simple and very interesting model of that kind is the one considered by van der Waals (72) in which the molecules are treated as spherical hard cores of diameter d with a weak but long-range attractive interaction such that the intermolecular potential is:

=-E / ~ 3 d < r < 6 d

=0 r > 6 d

69

(3.87)

The second virial coefficient for this system is b0{1-(e/kT)} but the dimensionality of the

cluster integrals is such that all higher virial coefficients are identical with those of the corresponding hard sphere system. Consequently, for this model system the virial series is also convergent for all fluid states, both gas and liquid, and the equation of state is analytic everywhere including at the critical point. However, for systems with an intermolecular potential o f f i n i t e range, it is known that the equation of state is non-analytic at the critical point and that the virial series must, therefore, have a finite radius of convergence on isotherms close to the critical temperature. The convergence of the virial series elsewhere for this class of system is not established theoretically owing, at least in part, to the absence of a complete set of virial coefficients. For Lennard-Jones molecules, virial coefficients up to the fifth have been computed (6) and found to diverge towards -oo as T ~ 0; the results are plotted in Figure 3.9. This behaviour is usually taken as evidence that the virial series does not converge for such molecules in the liquid state. However, a satisfactory virial expansion might exist for both gas and liquid branches below the critical temperature if the higher virial coefficients adopt finite positive values (73). The question therefore remains open.

.... ! . . . . . . . . 09 ~ - (''(~ O ---

~" -1

0

-~._ -2 D* B* .,.- >

0

"U

~ -4 n'-

-5 0.5 1 2 5 10

T Ot

Figure 3.9 Reduced virial coefficients for the Lennard-Jones T*= k T / s , B*= C/bo , C* = C / b g, D* = D / b 3 and E* = E / b 4 .

(12,6) potential (6).

70

3.4.2 Estimation of Virial Coefficients

For cases in which experimental values of B and C are unavailable, some method of prediction is required and for this purpose the principle of corresponding states is usually applied. In its simplest form, the principle applies to systems whose intermolecular pair potentials may be written in the form u(r )= e F ( r / or), where e and cr are scaling

parameters which characterise a particular substance and F is a universal function. Systems which obey this relation are said to be conformal. In all conformal systems to which classical statistical mechanics applies, the reduced second virial coefficient, B*= B/(ffT~Zt73), is a

universal function of T*= kT/e. Thus, the second virial coefficient of one conformal substance (labeled i) may be estimated from that of another (labeled 0) if the ratios si/e o and cri/cr 0 are known. From a theoretical point of view, the most satisfactory way of relating the scaling parameters to measurable properties is by means of the Boyle temperature and the so- called Boyle volume V B, equal to T(dB/dT) at T = T B. In terms of these quantities, B/V B is,

according to the principle, a universal function of T / T B and, if pair-wise additivity of the intermolecular forces is assumed, C/(VB) 2 is another universal function of T / T B. This method of selecting the scaling parameters has the disadvantage of requiting some measurements of B(T) in the first place. Furthermore, T B is inconveniently high for most substances.

Practical correlations of virial coefficients employ as scaling parameters the critical temperature T ¢ and the characteristic molar volume RTC/p ~, where pC is the critical pressure, and seek to i'epresent B ( f f / R T ~) and C(pURT~) 2 as universal functions of the new reduced

temperature T r = T / T ~. Although the principle, as stated above, applies to only a small number of simple fluids, Pitzer (74) was able to show that many different kinds of molecular complexity may be accounted for by the inclusion of a third parameter co which he called the acentric factor. This parameter is defined in terms of the vapour pressure p°, by the equation

co = -1-log,0 {pC (T r = 0.7)/pC} (3.88)

such that it is essentially zero for the simple fluids Ar, Kr and Xe. For other fluids, values between 0 and about 0.4 are usually found. The second virial coefficient is given in this extended principle of corresponding states by

B(pC/RT c) = B o + coB, (3.89)

where B 0 and B 1 are dimensionless functions of T r. Expressions for B 0 and B 1 w e r e given by Pitzer and Curl (75) in their original formulation but here the more recent correlations proposed by Tsonopoulos (76) for non-polar gases are given:

B o =0.1445-0.3300/Tr-0.1385/Tr2-0.0121/Tr3-0.000607/Tr 8 l

B, = 0.0637 + 0.331 / Tr 2 - 0.423 / Tr 3 - 0.008 / Tr s J (3.90)

As it stands, Equation (3.89) is useful only for essentially non-polar gases, but Tsonopoulos

showed (76) that the addition of the t e r m a/Tr 6 to the right-hand side of that equation

71

permitted the results for polar gases to be correlated also. A further term b/Tr 8 is necessary for

associating compounds. While there seems to be no simple universal correlation for the additional parameters a and b, correlations of the former in terms of the dipole moment may be established within certain classes of compounds such as ketones or ethers (76).

In order to apply these formulae to mixtures, some method is required for determining the

acentric factor coU and the pseudo-critical constants T/f and p~ for the unlike interactions. In

the present case, extended van der Waals one-fluid mixing rules are applied (76) in terms of which

co O'= I(coi "{" coJ)

Tiu ~ - ( 1 - ku)~/Ti~T~?

c c c c c c c c c Pu :4~. { ( P i V i / T i ) + ( p j V j / T j )} / {(V/C) 1/3 - ~ - ( V j )1/3}

(3.91)

where k u is a binary interaction parameter, which may be optimised against experimental data,

and V,. c is the critical molar volume of pure i. When an experimental value is unavailable,

Vic= (RTiCZ~/p~) may be estimated from the correlation of the critical compression factor

proposed by Lee and Kesler (77): Z/~ = 0.2905- 0.085coi. For interactions between polar and

non-polar molecules, the polar te rm a/Tr 6 is dropped while for polar-polar interactions an

arithmetic-mean combining rule is recommended for a. The third virial coefficients of pure non-polar gases have also been correlated using a

similar model by Orbey and Vera (78) and the results are:

C(pC/RTC) 2 = C o + co C 1 ]

C o = 0.01407 + 0.02432 / Tr 28 -0.00313 / Tr I°5

C 1 = -0.02676 + 0.0177 / Tr 28 + 0.040 / Tr 3 - 0.003 / Tr 6 - 0.00228 / Tr ~°5

(3.92)

Several methods have been proposed for the estimation of the interaction third virial coefficients C,jk in mixtures. Orbey and Vera (78) follow Chueh and Prausnitz (79) in proposing the relation

Cu~ = ( CuC,.k Cjk ) v3 (3.93)

in which C, 7 is evaluated from Equations (3.92) with the acentric factor and pseudo-critical constants pertaining to the binary pair i and j. In a recent test against accurate experimental results for binary mixtures (80), this method was found to give satisfactory estimates.

Although none of the formulae discussed in this section are very accurate, gas densities and partial fugacity coefficients estimated by such methods for non-polar and slightly-polar pure gases and mixtures may be accurate enough for many engineering purposes up to about two thirds of the critical density (78,80). Furthermore, as the method may be applied knowing only the critical constants and the acentric factor for each component, it is very easy to apply.

72

3.5 S U M M A R Y

In one sense, being restricted to gases at low and moderate densities, the virial equation of state is of rather limited application. Nevertheless, it has the merits of a sound theoretical basis, a large database of experimentally-determined virial coefficients, and the ability to describe mixtures exactly. The relations set out in Table 3.2 may be used to express a wide range of thermodynamic properties of pure gases and mixtures in terms of the virial coefficients, their temperature derivatives and the heat capacity of the ideal gas. The virial coefficients may be obtained from experiment, from knowledge of the intermolecular potential-energy functions or from the principle of corresponding states.

The relations between the virial coefficients and intermolecular forces is of real importance and has therefore been explored in detail. Equations (3.52), (3.54), and (3.55) may be used to determine B and C from Ul2 and Au 3 at temperatures where quantum corrections are unimportant. Quantum effects have also been considered above in sufficient detail to permit their calculation for the atomic and molecular systems of interest. Because such calculations can now be performed routinely, even for non-spherical molecules, it is perfectly feasible to represent the equation of state correct at least to the third virial coefficient by means of prescribed pair and triplet potentials.

R E F E R E N C E S

1. G. H/~ndel, R. Kleinrahm, and W. Wagner, J. Chem. Thermodyn. 24, 685 (1992). 2. D.R. Douslin, R.H. Harrison, R.T. Moore, and J.P. McCullough, J. Chem. Eng. Data 9, 358

(1964). 3. U. Setzmann and W. Wagner, 3". Phys. Chem. Ref. Data 20, 1061 (1991). 4. J.H. Dymond and E.B. Smith, The Virial Coefficients of Pure Gases and Mixtures, Clarendon

Press, Oxford (1980). 5. J.P.M. Trusler, W.A. Wakeham, and M.P. Zarari, Int. dr. Thermophys. 17, 35 (1995). 6. J.A. Barker, P.J. Leonard, and A. Pompe, dr. Chem. Phys. 44, 4206 (1966). 7. A. Michels, J. C. Abels, C. A. Ten Seldam, and W. de Graaff, Physica 26, 381 (1960). 8. E.A. Mason and T.H. Spurling, The Virial Equation of State, Pergamon Press, Oxford (1969). 9. C.M. Greenliefand G. Constabaris, dr. Chem. Phys. 44, 4649 (1966).

10. J.M. Prausnitz, R.N. Lichtenthaler, and E. Gomes de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed., Prentice-Hall, Englewood Cliffs (1986).

11. T.L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley, Reading (1960). 12. M. Modell and R.C. Reid, Thermodynamics and its Applications, Prentice-Hall, Englewood Cliffs

(1983). 13. J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird, Molecular Theory of Gases and Liquids (corrected

edition), Wiley, New York (1964). 14. J.E. Kilpatrick, J. Chem. Phys. 21,274 (1953). 15. J.E. Mayer and M.G. Mayer, Statistical Mechanics, Wiley, New York (1940). 16. J.E. Mayer, J. Phys. Chem. 43, 71 (1939). 17. K Fuchs, Pro¢. R. Soc. Ser. A 179, 408 (1942). 18. J.E. Mayer and M. G. Mayer, Statistical Mechanics, 2nd ed., Wiley, New York (1977). 19. W.E. Putnam and J.E. Kilpatrick, dr. Chem. Phys. 21, 1112 (1953). 20. Reference 13, Ch. 2. 21. G.C. Maitland, M. Rigby, E.B. Smith, and W.A. Wakeham, Intermolecular Forces: Their Origin

and Determination, Clarendon Press, Oxford (1981).

73

22. G.C. Maitland, M. Rigby, E.B. Smith, and W.A. Wakeham, Forces Between Molecules, Clarendon Press, Oxford (1986).

23. T. Kihara, Intermolecular Forces (English edition, S. Ichimaru, translator), Wiley, New York (1978).

24. C.G. Grey and K.E. Gubbins, Theory of Molecular Fluids, Vol. 1, Fundamentals, Clarendon Press, Oxford (1984).

25. Y. Midzuno and T. Kihara, J. Phys. Soc. Japan 11, 1045 (1956). 26. W. Ameling, K.P. Shukla, and K. Lucas, Mol. Phys. 58, 381 (1986). 27. R.A. Aziz and M.J. Slaman, Mol. Phys. 58, 679 (1986). 28. J.A. Beattie, J.S. Brierley, and R. J. Barriault, J. Chem. Phys. 20, 1615 (1952). 29. E. Whalley and W.G. Schneider, Trans. Am. Soc. Mech. Eng. 76, 1001 (1954). 30. N.J. Trappeniers, T. Wassenaar, and G.J. Wolkers, Physica 32, 1503 (1966). 31. Reference 13, Ch. 6. 32. Reference 8, p. 48. 33. T. Kihara, Adv. Chem. Phys. 1,267 (1958). 34. T. Yokota, J. Phys. Soc. Japan 15, 779 (1960). 35. J.G. Kirkwood, J. Chem. Phys. 1,597 (1933). 36. C.S. Wang Chang, PhD Thesis, University of Michigan (1944). 37. R.T. Pack, J. Chem. Phys. 78, 7217 (1983). 38. A. Pompe and T.H. Spurling, Aust. J. Chem. 26, 855 (1973). 39. B.H. Wells, Unpublished calculations. 40. R.A. Aziz and M. J. Slaman, Chem. Phys. 130, 187 (1989). 41. L. Monchick, Chem. Phys. Lett. 24, 91 (1974). 42. B. Schramm and U. Leuchs, Unpublished results (1979); Quoted in reference 4. 43. W.H. Stockmayer, J. Chem. Phys. 9, 398 (1941). 44. M.J. Offerhaus and J. de Boer, Proc. VII Int. Conf. Low Temp. Phys., Toronto (1960). 45. Reference 8, p. 46. 46. Reference 13, pp. 69-72. 47. M.E. Boyd, S.Y. Larsen, and J.E. Kilpatrick, J. Chem. Phys. 50, 4034 (1969). 48. J.W. Cooley, J. Math. Comp. 15, 363 (1961). 49. J.K. Cashion, J. Chem. Phys. 48, 94 (1967). 50. R.A. Aziz and M. J. Slaman, Metrologia 27, 211 (1990). 51. K.H. Berry, Metrologia 15, 89 (1979). 52. D. Gugan and D. W. Michel, Metrologia 16, 149 (1980). 53. R.C. Kemp, W.R.G. Kemp, and L.M. Besley, Metrologia 23, 61 (1986). 54. B.E. Gammon, J. Chem. Phys. 64, 2556 (1976). 55. G.S. Kell, G.E. McLaurin, and E. Whalley, J. Chem. Phys. 68, 2199 (1978). 56. L.A. Guildner and R.E. Edsinger, J. Res. Natl. Bur. Stand. A:80, 7.03 (1976). 57. F.C. Matacotta, G.T. McConville, P.P.M. Steur, and M. Durieux, Metrologia 24, 61 (1987).

(Corrected values of B given in reference 44). 58. R. Dashen, S-K. Ma, and H. J. Bemstein, Phys. Rev. 187, 345 (1969). 59. A.M. Dunker and R.G. Gordon, J. Chem. Phys. 64, 354 (1976). 60. R.J. Le Roy and S. Carley, Adv. Chem. Phys. 42, 353 (1980). 61. A.R.W. McKellar and H. L. Welsh, Can. J. Phys. 52, 1082 (1974). 62. A.E. Sherwood and J. M. Prausnitz, J. Chem. Phys. 41,413 (1964). 63. J.A. Barker and J.J. Monaghan, J. Chem. Phys. 36, 2564 (1962). 64. J.P.M. Trusler, W.A. Wakeham, and M.P. Zarari, Int. J. Thermophys. 17, 35 (1996). 65. E.J. Janse van Rensburg, J. Phys. A 26, 4805 (1993). 66. Reference 21, pp. 145-146. 67. M. Rigby, Mol. Phys. 66, 1261 (1989). 68. W.R. Cooney, S.M. Thompson, and K.E. Gubbins, Mol. Phys. 66, 1269 (1989).

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69. C. Vega, S. Lago, and B. Garzon, Jr. Chem. Phys. 100, 2182 (1994). 70. B.J. Alder and T.E. Wainwright, J. Chem. Phys. 33, 1439 (1960). 71. M.J. Maeso and J.R. Solana, Mol. Phys. 79, 1365 (1993). 72. J.D. van der Waals, On the continuity of the gaseous and liquid states (English translation, J. S.

Rowlinson, ed.). North-Holland, Amsterdam (1980). 73. J.H. Gibbs, I. Pavlin, and A.J. Yang, J. Chem. Phys. 81, 1443 (1984). 74. K.S. Pitzer, J. Amer. Chem. Soc. 77, 3427 (1955). 75. K.S. Pitzer and R. F. Curl, Jr., J. Am. Chem. Soc. 79, 2369 (1957). 76. C. Tsonopoulos, AIChEJ. 20, 263 (1974); ibid. 21,827 (1975); ibid. 24, 1112 (1978). 77. B.I. Lee and M. G. Kesler, AIChEJ. 21,510 (1975). 78. H. Orbey and J. H. Vera, AIChEJ. 29, 107 (1983). 79. P.L. Chueh and J. M. Prausnitz, AIChE J. 13, 896 (1967). 80. H.B. Brugge, L. Yurtlas, J. C. Holste, and K. R. Hall, Fluid Phase Equilibria 51, 187 (1989).

Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 75

4 CUBIC AND GENERALIZED VAN DER WAALS EQUATIONS

Andrzej Anderko

OLI Systems Inc. 108 American Road Morris Plains, NJ 07950, U.S.A.

4.1 Cubic Equations of State for Pure Components 4.1.1 Historical Perspective 4.1.2 Temperature Dependence of Parameters 4.1.3 Functional Form of the Pressure - Volume Relationship

4.2 Equations Based on the Generalized van der Waals Theory 4.3 Simple Equations Inspired by the Generalized van der Waals Theory 4.4 Methods for Extending Equations of State to Mixtures

4.4.1 Classical Quadratic Mixing Rules 4.4.2 Composition-Dependent Combining Rules 4.4.3 Density-Dependent Mixing Rules 4.4.4 Combining Equations of State with Excess-Gibbs-Energy Models

4.5 Explicit Treatment of Association in Empirical Equations of State 4.6 Equations of State as Fully Predictive Models

4.6.1 Correlations for Binary Parameters 4.6.2 Group-Contribution Equations of State 4.6.3 Utilization of Predictive Excess-Gibbs-Energy Models

4.7 Concluding Remarks References

76

Cubic and generalized van der Waals equations of state have been a subject of active research since van der Waals (1) proposed his famous equation in 1873. During the intervening years, they played a major role in the development of fluid state theories as well as in modeling fluid behavior for practical purposes. Currently, their development is motivated primarily by industrial needs for accurate process-design calculations and supported by steadily advancing computational techniques.

The cubic and generalized van der Waals equations are not the most appropriate models for the accurate representation of pure-fluid properties, because they usually lack the necessary flexibility in some regions of the phase diagram. Also, they are not the most useful methods for understanding the properties of fluids from a microscopic perspective. For such purposes, their theoretical background is usually insufficiently rigorous. However, cubic and, to a lesser extent, generalized van der Waals equations are the most frequently used equations of state for practical applications. This is due to the fact that they offer the best balance between accuracy, reliability, simplicity and speed of computation. Additionally, their success stems from the importance of multicomponent mixtures for many industrial applications. Cubic equations of state are usually the models of choice for phase-equilibrium computations for multicomponent mixtures.

The purpose of this chapter is to review cubic and generalized van der Waals equations of state as convenient practical tools for modeling the properties of multicomponent fluids. The subject is extremely broad and it is virtually impossible to enumerate all significant approaches that appeared in the literature during the last 130 years. However, an attempt will be made to identify the most useful equations of state as well as those that appear to be most promising for future research.

4.1 C U B I C E Q U A T I O N S O F S T A T E F O R P U R E C O M P O N E N T S

4.1.1 Historical Perspective

The first equation of state that was capable of reasonably representing both gas and liquid phases was proposed by van der Waals in 1873 (1). Although van der Waals derived his equation in an intuitive manner, he was able to create a model of fluid behavior that was later proven to be qualitatively correct. According to van der Waals' assumptions, molecules have a finite diameter, thus making a part of the volume v not available to molecular motion. This increases the number of collisions with the walls of the vessel and, subsequently, increases the pressure. Therefore, the actual volume available to molecular motion is v-b, where b is a characteristic constant for each fluid. On the other hand, intermolecular attraction decreases the pressure. It is reasonable to assume that the pressure decrease due to intermolecular attraction is proportional to the number of molecules in a volume unit and inversely proportional to volume. Therefore, the correction for intermolecular attraction becomes (-a/v2). It can be further assumed that the available volume (v-b) and the corrected pressure (P+a/v e) should obey the ideal gas law, so that

P + ( v - b) : RT (4.1)

This gives rise to a two-term equation for pressure, namely

77

R T a P = - - ( 4 . 2 )

v - b v 2

where the two terms correspond to repulsive and attractive contributions to pressure. Although the original van der Waals terms do not quantitatively represent the true repulsive and attractive forces, the concept of separating the repulsive and attractive terms in equations of state has proven to be extremely valuable for the representation of fluid properties. In fact, it is the cornerstone of the generalized van der Waals theory.

The original van der Waals equation has two features that make it very convenient for calculations. First, it can be rewritten as a cubic polynomial with respect to volume. Therefore, it can be solved analytically for volume or density. This feature has been retained by contemporary cubic equations of state. Second, the two van der Waals parameters a and b can be determined from critical-point coordinates by applying the critical-point conditions:

0/9 ~ 2 p -~-= ~--7- = 0 (4.3)

After simple algebraic manipulations, the values of the van der Waals parameters a and b at the critical point can be found as functions of the critical temperature Tc and pressure Pc:

2 2 27 R T c

a~ = ~ ~ (4.4) 64 p2

1 RT~ b~ = - ~ (4.5)

8P~

These values remain reasonable, although not accurate, in other regions of the P-V-T space. The van der Waals equation inspired a huge number of researchers to work on improved

equations of state. Partington (2) summarized the modifications proposed through the late 1940's. Here, we mention only two of the earlier researchers, whose equations can be regarded as precursors of modem cubic equations of state. In 1881, Clausius (3) replaced the volume in the van der Waals attractive term by (v+c), thus creating a three-parameter equation of state:

R T a P = (4.6)

v - b (v+c) ~

This approach resembles the volume-translation technique, which became popular roughly a century later. In 1899, Berthelot (4) introduced an equation with a temperature-dependent attractive parameter, i.e., aT = a/T. Several decades later, the concept of temperature-dependent attractive parameters proved to be essential for the practical success of cubic equations of state for phase-equilibrium calculations.

In the first half of the 20th century, the van der Waals equation and its modifications became gradually superseded by the virial equation and its empirical modifications. From the theoretical viewpoint, the statistical-mechanical foundations of the virial equation seemed superior. At the same time, empirical virial-type equations like the Benedict-Webb-Rubin EOS

78

(5) were shown to be empirically effective for pure fluids and promising for mixtures. However, the need for simple analytical tools for the calculation of fugacities for process design spurred a revival of interest in cubic equations of state. Redlich and Kwong (6) proposed the first cubic EOS that became widely accepted as a tool for routine engineering calculations of the fugacity. The Redlich-Kwong equation was constructed by postulating two reasonable boundary conditions in the low- and high-density limit. In the low-density limit, the equation was constrained to give a reasonable second virial coefficient:

constant B = b - .-.1/---------5--- (4.7)

I

At high densities, it was noted that the reduced volume at infinite pressure could be approximated by 0.26. This condition, together with the usual critical-point conditions, provided guidance for the functional form of the attractive term in which the v 2 term was replaced by v(v+b):

RT a P = ~ - (4.8)

v - b T1/Ev(v + b)

In analogy to the van der Waals equation, the parameters a and b were calculated from critical point conditions"

R 2 2.5 a = ~a T~ (4.9)

RT~ (4.10) b=f~b pc

where f2a and ~"2 b a r e numerical constants equal to 0.42747 and 0.0867, respectively. The Redlich-Kwong equation proved to be very successful for the calculation of the properties of gas mixtures. However, it was still not adequate for the modeling of both gas and liquid phases. The simple temperature dependence of the attractive parameter was insufficient for the representation of vapor pressures. Also, liquid volumes were not predicted with acceptable accuracy.

The success of the Redlich-Kwong EOS stimulated many investigators to introduce a number of improvements. The early modifications of the Redlich-Kwong EOS were reviewed by Horvath (7), Tsonopoulos and Prausnitz (8), Wichterle (9) and Abbott (10). Here, we mention the modification due to Zudkevitch and Joffe (11) because of its general character. In 1970, Zudkevitch and Joffe introduced temperature-dependent parameters a and b by constraining the quantities Oa and f2b to match vapor pressures and liquid densities along the vapor-liquid saturation line. In this way, the quantities f2a and f2b became temperature- dependent at subcritical conditions, whereas they remained fixed at T>Tc. Although an EOS constructed in this way exactly reproduces the vapor pressure and liquid density, it is inconvenient because of the lack of analytical expressions for ~a and £'2b and the need for extensive pure-component data.

79

Therefore, further research in this area focused on the development of fully analytical cubic equations of state. The developments concentrated in two areas: (1) Improving the temperature dependence of the attractive parameter to control the vapor- pressure predictions and (2) Improving the P(v) functional form to optimize the prediction of volumetric properties.

4.1.1 Temperature Dependence of Parameters

For the prediction of vapor pressures, it is entirely sufficient to introduce a temperature- dependent attractive parameter. Thus, it is perfectly possible (and, in fact, common) that an equation of state can correctly reproduce the vapor pressure even when it gives inaccurate PVT predictions. It is convenient to express the parameter a as a product of its value at the critical point (i.e., ac) and a dimensionless function of temperature a(T):

a=aca(T) (4.11)

In 1964, the first generalized a function was introduced by Wilson (12):

a = Tr[1 + (1.57 + 1.62~)T~-' ] (4.12)

where Tr is the reduced temperature and co is the acentric factor. This equation was established by forcing the EOS to give reasonable values of the terminal slope of the vapor pressure curve. However, this did not guarantee good predictions far away from the critical point and, therefore, the Wilson form turned out to be insufficiently accurate.

The first a function that gained widespread popularity was introduced by Soave in 1972 (13):

a = [1 + m(1- Trl/2)] 2 (4.13)

where m is a function of the acentric factor:

m = 0.480 + 1.574co - 0.175co 2 (4.14)

Soave developed his function by forcing the EOS to reproduce the vapor pressures at Tr=0.7. Therefore, the Redlich-Kwong EOS with Soave's a form (frequently referred to as RKS or SRK) accurately predicts the vapor pressures at reduced temperatures ranging from about 0.6 to 1.0. Since the a form is generalized in terms of the acentric factor, it is fully predictive for nonpolar compounds with acentric factors not exceeding about 0.6 (thus excluding heavy hydrocarbons).

The Soave function played a pivotal role in the development of cubic equations of state and greatly contributed to their acceptance as tools for vapor-liquid equilibrium calculations. It was used in conjunction with other equations of state such as those developed by Peng and Robinson (14), Schmidt and Wenzel (15), Patel and Teja (16), Adachi et al. (17) and Watson et al. (18). Only the dependence of m on co had to be changed to accommodate other equations of state. Further improvements of the a form focused on three problems:

80

1. Introduction of adjustable parameters to correlate the vapor pressure of any fluids, without the restriction to nonpolar fluids; 2. Improvement of the behavior of the a form in the supercritical region; 3. Development of generalized forms that overcome the limitations of Soave's function at low reduced temperatures and for large acentric factors.

The introduction of substance-specific parameters is necessary to obtain a satisfactory correlation of vapor pressures of polar or associating compounds. Several empirical extensions of the Soave function have been proposed for this purpose. Among the most significant modifications, Mathias and Copeman (19) proposed a three-parameter form:

a = [1 + Cl(1 - Trl/2)-~-c2(1 - Trl/2) 2 -t-c 3 (1 _ Trl/2 ])3 2 (4.15)

Soave (20) proposed a two-parameter equation:

oc=l+m(l+Tr)+n(Tr -1) (4.16)

Stryjek and Vera (21) developed an equation with one generalized (k0) and one adjustable (kl) parameter:

a = {1 + [ko(cO) + k1(1 + Trl /2)(0.7 - Tr)](1 - Trl/2)} 2 (4.17)

Androulakis et al. (22) analyzed several two- and three-parameter forms of the a function and recommended a three-parameter form:

a = 1+ d1(1- Tr 2/3) + d2(1- Tr2'3) 2 "~" d3(1- Tr2/3) 3 (4.18)

and its simplified version with d2=0. An optimum a function should be positive at all temperatures to prevent the attractive

parameter from switching from positive to negative values. It should decrease continuously with temperature and reach zero at infinite temperature. The original function of Soave does not satisfy these requirements because it becomes negative at some temperatures in the supercritical region and then increases. The above modifications of the Soave function may also share this deficiency for some parameter sets. The simplest way to obtain a qualitatively correct a function is to use an exponential form. Early exponential forms were proposed by Graboski and Daubert (23):

O~ -- C 1 exp(-c 2 T r) (4.19)

and Heyen (24):

a - e x p [ c ( 1 - 420)

81

Melhem (25) analyzed several forms and recommended an exponential version of the Soave (20) equation:

a =exp[m(1-T,)+n(1-Tr- ' ) ] 2 (4.21)

which has a similar accuracy, but extrapolates better above the critical temperature. Yu and Lu (26) developed a generalized exponential a form:

loglo a = M(oo)(A o + A IT r -I.- A2 T,2)(1- Tr) (4.22)

T w u et al. (27) derived a modified exponential a form and found that it is superior to other functions:

lna = N ( M - 1 ) l n T r + L(1- Tr TM) (4.23)

Although the additional adjustable parameters provide the necessary flexibility, they do not guarantee the accurate simultaneous representation of vapor pressure and derivative properties. Therefore, it is advisable to obtain the a form parameters by multiproperty regression. It is especially convenient to use vapor pressures and heat capacity departures simultaneously because Cp can be related to the second derivative of the vapor pressure, c f Mathias and Klotz (28).

Another important area of research to improve the a functions is the development of their extensions to heavy hydrocarbons, which are characterized by low reduced temperatures and high acentric factors. Carrier et al. (29) and Rogalski et al. (30,31) developed such a method in conjunction with the Peng-Robinson EOS (14). They divided hydrocarbons into thirteen groups and calculated the parameter b from additive contributions. The attractive parameter of a heavy hydrocarbon was expressed as

a= a(Tu)[1 +ml (1 - (T I Tb)l/2)-i t- m2(1 -T I Tu) ] (4.24)

where

m 1 = A m + B

m 2 = Cm + D (4.25)

The constants A, B, C, and D have universal values and, therefore, the only characteristic parameters are a(Tb) and m. The value of a at the normal boiling point Tb is determined from one vapor-pressure datum and m is calculated from group contributions. This fairly complicated technique gives very good results for vapor pressures of heavy hydrocarbons at low reduced temperatures.

A particularly simple and elegant approach has been proposed by Twu et al. (32), who observed that the a function should be linear with respect to the acentric factor, i.e.,

82

a = a (°) +co(a (1) -o~ (°)) (4.26)

This is in sharp contrast with the original Soave form, which is equivalent to a fourth-order equation with respect to the acentric factor. The linearity of the a form is remarkably good and makes it possible to make extrapolations for compounds with high acentric factors. Both terms a (°) and a (1) are expressed by the same function of reduced temperature, see Equation (4.23). The method is applicable to any cubic equation of state.

A different approach has been proposed by Dohrn (33), who developed a correlation for two-parameter cubic equations by using the liquid density at 20°C and the normal boiling point as characteristic parameters instead of To, Pc and co. While this approach may be advantageous for large molecules, it does not address the issue of diminished accuracy at low reduced temperatures.

Besides the generalizations for hydrocarbons and other nonpolar fluids, several attempts have been made to generalize the a form for polar fluids. Such attempts are hampered by the lack of a truly effective characterization parameter for polar fluids that would be comparable to the acentric factor for nonpolar fluids. For example, Guo et al. (34) utilized the dipole moment and Valderrama et al. (35) used the product cozc where zc is the critical compressibility factor. Although such correlations are adequate for selected groups of polar compounds, they cannot be generally valid in the whole universe of polar, associating and nonpolar compounds. Therefore, the use of individual parameters for polar and associating fluids should be regarded as the only safe approach.

In contrast to the attractive parameter, the repulsive parameter b of cubic equations of state is usually kept independent of temperature. A temperature-dependent parameter b usually does not lead to any significant improvement of the equation's accuracy. As shown by Salim and Trebble (36), a temperature-dependent b is even dangerous because it results in the prediction of negative heat capacities at constant volume when the pressure exceeds a certain value.

4.1.2 Functional Form of the Pressure-Volume Relationship

While an appropriate temperature dependence of the attractive parameter is sufficient for the accurate representation of vapor pressures, the functional form of the pressure-volume relationship is important for the prediction of volumetric properties. To keep the cubic form of the equation of state, at most five parameters can be used on an isothermal basis. Since the two-parameter Redlich-Kwong EOS gives a reasonable representation of fluid properties, it might appear that a five-parameter EOS would be vastly superior. Unfortunately, this intuitive reasoning is not true. In fact, the additional parameters can bring only a limited incremental improvement of the accuracy because of the fundamental limitations of the cubic form.

The most popular modification of the pressure-volume relationship, due to Peng and Robinson (14), does not involve any additional parameters beyond the original two. Peng and Robinson recognized that the critical compressibility factor of the Redlich-Kwong EOS, which is equal to 0.333, is greater than the compressibility factors of practically all fluids. Therefore, they postulated a function that reduces zc to 0.307:

RT a(T) P = ~ (4.27)

v - b v ( v + b ) + b ( v - b )

83

The Peng-Robinson EOS predicts better liquid volumes for medium-size hydrocarbons and other compounds with intermediate values of the acentric factor. However, it is worse than the Soave-Redlich-Kwong EOS for compounds with small acentric factors.

When a third parameter is introduced into a cubic EOS, the critical compressibility factor becomes substance dependent. Although a three-parameter equation can be forced to predict the correct critical compressibility factor, the isotherms at low and h i ~ pressures are then distorted much more than can be tolerated for the purposes of modeling PVT relations. Better overall results are usually obtained when the apparent (calculated) compressibility factor ~ is greater than the real one Zc. Early three-parameter cubic equations of state were proposed by Fuller (37) and Heyen (24). These two equations involve temperature-dependent co-volumes b, which adversely affects their accuracy for derivative properties.

A well-known three-parameter cubic EOS was proposed by Schmidt and Wenzel (15), who designed their equation as a form of interpolation between the Soave-Redlich-Kwong and Peng-Robinson equations. This interpolation ensures a better representation of liquid volumes over a reasonable range of acentric factors. Schmidt and Wenzel analyzed a general form

RT a(T) P = (4.28)

v - b v 2 + ubv + wb2

and constrained the parameters u and w by

u = 1- w (4.29)

Additionally, they related w to the acentric factor by w=-3co. The apparent critical compressibility factor was found to be a linear function of the acentric factor.

Other three-parameter cubic equations were proposed by Harmens ancl Knapp (38):

RT a(T) P = - - (4.30)

v - b v 2 + v c b - ( c - 1 ) b 2

and Patel and Teja (16)"

RT a(T) P = (4.31)

v - b v ( v + b ) + c ( v - b )

Patel and Teja treated the apparent compressibility factor as an adjustable parameter, which is temperature dependent for 0.9<Tr<l.0.

Yu and Lu (26) considered the four-parameter form proposed originally by Schmidt and Wenzel (Equation (4.28)) and arrived at a different constraint for the parameters u and w, i.e., u-w = 3. Their final equation takes the form

RT a(T) P = ~ - (4.32)

v - b v ( v + c ) + b ( 3 v + c )

where c = wb. Twu et al. (39) systematically analyzed the cubic equations of state on a u-w diagram with respect to the representation of volumetric properties. They checked 21 possible

84 combinations of u and w for a number of alkanes, alcohols and glycols. The optimum constraint was found to be u-w=4, which is closer to the result of Yu and Lu than to that of Schmidt and Wenzel. This gives the form

R T a(T) P = ~ (4.33)

v - b v(v + 4b) + c(v + b)

with the apparent critical compressibility factor optimized as a third parameter. In the development of these three-parameter equations, use was made of the fact that

acceptable representation of both low- and high-density regions requires that the apparent critical compressibility factor be treated as an empirical parameter. A simple, yet useful method to achieve this is the so-called volume translation. This concept, proposed by Martin (40) and refined by Peneloux et al. (41), is based on a translation along the volume axis, i.e.,

v~anslated _. Vioriginal -- C i (4.34)

If the translation parameter c for a mixture is related to composition by a linear mixing rule, the partial molar volumes are also translated and the fugacities are multiplied by a composition-independent coefficient so that the phase-equilibrium conditions (i.e., the isofugacity criteria) are not affected by the volume translation. Peneloux et al. (41) showed that this method markedly improves the prediction of liquid volumes except in the vicinity of the critical point. It should be noted that the original van der Waals EOS becomes, after volume translation, identical to the Clausius EOS (3). Martin (40) found the Clausius EOS to be the best cubic equation with respect to the representation of gas-phase properties. Joffe (42) and Kubic (43) used the Clausius EOS in a modified form. Salerno et al. (44), Czerwienski et al. (45) and Androulakis et al. (22) presented a translated van der Waals EOS, called vdW-711, that contains a temperature-dependent volume translation. The translating function was fitted to experimental liquid volumes including the critical volume and generalized in terms of the acentric factor.

A somewhat different volume translation was presented by Mathias et aL (46), namely

0.41 / V translated __ v°riginal + S -b f c 0.41 + 6 (4.35)

where 6 is the bulk modulus,

v2( 6 = - (4.36)

R T r

and the correction functionfi is chosen so that the actual critical volume is obtained:

f e -- Ve -- ( r e °figinal -k- S) (4.37)

Although a volume translation, defined in such a way, is very accurate, it introduces a thermodynamic inconsistency into the calculations because the bulk modulus is a function of

85

the equation of state and, therefore, the fugacity and volume calculations are not decoupled as in Peneloux's method. It is interesting to note that the concept of volume translation has even been used to develop an equation of state that reproduces the solid phase (47).

It should be noted that the invariance of the isofugacity condition with respect to the (original) volume translation is only one example of the useful mathematical properties of the cubic equations. In particular, Leibovici (48) showed that the properties calculated from the genetic Schmidt-Wenzel EOS (see Equation (4.28)) can be divided into two classes, called variant and invariant properties. The variant properties depend on the individual parameters whereas the invariant properties depend only on the combined quantity

l + u + w K = ~ (4.38)

( u + 2 ) ~

The difference of fugacities is an example of an invariant property. Other invariant properties include entropy, heat capacity at constant P or V, enthalpy of vaporization, internal energy and Helmholtz energy. On the other hand, volume is an example of the variant properties, which also include enthalpy, sound velocity, Joule-Thomson coefficient, Gibbs energy, isothermal compressibility and adiabatic compressibility. Thus, the volume translation can be applied to improve the representation of any variant property while keeping the invariant properties constant.

In addition to the three-parameter cubic EOS discussed above, several authors tried to improve the flexibility of cubic equations by using four or five parameters. A four-parameter form was used by Trebble and Bishnoi (49) and Salim and Trebble (50):

RT a(T) P = - - (4.39)

v - b v 2 + ( b + c ) v - ( b c + d 2)

Five-parameter cubic equations were analyzed by Adachi et al. (51) in the traditional two-term form, i.e.,

RT a(T)[v -c (T)] P = (4.40)

v - b [ v - b][v- d(T)][v + e(T)]

and by Kumar and Starling (52) in a genetic form, i.e.,

l + d, ( r )o + 4 ( r )o ~ z = ( 4 . 4 1 )

1 -i- d3 (T)p nt- d4 (T)p2 -t- d5 (T)p 3

The Kumar-Starling EOS was more recently modified by Chu et aL (53), who added additional temperature-dependent functions in the near-critical region and attempted to generalize the parameters for polar fluids.

The results obtained from the four- and five-parameter equations are only slightly better than those obtained from the three-parameter EOS. In most cases, the differences resulting from a more flexible P-V dependence are masked by the temperature dependence of parameters. It should be noted that, in contrast to other cubic equations, the original Kumar-

86

Starling EOS was not constrained at the critical point, which automatically improves the results in the far-from-critical regions.

The results of several authors indicate that cubic equations of state require at least three parameters for a reasonable representation of both vapor and liquid volumes. More than three parameters lead to only small incremental improvements. The attractive parameter has to be temperature-dependent. However, the temperature dependence of other parameters, such as the co-volume, may lead to spurious results. It appears that the property that can be most easily correlated using cubic equations of state is the vapor pressure. It is customary that cubic equations can predict vapor pressure within 1 - 2 % when their parameters are generalized in terms of To, Pc and ¢o. With parameters fitted to an individual fluid's properties, the deviations can drop to a fraction of a percent. The liquid and vapor volumes can be predicted within 3 - 4 % by the more advanced equations with at least three parameters. All cubic equations give a very poor performance in a relatively wide "near-critical" region' This is not only due to the fact that every cubic equation shows classical critical behavior. Another reason for this deficiency is the relative rigidity of the cubic form, which imposes limits on the quality of the representation of derivative properties such as residual enthalpy, entropy and heat capacity. Not surprisingly, the enthalpy of vaporization is usually the derivative property that is most accurately reproduced by cubic equations. This results from the fact that the enthalpy of vaporization is related to the slope of the vapor pressure curve by the Clausius-Clapeyron equation. Therefore, the relative error of the predicted enthalpy of vaporization is usually very similar to the relative error of vapor pressure. However, the residual enthalpies of the liquid and the vapor phases are usually predicted with a significantly lower accuracy. Second-order derivative properties such as the specific heat capacities Cp and Cv as well as the Joule- Thomson coefficient are predicted with a still lower accuracy, but the results are usually reasonable (i.e., deviations of the order of 10 - 15 % are typical). The equations that are correctly constructed (i.e., with carefully introduced temperature-dependent parameters) produce consistent results and anomalies are usually not detected. One of the few anomalies that is commonly detected in cubic equations of state is their failure to qualitatively reproduce Cv in the critical region, c f Salim and Trebble (50). In fact, all cubic equations of state give physically incorrect Cv versus density relationships within ca. 20-30 K from the critical point. Numerical values of deviations between experimentally determined properties (i.e., residual thermodynamic functions and P VT properties) and those predicted from selected cubic equations of state can be found in the papers of Trebble and Bishnoi (49), Salim and Trebble (50), Lielmezs and Mak (54), Mak and Lielmezs (55) and Solimando et al. (56) and in chapter 4 of Malanowski and Anderko (57).

For practical purposes, cubic equations can be easily tailor-made to fit specific properties in a specified range, e.g., by adjusting the parameters of the temperature-dependent attractive term or by making a volume translation. However, it should be noted that such fine-tuning may result in inaccurate extrapolations, e.g., in the supercritical region.

4.2 E Q U A T I O N S B A S E D O N T H E G E N E R A L I Z E D V A N D E R W A A L S

T H E O R Y

Although the original van der Waals equation was developed on the basis of simple molecular concepts, cubic equations of state can be regarded as nothing more than comprehensive empirical correlations of fluid properties. Although a cubic EOS can work very well as a whole, neither its repulsive nor its attractive terms reflect the physical reality.

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Therefore, numerous researchers focused their efforts on the development of equations of state with clear physical foundations.

The basic premise of the generalized van der Waals theory is the separation of the repulsive and attractive contributions to pressure. Originally postulated by van der Waals in 1873, it was theoretically established several decades later by Zwanzig (58), Hemmer et al. (59) and Barker and Henderson (60) for some model intermolecular potentials. Subsequently, more practically-oriented approaches were presented by Vera and Prausnitz (61), Sandier (62), Lee et al. (63), Lee and Sandier (64) and Abbott and Prausnitz (65). The development of equations of state based on the generalized van der Waals theory was strongly influenced by the progress in the theory of hard-body fluids, which are commonly used to model the effects of repulsive interactions. Since the structure of fluids has been shown to be determined primarily by repulsive forces, the equations of state for hard-body fluids can be conveniently used as reference terms for equations of state for real fluids.

Camahan and Starling (66) developed an accurate equation of state for hard spheres, i.e.,

1+~+~2 __~3 (4.42) Zrep = (1 _~)3

where ~ = zcpcr3/6 is the reduced density with a being the hard-sphere diameter. This equation has been extended to hard convex bodies by Boublik and Nezbeda (67):

1 +(3a -2)~ +(3a 2 - 3 a + 1)~ 2 - a 2 ~ 3 Zrep "- (1 - ~)3 (4.43)

where a is the nonsphericity parameter defined as a = RoSo/Vo, where R0, So and V0 are the mean curvature, mean surface and volume of the hard convex body, respectively. A different expression for the effect of hard-core geometry on molecular repulsions was formulated by Naumann et al. (68). Tildesley and Streett (69) obtained an accurate expression for dimers (or dumbbells). More recently, several authors (70-72) obtained equations of state for hard-sphere- chain fluids.

Several authors combined the rigorous hard-body repulsive terms with empirical attractive terms that had been used earlier in cubic equations of state. In early attempts to utilize the hard-core repulsive terms to construct equations for real fluids, the simple van der Waals or Redlich-Kwong attractive terms were employed as perturbations by Carnahan and Starling (73). The Carnahan-Starling - van der Waals EOS was further studied by Anderson and Prausnitz (74) and Dimitrelis and Prausnitz (75). More recently, the Carnahan-Starling EOS was combined with the Schmidt-Wenzel attractive term (76) and the Patel-Teja attractive term (77). The Carnahan-Starling EOS was also used with attractive terms expressed by truncated virial expressions (78-81). For example, Dohm and Prausnitz (81) formulated an attractive term that optimizes the representation of the critical isotherm for several fluids:

4a z = zrep (CS) - -~-~ r/(1 - 1.41r/+ 5.077/2 ) (4.44)

where 7/= b / 4v.

88

The combinations of the Carnahan-Starling repulsive term and various empirical attractive terms are not more accurate than comparable cubic equations with respect to the representation of pure fluid properties. They are usually reported to give better predictions of volumes in the high-density region, but this is mostly due to the fact that they are not constrained at the critical point, which facilitates the fitting of volumes. Thus, they are mostly useful for testing and developing mixing rules, for which a correct separation of repulsive and attractive terms may be advisable.

Further progress in the area of generalized van der Waals equations of state resulted from the use of molecular-simulation data, which made it possible to establish accurate attractive terms for selected intermolecular potentials. For example, Alder et al. (82) obtained an expression for the attractive term of a square-well fluid:

zat ~ =~~mD,, , , , (u /kT)"(v° /v~ (4.45) n,m

where u is the characteristic energy, v ° is the hard-core volume and Dnm are numerical coefficients. An alternative expression for the attractive term of a square-well fluid was proposed by Heilig and Franck (83), who constructed a Pad6 approximant from the second and third virial coefficients:

B1) Za~ (v 2 - v C / B ) (4.46)

where both B and C are expressed in terms of the square-well parameters. The theoretically based repulsive and attractive terms have been used by several authors to

construct generalized van der Waals - type equations of state. Chen and Kreglewski (84) and Kreglewski and Chen (85) combined Equations (4.43) and (4.45) into the BACK (Boublik- Alder-Chen-Kreglewski) equations of state. They determined the parameters Onto from the residual energy and volumetric data of argon and treated them as universal constants. Both hard-core volume and characteristic energy were assumed to be temperature-dependent:

v ° = v°°[1_ Cexp(-3u ° / kT)] 3 (4.47)

u / k - ( u ° / k)0 + / (4.48)

The BACK equation contains three parameters that have to be determined from experimental data, i.e., v °°, aand u°/k. The remaining parameters are assigned fixed values.

Beret and Prausnitz (86) and Donohue and Prausnitz (87) created the first generalized van der Waals-type equation of state that was designed for chains of segments. They utilized Prigogine's (88) arguments that the rotational and vibrational contributions to the canonical partition function can be factorized as

qr, v = (qr, v)ext (qr, v )int (4.49)

89

where (qr,v)ext and (qr,v)int are the external (density-dependent) and internal (density- independent) contributions to the partition function, respectively. The essential assumption was that the external contributions could be calculated in essentially the same way as the contributions of translational motion. Since the contribution of each translational degree of freedom is (qrepqattr) and the total number of external degrees of freedom is assumed to be equal to a substance-specific parameter 3c, the canonical partition function is postulated to be

1 V N Nc Q=-~.V(-~3-) (qrepqattr)(qr, v)int (4.50)

where A is the thermal de Broglie wave length. This partition function has two characteristic features. First, the ideal-gas term (V/A3) u is factorized so that the ideal-gas limit can be recovered (which is not the case in the original work of Prigogine). Second, the substance- specific number of external degrees of freedom in a system of N molecules, equal to N¢, replaces the number of molecules in the part of the partition function that accounts for nonideality (i.e., (qrepqattr)UC). The partition function factorized in this way gives rise, according to elementary statistical thermodynamics, to a general expression for the compressibility factor:

+Za°)

where Zrep is calculated from the Camahan-Starling equation corrected for the ideal-gas term and Zatt~ is obtained from the equation of Alder et al. (82). The complete equation was called PHCT (Perturbed-Hard-Chain Theory). Similarly to the BACK EOS, the PHCT equation contains three parameters for each substance: the characteristic temperature T* related to the depth of the potential well, characteristic volume v* related to the size of the segments and one- third of the external degrees of freedom c. These parameters were found to correlate smoothly within homologous series of hydrocarbons.

The PHCT EOS was further refined by Morris et al. (89), who replaced the attractive term based on the square-well potential by one based on the Lennard-Jones potential. Jin et al. (90) recast this equation in terms of group contributions. Cotterman et al. (91) improved the performance of PHCT at low densities by separating the attractive Helmholtz energy into high- and low-density contributions and applying a switching function to interpolate between them. Kim et al. (92) developed a simplified version of PHCT (SPHCT = Simplified Perturbed- Hard-Chain Theory) by replacing the complex attractive part of PHCT by a simpler expression that combines ingredients of the radial distribution function theory for square-well molecules with the lattice statistics for chain molecules:

cv*Y Zattr = --Z M ~ (4.52)

V+ v*Y

with

Y = exp(e.q / 2ckT) - 1 (4.53)

90

where ZM is the maximum coordination number, e is the characteristic energy per unit surface area and q is the normalized surface area..

Chien et aL (93) developed their COR (Chain of Rotators) EOS by adopting a different approximation for the chain-like structure of molecules than that used in PHCT and related models. They suggested treating the first segment of a chain-like species like a free molecule. The subsequent segments are assumed to rotate about their neighbors. Therefore, the configurational (i.e., excluding the internal degrees of freedom) canonical partition function is given by

Oconf Nc = Qtqr Qa~ (4.54)

where Qt is the translational partition function calculated in analogy to that of a spherical molecule, qr is the partition function of an elementary rotator and c is the number of rotational degrees of freedom of the chain. Qt is calculated from the Camahan-Starling expression whereas qr is evaluated by forcing Equation (4.54) to represent the partition ftmction of a hard dumbbell, i.e., Qdb 2N = Qtqr • The obtained expression is arbitrarily assumed to be valid for chains containing more than one segment. The Qattr contribution is calculated from the expression of Alder et al. given by Equation (4.45). The main difference between COR and PHCT is in the term that is not affected by the c parameter. In the case of PHCT, it is the ideal- gas term. In COR, it is the single-segment translation term.

Deiters (94) developed a different generalized van der Waals EOS by deriving a semi- empirical approximation for the radial distribution function for square-well fluids, involving corrections for nonspherical molecular shape, "soft" repulsive potential and three-body effects. The equation was shown to reproduce the P V T behavior in a large pressure range using only three characteristic parameters.

An important equation for fluids with chain-like molecules was proposed by Chapman et

al. (95) and Huang and Radosz (96). This equation, called SAFT (Statistical Associating Fluid Theory), is a combination of Wertheim's (97) perturbation theory for intermolecular association and chain formation with the usual expressions of Camahan and Starling and Alder et al. In the SAFT equation, the Helmholtz energy (and all thermodynamic functions that can be derived from it) is expressed as

A = A ig ..1_ A seg + A chain + A a~oc (4.55)

where the four terms on the right hand side represent contributions from the ideal gas, intermolecular interactions between segments, the formation of chains from segments and association. The segment Helmholtz energy is

A 'eg = r ( A hs + A disp) (4.56)

where r is the number of segments and A hs and A disp are the hard-sphere and dispersive contributions for a single segment, respectively. The A hs and A disp terms are calculated from the Carnahan-Starling and modified Alder et al. expressions, respectively. The A chain and A ass°c

terms are obtained from Wertheim's perturbation theory of association. For more information about this theory, the reader is referred to the chapter of this book that deals with associated fluids (Chapter 12).

91

More recently, Song et aL (98) presented a different equation, called PHSC (Perturbed Hard-Sphere Chain). It is based on Chiew's (72) Percus-Yevick integral-equation theory for chain molecules. The methods for treating chain molecules in both SAFT and PHSC are firmly based on statistical mechanics and are superior to those used in PHCT or COR. Both SAFT and PHSC have been shown to be very useful primarily for systems containing polymers even though they can also be applied to other systems (e.g., SAFT was used for heavy hydrocarbons). A more complete discussion of PHSC and SAFT can be found in the chapter on equations of state for polymer systems (Chapter 14).

Several authors attempted to improve the performance of generalized van der Waals-type equations for polar fluids by adding an additive term that explicitly accounts for anisotropic interactions that result from the presence of dipoles and quadrupoles. A perturbation theory for mixtures of multipolar fluids was developed by Gubbins and Twu (99) following an earlier work of Stell et al. (100). Vimalchand and Donohue (101) incorporated the perturbation expansion of Gubbins and Twu into the PHCT EOS. The resulting equation, called PACT (Perturbed-Anisotropic-Chain Theory) contains an additional Z ani term:

z = 1 + C(zreP + ziS° q_ zani) (4.57)

Similarly, Saager and Fischer (102) added dipolar and quadrupolar terms to the BACK equation. For the dipolar and quadrupolar terms, they used expressions fitted to molecular simulation data. Since the anisotropic term from the perturbation theory is quite complex, several simplified expressions have been developed. A particularly simple formula was proposed by Bryan and Prausnitz (103), who incorporated dipolar effects into a single term whose density dependence is identical to that of the Carnahan-Starling term:

zrep+dipolar ._ 1 + ft ' lr/+ ft2Jr/2 - ft31r/3 (4.58) (1-0) 3

where f i l l , f[21 and ft31 are functions of the reduced dipole moment. Brandani et al. (104) developed an equation of state on the basis of this term. Another simplification of the multipolar term was proposed by Sheng and Lu (105).

The main reason for introducing the multipolar terms into generalized van der Waals - type equations of state is to improve the predictions for mixtures. It has been found that smaller values of binary parameters are needed when the multipolar terms are taken into account and, in favorable cases, phase equilibria can be predicted without any binary parameters. The multipolar terms are, in general, not needed for improving the representation of pure fluid properties. It should be noted that the improvement for mixed fluids is usually limited to mixtures that contain multipolar and nonpolar, but not hydrogen-bonding fluids. However, the usefulness of the dipolar term has also been demonstrated for systems such as NaC1 + H20 or KC1 + H20 at very high temperatures (106).

A different, more empirical approach to decoupling the effect of polar interactions from the rest of the equation of state has been proposed by Lee and Chao (107). They postulated that the properties of hydrogen-bonding fluids could be treated using the dipolar-fluid formalism and separated the nonpolar and polar contributions to the pressure of water:

P - - Prep + Patt, np ~- Patt, p (4 .59)

92

where the subscripts np and p stand for the nonpolar and polar contributions, respectively. The nonpolar contribution was evaluated from the BACK EOS with parameters determined from the Lennard-Jones constants that represent the nonpolar interactions in water. The polar contribution was obtained by forcing the complete equation of state to be identical with the accurate multiparameter EOS developed by Keenan et al. (108). Later, Lee and Chao (109) and Peng and Chao (110) developed closed-form empirical expressions for Patt,p. The model was also extended to other polar fluids using a corresponding-states transformation. For mixtures, the model requires two binary parameters in the BACK EOS. The polar pressure is extended to mixtures using a corresponding-states-based technique. In this approach, the effects of hydrogen bonding are not treated explicitly and the polar term effectively describes both dipole-dipole and hydrogen-bonding interactions. The equation was shown to give good accuracy for correlating phase equilibria in aqueous hydrocarbon systems. It should be noted, however, that equations of state that explicitly take into account hydrogen bonding (described here in Section 4.5) can give similar results for aqueous hydrocarbon systems with only one binary parameter and have a simpler form.

The main virtue of the equations based on the generalized van der Waals theory is their ability to represent the saturated properties and P V T data outside of the saturation region using only a few (typically three) substance-dependent parameters. However, they contain a large number of universal parameters and their functional form is much more complex than that of cubic equations of state. Although the characteristic parameters of these equations are not correlated with the critical temperature, pressure and acentric factor, they usually follow well- established patterns. For example, the parameters of the SAFT EOS follow regular linear correlations within some homologous series of hydrocarbons (96). This makes it possible to apply the equation to heavy hydrocarbons, for which very little experimental information is available. It is becoming increasingly evident, however, that this family of equations is most useful for systems containing polymers as well as small molecules. For such systems, cubic equations of state appear to be not very well suited whereas models like SAFT and PHSC are at their best.

4.3 S I M P L E E Q U A T I O N S I N S P I R E D BY T H E G E N E R A L I Z E D V A N

D E R W A A L S THEORY

Several researchers have presented methods that attempt to combine the simplicity of cubic equations of state with the theoretical background of equations based on the augmented van der Waals theory. This approach has resulted in the development of models that include reformulated cubic equations, quartic equations (which still can be solved analytically) and higher-order polynomials. In the majority of models of this type, the Carnahan-Starling hard- sphere term is replaced by a simple approximation, i.e.,

Zrep : V + C

v - b (4.60)

This term can be used to approximate the Camahan-Starling expression by appropriately adjusting the parameter c or by expressing it as a function of b and other parameters.

93

An early example of such an approximation is the expression of Scott (111), in which the parameter c is related to the co-volume b, i.e.,

2 v + b = (4.61) ZreP 2V-- b

This equation was later combined with the Redlich-Kwong attractive term by Ishikawa et al.

(112) to yield a cubic EOS. A more complex cubic EOS was developed by Guo et al. (113) and Kim et al. (114), who simplified the COR EOS. The simplified equation, called CCOR (Cubic Chain of Rotators), contains analytical approximations for the hard-core and rotational- pressure contributions in COR:

p = RT(1 + 0.77b / v) _ c R (O.055RTb / v) _ a ( T ) _ bd (4.62) v - 0.42b v - 0.42b v(v + c (T ) ) v(v + c ( T ) ) ( v - 0.42b)

Unlike in most modem cubic equations of state, the critical compressibility factor in the CCOR EOS is constrained to match the experimental value. This appears to impair the equation's performance in some regions of P V T space.

Several authors developed quartic equations of state in the attempt to improve the prediction o f P V T properties. Among them, Kubic (115) developed a quartic equation of state by simplifying the Beret-Prausnitz generalized van der Waals partition function:

R T 1 .19cbRT a P = - ~ (4.63)

v v ( v - 0.42b) (v + d) 2

Soave (116) developed a quartic EOS by combining a modified form of Equation (4.60) with a genetic attractive term, i.e.,

b av t p = R T l + c - (4.64) v v - b ( v + d ) ( v + e )

Soave further reduced the number of independent adjustable parameters to three and found that the quartic equation was superior to cubic equations in the supercritical-fluid region. Another quartic EOS was developed by Shah et al. (117), who obtained a repulsive term by fitting molecular-simulation results for hard spheres:

R T f lk 1R T av + k o tic

P = (v - kob------ ~ + (v - k0fl) 2 - v(v + e) (v - kofl ) (4.65)

where ko, kl, e and/3 are coefficients regressed from simulation data. In the above equation, the first two terms on the fight-hand side form the repulsive contribution and the third term is an attractive contribution, obtained by fitting experimental data for argon.

An equation of a higher order was proposed by Anderko and Pitzer (118), i.e.,

94

+ a p r + ~p2 + ~p3 r (4.66) v bp~ r

where Pr is the reduced density and the parameters c, a, 13 and ~, are functions of reduced temperatures. Anderko and Pitzer found this equation to be very effective over large ranges of pressure and temperature.

From the practical point of view, one of the main reasons for using the equations based on the generalized van der Waals theory (whether simplified or not) is their superiority over cubic equations of state with respect to the representation of PVT properties, especially in the supercritical region and for dense liquids. This is usually achieved with a reasonable number of parameters. However, it should be kept in mind that most of the equations based on the generalized van der Waals theory are not constrained to reproduce the critical point (Deiters' equation (94), Soave's quartic equation (116) and the implementation of the SPHCT EOS by van Pelt et al. (231) are notable exceptions). This is due to the fact that their form is usually too complex to derive analytical relations for the critical point. As a result, the equations overshoot the critical temperature and pressure by considerable amounts. Such behavior is unavoidable when no near-critical corrections are introduced and is very similar for practically all equations that are not constrained at the critical point. This is illustrated in Figure 4.1 for methane. It should be noted that, if the equations were constrained at the critical point, the overall numerical deviations between experimental data and predicted values would be worse. Thus, an equation that is not constrained at the critical point will usually perform better when compared with a large set of experimental data points. Therefore, the frequently published comparisons between generalized van der Waals-type equations that are not constrained at the critical point and cubic equations (which are usually constrained) should be taken with a grain of salt. Additionally, it should be noted that the overprediction of the critical point coordinates is a serious disadvantage for process simulation because it may result in qualitatively incorrect phase equilibrium calculations. Such a disadvantage is usually not compensated by a better prediction of PVT properties in most regions of the PVT phase diagram. This is one of several reasons why cubic equations are, by far, the most popular equations of state for industrial process design.

4.4 M E T H O D S F O R E X T E N D I N G E Q U A T I O N S O F S T A T E T O

M I X T U R E S

Representation of mixed-fluid properties is the main, if not the only, purpose of using cubic and generalized van der Waals equations of state. Whenever only pure fluids are of interest, equations of this kind are not the method of choice. Here, we outline the various methods for extending the equations discussed above to mixtures.

4.4.1 Classical Quadratic Mixing Rules

The classical quadratic mixing rules can be easily deduced from the composition dependence of virial coefficients, which is known from basic statistical thermodynamics. If we

95

90

80

70

6O

n 50 o "It '"

40 13_

30

20

10

oC~ ~" • \

I o I I

cb I I

Q I

0 I !

O I 0 I ) I

I I I

0 5 10 15 20 25 30 P / mol d m 3

Figure 4.1 Vapor-liquid coexistence curve for methane calculated from a typical van der Waals-type equation of state by fitting its parameters to vapor-pressure and liquid-density data without imposing critical-point constraints (solid line). Experimental data are shown as solid circles. Also, the calculated critical isotherm is shown as a dashed line and compared with experimental data (hollow circles).

expand the original van der Waals EOS in a power series around zero density, we get

Z a z = 1 + (4.67) i=1 RTv

This allows us to identify the second, third, and higher virial coefficients:

B = b - a / R T

(4.68) C b ~ D b 3 = , = , e t c .

To maintain consistency with the quadratic composition dependence of the second virial coefficient, the parameters a and b can be, at most, quadratic functions of composition. However, the cubic composition dependence of the third virial coefficient imposes a more stringent restriction that the parameter b should be only a linear function of composition. Very

96

similar conclusions can be reached for other equations of state. Thus, the simplest (although not the only possible) function for the attractive parameter is given by

a = ~ ~ XiXja O. (4.69)

where the cross-term a• is related to the pure-fluid terms aii and aj~ by the geometric-mean rule with one adjustable binary parameter ku:

a u = (aiiab.) '/2 (1 - ku) (4.70)

The co-volume b is usually expressed by a linear mixing rule:

b = E x i b i (4.71)

However, many authors use a quadratic mixing rule for b in analogy with the mixing rule for a even though it violates the composition dependence of the third and higher virial coefficients. In addition to the results obtained from the virial equation, the classical mixing rules are supported by the results of Leland et al. (119,120), who derived them from the theory of radial distribution functions.

In the case of the generalized van der Waals equations, the mixing rules are either analogous to the quadratic mixing rules described above, or are derived from more sophisticated theories. For the ubiquitous Carnahan-Starling term, an extension to mixtures has been derived by Boublik (121) and Mansoori et al. (122):

1 + (3DE / F - 2)~ + (3E 3 / F 2 - 3DE / F + 1)~ 2 - ( E 3 / F 2 )~3 z = (4 .72 )

( 1 - - ~ ) 3

with F = ~Y.i~i 3, E = ~_,xi~i 2 and D = Exicri. Dimitrelis and Prausnitz (75) found that this equation is superior to the one-fluid approximation (essentially given by Equation (4.71)) when applied to the correlation of VLE in mixtures of nonpolar molecules that appreciably differ in size. The mixing rules for the attractive terms are equation-specific, but they generally follow the form of Equations (4.69) and (4.71).

The classical quadratic mixing rules are, in general, suitable for the representation of phase equilibria in multicomponent systems containing nonpolar and weakly polar components. Their performance has been extensively tested by Han et al. (123) for seven van der Waals-type equations including five cubic ones. It should be noted that very similar results are obtained from vapor-liquid equilibrium calculations using various equations of state. This indicates that the form of the mixing rules is more important than the particular P-K-T relationship embodied in an EOS.

There are very few examples that would indicate the superiority of generalized van der Waals-type equations over cubic equations when it comes to the correlation of phase equilibria in typical nonpolar systems. Peters et al. (124) obtained better results from the SPHCT EOS (92) than from the Soave-Redlich-Kwong EOS for the correlation of the phase behavior of mixtm'es containing short and long n-alkanes. Ashour and Aly (125) found that the correlation of VLE for mixtures containing CO2 and fatty-acid esters could be somewhat improved when

97

the Camahan-Starling-Boublik-Mansoori expression was used as a repulsive term. Such findings are corroborated, to a certain extent, by the study of the topology of phase diagrams obtained from the Camahan-Starling-Redlich-Kwong EOS by Kraska and Deiters (126) and the SPHCT EOS by van Pelt et al. (232). Although the global phase diagrams obtained from these equations are similar to those obtained from the van der Waals EOS when the components have equal sizes, significant differences appear when the sizes are different (e.g., a new tricritical line).

In most practical applications, a linear mixing rule is used for the co-volume b. However, several authors used a quadratic form for b even though it violates the cubic composition dependence of the third virial coefficient. In this way, a second binary parameter can be introduced. This parameter is usually found to be useful for correlating VLE in mixtures with components of very different sizes, such as hydrogen and hydrocarbons (127). Similarly, it usually improves results for solid - supercritical-fluid equilibria because of the large size differences in such systems. This was demonstrated by Caballero et al. (128) for a large number of solid + supercritical fluid systems.

An important application of cubic equations of state with classical quadratic mixing rules is the modeling of reservoir-fluid phase behavior and volumetric properties. Research in this field was initiated by Katz and Firoozabadi (129), who showed that the Peng-Robinson EOS can be used to compute high-pressure reservoir crude VLE by characterizing the heptane-plus fraction and using a methane - last fraction adjustable binary parameter. This subject was reviewed by Firoozabadi (130), who concluded that the prediction of VLE in reservoir fluids is influenced mainly by the binary parameters between C1, CO2, Nz and the last fraction. Cubic equations are, however, deficient in the critical region, where they have a tendency to fail to predict sharp changes of saturation pressures and liquid-dropout curves. In general, the use of equations of state for modeling reservoir fluids requires judicious tuning of both pure- component and binary parameters because both phase equilibria and volumetric properties are of interest. For example, Aasberg-Petersen and Stenby (131) obtained accurate volumes and liquid-dropout curves by using a volume-translated cubic EOS and proposing a characterization procedure for the C7+ fraction. The performance of various equations of state for modeling reservoir fluids was analyzed by Danesh et al. (132) and Anastasiades et al. (133).

Methods for improving the performance of cubic equations of state in the near-critical region will not be discussed here because they are described in Chapter 11 of this book. However, it should be mentioned that some techniques have been proposed specifically for near-critical hydrocarbon mixtures and may be applicable to reservoir-fluid simulation (134- 136).

In addition to phase equilibria, the van der Waals-type equations of state with classical mixing rules can be used to correlate thermodynamic functions such as excess enthalpy and excess volume. Several authors analyzed the prediction of excess enthalpies in mixtures containing nonpolar and weakly polar components (137-139). As shown by Casielles et al.

(137) and Zebolsky and Renuncio (138), even complex shapes of HE vs. x curves can be reproduced (e.g. curves in which HE changes from positive to negative values depending on composition). Also, excess enthalpies in ternary systems can be reasonably predicted from binary data. A simultaneous representation of VLE and heats of mixing is possible, although it has not been investigated extensively. A particularly accurate simultaneous representation of VLE and HE has been obtained by Abdoul et al. (218) using their group contribution method.

98

Van der Waals-type equations of state are also capable of representing excess molar volumes of fluid mixtures. For example, Serbanovic et al. (140) examined the application of cubic equations of state for the calculation of excess volumes in hydrocarbon mixtures and obtained satisfactory results. However, there is very little chance that excess volumes can be calculated simultaneously with vapor-liquid equilibria. The theoretical basis of the equations is too weak to predict consistently the different phenomena that manifest themselves in G E and V E. A successful simultaneous correlation can be obtained only in individual cases.

The classical quadratic mixing rules become inappropriate when applied to mixtures containing strongly polar and, in particular, associating components. A typical example of the failure of classical mixing rules for such systems is provided by mixtures of water and hydrocarbons. Fairly accurate results can be obtained only when different values of binary parameters are used for the water-rich and hydrocarbon-rich phases. This is, however, unacceptable because it leads to a thermodynamic inconsistency (141). A very convincing example of the failure of classical mixing rules for alcohol + hydrocarbon systems was presented by Trebble (142). Trebble used the Trebble-Bishnoi EOS, which makes it possible to use as many as four binary parameters by applying the quadratic mixing rules to its four constants. Even with four binary parameters, a satisfactory correlation of VLE for alcohol + hydrocarbon systems was not obtained. In the next three sections, we will discuss various techniques that have been used to create models that are applicable to strongly nonideal systems.

4.4.2 C o m p o s i t i o n - D e p e n d e n t C o m b i n i n g Rules

Since the classical quadratic mixing rules are not sufficiently flexible to correlate the phase behavior of mixtures containing strongly polar and associating components, several authors introduced empirical modifications to enhance their flexibility. These modifications effectively replace the composition-independent combining rule in Equation (4.70) with composition-dependent expressions.

Panagiotopoulos and Reid (143), Stryjek and Vera (21) and Adachi and Sugie (144) proposed three mathematically equivalent combining rules with two binary parameters. In the notation of Panagiotopoulos and Reid (143), the final mixing rule takes the form:

a = E E x i x j ( a i i a j j ) ' / 2 0 - k u + ( k i j - k j i ) x i ) (4.73)

Stryjek and Vera (21) also proposed another form, which they called a van Laar-type mixing rule because of a similarity with the van Laar equation for the excess Gibbs energy:

a- -EEx ix j (a i i a j j ) I / 20 -k i j k j i / ( x i k i j +xjkji)) (4.74)

Kabadi and Danner (145) proposed a two-parameter equation designed specifically for water + hydrocarbon mixtures:

2 [(a~,a22),/2 (1 - k12 ) + x, G(1 - Tr~ s )] a = x2all + x 2 a22 + 2xlx 2 (4.75)

99

where the subscript 1 denotes water. Correlations were established for the parameters k12 and G within homologous series of hydrocarbons. Michel et aL (146) and Daridon et al. (147) developed alternative correlations for mixtures containing water and hydrocarbons.

Schwartzentruber et al. (148) proposed a three-parameter expression to further enhance the flexibility of the Panagiotopoulos - Reid mixing rule:

a = Z Z x i x j ( a i i a j j ) ' / 2 I I - k u - l u m U x i - m j i x j ] i j mo.Xi + mjixj (4.76)

w h e r e kji = ko. , lji = -lij, mji "-- 1 - m U a n d kii = lii = 0. T h i s expression reduces to t h e

Panagiotopoulos-Reid mixing rule when m,j = 0. In general, the composition-dependent combining rules constitute the simplest method for

applying equations of state to complex systems. Their accuracy for binary systems is usually very satisfactory as shown by Sandoval et al. (149), Margerum and Lu (150) and others. However, the mixing rules described above suffer from a very serious deficiency, which was discovered for the first time by Michelsen and Kistenmacher (151). They are not invariant to dividing a component into a number of identical subcomponents. If a binary mixture with composition (Xl, x2) is treated as a temary system with composition (Xl,X2,X3), where the temary is formed by dividing component 2 into two pseudocomponents with identical properties, a different value for the parameter a will result. Therefore, the calculated properties of a mixture will depend on the number of pseudocomponents, which is in contrast to experimental evidence.

Several attempts have been made to eliminate this inconsistency while retaining the flexibility of the composition-dependent combining rules for mixtures. Mathias et al. (152) proposed a mixing rule that overcomes the problem of invariance:

/ 3 1/3 a = Z Z x i x j ( a i i a j j ) l / 2 ( 1 - k u ) + Z x i xj[(aiiajj)l/2]l/3lji

i j i (4.77)

The first term is the classical quadratic mixing rule. The second term introduces a cubic composition dependence, when / j i - - -ltj or a quartic one, when /ji ~: -ltj. It can be shown that Equation (4.77) reduces to the Panagiotopoulos-Reid mixing rule when lji = -lij. Twu et al. (27) proposed a somewhat similar mixing rule:

E 13 Z HuUS Gu,/3(agiaii) '/6 xj

a = Z Z x i x j ( a i i a j j ) l / Z ( l _ k o . ) + Z x i J i j i ~'~/~GO.X j

J

(4.78)

where

H o =(kj/- k0)/:r (4.79)

100

G u = exp(-/3~/H~/) (4.80)

This mixing rule can be used either in a two-parameter version (i.e., with only k O. and k j i

adjusted) or in a four-parameter version (i.e., with/3/j and/3ji adjusted in addition to k O. and kji). In both Equations (4.77) and (4.78), an improvement for multicomponent mixtures was obtained by introducing more symmetry into the non-quadratic part of the mixing rule, which bears resemblance to the composition dependence of the third virial coefficient. Another method for overcoming the Michelsen-Kistenmacher consistency problems was proposed by Schwartzentruber and Renon (153), who also introduced more symmetry into the cubic composition dependence of the mixing rule.

Figure 4.2 illustrates the performance of several composition-dependent combining rules for the strongly nonideal ternary system methanol + cyclohexane + hexane. It should be noted that all combining rules are capable of correlating vapor-liquid equilibria for the binary subsystems with reasonable accuracy. However, a much more stringent test is provided by the prediction of VLE in the ternary system using the parameters obtained from binary data. In this case, the three mixing rules that suffer from the Michelsen-Kistenmacher inconsistency (i.e., Panagiotopoulos-Reid, Stryjek-Vera and Schwartzentruber et al.) show a large loss of accuracy for the ternary system. Similar conclusions were also reached for other systems in the more comprehensive studies by Knudsen et al. (154) and Malanowski and Anderko (57). However, the mixing rule of Twu et al. (27) shows a more satisfactory behavior. Although it loses accuracy for the ternary system when only two binary parameters (i.e., k u and kji) are regressed from binary data, it becomes more accurate in its four-parameter form. The parameters of all the mixing rules discussed above usually show appreciable temperature dependence. The temperature dependence of these mixing rules cannot be established a prior i and has to be determined from experimental data at several temperatures.

4.4.3 Density-Dependent Mixing Rules

It is evident from the functional form of the composition-dependent combining rules that they do not reproduce the quadratic composition dependence of the second virial coefficient and, thus, violate basic statistical thermodynamics. This is indicative of their weak theoretical background. Therefore, several authors came up with a theoretically appealing concept that better results could be obtained if the constants of a selected equation of state were allowed to depend not only on composition and, sometimes, temperature, but also on density. To illustrate this concept, we consider the mixing rule of Luedecke and Prausnitz (155), who postulated that the attractive parameter can be written as

a = Z Z x i x j a u +a "c (4.81)

where the first term is identical to the classical quadratic mixing rule, a/j is equal to (aiiajj)l/2(1- k,j) and anc is a contribution of noncentral forces which are due to differences in polarity, size and shape of molecules. An expression for this term is constructed by analyzing boundary conditions. First, since anc refers only to contributions from unlike pairs, then:

l i m a ne --- 0 (4.82) x i ->0

101

Correlation of binary vapor - liquid equilibria Percent deviations

6.00 1 4.00 ---]

cyc lohexane + hexane 2.oo

o.oo . . . . ~ . . . .

m e t h a n o l +

cyc lohexane

6.00 1 4.00 2.00 -~ 0.00 r][ ]DDr D

m e t h a n o l +

hexane

6.00 4.00 2.00 o.oo __1

PR SV SGR AEOS SRKS2 SRKS4

Prediction of ternary vapor liquid equilibria

m e t h a n o l +

cyc lohexane +

hexane

8.00

00o 4.00

2.00 D 0.00

PR SV SGR AEOS SRKS2 SRKS4

Figure 4.2 Application of several composition-dependent combining rules to the correlation and prediction of VLE in the ternary system methanol + cyclohexane + hexane. Vapor-liquid equilibria in the three binary subsystems are correlated and the parameters are used to predict VLE in the ternary system. The following models are used in conjunction with the Peng-Robinson or Soave-Redlich- Kwong EOS: PR - Panagiotopoulos and Reid's mixing rule; SV - Stryjek and Vera's van Laar-type mixing rule; AEOS - Anderko's EOS incorporating association; SRKS2 - Twu, Bluck, Coon and Cunningham's mixing rule with two binary parameters; SRKS4 - the same model with four binary parameters.

Second, it can be assumed that, as temperature rises or as density falls, the importance of noncentral forces declines:

lim a ne = 0 (4.83) p/RT->O

The simplest non-quadratic approximation that is consistent with these boundary conditions introduces cubic composition dependence and linear density dependence:

102

a nc : ( p / R T ) Z Z x i x j ( x i c i o ) -t-xjcj(i)) (4.84)

where ci(j) is a binary parameter that reflects noncentral forces when molecule j is infinitely dilute. The resulting mixing rule is

a = Z Z x i x j ( a i i a j j ) ' / 2 ( 1 - k o . ) + ( p / R T ) E E X i X j ( X i C i ( ] ) -I- XjCj(i) ) (4.85)

and contains three binary parameters. In the low-density limit, it correctly reproduces the quadratic composition dependence of the second virial coefficient. While this mixing rule accurately correlates binary VLE and LLE, it was found to be inaccurate for ternary systems.

Following the early studies of Whiting and Prausnitz (156) and Mollerup (157), most density-dependent mixing rules were derived using the local composition concept to represent mixture nonideality. These methods have been reviewed by Danner and Gupte (158) and, later, by Anderko (159). We will not discuss them here because they have not been proven advantageous for practical calculations. The density-dependent mixing rules do not improve the results of phase-equilibrium computations over the density-independent, composition- dependent combining rules. The loss of accuracy for ternary and multicomponent systems, which has been discussed in Section 4.4.2, is also shared by the density-dependent mixing rules (25). At the same time, the density-dependent mixing rules eliminate the third-order density dependence of cubic equations of state and impose a higher-order dependence. Also, there are alternative techniques for satisfying the second-order composition dependence of the second virial coefficient, which will be discussed in Section 4.4.4.

4.4.4 Combining Equations of State with Excess-Gibbs-Energy Models

Excess Gibbs energy (gZ) models based on the local composition concept have proven very useful for the correlation of low-pressure phase equilibria. Therefore, much effort has been focused on combining them with equations of state. A detailed discussion of the combined gE-EOS models is provided in Chapter 9 of this book. However, it is worthwhile to mention them here in order to provide an appropriate perspective for other methods discussed in this chapter.

Standard thermodynamics defines a simple relationship between the excess Gibbs energy and fugacity coefficients, which are calculated from an equation of state:

gE(T,P,x)=RTOn~= -Zxiln~i) (4.86)

where ~ x and ~bi are the fugacity coefficients of the mixture and the i-th pure component, respectively. This relationship was employed by several authors to combine excess-Gibbs- energy models and equations of state so that the desirable features of both classes of models can be utilized.

Vidal (160) derived the infinite-pressure limit of the excess Gibbs energy calculated from the Redlich-Kwong EOS with quadratic mixing rules. To arrive at an explicit expression, Vidal assumed that v = b and v E = 0 at infinite pressure.

gE (p = oo) = ln21- a / b + Z xiai / bi I

103

(4.87)

It is interesting to write this equation specifically for a binary system when the binary parameter k O. is equal to zero:

/[/ /2/ /2] 2 gE(p=oo)=ln2(Xlb11/( xzbz2 all/bll -- a22/b22

t , ~ ) t , b bll b22 (4.88)

In this equation, xibii/b are the infinite-pressure volume fractions and the terms [(aii/bii)ln2/bii] 1/2 can be identified with solubility parameters. Therefore, Equation (4.88) becomes essentially identical to the well-known Hildebrand-Scatchard regular solution theory:

gE(P: ~) = b*,02(6, -62) 2 (4.89)

where (I) i and ~i are the volume fractions and solubility parameters, respectively. This explains the inherent limitations of quadratic mixing rules because it becomes evident that they are applicable only to mixtures whose excess properties can be approximated by the regular- solution theory. While this is a good approximation for hydrocarbon mixtures, it cannot be applied to systems containing strongly polar or associating components.

Vidal's (160) derivation can be generalized to other equations of state as shown by Huron and Vidal (161):

gZ(P: m) = A( -a /b+ Zxiai /bil (4.90)

where A is a characteristic parameter for each cubic EOS used. This equation suggests a mixing rule for the attractive parameter in cubic equations of state:

a :b~_axiaii/bii-gE(P=c~)/A) (4.91)

Any excess-Gibbs-energy model can be selected to represent the infinite-pressure excess Gibbs energy. Huron and Vidal (161) suggested using the NRTL equation. It should be noted, however, that the excess Gibbs energies at finite and infinite pressure are not identical and cannot be used interchangeably. Therefore, the parameters of the Huron - Vidal mixing rule have to be determined by fitting the complete equation of state to phase-equilibrium data. Parameters of the original NRTL model cannot be used for that purpose. The Huron - Vidal mixing rule was shown by several authors to give results similar to those from the NRTL model. In particular, it makes it possible to predict multicomponent phase equilibria from binary data with satisfactory accuracy (162,163). It was also shown to compare favorably with several composition-dependent combining rules by Knudsen et al. (154).

104

Kurihara et al. (164) proposed a modification of the Huron-Vidal mixing rule that separates the classical quadratic term from a "residual" term, which is calculated from the Wilson equation:

a = ~ ~ x i x j ( a i i a j j ) 1/2 - (b/ln2)g E(resiaual) ( P = oo) (4.92)

Tochigi et al. (165) used this equation to predict temary vapor-liquid equilibria from binary data and obtained satisfactory results.

More recently, the focus of research in this area shifted to developing mixing rules that not only share the numerical properties of equations for the excess Gibbs energy, but also can be used with exactly the same parameters as the original excess-Gibbs-energy models. For this purpose, the infinite-pressure limit was abandoned in favor of a more realistic zero-pressure limit. An early method for accomplishing this was proposed by Mollerup (166). It should be noted that the zero-pressure limit creates problems whenever the equation of state does not predict a liquid root, which is needed to calculate the liquid-phase fugacity coefficients and, subsequently, the excess Gibbs energy. For example, the zero-pressure liquid root of the van der Waals EOS exists when a/RTb > 4. In other cases, it has to be determined by suitable extrapolation. For this purpose, two techniques have been proposed by Michelsen (167) and Heidemann and Kokal (168).

Michelsen (167) applied the Huron-Vidal technique of matching the excess Gibbs energy using the reference pressure of zero. In contrast to the Huron-Vidal mixing rule, the expression obtained is not explicit:

gE q ( a ) = ~_ziq(aii)+--~--f+ ~izi ln( b ] (4.93)

i • ~ b i i )

In this equation, a is a shortcut notation for a/bRT and the function q(a) is given, for the Redlich-Kwong EOS, by:

ln( u+ 1 t q(a) = - 1 - ln(u- 1)- a ,,---~- (4.94)

with

u 1( ) = = a - 1 - ~/a 2 - 6a + 1 (4.95)

These equations cannot be used to evaluate the mixing rule under all conditions because the function q(a) is defined only for a>3+23/2. To maintain continuity for all values of a, Michelsen suggested two approximations. In the first approximation, a linear relation was used:

q(a) = qo + ql a (4.96)

This leads to an explicit mixing rule:

105

1EgE a ~ = ~ Z i a i i n t- n t- Z i In

i -gl- • (4.97)

where ql has a universal recommended value. Due to its similarity to the Huron-Vidal mixing rule, this equation was called the Modified Huron-Vidal First-Order mixing rule (MHV1). In the second approximation, Michelsen employed a quadratic approximation q(a)=qo +ql a +q2 a2, which yields the Modified Huron-Vidal Second-Order (MHV2)

mixing rule:

( ~ i ) ( 2 ~ 23 gZ ~i b (4.98) ql a,~,, - . ziaii -]- q2 a ~ - . z i a i i = --~ + . z i ln-~-//

where ql and q2 are universal parameters. Dahl and Michelsen (169) demonstrated that the MHV2 mixing rule gives very good predictions of high-pressure VLE using parameters obtained from low-pressure VLE regressions. In spite of its success, it should be noted that MHV2 requires a relatively arbitrary quadratic extension for the calculation of volumes. Also, similarly to the Huron-Vidal mixing rule, it fails to satisfy the quadratic composition dependence of the second virial coefficient.

Heidemann and Kokal's procedure (168) is, in spirit, similar to Michelsen's because it also involves the use of an auxiliary function q(a). Since this function is cubic, the amix parameter has to be found in an iterative way. However, it has similar properties. More recently, Boukouvalas et aL (170) proposed a linear combination of the algebraically similar MHV1 and Huron-Vidal mixing rules and obtained an improvement for mixtures containing components with appreciably different sizes. This combination is purely empirical because the MHV1 and Huron-Vidal mixing rules have been derived at different reference pressures and the reason for the improvement is unclear.

A different approach has been proposed by Wong and Sandler (171). Instead of trying to match the zero-pressure excess Gibbs energy, Wong and Sandler observed that the excess Helmholtz energy is practically independent of pressure, which makes it almost the same at low and infinite pressure. At the same time, the zero-pressure Helmholtz energy is equal to the zero-pressure Gibbs energy. This makes it possible to identify the low-pressure excess Gibbs energy calculated from an activity-coefficient model with the infinite-pressure excess Helmholtz energy calculated from an equation of state:

E (T,P = oo, x)= G(T,P(low),x) AEos (4.99)

Additionally, Wong and Sandler utilized the quadratic composition dependence of the second virial coefficient to relate the mixed-fluid a and b parameters to their pure-fluid counterparts:

B(T,x) =b- a = ~-,~-,xixjIbi j aij I (4.100) RT - - -~)

106

where the cross-term for i¢j is related to the pure-fluid terms by a combining rule, i.e.,

b~i - - ~ = -2 bii RT) (4.101)

with an additional binary parameter k, 7. The two conditions given by Equations (4.99) and (4.100) give the following mixing rules:

D amix = Q (4.102) RT 1 - D

O b m = ~ (4.103) 1 - D

where

/ a/ Q'--22XiXj b - - ~ o " (4.104)

a i Ge(x) = + ~ (4.105)

O Z X i b i R T crRT

and cr is an EOS-specific constant.

The Wong-Sandler mixing rule contains one additional binary parameter k U (cf Equation (4.101)), which has to be regressed, preferably using low-pressure data. Alternatively, it can be obtained from the same excess Gibbs energy model that was used to formulate the mixing rule. Therefore, the application of this mixing rule is not automatic even though the parameters of the excess Gibbs energy model can be used unchanged. Wong et al. (172) showed that the Wong-Sandler mixing rule remains accurate even when it is used at pressures and temperatures that are very remote from the conditions of the low-pressure VLE data that were used to regress the parameters. Huang and Sandier (173) compared the MHV2 and Wong-Sandler mixing rules and found that the Wong-Sandler formulation is more accurate, especially at high pressures and temperatures. Also, Escobedo-Alvarado and Sandier (236) studied the application of the MHV1 and Wong-Sandler mixing rules to liquid-liquid equilibrium computations and concluded that the Wong-Sandler rule is better for predicting high-pressure LLE from low-pressure data although both mixing rules are appropriate for correlating the data.

An extension of the Wong-Sandler mixing rule was proposed by Chou and Wong (174), who imposed the correct composition dependence on the third virial coefficient. Since the mixing rule for the third virial coefficient contains triplet interaction parameters, Chou and Wong utilized them to introduce ternary parameters for fitting ternary phase equilibrium data. It should be noted that this kind of additional flexibility limits the predictive character of the model.

107

In their original forms, the combined GE-EOS models do not reduce to the classical quadratic mixing rules, which makes it difficult to use them in a unified computational environment for both hydrocarbons and polar and associated chemicals. This is important in the context of process and reservoir simulation, where users have a tendency to rely on the well-established quadratic mixing rules for hydrocarbon systems. Therefore, Orbey and Sandier (237) and Twu et al. (238,239) developed modifications of the Wong-Sandler mixing rule so that the classical quadratic mixing rule can be recovered. The modification of Orbey and Sandier (237) is based on a reformulation of the excess Gibbs energy model so that it can reduce to the classical mixing rule. The approach of Twu et al. (238,239) relies on a different concept. It is based on equating the difference between the excess Helmholtz energies for the complex fluid and a classical van der Waals fluid with an analogous difference for the Helmholtz energy departure, i.e.,

A E E - Avd w = AA-/~vdW (4-105a)

where the subscript vdW denotes a van der Waals fluid for which the classical quadratic mixing rule is used. This expression makes it possible to define a mixing rule that combines the quadratic mixing rule with an excess Gibbs energy model.

The mixing rules based on excess Gibbs energy models have gained considerable popularity for practical calculations. This is due to the fact that they share their features with the activity coefficient models such as NRTL or UNIQUAC, which have been commonly used for engineering calculations for almost three decades, especially in the context of process simulation.

4.5 E X P L I C I T T R E A T M E N T O F A S S O C I A T I O N IN E M P I R I C A L

E Q U A T I O N S O F S T A T E

Molecular association and solvation strongly affect the thermophysical properties of pure and mixed fluids. In the case of pure-fluid PVT properties, association significantly increases the critical temperature, makes the second virial coefficient more negative and affects the shape of the vapor-pressure curve. For a number of compounds, such as carboxylic acids or hydrogen fluoride, association drastically reduces the gas-phase compressibility factor even at low densities. Derivative properties such as enthalpies are also strongly influenced. While the behavior of vapor-pressure curves can be adequately modelled up to the critical temperature using the empirical techniques described in Section 4.1, the effect of strong association on the gas-phase P V T properties makes it necessary to account for association in the equation of state. In addition to PVT properties, association profoundly influences the nonideality of mixtures, which affects phase equilibria. In principle, the multiparameter mixing rules discussed in Section 4.4 are sufficiently flexible to correlate the phase behavior of strongly nonideal, associated systems. However, these methods require a substantial number of binary parameters (typically 2-4 per binary) and are, therefore, only correlative. It is usually difficult to extrapolate their parameters beyond the range of experimental data and the prediction of phase equilibria in multicomponent systems from binary data is frequently unreliable. Therefore, several methods have been devised to incorporate association explicitly into the framework of equations of state.

108

The possibility of representing phase equilibria by postulating that molecules associate or solvate to form new species has been recognized for a long time (175). The chemical theory has been shown by many authors to be particularly useful when applied to mixtures containing one associating compound and some inert solvents. In this ease, the chemical approach made it possible to obtain an accurate representation of phase equilibria with a limited number of physically meaningful parameters. However, the application of the chemical theory to systems containing any number of associating species encountered substantial difficulties. The major drawback of the chemical theory has been its specific character, restricted to particular classes of mixtures, and the difficulty of extending the models that are valid for binaries to multicomponent mixtures. In this section we will show how these and other problems were treated by various investigators. Our discussion will be limited to the application of the chemical theory to the modeling of associated fluids. Statistical-mechanical methods for treating association will not be discussed because they are described in Chapter 12 of this book.

In general, there are two methods for incorporating association equilibria into an equation of state. The first one is conceptually simpler, but it is more difficult to use, as we will show later. It consists in simultaneously solving chemical equilibria between associated species and physical (phase) equilibria for all species in the mixture. A pure associated compound is assumed to be a mixture of a few associated species Ai, each of them containing i monomeric units in a multimer. Once the associates are selected, they are treated as pseudocomponents and the system of equations to be solved is as follows:

Chemical equilibria for the reaction iA1 - Ai in each phase:

xi qg i (4.106) g i = i ni-1 i

X l l " (D1

Phase equilibria:

fiPhasel (T~ P, x) =ft" phase2 (T~ P, x_) (4.107)

where the superscripts stand for any two phases in equilibrium. The association constant Ki depends only on temperature because the standard state for the Ai species in both the gas and liquid phases is pure Ai in the ideal-gas state. These equations can be solved numerically provided that the fugacity coefficients of each species are calculated from an equation of state that is suitable for multicomponent mixtures. For this purpose, Wenzel et al. (176) used the Schmidt-Wenzel (15) cubic EOS whereas Grenzheuser and Gmehling (177) employed the PHCT EOS (87). In general, the most difficult problem in this approach is the assigmnent of associated species, the number of which has to be minimized. For example, if we assumed the continuous linear association model, the number of equations would be infinite. Therefore, Grenzheuser and Gmehling (177) restricted their calculations to dimers, which makes their method applicable primarily to carboxylic acids. On the other hand, Wenzel et al. (176) allow for the existence of higher multimers. For example, methanol is assumed to be a mixture of monomers, tetramers and dodecamers. The selection of multimers is arbitrary and results from a compromise between the requirements of reproducing the data and minimizing the number of adjustable parameters. Nevertheless, the number of adjustable pure-component parameters is very large (three parameters for each mono- or multimer and two or three for each reaction)

109

and their determination is by no means simple. To determine pure-component parameters, Wenzel et al. (176) utilize not only pure-component data (i.e., vapor pressure, liquid and vapor volumes and critical properties), but also binary VLE data.

Extension of this scheme to mixtures is straightforward, provided that the mixture contains only one associating component. In this case the chemical equilibria are not changed and the number of phase-equilibrium equations only increases by the number of inert components that are present in the mixture. However, in the case of mixtures containing two or more associating components, a problem of cross-associates appears. Since it is impossible to deduce all parameters for cross-associates from binary data, only arbitrarily selected cross- associates (usually cross-dimers) are allowed for.

The computational scheme outlined above makes it possible to represent VLE and LLE with good accuracy using only one binary parameter as shown by Wenzel et al. (176) and Kolasinska et al., (178). Lang and Wenzel (179) were even able to simulate a third phase for a pure component by assuming the existence of a large oligomer (e.g., a 25-40 mer). The parameters of that oligomer were adjusted to match the properties of the solid phase. This approach can be used as an alternative to introducing an additional physical term to simulate a solid phase in an equation of state as done by Schmidt and Wenzel (230).

However, the good features of this method for introducing association equilibria have been achieved at a considerable cost. First, the determination of pure-component parameters is cumbersome and no clear-cut algorithm exists for guessing which associates should be assumed to provide a good representation of phase equilibria. This can be done only by trial and error. Second, this approach cannot overcome the traditional deficiency of association models, i.e., the difficulty of formulating a consistent model for systems containing any number of associating and inert components.

A successful attempt to solve these problems has been made in the development of the second method for incorporating association into an equation of state. This method is based on solving the chemical-equilibrium expressions for an assumed association model analytically in order to derive an explicit equation of state with association built-in. The first model of this type was proposed by Heidemann and Prausnitz (180), who combined an EOS with the continuous linear-association model. According to this model, the equilibrium constants for the consecutive association reactions Ai + A1 = Ai+l (i = 1 . . . . oo) are equal to each other (i.e., Ki,i+l = K). Heidemann and Prausnitz used a generalized van der Waals EOS:

p R T a = v Zrep(~)- ~-T1--iat t (~) (4.108)

where a and b are the generalized van der Waals parameters andZrep and 1-latt are functions of the reduced density ~. For associated species, the classical quadratic mixing rules have been used:

a = E Z x i x j ( a i a j ) '/2 (4.109)

b : Z xibi (4.110)

with the following combining rules for the multimers' parameters:

110

a i = i 2 a l (4.111)

b i = ib, (4.112)

Heidemann and Prausnitz obtained the remarkable result that, with these combining rules, the reduced density and, subsequently, the Zrep and I-latt functions are independent of association. The effects of association enter the equation of state only through the factor nT/no, where nT is the total number of moles of the associated species and no is the number of moles of the compound in the absence of association. Then, the van der Waals parameters take the form:

a = (n o / n T)2 a, (4.113)

b = (n o / n r)bl (4.114)

and the equation of state becomes

p = n r RT~ a n o b Zr~P(~)--~ -1-Iatt(~) (4.115)

After evaluating the material-balance and chemical-equilibrium equations, the factor nT/no becomes

n__ L = 2 (4.116) n o 1 + (1 + 4 R T K e g / v) 1/2

where

g = f[(Zr~ p - 1) / ~]d~ (4.117) o

The expression for nT/no can be easily generalized for mixtures containing one associating component A and any number of inert ones:

n T 2xA ~ = + l - x A (4.118) n o 1 +(1 + 4 R T K e g x 4 / 1 ) ) 1/2

It should be noted that the final equation of state is strongly dependent on the combining rules for the parameters a and b of the multimers. These combining rules were tested by Wenzel and Krop (181), who analyzed the a and b parameters calculated for a series of hydrocarbons using the Peng-Robinson EOS (14). They found that the multipliers i 2 and i in Equations (4.111) and (4.112) are reasonable, but not very accurate. A closer inspection of the a and b values would suggest the multipliers i e where the exponent e lies between 1 and 2 (closer to 2 for a and closer to 1 for b). However, it is not worthwhile to establish a more accurate approximation for the parameters a and b of hydrocarbons for two reasons. First,

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hydrocarbons provide only a very rough approximation of the properties of multimers and, second, any fractional exponent would preclude any further analytical derivations. It should be realized that the combining rules for the associates are purely hypothetical and serve merely as a tool for deriving a convenient closed-form equation of state.

The original Heidemann-Prausnitz (180) technique yields equations of state that can be very accurate for the calculation of PVT properties of pure fluids. This has been demonstrated by Twu et al. (182,183), who applied this technique in conjunction with the Redlich-Kwong EOS to carboxylic acids and hydrogen fluoride. They achieved a very good representation of gas-phase PVT properties, which are extremely nonideal for these fluids. However, Twu et al. did not obtain any simplification for mixtures and continued to use a multiparameter mixing rule with composition-dependent combining rules for mixtures containing HF and acids. Thus, the original Heidemann-Prausnitz method is fully successful for PVT properties, but still requires multiparameter mixing rules for mixtures. Therefore, several authors introduced modifications to develop a more convenient equation for mixtures.

Ikonomou and Donohue (184,185) used the concept of Heidemann and Prausnitz in conjunction with the PACT equation of state given by Equation (4.57). They assumed that the mixing rules for the associated species are analogous to those used for nonassociating components. As with the Heidemann-Prausnitz EOS, the key step was to assign appropriate combining rules for the associated species:

3 3 O" 0. = 0"11

rj = jr~

qj = Jql

(4.119)

The form of the resulting equation of state, called APACT (Associated Perturbed-Anisotropic- Chain Theory) is significantly different from that obtained by Heidemann and Prausnitz in that it contains the ratio (nT/no) as a separate term:

n T PACT PACT (4.120) Z = ~ + Zre p + Zattr

n 0

This is a consequence of a different equation of state. In particular, it can be shown that the presence of the Prigogine parameter c, which multiplies the nonideal part of the equation, gives rise to the separation of the (nT/no) term from the rest of Equation (187). The factor (nT/no) is similar to that obtained by Heidemann and Prausnitz except for the term g(O, which is equal to zero in APACT. Thus, for a mixture containing one associating component and any number of inert ones, the factor (nT/no) becomes:

n T 2 x A

no I+(I+4RTKxA/v)l/2 +1 x A (4.121)

This equation was later extended to systems with multiple associating components (186). It should be noted that an analytical solution for (nT/no) cannot be found when more than one

112

associating component is present. Therefore, Ikonomou and Donohue (186) and Economou et al. (189) solved the chemical equilibria and material balances numerically and, subsequently, devised an analytical approximation. Economou and Peters (233) also developed a specialized version of APACT for systems containing hydrogen fluoride. The method of Ikonomou and Donohue was also used by Elliott et al. (188) in conjunction with a much simpler equation of state that approximates the Perturbed-Hard-Chain Theory. A somewhat similar model has also been proposed by Deiters (190).

Anderko (191-195) developed an equation for associated mixtures that combines a cubic EOS with a chemical term that accounts for association. Initially, Anderko (191) added a chemical term z (~h) to the usual (or physical) compressibility factor, i.e.,

z = z ~h) + z (oh) - 1 (4.122)

by generalizing the observation that the second and higher virial coefficients could be separated into physical and chemical contributions (196,197). Later, Anderko (195) rederived Equation (4.122) by applying the Heidemann-Prausnitz technique with different combining rules. The term z (oh) is equivalent to the ratio n+/no, i.e.,

n T z (oh) = ~ (4.123) n o

whereas the term z (ph) is an equation of state for the hypothetical monomeric fluid and can be expressed by any cubic equation of state. Most calculations were performed, however, using the Yu-Lu (26) EOS because of its accuracy for liquid-density calculations. The combined model was called AEOS (Association + Equation of State).

For the continuous linear association model, the expression for z (oh) is identical to Equation (4.121) obtained by Ikonomou and Donohue. To extend the applicability of the equation to systems containing any number of associating components, Anderko (193) developed a multicomponent analog of the continuous linear association model. The model is based on the assumption that, 1. Only linear multimers containing any possible combinations of monomeric units occur in the mixture. There is no upper limit on the size of the multimer and 2. The multimers are formed in consecutive association reactions. For each of the elementary reactions the equilibrium constant depends on the chemical identity of the monomers that form the bond but not on the number of monomers that constitute the multimer. In this scheme a system containing n associating components is characterized by n self-association constants gii and n(n-1)/2 cross-association (solvation) constants Kij (i~j).

The derivation requires certain approximations because the system of chemical equilibrium and mass-balance equations cannot be solved analytically. The final result is:

= nT (4.124)

no i=1 1 + [1 + 4 R T ( ~ KuxAo )) / v] 1/2 k=, j=l

where A (i) and B (k) denote the i-th associating and k-th inert components, respectively. This equation satisfies three important boundary conditions:

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1. If the number of associating components is equal to one, Equation (4.124) reduces to Equation (4.121) for a mixture with one self-associating component. 2. When all self- and cross-association constants are equal to each other, the associated species become indistinguishable. Then, the expression for z (oh) reduces to the pure-component case. 3. In the low-density limit, the correct composition dependence of the chemical contribution to the second virial coefficient is recovered, i.e.,

B(Ch) = ££xAo)xAo~(-RTKij ) (4.125) i j

Although the continuous linear-association model is successful for a variety of associating compounds (e.g., alcohols, phenols, amines, etc.), some compounds require different association models. An important example is water because the structures of water clusters are three-dimensional rather than linear. Another example is hydrogen fluoride, which preferentially forms associates of some intermediate sizes (such as 5, 6, 7,... mers) rather than long chains. At the same time, the analytical solution of the chemical-equilibrium and mass- balance expressions is very difficult, if not impossible, for more complex association models. To overcome this difficulty, Anderko (195) noted that the nT/no term is always an algebraic function of the product RTKxA/v for any one-equilibrium-constant association model. In the simple case of a mixture containing one associating component, the term nT/no is

nT F( RTKx A ] (4.126) = x A + 1 - x A

no x , V

where F is an algebraic (even though unknown a priori) function of the product RTKxA/v. This equation is valid irrespective of the characteristics of the association model. If, for brevity, we use the symbol q to denote the product RTKxA/v, then the function F(q)=2/[l+(l+4q) 1/2] will give us the expression for the continuous linear-association model.

Lencka and Anderko (198) and Anderko and Prausnitz (199) utilized this result to arrive at a closed-form expression for hydrogen fluoride. The preferential formation of intermediate- size associated HF species can be expressed using a Poisson-type distribution function (199):

K~J -1

Kjj+I = K ~ (4.127) j!

which can be compared with the simple relationship Kjj+a=K for the continuous linear model. For this model, the values of nv/no were found numerically and closely approximated using the function

8

1 + Z akq k k=l (4.128) F(q) = (1 + q)8

where ak are numerical coefficients and q denotes, again, the product RTKxA/v. For water, Anderko (195) developed a simple function:

114

1 F(q) = (4.129)

1 + q + a q 2

where a is a numerical constant. This function was later empirically improved by Shinta and Firoozabadi (200):

(4.130) F(q) = ~f~ + q + 3q 2

where ~ and 13 are, again, numerical constants. For any association model, the final AEOS equation of state contains five parameters for

each associating compound, i.e., the enthalpy and entropy of association to evaluate the temperature dependence of the association constant and the apparent critical temperature, critical pressure and acentric factor of a hypothetical monomer. The latter three parameters are used to calculate the physical contribution to the compressibility factor (zCPh)), which is expressed by a generalized cubic EOS. All of these parameters are fitted to match pure- component vapor pressure, liquid density and (optionally) critical coordinates. Alternatively, the equation-of-state parameters can be evaluated locally for each temperature of interest using vapor-pressure and liquid-density data generated from another equation of state as discussed by Anderko (201).

When applied to mixture calculations, the AEOS model is capable of reproducing VLE and LLE in strongly nonideal systems using only one binary parameter. This parameter is introduced in the most traditional way, i.e., by applying the classical quadratic mixing rule to the a parameter in the cubic equation of state that represents the physical contribution to the compressibility factor z (ph). For example, Shinta and Firoozabadi (200) demonstrated that AEOS is very effective for the simultaneous correlation of VLE and LLE in aqueous hydrocarbon systems. Similar results were obtained by Lencka and Anderko (198) and Anderko and Prausnitz (199) for mixtures containing hydrogen fluoride and various chlorinated and fluorinated hydrocarbons. Anderko and Malanowski (194) showed very good results for mixtures containing alcohols and phenols. Since only one binary parameter is needed, there are no problems with the intercorrelation of binary parameters. This enhances the accuracy of the model for multicomponent systems. As shown in Figure 4.2, AEOS predicts the VLE in the ternary system methanol + cyclohexane + hexane with a very good accuracy, which is similar to the accuracy of VLE correlation for the binary subsystems. Shinta and Firoozabadi (200,202) showed that AEOS accurately predicts the phase behavior of multicomponent systems encountered in reservoir engineering (i.e., water + H2S + CO2 + hydrocarbons). Also, they demonstrated that the behavior of water + reservoir-crude systems (e.g., in high-pressure steam distillation) can be modeled with the same degree of simplicity as for dry reservoir-crude systems (202). In many cases, AEOS is purely predictive, i.e., good estimates can be obtained with the binary interaction parameters set equal to zero. This has been demonstrated for the industrially important systems containing HF and halocarbons (198, 199). For example, Figure 4.3 shows the representation of VLE and VLLE for the HF +

I : I I

20 / ~ . . . . . . . ~ . . . e . . ~ . . . . . . . . . .

' o

~ 1 5

m lO

o i i i i o.o 0.2 0.4 0.6 0.8 1.0

X H F

12 • . . . . . . . . .

lO

8

7:::, 6

n

4

2

0 0.0 0.2 0.4 0.6 0.8 1.0

X H F

115

Figure 4.3 Vapor-liquid-liquid equilibria calculated for the systems HF + CF3CC13 (upper graph) and HF + CHF2C1 (lower graph) using the AEOS equation (198) without any binary parameters (dotted lines) and with one regressed binary parameter (solid lines). Experimental data (solid circles) are from Knapp et al. (234) and Wilson et al. (235).

CHF2C1 and HF + CF3CC13 systems without any binary parameters (dotted lines) and with one regressed binary parameter (solid line).

The main difficulty associated with the use of equations of state based on the chemical theory is the determination of pure-component parameters because five parameters have to be fitted to vapor-pressure and liquid-density data and, most importantly, their values have to be physically meaningful. However, this difficulty is compensated by the simplicity of calculations for mixtures.

116

4.6 E Q U A T I O N S O F S T A T E AS F U L L Y P R E D I C T I V E M O D E L S

Most equations of state contain some binary parameters that have to be determined from experimental data. Therefore; numerous methods have been developed to enhance the predictive capability of equations of state so that they can be used without having to regress experimental data. The simplest way to achieve this is to develop correlations for the binary parameter k o. for separate families of mixtures that contain one common component. A more general approach is to create equations of state with parameters calculated from group contributions. A third possible approach is to combine equations of state with existing predictive group-contribution methods such as UNIFAC.

4.6.1 Correlations for Binary Parameters

Correlations for binary parameters in equations of state can be developed for homologous series of mixtures, i.e., mixtures containing one common component and a number of components that belong to a certain family. Not surprisingly, most such correlations have been developed for mixtures of hydrocarbons and a few selected nonhydrocarbons such as carbon dioxide or nitrogen. To develop such correlations, it is necessary to determine the binary parameters for individual binary pairs and find appropriate independent variables for the correlation.

For example, Valderrama et aL developed correlations for binary pairs containing hydrogen sulfide (203), carbon dioxide (204), hydrogen (205), nitrogen (206) and nonpolar compounds. They used a temperature-dependent function for the k,j parameter:

k o. = A - B / T w (4.131)

with the parameters A and B correlated with the acentric factor:

2 (4.132) A = A o + A~co j + A2co j

2 (4.133) B=B 0 +BI(.O j +B2(-o j

In this correlation, the subscript j denotes the nonpolar component that forms a binary system with either H2S or CO2 or H2 or N2.

Kordas et al. (207) and Avlonitis et al. (208) developed similar, but somewhat more elaborate correlations for systems containing carbon dioxide and nitrogen, respectively. Correlations for CO2 - containing mixtures were also developed by Kato et al. (209) and Moysan et al. (210). Nishiumi and Gotoh (211) obtained a correlation for systems containing hydrogen and Schulze (212) generalized the parameters for helium-containing mixtures. Another correlation was proposed by Gao et al. (213) for light hydrocarbon systems. More recently, Coutinho et al. (214) proposed a correlation for binary parameters on the basis of the theory of dispersion forces and applied it to systems containing CO2 and hydrocarbons.

It should be noted that such correlations can be obtained only when a single binary parameter is used for each binary pair. If two or more parameters are used, they usually become strongly correlated with each other which makes it practically impossible to correlate

117

them with parameters such as Tc and co. Therefore, correlations of this kind are used only in conjunction with the classical quadratic mixing rule with one binary parameter. In particular, they can be established when the one-parameter quadratic mixing rule is used for equations of state based on the chemical theory. This has been demonstrated by Anderko and Malanowski (194), who obtained correlations for homologous series of mixtures containing methanol and phenol.

4.6.2 Group-Contribution Equations of State

A much more general approach is the development of equations of state in which all parameters are calculated from group contributions. Generalization of the classical quadratic mixing rules in terms of groups rather than components is relatively straightforward even though it requires large-scale parameter regressions. For example, Pults et al. (215,216) presented a generalization of the classical quadratic mixing rule in terms of group contributions:

NcUc No~ a = Z Z X i X j Z Z V i m V j n q m q n a m n (4.134)

i j m n

Uc U~

b : Z x i Z v i m b m (4.135) i m

where the number of groups m in molecule i is given by Vim and the quantities Nc and NG represent the number of components and the number of groups, respectively. The quantity qm is a reduced area for group m, normalized to a value of 10 for methane. Pults et aL applied these mixing rules to a simplified version of the COR EOS (93) obtained by combining the COR repulsive term with a Redlich-Kwong attractive term. Since the mixing rule is a group- contribution version of the classical quadratic mixing rule, the equation is limited to nonpolar components. Another equation of state for nonpolar fluids was proposed by Georgeton and Teja (217), who rewrote the Patel-Teja EOS (16) in terms of group contributions.

A more elaborate group-contribution EOS for hydrocarbons and nonpolar fluids was proposed by Abdoul et al. (218). Their model is based on the combination of a cubic equation of state with an excess-Gibbs-energy model. The excess Gibbs energy is defined at a constant packing fraction rl=b/v. The same packing fraction is used for pure components and mixtures so that the excess functions can be unambiguously calculated. Apart from the constant- packing-fraction assumption, the method of Abdoul et al. (218) is similar to the Huron-Vidal technique (161). To calculate the properties of mixtures, an excess Helmholtz energy is added to pure-component Helmholtz energies calculated from an equation of state. The excess function is given by

A E (T,~l,x) = N R T ~ X i In ~- + E(T,x)~ (7/) (4.136) i

The first term results from mixing at the constant packing fraction and corresponds to a combinatorial contribution to the excess functions. The term ~(r/) is chosen in such a way that

118

the same P-V relationship applies to mixtures and pure components. Since Abdoul et aL use a translated Peng-Robinson EOS, the term ~(r/) is equal to [ln(l+~,r/)]/?. The term E(T,x) introduces the actual excess function and is calculated from

E ( T , x ) = E 1 ( T , x ) + E 2 ( T , x ) (4.137)

where E1 follows from Guggenheim's quasi-lattice model and E2 is an empirical correction for chain-length differences. E1 is given by

b ~ ~ x i b i xjbj Eij (T) E1 = -2 i=l j=l b b

(4.138)

with

1 N N

E O. (T) = --~ Z Z (aik - {~jk )(ai, -- ajl )Akl (T) k=l l=1

(4.139)

where a/k is the relative amount of group k in molecule i and Akl is the interaction parameter between groups k and l. The parameters Akt are calculated from temperature-dependent group contributions. In view of the composition dependence of El, this method is suitable only for mixtures containing nonpolar or weakly polar compounds. For these classes of compounds, this method is very accurate because it predicts not only phase equilibria, but also heats of mixing and volumetric properties. Following the original work of Abdoul et al., Le Roy et al. (219) expanded the group contribution table to improve the prediction for systems containing heavy hydrocarbons. Also, Fransson et al. (220) extended the method to chlorinated and fluorinated hydrocarbons.

A group contribution method that is not limited to mixtures of nonpolar and weakly polar components has been developed by Skjold-Jorgensen (221,222). In this equation, the residual Helmholtz energy is calculated as a sum of a free-volume term and an attractive term. The free-volume term is obtained from the Carnahan-Starling EOS with the Boublik-Mansoori mixing rules whereas the attractive term is a group-contribution version of a density-dependent mixing rule with the NRTL equation used as the excess-Gibbs-energy model. The Skjold- Jorgensen model was specifically designed for the calculation of gas solubilities in solvents of different chemical character. The original group contribution tables were later extended by Wolff et al. (223) and Pusch and Schmelzer (224).

The existing group-contribution equations of state gained only limited popularity for two reasons. First, their group contribution tables are not as extensive as those of the existing excess-Gibbs-energy group contribution methods (i.e., UNIFAC and ASOG). Second, end users in industry tend to prefer to use the well-known and extensively tested UNIFAC method. Since UNIFAC can be coupled with cubic equations of state in an essentially unchanged form, the resulting UNIFAC-based equations tend to be more appealing. However, equations of state with independent group contributions may be advantageous for mixtures containing components with strongly differing volatilities, for which the UNIFAC parameters may be missing or less reliable.

4.6.3 Utilization of Predictive Excess-Gibbs-Energy Models

119

Because of the great popularity of the UNIFAC group-contribution model for chemical- engineering calculations, several authors developed methods for combining UNIFAC with cubic equations of state so that the group-contribution tables can be used unchanged. An early method for achieving this, called UNIWAALS, was proposed by Gupte et al. (225), who developed a technique for matching the excess Gibbs energy at zero pressure obtained from the van der Waals EOS with that from UNIFAC. The technique of Gupte et al. suffered from a problem that precluded it from generating the critical point. UNIWAALS was later improved by Gani et al. (226) to remove this deficiency. However, the resulting EOS was difficult to use because the pressure and volume were related through a differential equation rather than the customary cubic equation.

More recent procedures developed by Heidemann and Kokal (168), Michelsen (167) and Wong and Sandier (171) made it possible to incorporate UNIFAC into equations of state in a fully consistent way. These procedures have been described in Section 4.4.4. Their usefulness has been extensively demonstrated. In particular, Dahl and Michelsen (169) and Dahl et al. (227) used Michelsen's MHV2 mixing rule (167) and obtained good prediction of high- pressure VLE. Holderbaum and Gmehling (228) applied a slightly modified MHV2. Orbey et al. (229) used the Wong-Sandler mixing rule and obtained good predictions of both VLE and LLE. In all cases, the UNIFAC group-contribution parameters were used unchanged.

Although these methods make it possible to use the existing UNIFAC parameters, it should be noted that new parameters have to be regressed for numerous systems of interest. This is due to the simple fact that components that occur in high-pressure VLE do not necessarily occur in low-pressure VLE. For example, Holderbaum and Gmehling (228) had to regress additional group-contribution parameters that were necessary for calculating gas-gas and gas-alkane phase equilibria. Another problem associated with this approach is the temperature range for which the UNIFAC parameters are valid. Since the UNIFAC parameters were determined from low-pressure VLE, this temperature range is usually very limited. As we go to high-pressure VLE, the high pressures are frequently accompanied by high temperatures. Therefore, an extrapolation of the parameters is necessary. This extrapolation may be particularly doubtful if temperature-dependent UNIFAC parameters are used (i.e., the so-called Dortmund and Lyngby modifications of UNIFAC).

4.7 C L O S I N G R E M A R K S

Cubic equations of state are currently the most widely used models for a variety of practical applications, including chemical-process and reservoir simulation. Cubic equations for pure components have reached maturity and no substantial progress can be expected in this area. Reliable techniques have been established to represent vapor pressures of both nonpolar and polar or associating compounds. The representation of volumetric properties has been improved within the limits imposed by the cubic form. The only exception is the representation of properties of heavy hydrocarbons and ill-defined compounds, which is still an area of active research.

In contrast to pure components, the representation of mixed-fluid properties by using cubic equations of state is still in a state of flux. While the classical quadratic mixing rules are the method of choice for mixtures containing nonpolar or weakly polar components, several alternative methods are available for mixtures containing strongly polar or associating

120

components. The composition - dependent combining rules provide the simplest successful method. They are capable of satisfactorily representing the properties of binary mixtures, but may lose accuracy for multicomponent mixtures. This may be due either to their internal consistency problems (i.e., the Michelsen-Kistenmacher syndrome) or the simple fact that empirical parameters regressed for a multiparameter model may be strongly correlated with each other and, therefore, not always suitable for extrapolations to systems with a larger number of components. An alternative method is based on the combination of excess-Gibbs- energy models with equations of state. Such combined models have essentially the same properties for multicomponent mixtures as the excess-Gibbs-energy models. Thus, they are usually fairly reliable for the prediction of multicomponent phase equilibria from binary data, except for the most strongly nonideal systems. An important advantage of the more recent models of this type (such as the MHV2 or Wong-Sandler mixing rules) is the possibility of using the existing parameters of excess-Gibbs-energy models without having to regress them specifically for the combined EOS+G E model. This feature has been extensively utilized to extend the applicability of the UNIFAC group-contribution method to high pressures and temperatures.

Substantial progress has been achieved in the generalized van der Waals equations of state. Their development has been influenced, in particular, by the progress in the statistical mechanics of chain-like molecules. It is becoming apparent that the more recent equations of this kind (such as SAFT or PHSC) are particularly suitable for polymer systems. Another area in which the generalized van der Waals equations may be somewhat superior to cubic equations is the modeling of ill-defined mixtures, for which special parameterization techniques are necessary. However, cubic equations of state are well entrenched in the modeling of phase equilibria and thermodynamic properties for hydrocarbon processing and the manufacture of commodity chemicals. It is extremely unlikely that they will be superseded in these areas by models with a better theoretical background.

A lot of attention has been devoted in this review to methods that combine simple equations of state with chemical association models. This is justified by the fact that these methods significantly improve the performance of equations of state for associated mixtures without resorting to multiparameter, empirical mixing rules. Also, they are only moderately more computer-time consuming than regular cubic equations of state. The only major shortcoming of the equations of state based on the chemical theory is the relatively cumbersome procedure for evaluating both the chemical and physical pure-component parameters from the same experimental data (i.e. vapor pressures and densities). It should be noted that the equations of state based on the chemical theory face a strong competition from associated-mixture models based on the thermodynamic perturbation theory, which are reviewed in Chapter 12 of this book. The statistical mechanics-based equations of state (such as the SAFT equation) have been proven to be very valuable for systems, in which the strength of association varies from weak hydrogen bonding to covalently-bound polymers. However, the models based on the chemical theory give very good results for several important systems, such as mixtures containing hydrogen fluoride or hydrocarbon-water reservoir systems.

At present, most researchers in this area believe that further progress will result from developments of statistical thermodynamics. At the same time, almost all calculations in industrial practice are performed using empirical, mostly cubic, equations. For practical applications, equations of state with a better theoretical background have been shown to be superior to the cubic equations only in a few niche areas such as polymer systems and electrolyte mixtures at high pressures and temperatures. It can be expected that the slow

121

process of merging theoretical concepts with empirical techniques will lead to improvements in the future.

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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 127

P E R T U R B A T I O N T H E O R Y

TomA~ Boublik

Faculty of Science Charles University of Prague 128~ 0 Prague, Czech Republic boublik@natur, cuni. cz

5.1. Introduction 5.2. Basic Concepts of Perturbation Theory 5.3. Perturbation Theories of Pure Simple Fluids

5.3.1 Van der Waals Equation of State 5.3.2 Equations of State and Radial Distribution Functions of Hard Spheres 5.3.3 Perturbation Expansion for Fluids with Infinitely Steep Repulsions 5.3.4 Second Order Perturbation Term 5.3.5 Perturbation Expansion for Fluids with Soft Repulsions 5.3.6 Quantum Effects

5.4. Mixtures of Simple Fluids 5.4.1 Mixtures of Hard Spheres 5.4.2 Perturbation Expansion for Mixtures with Infinitely Steep Repulsions 5.4.3 Perturbation Expansion for Mixtures with Soft Repulsions

5.5. Perturbation Theories of Molecular Fluids 5.5.1 Pair Potential of Molecular Fluids 5.5.2 Equations of State and Distribution Function of Hard Nonspherical Molecules 5.5.3 Perturbation Expansion for Pure Molecular Fluids

5.5.3.1 Fluids of Kihara Molecules 5.5.3.2 Fluids of Molecules with Multicenter Pair Potential 5.5.3.3 Fluids of Molecules with Electrostatic Interactions

5.5.4 Water 5.6 Mixtures of Molecular Fluids

5.6.1 Mixtures of Nonelectrolytes 5.6.2 Mixtures of Electrolytes and Aqueous Solutions

5.7. Conclusions References

128

5.1 I N T R O D U C T I O N

Perturbation theories of fluids belong to a class of modern statistical-thermodynamic theories in which the structure of a studied system is not a priori determined (e.g. by assuming some kind of a quasi-lattice) but is characterized by a set of distribution func- tions. In a system with given intermolecular forces the distribution functions express average numbers of pairs, triplets etc. as functions of molecular distance(s), orientation coordinates and state conditions.

The basic relationship of statistical thermodynamics (2, 3, 4) is the expression relating the Helmholtz energy, A, to the canonical partition function, Q, of a thermodynamic system

A = - k B T In Q

where kB is the Boltzmann constant and T the absolute temperature. If Ej is the energy of quantum state j of the studied system, then

(5.1)

Q = ~ e x p ( - E j / k B T ) (5.2)

and the probability of finding the system in quantum state i is equal to

e-Ei/kBT Pi - (5.3) Q

a s

In fluids composed of N identical molecules, the partition function Q can be written

1 ( v ) N Q = ~ . ZN (5.4)

where ZN -- ~((V) "'" / e-UN(rl'r2""rN)/kBT drldr2""drN

and the Helmholtz energy

1 ( v ) N A = - k B T In ~ ZN

(5.5)

Here V denotes the volume of the system, q the molecular partition function (containing contributions of translation, rotation, vibration, etc. of a single independent molecule), ZN the configuration integral and UN the potential energy of the studied system. In the simplest case ZN depends only on the coordinates rl,r2, ...,rN. It is obvious, that, if UN -- O, then the configuration integral equals V N and Equation (5.6) reduces to the relation valid for the ideal gas. In the general case (UN ¢ 0) the determination of the configuration integral represents a formidable task.

The potential energy, UN, can be expressed as a sum of intermolecular pair potentials, uij, of all the couples i - j , triplet potentials (given by the difference of interaction energy of a triplet and three pair potentials), etc. Higher potentials than triplet ones are of negligible magnitude; quite often also triplet interactions are neglected and only pair potentials are considered (1). Then in terms of 'effective' pair potentials

(5.6)

129

= (5.7) j>i>l

If the pair potential depends only on the center-center distance, rij, of molecules i and j, the system is called a simple fluid.

A great variety of pair potentials of simple fluids has been proposed to this date [see reference (5)]; in this chapter we will consider only the simplest ones, namely that of hard spheres defined as

× v :3

f

Figure 5.1 Hard sphere, square-well and Lennard-Jones pair potentials.

u(r) = oc for r < d

= 0 for r > d

the square-weU potential oc r K d

u ( r ) = - e for d < r K v d

0 r > v d

and the Lennard-Jones 12-6 potential

u(r) : 4e

(5.s)

(5.9)

(where d stands for the hard-sphere diameter, "y scales the width of the well and e and a are the energy and length parameters). The potentials are schematically depicted in Figure 5.1; parameters of the LJ pair potential for several substances are listed in Table 5.1.

In the following text dealing with simple fluids we will limit ourselves to these three model potentials, because (i) the use of more sophisticated potentials usually brings only slight additional problems in formulation of theoretical expressions, (ii) application

130

Table 5.1 Parameters of the Lennard-Jones 12-6 potential.

Compound e/kB (K) a (rim) Ne Ar Kr Xe CH4 CF4

34.2 0.2860 119.8 0.3471 164.0 0.3827 222.3 0.4099 148.9 0.3783 151.5 0.4744

of the LJ potential often yields sufficiently accurate results and (iii) the use of complex potentials is connected with increasing amount of computational time and often results in only modest improvement of the calculated values.

2

1

0 3 4 5 6 7

r

Figure 5.2 Radial distribution function of simple fluids.

As already mentioned, modern theories of fluids do not consider a priori structure of fluids but describe the structure by a set of distribution functions. If we assume the potential energy UN to be pair-wise additive [see Equation (5.7)] then the distribution function of the second order, p(2)(rt, r2), defined as

p(2)(rt r 2 ) = N ( N - 1) f f

(where/~ = 1/kBT) and the radial distribution function, g(rt2), given by

(5.11)

g ( ~ ) = p(~)(~l, r~)/p ~ (5.12)

play the most important role in the theory of fluids. The product of the radial distribution function (rdf), bulk density (p) and volume element gives the average number of centres of

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molecules in a given elementary volume (with distance r12 from the test-particle centre). The rdf as a function of distance is depicted in Figure 5.2

The knowledge of rdf's makes it possible to determine all the thermodynamic functions of pure fluids and mixtures. Thus, for pure monoatomic fluids (where only translation motion contributes to q) we can write the following relationship for the internal energy (3)

U 3 p f oo NksT = -2 + 2kBT .u u(r)g(r)4~rr2dr (5.13)

The compressibility factor is given by

PV = 1 2~p fo oo du(r) NksT - 3kB-----T _ dr g(r) r3dr (5.14)

For the chemical potential it holds

# = In A3p + 4~p f01 f0 °°

0u(r, ~) kBT ~BT O~ 9(r, ~) r2dr d~ (5.15)

and the isothermal compressibility is

( Op ) = l + 47rp f [g(r) _ l]r2dr (5.16) kBT -ffP T,V

Here A = (h2/2rmkBT) 1/2 is the de Broglie wave length and ~ a coupling parameter (2). Combinations of the above equations then give relations for all the other thermodynamic functions. Unfortunately, determination of distribution functions is not an easy matter even in the case of relatively simple pair potentials, as they depend (besides the explic- itly given dependence on distance) also on density and temperature (and in the case of mixtures on composition). Actually, only the distribution functions for systems of hard spheres (and to some extent of the other hard bodies) and of LJ molecules are available (see Section 5.3.2.).

It is a great advantage of the perturbation theory that it allows us to exploit our rather limited knowledge of distribution functions (and equations of state) to describe the behavior of real systems, i.e. systems with more sophisticated intermolecular interactions.

5.2 BASIC C O N C E P T S OF P E R T U R B A T I O N T H E O R Y

Application of the perturbation technique is known from different fields of physics (e. g. quantum mechanics, astronomy) where some property of a studied system is expressed on the basis of the known property of a reference employing an expansion in a series; the expansion is usually limited to the first or second order.

In statistical thermodynamics we usually work within a canonical ensemble where the basic relation is that between the Helmholtz energy, A, and the partition function, Q, [cf. Equation (5.1)]. Therefore, the Helmholtz energy is the property for which perturbation expansion is considered first. In his pioneering work Zwanzig (6) assumed the potential

132

energy, UN, to be given by a sum of the potential energy of a reference, U0, and the (much smaller) perturbation energy, Up. Since the reference and the studied systems do not differ in number of molecules and molecular partition function, q, we can write the following expression for the difference in the Helmholtz energies, A and A °, of the studied and reference systems

A - A o

kBT 1

- - ln_~Nf. . . fe-Z(Vo+Up)drldr2. . .dr N -

_ l n f . . . f e - ~ U p e-~U° ZO drldr2...drN

Realizing the fact that the ratio e-BUo

z o =7'0

equals the probability density of a configuration of the reference system, and expanding exp(-/~Up) in a series we have

knT = - I n 1 - ~(Up)o + ~-. (U~)o + ... (5.18)

where the symbol 00 denotes properties averaged over all the configurations of the refer- ence system. Expanding the logarithmic function in series one gets

A - A ~d A ° - A ~d ( U p ) o (U~)o - (Up)o 2 = ~ + ... (5.19)

kBT kBT kBT 2!(kBT) 2

Here A id stands for the ideal-gas Helmholtz energy. We thus introduce residual functions of the studied and reference systems which are quantities often used in the thermodynamics of liquids, simply linked with equations of state.

Another way leading to Equation (5.19) consists in the use of a coupling parameter, ~, scaling the perturbation energy,

UN = U0 + (5.20)

Writing OA +...

( A - Aid) - (A ° - Aid) + -~- 0 0

and letting ~ - 1, we find that the single derivatives are equal to the first, second, etc. perturbation terms:

1 At = (Up)o A2 = 2!(kBT)((U~)o - (Up)~) (5.22)

etc. It is, however, also possible to write (7)

A - A id A 0 - A id f l (Up)~ NkBT -- NkB---------~ + Jo NkBTd~ (5.23)

133

express (Up)¢ in terms of the radial distribution function and consider an expansion of g(r, ~) in terms of ~. A slightly more involved perturbation method, the blip-function expansion will be discussed later.

In the case of the pair-wise additive perturbation energy we can express Up in Equation (5.22) in terms of the perturbation pair potentials, Up; after integration, N ( N - 1)/2 equal contributions result. Therefore

A, N ( N - 1) 1 : 2 f fu,(r12)Podr dr#r3...drN = -~ f fup(rx2)p(o2)(rl,r2)drldr2 (5.24)

Transformation of variables [rx, r2] -+ [r~, r12] and integration over rl gives

1 f0~ A1 = -2 NP up(r12)go(r12)47rr22dr12 (5.25)

(where the index 0 denotes properties of the reference). It is obvious, that the knowledge of the course of the reference rdf allows us to evaluate the first-order term in the expan- sion of the Helmholtz energy for any pair potential of simple fluids. Determination of the second-order term is more difficult; we postpone the discussion of its evaluation to Section 5.3.4

5.3 P E R T U R B A T I O N THEORIES OF PURE SIMPLE FLUIDS

5.3.1 Van der Waals equation of State

Van der Waals' equation of state is a prototype of a class of cubic equations of state widely used in thermodynamics and chemical engineering. Here we will show its deriva- tion within the perturbation approach under the condition of low density. Discussion of approximations used to develop the van der Waals equation may serve to put limits on its applications and also to point out goals of the theory.

In the derivation we will consider the first-order perturbation expansion for the Helmholtz energy

A - A id Ao - - A id

NkBT NkBT p O0

+ 2ksT fo Up(r)g°(r)4~rr2dr (5.26)

Let the intermolecular forces be characterized by the square-well potential (which very roughly describes repulsive and attractive interactions). The obvious choice for the refer- ence is the hard-sphere (hs) system. Available equations of state (EOS) and methods to determine the rdf will be discussed in the next section. At low densities we can find the expression for ZN of hard spheres when we realize that the integrand in Equation (5.5) possesses two values - zero (in the case where at least two spheres overlap) and one (in all other cases); each molecular center can thus move in a 'free' volume, i.e. the volume of the system lessened by a small volume around each sphere, excluded for centers of the other particles. For each pair of molecules (with hs diameter, d) this excluded volume at

134

low density amounts to 4rda/3 so that the total free volume, V}, is

Vf = V - 2~rNd3/3 = V - Nb

The residual Helmholtz energy of the reference is thus

kB------T-~ = - N In -- = - N ln(1 - pb)

(where p- - N/V) . In the first-order perturbation term, A~, the perturbation pair potential is

(5.27)

u p ( r ) = - e for d_<r_<Td

0 r > 7d (5.28)

and the rdf possesses at low densities two values; zero for r < d and one elsewhere. After substitution we have

Defining parameter a as

A1 2 e kBT = - 3 7tWo (k--~) d3(73 - 1) (5.29)

we have

Finally

2 3 a = -~Tred (9 ' 3 - 1) (5.30)

A1 pa = (5.31)

NkBT kBT

A - A id pa NkBT = - ln(1 - pb) kBT (5.32)

The equation of state follows from the thermodynamic relation

O ( A - Aid) P Op = Z - 1 (5.33)

where Z --- P V / N k B T is the compressibility factor. By differentiating Equation (5.32) we obtain the van der Waals equation of state,

P V 1 pa = (5.34)

NkBT 1 - pb kBT

From the derivation of Equation (5.34) one can see that the form of the van der Waals equation does not depend on details of the form of the assumed pair potential. It should be emphasized, that we obtained the EOS under the assumption of low density and with neglection of the second- and higher-order perturbation terms. If these terms are taken into account the simple dependence of the attractive term on temperature would change. (Such a change is expected taking into consideration the temperature dependence of the

135

second virial coefficient). Moreover, the dependence of a and b on density would change. (For the density dependence of the attractive term, however, the dependence of the rdf on density is more important.)

As far as the repulsive term ( 1 - pb) -1 is concerned, at higher densities the free volume decreases more slowly due to overlapping of the excluded volumes around neighbouring molecules. These arguments give some evidence in favor of modifications of the van der Waals equation of state (and other cubic EOS) in which a, b are considered as density and temperature dependent quantities. (See Figure 5.3)

J i

a a

i a

a S

e S

vdW ,, J

i

- 1 , I i I

0.0 0.1 0.2

Figure 5.3 Compressibility factors from the van der Waals EOS and perturbation theory, • - simulation data.

From the brief outline of the formulation of the perturbaton expansion for the Helmholtz energy and consequent determination of the compressibility factor (or internal energy and chemical potential) it is evident that (i) a system of hard spheres represents a natural choice of the reference in the case of potentials with a harsh repulsive branch, (ii) fair knowledge of the equation of state - or an expression for another thermodynamic function - of the hard-sphere system and its rdf is necessary in order to get good pre- diction of the behavior of the fluids studied. In the case of the pair potential with soft repulsion a system of representative hard spheres for the given reference soft spheres is sought and thermodynamic functions are then determined from the hard sphere EOS.

As a consequence of the mentioned choices of the reference system, the zeroth- and first-order terms possess large (possitive and negative) values which yield relatively small residual thermodynamic .functions. It is thus obvious, that accurate knowledge of the hard-sphere EOS and the corresponding rdf is essential for the perturbation approach to yield an accurate description of real systems. Therefore, we will briefly discuss proper- ties of the hs system in the next section and those of nonspherical ones in the further part.

136

5.3.2 Equations of State and Radial Distribution Functions of Hard Spheres

The system of hard spheres (hs) is the simplest possible model of fluids in which only the repulsive forces a r e - in the simplest way - taken into account. Because of the form of the hs pair potential, Equations (5.13) - (5.15) simplify considerably: internal energy is given only by the contribution of translational motion, U - ~NkBT. Due to the fact that the derivative of the pair potential multiplied by exp(-~u), [cf. Equation (5.43)], gives the Dirac function 5(r - d) the compressibility factor is

P v NkBT = 1 + ~rpd3g(d) = 1 + 4yg(d) (5.35)

where y stands for the packing fraction, y = rpd3/6, and g(d) is the value of the rdf at r - d. In the expression for the chemical potential a new coupling parameter, ¢, can be introduced which scales the size of the test particle (e.g., molecule 1); the lower and upper bounds of the first integral of Equation (5.15) are chosen in such a way that for ~ - a full decoupling occurs and for ~ - b the full size of the test particle is obtained, (i.e. that of the other hard spheres). Then

/. # = l n A a p + 4 r p g(¢,¢)¢2d¢ = l n A 3 p + 4 r p d 3 bG(x)x2dx (5.36) kBT

where G(~) stands for g(~, ¢). In the case of ~ = b, G(b) = g(d). It is obvious that the value of the rdf at r /d = 1 yields the corresponding EOS im-

mediately; on the other hand knowledge of the Equation of state allows formulation or improvement of expressions for the rdf.

There are several sources of our knowledge of the radial distribution function and Equation of state of hard spheres:

• Pseudoexperimental (simulation) data. These have been determined either by the Monte Carlo or Molecular Dynamics technique. Both the methods represent very powerful tools of theory and the number of simulation studies of different physical phenomena is ever increasing. Simulation techniques are described in several mono- graphs and textbooks (8, 9). Hard-sphere simulation data published before 1986 were collected in references 10,11.

• Integro-differential methods (see reference 12). When p(2)) in Equation (5.11) is differentiated with respect to the coupling parameter (Kirkwood) or the gradient of p(2) is taken (BBGY) differential Equations are obtained which - after rearrangement - can be integrated again. As a result, the rdf (of the second order) is expressed in terms of an integral containing g(3)(rl, r2, r3). It is possible to repeat the operation for g(3) and to obtain an expression for g(3) in terms of g(4) etc.; a hierarchy of integro-differential equations results. Usually this hierarchy is broken after the first step assuming (13)

g(3) (rl, r2, r3) - g(r12)g(r13)g(r23)

Solutions of the integro-differential equations with this 'superposition approxima- tion' yield in both cases (Kirkwood, BBGY) values of the rdf, which differ con- siderably from simulation data at high densities. If, however, the superposition

137

approximation is used for g(4) very accurate rdf values are obtained at the cost of extensive computing (14).

• Integral equations. These originate from the Ornstein-Zernike (15) equation (OZ) for the total correlation function, h(r) - g(r) - 1

f

h(r,2) = c(r,2) + p J c(r13)h(r23)dr3 (5.37)

where the direct correlation function, c(r), is defined by this relationship. The essence of introducing c(r) is in the fact that the range where c(r) is non-zero is similar to that of the pair potential; the direct correlation function can thus be easily a subject of simple approximations. These approximations and methods of solution are discussed in detail in Chapter 6. For perturbation methods the results found for the Percus-Yevick (BY) approximation (10,16) are of special importance. The solution of the OZ relation in the PY approximation (16,17) yields two forms of the equation of state, the v-form [resulting from Equation (5.14)] and c-form [from Equation (5.16)]:

P V 1 + y + y 2 - c - form (5.38)

N k s T (1 - y)a

and P V 1 + y + y2 _ 3ya

= v - form (5.39) N k s T (1 - y)3

Moreover, the Laplace (and Fourier) transform, g(s), of the product xg(x), becomes

sL(s) (5.40) G(s) = 12y[L(s) + S(s)e 8]

Here L(s) and S(s) are simple polynomials in s. Performing an inverse transform one obtains values of g(x) at given reduced distances x = r/d. As the c- and v- forms lie above and below simulation data, a better EOS will result by taking some kind of average of Equations (5.38) and (5.39). If (2/3) of the c- and (1/3) of the v- form are considered, the widely used Carnahan-Starling (CS) EOS (18) is obtained

P V _- l + y + y2 - y3 (5.41) N k B T (1 - y)3

While CS or Kolafa (see later) EOS's agree very well with simulation data at packing fractions y _ 0.5 (see Figure 5.4) but disagree at higher densities, the course of the PY rdf disagrees slightly with simulation data in the interval x E (1, 1.6) even for y < 0.5. Several approximations were proposed to get equal c- and v-forms of the EOS and improve the predicted behavior of the rdf. Recently, a modification of the Laplace transform [cf. Equation (5.40)] was proposed (19) with two coefficients adjusted to exact results at low densities. The modification gives the rdf vs distance dependence accurately and an EOS valid even at y > 0.5. In practical applications,

138

lO

8

I--- 6 n," > Q. II

rq 4

| A | | I |

y Figure 5.4 Compressibility factor of hard spheres; - CS EOS, • - simulation data.

however, a semiempirical correction is employed (20) where g(x) is calculated at slightly lower density yc - y(1 - y/16) and Ag for x E (1,1.6) is calculated from

= A 1) (5.42) X

• Scaled particle theory (SPT) (21). This theory employs the coupling parameter which continually changes the diameter of a test hs particle in Equation (5.36).

The distribution function G(() is expressed as a second-order polynomial, three coefficients of which are determined from the conditions of continuity of G and its derivative at ~ = 1/2 and the condition (P/kBT) = pG(oc). The SPT EOS is identical with the PY c-form. In spite of the fact that additional conditions on G(~) are available, which makes it possible to consider a higher order expression, attempts to improve the SPT EOS have not been successful. The SPT gives - besides EOS - only the contact value of the rdf.

• Background correlation method (22). From the solution of integro-differential equa- tions (and other sources) it follows that at low densities g(x) = exp[-~u(x)]. It is thus useful to express the rdf as a product of the exponential function and the background correlation function, Y(x),

(5.4a)

(x = r/d). In the case ofhs , g(x) = r ( x ) f o r x >_ 1. In contrast to therdf , the background correlation function (of hs and other pair potentials with infinitely steep repulsions) is continuous at x = 1. Several exact relations were found (23, 24) allowing expression of Y(x) in terms of the residual chemical potential of pure hard spheres and that of a hard dumbell, with the site-site distance x (see Section 5.5.2) infinitely diluted in hard spheres

139

In Y(z) = 2/~# hs - / ~ # ~ (5.44)

The residual chemical potential of hs can be determined from any accurate hs EOS and/3#~ from the EOS of a mixture of hard non-spherical bodies (see Section 5.5.2); finally

x 3 y2 3x 2 Y ( 7 - 6x + -~-) + (15 - lSx + 3x 3) In g(x) -- - l n ( 1 - y) + (1 - y) 2(1 - y)2 ~-

+ (1 - y)3 ~ + -~-) (5.45)

In the limit of low densities the exact expression on the interval x E (1,2) can be obtained

X 3 X 3 g(x) ~ exp[y(8 - 6x -+- --~-)] -- 1 + y(8 - 6x -+- --~) (5.46)

Due to reasons connected with the geometry of the hard dumbbells, Equation (5.45) gives values of the rdf only for distances smaller than that of the first minimum. To extend the range of distances it was proposed recently (25) to use a formula for damped oscillations, i.e.

g ( x ) -- A e -B(x-xmin) cos ?r(x - Xmin)/C ( 5 . 4 7 )

where constants A and c are determined from the conditions of continuity of g(x) and its derivative at x = Xmin; B is a function of y. In Figure 5.5 a comparison is given of the rdf calculated from Equations (5.45) and (5.47) with simulation data; a fair agreement is seen.

• Virial coefficients. For hard spheres the first few virial coefficients, Bi, axe known either from theory ( B 2 - B4) or numerical calculations ( B 5 - Bin) (10). These coefficients allow one (i) to judge the accuracy of the approximate EOS, (ii) to formulate new EOS's. There are two routes to get the EOS: (i) to consider the so-called Pad~ approximant p(m, n)

P V 1 + a~y + ... (5.48) N k B T = p ( m , n) = 1 + bly + ...

with coefficients a,~, bn determined from the individual virial coefficients after expan- sion of the last Equation, (ii) to resum all the virial coefficients whose approximate expression in terms of n (where n denotes the order of the virial coefficient) is avail- able. Thus, Carnahan and Starling found (18) that the nine virial coefficients in the expansion of P V / N k B T in terms of y, see reference 10, can be approximately written (assuming rounded values)

Bi+l = i(i + 3) (5.49)

140

6

5 * p = 0.925

X "~3

2

1

15 i i f1.0 1. 2.0 2 .5 3.0

× Figure 5.5 Radial distribution function of hard spheres; - calculated, • - simulation data.

Resumming then the virial expansion

P V 1 + y + y2 _ y3 N k B T = 1 + ~ i(i + 3)y = (1 - y)3 (5.50)

i > 0

i.e. the CS EOS is obtained. Kolafa (10) improved the approximation for Bi by taking

5 B i = ~ ( i 2 + 3 i - 6 ) fo r i>_3 (5.51)

and obtained

P V 1 + y + y2 _ 2y3/3 _ 2y4/3

N k B T (1 - y)3 (5.52)

This equation is in even better agreement with simulation data than the CS EOS.

The deficiency of these equations is that a singularity is found at y - 1, whereas the close-packed value for the cubic lattice is ~ 0.74. Several hs EOS have been proposed which predict correctly the P - - V - - T behavior up to y ~ 0.74; because of their complicated form they have not been used in the perturbation approach.

5.3.3 P e r t u r b a t i o n E x p a n s i o n for Flu ids wi th Inf in i te ly S teep Repu l s ions

Systems of this type can be represented without loss of generality by the system of square-well molecules. The choice of the reference is unambiguous - it is the hard-sphere system with the hs diameter d. For the reference residual Helmholtz energy one can use the expression derived from the CS equation

141

A0 - A id y ( 4 - 3y)

NkBT (1 - y)2 (5.53)

(A slightly more complicated expression follows from the Kolafa EOS.) By differentiating Equation (5.53) with respect to p and multiplying by p (or equally well differentiating with respect to y and multiplying by y) one finds

Z 0 - Z id - - Z 0 - 1 = 4 y - 2y 2 (1 - y)a = 4ygo(d) (5.54)

where Z stands for the compressibility factor; the equation can be obtained directly from the CS EOS. Summation of Equations (5.53) and (5.54) yields the reduced residual Gibbs energy or reduced residual chemical potential of the reference

Go - G id A#0 8y - 9y 2 + 3y 3 = = (5.55) NkBT kBT (1 - y)3

Thus, all three functions of the reference system are available in simple analytic forms (the same is true also for the residual reduced entropy, the absolute value of which is equal to the residual Helmholtz energy due to the fact that the diameter and, therefore, the packing fraction are independent of temperature).

In the first-order term, A1, it is usually advisable to express the rdf in terms of the total correlation function, h, i.e. g - 1 + h, and write

A1 2rp o¢ 2rp fo¢ NkBT -- kBT fd Up(r)g°(r)r2dr = [dJ~ Up(r)r2dr + fd up(r)h°(r)r2dr] (5.56) -

The former integral can be determined analytically while the latter (in the general case) by numerical integration.

In the case of the square-well system, the perturbation potential is

After substitution we obtain

up(r) - -e f o r d < r < T d

0 elsewhere (5.57)

A1 NkBT

2rpd3 [ (73 - 1 )+ 3fl'Yho(x)x2dxJ 3 T*

(5.58)

(Here T* = kaTie is the reduced temperature and x = r/d). To get some feeling about the integral in the last equation we can consider the low-density limit and substitute from Equation (5.46); then

A1 NkBT T* (./3 _ 1) + y(873 - ~'), + ~'), - )

The first-order contribution to the compressibility factor, Z1, is

(5.59)

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[ (5.60) OA1 = - 4 ~ , ( 7 3 - 1 ) + 3 Y Oy Z1 = p Op

In our special case

1 0 _ ~ ] Z1 = - 4 ~ , (,y3 _ 1)+ 2y(8"y 3 - 974 + ~7 ) (5.61)

It should be mentioned that the EOS obtained as a sum of the above expression for Z0 and the last equation is valid only at low densities. It represents an improvement over the statistical analogue of the van der Waals EOS. Comparison of the compressibility factor calculated from the van der Waals EOS with that obtained from Equations (5.53) and (5.60) vs y is given in Figure 5.3. As in the case of the reference, the first-order pertur- bation contribution to the residual chemical potential is given by the sum of expressions for A1 and Z1.

It is obvious that the same approach can in principle be used to determine A1 and Z1 in the case of systems with other pair potentials of the same type. Thus, if the Sutherland potential, defined by Equation (5.62) is considered

u(r) = c¢ f o r r < d

- e ( d / r ) 6 r > d (5.62)

the first part of Equation (5.56) is - 4 y / T * . Generally, the remaining integral can and should be determined numerically.

There is, however, another route for determining A1, in which it is assumed that the function XUp(X) is the Laplace transform of some (for a moment unknown) function H(t). After substitution in the first-order integral, one has

/0 /1 /0 /0 XUp(X)xgo(x)dx = e- t*bl( t )xgo(x)dtdx = ld(t)~j(t)dt (5.63)

The Laplace transform, g (t) is available for the PY approximation of the OZ equation and the inverse transform of functions like x -6 is readily available, too. Numerical evaluation of the integral in the last equation should be performed stepwise because of the character of the function G(t) and corrections to PY rdf should be considered. In passing we note that the method can be used also for the direct determination of Z1.

5.3.4 Second-Order Perturbat ion Term

The second-order perturbation term in the expression for the Helmholtz energy, A2, is given by fluctuations of the perturbation energy due to fluctuations in configurations of the reference. Employing the grand canonical partition function Barker and Henderson (26) derived an exact relation

143

1 A~ = ~Npf vo(X,2)u~(X,2)d2 + Np ~ f f g~)(x,2,a)up(X,2)~(x,a)d2d3

f f[,0(')(1, 2, z, 4 ) - go(X, 2)g0(3 , 4)]up(X, 2)Up(3, 4)d2d3d4

(where 1,2,... stand for rx,r2, ...). One can see that distribution functions g0 (4) and g0 (a) must be known in order to determine A2 exactly. At present, however, these functions are not available. We may employ the superposition approximation for g0 (4) and g~3) in terms of go; complicated integrals result (which form the basis of the ORPA and EXP expressions - see Section 5.3.5). Alternatively, we may turn to simple approximations containing only the rdf of the second order, go- Such approximations were proposed by Barker and Henderson (26) who wrote Up as a sum of square-well contributions, Up, and assumed that only fluctuations in the number of couples with the same distances are correlated and contribute to (U~) - (Up)~. From these considerations they obtained

A2 f0 °~ ~(~)~0(~):d~ (m.c.) (5.65) NknT - - -/~rP (Op/OP)° up

The derivative (Op/OP)o can be determined from any accurate EOS for hard spheres. Equation (5.65) is called the 'macroscopic compressibility' approximation (m.c.) as the relation for fluctuations in the macroscopic hs system is employed. The authors proposed also a slightly modified 'local compressibility' approximation (1.c.)

/0 (,c) Up A2 -fllrp \ OP )o

(/3 -- 1/knT). For the CS EOS the expression for kBT(Op/OP)o is

( Op ) = ( l - y ) 4 k n T \ z ~ 0 l + 4 y + 4 y 2 - 4 y 3+y4 (5.67)

It is apparent that Equation (5.67) tends to 1 at very low density, whereas, for high densities, it is of the order 10 -2 . This behavior is in good qualitative agreement with the fact that the third-order polynomial (in the reciprocal temperature) is necessary in order to express the temperature dependence of the second virial coefficient with suffi- cient accuracy, whereas at high densities the mean-field term is large but all higher-order terms small. The predicted A2 vs y dependence, however, differs from the pseudoexperi- mental data. This difference affects considerably the accuracy of the second-order terms in expansions of the compressibility factor and residual chemical potential. In order to improve on the B-H approximation, Zhang (27) took into consideration also correlation of the number of molecules in different shells with the common central molecule. As a result, the second-order term reads as

A2 NkBT = -131rp (Op/OP)o (1 + 2Ky 2) fo °~ U2p(r)go(r)r2dr (5.68)

144

Better agreement with the simulation data of the second-order term for the SW system was found.

Concluding this part we note that within the framework of BH approximation an ex- pression for the third-order perturbation term was also derived (28); it is, however, of limited practical importance.

5.3.5 Perturbation Expansion for Fluids with Soft Repulsions

Pair potentials, such as the square-well or Sutherland models, characterize repulsive interactions of molecules in an oversimplified way. Actual repulsive interactions are soft, expressed by the power law or exponential function of distance. The Lennard-Jones pair potential with its r -n terms represents the interactions of simple fluids well and is widely used. In this part, we will consider the LJ 12-6 model to show some details of the perturbation approach when dealing with systems of realistic pair potentials.

The first question is the choice of the reference system. It is known that repulsive interactions determine the 'volume' (and 'shape') of individual molecules and consequently the structure of fluids; it is thus of special importance to define properly the range of action of repulsive forces. At present, two definitions of this range - due to Barker and Henderson (BH)(29) and due to Weeks, Chandler, and Andersen (WCA) (30) are widely used. In the former case the reference pair potential is

uo(r) = 4e - for r _ a

0 r > a (5.69)

(where a is the distance where ULj -- 0). In the latter case the reference potential is given by the prescription

uo(r) = 4e - + e for r _~ rmin

0 r > rmin (5.70)

Here rmin stands for the distance of minimum on the potential function; for the LJ po- tential rmin -- 21/6a.

Thermodynamic properties of the reference in both cases are determined from the EOS of the representative hard spheres. For the BH definition of the reference soft spheres, the diameter d of the representative hs is calculated from

dBH f01 = c = (1 - e-~U°(~))dx (5.71) (7

where x - r / a . In the WCA approach the diameter of the representative hard spheres follows from the

blip-function expansion. The blip function, Ae, is a difference of the Boltzmann factors for the reference and representative (hs) potentials, u0 and u hs

145

Ae = e -~° - e -~hs (5.72)

We can write (31) e - ~ = e -~h" + ~Ae (5.73)

For ~ = 0 we get the Boltzmann factor of hs, for ~ - 1 that of the reference. Let us consider an expansion of the Helmholtz energy, A0, in ~ about ~ -- 0

Ao=AhS+(O-~~)¢=o + ... ( 5 . 7 4 )

When performing differentiation we obtain N(N- 1)/2 terms which yield after integration equal contributions. Thus

OflA N ( N - 1 ) f f[ff_~ e-~U(~) e~U(~)e-~UN drldr2dra..drN (5.75)

The derivative in brackets yields Ae; considering the definition of p(2) and realizing that exp (flu)g is the background correlation function, Y, we have

( OflA ~ _ Np yhs 2 (5.76/

Putting this term equal to zero, the Helmholtz energy of the reference is given (up to the second-order term) by A hs of hard spheres whose diameter, dwcA, is given by the prescription

formln[e--~UO__ e--t~uhs] yhs d r - 0 (5.77)

The reference rdf is then

go(r) -- e-~°(~)YhS(r) (5.7s)

In the case of the BH variant the representative hs diameter depends only on temper- ature, in the latter case both on temperature and density. In the hybrid approach the WCA definition of the range of repulsive forces is combined with an analogue of Equation (5.71); dhs depends only on temperature.

If the value of the hs diameter (or c) of the representative system is known, one can find its packing fraction yhs _ ~pc3a3 and determine the reference residual Helmholtz energy from Equation (5.53). For the WCA approach the formulas given above can be used with a slight modification. Due to the dependence of y on temperature and density, the consistent values of the reference compressibility factor and residual chemical poten- tial for WCA are determined by numerical differentiation.

The perturbation pair potential is given by the difference of the original and reference potentials. In the case of the BH variant, up is zero for r < a and Up - U L j for r > a. In accordance with this definition and the fact that dBH < a we see that

146

A1 = 2r/~p I NkBT Ja Up(r)go(r)r2dr- 27r/~p [~°°Up(r)r2dr ~ faCCUp(r)ho(r)r2dr] (5.79)

The first integral in brackets equals 8 a -6ea . When determining the remaining integral, we have to realize that ho depends on the variable r/d; the integral can be expressed in two different ways

oo( ) ( I ) 4ecaa 3 = 4ea 3 _x -10 _ x-4ho_x cdx = I /c

After substitution one has

[(xc) - 1 2 - (xc)-6]ho(x)x2dx (5.80)

NkBT = T* --9 + 4 (x -1° - x-4)ho(x/c)dx (5.81)

The last integral is determined numerically either by calculation of single values of h0 - h as on the short interval, e.g. x E (1,4), or employing the Laplace transform g(t) for the PY approximation.

The first-order perturbation term in the expansion for the compressibility factor is obtained from the derivative of A1 with respect to p; the first-order contribution to the residual chemical potential is given by a sum of expressions for A1/NkBT and Z1.

In the case of the WCA choice the perturbation potential is

Up(r) -- --e for r ~ rmin

ULj(r) r > rmin (5.82)

Then

31 __ /~p ~O°° Up(r)e_~uo(r)ghs(r)dr : ~~_ [_~ ~ormi'e_~6uoghSdr _[_ ~r~iC~ uLjghSdr] NkBT 2

(5.83) From Equation (5.77) we see that the first integral of the last equation equals

-fie fd r=i~ yhSdr

One can thus write

A1 NkBT

21rpaa [ fc°° Up(X)x2dx "b /c°° Up(x)hhS(x/c)x2dx kBT

Performing integration in the first term one obtains

(5.84)

A1 2rpaa I 8_~92 c 3 1 ~ °° ] NkB-----T = T* - - - + -3 + -e Up(x)hhS(x/c) x2dx (5.85)

The expression for the compressibility factor follows from the derivative of A1 with respect to p, usually performed numerically because of the dependence of the hs diameter- or c

147

i

[

0

Figure 5.6 Compressibility factors of the LJ system at T* =1.15; - perturbation theory (WCA), • - simulation data.

- on density. In Figure 5.6 a comparison of the calculated compressibility factor of the LJ system with MC data is given.

The determination of the second-order perturbation term in the case of the BH approximation does not bring any greater problem than that for fluids with infinitely steep repulsions. In the WCA approach the second-order term is seldom considered because of its relatively small magnitude in comparison with the first-order term at high densities. Instead, more general approaches - ORPA (optimized random-phase approximation) and EXP - were formulated (32,33). In these versions the terms of the perturbation expansion were expressed as sets of diagrams and the Fourier transforms of single classes (e.g. ring diagrams) resummed. Thus, to the pair potential of the studied system another potential ut with an infinitely steep repulsive branch is attached, such that

a t ( r )=oc f o r r _ d and ut ( r )=up f o r r > d

Because of the behavior of u as we can also write

u~(r) = u~S(r) + u~(r) (5.86)

Considering the Fourier transform of the potential up(x), fi(k), and denoting by S the structure factor [S = 1 + ph(k)], the Fourier transform of the renormalized potential, C(k), is

O(k) = 1 +

Its inverse transform, C, then appears in equations for the function g,(r) in the ORPA and EXP approximations

gt(r) = 9b~(r) - C(r) ORPA (5.88)

148

In ORPA the residual Helmholtz energy is

EXP (5.89)

A - A id A h s - A id

NkBT NkBT NkBTA1 + 4-~pl f {ln[l+pShS(k)fl~(k)]_pShS(k)/~(k)}k2d k (5.90)

where the first-order term is given by Equation (5.85). From Equation (5.86) for ut(r) it is obvious that the form of the potential up(r) for r E (0, d) is irrelevant for the behavior of ut; however, it affects the magnitude of the Fourier transform. It is therefore possible to optimize the form of Up on the interval r < d in such a way that the functional derivative

6(A/NkBT) 5%

= 0

(In EXP a further class of diagrams is considered and an additional term appears in the expression for A - Aid).

Thermodynamic functions calculated from these approximations agree very well with pseudoexperimental data, but the determination of state points is often quite time con- suming. Both the mentioned methods are more suited for description of solutions of electrolytes.

5.3.6 Q u a n t u m Effects

So far we have assumed that the kinetic part of the total partition function can be separated from the configurational part and expressed via the partition function of the ideal gas. This is generally correct; however, for some compounds belonging to the class of simple fluids (such as light rare gases) this factorization is not fully justified even at temperatures close to their critical values and a correction term should be added to take into account these quantum effects. The theory (34) leads to a series expansion of the Helmholtz energy in powers of A 2, where A = (2~rmkBT/h2) -1. If the first-order expansion is considered, then the correction term in the expression for the Helmholtz energy is

A 2 N f V2 481rkBT V- g(r) u(r)dr

In the case of potentials with soft repulsions the correction for the quantum effect can be accounted for by using an effective hard-sphere diameter slightly larger than the real one.

5.4 M I X T U R E S OF S I M P L E F L U I D S

5.4.1 M i x t u r e s of Hard Spheres

Sources of our knowledge of the behavior of hard sphere mixtures - i.e. systems of hard spheres of two (or several) different diameters, are similar to those for pure hs fluids. Within the scaled particle theory (21), and from the OZ equation in the PY approximation (35), the following equation of state (PY-c) was derived

149

P V NkBT

whereas the PY-v form is

1 3~1{2 3{ "a = 1 - { } {o(1-{)2 + { o ( 1 _ { ) 3 P Y - c (5.91)

Here

P V 1 3~2(~ + ~1) N k B T - 1 - ~ + ~0(1-~)2 P Y - v (5.92)

~k = 6 p ~ x i d ~ k = 0 , 1 , 2 , 3 ~=~3 i

The extended CS equation was formulated by the present author (36) and Mansoori et al. (37). It is

P V 1 3~1~2 ~3(3 - ~) NkBT = 1---~ ÷ ~0(1 - ~)2 + ~0(1 - ~)3 BMLCS (5.93)

The extended Kolafa EOS (38) reads

2~2~ P V 1 3 ~ 2 ~ ( 3 - ~ - 5 j (5.94)

N k B T = 1 - ~ t ~0(1-~)2 + ~0(1-~)3

Both the BMSCL EOS and Equation (5.94) describe well the behavior of mixtures of hs that do not differ extremely in the ratio of their diameters. However, for the extreme case of the so-called colloidal limit (infinitely diluted mixture of a big molecule in small molecules) some corrections (39,40) have to be made.

Within the PY approximation expressions for the Laplace transform of rdf's of the binary mixture were also determined (35) which were more complicated than those of pure fluids; they will not be reproduced here. Also Equation (5.45) was extended to mixtures. For the rdf of a pair i - j in the mixture of the given composition it holds that

Y ~ * * *

In gij(x) - - ln(1 - y) + (1 - y)(3Rmsm ÷ 3S~r~ + V~pm)

y2 _ _ , , , , , , y (2- y / 3 ) , , , ( 5 . 9 5 ) + 2(1 - y)2 (3Qmsm ÷ 6S~sm ÷ 6V~rmsm) ÷ (1 -- y)3 VmSm

Here R*, Q~, S~, V,~ denote geometric properties of a hard dumbbell (see Section 5.5.2) and r~, s~, v~ geometric properties of a given mixture (calculated from the properties of pure components). Values of rdf's for distances larger than the distance of the first mini- mum are determined from Equation (5.47). The method allows the determination of rdf's

$ * * * in mixtures with an arbitrary number of components, rm, sin, vm, p,~ remain unchanged for all the different pairs; R*, Q~, S~, V,~ are characteristics of the hard dumbbell composed of hard spheres i and j with the site-site distance equal to 1 = xdij.

In the end we would like to stress the fact that all the discussed relations are valid for the so-called additive hard spheres where the cross diameter is given by the arithmetic mean, dij = (dii + djj)/2. Mixtures of nonadditive hs's are considered when dealing with metallic systems.

150

5.4.2 P e r t u r b a t i o n E x p a n s i o n for M i x t u r e s w i t h Infini tely Steep Repuls ions

In this part we will briefly discuss methods used to determine thermodynamic functions of mixtures, the molecules of which interact via a square-well pair potential (as a prototype of potentials with an infinitely steep repulsive branch).

Usually it is assumed that the square-well cross parameter, d i j - (dii ÷ djj)/2, so that the EOS and rdf formulas for additive hard spheres can be employed. The reference residual Helmholtz energy is then

- 1 l n ( 1 - ~ ) + ~ + ) 3~t~2 ~3 (5.96) Ao~ A~ d ( ~ ]

NkBT = \ ~ o ~ ~ - ~ o ( t - ~ ) ~ o ~ ( 1 - ~)~ The first-order term is

Als NkBT

27rp fd °° , kBT ~i ~ • is UPij(r)g°ijr2dr

kBT ~i" ~" xixjeij i~ UPij(r)r2dr + i, UPij(r)hoijr2dr

= -27rp ~ y~ x i x j ~ (73 - 1) + hoijx2dx i j J1

(5.97)

Here xi stands for the mole fraction, u* = u/e and T/~ = kBT/eij; values of e12/kB and 712 are usually determined from some combining rules.

For the second-order perturbation term the simplified relation

A2s _ _Tr ( Op ) ~ ~-~xixj fa °° Upij(2 r)goij(r)r2dr (5.98) NkBT - ~ o '~

is the only one used although the exact extension of the BH approach was derived, as well. The derivative (Op/OP)o is available from the EOS for mixtures.

The compressibility factor follows from the derivative of A s - A~ d with respect to p; the leading part in the first-order term can be written analytically. The same situation occurs when we determine internal energy from the derivative of A, - A~ with respect to temperature.

5.4.3 P e r t u r b a t i o n E x p a n s i o n for M i x t u r e s w i t h Soft Repuls ions

In analogy to pure fluids, the mixture of LJ molecules is a typical example of multi- component systems with soft repulsions. Soft spheres with interactions defined either by the BH or WCA decompositions form the reference system. Its thermodynamic functions can be calculated from the equation of state for mixtures of additive hard spheres. In a binary system we have three pairs of parameters eij/kB and aij. For each pair of parame- ters one can determine a value of D 0 (in the case of like pai rs- representative diameters), either from the BH or WCA prescriptions. When employing an EOS for additive hard spheres we take du - Du and dij = (du + djj)/2.

The reference term in the perturbation expansion for A, - A ~ possesses the same form

151

as in the case of the square-well potential. In the first-order contribution, an additional term appears as a correction for the fact that the thermodynamic functions of the reference are determined from the EOS of additive hs's. Then

Als 27rp fa g N k B T = kBT ~ ~ xixjaij ghS(x/cij)Upij(X)x2dx + Aa c°r (5.99) ij

(here c~ij = 1 in the BH and c~ij = cij in the WCA variants). The correction term Aa c°r in the BH approach is

AaCOr 2 hs = -47rpxix jd i jg (dij)[dij - Dij] i ¢ j (5.100)

In the WCA approximation a similar term is usually neglected. Further thermodynamic functions, such as the compressibility factor, internal energy,

etc. follow from the corresponding derivatives. In passing we note that the ORPA version was also extended to mixtures and applied

to a special kind of solutions.

5.5 P E R T U R B A T I O N T H E O R I E S OF M O L E C U L A R F L U I D S

5.5.1 Pair Potent ia l s of Molecular Fluids

Molecular fluids are composed of molecules whose pair interactions are described by potentials of the form u - u(r, Wl, w2), where wi stands for a set of three angles (or, in the special case of axially symmetric molecules, of two angles: wi = 0i¢i; f dwi - 1). Pair potentials of this type describe (i) repulsive and dispersion forces of non-spherical molecules, (ii) electrostatic interactions.

We focus here on the interaction of permanent multipole moments such as dipoles, quadrupoles, etc. Electrostatic pair potentials (ES) can be written in the general form (31,41)

u(rx2, x, 2) = ¢2) li=O lj--O m

or u(r~2,w~,w2) = X r -n O(w~,w2) (5.101)

In this general expression Sh,~(0i , ¢i) stands for spherical harmonics; the coefficients X contain values of two multipole moments, n possesses integral values and ~ is a function of angles 01, 02 and ¢12.

For two permanent dipole moments, #1 and #2,

X = -#1#2, n = 3, • = 2 cos 01 cos 02 - sin 01 sin 02 cos ¢12 (5.102)

for two permanent quadrupole moments, Q1 and Q2,

3 X = ~Q1Q2, n = 5,

(I) = 1 - 5(cos 2 01 + cos 2 02 + 3cos 2 01 COS 2 02) 2r- 2(sin 01 sin 02 cos ¢12 -- 4cos 01 cos 02) 2 (5.103)

152

for permanent dipole and quadrupole moments, #1 and Q2, resp.,

3 X = ~#lQ2, n = 4,

Z

(I) - cos 01 (3 cos 2 02-1) +cos 02(3 cos 2 01-1) -2 (cos 01-cos 02) sin 01 sin 02 cos ¢12 (5.104)

Interaction of permanent and induced dipole moments can be expressed as

1 u(r, O)l,OJ2)= --2 r -6 [o~1~22(3 COS 2 02-~- 1 )+ a2#~(3 cos 2 O1 ~- 1)]

and similarly for quadrupoles

(5.105)

O u(r, wx,w2) = -~r -S [~1Q~(5 cos 4 02 - 2cos 2 02 + 1) + c~2Q~(5 cos 4 01 - 2cos 2 01 + 1)]

(5.106) (where a stands for polarizability). To the class of ES interaction models can be added the overlap potential (which describes interactions of slightly nonspherical molecules, charac- terized by parameter 5)

u ( ~ , ~ , ~ ) = 4~(~- )~ (3 cos ~ o~ + aco~ ~ o~ - 2) (5.107) r -

and the Ganssian-overlap potential (based on the idea of the Gaussian distribution of interaction sites within two molecules, with the hs or LJ interactions between sites) (42) which reads

u(r,2, wl, w2) -- 4% - (5.108)

Here both eg and ag are functions of wl and w2. In the simplified version (GOCE) % is assumed to be orientationally independent; for ag(wl,w2) we obtain

°~(~" ~ ) 1 - x + (5.109) o0 1 + ~(u , , u~) i : } ~ g :

(where ul, u2, T stand for three unit vectors characterizing orientation of main axes an the centre-centre distance). The parameter X is defined as X - ( a2 - 1)/( ~2 + 1); and a0 are characteristics of the spatial distribution of sites. For the special case of the hard Gaussian-overlap model [where u g ( r 1 2 , w l , w 2 ) - - oo for r12 _< ag(Wl,W2) and zero elsewhere] the GOCE interactions correspond closely to those of two hard ellipsoids of revolution with the short axis equal to ao and the axes ratio equal to n. In passing we note that the modified G aussian-overlap potential is used in simulation experiments to substitute for interactions of hard dumbbells.

Intermolecular interactions of nonpolar nonspherical molecules are characterized most often either by the multicenter model or by the Kihara generalized potential. In the former case the pair potential is assumed to be given by a sum of interactions between individual sites on the first and second molecules of the given pair:

153

?~(r12, ~.dl, ~d2) -- Z Ua~(r~2~) (5.110)

where r~2 ~ stands for the distance of the c~-site on molecule I from the fl-site on molecule 2; ua~ can possess e.g. hard sphere or 12-6 functional forms. The multicenter potential model is close to the chemical view of structure of molecules and easy to apply in simulations of molecules with one- or a few types of sites. On the other hand the derivation of the rdf faces considerable difficulties.

In the case of the Kihara pair potential a hard convex core is ascribed to given molecules; the pair potential can generally be written as

U(r12,031,O32) = U(8) (5.111)

where s is the shortest surface-surface distance between the cores of two interacting molecules; u can again possess different functional forms; in case of the 12 - 6 form we have

(:/'] = = - ( 5 . 1 1 2 )

The Kihara generalized pair potential can be considered as some kind of average over

Table 5.2 Parameters of the Kihara pair potential for hydrocarbons.

System e/kB (K) a (rim) l(nm) ethane propane butane pentane hexane

366.76 0.3057 0.3098 471.26 0.3193 0.4309 532.50 0.3314 0.5376 641.20 0.3409 0.6469 709.87 0.3486 0.7586

different orientations of a couple of molecules; its advantage is good characterization of the repulsive forces by suitable choice of the size and shape of the ascribed hard convex core. The use of the Kihara potential brings difficulties in the simulation processes; on the other hand its use simplifies expressions in perturbation expansions. Parameters e/kB, a and length of the rod-like core, l, of several hydrocarbons are listed in Table 5.2.

Both the multicenter and Kihara models can be combined with the electrostatic con- tributions to obtain potentials characterizing molecular interactions of polar (dipolar, quadrupolar, etc.) nonspherical molecules. A special case of such models is the Stock- mayer potential which combines LJ and dipole-dipole expressions to describe behavior of spherical molecules with the embedded dipole,

= + (5 .113)

Similar expressions can be written for non-spherical molecules with any of the ES inter- actions.

154

5.5.2 Equations of State and Distribution Functions of Hard Nonspherical Molecules

A considerable amount of pseudoexperimental data has been published up to the present time on the equilibrium behavior of hard nonspherical molecules, especially of hard dumbells and other fused-hard-sphere models (FHSM) and of prolate hard sphero- cylinders. The data include the compressibility factor, chemical potential and distribution functions (often in the form of spherical-harmonic expansion coefficients). Also data for the second, third and fourth virial coefficients are at our disposal (10,11). All these data indicate the fact that the reduced virial coefficients (i. e. values divided by the correspond- ing powers of volume) and consequently the compressibility factor depend considerably on the shape of the studied molecules.

Extension of the hard-sphere equations of state to nonspherical hard-body systems represents a formidable task. At the same level of approximations as considered for hs only, the EOS for the hard convex body system was derived within the scaled particle theory (43,44). The SPT method employs the fact that the mean volume V~+b of two hard convex bodies a and b in contact is (45)

V.+b = y~ + s.P~ + &R. + Yb (5.114)

where Vi, Si, Pu are geometric characteristics of the given hard convex body (hcb), namely the volume, surface area and the mean-curvature integral divided by 4r. If the dilatation coefficient (~) is used as the coupling factor, it is possible to determine from an expression similar to Equation (5.114) the mean volume Va+b even for negative values of ¢ and consequently the distribution function Ghcb(¢) for ~ E (--1, 0). From the conditions of continuity of G hcb and its first derivative at ¢ = 0, and from the expression for Ghcb(¢ --+ O0) three parameters in the second-order polynomial can be determined. Knowledge of G hcb allows one to write the hcb EOS

P V 1 3c~y 3c~2y 2 - { ~ + ~ SPT (5.115)

NkBT 1 - y (I - y)2 (i - y)3

where o~ - P u S i / 3 V i is the parameter of nonsphericity. By modifying the expression for Gcontact in the same way as was done in the case of hard sphere mixtures, an extension of the CS equation has been formulated (47)

P V 1 3(~y c ~ 2 y 2 ( 3 - - y) = ~ + ~ + ISPT (5.116)

N k B T 1 - y (i - y)2 (I - y)3

On the basis of the analysis of the viria] coefficients of hard convex bodies the present author (I0) proposed the expression

P V = 1 ~ 3c~y + y213c~2(1 - 2y) -+- 5c~y] MSPT (5.117) N k B T 1 - y (1 - y)2 (1 - y)3

Comparison of the compressibility factor for hard prolate spherocylinders (7 - 2) calcu- lated from MSPT with simulation data is given in Figure 5.7.

It was found that the third and higher virial coefficients for prolate and oblate hcb's

155

Prolate spherocylinders 7=2

/

0 , i . . . .

Y

Figure 5.7 Compressibility factors of hard prolate spherocylinders (~,=2); - calculated from MSPT, • - simulation data.

of the same parameter a differ slightly in their magnitude. The same is true for the com- pressibility factors at the same y and a. Therefore, a second nonsphericity characteristic has been sought and further parameters have been introduced. None of the (correspond- ing) EOS with 2 nonsphericity parameters has been used in the perturbation theory.

Whereas the characterization of the P - V - T behavior of hcb systems from the recent EOS is fairly accurate (see Figure 5.7), knowledge of the distribution functions is unsat- isfactory. The most important function is the average correlation function, ghcb(s). It is defined as an average over all orientations of the product of the molecular distribution function and surface area, divided by the mean surface area. From the fact that hard spheres are a special case of hcb (with a = 1) and from semiempirical arguments, the following relation was proposed (48)

a(x- 1) ) (5.118) hhCb(x) = O h hs 1 + 1 + Rci + Rcj

Here Rci stands for the mean curvature integral divided by 47r of the i-core, @ = shcb/Seh~v and shay is the surface area of the "equivalent" hard sphere with its volume equal to that of the hcb considered.

To determine the distribution function of the hard convex body system, one can also apply a method originally developed for the FHSM fluids (see the section dealing with hard dumbbells and FHSM). Determination of the perturbation terms is, however, more difficult.

All the above relations can be extended readily to mixtures. From the SPT, it follows

P V 1 3a.sy 3flsy 2 = ~ + SPT (5.119)

N k B T 1 - y (1 - y)2 (1 - y)3

156

where

and

Similarly

rs qs 2

a S = 3 p y ~ ' = 9 p y 2

y = p E z , y~, r = p E z , P~, s = p E z , S~, q= p E z , R~ i i i i

and

P V 1 3c~sy ]~sy2(3 - y) = } ~ + ISPT (5.120)

N k B T 1 - y (1 - y)2 (1 - y)3

P V 1 3a, y y213/~(1 - 2y) + 5a~y] - { + MSPT (5.121)

N k s T 1 - y (1 - y)2 (1 - y)3

In the case of hard dumbbells and other FHSMs, the situation is less satisfactory. On the basis of simulation data, Tildesley and Street (49) suggested for the system of hard dumbells

P V 1 + (1 + UL + VL3)y + (1 + W L + X L 3 ) y 2 - (1 + Y L + ZL3)y 3 N k B T = (1 - y)3 (5.122)

where L - I /a , I is the site-site distance and U - Z are empirical coefficients. The equation yields very accurate values of the compressibility factor for homonuclear dumbbells with L < 1. For systems of FHSM of arbitrary shape the ISPT variant of the hcb EOS is also employed; the Boublik-Nezbeda (B-N) rule (50) to determine a is used

o 3 -- FHSM

where the hard convex body considered just envelopes the given FHSM. In passing we note that several EOS have been suggested to describe the P- V- T

behavior of chain molecules - models of polymers (51-54). The compressibility factor of a hard-chain of m segments is

[ OlnghS(d)] (5.123) Z hch = m Z hs - (m - 1) 1 + y 0 ~

This EOS describing the behavior of associating fluids is often combined with van der Waals attractive term and the resulting SAFT equation used to desribe the behavior of alkanes, polymers, etc.. A similar equation of state also valid for the "pearl-necklace" chains was derived by Chiew (55) solving PY-OZ equations for mixtures under conditions following from the pearl-necklace structure; within this approximation it holds

1 + y /2 (5.124) Z h¢h = m Z h s - ( m - 1)(1 - y)2

Both Equations (5.123) and (5.124) consider one kind of segments; extension to different segments in a chain was proposed e.g. in reference 54.

157

Two types of distribution functions can be used in perturbation expansions to charac- terize the structure of fused-hard-sphere models: (i) the molecular distribution function gFHSM(rx2, W~,W2) and (ii) the site-site correlation function gss(ra~). For homonuclear dumbbells radial slices of the molecular distribution function at specific orientations (par- allel, end-to-end) can be calculated via the method proposed for the background cor- relation function. More often, however, gFHSM is expressed in terms of the background correlation function,

gFHSM(r12, Wl, W2) -- exp[--~UFHSM(r~2, 031, 032)] YFHSM(r12,031,032) (5.125)

KShler (56) and Fischer (57) assumed that the effect of the nonsphericity of a FHSM is absorbed mainly in the exponential function, whereas the background function depends mainly on the centre-centre distance. Thus, we can substitute for YFHSM the background function for soft spheres with potential u'(r12, T*) obtained from

uS(r12, T*) - - -- - l n ( e -u~"sM(r'2'~l'w2)/ksT\ /w,w2 (5.126)

kBT

Then, gFHSM(r12, Wl,W2) = e -t~u~'ssM(r'2'~''~'2) YS(r12, T*) (5.127)

Values of y8 for the soft-sphere potential u 8 are obtained from the solution of the OZ equation in some approximation such as, e.g., PY. It is obvious that the method is limited to dumbbells (and linear FHSM) with small site-site distances. The site-site correlation functions of chain molecules are - as mentioned above - available from the Chiew method.

In passing we note that all the above expressions were suggested for isotropic systems; characterization of nematic, smectic, etc. fluids is beyond the scope of this Chapter.

5.5.3 Perturbat ion Expansion for Pure Molecular Fluids

5.5.3.1 Fluids of Kihara Molecules

In the general case, the pair potential and molecular distribution functions depend on a set of (at least three) angles in addition to the dependence on the centre-centre distance. Therefore, formulation of the perturbation expansion and evaluation of the single perturbation terms are more tedious than in the case of simple fluids.

The perturbation theory of the Kihara molecules (59) can be considered as an extension of the theory of simple fluids to systems where the molecular core is not a point but a rod, convex figure or convex body. In the simpler BH-like method the soft convex bodies - parallel bodies (e.g., bodies enveloping individual cores with the soft 'thickness') ascribed to the studied molecules - form the reference system. The pair potential, u0, is given by Equation (5.112) for s <_ a. If the same coupling parameter as in the BH method was assumed, it was found that

2~ = (1 - e - ~uo ) dr (5.128)

158

where ~ stands for the thickness of the parallel hcb to the given core; knowledge of enables one to define the representative hard convex body. Its geometric characteristics axe

and

R4 - Rci + ~, Si -- Sci + 87rRci~ + 4 ~ 2, Vi - Vci + Sci~ -P 4~rRci~ 2 + 4~~3/3

y = pVi a = P~Si/3Vi

The reference residual Helmholtz energy, A 0 - Aid, can be calculated from the ISPT or MSPT EOS's. In the former case

in the lat ter case

Ao - A id = (a2 _ 1)ln(1 - y) + (~2 + 3c~ - 3o~y)y N k B T (1 - y)2

(5.129)

A0 - A ia N k B T - (6~2 - 50~- 1)ln(1 - y) + 2(1 - y)2

(12c~ 2 - 4 a ) y - (15~ 2 - 9(~)y 2

The first-order per turbat ion term for moiecular fluids is generally given as

(5.130)

At _ p oo N k B T - 2kBT fo //up(r12,wl'w2)gO(r12'wl'w2)drdwldO)2 (5.131)

In the case of the Kihara molecules both the pair potential and the average correlation function depend only on the shortest surface-surface distance s; for hcb it holds that

(5.132)

where $1+2+s is the mean surface area of convex bodies I and 2 with the constant surface- surface distance s. This mean surface area can be written as

$1+2+8 = St+2 + 8rRt+2s + 41rs 2 (5.133)

where S1+2 = St + $2 + 8rR1R2 and Rt+2 = Rt + R2 (5.134)

After substi tution and rearrangement one obtains, for the pure Kihara fluid in the BH-like aproximation, (59)

A1

N k B T ~1 °° 8* [8,2 2rpaa Up(8*)ghCb(-) -+-4R's* + 2 ( 3 " / 4 r + R*2)]ds *

kBT c (5. 35)

where s* = s /a , R* = R / a , S* = S / a 2 and c = 2~/a. In the case of the LJ fluid R* - S* - 0 and the BH relation for simple fluids is recovered. It is obvious that one can write ghcb _ 1 + h hcb and express the first-order perturbat ion integral as a sum of two parts, the larger of which can be determined analytically by

159

NkBT = T* + --~-R + ~-~(S*/4r +

27rp* fo~ Up(S,)hhcb(S)[s, 2 + 4R's* + 2(S*/4r + R*2)]ds * (5.136)

Within the BH-like method, one can also determine the second-order perturbation term in the m.c. approximation; the corresponding expression will not be given here.

For systems of Kihara nonspherical molecules one can consider also the WCA-like demarcation of the regions of repulsive and attractive forces. Determination of the temperature- and density-dependent thickness from the extended version of Equation (5.77) is more difficult due to the form of ghcb and the fact that &+2+8 has to be consid- ered in the corresponding integral for the determination of ~WCA. Quite often a 'hybrid' version is used instead of the WCA.

5.5.3.2 Fluids of Molecules with Multicenter Pair Potentials

Application of the perturbation theory to systems of diatomic molecules or, more generally, to molecules with a multicenter pair potential (mC) is less unambiguous than is the case for the Kihara fluid. In the first place the demarcation of the range of repulsive forces causes problems. For example, if the WCA-like division of the pair potential is considered, we have to realize that, at the distance of the potential minimum of the total pair potential, some of the contributing site-site pairs will already reach distances where attractive forces prevail, whereas other pairs will contribute by strong repulsions. An analogy of the BH division would be, however, even more problematic. Thus, the WCA-like division is the only one used for systems with a multicentre type of molecular interaction; the reference system is defined by

Uo(rl2,031,032) -- umC(rl2,031,032) -- umC(rl2min, 031,032)

0 r12 > rl2mi n

The perturbation pair potential is then

for r12 _~ rl2min

(5.137)

= - ( 5 . 1 3 8 )

In the perturbation theory proposed for systems with multicentre pair potentials, the KShler-Fischer approximation (56,57) for the molecular distribution function, Equation (5.127), would be used; the hs diameter of a site in a homonuclear molecule then follows from the condition

fo ° T . ~ r2 - AeoYS°ft(r12, ) 12ar12 (5.139)

where Ae0 denotes the blip function for the reference and fused-hard-sphere model and ysoft is the background correlation function of the soft-sphere multicenter model. Knowl- edge of the hs diameter of the representative hard body (e.g., hard dumbbell) makes it possible to determine thermodynamic functions of the reference by employing, e.g., ISPT

160

EOS; for the residual reference Helmholtz energy, one then obtains Equation (5.129). The first-order term is

A1 2rp oo N k B T = k B T f o //Up(r12,Wl,W2) e-U°(r12'wl'w2)/kBTysoft(r12,T*)r212dr12d~1dw2

(5.140) The manifold integral is solved - after determining yso~ _ by a numerical method (e.g., that due to Conroy).

It is obvious that the determination of the first-order term in this case is a time- consuming procedure; quite often only a limited number of values is determined and A1 is expressed in the form of a polynomial. Further thermodynamic functions are then calculated from derivatives of this formula.

5. 5. 3. 3 Fluids of Molecules with Electrostatic Interactions

Fluids of this class contain spherical or nonspherical molecules with permanent and/or induced dipoles, quadrupoles, etc. An example of the spherical polar molecule is the model of H C1, of the latter group a model of ethyl chloride. Intermoleculax interactions in the former case axe characterized usually by the Stockmayer potential, in the latter case, e.g., by the two-center LJ plus dipole-dipole pair potential.

The Stockmayer potential is a prototype of potentials which can be written as

~zA(rl2, St)l, 502) -- ~o(r12) + ~a(r12, 031, 032) (5.141) where ua denotes the anisotropic part of the potential; the reference potential, uo, is defined as (41)

uo(.1.) = f f (5.142/ Considering the A-expansion one finds

(0u(ri2, We, w2) ) =0 (5.143)

Thus, the first non-zero contribution is given by the second-order perturbation term which for multipole-multipole interactions reduces to the form

A2a ~ 2 p / 2 ,w2)>Wl•2 g0(r12)drl2 (5.144) =

NkBT 4 For electrostatic interactions it is necessary to consider at least the third-order perturba- tion expansion. For A3a one obtains

A3a _ 1 (AaA + AaB)-- ~13--~ ' f<ua(r,2,Wl,W2))~,o~2 go(r12) dr12

NkBT NkBT 3p2

/ / / (~a( i , 2)Ua(X, 3)Ua(2, 3))WlW2Wa g0( i, 2, 3)dridr2dr 3 (5.145) (In the expression for AaB, symbol 1 stands for rl ,wl, etc.). When substituting for ua the potential, Equation (5.101), one finds that only function (I) depends on the mutuaJ

161

orientation of two molecules. After averaging over all the orientations wlw2, a constant factor results in the second and AaA terms. Distribution functions go(r) and g0(123) correspond to the reference system of LJ molecules in the case of the Stockmayer potential and hs in the case of hard spheres with embedded multipole moment. Evaluation of the integral from the product go r - " is straightforward in case of the hs reference system; for the LJ reference, parametric expressions are available in the literature (58). To determine AaB, the superposition approximation is used (60).

In the case of electrostatic forces, the perturbation expansion converges slowly for larger values of the multipole moments; determination of perturbation terms higher than the third is, however, prohibitively difficult. To improve the quality of prediction, Stell (61) suggested writing the contribution of the anisotropic part of the potential in the form of Pad~ approximant

A - A0 A2 1 = (5.146)

NkBT NkBT (1 - As) A2

Due to the factorization of the perturbation integrals, the differentation with respect to density or temperature is easy and one can quickly obtain the contribution to the compressibility factor and internal energy.

The perturbation theory for nonspherical dipolar, quadrupolar, etc. molecules follows ideas discussed for the case of spherical polar particles. Polar nonspherical molecules have been modelled as hard dumbbells, hard prolate spherocylinders, two-centre LJ or prolate Kihara models with permanent point dipoles, quadrupoles, etc. embedded in the centre of the molecule. For such models the first-order ES perturbation terms are zero and the reference thermodynamic functions are determined either from the hard body EOS or from the perturbation expansion for the nonpolar part of the potential (e.g., the Kihara fluid).

In the second-order perturbation term (62)

A2a - - p X 2 /oC~r-2n((I)(O)l 032)go(r, w1 w2))w,w2r2dr (5.147)

NkBT (knT) 2 ' '

however, the molecular distribution function, go, of the reference (for single orientations) is not available. In the beginning of this section the similarity of the hard gaussian- overlap model and hard ellipsiods of revolution was mentioned; one can expect that such similarity holds also between Kihara general and GOCE molecules. This is the basis for the approximation

go( , where x in the GOCE expression is defined as x = r/a(wx,w2). Substituting for go and changing variables yields

Nk T f f (I)2dwld O2f 0 gGO(x)x-2nX2dx (5.148)

Defining X~ -- X/kBTa~ it holds

162

A2a = ~r "~3X*2j~ ~0 c~ x-2r*+2dx

The manifold integral, Jk, over angular coordinates depends only on a, as follows from Equation (5.109). The A3A term is

AaA = 1 .Pao3X~aKa(a:)foo=gGO(x)x_a,,,+2dx NkBT 3

(5. 50)

For given a, integrals Jk and Kk can be evaluated from Pad~ approximants available in the

6 N

5

4 I I 0 1 2 3

Q"2

Figure 5.8 Compressibility factor of the quadrupolar Kihara fluid of the reduced length l* = 0.8118 at temperature T* -- 1.1 and density p = 0.36. - theory, * - simulation data.

literature (62). For A3B, a semiempirical parametric equation is to be used. To calculate the total electrostatic contribution to thermodynamic functions, Pad6 approximants are formulated from the second- and third-order terms. The dependence of the compressibility factor of quadrupolar hard dumbbells on the quadrupolar moment at constant density is compared with MD data in Figure 5.8.

In passing we note that, in principle, the approach discussed for nonpolar multicenter models of molecules can be used for electrostatic contributions, too. Thus, hard dumbbell or two-center LJ molecules with embedded point dipoles or quadrupoles can be considered as models of dipolar and quadrupolar molecules; the approach of averaging the product of the Boltzmann factor and potential over all the orientations, which has been mentioned, is applicable as long as the site-site distance is small.

Another way of evaluating the contribution of electrostatic forces is to consider an expansion of the molecular distribution function of the reference in spherical harmonics (an expansion similar to that given in the beginning of this section for pair potentials) and then evaluate individual terms; due to the properties of spherical harmonics simple expressions for these terms result. The lowest coefficient in the g-expansion equals rdf go. This very general method faces two problems: slow convergency and the fact that higher coefficients of the expansion are available only from numerical methods.

163

5 . 5 . 4 W a t e r

Due to the importance of water for life on earth the equilibrium behavior of water has been studied by a large number of scientists. It appears, that due to its ability to form hydrogen bonds, and due to substantial polar effects, water represents a special case of molecular fluids which shares the traits of both the polar molecule and associat- ing molecule systems (see Chapter 12). To interpret the equilibrium behavior of water, several kinds of pair interaction potentials (some of them quite sophisticated) have been introduced and used in a large number of theoretical and simulation experiments (see Chapter 12.)

Applications of the perturbation theory to describe the behavior of water depend on the range of temperatures and pressures considered. It was found (63) that at high tem- peratures and pressures (P ~ 100 MPa) the perturbation expansion of simple fluids yields a fair description of the system. At low pressures and room temperatures, however, such a method is inappropriate and one has to use a perturbation method in which the reference includes a model of hydrogen bonding also. The use of such a reference has been proposed by Kolafa and Nezbeda (64) who introduced the 'primitive model of water'. Within this model the water molecule is assumed to be a hard sphere with four tetrahedrally arranged interaction sites, namely two of the H- and two of the e-type. The square-well potential describes the interaction between H- and e- sites (resulting in the formation of the hydro- gen bond) whereas zero potential energy corresponds to interaction of pairs of like sites. Applying the results of the Wertheim theory of associating molecule fluids (52), one can express the compressibility factor (considering the CS EOS) as

Z P M w = ZhS .Jr ZaSS _ i + y + y2 - y 3 2y(1+2c) ( O c ) (5.151) - ( 1 - y ) a - ( 1 + c ) 2 - -Oyy T

where

{1 cl( i - y /2) - 4"5c2y(i + Y) } ~ / 2 1 (5.152) c = -~ + 48y[e 1/T* - 1] (1 - y)3 2

and Cl and c2 are two constants. The Helmholtz energy follows from the corresponding integral. The distribution function for this primitive model of water is not available; thus, a van der Waals attractive term with an adjustable parameter is added to the reference term, instead of the correct perturbation term(s), to characterize the effect of the attractive forces in water.

5 .6 M I X T U R E S OF M O L E C U L A R F L U I D S

5 . 6 . 1 M i x t u r e s o f N o n - E l e c t r o l y t e s

The perturbation theory of mixtures of molecular fluids combines ideas and approx- imations of the approaches for pure molecular fluids with those for mixtures of simple fluids. It is practical to consider first the mixtures of nonpolar nonspherical molecules and only then mixtures of polar molecules.

In the case of mixtures of Kihara molecules, the BH-like method of dividing the pair

164

potential into reference and perturbation parts and determining the respective hcb thick- ness is generally used; alternatively the hybrid approximation is considered. In the WCA expression for thickness, ~, the background correlation function depends not only on tem- perature and density but also on the relative size and shape of representative bodies.

Once the thicknesses of the representative hcb's are known the reference residual Helmholtz energy is calculated, e.g., from

A0, - Aid NkBT - (6~s - 5c~s - 1)ln(1 - y) -{- 2(1 - y)2

(12/~s- 4c~s)y- ( 1 5 ~ - 9a,)y 2 (5.153)

The first-order perturbation term is

_ p f o o • • •

NkBT - 2kBT ~ ~ xixja~ . . Up(S*)ghCb(s /c)S~+j+sds

pa 3 12 hcb * * (5.154)

where the asterisk denotes the reduced quantities. The average correlation function is available from Equation (5.118) where O is given by sums of surface areas for the pair of molecules considered. When determining the excess thermodynamic functions (which depend on subtle differences in expressions for a mixture and pure components) it is desirable to include the second-order perturbation term. However, its rough form can lead, in some cases, to relatively larger differences between the calculated and experimental values.

Only the first-order perturbation expansion is used in the case of mixtures of molecules interacting via multicenter pair potentials. Thus, binary mixtures of L J- plus two-center LJ molecules or mixtures of two two-center LJ compounds were studied (65). The WCA- like division of the pair potential was used and the first-order integrals determined in a similar way to that used in the case of pure fluids. Excess thermodynamic functions of several binary mixtures of homonuclear molecules were determined. In spite of the fact that the (less well known) second-order term is negligible in the WCA-like variant, the calculated values of the excess thermodynamic functions agreed with experimental data in the majority of cases quite well for mono- and diatomic molecules of the moderate nonsphericity. This is apparent from Table 5.3. where some results of Bohn et al. (65) are listed.

Description of the equilibrium behavior of mixtures of polar nonspherical molecules represents a difficult task for the theory of fluids. For applications in chemical engineering an extension of the approach described previously for the pure fluid of polar Kihara (rod- like) molecules was proposed (66). In this method the BH-like demarcation of the range of the predominant effect of repulsive forces is used and thicknesses of pure representative hcb's evaluated accordingly. The second- and third-order electrostatic perturbation terms are determined after substitution of the GOCE distribution function for ghcb(r, Wl,W2) of the reference mixture. The corresponding GOCE bodies have the same volume and parameter of nonsphericity as the reference bodies. An arithmetic mean for a0 and ~ was assumed when evaluating the contribution of the unlike pairs. In the process of calculation

A1

165

Table 5.3 Excess thermodynamic functions (in J tool -1 and cm3mo1-1) from the perturbation theories for 2cLJ and Kihara (Kih) mixtures.

System expt Kih 2cLJ CH4-N2(91 K) AG E AH E AV E

Xe-C2H4(ll6 K) AG E AH E AV E

O2-N2(78 K) AG E AH E AV E

C2H6-C2H4 (161 K) AG E AH E AV E

=0.960 ~=0.0.956 170 170 170 138 164 148

-0.54 -0.39 -0.55

209 255

~=0.976 ~=0.967 209 209 181 219

-0.46 -0.49

~=1.003 ~=1.004 40 4O 4O 60 41 34

-0.25 -0.25 -0.24

~=0.984 ~=0.984 99 99 99 193 132 143 0.16 0.10 0.12

of the third-order term the A3B contribution has been neglected because of problems with the distribution function g(a)(1, 2, 3) for hard nonspherical molecules of different size and shape.

In the case where the hard-convex-body mixture serves as a reference, one can write the second-order ES term as

s

NkBT - pEE 3 [ 1

= xixja°ijXnijJkiJ(~iJ) 2 n - 3 -~- 1 hs -2n+2 (5.155)

where X*ij stands for the reduced coefficient of the ES pair potential with r -n dependence on distance; the integral in Equation (5.155) can easily be determined numerically.

5.6.2 M i x t u r e s of E l e c t r o l y t e s and A q u e o u s So lut ions

Description of the equilibrium behavior of aqueous solutions of salts or, more generally, solutions of strong electrolytes differs from that of nonelectrolytes, because, in the former case, the structure of the electrolyte systems depends more pronouncedly on the attractive electrostatic forces than in the case of nonelectrolytes. Due to this fact a special theory of electrolytes has been developed and applied to characterize the equilibrium behavior of ionic systems (see Chapter 16.). Only relatively recently has the general perturbation

166

theory been considered for the simplified models of aqueous solutions of electrolytes. The extended EXP variant, c.f. Equation (5.90), of the perturbation theory was used (67) to determine the behavior of the uni-univalent model of electrolyte solutions; fair agreement with simulation data was found. Another variant of the perturbation theory, in which resummation of terms of different orders is considered (similarly to the EXP variant), was proposed for mixtures of electrolytes by Henderson(68). The resummed terms contain only the perturbation pair potential (i.e., the product of 1 × Up where 1 comes from the expression for the reference distribution function, gO _ 1 ÷ h °) whereas contributions to the individual perturbation terms containing integrals from h ° ( r ) u p ( r ) - which in contrast with the previously mentioned terms are convergent - are determined individually. This variant makes it possible to take into account the effect of properties of the solvent on the thermodynamic functions of ionic fluids.

Aqueous solutions of nonelectrolytes were studied by Nezbeda et al. (69) who deter- mined the pressure of the reference from a sum of the EOS for a mixture of hard spheres of different diameters (or a similar EOS of hcb's) plus the associating-molecule term for water. A van der Waals term for mixtures takes into account the effect of all kinds of attractive forces.

5.7 C O N C L U S I O N S

In this Chapter an outline of the perturbation theory and some applications have been given. We limited ourselves to simple and easily tractable methods that will allow chemical engineers to apply these expressions of statistical thermodynamics to calculate different characteristics of the phase equilibria of pure fluids and their mixtures. It was our aim to show that the use of expressions of the perturbation theory brings only a slight increase in the complexity of the procedures in computing programs (in comparison with those for cubic equations, most often used for these calculations); these changes are often marginal considering the present level of computing technique.

There are, of course, more fundamental methods available in the literature that allow very detailed characterization of the intermolecular interactions of molecules and sophisti- cated methods for evaluating thermodynamic functions; these approaches are often quite time-consuming, comparable in this respect with simulation experiments. Such methods (including those with corrections for quantum effects) axe of great importance for the systematic study of fluid behavior within physics, but are less suitable for chemical engi- neering studies.

Perturbation theory yields very powerful and versatile approaches that enable the study of diverse problems of the thermodynamics of fluids and phase equilibria. A great ad- vantage of this theory is the fact that any improvement in our knowledge of the model systems results in an improvement of the accuracy of perturbation theory for calculations of all the properties of real fluids.

Acknowledgement. The author is indebted to Professors U.K. Deiters and J. Fischer for materials and notes for this Chapter and to Dr. K. Aim for reading the manuscript.

167

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42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68.

69.

C.G. Gray and K.E. Gubbins, Theory of Molecular Fluids. Vol.l: Fundamentals, Clarendon, Oxford (1984). B.J. Berne and P. Pechukas, J. Chem. Phys. 56, 4213 (1972). R.M. Gibbons, Mol. Phys. 17, 81 (1969). T. Boubllk, Mol. Phys. 27, 1415 (1974). T. Kihara, Adv. Chem. Phys. 5, 147 (1963). T. Boubllk, J. Chem. Phys. 63, 4084 (1975). T. Boublik, Mol. Phys. 42, 209 (1981). T. Boublik, Mol. Phys. 51, 1429 (1984). D.J. Tildesley and W.B. Street, Mol. Phys. 41, 85 (1980). T. Boublik and I. Nezbeda, Chem. Phys. Letters, 46, 315 (1977). R. Dickman and C. Hall, J. Chem. Phys. 89, 3168 (1988). M.S. Wertheim, J. Star. Phys. 42, 477 (1986). W.G. Chapman, G. Jackson, and K.E. Gubbins, Mol. Phys. 65, 1057 (1988). T. Boubllk, C. Vega and M. Diaz-Pena, J. Chem. Phys. 93, 730 (1990). Y.C. Chiew, Mol. Phys. 70, 129 (1990). F. KShler, N. Quirke, and J.W. Pertain, J. Chem. Phys. 7'1, 4128 (1979). J. Fischer, J. Chem. Phys. 7'2, 5371 (1980). M. Luckas, K. Lucas, U. Deiters, and K.E. Gubbins, Mol. Phys., 57, 241 (1986). T. Boublik, J. Chem. Phys. 87, 1751 (1987). G.N. Patey and J.P. VaUeau, J. Chem. Phys. 64, 170 (1976). J.C. Rasaiah, B. Larsen, and G. Stell, J. Chem. Phys. 63, 722 (1975). T. Boublik, Mol. Phys. 73, 417 (1991). E.U. Franck, Pure Appl. Chem. 59, 25 (1987). J. Kolafa and I. Nezbeda, Mol. Phys. 61, 161 (1987). M. Bohn, J.Fischer, and F. KShler, Fluid Phase Equilib. 31, 233 (1986). T. Boubllk, Mol. Phys. 90, 585 (1997). H.C. Andersen and D. Chandler, J. Chem. Phys. 55, 1497 (1971). D. Henderson, in :Molecular Based Study of Fluids. (J.M. Haile and G.A. Mansoori, eds.), American Chemical Society, Washington (1983). I. Nezbeda., W.R. Smith, and J. Kolafa, J. Chem. Phys. 100, 2191 (1994).

Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 169

6 E Q U A T I O N S O F S T A T E F R O M A N A L Y T I C A L L Y S O L V A B L E I N T E G R A L E Q U A T I O N A P P R O X I M A T I O N S

Yu. V. Kalyuzhnyi ~b and E T. Cummings a

aDepartment of Chemical Engineering University of Tennessee Knoxville, TN 37996-2200, USA and Chemical Technology Division Oak Ridge National Laboratory Oak Ridge, TN 37831-6181, USA

blnstitute for Physics of Condensed Matter Svientsitskoho 1, 29001 Lviv, Ukraine

6.1

6.2

6.3

Introduction 6.1.10rnstein-Zernike Equation and Closure Relations 6.1.2 Short Review of the Recent Progress in Integral-Equation Theory Baxter/Wertheim Factorization of the Ornstein-Zernike Equation 6.2.1 One-Component Monatomic Fluids 6.2.2 Multi-Component Monatomic Fluids 6.2.3 Remarks Equation of State for Analytically Solvable Models of Simple Fluids 6.3.1 Hard-Sphere Fluid

6.3.1.1 Solution of the PYA for the Hard-Sphere Fluid and Fluid Mixtures

6.3.1.2 Equation of State for Hard-Sphere Fluid and Fluid Mixtures 6.3.1.3 Semiempirical Correction to the PYA: The Generalized MS A

6.3.2 Adhesive Hard-Sphere Fluid 6.3.2.1 Solution of the PYA for the Adhesive Hard-Sphere Fluid and

Fluid Mixtures 6.3.2.2 Equation of State for the Adhesive Hard-Sphere Fluid and

Fluid Mixtures 6.3.2.3 Percus-Yevick Theory for the Hard-Sphere Chain Fluid

6.3.3 Hard-Core Yukawa Fluid

170

6.3.3.1 Solution of the MSA for the Hard-Core Yukawa Fluid and Fluid Mixtures

6.3.3.2 Equation of State for the Hard-Core Yukawa Fluid and Fluid Mixtures

6.3.4 Charged Hard-Sphere Fluids 6.3.4.1 Solution of the MSA for the Primitive Model of Electrolyte

Solutions 6.3.4.2 Equation of State for the Primitive Model of Ionic Fluids and

Fluid Mixtures 6.3.4.3 Corrections to the MSA: the Generalized MSA and the

Associative MSA 6.4 Equation of State for Analytically-Solvable Models of Molecular Fluids

6.4.1 Interaction-Site Model Fluids 6.4.1.1 Site-Site Omstein-Zemike Equation and Closure Conditions 6.4.1.2 SSOZ-PYA for the Fluid of Fused Hard-Sphere Diatomic

Molecules 6.4.1.3 SSOZ-MSA for the Fluid of Fused Charged Hard-Sphere

Diatomic Molecules 6.4.1.4 Corrections to the SSOZ Formalism: Chandler-Silbey-Ladanyi

Equation 6.4.2 Hard Spheres with Dipolar Interactions

6.4.2.1 Analytic Solution of MSA for Dipolar Hard-Sphere Fluid 6.5 Equation of State for Analytically Solvable Models of Associating Fluids

6.5.1 Two-Density OZ Equation and Closure Conditions: Relation to the CSL equation

6.5.2 Two-Density PYA for Dimerizing Hard-Sphere Models 6.5.3 Two-Density PYA for the Smith-Nezbeda Primitive Model of

Associating Fluids 6.5.4 Two-Density MSA: RPM of Electrolytes and Dimerizing HCY Fluid

6.6 Conclusion Appendix: Summary of the Invariant Expansion Method for Linear

Molecules References

6.1 INTRODUCTION

Modern statistical-mechanical theories of liquids and dense fluids are based on the calculation of distribution functions, which describe the average structure

171

of the system. The major theoretical question to be answered is: Given the inter- molecular potential between the molecules in the system, what will the structure of the system be at a given thermodynamic state point? Usually it is assumed that the intermolecular potential is pair-wise additive, with the consequence that only the pair structure is needed to describe all the thermodynamic properties of inter- est. Although the assumption of pair-wise additivity is not valid at high densities and low temperatures, it proves to be a good approximation in most instances. Moreover, intermolecular pair potentials are frequently parameterized to take into account higher-order interactions.

In this chapter, we will focus on the integral-equation approach to determining the structure of a fluid and therefore its thermodynamic properties and equation of state. For a fluid composed of spherically symmetric molecules, the princi- pal quantity calculated by an integral-equation-based theory is the radial distribu- tion function, 90(r), which is proportional to the probability density of finding a species i and a species j molecule separated by distance r and is normalized so that 9ij(r) --+ 1 as r --+ ec. The radial distribution function is a statistical measure of the average structure in the fluid, and knowledge of this function permits all thermodynamic properties to be determined for a system interacting by pair-wise potentials. The quantity pjgij(r), where pj is the number density of species j molecules, can be regarded as the local number density of species j molecules at a distance r from a species i molecule. For an ideal gas, 9ij(r) = 1 for all r; hence it makes sense to define a 'residual' function, called the total correlation function, given by hi j ( r ) = 9i j (r) - 1. This function can be thought of as a measure of the correlation (non-randomness) in the relative positions of two molecules.

There are three routes to thermodynamic properties from distribution-function theories. We will quote these formulae here without proof, but note that a partic- ularly transparent derivation is given by Baxter (1). The compressibility route to the thermodynamic properties of an m-component mixture is given by

] m / 1 0 P - 1 - ~ pk Cik(r)dg" (6.1)

kBT Opi T k=l

where P is the pressure, k B is Boltzmann's constant, Pm is the number density of a species m, T is absolute temperature, c~j(r) is the direct correlation function between a species i and a species j molecule separated by distance r defined in section 6.1.1 below. The virial route is given by

P 27rp m m foo pksT = 1 3ksT i~1 ~ xixj Jo 9ij(r) duij(r) r3dr (6.2)

• = j=l dr

172

where uij(r) is the pair potential between a species i and a species j molecule, p is the total number density and xi is the mole fraction of species i, i.e.,

m Pi P - - ~ Pk , Xi - - - -

k=l P

Finally, the energy route to the thermodynamic properties is given by

u res

NkBT 27rp m m f ~ ksT ~ ~ xixj Jo 9ij(r)uij(r)r2dr

i--1 j--1 (6.3)

where U res is the residual internal energy over the ideal gas at the same tem- perature and composition. In general, there are three ways to obtain the pair- correlation functions:

1. radiation scattering experiments, such as neutron and X-ray scattering; 2. molecular simulations such as molecular dynamics and Monte-Carlo

simulation; and 3. theoretical approaches such as integral-equation methods and perturba-

tion theory. The main advantage of the first two methods is that both of them are, in principle, exact: scattering experiments are exact for a particular substance while molecular simulation is exact for a particular model. However this immediately suggests their limitations. Since both of them are essentially experimental methods, the structural results obtained by the methods are restricted to a particular thermody- namic state point at which they were measured. Thus these methods do not have the predictive ability of theory.

The integral-equation approach is a theoretical method for the calculation of the distribution functions and is not subject to these limitations. However, unlike molecular simulation, integral-equation theories necessarily involve approxima- tions as noted below and consequently the results are not exact. For this reason, they are frequently referred to as integral-equation approximations (IEAs). An important fact to bear in mind about integral equation approximations is that the approximations inherent in the correlation functions result in inconsistency be- tween the thermodynamic properties predicted by the three different routes. This is especially important in considering the critical behavior of integral-equation approximations. An additional source of uncertainties in the predictions of the IEA is insufficient information concerning the precise form of the intermolecular forces in real fluids. [The same problem arises when the results of molecular sim- ulations are compared directly with experiment.] In the case of theoretical studies

173

the latter problem can be bypassed by assuming a certain reasonable potential model, which correctly reproduces the most characteristic features of a particular class of fluids. The accuracy of the results obtained with such a model is quite sufficient for most theoretical studies, which have as their aim understanding the origin of fluid-phase properties. However, engineering applications frequently re- quire a much higher degree of quantitative accuracy, so that the direct application of IEAs in such practical calculations is of limited use. In practice, the desired accuracy can be reached by using empirical formulae (in which high-order poly- nomials are fitted to experimental data) or semi-empirical methods (in which the molecular parameters in a theoretically derived equation of state are fitted to ex- perimental data). The predictive capability of such methods strongly depends on the quality of the theoretical basis used. Purely empirical methods without any reasonable theoretical basis are little more than convenient interpolation schemes, the results of which cannot be used to extrapolate with confidence outside the in- terpolation region. This was clearly demonstrated by Henderson (2) in his analysis of the semi-empirical equations of state commonly used in chemical engineering calculations. A good example of a very accurate semi-empirical equation of state (EOS) for the Lennard-Jones (LJ) fluid developed on a sound theoretical basis is the EOS proposed recently by Kolafa and Nezbeda (3).

IEAs can play an important role in the formulation of semi-empirical equa- tions of state by providing the theoretical basis and functional form for equations of state which can then be used to correlate experimental data. In particular, ana- lytically solvable IEAs are therefore of greatest interest for this purpose.

The goal of this Chapter is to introduce the reader to some of the recent ad- vances in the analytic solution of IEAs for molecular fluids and to demonstrate their ability in predicting thermodynamic properties of physical systems focusing mainly on the results for the EOS. Analytic solutions are now available for a num- ber of rather non-trivial Hamiltonian models of dense fluids and liquids which, in particular, can undergo the liquid-gas and fluid-fluid phase transitions. [A de- scription of the solid phase, and hence fluid-solid transitions, is precluded from IEAs by the disordered fluid assumption, Equation (6.18).] We illustrate the re- sults obtained for thermodynamic properties by presenting the liquid-gas phase diagram. The paper is organized as follows. In the remainder of this section, we introduce the Ornstein-Zernike equation and its closure relations and discuss briefly the recent progress achieved in integral-equation theory. In Section 6.2, we review the Baxter Wiener-Hopf technique for pure fluids and mixtures used to solve IEA analytically. Section 6.3 contains a review of applications of the Baxter technique to simple fluids. This includes sections on the hard-sphere fluid and its

174

mixtures, adhesive hard-sphere fluid and its mixtures, the hard-core Yukawa fluid and the charged hard-sphere fluid. In Section 6.4, we review analytical solutions of IEAs for non-spherical molecules, including interaction-site-model fluids and fluids composed of hard spheres with multipolar interactions. In the last section, Section 6.5, we discuss IEAs for associating fluids and present recently obtained analytic solutions for a number of simple models of associating fluids.

6.1.10rnstein-Zernike Equation and Closure Relations

To calculate the total correlation function, Ornstein and Zernike introduced a second function via the equation that bears their name, the Ornstein-Zernike (OZ) equation. This second function, called the direct correlation function, is less singular at the critical point (i.e., does not become long-ranged in the sense that the volume integral remains finite) than the true function of interest, the total correlation function. The OZ equation for an m-component mixture is given by (4)

m

hij(r) - cij(r) q- ~ Pk /Cik(l~)hkj(I ~'- ~)d~ (6.4) k-1

This equation can be interpreted as follows: the total correlation function between species i and species j molecules at a distance r apart, h#(r), can be broken up into a direct part, e~j(r), and indirect part ~=~ Pk f c,k(]~)hkj(]~"- ~)d~. The form of the latter term indicates that indirect correlations take place when the species i molecule, located at the origin, is directly correlated with a species k molecule, located at the position g, which in turn is totally correlated with a species j molecule located at ~'. For a pure (i.e., one-component) fluid, Equa- tion (6.4) simplifies to

h(r) - c(r) 4- p f c( ] 8--] ) h( ]~' - (6.5)

To have a closed mathematical problem, an independent relation between the direct and total correlation functions is required. A formally exact independent relationship can be written down using the language of graph theory, which is beyond the scope of this review. The total and direct correlation functions are expressed in terms of an infinite sum of graphs ordered by density, the direct- correlation-function graphs being a subset of the total correlation graphs, as is clear from the OZ equation. The difference between the two functions can there- fore be expressed as a graphical expansion in which the graphs have well defined

175

topological features. The graphical definition of this difference amounts to the for- mal, exact independent relationship between the two correlation functions. The reader interested in learning more about the graphical expansions is referred to the monograph by Hansen and McDonald (5) for a complete and very understand- able account of the graph theoretical developments pioneered in rigorous form by Stell (6,7). The net result can be written as

hi j ( r ) - ciy(r) - log[hi j (r ) + 1]-~ k B T

B i j ( r ) (6.6)

where, because of their topological structure, the graphs in the function B i j ( r ) are often referred to as the bridge diagrams. Choosing B i j ( r ) - 0 yields the hypernetted chain approximation (HNCA) (5),

while rewriting Equation (6.6) as

#ij (r) -- exp

k B T

u ij ~ r~ ] + B i j ( r ) + h i j ( r ) - c i j (r)]

k B T J

,.~ exp m U ij ~ r~ ]

+ k B T j " '

(6.7)

g q ( r ) - exp [ - U ~ j ( r ) l [g~j(r) - (6.8)

An important correlation function from the point of view of solvent effects on chemical reaction is the cavity-distribution function, y i j ( r ) , defined by

Yij(r) -- exp L k s T 9i j (r ) (6.9)

For systems without hard cores (that this is not the case for hard-core systems is a subtle point not often appreciated (6)), Equation (6.8) can be written as

(6.10)

setting B i j ( r ) - O, and neglecting the higher order terms in the expansion of the exponential yields the Percus-Yevick approximation (PYA) (8,9)

176

For hard-core systems (i.e., systems for w h i c h uij(r) - - ~ for r < aij), all the IEAs satisfy the exact core condition 9ij(r) = 0 for r < ai3. In addition, we have the exact asymptotic result for the direct correlation function given by (10)

For hard-core systems, we can combine the exact core condition with the assump- tion that the asymptotic form holds for all intermolecular separations outside the hard core. That is, we define the closure

hij(r) - - 1 r < aij (6.11) cij(r) - - u i j ( r ) / k B T r > aij

which is known as the mean spherical approximation (MSA) (9). The advantage of the MSA is that it is analytically solvable for a broad class of pure fluids and mixtures, as we shall describe below.

Many other integral-equation approximations have been defined, and it is be- yond the scope of this Chapter to attempt to review them here. The monograph by Hansen and McDonald (5) contains a comprehensive discussion of more re- cent developments. We mention two others briefly here: the soft mean spherical approximation (SMSA) can be defined for systems without hard cores by break- ing the intermolecular potential up into a reference part, u°j(r), and an attractive perturbation part, ui~(r ), and is given by (11)

eli(r) - 9ij(r) 1 - exp u°j(r) UiJ (6.12) kBT kBT

The hybrid MSA (HMSA) (12) is defined as a combination of the SMSA and the HNCA, and is defined by

1 In 1 - s + sg i j ( r ) exp [ k s T k s T (6.13)

where s is an adjustable parameter. In the limit s = 1, Equation (6.13) reduces to the HNCA while for s = 0 it yields the SMSA. The parameter s is adjusted to yield consistency between the energy and compressibility routes to the thermodynamic properties. The application of the SMSA and HMSA, as well as other IEAs, to supercritical fluids was recently reviewed by Cummings (13).

6.1.2 Short Review of the Recent Progress in Integral-Equation Theory

177

The OZ equation, Equation (6.4), together with the relation Equation (6.6) is a formally exact recipe whereby the structure and thermodynamics of a fluid can, in principle, be calculated if the intermolecular interaction potentials are known. In fact, however, exact calculations have been carried out for only a very limited number of highly-simplified model potentials. One example is the one- dimensional hard-sphere system (14,15), for which the PYA, Equation (6.8), ap- pears to be exact (16). The study of most systems of practical interest requires the use of a variety of different approximations. The success of a given lEA depends on its ability to reflect the most specific features of the intermolecular interactions as well as on its ease of implementation.

In general the different parts of the total potential cause different physical effects. The short-range steeply repulsive part of the potential gives rise to the excluded-volume effects and provides the main contribution to the entropic and structural properties of the system. The long-range interactions, such as attractive dispersion forces and multipolar forces, vary with the distance more smoothly and contribute mainly to the energetic properties. The long-range Coulombic in- teractions cause the collective behavior of charged molecules which results in screening. Short- and long-ranged effects differ fundamentally and thus require different approximations in their description. Although this idea was formulated by van der Waals a long time ago, significant progress in its application became possible only in the last three decades primarily due to the development of the PYA and HNCA integral-equation theories. The PYA appears to be quite accurate in describing excluded-volume effects (17), while the HNCA has proven to be rel- atively successful in accounting for the long-range part of the potential (18-22). Finally these developments result in the implementation of the van der Waals' idea in the form of the reference hypernetted-chain approximation (RHNCA) of Lado (23-26), which is perhaps the most accurate integral-equation theory for simple fluids. Recent progress extending Lado's ideas can be found in the work of Labik et al. (27-29) and Pospi~il et al. (30,31) who developed an accurate description of the molecular reference system for fluids with nonspherical hard-core interaction, and of Lomba et al. (32) and Lombardero et al. (33,34) who used the RHNCA to obtain results for the properties of the LJ and polar molecular fluids.

An alternative approach, which can be also viewed as an implementation of the van der Waals' idea of using different approximations for different parts of the potential, has been initiated by Rowlinson (35) and developed in later studies by Hutchinson and Conkie (36), Rogers and Young (37), and Zerah and Hansen (38). This is based on the idea of constructing a closure which interpolates between two different closure conditions, for example PYA for short-range and HNCA for

178

long-range (37). A mixing parameter in the closure is defined to enforce equality of the virial and compressibility equations of state. An important advantage of this approach is the elimination of the intrinsic thermodynamic inconsistency of the traditional IEAs.

The difficulty encountered in use of the above discussed IEAs is the need for numerical methods for the solution of both the RHNCA and the self-consistent theories. However, the numerical methods are much less time consuming than the corresponding computer-simulation calculations. One can expect significant progress in the future through the idea of self-consistently combining different ap- proximations for different parts of the potential in the context of analytical tech- niques discussed in this review.

All the IEAs discussed above are generally applicable only to the case of a pair potential with attractive interactions of moderate strength (i.e., characteristic attractive energies of the order of several kBT at most). Attempts to apply them in the case of associating fluids have met with limited success. The latter class of fluids is characterized by strong short-range attraction with the absolute value of the attractive potential well varying from 3 - 5kBT up to 102-3kBT. This short- ranged attractive part of the potential causes the quite different physical effect of clustering of particles and thus requires the use of an appropriate approximate description. The regular IEAs are closely related to the Mayer density expansion, or p-expansion. For strongly associating liquids, there is a severe problem with this expansion: an inf ini te number of terms must be included to reproduce the correct low-density limit (39,40). This can be demonstrated by using the following simple argument (40). Consider the virial expansion for the pressure. In the low- density limit, the power series in the activity z for the pressure P and density p may be terminated at a second order to give

P = z + b2z 2, p - z + 2b2z 2, (6.14)

k B T

where b2 is the second virial coefficient. Elimination of z from these two equations yields

P 1 (1 + 8b2p) 1 / 2 - 1 = - + . (6.15)

k B T p 2 8b2p

The more familiar corresponding virial equation of state is obtained from Equa- tion (6.15) by expanding the square root in the power series of p to obtain

P = 1 - b2p + 462p 2 - 20b~p 3 + .... (6.16)

k B T p

179

For systems with weak interactions, b2 is finite and both Equations (6.15) and (6.16) reproduce the ideal gas EOS, P / k B T p = 1, in the p --+ 0 limit. However in the limit of strong attraction between pairs, b2 ~ cx~ and the correct equa-

l follows only from Equa- tion of state for an ideal gas of dimers, P / k B T p - -~, tion (6.15), and cannot be reproduced by Equation (6.16) with a finite number of terms. Hence, the usual OZ closures, with their connection to the p-expansion, are not designed to treat strongly associating systems, and cannot be expected to systematically produce good results.

Inspired by early attempts to model association effects using the (41-44) PYA, Wertheim developed an integral-equation theory for strongly associating fluids (45-48) which was subsequently extended by others (40,49,50). This the- ory is based on the multidensity formalism, in which the descriptions in terms of the activity and density expansions are combined. Such a combined description allows one to introduce different approximations for different parts of the poten- tial. For the short-ranged strongly attractive part an activity expansion is used as a starting point, while the rest of the potential is taken into account by means of the standard Mayer p-expansion. Thus the short-ranged repulsive and long-ranged parts of the potential are described using approximations similar to those used in standard integral-equation theory, while for the short-ranged attractive part new approximations, based on the multidensity descriptions, have been proposed. The theory appears to be quite successful in the description of a number of different models of associating fluids (40,50,51-57). Details of the theory and its applica- tion in the case of analytically solvable models are discussed in Section 6.5 of this review.

We conclude this section by noting that the above short discussion cannot be considered a comprehensive review of recent progress in the field of integral- equation theory. We discuss here only those recent developments which, in our opinion, are the most promising for the future development of equations of state. In all of these theories, a reasonable compromise between the accuracy and the simplicity of the approximation used is achieved on the basis of van der Waals' concept of utilizing different approximations for different parts of the potential.

6.2 BAXTER/WERTHEIM FACTORIZATION OF THE ORNSTEIN-ZERNIKE EQUATION

Historically, the first IEA to be solved analytically was the PYA for the hard sphere fluid (58,59) just under forty years ago. It is difficult to overstate the im- portance of this analytic solution in the subsequent development of the statistical-

180

mechanical description of fluid systems. For example, the development of pertur- bation theories relied on the availability of analytic expressions for the structural and thermodynamic properties of the hard-sphere fluid which served as the refer- ence system (5).

Wertheim (60) and Baxter (61) subsequently developed general techniques for the analytic solution of IEAs based on factorization of the OZ equation in the Laplace domain (in Wertheim's case) and in the Fourier domain (in Baxter's case). Baxter's factorization, based on the Weiner-Hopf method (62), has proved to be the technique most frequently used by later authors for solving IEAs analytically. The basic rationale behind the Weiner-Hopf technique is to examine the integral equation in Fourier k-space, and to seek a factorization of the transform of one of the required functions into either a sum or product of functions which have identifiable analyticity properties in separable regions of the complex k-plane. The exact manner in which this is done is very much dependent on the specific problem; however, the general technique is a standard one, which can be found in any advanced textbook on integral equations (63-65). The key contribution of Baxter (61) was in recognizing that the Weiner-Hopf techniques developed for analyzing one-dimensional equations similar to the OZ equation (for example, (66)) could be successfully carried over into three dimensions. Essentially, this stems from the observation that for n odd, the n-dimensional Fourier transform of a function f(r-') (where g' is an n-dimensional vector) which depends only on It-q, reduces to a one-dimensional Fourier transform after performing the angle integrations. Once the problem is cast in the language of one-dimensional Fourier transforms, the factorization proceeds in a standard way, a fact pointed out by Baxter (61).

6.2.1 One-Component Monatomic Fluids

The Baxter Weiner-Hopf factorization (which we shall simply refer to as the Baxter factorization) is based on only two assumptions. First, the direct correla- tion function is assumed to be finite ranged,

c(r)=O, r > R (6.17)

This is the case for the PYA and the MSA applied to a finite-ranged potential (u(r) = 0 for r > R). The hard-sphere fluid, described below in Section 6.3.1, is clearly a special case (R = a) of this general assumption. Second, the fluid is assumed to be disordered. This means that the total correlation function is not

181

long-ranged (as it is at a liquid-gas critical point or in the solid phase), i.e.,

f h(r)d~' < (6.18) cx:).

Multiplying both sides of the OZ equation, Equation (6.5), by exp(ik • ~) and integrating with respect to ~' over all space, we find that,

h(k) - ~(k) + p]z(k)~(k) (6.19)

where ]z(k) and ~(k) are the three-dimensional Fourier transforms of h(r) and c(r), depending only on the magnitude k of the vector k. Equation (6.19) may be rearranged into the form

[1 + ph(k)][1 - p(?(k)]- 1 (6.20)

which is more suitable for factorization. It is possible to prove, by using Equations (6.17) and (6.18) and standard prop-

erties of integrals in the complex plane, that 1 - pO(k) can be factorized into a product of the form (61)

1 - p~(k) - Q ( k ) Q ( - k ) (6.21)

where Q(k) is regular, has no zeros in the upper half plane and approaches unity as Ik[ -+ ec. These properties permit one to show that Q(k) is essentially the one-dimensional Fourier transform of a real-space function q(r), defined by

1 o~ 27rpq(r) - -~ f ~ e -ik~ [ 1 - Q(k)] dk (6.22)

with the following properties

q ( r ) - O for r < 0 and r > R (6.23)

One then writes the OZ equation, Equation (6.19), as a set of two equations

1 - p5(k) - Q ( k ) Q ( - k ) (6.24)

Q(k) [1 + p i t ( k ) ] - [(~(-k)] -1 (6.25)

and uses the properties of (~(k) and ~)(-k) in the respective half planes to perform the one-dimensional Fourier inversion of these two equations, obtaining

rc(r) - -q ' ( r ) + 27rp q ' ( t ) q ( t - r)dt (6.26)

182

fo R rh(r) - -q ' ( r ) + 27rp q(t)(r - t )h( r - t l)dt (6.27)

As a result the initial OZ equation, Equation (6.19), has been rewritten as a pair of equations in which the functions h(r) and c(r) are decoupled. These equations are the principal results obtained by Baxter (61) and yield a particularly useful form for the OZ equation. First they indicate that, if c(r) is finite-ranged, knowledge of c(r) and h(r) on that range alone is sufficient to determine h(r) for all r. This result does not follow from the OZ equation itself, and is a simplification that comes out of the Weiner-Hopf analysis, Equations (6.21)-(6.27). Second, it is also clear that computing the Baxter q-function on the range (0, R) is sufficient to determine h (r) for all r.

A similar factorization can be carried out for the multicomponent version of the OZ equation, Equation (6.4), as will be demonstrated in the next section.

6.2.2 Multi-Component Monatomic Fluids

The factorization, described above, was extended to mixtures by Baxter (67). We assume, as in Section 6.2.1, that the c4j(r) are finite-ranged, that is

(Ri + Rj) (6.28) c # ( r ) - O, r > R~j = 2

where R/, i = 1 , . . . , m are some additive range parameters. Under this assump- tion, Baxter (67) was able to argue that a Weiner-Hopf factorization of the OZ equation, Equation (6.4), should be possible. The result of the factorization is

m

rcij(r) - -q~j(r) + 27r ~ Pk f qki(t)q'kj(r + t)dt k=l

(6.29)

rhij(r) - -q~j(r) + 27r ~ Pk qik(t)(r -- t)hkj(Ir -- t l)dt (6.30) k= l ik

_ 1 (Ri - Rj) the integration in Equation (6.29) is over the range where Sij -~

Ski < t < min[Rki, Rkj -- /'] (6.31)

and the Baxter q-functions satisfy

qij(r) = O, r < Sij, r >_ Rij (6.32)

183

Hence the OZ equation for mixtures has been transformed into a form which de- couples the hij(r) and cij(r) functions. In addition, for Sij < r < Rij , Equa- tion (6.30) involves each function hkj(s) only over the range (0, Rkj) . Thus, as in the case of a one-component system, knowledge of the Baxter q-function in the range (0, Rij) enables one to calculate hij(r) in the whole range of r.

6.2.3 Remarks

As we shall see below, the factorization method has permitted the analytic so- lution of IEAs for many important model fluids, beginning with the hard-sphere fluid. In some cases, the analytic solution is in closed form, with expressions for all the quantities of interest given explicitly in terms of the thermodynamic state variables, temperature, density and composition. In other cases, the analytic so- lution reduces to a set of simultaneous non-linear algebraic equations which must be solved numerically. In the latter case, the calculation of thermodynamic prop- erties on the liquid side of the gas-liquid phase transition can be difficult, if not impossible, due to limitations on integrating through the two-phase region which typically contains a region of no real solutions to the algebraic equations. Finally we comment that, in all cases studied to date, a variation on Perram's method (68) permits calculation of the radial distribution function. In some cases, such as the hard-sphere fluid in the PYA, the radial distribution function is known analytically in closed form (69). In this review, we shall regard an integral-equation approx- imation as solved analytically if an explicit analytic expression for the Baxter q-function can be obtained.

6.3 EQUATION OF STATE FOR ANALYTICALLY SOLVABLE MODELS OF SIMPLE FLUIDS

In this section, we shall consider simple fluids (i.e., composed of spherically symmetric molecules) for which the PYA or the MSA is analytically solvable. The PYA is solved analytically for the hard-sphere (section 6.3.1 and adhesive hard- sphere (section 6.3.2) fluids, while the MSA is solved analytically for the hard- core Yukawa fluid (section 6.3.3) and the charged hard-sphere fluid (section 6.3.4).

6.3.1 Hard-Sphere Fluid

The hard-sphere fluid (HSF) plays a central role in the modern molecular the- ory of fluids because of its use as the reference system in many perturbation the- ories. The intermolecular pair potential for the pure HSF is given by

u(r) = ~ r < cr = 0 r > cr (6.33)

184

where cr is the diameter of the hard sphere. For mixtures, the HSF is defined by

ui j ( r ) - c~ r < aij (6.34) = 0 r > O'ij

where aij is the distance of closest approach of the centers of a species i and a species j hard sphere, and so is related to the individual diameters ai and aj by

ai + aj (6.35) O'ij = 2

The analytic solution of the PYA for the HSF and HSF mixtures led to the devel- opment of the highly accurate analytic expressions for the HSF structure (70) as well as an accurate equation of state for HSF mixture (71). In the following, we derive the analytic solution of the PYA for the pure HSF, for HSF mixtures, and their thermodynamic properties and discuss corrections to the PYA HSF structure and thermodynamics.

6.3.1.1 Solution o f the PYA f o r the Hard -@here Fluid and Flu id Mixtures

For the PYA applied to the one-component hard-sphere fluid, we obtain

h(r ) - - 1 r < a

c ( r ) - 0 r >

(6.36)

(6.37)

so that c(r) is finite-ranged and the factorization of Section 6.2.1 may be used. Substitution of Equation (6.37) into Equation (6.27) yields a Fredholm integral equation for q(r) of the first kind with degenerate kernel. Hence, using continuity of q(r) at r = R = a, we obtain (61)

where

q(r) - ~ a ( r 2 - cr 2) + b(r - or), 0 < r < cr (6.38)

f0 O" t" a -- 1 - 27rp q ( t )d t - Q(O)

b - 27rp tq ( t )d t

(6.39)

(6.40)

185

Equations (6.39) and (6.40), when combined with Equation (6.38), yield two lin- ear equations for a and b, which may be solved easily to obtain

(1 + 2r/) -3r/o- a = b = (6.41)

(1 - r/) 2 ' 2 ( 1 - r/) 2

where ~7, the volume fraction occupied by the spheres, is given by

7I" 2 ~7- -~pcr (6.42)

This completes the solution of the PYA for the one-component hard-sphere sys- tem.

The PYA for hard spheres was first solved by Wertheim (58,72) and Thiele (59) by Laplace transform techniques. Their methods, and also Bax- ter's transformation, (73), of the OZ equation, formally gives a solution for h(r) through its Fourier transform. Baxter's factorization, on the other hand, yields a real-space equation for h(r) , Equation (6.27), which, for a given value of r, re- quires only the values of h(t) on the range r - cr < t < r. This permits a simple numerical calculation of the total correlation functions for arbitrary separations via the method due to Perram (68). This numerical method supplements the exact formulae for 0 < r < 5(r obtained by Smith and Henderson (69) by inversion of the Laplace transform of 9(r).

The PYA for a mixture is given by

c ~ ( r ) = [1 - e x p ( u ~ ( r ) / k B T ) ] g ~ ( r ) (6.43)

For a mixture of M species of hard spheres with additive diameters, the PYA, first solved by Lebowitz (74) by Laplace transform methods, may be written as

h ~ ( r ) = - 1 O < r < a ~ (6.44)

c~ (~ ) = o r > ~

where a ~ _= R ~ = (cr~ + a~) /2 and u ~ - S ~ = ( a ~ - a~)/2. The conditions for Baxter's factorization are now met, and, analogously to the single-component case, we find that (67)

2 (6.45) q~z(r) - a~(r 2 - cruZ) + b~(r - cruZ), u~Z < r < cruz

where 1 - ~3 + 30a~2

as = (1 - ~3) 2 (6.46)

186

I0

• , , ' i . . . . i . . . . i . . . . i . . . .

. . . . . P~n_v I / / /

. . . . . . . . . /A.;:,

.../Z...ff ;"

0 0.0 0.1 0.2 0.3 0.4 0.5

Figure 6.1 The equation of state of the hard-sphere fluid obtained from the PYA compressibility (labelled PYA-C), PYA virial (PYA-V), HNCA compressibility (HNCA- C) and HNCA virial (HNCA-V) equations of state compared with the essentially exact Carnahan-Starling equation (CS). After Hansen and McDonald (5).

and

O-2 b , ~ = - 3 ~2 (6.47)

2(1 - ¢ 3 ) ~

M 7/

~j - ~ ~ p.ycr~ (6.48) "),=1

Perram's algorithm (68) may also be generalized to mixtures enabling the compu- tation of the hard-sphere correlation functions 9~(r).

6.3.1.2 Equation of State for Hard-Sphere Fluid and Fluid Mixtures

Combining Equation (6.39) with Equation (6.1), and integrating with respect to p, we find that

pc 1 + r /+ 7? 2 - (6.49)

pksT (1 - ~7) 3

where pc denotes the pressure obtained from the compressibility equation of state. The virial equation of state is given by

pkBT P = 1 3kBT27rp r3 d ) g(r)dr - 1 + --if-- y(r)dr (6.50)

187

where f ( r ) - e x p [ - u ( r ) / k u T ] - 1 is the Mayer f-function and y(r), the cavity distribution function given by Equation (6.9), is continuous. For hard spheres, d f ( r ) / d r - 6(r - or), so the virial pressure is given by

pv = 1 + 4qg(a +) - 1 + 2q + 3q 2 (6.51) pkBT (1 - q)2

where, from Equation (6.27),

b 1 + rl/2 g(cr +) - h(cr +) + 1 - a + - - (6.52)

cr (1 - q)2

As is evident from Figure 6.1, the compressibility pressure overestimates the true pressure while the virial pressure underestimates it, with the former being the more accurate of the two. The need for an accurate equation of state for hard spheres was met partially by the Ree-Hoover (75) Pad6 approximant (76). In recent years, however, the more popular equation of state for hard spheres is that developed by Carnahan and Starling (77). These authors fitted an integer-valued function B(n) to the known n-th virial coefficients, then extrapolated the series to an infinite number of terms to obtain

P 1 -]- 7]-]- 7] 2 --[- 7] 3 = (6.53)

pkBT (1 - q)3

which is comparable in accuracy to the Ree-Hoover Padd approximant. The Carnahan-Starling (CS) pressure (denoted by p c s ) was subsequently found to be simply related to the PYA compressibility (pc) and virial (pv) pressures (cf. Equations (6.49) and (6.51) by (78)

pCS 2pc 1pv - + (6.54)

3 3

This is consistent with the observation that the compressibility and virial pressures bracket the computer simulation values.

In the case of the multicomponent hard-sphere system the compressibility pressure pc is given by

OP = 1 - 4 7 r

Op~

which, on solving for pc, yields

p c _ _ T r ( ~0

kBT 6 1 - ~3

M

E p, fo #=1

(6.55)

3~'1~'2 3~'2 + (6.56) ( 1 - [3) 2 ( 1 - [3) 3

188

and the virial pressure, pv, is given by (79)

kBTPV M ( 27r ) Pc 18 ~3~3 = ~1 p~ + ~ PZ9~Z(cr~+~) - (6.57) = Z=I kBT 7r (1 - ~ 3 ) 3

The same observations regarding the PYA for single--component hard spheres may be made for mixtures: the machine-simulation pressures are bracketed by P: and pv. The Camahan-Starling equation of state (Equations (6.53) and (6.54) has been generalized to mixtures as the Mansoori-Carnahan-Starling-Leland (MCSL) (80) equation of state, whose form is the same as that given by Equation (6.54).

6.3.1.2 Semiempirical Correction to the PYA: The Generalized MSA

The main deficiencies in the PYA 9(r) are the lowered first peak which, from Equation (6.50), will yield an underestimation of the vifial pressure, and the fact that the PYA 9(r) is slightly out of phase with respect to the simulation results, causing an overestimate of the main peak of the structure factor. The need for accurate correlation functions for the hard-sphere fluid has been met by the semi- empirical correction to the PYA 9(r) due to Verlet and Weis (70). Verlet and Weis corrected the underestimation of the first peak by adding a very short-ranged contribution ("~ 0 for r > 1.6a); the correlation function is adjusted to be in phase with the simulation results by using the PYA radial distribution function at a slightly modified density chosen to minimize the squared difference between the simulation and theory over the range 1.6or < r < 3or.

A em(r_~) [m(r or)] (6.58) _ gp (_,r _ c o s -

o r

where the parameters a and m are adjusted to yield the CS pressure and compress- ibility and ~7' - 6P[ 1 - ~r/] 3- The corresponding semiempirical expression for the direct correlation function was obtained by Grundke and Henderson (81).

c(r) = Cpv(r)[1 -- 0.127(po-3) 2] r < o- (6.59)

c ( r ) _ _B e2O(1_~) r > a (6.60) r

where parameter B is chosen from the condition imposed on the jump discontinu- ity of c(r) at r = a, viz. c(a +) - c(a-) = 9(cr+).

Finally Waisman (82) introduced the concept of the generalized MSA (GMSA) for hard spheres by combining the hard-core condition of h(r) with a

189

Yukawa form for c ( r ) outside the core, the latter being an ansatz for the correct hard-sphere c(r). Thus, the GMSA is defined by the closure

h ( r ) - - 1 r <

(6.61) ~(,) - K ~ - ~ / , / - - ~ ) / ( ~ / . ) , >

Note, that the latter equation has a form similar to that assumed by Henderson and Grundke. The parameters K and ~ are now regarded as density-dependent parameters whose values are set by the following two conditions,

b 1 - q / 2 g ( ~ + ) - a + - + 9 d ~ = ( 6 . 6 2 ) o- (1 - q)2

a2 = (1 + 2q) 2 - @ ( 4 - q) (6.63) ( 1 - ~ ) 4

where a, b, fl and d are parameters defined in Section 6.3.3 below. Equation (6.62) requires that the vifial pressure in the GMSA be equal to the Camahan-Stafling pressure, and Equation (6.63)requires that the compressibility in the GMSA match that of the Camahan-Starling equation. Thus, the two equations demand thermodynamic self-consistency in the GMSA (equivalence of compressibility and vifial pressures) plus agreement with the Camahan-Starling equation of state.

An analytical solution of the closure of the type Equation (6.61) is obtained in Section 6.3.3. Here we only note that GMSA for most densities pGa _~ 0.7 is in error by less than 1%, and at worst is in error by no more than than 3% (83). This is a significant improvement over the PYA.

Thus, by combining the Camahan-Starling equation for thermodynamics with the Verlet-Weis or GMSA-coGected PYA correlation functions one obtains a fully analytic and very accurate theory of the hard-sphere fluid. The availability of these analytic expressions has made a tremendous impact on the theory of fluids, since most perturbation theories ultimately require easily-calculated ha[d-sphere properties to make them computationally feasible.

6.3.2 Adhesive Hard-Sphere Fluid

So far the models considered in previous sections take into account only the ef- fects of excluded volume, the main contribution to structural and thermodynamic properties of dense fluids. However, in addition to short-ranged strongly-repulsive

190

interactions, the pair potential for any real fluid generally includes attractive in- teractions. A simplified model for a real fluid, which includes both types of inter- action, is the hard-sphere system with a square-well attraction. This potential can be defined by

u s w F ( r ) = ~ r < ~ l

= - e al < r < a (6.64) = 0 r > a

which has the Mayer f-function given by

f ( r ) - - 1 r < c r l = e x p ( e / k s T ) - 1 or1 < r < a (6.65) = 0 r > c r

Attempts to solve the MSA and PYA for this potential analytically using the fac- torization technique have not been successful. Baxter proposed an analytically tractable limit (al --+ a - while keeping the second virial coefficient fixed) of the square-well fluid (84), the so-called adhesive-hard-sphere fluid (AHSF). The AHSF is defined by its Mayer f-function

f ( r ) - - l + ( a / 1 2 z ) 5 ( r - a - ) r < cr

= 0 r > a (6.66)

The parameter r is a dimensionless measure of temperature and 5(x) is a Dirac delta function (85). The actual connection between r and temperature for a model of a given fluid can be established by equating the second virial coefficient for the AHSF with that of a square-well fluid (86). The PYA for the AHSF was initially solved by Baxter (84).

6.3.2.1 Solution o f the PYA f o r the Adhesive Hard-Sphere Fluid and Fluid

Mixtures

Within the PYA for the AHSF, c(r) - 0 for r > cr so that Baxter's factoriza- tion, Equations (6.26) and (6.27), is applicable with R - a. We require h(r) on the range (0, a), and from the cluster expansion for h(r) we expect that

~(~- ~-), h(r) - - 1 + --~ 0 < r < a (6.67)

191

where A is an, as yet unknown, dimensionless parameter. Substituting Equa- tion (6.67) into Equation (6.27), we obtain

Aa2[1 - 0(r - a-)] , q(r) - a(r 2 - o r 2) + b(r - cr) + 0 < r < c ~ (6.68)

where a - Q ( O ) - ( l + 2 r / - p )

( 1 - r/) 2

where O(x) is the Heaviside function,

b - 3 ~ + # , - = (6.69)

o- 2(1 - r/) 2

O ( z ) - 0 z < 0 = 1 x > l

and # - At/(1 - r/)

The parameter A follows from the quadratic equation

(6.70)

0 (I-~) ~ A+ (I-~)~-

which is obtained from the PYA-closure conditions written in the form

9 ( r ) - [1 + f ( r ) ]y ( r ) (6.72)

where y(r) = 9(r) - c(r) (6.73)

The analytic solution of the PYA for the AHSF mixture, developed indepen- dently by Perram and Smith (87) and Barboy (88), is a straightforward general- ization of the two analyses given here and in Section 6.3.1.1. The AHSF-mixture model assumes that the Mayer f-functions for the mixture are given by

f ~ ( ~ ) - - 1 + ( ~ o ~ / 1 2 ~ . ~ ) 6 ( ~ - < ~ ) ~ < ~

= 0 r > cr~/~ (6.74)

where c ~ is defined in Section 6.3.1 and ~-~ is both a measure of the temperature and the relative strength of the c~/9 attraction. [In fact, one can make the identifi- cation ~-~ - e ~ - , where ~- is the dimensionless temperature of the mixture and

192

c ~ is related to the well-depth of the aft interaction.] Correspondingly, the total correlation functions will be given by

- h ~ z ( r ) - - 1 + 12 r < cr,~/~ (6.75)

where the A~ have yet to be determined. The condition on c~(r) is the same as that for hard spheres given in Equation (6.44). The mixture factorization can then be applied and Equation (6.30) yields

a q~z(r) -- ~(r 2 - a2Z) + ba(r - cruz) -f A~Za2Z12 - ~ , u,~ < r < o-,~ (6.76)

where 1 - ~c3 + 3a~2 X~

- (6.77) as = (1 - ~3) 2 1 - ~a

2 3cr a ~2 aaXa b~ = - ÷ (6.78)

2 ( 1 - 2 ( 1 -

M 71"

-y--1 (6.79)

The parameters A~ are the solutions of coupled quadratic equations obtained, as above, by application of the PYA closure:

A.~-.~ - a . + + ~ z_, P-r (6.80) Oafl 7=1 O'a#

6.3.2.2 Equation of State for the Adhesive Hard-Sphere Fluid and Fluid Mixtures

The compressibility pressure is computed using the variational principle de- rived by Baxter (89) and is given by (84)

pc 1 + 7 + 72 - p(1 + r//2) + #3/367

pkBT (1 - 7) 3

The virial pressure is given by

(6.81)

p v

pkBT

[ II 1 1(2 ÷ 107)# ÷ #2 ( 1 _ 7 ) 2 1 + 2 7 +372 ~

193

Table 6.1 Critical constants from the compressibility and energy equations of state for the adhesive hard-sphere fluid. From Barboy and Tenne (90).

Property Compressibili ty EOS Energy EOS Tc 0.5858 0.7110 ~/c 0.1213 0.32

Pc/kBTcpc 0.3794 0.32

#3 1 _ r/ 1 - 5 r / - 5r/2 + 6 r / # - #2 + ~ (6.82) 3T(1 - ~7) 3

In the case of the energy EOS, it should be noted that it includes the hard-sphere pressure pHS as the arbitrary constant of integration and thus the pressure isotherm is not uniquely defined since it depends on the choice of the expression for the hard-sphere pressure.

R e _ p g S 6 ~ [ l n )~T(1 - r/) 4 r / - 1 pkBT = ( l - r / 2 ) ~ 3(2+~7) + 2 r/ f()~,r/)] (6.83)

where

f0 :~ d)~ f()~, r]) - )~ - 12~7 + 18 + 12(1 - r/)/r/ (6.84)

The latter integral can be evaluated analytically (90). Similar expressions for the EOS are also available for the mult icomponent case (90).

An important feature of the PYA for the AHSF is its prediction of a liquid-gas phase transition. After analyzing the quadratic equation, Equation (6.71) for A

^

and studying the locus of the points where a - Q(k = 0) = 0, which defines the spinodal curve, one can identify two temperatures, 7-1 and Tc, which have the following properties: (i) for 7- > 7-1, there are real solutions of Equation (6.71) for the whole range of the density, while for 7- < T1 there is a range of r] for which there are no real solutions. There is a density ~7~ = ~TG(T~) = ~]L(T1) where two possible real solutions from the low-density side ~TG and high-density side r/L merge; (ii) for 7- > ~-~, a > 0 for all ~7 and at 7- = ~-~ the equation a = 0 has a double root in density, r/~.

These two temperatures and densities coincide ~-1 = 7-~ = (2 - v /2) /6 = 0.0976, ~7~ = ~7c. Thus the point (7-~, r/~) can be identified with the critical point and its coincidence with (T1, r/i) means that the compressibili ty equation of state

194

1.2 . . . . i . . . . i , ' , , I . . . .

l[ ..... --Energycompressibility [ 1.0 t ] " , .

~/'rc 0.60"8 ,' , • ",

0.4

0.2 ",\

0.0 ' ' ' ' ' . . . . ' . . . . ' ' " ' ' '

0 1 2 3 4 P/P~

Figure 6.2 Vapor-liquid phase envelopes for the AHSF from the compressibility and energy equations of state. Note that the temperatures and densities are given in reduced units. After Barboy (90).

for the AHSF yields classical critical exponents (91), although as pointed out by Baxter (84) and other authors (92), the compressibility equation of state is not of the classical-scaling form near the critical point. Because of thermodynamic inconsistency the energy EOS, Equation (6.83), gives different values of the crit- ical parameters Tc and qc (see Table 6.1). The energy EOS also yields classical exponents but the EOS is of the classical-scaling form near its critical point.

In Figure 6.2, we present the phase diagram for the AHSF obtained by using the Maxwell construction

-

#(~-, ~G) - #(7-, 7/L) (6.85)

where r/G and qL denote the equilibrium densities of gas and liquid, respectively, and # is the chemical potential. It should be noted that the # used in this equation is thermodynamically consistent with the corresponding volumetric EOS (i.e., ei- ther the compressibility or energy expressions for both P and # are used in the equations to obtain the phase envelopes shown in Figure 6.2).

One cannot obtain solutions to Equation (6.85) from the virial EOS, Equa- tion (6.82). Solutions to Equation (6.85) can be found for the compressibility

195

EOS, but only in the range of temperatures 0.292rc < 7 < re, while solutions to the energy EOS can be obtained for all r < r~. The limitations on the compress- ibility and virial coexistence regions come about due to intersections between the coexistence region and the region inside which no real solutions exist.

6.3.2.3 Percus-Yevick Theory for the Hard-@here Chain Fluid

An interesting application of the AHS fluid model for the description of a fluid composed of athermal freely-jointed tangent hard-sphere chains has been pro- posed by Chiew (93-95). This is based on the "particle-particle" concept (96,97) and on the solution of the PYA for the AHSF.

The hard-sphere chain fluid consisting of Nc chains each composed of n hard- sphere units is recovered by the N-component (N = nNc) AHSF with the ap- propriate constraints imposed on the adhesive (frequently referred to as "sticky") interaction between different species. All N species are separated into Nc groups each consisting of n species, and adhesive interaction is allowed only between the hard spheres belonging to the same group. In addition, the adhesive interactions in the group are valid only between the particles of the first and second, second and third ..... n - 1 and n-th species of the group. The stickiness parameter, A~, Equation (6.75), is chosen from the condition that the number of nearest neighbors of a certain species bonded to the particle of the corresponding species is equal to unity

47rp~ fo ~ r2 g~(r) d r - 2~c~3~]3(~) 3 - 1.

where r/~ -- (Tr/6)p~. Moreover, to ensure that only linear chains are formed in the system, each species is represented by only one particle. The corresponding set of OZ equations, together with the PYA closure conditions can be solved in a closed form yielding an expression for the q-function similar to that presented in Section 6.3.2.1. In the simplest case of the system of one-component homonuclear chain molecules the compressibility EOS becomes

1 P 1 3q 3q 2 n - 1 1 + ~q = + + ] (1 - r/) 2 (6.86) pkBT [1 - r/ (1 - q)2 (1 - r/) 3 n

One can easily see that the terms in the square brackets represent the PYA com- pressibility pressure for the hard-sphere system, while the last term represents the contribution to the pressure due to the effects of bonding. Similar expressions were derived also for the heteronuclear chain fluids, for mixtures of homonuclear

196

and heteronuclear chains, and for homonuclear chains in a hard-sphere solvent. In the case of homonuclear chains of length n - 4, 8, 16, the results of the present theory have been compared with computer simulation results (98). The agreement between theory and simulation appears to be very good.

6.3.3 Hard-Core Yukawa Fluid

The hard-core Yukawa fluid (HCYF) is a model for a simple fluid that consists of a hard-sphere repulsion and a sum of tails of Yukawa form. The pair potential for the simplest one-tail one-component version of the model is given by

u H c y F ( r ) - - ~ r < cr

(6.87) = - D e - z ( r -~) / r r > a

Thus, unlike the case of AHSF, the attractive part of the pair potential for HCYF has two parameters, one defining the depth of the potential (D) and the other defining its range (z).

Although there is no real system which has the interparticle potential of the HCYF form, it takes into account the basic elements of the interparticle interaction in real potentials, i.e., short-range repulsion and long-range attraction. Therefore in the case of systems with strong attractive interactions, the HCYF is obviously a better candidate than the hard-sphere system for the role of the reference system in a perturbation theory (99). This, together with the availability of the analytic description via MSA, are the main reasons for the considerable interest in the HCYF.

6.3.3.1 Solution o f the MSA for the Hard-Core Yukawa Fluid and Fluid Mixtures

The MSA for the potential model, Equation (6.87), yields the following con- ditions on the total and direct correlation functions

h(r) - - 1 r < cr

c(r) - Le-Z(~-~)lr r > a

(6.88)

where L = D / k B T . The OZ equation with these closure conditions was originally solved by Waisman (82) using Laplace-transform methods to reduce the problem to a set of four non-linear algebraic equations. Waisman used this analysis to de- velop the generalized MSA for the hard-sphere fluid described in Section 6.3.1.3.

197

On the basis of a graphical analysis of the MSA, Henderson et al. (100) de- veloped low-density and high-temperature series expansions for the solutions to Waisman's equations, thus making it possible to distinguish the physically correct solution. HCye and Stell (101) simplified Waisman's equations significantly and identified the physically correct solution. HCye and Blum (102) studied the MSA for the HCYF using the Baxter factorization technique and reduced the problem to the solution of a single quartic equation. Cummings and Smith (103,104) also used the Baxter factorization approach, but were able to find a simplified quar- tic equation which can be easily analyzed to determine the physically meaningful root. In this section, we follow the analysis of Cummings and Smith.

The Baxter q-function, introduced via the factorization Equation (6.21) and relation Equation (6.22), is found to be similar in functional form to re (r ) in Equation (6.88), i.e.,

q(r) - qo(r) + a e -z (r -a) (6.89)

where qo(r) - 0 for r > cy and the expression for the parameter, a

L a - z Q ( i z ) (6.90)

which defines the long-range asymptotic form of q(r), is obtained using the long- range behavior of the direct correlation function c(r), Equation (6.88), and ana- lytical properties of the function (~(k). The factorized version of the OZ equation in r-space now reads

rc ( r ) - - q ' ( r ) + 27rp q ' ( t ) q ( t - r ) d t r > 0 (6.91)

/o r h ( r ) - - q ' ( r ) + 27rp q ( t ) ( r - t ) h ( I r - t l ) d t r > 0 (6.92)

To determine the functional form of q0 (r), we consider the latter of these equations on the domain 0 < r < or, which gives

qo(r) -- -~(r 2 - cr 2) + b(r - or) + ad[1 - e - z ( r - a ) ] (6.93)

where

fO ° a - Q(O) - 1 - 27rp q o ( t ) d t - 27rpaeZ~

Z

fO ° b - 27rp tqo( t )d t +

21rpae z~

Z 2

(6.94)

(6.95)

198

27rp d = 1 - .~(z) (6.96)

z

and .~(s) is the Laplace transform of rg ( r ) ,

{7(s) -- fo ~ r g ( r ) e - ~ r d r

The set of equations (6.90), (6.94-6.96) defines the unknown parameters a, a, b and d. An additional equation for these parameters follows from Equation (6.92) after rearranging it in terms of the function 9(r ) = h ( r ) - 1,

/or r g ( r ) -- ar + b + adze-Z(r-~)27rp q ( t ) ( r - t ) g ( r - t ) d t (6.97)

and, performing the Laplace transform of both sides of this equation, we obtain

r(a _jr_ 80")-]'-s 2-b ad] e-zo- O(z) - tV'I J 1 - 27rp~(z) (6.98)

Finally, after performing the integration in Equations (6.94) and (6.95) and some algebraic manipulations, the set of equations (6.90), (6.94), (6.95) and (6.98) re- duces to the following quartic equation

-36r/2/34 + X ~ 3 -- 12~7K/32 + K D f l - K 2 = 0 (6.99)

where/3 = a / a 2 and

18~ 2 q

[2 - ¢ - 2 ~ - ~ - ¢~-~] 6r/ [2 - ~2 _ 2 e - ~ _ 2~e-~]-~2( 1 _ r/) 2 D

72r/2 [ 2 - ~ - 2 e - ' ] - 216r/3 2 [ 4 - 4 ~ + ~ 2 - 4 e -~ +~2e - ' ] e - -6'qzr {2(1 _ r/) ~ 4 ( 1 - - 77)

72~72 [2 _ ~ _ 4e-¢ + 2e-2¢ + @-2¢] F - -6 (1 - e-~)2~7 + {2(1 _ 77)

216~73 [4 - 4{ + {2 - 8~-¢ + 2{:~ -¢ + 4~ -~¢ + 4{~ -:~ + {:~-:¢]

~ ~ 4 / 1 ~7i 2

36~72 [2 - 2 { - 2~-~ + {:~-¢] X - 6~7{e -{ - {~ (1 - 77)

- 108r/~ [2 - ~ " - 2 e - ~ - ~ 9 e - ~ ] , ~ ( 1 - , ~ ) ~. I. J

199

with ~ = zcr. The solutions of this quartic equation and their relationship to phase behavior are described qualitatively in section 6.3.3.2 below. Equation (6.92) can be written

fo G rh(r) - 27rp qo(t)(r - t)dt + 27rpae-z(r-G)eZSsh(s)ds

27rpc~e-Z(r-G) +adze -z(r-G) +

z 2

which permits calculation of the total correlation function using an extension of Perram's algorithm (83).

The solution of the MSA for the Yukawa fluid can be extended to the case in which the potential outside the hard-core region is given by a sum of Yukawa terms and to the multicomponent case. The solution of the MSA for a one- component system with two Yukawa tails was obtained by HCye, Stell and Wais- man (105) using Laplace-transform methods. For the general case of a multi- component hard-sphere system with MSA closure involving n Yukawa terms, the formal solution via the Baxter factorization method was developed by HCye and Blum (106-108) and Cummings (83). We will briefly outline the HCye-Blum so- lution here. We begin with the MSA closure conditions

h~m(r ) = - 1 r < cr~ m

n . ( i ) _ z i ( r _ a ~ Z ) / r r > (Tam c.m(r) - Ei=l L.me (6.100)

which extend Equation (6.88) to the multicomponent multi-Yukawa case. Here L~ m = D~m/kBT. An analysis similar to that carried out for the one-component case leads to the following expression for the Baxter q-functions,

q~m(r)_ ,~(o) k 1 (i) -z~(r-~.~) ~ (r) + - % ~ e i=1 zi

(6.101)

where

q(O) 1 n ~(i)

~m(r ) _ _~a~m(r_o. m)2+b.m(r_G.m)+ y~. ~ (e -z { ( r -~ .~)_ l ) i . m < r < o.m i=1 Zi

(6.102) q(0) ( r ) - 0 r > cr~ m (6.103) am

,~ (i) ~(i) satisfy a The unknown coefficients of the q~m(r), i.e. a~ m, b~ m, ~ m and ~ m ' set of nonlinear algebraic equations, which can be reduced to a set of equations

200

,, (i) ~(i) involving only the unknowns ~ and Note the slightly different functional ~ " O~/~ '

form for q(0) ~ (r), which corresponds to a minor redefinition of b~. The slightly different form for the Baxter q-function has some advantages for more complex closures such as the multi-component multi-Yukawa system considered here. This cumbersome set of equations is not reproduced here and the reader is referred to the original publications (106-108) for details, because it is beyond the scope of this review to report further on these developments. We shall complete this section by presenting briefly the solution for the model of a mixture which exhibits phase separation.

The model, which was introduced by Waisman (109), consists of an equimolar binary mixture (px = p2 = p/2) of species 1 and 2 molecules which are equal- diameter and have pair potentials

U i j ( T ) - - CX:) r < (71

= ( - 1 ) i + j + l D e - Z ( r - a ) / r r > O

(6.104)

1[hll (r) + hl2(r)] , h ( ~ ) -

-h(r)- -~l[hll (r) - h12 (r)],

1 c ( r ) - - ~[Cl1(7") -~- C12(T)]

1 - klX( ) -

we find that h(r) and c(r) satisfy the usual pure-fluid OZ equation with closure given by Equation (6.37). Thus, h(r) and c(r) are the same as for the hard-sphere fluid in the PYA at number density p. The h(r) and e(r) also satisfy the usual pure-fluid OZ equation but with closure

h(r) - 0 r < cr

~(r) - Le -Z ( r -~ ) l r r > cr

(6.105)

This is a coreless version of the HCYF, and the same analysis as that given in Section 6.3.3 remains valid, except that the a and b terms in the q(r) function are zero, which yields

-qo(r) =/3dcr 2 [ 1 - e-¢(r/~-l)] , 0 < r < o- (6.106)

so that like species interact via an attractive Yukawa potential and unlike species via a repulsive Yukawa potential. Introducing the quantities

201

where/3 is the solution of the quartic equation

36~2fl 4 -- 6~7@-~fl 3 + 12r/K/32 - K~fl + K 2 - 0 (6.107)

The solutions of this quartic equation are described qualitatively in section 6.3.3.2 below.

6.3.3.2 Equation of State for the Hard-Core Yukawa Fluid and Fluid Mixtures

The compressibility pressure of the HCYF in the MSA is calculated by inte- grating the equation

1 Opc[ - a 2 (6.108) kBT Op ]T

The virial and energy routes to the thermodynamic properties follow from the expressions derived in the case of MSA by HCye and Stell (110)

pv = 1 + 4~Tg(cr +) + J (6.109)

pkBT pe

= 1 + 2r] r _ _ /g2 (c r+) - - -'~[gHs(cr+)]2) -k- J (6.110) pkBT

where f oo j _ 27cp ra du(r)

3kBT dr g(r)dr

and the H S subscript refers to hard-sphere quantities. For the case of the Yukawa fluid, J can be written in terms of ~(s) as

J-4~TKe~[zd{7(s) I - ~(z)] (6.111) ds s:z

Similar expressions can be written also for the general case of a multicomponent multi-Yukawa system. Here the inverse isothermal compressibility is defined by

1 Op c aa~ 2 - ~ x~(--~ ) (6.112)

kBT Op y

while the virial and energy pressure follow from the corresponding expressions of HCye and Stell (108,110), i.e.,

pv 2~ x~x~%~g~(%~) + J (6.113) 1 + --=-p ~-" a +

pkT = 5 " - - - " az

202

R e

pkBT 3 = pUS + 3PEX~Xza~z{[g~z(a+Z)]2 [g~H~(a~Z)]2} + J

a~

where x~ is the mole fraction of the species a,

(6.114)

271

a ~ i s - - z i

The thermodynamic properties of the Yukawa fluid have been discussed at length by Henderson et al. (111), Cummings and co-workers (103,104) and Arri- etta and co-workers (112,113), and its critical properties by Cummings and Stell (91). From the expressions for the pressure, Equations (6.108)- (6.114), it follows that the thermodynamic properties of the HCYF are defined by the parameters in the Baxter q-function. These parameters are determined by a set of nonlin- ear algebraic equations, which in simple cases can be fully analyzed, providing some insight into the peculiarities of the thermodynamic behavior of the model. In the case of a one-component one-Yukawa fluid, the set of equations reduces to a single quartic equation, Equation (6.99). For K > 0 (attractive potentials), this equation has two real and two complex roots. To leading order in density, the two real roots are given by (103,104)

K ~e -~ 91 -- ~ "ll- 0(77), /~2 = - - ~ + 0(7/0). (6.115)

Clearly, ill, the smaller of the two roots, is the physically meaningful root since this is the root which goes to a finite value as r/--+ 0 and to zero as T ~ oo, thus yielding the correct low-density and hard-sphere limits respectively. This is strik- ingly similar to the behavior of the two roots of the quadratic equation obtained for the AHSF in the PYA described in Section 6.3.2. Thus, as in that section, we can compute the locus of two lines: L1, the line along which a -+ 0 and the compress- ibility diverges; and L2, the line along which the two real roots of Equation (6.99) are equal. We can also compute, for a fixed value of ~, the dimensionless critical temperature (lIKe) of the HCYF as the temperature below which there is a range of r/for which a < 0. Likewise, l /K1 is a value of the dimensionless inverse tem- perature below which there is a range of ~/where Equation (6.99) has no real roots. As in Section 6.3.2, there are corresponding densities, r/c and ~71. Unlike the case of the AHSF, the critical temperature and density (1/Kc, r/c) do not coincide with

203

1 . 4 . . . . i . . . . ! . . . . ! . . . . i . . . . i . . . . ! . . . . i . . . .

1"3 I

1.2 "" T*

1.1

1.0 N

O. 9 "~,

0.8

0.7 0.0 O.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

p*

Figure 6.3 The liquid-vapor phase envelope for the Lennard-Jones (LJ) model fluid calculated from the MSA for the HCYF (solid line), from the PYA for the LJ fluid using the energy EOS (dashed line) (114), from the RHNCA approximation (dotted line) (115), and from Monte-Carlo simulation (diamonds) for the LJ fluid (116-118).

(1//£1, ~]1). The consequence is that for the HCYF the thermodynamic behavior near the critical point of the compressibility equation of state yields spherical- model exponents (7 - 2, 6 - 5) rather than classical critical exponents (91). A similar analysis can be carried out for the case of the two-component HCYF con- sidered in the previous section. Again, for Equation (6.107) there is a locus of

t . .

points in (/3, r/) space along which Q0(0) - 0. Thus, as one increases the den- m

sity along an isotherm for this fluid, one reaches a spinodal point at which h(r) becomes long ranged, and the fluid phase separates. Since there is no phase be- havior associated with h(r) and c(r) (they are simply hard-sphere quantities) the separation can be regarded as a fluid-fluid phase separation.

There is a sizable literature dedicated to the investigation of the analytical structure of the MSA for HCYF (99,101,103-105,112,119-127) and to its appli- cation to the description of different systems and physical phenomena, (128,129). However, it is beyond the scope of our study to discuss and analyze all possible applications of the HCYF. Rather, we will conclude this section by demonstrat- ing the possibilities of the HCYF and MSA in the description of a system with a realistic potential. In Figure 6.3, we present the liquid-gas phase diagram for

204

the LJ fluid (116-118). We also present the RHNCA and PYA results obtained from the energy route for the same LJ fluid (114,115) and results of the MSA for the corresponding HCYF (130). The parameters of the HCYF model have been chosen to approximate the LJ potential. The agreement between the MSA results and the results of the MC simulation is in general good and is comparable to that for the PYA and RHNCA.

6.3.4 Charged Hard-Sphere Fluids

An important model, which belongs to the class of analytically solvable mod- els, is the primitive model for ionic fluids. It is defined as an n-component mixture of charged hard spheres of species 1, ..., c~, ..., n with number densities p~ and the following pair potentials

u ~ ( r ) = cc r < cr~

- r > o - ~ ( 6 . 1 1 6 ) er

where e is the elementary charge, Z~ is the charge of the ion of a species in electron units and c is the dielectric constant of the continuum. In spite of its sim- plicity this model has been widely used to study ionic solutions (in which case represents the solvent dielectric constant), molten salts and liquid metals. Unlike the point-charge model used in the classical Debye-Htickel theory of electrolytes, the primitive model takes into account the effects of excluded volume. This, to- gether with the availability of an analytical solution in the MSA (which properly describes the hard-sphere repulsion), are the main reasons for the considerable in- terest in the primitive model, Equation (6.116). The MSA for the primitive model satisfies exact Onsager bounds (131) on the Helmholtz energy and on the internal energy, as well as the Stillinger-Lovett moment conditions (132).

In this section we present the analytic solution of the MSA for the primitive model (PM) of electrolytes and the EOS which follows from it. At the end of this section we briefly discuss corrections to the MSA which extend its applicability to ionic systems under the conditions of high coupling (low temperature or low dielectric constant) and low density.

6.3.4.1 Solution of the MSA for the Primitive Model of Electrolyte Solutions

We consider first an analytical solution of the MSA for the restricted primitive model (RPM) of electrolytes. This model consists of a mixture of two species

205

of equal diameter, equal charge (-- q - Z e , where Z is the valence and e is the electronic charge) hard spheres. In order to solve the MSA for this model, we begin by defining a slightly different model, v i z

~ ( ~ ) - ~ ,. <

= ( - 1 ) ~ + Z q 2 e - Z ( ~ - ~ ) / c r r > cr (6.117)

so that the RPM is recovered in the limit z --+ 0. Just as in Section 6.3.3.1, we define the correlation functions h ( r ) , c(r), h ( r ) and e(r) and find that the mixture OZ equations separate into two OZ equations, one for h ( r ) - c(r), the other for _ R

h ( r ) - -~(r). The h ( r ) and c ( r ) are given by the PYA for hard spheres; the h ( r )

and ~(r) satisfy the usual OZ equation with closure

(6.118)

m

h ( r ) - 0 r < cr

q2 - z ( r - a ) / r r > (r - e ( r ) = - ~ k ~ e

We are interested in the limit z --+ 0, in which case Equation (6.107) becomes

- - 0 ~ / 3 - - 6r/ (6 .119)

where q 2

K - (6.120) c k B T ( y

Equation (6.119) leads to expressions for cij(r) that agree with those obtained by Waisman and Lebowitz (133,134) using Laplace transform techniques. Here the Baxter q-function, ~0(r), has the form

-qo(r) -- I t~ ( r - or) (6.121)

where

~2 _ 4 7 r p K , 27rpI - l + c r F

2For - (1 + 2ha)1 _ 1

The analytic solution of the MSA has been extended by Blum (135) by ap- plying the Baxter factorization technique to the general case of any numbers of components with arbitrary charges (subject to electroneutrality) and hard-sphere

206

diameters. In this general case it is not possible to decouple the initial set of the OZ equations into independent sets of equations by using functions defined as in section 6.3.3.1. The method of solution is quite similar to that used to solve the MSA for the multicomponent HCYF. For the general case the MSA closure conditions are

h a z ( r ) - - 1 , r < a~Z

e2ZaZ~ e -z(r-a~) r > cr~/~, (6.122) c.~(r) - eksT

where we consider the limit z --+ 0. The system satisfies the condition of charge neutrality

n

~_, p~Z~ - 0 (6.123) a = l

The final solution of the problem can be expressed in terms of only one parameter, F, which is the solution of the algebraic equation

4F 2 - a ~ p~ OL=I 2A(1 J

(6.124)

where rc p~a a l ~ p ~ a ~ Z ~ ~ - 1 +

Pn - ~ 1 + a . F ' ~ 1 + a . F '

6 ' ckBT

The Baxter q-functions are given by

12(r/~r/z)½q~z(r) - 12(rl~r/z)½q(°)(r)- Z~az e -~('-~"~) af~ (6.125)

(o) l ( r )2q~

" (1 + q ~ -

q ~ - -~ 2 a ~ + a~a~2 ~ - -~ Da~a~,

4F 2 a 2 2AZ~ - rcPna~ D - a 2 , a s - 2F 2 A ( l + a ~ r )

in the limit (# --+ 0).

207

6.3.4.2 Equation of State for the Primitive Model of lonic Fluids and Fluid Mixtures

The thermodynamic properties of the primitive model of ionic fluids are calcu- lated using the scheme proposed in (136,137). In the case of a restricted primitive model, Equation (6.117), the virial and energy routes to the thermodynamic prop- erties yield

pv _ PI~S 1 E - E id

pkBT 3 pkBTV

pe __ Pus

pkBT

(6.126)

F3 = (6.127)

37rp

where E - E id is the residual internal energy

E - E id q2 F

pkBTV kBTe 1 + crF

In the general case of the unrestricted primitive model, Equation (6.116), we have, (136,137), that

p v _ PIYlS 1 E - E id = - (6.128)

pkBT 3 pkBTV

where

p e PHS F3 o~2 p2

pkBT 37rp 8p A2 (6.129)

_ _ _ n 71" 2 E E id e2 ~ 1 PaZ2 -Jc" kBrV - -ekB-------T [r 1 + o-~r - ~ aPx]

The compressibility pressure follows from the integration of the expression

kBT

n OP ~ _ _ 1 Q2

Op r 4~o 7r2~=1 ~ p~ ~ (6.130)

where -~-2~ [1 + (2o~ ~ ~ 1 Q~ - + ~a~P~]

OZ 2 71"

~° = 2 r (1 + ~ r [z~ - P . ~ ~-£]

The thermodynamic and structural properties of the primitive models, both re- stricted and unrestricted, of ionic systems predicted by the MSA have been studied

208

1 . 4 , , , , , , , i , , , , , , , , , , , , , , , , , , ,

1 . 3

1 . 2

1 . 1

1 . 0

0 . 9

0 . 8

0 . 7

0 . 6

0 . 0

1-1 s a l t

1 -2 S a l t

1-3 S a l t

/ f H N C A

• M S A

, , , I , , , I , , , I , , , I , , , I , , , I , , ,

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4

c ( m o l / l i t )

Figure 6.4 Comparison of the osmotic coefficient of the primitive model calculated from the MSA (solid circles) and the HNCA (lines). For the 1-1 and 1-3 salts, the cationic and anionic diameters are equal (3.6~ and 2.14 respectively). For the 1-2 salt, the cationic diameter is 3.6~ and the anionic diameter is 5.76~. After Blum (138).

in detail (139-149). The results of these analyses show that the MSA gives accu- rate predictions for the thermodynamics of systems with low values of the inverse temperature/3* - o~21Z+Z - ] and/or at high densities and is unsatisfactory at high values of/3* and low densities, which corresponds to high charge, low tempera- ture, low ~ or any combination of these conditions. It is also found that the energy EOS provides the best predictions for thermodynamic properties. In Figure 6.4, the osmotic coefficients for 1 - 1-, 1 - 2- and 1 ,3 -va lence salt solutions obtained from the energy EOS (138) are presented and compared with HNCA results, the latter being in good agreement with Monte-Carlo simulation results. One can eas- ily see that the agreement becomes less satisfactory with increasing charge. In the case of 2 - 2 and higher-valence electrolyte solutions, the results are rather poor as is evident from Figure 6.5.

In Figure 6.6, we compare the phase envelope of the restricted primitive model obtained from the MSA using the energy EOS with that obtained by molecular simulation (154,155). There is clear evidence that the MSA result differs signif- icantly from the simulation result with a critical temperature which is almost a factor of two larger than the simulation result. This is not surprising given that

209

1.0

0.9

0.8

0.7

0.6

0.5

. . . . i . . . . i . . . . i . . . .

~ , . . • M o n t e C a r l o

" ~ , . ,. - . . . . M S A

" ~ , _ ~ ,. ~ A M S A

" - . . . ~ • - - - - - H N C A n d e r s o n

• ~ ~t d

I , ~ i i I , i , , I , , i

-3 -2 -1 log(c)

Figure 6.5 Osmotic coefficient of the restricted primitive model at/3* = 6.8116. The solid line refers to ASMA results obtained from the energy equation. After Kalyuzhnyi and Holovko (55).

the phase envelope of the restricted primitive model is at very low temperature and density, precisely the domain on which, as we have already noted, the MSA becomes quite inaccurate.

Because of the rather high accuracy of the MSA for 1 - 1 and 1 - 2 electrolyte solutions, a number of researchers have modeled the properties of real electrolyte solutions (144-147,156-158) using the MSA as the basis for the thermodynamic properties.

In general, unlike the case for thermodynamics, the MS A does not appear to be a reliable source for structural properties. Comparison of MSA radial distribu- tion functions with Monte-Carlo simulation results yields rather poor agreement. In particular, the pair-distribution function for the equally charged ions becomes negative at short distances, which is clearly an unphysical result. This behavior is a consequence of the linearity of the MSA.

6.3.4.3 Corrections to the MSA: the Generalized MSA and the Associative MSA

As was already noted in the previous section, the predictions of the MSA be- come rather poor for high values of/3* and low concentration. Several different schemes have been proposed to extend its range of applicability. To correct the

210

0.08

0.07

0.06

T*

0.05

0.04

• ' ' 1 ' ' ' 1 ' ' 1 ' ' " 1 ' • ' 1 ' ' ' 1 ' '

ss

- . S#S s ~ /s tl i |

1E-0()7 1E-005 0.001 0.1 Reduced density

Figure 6.6 Phase envelope of the RPM as predicted by the AMSA (solid line), MSA (dashed line 1), MSA3 (dashed line 2), the version of the Fisher-Levin theory (150) with the consistent account for the hard core contribution (dashed line 3) and by Monte-Carlo simulation (symbols) where 7* = 1//3". AMSA results are after Kalyuzhnyi (151), MSA3 results are after Stell et. al (152) and results of the Fisher-Levin theory are af- ter Guillot and Guissani (153). MC results are after Caillol (154) (solid diamonds) and Panagiotopoulos (155) (open cirles).

MSA for structural properties and for thermodynamic consistency, the concept of the GMSA (82,159), has been used by several authors (160-162). An impor- tant feature of this approach is that the GMSA preserves the fulfillment of the Stillinger-Lovett moment conditions (132). Another possibility for the correc- tion of the MSA is the EXP approximation of Andersen and Chandler (163,164). Although the EXP approximation leads to an overall improvement for the struc- ture results, it violates the Stillinger-Lovett moment conditions. In addition, t h e thermodynamics of the EXP approximation cannot be expressed in analytic form. Larsen et al. (165) proposed the so-called truncated-gamma second-order approx- imation (TG2A) in which the MSA Helmholtz energy is corrected by adding the second-order chain diagram. This leads to an analytical expression for the osmotic coefficient,

~TG ~___ (~MSA __ St (6.131)

211

where s ' - ~(1+ ~) X

12s(1 + 2~)

{20z 2 - 12z + 3 + [4(4z - 1) - sin(2z) - (8z - 1)cos(2z)] e -2z }

and z - o-F. This approximation extends the application of the MSA over a relatively large region of/3* and concentrations.

The poor performance of the MSA is manifested in systems with large val- ues of/~*, which means that such systems are characterized by an appreciable degree of ionic association. Thus, one mechanism for extending the applicabil- ity of the MSA is to explicitly account for association effects. Several studies have been initiated with this aim. These studies fall into three general categories. In the first category, the corrections to MSA are introduced by adding an addi- tional adhesive term into the closure conditions (see reference 166 and references therein). Following the concept of the GMSA, this adhesive term can be chosen to reproduce additional information about the system, such as known thermo- dynamic behavior. However, as was already noted in Section 6.1.2, the regular OZ closure approximations, with their close relation to the Mayer p-expansion in terms of a density, are not well suited to treat highly associating systems. In the second category, a description in terms of both the density and the activity is used. This type of approach was initiated by Bjerrum (167) who combined the law of mass action with the Debye-Htickel theory of electrolytes. Several [168, 169, 170, 171,172, 173, 174, 175, 152] authors (152,168-175)extended this concept of correcting the Debye-Htickel theory for the effects of ionic association but used the MSA solution in place of the Debye-Htickel theory.

As noted in Section 6.1.2, a consistent integral-equation theory for associat- ing liquids has recently been proposed (45-48). This is based on the multidensity formalism in which the description in terms of activity and density expansions are combined. In references 40, 49 and 50, the multidensity formalism was re- formulated and presented in a form suitable for the study of association effects in ionic fluids; this is the third category of approaches to incorporating associ- ation. The simplest two-density version of the latter theory has been applied to correct the MSA predictions for thermodynamic properties of the RPM of elec- trolytes (55) and for its liquid-gas phase diagram (151). An analytic solution for the two-density MSA (or associating MSA (AMSA)) (53,176) was obtained and closed form analytical expressions for the thermodynamic properties has been derived (151,177,178). We will present here only the final results for the pres- sure of the 2-2 electrolyte solution and liquid-gas phase diagram of the RPM as

212

predicted by the AMSA. These are shown in Figures 6.5 and 6.6 where the re- suits of the theory(55,151,178) are compared with predictions of the computer simulations, regular MSA and other currently available theories. The agreement between AMSA, the cluster theory of Tani and Henderson (169) and the Monte Carlo results in the case of the osmotic coeficcient is reasonably good, especially at lower concentrations, while the MSA results are rather poor. The prediction of the AMSA for the coexistence curve is more accurate than those of the MSA and pairing MSA3 of Zhou et al.(152) and is of the same order of accuracy as that of Fisher and Levin with a consistent account of the hard core contribution (153). Thus, use of the AMSA enables the range of applicability of analytic theory to be substantially increased.

6.4 EQUATION OF STATE FOR ANALYTICALLY-SOLVABLE MODELS OF MOLECULAR FLUIDS

Thus far in this chapter, we have considered IEAs for monatomic fluids- i.e., fluids for which the intermolecular pair potential is spherically symmetric and thus depends only on the distance separating the two molecules. More complex models are required to describe molecular fluids. In general, each polyatomic molecule has 3n degrees of freedom where n is the number of atoms in each molecule. These include the translational, rotational and intemal degrees of freedom. In most cases of interest, the intramolecular interactions are much stronger than in- termolecular interactions and, to a high degree of accuracy, one can neglect the intramolecular degrees of freedom and assume that the molecules are rigid. The pair potentials, and thus the pair-correlation functions, for rigid nonspherical par- ticles are functions of the coordinates which specify the positions and orientations of the two molecules.

There are two formalisms for describing the structure and thermodynamic properties of molecular fluids. The first, the so-called molecular formalism, is based on the extension of the OZ equation and closures used for monatomic fluids by explicitly taking into account the orientational dependence of the pair potential and correlation functions. Although such an extension is formally rather straight- forward, the resulting IEAs are much more difficult to handle both numerically and analytically. In addition, to perform calculations in the molecular formalism, one needs to expand the potential and correlation functions using the invariant expansion (see the Appendix). Truncation of the invariant expansion at a finite number of terms leads to additional approximations beyond those of the IEAs themselves. The second formalism, the so-called interaction-site formalism (ISF),

213

was developed by Chandler and co-workers(179,180). Some important limitations of early versions of the ISF are discussed by Cummings and co-workers (96,181- 183). Within the ISF it is assumed that the intermolecular pair potential can be ex- pressed as a sum of spherically-symmetric site-site potentials and that the structure and thermodynamic properties are described in terms of the site-site correlation functions. In spite of the limitations imposed by these assumptions in the model, this approach is quite general and can be applied to a wide range of molecular fluids. The sites are usually associated with the atoms comprising the molecules. Because of the spherically-symmetric character of the site-site interactions- and thus the site-site correlation functions - the corresponding IEAs are much eas- ier to handle than their molecular-formalism counterparts. However, on the other hand, the site-site correlation functions contain less information about the struc- ture of the system than the molecular-formalism correlation functions. Knowledge of the molecular-formalism correlation functions allows one to calculate any site- site correlation function, while it is not possible to construct molecular-formalism correlation functions rigorously from site-site correlation functions.

In Section 6.4.1 below, we discuss analytically solvable models for molecular fluids within the ISF, while in Section 6.4.2 models described by the molecular IEA will be considered. For the sake of simplicity, in each case the formal results will be presented for the one-component case, though extensions to mixtures are possible.

Several reviews have appeared on the statistical mechanics of molecular fluids in general (184) and on interaction-site fluids in particular (185) Accordingly, our presentation of formal results will be very brief and we will focus mainly on the recent developments in this field.

6.4.1 Interaction-Site Model Fluids

As was already noted above, within the ISF one assumes that the intermolec- ular pair potential u(12) can be represented as a sum of the site-site spherically- symmetric potentials u~z(r~z)

u(12) = ~ u~z(r~z) (6.132) az

where 1 and 2 stand for the spatial and orientational coordinates of molecules 1 and 2, r ~ is the distance between the site a in molecule 1 and site/3 in molecule 2. The structural properties of the model are described by the site-site pair- distribution functions 9~(r), which are proportional to the probability density

214

of finding a site a of one molecule and site/3 of another molecule separated by the distance, r, regardless of the orientations of the two molecules. Thus the site- site correlation functions contain less information on the structure of the system than the molecular-formalism correlation functions. However, unlike the latter, they can be compared directly with the results of X-ray- and neutron-scattering experiments.

The thermodynamic properties of the system can be obtained from the site-site pair correlation functions using the energy route, i.e.,

ures m f0cx ~ N = 27rp~ 9~( r )u~( r ) r 2 dr (6.133)

where U res is the residual internal energy, m is the number of sites in the molecule and p is the molecular number density, or the compressibility route

Op f0 ~ kBT ~ - 1 ÷ 47rp [ 9 ~ ( r ) - 1]r 2 dr (6.134) V,T

where a and/3 denote any pair of molecular sites. Note that Equation (6.134) im- plies that the volume integral of site-site correlation functions h~z(r) = g~z(r)- 1 is independent of both a and/3. This independence, known as the compressibility theorem, causes singular behavior in the site-site direct correlation functions de- fined by the site-site OZ equation given below and lies at the heart of many of the difficulties associated with the ISF (96,181-183).

The virial route to the pressure cannot be expressed in terms of site-site cor- relation functions only. Additional information is needed concerning the orienta- tional structure of the system.

6.4.1.1 Site-Site Ornstein-Zernike Equation and Closure Conditions

An integral-equation theory for calculating the site-site correlation functions has been proposed by Chandler and Andersen (186). It is based on the so-called site-site Ornstein-Zernike (SSOZ) equation, which can be derived using the same arguments as those utilized in the case of the OZ equation in Section 6.1.1. The total site-site correlation function h~(r) = g~(r) - 1 can be decomposed into a sum of direct correlations c~z (r) and all possible indirect correlations presented by the convolution integral

h~z(r) - ~ /w~(r~.y)c~(r~)w~z(rsz) d~'~d~'5+ ,y5

215

+p ~ f w~(r~)c~6(r~6)h6z(r6z) d -~ d -~6 (6.135)

Here, unlike the case of the regular OZ equation, the correlation is propagating also via the intramolecular distribution function

1 w~z(r) - 47rL~ 6(r- L~Z) (6.136)

where L ~ is the distance between the sites a and /3 in the same molecule. The same SSOZ equation (6.135) can be obtained using different arguments (96,187,188)

The closures typically used for the SSOZ equation (6.135) are straightforward extensions of the PYA, HNCA and MSA (186,189-194). They can be presented in terms of the site-site correlation functions as follows

g~(r) - [9~(r) - c~(r)]e -~(r)/kBT, (PYA) (6.137)

g~(r) - e - u ~ / k B T + h ~ ( r ) - % ~ ( r ) , (HNCA) (6.138)

h~z(r)--1, r < a~Z

c~z(r)- -u~z(r)/kBT, r > cruZ, (MSA) (6.139)

In the case of the MSA, it is assumed that the site-site potential involves the hard- sphere site-site interaction.

These IEAs appear to be relatively successful in predicting the short-range structure of a number of different site-site models. However there are several im- portant defects in the SSOZ formalism. As in the case of the usual OZ equation, the SSOZ equation, equation (6.135), can be viewed as a rigorous equation which defines the site-site direct correlation functions c~z(r). However such a direct correlation function does not have the same status as that of the direct correla- tion function introduced in Section 6.1.1. The latter represents a specific subset of the diagrams contributing to the total pair-correlation function which is quite different from the former case. The result is that the exact site-site direct corre- lation functions, as defined by (6.135), do not in general, satisfy, the long-ranged asymptotic relation c~(r) -+ -u~(r)/kBT as r -+ oc as is embodied in the closure conditions (6.137)-(6.139) (96,183,184). The use of the PYA, HNCA and MSA in the SSOZ formalism can lead to serious errors in the prediction of long-ranged orientational quantities such as the dielectric constant and the Kerr constant (96,183,184). In addition, the site-site correlation functions as predicted

216

by the SSOZ theory depend on the presence of 'auxiliary' sites, which do not contribute to the site-site potential (181,195).

Analytical solutions of the SSOZ equation have been obtained for a number of different simple models. The SSOZ-PYA has been solved for the symmetric n-atomic fused-hard-sphere fluids (in which each molecule consists of n equal- diameter fused hard spheres in geometries of maximum symmetry such as equi- lateral triangles and regular tetrahedrons) (196-200) and their mixtures with the hard-sphere molecules (201). The SSOZ-MSA was solved for the polar version of the n-atomic symmetric fused-hard-sphere molecules (194), for their mixtures with hard-sphere ions (202), and for the fused-hard-sphere diatomics with Yukawa attraction (203-205). A general analysis of the class of analytically solvable site- site models is provided by Cummings and Stell (96).

In the next two sections we will briefly present the results of the analytical so- lutions of the SSOZ-PYA and SSOZ-MSA for the symmetric fused-hard-sphere nonpolar and polar diatomic molecules. Analytical solutions for these models were recently reviewed in reference (185) and we refer the readers to this publi- cation and to the original papers for more details.

6.4.1.2 SSOZ-PYA for the Fluid of Fused Hard-Sphere Diatomic Molecules

Let us consider a fluid consisting of diatomic molecules with a hard-sphere site-site interaction of diameter R = 1 for each site and intramolecular distance between sites L. Since the two sites are equivalent the SSOZ equation (6.135) for this system can be written as a single scalar equation

h(k) - (1 + cb(k))~(k)(1 + cb(k)+ 2ph(k)) (6.140)

where h(k) and ~(k) are the Fourier transforms of the total and direct correlation functions respectively, cb(k) -- sin kL /kL and p is the number density of the molecules. The SSOZ-PYA for the present system takes the form

h(r) = - 1 , r < 1

c(r) = 0, r > 1 (6.141)

where for convenience the hard-sphere diameter has been taken to be unity. The Baxter q-function can be introduced in a way similar to that discussed in Sec- tion 6.2.1. The final result for the case, L - ½, is

1 q(r) -- ~ (--anr + -~an -- bn) -q- Re(Poe-it)-+ -

n=O

217

(9O + Z R~{[~

e-iA,~ e-i,kn r 1 dn(iAn + e- l ix")] l~_ )~, 0 < r <

71---7-- n--1 1

q(r) - Z (a~r + -~a~ + bn) + R ~ ( i P o ~ - ~ r ) + n-O

oo i)~n e-i)'" Ca 1 - i/~ne½ ix" 1 + n=lE/~C{[ ~ =-~n 71"/9 t-( 1 - /~n2 )dn]e- i )~nr} , -2 < r < 1

oo q ( r ) - -~pl n~ Re(~ne_iX.~ ), r > 1 (6.142)

where ~,~ - i[©'(k)Q(-An)] -1, Q(k) - 1 - 27rp f ~ q(r)e ikr dr, An is the solution of the equation 1 + &(An) - 0 and

an -- --4/~C(/~n , bn -- -41i~e[~nn (1 n I- i)kn)e_,-iA'~],

/oo d n - - 4 R e ~ rg(r)e -ix'~ dr

The parameters a0, b0, P0, dn and ~ are obtained from the solution of a set of nonlinear algebraic equations (196). To utilize this solution one needs to truncate the infinite sums in the above expressions. The investigation of the influence of the number of terms included in Equation (6.142) on the solution of the SSOZ, equation (6.140), has been carried out (198-200). It was demonstrated that, in the region of liquid-state densities, the simplest so-called zero-pole approximation (ZPA), in which all the terms with n > 0 are neglected, yields accurate predictions for structural properties. The solution has also been applied to the case of different values of L and for models consisting of three or four hard spheres fused at a distance L to form an equilateral triangle or a regular tetragon.

The EOS for these models can be obtained by integrating equation (6.134) or equivalently from the expression for the bulk modulus

1 OP - 1 - p ~ f c~z(r ) d F - 2[Q(k - 0)] 2 (6.143)

k sT Op T aZ

The thermodynamic properties of the present model, as predicted from the compressibility route by SSOZ-PYA, have been studied by Lowden and Chan- dler (190). In general the results for the pressure do not agree satisfactorily with the corresponding Monte-Carlo simulation results. This conclusion is valid also

218

for the LJ diatomics studied using the SSOZ-PYA (206) and the SSOZ-HNC ap- proximation (207). In contrast, the energy route, equation (6.133), to the SSOZ- PYA pressure for the LJ diatomics gives reasonably good agreement at liquid- state densities (208). Thus, for the present model, the SSOZ-PYA, while being relatively accurate in predicting the structural properties, is not a reliable method for calculating thermodynamic properties. This situation is different in the case of the SSOZ-MSA which will be considered in the next section.

6.4.1.3 SSOZ-MSA for the Fluid of Fused Charged Hard-@here Diatomic Molecules

For the polar molecular fluid with molecules composed of two equal-diameter oppositely-charged hard spheres (the polar homonuclear hard-dumbbell fluid), the SSOZ equation, Equation (6.135), can be decoupled into two independent scalar equations by introducing the sum and difference correlation functions

1 h~(r) - -~(h++(r) + h+_(r)),

1 hd(r) -- -~(h++(r)- h+_(r)) (6.144)

The corresponding SSOZ equation and closure conditions for the sum correla- tion functions coincide with the SSOZ equation and the SSOZ-PYA considered in the previous section, while the SSOZ equation and closure conditions for the difference correlation functions are

[1 - & ( k ) + 2phd(k)][(1 - & ( k ) ) - 1 - 2pOd(k)] = 1 (6.145)

and hal(r) = O, r < 1

q2 c d ( r ) - kBTr' r > 1 (6.146)

where the hard-sphere diameter is taken to be unity. The factorization of this equation has been carried out by Morriss and Perram (194). The final expression

1 for the q-function for the system with L - ~ is

q(r) = 2ZJo + p cosr + v sin r, 1

O < r < - 2

Z q(r) = -2ZJo + -27rp

1 1 - p s i n ( r - ~) + v cos(r - ~),

1 - < r < l 2

219

q(r) = 0, r > 1 (6.147)

where z = 6 / L 2 + 87rp/32q 2 and the parameter Jo = f ~ rhd(r) dr is the solution of the quadratic equation

z 576~72z(1-C)J~ + 1 2 ~ 7 ( 2 z ( S + 3 C - 3 ) + ( 2 - C ) ] J o + - ~ ( 5 - 3 S - 4 C ) + ( C - 1 ) - 0

(6.148) 1 1 where ~ - 7rp/3, C - cos 7, S - sin 7 and expressions for the remaining param-

eters are presented in reference 194. Here the analogue of the ZPA discussed in the previous section is used. The thermodynamic properties of this model have been analyzed using the energy route by Morriss and Isbister (209,210). The residual internal electrostatic energy, U res, is related to the parameter J0 by ures /NkBT = 24r//3J0q 2. Using this expression and the 'charging' procedure for calculating the residual Helmholtz energy (209,210), the following expression for the pressure has been derived

pe Po 1 12~flq 2 = + G'(z) + G(zo) - G(z)] (6.149)

pkBT pkBT 24r/(1 - C) [ z

where

Z 2 2 1C)-}- (C1z -+-Co) 3/2 G(z) - - - ~ - ( 3 C - 3 + S ) - z(1 - ~ ~ -

C1 - - ( C - 2 ) ( 1 - S ) _71_ C 2 C o - (1 - 1 , 7C) 2 and/90 is the pressure of the corresponding uncharged version of the model. This equation, together with the corresponding expression for the Helmholtz energy, has been used to predict the vapor-liquid phase separation which appears due to the dipole interaction (209) (see Figure 6.7). The analysis shows that this phase separation occurs at tem- peratures much lower than that of the corresponding non-polar LJ diatomics. In addition the analytical solution of the SSOZ-MSA was also used to study the liquid-liquid phase separation (210) which occurs in a mixture of polar and non- polar hard dumbbells.

6.4.1.4 Corrections to the SSOZ Formalism: Chandler-Silbey-Ladanyi Equation

The success of the SSOZ formalism is mixed. In spite of its ability to give reasonably accurate predictions for the short-ranged structure, it fails to describe certain angular correlations (such as the Kerr constant and dielectric properties). There are other deficiences of the SSOZ theory, such as the dependence of the

T * x l 0

1 .30 . . . . . . . . . . . . . . , . . . . , . . . .

1 .10

1 .20 , Coexistence

- - - Spinodal • CP-DHS

i

1 . 00

0 . 9 0

0 . 8 0

0 . 7 0

220

, , , , I , , , , ! , , , , i , , , , i , , , ,

0 . 0 5 0 .1 0 . 1 5 0 . 2 0 . 2 5

11

Figure 6.7 The phase envelope for the dipolar hard-dumbbell fluid with elongation L = 0.Scr. The full line is the coexistence curve and the dashed line is the spinodal curve. The point above the curves is the critical point of the dipolar hard-sphere fluid with charge separation 0.Set. After Morriss and Isbister (209).

results on the presence of 'auxiliary' sites, and the thermodynamic inconsistency between different routes to the EOS, although the latter problem is shared by all IEAs except those which are specifically designed to eliminate this problem.

A number of attempts have been made to correct the defects in the theory. These developments follow two alternative lines. The first one, initiated by Cum- mings and Stell (96) and utilized in later studies (211,212), is based on the idea that SSOZ equation, Equation (6.135), is a definition of the direct site-site correla- tion functions, for which more appropriate closures should be used. In particular, the analysis carried out in references 182, 183 and 213 shows that, to correct the results for the dielectric properties, it is necessary to assume asymptotic behavior for the direct correlation functions which is different from that which follows from the regular SSOZ closure conditions, Equations (6.137)-(6.139). Unfortunately, this asymptotic form depends on the particular geometry of the site-site model used and quite often, due to its complicated functional form, is extremely difficult to treat as a direct correlation between the particles. This is in direct contrast to the case for simple fluids or to the case for the molecular formalism where the short range and simple asymptotic behavior of the direct correlation functions is

221

the key to the success of the IEAs away from the critical point. The alternate possibility is to correct the SSOZ equation itself, rather than

to use more sophisticated closure conditions. This possibility can be realized by constructing an OZ-like equation in which the direct site-site correlation functions consist of a subset of diagrams from the set contributing to the total site-site pair- correlation function. Such a modified version of the SSOZ equation, the so-called Chandler-Silbey-Ladanyi (CSL) equation has been proposed by Chandler et al. (214,215). We will discuss the CSL equation using the notation of Rossky and Chiles (216) who reformulated this equation and presented it in a convenient form. In this formalism the total site-site correlation function h ~ ( r ) is written as a sum of four terms

hoL~(r) -- ho~alOB (r) + hola~ s ( r ) + hoLs~ a (r ) + hoLstOs ( r ) (6.150)

The subscripts s or a indicate whether or not the indirect correlation of the corre- sponding site starts with the intramolecular distribution function. The correspond- ing site-site direct correlation functions are defined via the following OZ-like in- tegral equation

(6.151)

which is a slightly modified version of the CSL equation presented by Rossky and Chiles (216). Here

li~ - (h~°~a

S~ __(^1 a2a~

hoLs~s" ' Ca~ -- CoLs~a Co~s~s '

o) 1 ' P ~ - p 0 ' c o ~ - k L ~

and Equation (6.151) is written in the Fourier k-space. The analogues of the regular HNCA, PYA and MSA, which couple the direct

and total site-site correlation functions are (i) HNCA (216)

90la~a (r ) -- hoLal~a (r ) + 1 -- ~ -I~U~13(r)-jl-tc~al~a(r)

has l3 a ( r ) - - haal3 s ( r ) - ga a~ a (r)taaZ~ (r),

h~z~ (r) = g~a"o ( r ) ( t~ ,~ (r) + t~oz ~ (r)t~,Za (r)) (6.152)

222

(ii) PYA (216) COta~ a ( r ) _7_ faz(r)[1 + tOla~ a (r)]

C . ~ (r) -- c.~Z. (r) -- f . z ( r ) t . ~ a ( r ) , ¢c~.~. ( r ) - - f . z ( r ) t . . z . (r)

(iii) MSA (53)

(6.153)

= r <

ca,f ly(r)---(~ia(~jaflUa~(r) r > R~Z (6.154)

The PYA and HNCA for a fluid of LJ homonuclear diatomics have been solved numerically by Rossky and Chiles (216). Although the results for the structure appear to be qualitatively correct, they are less accurate than the corresponding results of the SSOZ-PYA. Recently, taking into account the bridge diagrams, the HNCA for the same LJ model has been solved by Attard (217). In general the results obtained are in better agreement with computer-simulation results than those of Rossky and Chiles.

Analytical solutions of the PYA and MSA for a number of simple models with hard-core short,range interactions are possible within the CSL formalism. How- ever most of the solutions were obtained within more general integral-equation formalism for associating fluids, for which the present CSL theory is a limiting case of complete association. The only exception is the solution of the PYA for the multicomponent molecular fluid with hard-sphere site-site interaction (218). Nevertheless, the general scheme of the solution is quite similar to that of the associative PYA for the multicomponent dimerizing hard-sphere system (57). Ac- cordingly, we postpone the discussion of these solutions to Section 6.5, in which the integral-equation theory for associating fluids is presented. We only note here that the solution obtained in reference 218 was extended to the case of flexible molecules and a closed form analytical expression for the compressibility pres- sure was derived. In the case of the linear flexible-chain model polymer system, this equation of state coincides with the equation of state derived by Chiew (93) (see Section 6.3.2.3 of the present review).

6.4.2 Hard Spheres with Dipolar Interactions

Despite the difficulties associated with the molecular formalism noted above, it has nonetheless proved to be a useful approach to certain classes of molecular fluids. In particular, fluids which can be modeled as spherical in shape with the anisotropy in the pair interaction resulting from multipolar forces (such as dipole

223

and/or quadrupole forces) have proved amenable to the molecular formalism, and many technologically important substances, such as water, can be approximated by such a model. In fact, a large measure of our current understanding of the origin of the static dielectric properties of water at ambient conditions comes from the numerical solution of the HNCA for water modeled as a Lennard-Jones sphere with a polarizable dipole and tetrahedral-quadrupole interactions (219).

The number of molecular-formalism models for which IEAs are analytically solvable is limited, but includes models for polar fluids, such as dipolar hard spheres (220,221) and hard spheres with arbitrary multipole moments (222-224), and simple models for water (225,226). To give some indication of how an IEA is solved analytically within the molecular formalism, we present the analytic solu- tion of the MSA for dipolar hard spheres.

The microscopic structure of dipolar hard spheres (DHS) is given in terms of the molecular pair-distribution function 9(12) which is proportional to the proba- bility density of finding a molecule of type i with center of mass located at ~'1 and with orientation ~1 [equal to (01, ¢1) for a linear molecule and equal to the Euler angles (01, ¢1, X1) for a nonlinear molecule] and a molecule of type j with center of mass located at ~'2 and with orientation ~72. [We use the numeric symbol, z, to symbolize collectively ~'z and ~z, z = 1, 2, etc.] The molecular pair distribution function is related to the molecular total correlation function by the usual relation- ship h(12) - 9(12) - 1 and the molecular direct correlation function c(12) is then defined by the molecular OZ equation

h(12) - c (12)+ ~-~p f c(13)h(32)d73d~3 (6.155)

where ~ - f dw - 4~- for linear molecules such as DHS, 87v 2 for nonlinear molecules. For DHS with hard-core diameter a, the intermolecular potential u is given by

u(12) - ~ r~2 <

(6.156) = - >

I+'12[ 3 le'1215

Here, g~ is the unit vector lying along the direction of the dipole in molecule i and rq2 is the vector joining the centers of mass of molecules 1 and 2.

6.4.2.1 Analytic Solution of MSA for Dipolar Hard-Sphere Fluid

224

For the dipolar hard spheres (DHS), the MSA yields

h(12) = - 1 r12 < cr

(6.157) C ( 1 2 ) - - ~t2 [ 8"1 "8'2 38"1"r'12 8'2 "r'12 ]

- - - - k B T 11~'121 a - - 1~121 s r12 > O

The OZ equation, Equation (6.155), combined with the closure Equation (5.157), was first solved analytically by Wertheim (220). We present here the analytic solution obtained using the technique of Blum and co-workers (222-224). In contrast to the solution of Wertheim, which exploits particular simplifications peculiar to the dipole-dipole interaction, the method employed here is readily generalized to include higher-order multipole and non-spherical overlap interac- tions (221,226,227).

In terms of the rotational invariants defined in the Appendix, the pair potential for DHS is given by

u(12) - .000~.000 ~112~i-)112 '/Z WOO ((.dlCd2&Jr)n t- =:00 ((M1Cd2(Mr) (6.158)

where u°° ° ( r ) - c o r12 < c~

= 0 r12 > cr (6.159) u l 1 2 ( r ) - - - - ~ i ~ r / ~ 3 r12 > a

where # is the dipole moment of the DHS. Consequently, there are only three expansion coefficients of h(12) and c(12) which are non-zero in the MSA [mnl

= 000, 110 and 112] and the closures on them are given by

]~°°°(r) - ---1 r < cr ~oOO(r) - 0 r >

h 1 1 ° ( r ) - - 0 /" < (y

~11°(r) - -0 r > (7

h ~ l ~ ( r ) - 0 r <

o l 1 2 ( r ) - - - - k B T r 3 r > cr

(6.160)

Of the functions . ~ n(r) (9 r = J or S) defined in the Appendix, only three are distinct and non-zero" .~0°(r), 0vii(r) and 9r111 (r) = .~.1] (r). Thus, according to equation (A12), we obtain three distinct OZ equations:

HO0(k) - CO0(k) + pH°°(k)C°°(k) H l l ( k ) - Vi i (k) --[- pHl l ( k )V l l ( k ) HI l (k) - Vi i (k) -Jr- pH11(k)CIl(k)

(6.161)

225

To proceed further, we need to obtain closures on J~n(r) and S2n(r ) in real space. From Equation (6.232) in the Appendix, we have

m n S x (r) -- 0 r > o- for rnnx -- 000,110,111 (6.162)

and from Equation (6.230) we obtain

m n m n ~ m n 2 J~ (r) - /3~, o + Px,2 r r < a for m n x - 000,110,111 (6.163)

where

00o - booo, ° =

~x1,% - - ( _ ) x [ ( 1 1 0 )b l l0 1 ( 1 1 2)b~12 ] X - X 0 - 2 X - X 0 (6.164)

/31,1 __ (__)x3_ ( 1 1 2)5112 2 X - X 0

In these equations,

- 27r foo ~ ]~mnt(r)rl-Pdr (6.165) b ; n*

We now note that each of the three OZ equations given in Equation (6.161) and the closures given in Equation (6.164) have the same genetic form given by

subject to the closure

H(k) - C(k) + pH(k)C(k) (6.166)

J(r) - /3o+/32r 2 r < c r

S(~) - o ~ >

(6.167)

However, there is a crucial difference between the mn = 00 problem and the mn - 11 problems: in the former case, 132 is known explicitly (= 70 whereas in the latter case (for X = 0 and 1)/32 will be determined by the closure relation. Despite this difference, it is expedient to perform the Baxter factorization on the problem as stated in Equations (6.166) and (6.167) and then to consider the specific cases.

Solution of generic problem. Since the S~n(r) are finite ranged, implying that C~n(k) is the Fourier transform of a finite-ranged function, the OZ equation, Equation (6.166), can be factorized in the usual fashion yielding.

1 - pC(k) - Q(k )Q( -k )

1 + pH(k) - [ Q ( k ) Q ( - k ) ] -1 (6.168)

226

fo (y Q ( k ) - 1 - p q( t ) e x p ( i k t ) d t (6.169)

The real-space function q(r) satisfies q(r) - 0 for r < 0 and r ___ or. One should note that the symmetry of the harmonic coefficients used in the dipolar- hard-sphere mixture problem (m - n for all coefficients) implies the following relationships between the H, C, J and S quantities"

H ~ n ( k ) - 2 f ~ c o s ( k r ) J ~ n ( r ) d r

(6.170) c ~ n ( k ) = 2 f ~ c o s ( k r ) S ~ n ( r ) d r

These equations are special cases of Equations (6.229 and (5.231). Inverting Equation (5.158) to real space yields (see the Appendix)

J ( r ) - q(r) + J ( l r - t l ) q ( t ) d t (6.171)

Substituting Equation (6.167) into Equation (6.171) yields

1 q(r) - -~a(r - a ) r + b(r - a) (6.172)

Substituting Equations (6.167) and (6.172) into Equation (6.171) and solving for a and b, we obtain

2/32 (1 + 3eaa/32) b /32 (6.173) a ~ - - - - -

Solu t ion o f the m = n = O, X = 0 problem. In this case, the/32 = ~ is known explicitly yielding

2~r (1 + 2r/) b 7r a - (1 - ~7) 2 ~ = (1 - r/) (6.174)

where ~7 - (Tr/6)a~. Taking into account the slight differences between the def- initions of q(r ) , a and b used here and in Baxter's solution of the Percus-Yevick approximation for hard spheres, Equations (6.38) and (6.39), it is readily seen that the results for a and b are consistent. Hence, the m = n = X = 0 problem reduces to the Percus-Yevick approximation for a hard-sphere fluid as it should.

Solut ion o f the m = n - 1, X - O, 1 problems . We begin by noting that

90,2 - - 271"~, 91,2 -- --Tr~ (6.175)

227

where

Following the general analysis above, we find that

(6.176)

11 1 qx (r) - -~ax(r - cr)r + bx(r - or) (6.177)

where 47r~ (1 + 4 ~ ) 27c~ (1 - 2 ~ ) / (6.178)

a o = (1-2~/~) 2 a l - - ( l+ r ]~ )

Hence, the solution of the problem reduces to finding ~ which comes from consid- ering the closure o n O l 1 2 ( r ) for r > o-. From Equation (6.235) of the Appendix, for r > cr we obtain

c l 1 2 ( r ) - - ~ / ~ [ f o o a ~ 1 1 ( r l ) d r 1 - f o ~ l l l ( r 1 ) d r l ] r -~ 27r ( 6 . 1 7 9 )

upon evaluation of the 3 - j symbols. Combining this equation with Equa- tion (6.160) we obtain after some manipulation

( )2 3y - ao _ a l = (1 + 4r/~) 2 _ (1 - 2r/~) 2 -27r~ (1 -- 2T]~) 4 (1 -1- T]~) 4

(6.180)

where 47~p# 2

Y - 9kBT

The residual internal energy per molecule, ures/N, is given by (138)

(6.181)

ures 1 m~nl (--)l f°e~tmnl ~mnl N = -~47rp 2l + 1 (r) (r)r2dr (6.182)

which simplifies in the present case to

u res

N = -3y~ (6.183)

This completes the analytic solution of the MSA for DHS. The energy equation of state for DHS in the MSA predicts a liquid-gas phase transition. This is a some- what artificial system since the sole source of the attraction giving rise to the phase

228

transition is the dipole-dipole interaction. Nevertheless, the MSA provides an an- alytic expression for the state dependence of the dipole-dipole contribution to the internal energy via Equation (6.183). Cummings and Blum (221) extended this analysis to dipolar hard-sphere mixtures and performed Monte-Carlo simulations to compare with the MSA results. They found that the MSA underpredicts the magnitude of both the dielectric constant and the residual energy, U res. However, the importance of the analytic solution of the MSA for DHS cannot be under- stated: It inspired much of the subsequent work leading to microscopic theories of the dielectric constant and provided the basis for more accurate equations of state for DHS. Many of these developments are reviewed by Stell et al. (228).

6.5 EQUATION OF STATE FOR ANALYTICALLY SOLVABLE MODELS OF ASSOCIATING FLUIDS

During the last decade, significant progress has been achieved in the devel- opment of statistical-mechanical theories for associating fluids. Associating flu- ids are characterized by the formation of long-lived clusters of particles due to strong short-ranged, frequently directional, attractive interactions, such as hydro- gen bonding and ion-pairing. Theories for associating fluids date back to the work of Bjerrum (167) and include an important study by Ebeling and Grigo (168) using the 'chemical approach' to treat ionic association as a chemical reaction. Impor- tant milestones in the development of the statistical-mechanical theory of associ- ating fluids are Andersen's work on the cluster expansions for hydrogen-bonded fluids (229-231) and the subsequent general formalism developed by Chandler and Pratt for chemically associating fluids (232). However, these developments do not readily lend themselves to the accurate description of the structure via integral-equation approximations.

Two alternative versions of the integral-equation formalism for associating fluids have recently been proposed. The first one, which was initiated by Cum- mings and Stell (41,42), consists of utilizing the regular integral-equation closure approximations for studying model systems with strong attractive interactions. The shielded-sticky-shell (SSS) versions of this type of model are considered in (41,171,233) and is defined by the two-component mixture of hard spheres of equal diameter cr with hard-sphere interaction between the particles of the same species and hard-sphere and sticky interaction (of the same functional form as the attractive part of the adhesive-hard-sphere fluid interaction) between the particles of different species. The sticky interaction is placed inside the hard-core region and is effective at a distance L < ~r between the particles. For L _< cr/2 only

229

dimers can be formed in the system, while for L > a/2 linear and branched poly- merization is possible (171,233). The theory was applied to several similar model systems (43,44,211,234,235).

However, since the regular IEAs are not well suited for treating strongly asso- ciating fluids, this approach has several deficiencies. In particular, it is believed that the IEAs used are not faithful to the law of mass action in the low-density limit (236). In this study and in the later studies of Zhoe and Stell (237,238), the theory was corrected for this deficiency and leads to results of reasonable accu- racy (239-241). However, the corrected version of the theory requires, as an input, knowledge of thermodynamic properties of the system. Since the purpose of this review is to demonstrate the abilities of the IEA in predicting thermodynamic properties, we will not discuss this approach here any more.

The second version of integral-equation theory for associating fluids was ini- tially developed by Wertheim (45-48). The theory is based on a combined de- scription, in terms of activity and density expansions, leading to the multiden- sity integral-equation formalism. There is a close structural similarity between the multi-density integral-equation theory and the regular single-density integral- equation theory, thus permitting the derivation of an OZ-like integral equation and analogues of widely-used closure conditions. Originally, the theory was de- veloped for associating fluids modeled by hard spheres with off-center sites in- teracting via a strongly attractive short-ranged potential responsible for associa- tion. More recently, the multi-density theory has been reformulated and extended to the case of associative potentials of any symmetry (including the spherically- symmetric case) by Kalyuzhnyi et al. (40,49). Analytical solutions of the sim- plest two-density versions of the IEA for both theories have been obtained for a number of simple models of associating fluids. These will be discussed in the succeeding sections. It is important to note here that Wertheim's formalism for associating fluids (45-48) forms the theoretical basis for a highly successful engi- neering equation of state, the statistical associating-fluid-theory (SAFT) equation of state (242-244).

6.5.1 Two-Density OZ Equation and Closure Conditions:Relation to the CSL equation

The two-density version of the theory begins by expressing the pairwise po- tential energy uab(12) as a sum of two terms

U~b(12) - - " (n)(12)-}-~ (a) ~b ~b (12) (6.184)

230

where 1 and 2 stand for the spatial and orientational coordinates of the two par- ticles, ~ (a) and ~ (n) ~ab ~ab are the associative and non-associative parts of the total po- tential respectively, and the subscripts a and b denote the particle species. Fol- lowing the separation of the potential into associative and non-associative parts, Equation (6.184), the total number density of the system p~ is separated into two contributions, the density of 'non-associated' particles p~ ('monomers') and the density of 'associated' particles p~

Pa - P'~ + P~ (6.185)

and the pair-correlation function h~b now becomes a sum of four terms

hab(12) -- h~b(12)+ xah~0b(12)+ xBh~(12)+ XaXBh~(12) (6.186)

where xa -- P~/Pa, and the lower subscripts in the functions hi~b(r) indicate the degree of association of the corresponding particles.

The two-density analogue of the OZ equation is

hab = Cab + E Cad * Pet * hdb (6.187) d

where • denotes the convolution and

ho0- ii , Ca --%0o c10 p~ 0

Here the c ab functions are analogues of the direct correlation functions. To complete the theory one needs to assume an approximate closure condition,

~b and densities p~ and p~. which provides the additional relation between h~ b, c~j In the present theory, there are two approximations. The first is similar to those used in the usual one-density integral-equation theory and follows from the ap- proximation for the 'bridge function.' The second approximation is specific to the multi-density description and is related to the choice of the maximum number of particles which can be simultaneously bonded by the associative part of the poten- tial. Assuming the simplest 'dimer' approximation, in which not more than two particles can be bonded simultaneously, and making use of the close similarity between the structure of the multidensity and of the usual one-density integral- equation theories, we can derive the two-density analogues of the widely-used single-density IEAs, the HNCA (46)

90abo -- e-u(~) /kBT+t~bo,

231

g ~ ~ _ab.ab ab _ab.ab 900~01, glo - - gOO~lO,

_ ÷ab÷ab ~(a) g ~ goabo(t~ -t-~01~10 + Jab ) (6.188)

ab ab ab ab ab the PYA (46) where 9ij - - h i j + 5oi~oj and t i j - - h i j - cij ,

ab ab ab Yij - - gij -- Cij (6.189)

where the analogues of the regular cavity correlation functions, ~b y~j, are defined by

ab __ eU(ab ) /kBT ab ,~ g(a) ab gij (Yij + (~il YO0)" UjlJ ab

and the MSA (53,55)

ab dli(~lj f a b ( a ) (n) Cij - - --(~Oi(~OjUab / k s T , for r > Crab ,

ab _ hij -5oi5oy, for r < aab, (6.190)

where 5ij is the Kronecker delta, aab = (cr~ + 08)/2 and era is the hard-sphere diameter.

Within the same 'dimer' approximation, the relation between the 'monomer' and total densities is given by

Pa -- P~ + 47Cp~o ~ pbo f go~(12) r(a) Jab (12)d l (6.191) B

where g~b _ hi? + (~0i50 j and ~e(a) u(a)/k - e- ~b / B~ _ 1 The latter relation can be , J ab viewed as the analogue of the mass-action law.

In the more general case of 'trimer' and higher-order approximations an infi- nite number of terms involving the higher-order correlation functions will appear in the closure conditions. However, there are several models for which the 'dimer' approximation appears to be either exact or quite accurate.

Since the pair potential, Equation (6.184), depends on the mutual orientation of the two particles, all the partial correlation functions, involved in the OZ equa- tion, Equation (6.187), are also orientationally dependent. This makes the solution of the OZ equation, Equation (6.187), rather difficult. However, in most applica- tions it is assumed that only the associative part of the potential is orientationally dependent, while the nonassociative part is spherically symmetric. This, together with the specific structure of the OZ equation, Equation (6.187), makes it possible to orientationally average Equations (6.187), (6.191) and the closure conditions, Equations (6.188)-(6.190). The orientationally-averaged version of the OZ-like

232

equation with closure conditions is equivalent to Equations (6.187)-(6.190), in which the orientationally-averaged correlation functions ~f (r) and h~ f (r) replace the original correlation functions c~b(1 2) and h~b(12)

Thus far, the solution of the OZ equation, Equation (6.187), has been ob- tained only for its orientationally-averaged form. In addition, we note that the two-density version of the theory, developed in references 40 and 49 to study the association effects in fluids with spherically-symmetric potentials, is equivalent to the orientationally-averaged version of the present two-density theory.

Finally, it is interesting to note that in the complete-association limit the present orientationally-averaged two-density version of the integral-equation the- ory reduces to the CSL equation discussed in Section 6.4.1.4 of this review. This can be easily demonstrated by considering the models with associative interaction presented by the sticky interaction. For the sake of simplicity we will show this for the one-component system. In this case the Mayer function for the associative po- tential will be substituted by the delta-function term f(a)(r) = BS(r - L), where B is the stickiness parameter and L is the distance at which bonding between par- ticles occurs. The direct contribution from the delta-function, which appears in the total and direct correlation functions, hij(r) and c~j(r), can be separated out

hij(r) = Hij(r)+SiXSj lByoo(L)5(r-L) , cij(r) = Ci j (r)+Si lSj1Byoo(L)5(r-L)

and the OZ equation, Equation (6.187), can be renormalized with respect to this contribution. This yields the following renormalized OZ equation written in ma- trix form in Fourier k-space

_ ~ r l ~ r T + ~v~(~Vfl (6.192)

where I=I and (~ are the matrices containing the elements which are the Fourier transforms of the correlation functions H~j(r) and C~j(r) and

I?Vij - ~ij + ~ilSjO47rL2Bpo sin kL kL Yoo(L) (6.193)

The relation between the densities, Equation (6.191), now takes the form

P - Po + 47rL2p~Byoo(L) (6.194)

In the complete-association limit B --+ ee and from Equation (6.194) we have

p2oB - P (6.195) 47rL2Yoo(L)

233

Substituting this into Equations (6.193) and (6.192) and multiplying both sides by the matrix with the elements Xij - 5io5jo + 5i15jlpo/p we finally arrive at the CSL equation

I=I - W ( ~ W T + ~q(~p~2I (6.196)

where ~ r - X W X - 1 , ~6 - X- lpX -1 and the elements of the matrices ISt - XI2IX and (~ - X(~X are the partial correlation functions as defined by the CSL formalism (214,216). From Equations (6.188)-(6.190)one can easily see that the closure conditions for the functions Cij(r) and Hij(r) coincide with the corre- sponding closure conditions, Equations (6.152)-(6.154) for the CSL equation. A similar analysis can be carried out for 'trimer' and higher-order approximations. In this case, the two-density OZ equation, Equation (6.187), reduces to the CSL equation for the molecular system with the corresponding number of sites in the molecules (218).

As in the case of the one-density IEA, analytical solutions for the present two- density IEA have been obtained for simple models with the nonassociative poten- tial presented by the hard-sphere, charged-hard-sphere or hard-core Yukawa fluid models with the sticky potential for the associative interactions. This includes the one-component and multicomponent dimerizing hard-sphere models (51,57), dimerizing RPM and PM of electrolytes (53,176), dimerizing HCYF model (245), and the Smith-Nezbeda (SN) primitive model for associating fluids (54). Recently, analytical solutions where obtained for polymerizing models (130,246-251) using the multi-density formalism with the number of densities higher than two (47- 49) and their thermodynamic properties, in particular equation of state, has been analysed (252,253). However, due to the lack of space we will not discuss these solutions here, refering the reader to the original publications.

6.5.2 Two-Density PYA for Dimerizing Hard-Sphere Models

The two dimerizing hard-sphere models we will consider in more detail are the shielded-sticky-shell (SSS) (41) and shielded-sticky-point (SSP) (45,236) models of associating fluids. The former model, which was discussed at the beginning of this section, consists of the two-component mixture of hard spheres of equal diameter cr and with sticky interaction between unlike species effective at a dis- tance L < cr between the two particles. For the values of L from the interval 0 < L < ~r/2 only dimers can be formed in the system. The SSP model can be viewed as a limiting case of the SSS model, in which the sticky area is re- duced to a single sticky point (171). For this model the associative interaction is

234

orientationally dependent

~tSSP a ab (12) -- uHS(r) + u.b(12 ) (6.197)

and bonding between the particles occurs at a distance L if they have an appropri- ate mutual orientation. This, together with the hard-core repulsion, provides the saturation of the associative potential on the dimer level at any values of L. The unshielded one-component version of this model (L = (aa + OB)/2 , where era is the hard-sphere diameter) has been used by Wertheim (45,51), while the shielded version of both models was proposed by Cummings and Stell (41) (SSS) and Stell and Zhou (236).

Analytical solutions of the two-density PYA have been obtained for both SSS and SSP versions of the dimerizing hard-sphere models (51,240,254,255). In the latter case an orientationally-averaged version of the two-density PYA was uti- lized. Since there is close similarity in the solutions for both types of the model we will present here only the solution for the SSP model.

An orientationally-averaged version of the two-density PYA for the one- component SSP model can be written in the form

hij(r) = -5oiSojBVoo(L)5(r- L), r < (7

cij(r) ----- 0, r > o (6.198)

The corresponding two-density OZ equation can be factorized using the Baxter factorization method. The resulting q-functions are

1 r2 qio(r) -- -~ai + bir + Ci

qii (r) -- D}n)r 3 -+- Aln)r 2 + B}n)r 4- C~ n),

where In denotes the following intervals

1-L], 1],

I1 = [0, 1 - L], I 2 = [ 1 - L , L ] , I3 = [L, 1],

for r C In (6.199)

o" f o r L < -

2 o"

f o r L > - 2

the constants D} n), AI '~) and B} n) are functions of the unknowns ai, b,, c,, C~ n) and Yoo(L), which follow from a set of algebraic equations. For the monomer density, P0, we have

- 1 + (1 + 167rByoo(L)L2p) 1/2 Po - 87rByoo ( L) L 2 (6.200)

235

The set of algebraic equations for the solution parameters can be solved in a closed form for L - cr (51)

1 + 2r/ -3r/o. o .2 ao = (I - r/) 2' bo -- 2(1 - ~])2' Co -- 2(1 - q) '

27rpod11 7rpod11 1 -}- 1 a l = b l - ~ , c 1 - 0 , y o o ( l + ) _ gr/

r / - 1 ' 1 - r/ (1 - r/) 2'

P o - - 1 + (1 + 167rpd11) 1/2

87rdll

where r / - 7rp~73/6 and dll - o.B9oo(a+). The compressibility pressure for the one-component SSP model follows from

the two-density analogue of the PYA compressibility pressure (256) derived by Wertheim (51 )

p c

kBT = ~ Pijq~}(O) - 27r y~ ~ Pijqmj(O)ptmq~m(O)+ ij i j l m

4 +57c2 ~ Y~. ~ Pijqlj(O)ptmqkt(O)Pknqin(O) -- -27rL2Bp2y2o(L) (6.201)

i j l m k n

The expression for the virial pressure (240,254) follows from the standard relation, Equation (6.2),

p v

kBT 2 2

= p + ~lrp 9(1 +) - 7rp~L2B{3yoo(L)+ Lyoo(L)} (6.202)

where the expression for y/~0(L) is given in reference 240. Again, for L - cr the resulting EOS can be obtained explicitly as (51),

pc _ _ 1 + r / + r/2 _ (1 + lr/)(1 - x) (6.203) pkBT (1 - r/) 3 2(1 - 7/) 2

pv 1 + 2 r /+ 3~ 2 (1 + 2q)(1 - x) q(1 - x) 2 1 1 pkBT (1 - q)2 2(1 - r/)(1 + gq) 4(1 + 7q)

(6.204)

where x - p0/p. These expressions are valid for any degree of association, including the limit

of complete association, B --+ ec. As was argued in Section 6.5.1, in this limiting

236

P/PdkB T

2 0 . 0 . . . . . . . . . . . . . . . . . . . . . . . t~ , , , , , ]

o TS, L=a i I /1 -- -- -PY-C, L=o I 15.0 . . . . . PY-V, L=cr ~ / "

[] MC, L--0.6o I ee ,, • PY-C, L=0.6o I • tz ,,"

......... PY-V, L=0.6cr , ' . . " / /

1 0 . 0 / / , .....

~D • . ..."

le, • [] . ' " ,~ • . .

5 0 ,,~'" . . ...... q

4

0 . 0 . . . . ' ' . . . . . . . . ' . . . . ' . . . . ~ . . . . .

0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6

p °.3

Figure 6.8 Pressure for the homonuclear hard-dumbbell fluid calculated from the complete-association limit of the SSP model for two bond lengths, L -- 0.6or and L -- or. The compressibility (labelled PY-C) and virial (PY-V) pressures are labelled accordingly. For L = a, the results are compared to the Tildesley-Street (TS) equation of state while for L -- 0.6a comparison is made with Monte-Carlo (MC) results.

case the present solution of the two-density PYA reduces to the solution of the CSL equation supplemented by the corresponding PYA-like closure conditions.

In Figure 6.8 we present the comparison of the results for the compressibility and virial pressure of the SSP model in the complete association limit at L -- 0.60- (254) and L -- cr (51) with Monte-Carlo and Tildesley-Streett empirical EOS results (257). A comparison of the results for the pressure as predicted by the two-density PYA and Monte-Carlo simulation results for the SSS model at L -- 0.42a and L = 0.3a under the condition of partial association (K0 = 4 7 r L 2 B =

33) (240) is presented in Figure 6.9. Predictions of the two-density PYA for both models are in good agreement with the simulation results as long as the density of the system is relatively low or, at high densities, when the hard-core volume of a dimer is not substantially less than that of the two free monomers from which it is formed.

The above method of solution has been extended and applied to the general case of the n-component SSP model in which bonding occurs at the contact dis- tance between the hard spheres of species a and b (57) (i .e. , Lab = crab).

237

7 . 0 , , , , , , , , , , , , , , , , , , , , , , ,

6 . 0

5 . 0

P c Y 3 / k B T 4 . 0

3 . 0

2 . 0

1 . 0

0 . 0

0 . 0 1.2

. . . . . A P Y - V , L---0.42~

- - - - - A P Y - C , L---0.42~

o MC, L---0.42~

. . . . . . . . . A P Y - V , L = 0 . 3 ~ f /

~ A P Y - C , L-----0.3~ / o •

[] M C , L=0.3c~ o ~ ' o • f • •

/. -": ,1..s, •

i , i , , , , , , , i , , , i i , , ,

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

p (~3

Figure 6.9 Pressure for the SSS model for the homonuclear hard-dumbbell fluid cal- culated for two bond lengths, L -- 0.42cr and L = 0.3a, under the condition of partial association (K0 = 47rL2B -- 33) (240). The results for the APY compressibility (la- belled APY-C) and APY virial (APY-V) pressures are labelled accordingly. The results are compared to Monte-Carlo (MC) simulation results.

6.5.3 Two-Density PYA for the Smith-Nezbeda Primitive Model of Associating Fluids

An interesting extension of the two-density formalism has been proposed by Wertheim (52) in his study of the Smith-Nezbeda (SN) primitive model of associ- ating fluids (258). The model consists of hard spheres, each having one attractive site located on the surface. The attraction acts only between the site of one sphere and the center of another sphere, and is given by a square-well potential. It thus may be considered to mimic the behavior of a hydrogen bond. The pair potential of this model is

u(12) - UHS(r) + u01(12)-+- Ulo(12) (6.205)

where U H S is the hard-sphere potential of diameter cr = 1 and U 0 1 and ul0 are the center-site and site-center potentials respectively. The appropriate choices for the parameters of the square-well potential guarantee that saturation conditions, in which the site can be bonded to only one center, are satisfied. Unlike the dimer- ization case, the present model allows polymerization. However it appears that

238

the two-density theory can be modified to include the present SN model (52). The theory results in the OZ-like integral equation which formally coincides with Equation (6.187). As in Equation (6.187), the lower indices denote the bonded states of the particle. However, now the bonded states are defined only for the sites, while for the centers the theory does not distinguish between bonded and un- bonded states. This results in the following orientationally-averaged two-density PYA

< 1

cij(r) - (1-Sij)goo(r) f(a) (r)+SilSjl {291o(r) f(a) (r)-goo(r)[f(a) (r)] 2} (6.206)

and relation between the densities fl/2+A

P -- P0 + 47rp0 [pgoo(r) + pogol(r)]f(a)(r)r 2 dr (6.207) dl where A is the width of the square-well potential, F (a) is the angle average of the Mayer function, fox, corresponding to the hydrogen bond. These relations are quite different from the corresponding two-density PYA expressions for the dimerizing fluid.

To make the present PYA analytically solvable, the original SN model must be modified by substituting the Baxter sticky interaction for the square-well interac- tion. Applying the Baxter factorization method to this model (54), we can derive the Baxter q-function given by

qij(r) - (~air 2 + bir)5oj + dij (6.208)

The unknown parameters of the problem ai, b~, dij and P0 follow from the solution of a set of eight linear equations and one nonlinear equation (54). As in the case of the dimerizing hard-sphere system, the analogue of the Baxter expression for the compressibility pressure can be obtained (52,256). However only the virial EOS was analyzed using the analytical solution, (54). The latter was calculated from the following expression

4 fl/2+~ /3PV = 1 + ~Trpg(1 +) + ~Trp0 [g00(r) + ~glo(r)][f(~)(r)]'r 3 dr (6.209) P

In Table 6.2, we present a comparison between the pressure calculated from Equa- tion (6.209) and that obtained using Monte-Carlo simulation for the original SN model with finite square-well associating potential. The largest discrepancy be- tween theory and simulation is at high densities and approaches 10%, suggesting that the agreement between theory and simulation is quite satisfactory.

239

Table 6.2 The compressibility factor, ffp~T, of the Smith-Nezbeda primitive model of associating liquids (54). For each temperature the first row is the analytical result from the virial EOS, the second row is the simulation value. Temperature and density are given here in units of e/lc and 7ro3/6 where e is the depth of the square-well associative

kBT/e 7rpo3/6 0.2 0.3 0.4

0.5 2.107 3.262 5.313 2.05 3.39 5.85

0.3 1.695 2.699 4.629 1.61 2.79 5.18

0.2 1.314 2.320 4.272 1.32 2.68 4.92

0.15 1.200 2.222 4.189 0.97 2.55 4.75

potential.

6.5.4 Two-Density MSA: RPM of Electrolytes and Dimerizing HCY Fluid

The analytical solution of the two-density MSA has been obtained for the dimerizing RPM of electrolytes, defined as a charged hard-sphere version of the two-component SSP model at L = R (53) and for the same dimerizing RPM of electrolytes with an additional Yukawa interaction (55).

The latter model is more general and therefore we will briefly discuss its so- lution within the two-density MSA (or associative MSA (55)). In the closure conditions, Equation (6.190), the nonassociative part of the potential is defined to be the sum of the Coulomb and Yukawa potentials. Similar to the case of the dimerizing hard-sphere system, the Mayer function for the associative potential in Equation (6.190) is modeled by the Dirac delta function. The method of so- lution of the present AMSA is similar to that of GMSA for the primitive model of electrolytes. The solution results in an analytical expression for the Baxter q- functions with parameters determined by a set of nonlinear algebraic equations. We will not present the analytical expressions here for either the q-functions or the set of algebraic equations because the resulting expressions are too tedious: We refer the interested reader to the original publications (55). We present here

240

the expressions for the EOS obtained from the virial route

p v v

N k B T

where

1 -- 527rP[9~°s) (or+) + 2x9;~) (a+) + x29~1s)(or+)] + 3 UNkBT- Uid

U - V id 9 VkB------T- = 2 z*(go + x J1),

1

Ji = pJoi + poJ l i ,

P = P + + P - ,

1 +_

po - P+o + Po, x - x+ + x_

and the energy route

(6.210)

0 N ( P e - PHS) -- --p(1 -- p--4--)[A~ + A~2]

op - (6.211)

where A (a) 1 1

V k B T = p ln x - ~po + ~p,

A (n) is the Helmholtz energy of the system interacting only via the nonassociative

part, u~b ), of the total potential. The parameters, Ji, follow from a set of nonlinear algebraic equations. The above solution of the two-density MSA was used to take into account the effects of association in the RPM model for a 2-2 electrolyte solution. With this aim the total potential of the model was split into associative and nonassociative contributions,

- - - - + (6.212)

where UHS and u ~ ) are the hard-sphere and Coulombic potentials, respectively, and o (a) and° (n) U'ab ~ab are the associative and non-associative parts of the total potential, respectively, and the subscripts a and b denote the particle species.

To obtain an analytical solution, the associative part of the potential is chosen as a sum of Yukawa terms (55) with the parameters calculated from an extremum condition on the Helmholtz energy. In addition, the Mayer function which arises for this associative potential is substituted by a delta function. An analysis of the results obtained is contained in Section 6.3.4.3.

241

1 . 5 , , , i , , , i , , , i , , , i ,

f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

1.4 ,.-" "'"'..

13 I / ' " .- . . . . .. "'"'.... ":. / I ~ ~ , ".,.

T* i / <> ~ ": I N :.,

1 1 t

' ,,.,. 1 0

0 9

0.8 0.0 0.2 0.4 0.6 0.8

p*

Figure 6.10 Liquid-gas phase envelope for the dimerizing hard-core Yukawa fluid and the hard-core Yukawa fluid (HCYF). The solid line is the phase envelope for the HCYF from the MSA. The dashed and dotted lines are the phase envelopes for the dimerizing HCYF obtained from the AMSA when the square-well depth is ten times (dashed line) and fifteen times (dotted line) the well depth of the Yukawa interaction. The symbols are Gibbs ensemble Monte-Carlo simulation results for the phase envelope of the HCYF given by Smit (259).

Another application of the present solution is related to the study of the ther- modynamic properties of the dimerizing hard-core Yukawa fluid. The solution of the two-density MSA for this model can be obtained as a particular case of the solution discussed above by setting the ionic charge to zero. In Figure 6.10, we present the liquid-gas phase diagram for the dimerizing hard-sphere Yukawa fluid as predicted by the two-density MSA from the energy route to thermody- namics (55). Comparison with the corresponding phase diagram for the nonasso- ciating hard-sphere Yukawa fluid shows that the liquid-gas phase transition for the dimerizing hard-sphere Yukawa fluid occurs at higher temperatures.

6.6 CONCLUSION

It is just mder forty years since the first analytic solution of an integral- equation approximation for a non-trivial model of fluids (the PYA for the hard- sphere fluid) was reported. Since then, analytic solutions of integral-equation

242

approximations have been used to develop equations of state for simple, molec- ular and polymeric fluids, for systems undergoing chemical reaction (including polymerization) and for electrolyte solutions. They have also been used to de- velop molecular expressions for adsorption isotherms in non-uniform systems, a topic not covered in this review. Furthermore, they have provided the input to perturbation theories of fluids and led to the development of the modem theory of the dielectric constant. In short, despite their limitations, integral equations have played an invaluable role in furthering our knowledge of the molecular ba- sis for thermodynamic behavior and providing the insight and mathematical ex- pressions necessary to develop accurate equations of state. Recent developments in associating-fluid models and their relationship to polymeric fluids suggest that integral-equation approximations, and the analytic solutions thereof, will continue to make important contributions in the coming decades.

ACKNOWLEDGMENTS

The authors are grateful to their many colleagues with whom they have had the pleasure of interacting over several decades in developing analytic so- lutions to integral-equation approximations, including Lesser Blum, Miroslav Holovko, Gary E Morriss, Ivo Nezbeda, Edgar R. Smith, George Stell and Michael Wertheim. The authors gratefully acknowledge support of this research and the preparation of this review by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy. The authors are especially grateful to Ariel Chialvo for his helpful comments on an earlier draft of this chapter.

APPENDIX: SUMMARY OF THE INVARIANT EXPANSION METHOD FOR LINEAR MOLECULES

In this Appendix, we briefly summarize the invariant expansion introduced by Blum (222-224) as a generalization of the method employed by Wertheim (220) in solving the MSA for dipolar hard spheres. In general, the invariant expansions of the functions h(12) and c(12) would be given in terms of rotational invariants

I" m n l q?u~" (W1W2Wr) defined by "" m n l

--[(2m+1)(2n-+-l)] 1/2 Z ( m rt l ) m u'~"~" #' U' A' D~'u'(w1)D~'~'(w2)Dt°)"(Wr) (6.213)

243

m ( ) In this equation, Du,(w ) is a Wigner rotation matrix, m n l is a 3 - j p u A

symbol in the notation of Edmunds (260) and ~1, ~22 and Wr are, respectively, the Euler angles specifying the orientation with respect to an arbitrary set of axes of molecules 1 and 2 and of the vector g'12 joining their centers of mass. For linear molecules, however, we require only # = u = 0 terms in the rotational invari- ants (222), so in all the succeeding equations the pu subscript on the expansion coefficients of the correlation functions will be suppressed on the understanding that this implies #u = 00. Thus, we define the invariant expansions of h(12) and c(12) to be given by

h(12) - ~ ]~mnt ^mnZ (r) ¢b00 (~Zl~Z2~Zr) (6.214) mnl

C(12) -- Z cmnl(r)~r~onl(('dl('d2OJr) (6.215) toni

The rotational invariant expansion is derived in reference 233. The Fourier trans- form f(12) of the general function f(12) [= h(12) or c(12)] is given by

f(12) - ~ fmnl(k)( f )mnl :tO0 (Cd1032~k) (6.216) mnl

where fm~t(k) - 47ri t fo~Jt(kr)fmnZ(r)r2dr (6.217)

where jr(x) is the lth order spherical Bessel function. In Equation (6.216), cok is the orientation of the vector k in the Fourier transform

37(12) - f exp(if~. 7~2)f(12)d~'12. (6.218)

Expansions (6.214), (6.215) and (6.216) are referred to as laboratory-frame (LF) expansions by Gray and Gubbins (184); the g-frame (RF) (also known as intermolecular-frame) expansion of f(12) (f = h, c) corresponds to the LF expan- sion when the z-axis of the laboratory frame is chosen to lie along the vector ~2 joining the centers of mass of the two molecules, thus yielding

f ( 1 2 ) - E(-)x f~(r)[(2m + 1)(2n + 1)]I/2D~(wl)Donx(w2) m n

x

(6.219)

where m+n

Z t=lm-nl

m

x n

-)(. l fmnl o) (r) (6.220)

244

This equation is derived by noting that, for this choice of axes, Or - 0 so that Dl0:~ (wr) -- 3t0 where 3ij is the Kronecker delta function. From the orthogonality properties of the 3 - j symbols, it follows that (222-224) an "inverse" relation to Equation (6.220) exists and can be written in the form

inf(m,n) front(r)--(21+ 1) E (_)x ( m n l ) mn

x=_inf(m,n) ~ -X 0 fx (r) (6.221)

We can similarly write down a k~--frame (KF) expansion for f(12) by choosing the z-axis of the co-ordinate system to lie along the k:

f(12) - - E(-)xF~n(k)[(2m + 1 ) ( 2 n + 1)]'/2D~(w1)Don_x(W2) mre X

(6.222)

where

m+n

Z l=lm-nl X -X 0

(6.223)

m + n ( ) f o C X ~ = ( _ ) x 4 ~ ~ m n l it jt(kr)fmnt(r)r2dr l=lm-nl X - X 0

(6.224)

and

inf(m,n) Z o (k) (6.225)

X---inf(m,n) ~ - ~ 0 -

Note that w1 and w2 in Equations (6.214), (6.220) and (6.222) are all different: in Equation (6.214), Wx and w2 are measured with respect to an arbitrary laboratory frame; in Equation (6.220), they are measured with respect to the ~'-frame and in Equation (6.222) with respect to the k-frame.

The utility of the KF expansion results from the fact that the molecular OZ equation can be written in a particularly simple form in terms of the KF harmonic expansion coefficients as (184,222-224)

mn mn n 1 H x (k) - C x (k) + p ~ --X[-[mnl ( k ) C ' ~ (k) nl

(6.226)

245

This equation bears a striking resemblance to the OZ equation for a mixture of spherically symmetric molecules; this similarity is exploited in Section 6.4.2.1 to solve the MSA analytically for dipolar hard spheres.

The final result which is required is the equation relating fmnl(r) to Uxn(r), the real-space function whose Fourier transform is F~xn(k ). From Blum and Tor- ruella (222), we have

fm"(r) = (-)z fn~Z(r). (6.227)

The lth order spherical Bessel function can be related to Pl (x), the Legendre poly- nomial of order l, by

1 fo I [eikrt )l -ikrt]d t jt(kr) - ~ P~(t) + ( - e (6.228)

By substituting this relation into Equation (6.224), one obtains

/7 Hxn(k) - [eikrJxn(r ) + e-ikrJxm(r)]dr (6.229)

where

m+n ) fr (< ) jxn(r) _ (_)x2r c ~ ( m n 1 ~ r £mr~t (6.230) l=lm-,~l X --X 0 Pt (rl)rldrl.

Similarly, for the direct correlation function, one finds

/7 mn _ _ mn S X (r)]dr (6.231)

where m+n

Sxn ( r )_ (_ )x2r r ~ ( m n l-Ira-hi X -X

1 oo T O)fr P l ( < ) ~mnl(rl)rldrl" (6.232)

The functions F~n(k) (F = C, H) satisfy the symmetry (224)

(6.233)

The "inverse" of Equations (6.229) through (6.232) is given by (224)

fmnl(r ) -- 2 l + 1 inf(m'n) ( ) f0 ~

X=-inf(m,n) X --X 0 ( - ~ (rl)

(6.234) 1 1 p , , _

X { !(51(r-r re)-- 7~P/(1)5(r -r l )+ r3 l ( ~ ) O(r rl)} drl (6.235)

where (f, 9 r) - (h, J) or (c, S) and 0(x) is the Heaviside function.

246

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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 255

7 Q U A S I L A T T I C E E Q U A T I O N S O F S T A T E F O R M O L E C U L A R

FLUIDS

N.A. Smirnova and A.I.Victorov

Department of Chemistry St.Petersburg State University St.Petersburg, 198904 Russia

7.1 Introduction 7.2 Basic Features of Lattice Theories; Structure of a Quasilattice EOS 7.3 Influence of Molecular Size and Shape 7.4 Contact-Site Models for Fluids with Strong Directional Attractive Interactions 7.5 Results of Thermodynamic Modeling by the Quasilattice EOS 7.6 Conclusions References

256

7.1 INTRODUCTION

Lattice theories of fluids can be traced back to the nineteen-thirties (1-4) when Eyring, Frenkel, Lennard Jones, and Devonshire showed that lattice models can serve to obtain an equation of state (EOS) and to describe the thermodynamic behavior of a dense fluid composed of similar sized spherical molecules. For a long time, however, the lattice models belonged rather to the realm of physics, and their application for practical purposes had not been started in chemical engineering. The wide practical use of quasilattice models was first due to the works of Flory (5,6), Guggenheim (7), Barker (8), and others on the description of the excess thermodynamic functions of liquid solutions. These models allowed a description of the entropy effects in their relation to the differences in molecular size. The approaches also accounted for the contribution of intermolecular attractions to the excess thermodynamic properties of a solution. However these approaches were incapable of describing the volume effects, nor did they lead to an EOS, because a model of a rigid, incompressible lattice had been utilized. Thus, until the nineteen-fillies the existing quasilattice EOS's were restricted to systems of nearly spherical molecules and found little practical application. The interest in the quasilattice EOS's was animated after the work of Prigogine (9), in which the ideas of the lattice approach were combined with the corresponding-states theory, and an EOS was constructed for solutions containing chain molecules. Further development of this approach is due to Flory et al. (10,11), and to some more recent studies (12-15). The fragments of the Prigogine and Flory theory are present in many contemporary EOS's, e.g., PHCT(16), APACT (17) and some other (18). However the Prigogine approach is originally designed to model the liquid, not the gas and, hence, does not allow a description of liquid-gas equilibria.

The practical application of quasilattice theories for modeling of molecular fluids, both in the liquid and in the gas states, was started by the works of Sanchez and Lacombe (19,20), who first derived a lattice gas EOS for a fluid composed of molecules of arbitrary size on the basis of Flory's statistics for mixtures of r-mers (5). Among the numerous modifications of the EOS proposed later, the most general versions are related to the implementation of the molecular contact-sites approach (21-24) and the quasichemical approximation (21,23-25). These made it possible to describe the orientation effects and the association and to formulate a group- contribution version of a lattice-gas EOS. At present the applicability range of lattice-gas EOS's is rather wide. Certainly they should be considered as semiempirical, the lack of rigor in their theoretical derivations being to a large extent compensated by the introduction of various adjustable parameters to perform calculations. The merit of lattice models is the simple way they provide to reflect the molecular characteristics of the system, in particular, strong interactions such as repulsion or hydrogen bonding.

A common weak point of all these models (leading, e.g., to their inability to describe the compressibility factor of some model fluids correctly) is embedded in their discrete-space provenance. Recently, new continuous-space versions of the lattice-gas EOS's were developed (26,27) which give a quantitative description of computer experiments on long chain (and short chain) molecular fluids. This work has inspired the appearance of a new family of lattice-gas EOS's, mainly for hard-chain fluids (28-31).

In what follows we shall discuss the basic features of the hole-lattice approach to an EOS. A variety of lattice-gas EOS's of practical importance will be surveyed. The influence of the molecular size and shape, the orientation effects and association between molecules will be considered in various approximations. For associating fluids a comparison with "chemical

257

equilibrium theories" (32) and with the Statistical Associated Fluid Theory (SAFT) (33) is drawn. The results on fluid-phase equilibria modeling with the aid of different EOS's are given. The applicability of the EOS's and the trends and perspectives in their further development are discussed. In the present paper we will leave apart the consideration of polymer systems. A separate chapter is devoted to this large and very important field. Critical phenomena are also beyond the scope of our discussion, though many useful results have been obtained in this area with the aid of the lattice approach.

7.2 BASIC FEATURES OF LATTICE THEORIES; STRUCTURE OF A QUASILATTICE EOS

Lattice models attribute a quasicrystalline structure to fluids. This implies that the system volume is divided into cells, whose size is of the order of intermolecular distances in a liquid. In the so-called cell theories every cell is occupied by a molecule, whereas in hole (or lattice gas) theories some vacant cells are allowed. Consequently, the lattice-gas theories are capable of modeling both liquid and gas states of a fluid, and the cell theories apply exclusively to the liquid state.

The quasilattice approach is used to evaluate the interaction energy and the configurational partition function. It is not concerned with the thermal motions of molecules and, hence, does not impose any restrictions on their movements throughout the system volume. The evaluation of the configurational partition function reduces to the estimation of the interaction energy for different ways of distributing the molecules over a perfectly ordered array of the lattice cells, and, to the calculation of the configurational integral of a single molecule within the volume of a cell, which is called free volume. Since the number of ways to distribute the molecules over the discrete array of cells becomes countable, it seems to be much easier to estimate the probability of any particular configuration of cells, and, hence, to evaluate the partition function, rather than taking the configurational integral over the continuous set of spatial coordinates. However reasonable these arguments might seem, our ability to get rigorous results, even for this system with countable states, still remains essentially limited to what is called the two-dimensional Ising model (34), and we shall not consider these theoretically refined approaches. The shortcomings of a quasicrystalline approach in attributing too much order to a fluid structure are obvious. The quasilattice models consider only a part of the continuous configurational space of a real fluid and their accuracy is determined by how completely they represent the configurations giving significant contributions to the statistical averages. Nevertheless, the approach served to establish the approximate relations between the characteristics of intermolecular interactions and the thermodynamic behavior of fluids and has proved impressively fruitful in the derivation of various simplified equations of state for complex molecular fluids.

The main goal of the cell theories of liquids is to find the free volume, and it is evaluated by writing a mean-field interaction energy between a particle and the environment. The dependence of the free volume on temperature and density determines the equation of state of a fluid. In the case of chain molecules their segments are accommodated in the cells, each having an effective number of external degrees of freedom, which is less than three to account for the connectivity of the molecule. There has been substantial progress in the development of cell- lattice EOS's, in particular, new versions of them have been created, which can be applied to associated solutions (13,15). Nevertheless, the leadership in the modeling of thermodynamic

258

properties and phase equilibria for practical applications belongs to lattice-gas EOS's, and we turn to discuss these equations now.

In the classical versions of lattice-gas models, all the molecules are located at the lattice sites (there is no free-volume term, the models are not concerned with the displacements of molecules within the cells), and thus the thermodynamic properties of a fluid are determined solely by the lattice statistics. To bring into consideration the density of the system, a certain fraction of lattice sites is allowed to be unoccupied (occupied by holes), which leads to the dependence of thermodynamic properties upon the volume, and, consequently, to the EOS. If we write the system volume, V, as

n

V = vh(N 0 + Z r i N i ) (7.1) i=1

where n is the number of components, N O is the number of holes, N i is the number of

molecules of type i occupying r i lattice sites each, and assume that the volume per lattice site, • v h , is constant, then, for the chemical potential of the component i, we get:

/ y / ) - - IAi -- =~.l i - ril.t o ( i , j = 1,2,...,n) (7.2) T , V , N j¢ i

with

la k = ( k , l = 0,1,2,...,n) (7.3) T, N t , k

Here index 0 refers to holes and A is the Helmholtz energy of the system. Equations (7.2) and (7.3) relate the thermodynamic functions of an n-component system to the properties of a lattice fluid having n+l components, one of which represents the vacancies. The system pressure, P, is connected to the quantity/Y0 for holes:

~t o = - V h P (7.4)

and the condition of equilibrium between phases (~) and (13) can be expressed as:

fi!a) = fi}/3) (i=0,1,.. . ,n) (7.5)

which includes the condition of mechanical equilibrium of the lattice fluid (p (a ) = p(I3)) .

Thus, for any rigid lattice model (i.e., one without vacant lattice sites), a corresponding lattice gas EOS can be immediately generated by means of Equations (7.3) and (7.4).

The most typical approximation in the existing versions of the lattice-gas theories is that intermolecular repulsion and attraction are treated independently. The repulsion is connected to the molecular size (represented by the number of lattice sites, r i, needed to accommodate a

259

molecule without overlap with the others), and is reflected in the lattice combinatorics. The attractive contribution is embedded in the residual terms (normally only the nearest-neighbor interactions are taken into account). This leads to a configurational partition function Q which can be written as a product of two terms:

Q = QcombQres (7.6)

and to the EOS in the form:

P = Prep + eres (7.7)

where

Prep:-k(6° In Qc°mb t gV T, Ni

/Ores=-kT(O In Qres~ (7.8) ' OV JT, Ni

Here Prep is the repulsion part of the pressure; this basic structure of the formulae corresponds to what is called the generalized van der Waals model in the nonlattice theories of fluids.

Further approximations are introduced to perform a calculation of the combinatorial and residual terms, in which the molecular nature of a fluid should be specifically reflected. In the first place, due consideration of the molecular sizes and shapes affects the expressions for Qcomb, while the existence of strong preferential attractions (specific interactions, like hydrogen bonding) must be accounted for in Qres. For a system of equal-sized molecules occupying one lattice site each, the derivation of Qcomb becomes trivial and corresponds to the entropy of an ideal solution. Even with this simplest combinatorial (repulsion) term in the EOS, and assuming complete randomness in the distribution of the nearest neighbors for the calculation of Qres (i .e. , no correlation between interacting pairs, the so-called Bragg-Williams or molecular field approximation), it has proved possible to reproduce basic features of liquid- gas phase behavior for a binary mixture, including liquid-gas and liquid-liquid equilibria, positive and negative azeotropy, and gas-gas equilibria of the first or second kind (35). In this simplest version of the hole model the type of phase behavior is completely determined by the relative values of nearest-neighbor interaction energies between like and unlike molecules (Ui j ) , and the various phase diagrams have been classified in terms of a conventional lattice theory parameter, the interchange energy: co o. = U U -(Uii .-[-Ujj)/2. If a neighboring site is

vacant (occupied by a hole), no interaction is assumed, so for the hole-molecule interchange energy: COoi = - U i i / 2 . Too much symmetry introduced into the model by the equality of the

molecular sizes and by the requirement of complete randomness prevents it from being used for a quantitative description of real mixtures. The advantages of the quasilattice approach were much more recognized in connection with the mixtures of molecules differing strongly in size, after the pioneering work by Flory (36).

7.3 INFLUENCE OF MOLECULAR SIZE AND SHAPE

The central quantity in the Flory model for a mixture of flexible and nonattracting r-mers with monomers (36) is the probability, that an r-mer molecule may be inserted without overlap

260

into a lattice occupied partly (and randomly) by other r-mers (26). This insertion probability, Pr , can be related in a straightforward manner to the chemical potential (37) and to the lattice gas EOS (26):

&.v, I f 1

[1 l n P r ( f , r ) ] + j ln - - - P r ( f ' , r ) d f ' R T r r o

(7.9)

where f is the lattice occupation fraction and v*= N A V h is the molar value of volume per

lattice site and N A is Avogadro's number. The Flory approximation (36) treats the occupied sites as if they were scattered randomly

over the lattice and disregards completely the chain molecule structure. This yields a very simple expression for the insertion probability:

P r ( f , r) = [ P r ( f ,1)] r = (1- f )r (7.10)

and corresponds to a random walk on a lattice with infinite coordination number, since it is assumed that there is always a vacant neighboring lattice site available to accommodate a consecutive segment of a r-mer chain, with the probability being proportional to the average fraction of holes. The extension to mixtures of the chains of varying chain lengths is straightforward. From Equations (7.10) and (7.9) we get the repulsion part of the EOS:

repV" (r ) - - I n ( 1 - f ) - f 1 - 1 (7 .11) R T

The Flory approach has been used by Sanchez and Lacombe (19,20) to obtain the first practically-sound, hole-lattice-theory EOS (often called the lattice-fluid theory, LF). LF has the repulsion part given by Equation (7.11) and the residual part obtained in the Bragg-Williams approximation. This equation has become rather popular since it has been successfully applied for the description of phase equilibria in pure and mixed fluids containing light and heavy compounds (20,38), and polymers (20,39). It can describe liquid-vapor equilibrium and liquid- liquid miscibility gaps with a lower or an upper critical consolute point, or with both. The main disadvantage of this EOS is its inability to describe systems with strongly polar compounds, which is obviously due to the fact that its residual contribution has been obtained under the assumption of complete randomness. Another EOS, which also has the Flory repulsion part, but takes into account the nonrandomness by means of an empirical correction parameter, is the KK (Kleintjens and Koningsveld) model (40). The KK EOS is rather flexible and is presently among the best; however, the correction brings too much empirism into the EOS. Less empirical assumptions to account for strong attractive interactions will be considered in the next section.

The Flory approximation to evaluate the combinatorics seems itself to be a very rough one. A certain improvement over this is attained in the Flory-Huggins theory (41), which corresponds to a first-order Markovian process of placing chain segments onto a lattice with a finite coordination number. The insertion probabilities are estimated then by:

P r ( f , r) = (1- f )[P(k I k - 1)] r-1

261

(7.12)

where P(k I k - 1) is the probability, that a randomly chosen lattice site, k, is vacant, given that

the adjacent site, k-1, is also vacant. This implies a difference in the joint probabilities if a site, k-1, is occupied by an r-mer segment, or is vacant. The probabilities are estimated taking into account the difference between the available lattice sites next to the terminal and to the internal segments of an r-mer chain. Essentially the same lattice statistics have been considered by Chang, Miller, and Guggenheim (7). The repulsion part of the EOS takes the following form:

PrepV • z I q-r 1 - I n ( l - f ) + l n l + f (7.13) RT 2 r

Here z is the lattice coordination number, and q is an r-mer surface area parameter such that zq is the number of lattice sites neighboring an r-mer molecule. The generalization of the approach to the multicomponent case of flexible r-mers of arbitrary chain lengths has been made by Guggenheim (7). Equation (7.13), unlike Equation (7.11), incorporates the parameter q, which depends on the structure of a chain molecule. However, the Miller-Guggenheim- Huggins statistics does not apply when some of the r-mers contain closed rings. An empirical extension of this approach for more general molecular geometry, including both chain-like and ring-like molecules, has been developed by Staverman (42). His expressions contain a shape parameter, the so-called bulkiness factor 1:

I~D Z -~(q - r) - ( r - 1 ) (7.14)

The Staverman combinatorics leads to an EOS of the form:

repV* Z I q r ] ' - I n ( l - f ) + In l + f + f (7.15) RT -2 r r

which reduces back to Equation (7.13), when there is no ring-like structures in the system ( l = 0). Equations (7.13) and (7.15) have been incorporated into some practically sound EOS's (23-25,43-46) to account for the molecular shape.

The basic deficiency of the EOS's considered so far is that all of them give logarithmic dependency of the athermal (repulsive) compressibility factor on the occupation fraction, this dependency being much too weak in comparison with the results of computer simulation for nonattracting flexible r-mer chains (26,27). The lattice gas EOS's dramatically underestimate the compressibility factor for a hard-sphere fluid (r=l) and also give too small an effect of chain length on the EOS (Figure 7.1). This major drawback of the lattice-gas EOS's has been overcome (26,27) by the invention of a continuous-space generalization of the Flory method. The molecular chain structure is taken into account by exclusion volumes, saying that because of the overlap of the r-mer segments somewhat less space is needed to accommodate a segment of an r-mer, than a monomer, in a fluid. In this approach, called the Generalized Flory Dimer EOS (GFD) (27), the insertion probability for an r-mer is expressed as:

262

3 0 i I i t

O . (1)

20

10

0.0 0.0

r=4 ° r=2 -/

- [] / / / -

/ r = l

[ ]

~ I"/*/

i I i I t

0.2 0.4 O.B f

Figure 7.1 Compressibility factor Zre p v e r s u s the reduced density (occupation fraction) for hard-chain fluids of various chain length, r. Computer simulation results: asterisks, triangles (124) and open squares (26). The LCT EOS (52) [Equation (7.18)]: full squares. The lattice gas EOS [Equation (7.13)]: dotted lines. The SAFT EOS: dashed lines. The 'quasichemical dimerization' approach [Equations (7.24), (7.25)]: solid line.

Pr(f,r)= p~(f)--Yr p2 (f)Yr +1 (7.16)

where P1 and P2 are the probabilities to insert a monomer into a fluid of monomers and a dimer into a fluid of dimers, Yr is the quantity related to the exclusion r-mer volume and determined by purely geometric arguments (27). The EOS written in terms of the compressibility factors has the following form:

ZGFD(f, r) = (Yr + 1)Z(f,2) -YrZ(f,1) (7.17)

where Z ( f , 2 ) , Z(f,1) are the compressibility factors for fluids composed of dimers and of

monomers, respectively, for which the Tildesley-Streett EOS and the Camahan-Starling EOS are used in this continuous version of the theory (27). General recursion formulae have also been derived (27), which relate the EOS for an r-mer to the EOS's for (r-l)- and (r-2)-mers. The GFD EOS gives excellent agreement with the computer simulation results for r-mers of various chain lengths, up to very long ones (r=201) (47). Definitely, these continuous generalized formulations opened a new period in the development of the quasilattice equations, and much of the effort has been directed to creating and testing new versions of them (28-31). However, there is some evidence that the approach does not provide correct values for the insertion probabilities, and that the success of the EOS is merely due to a cancellation of errors (48).

263

Not only the chain length and the shape of the molecules, but also their rigidity affect the fluid structure and thermodynamic behavior. It is well known, that anisotropic hard-core molecules (such as rods, disks, etc.) can under certain conditions form nematic liquid crystals, that is the fluids having long-range orientational ordering. The Flory method has been utilized by Di Marzio (49) to describe the statistics of an anisotropic fluid formed by rigid r-mer rods. A useful extension of this method was elaborated recently for a multicomponent lattice gas of hard rectangular parallelepipeds (50). The approach proved to be successful in explaining the effects of molecular shape on phase diagrams for mesogen - nonmesogen systems, and in studying the influence of molecular biaxiality on phase equilibria for binary liquid crystal mixtures (51 ).

Recently a new method for considering lattice statistics has been proposed which gives formally exact results (52-54). The characteristic feature of this so-called Lattice Cluster Theory (LCT) is that, unlike other mean-field methods considered so far, it describes the multiparticle (two-particle, three-particle, four-particle, and so on) correlations. The LCT provides a systematic method for evaluating corrections for many body correlations to the zeroth order uncorrelated mean-field approximation. The molecular shape and rigidity are also taken into account. The theory reduces, under certain conditions, to the Flory statistics for fully flexible r-mers, and to the Di Marzio formulae for the case of rigid rods (52). It has been applied to describe a system of hard rigid rectangular mesogens, which may exhibit isotropic liquid and nematic, and also discotic nematic liquid crystalline phases (54). The LCT has been used to model polymer systems at low pressures (55) and to describe liquid-liquid equilibrium (56). In both cases good agreement with experimental data has been achieved.

Because the LCT gives formally correct results, it can serve to illustrate the principle limitations of lattice gas EOS's. From the LCT corrections to the Flory mean-field Helmholtz energy (52) one can obtain the pressure of a lattice fluid as a power series in the occupation fraction and the space dimensionality, d :

- l n ( 1 - f ) - f - - R r -~-

4

- 3 7 - 4a '~

_ ( ~ 3 2 (2 r -1 )2 (d -1 ) f4+o( f sd_3 ) 24d 3

(7.18)

At the same time, expanding the second logarithmic term in the mean-field EOS, Equation (7.13), we get:

R T - I n ( l - f ) - f - f - -~2--(@/3 4f----~3 + O(f 4) (7.19)

z 3z 2

Comparing Equations (7.19) and (7.18) for a three-dimensional (d=3) cubic lattice with z=6 we find that they are identical up to the terms includingf . This implies: (1) that the Guggenheim- Miller-Huggins EOS is a real improvement over the Flory EOS, Equation (7.11), and (2) even the exact results for a lattice fluid deviate significantly (Figure 7.1) from the computer simulation data for continuous fluids of nonattracting r-mers (26). Thus, the main limitation in describing the repulsive part of an EOS is caused not by poor lattice statistics, but rather by the introduction of the lattice itself. Nevertheless, as we will see later, the quasilattice EOS can be

264

successfully applied for phase equilibria modeling, apparently because of a cancellation of errors in the repulsive and the attractive parts.

fortunate

7.4 CONTACT-SITE MODELS FOR FLUIDS WITH STRONG DIRECTIONAL ATTRACTIVE INTERACTIONS

Today there are three basic classes of EOS's for associated fluids: the chemical-theory EOS (combination of the law of mass action with an EOS selected to model physical interactions between the molecular species present in the fluid (17,57)), the SAFT (33,58,59) (an approach based on the perturbation theory expansion for the Helmholtz energy of an associated fluid (60), see Chapter 12 of this volume), and the lattice quasichemical EOS (21,23-25,44,45). The two latter approaches utilize a model according to which a molecule has different contact sites (functional groups). Some particular kinds of contact sites can participate in strong short-range attractions leading to the formation of aggregates, and some cannot. In this way the preferential molecular orientations due to association are reflected. The quasichemical approximation is one of the theories most commonly used to account for nonrandomness in lattice models.

The quasichemical approach was originally proposed by Guggenheim (7) and generalized later by Barker (61) and Kehiaian et al. (62) for the case of molecules composed of an arbitrary number of different functional groups. The expression for the residual contribution to the configurational partition function of a lattice fluid, Ores, reflects the effects of nonrandomness and can be written as:

Ores = (zA, s)! , (zA, t / 2)!

1

S s , t

(7.20)

where U c is the configurational energy of the fluid, zA,t is the number of pairs between groups of kind s and of kind t in a totally disordered random state, and zA, t is the most probable number of pairs for the fluid of interest, this number corresponding to the maximum term of the configurational partition function, Equation (7.6). The equations to find these latter numbers have the following form:

12ostl = 4 exp - (7.21) (zAsszAtt ) R T

This resembles the equation of the law of mass action for a 'chemical reaction' involving the formation of s-t contacts from s-s and t-t contacts, wherefrom the term 'quasichemical' originates. Typically, the condition of conservation of the numbers of contacting lattice sites applies for various lattice models, which allows one to rewrite Equation (7.21) in a more convenient form:

265

X~ ~ a,Xt r/s, = 1 (7.22) t

Here (~t is the surface-area fraction of groups of type t in a fluid, ~st-- exp(-cos,/RT), and X i is

the target variable connected to the numbers of pairs by:

Ast = AasatXtX~ fist (s :/: t) As, = A (a,X~ )2 /2 (7.23)

where A is the total surface area of all the species in the fluid. The quasichemical approximation has been carefully analyzed (63-65) and criticized for

giving poor pair probabilities for energetically strongly unfavorable contacts (large positive COst); however, it gives reasonable results in the case of predominantly associating pairs, as we will see below, and still remains one of the most popular and useful tools to account for specific interactions. A limiting case of a very strong association has been considered by applying the quasichemical approximation to model covalent bonds between segments of an r- mer by an infinitely intensive attraction between corresponding contact sites of monomers (66). Thus a monomer + r-mer binary mixture has been treated as a fluid of (r+l) monomeric particles with associating contact sites. This approach leads to the same expressions for the activity coefficients as for the original monomer + r-mer binary mixture (66), provided that the Guggenheim approach to the combinatorics is used for the athermal terms. The same method can serve to obtain the corrections to the EOS resulting from the connectivity of the molecules. If we consider, for example, a lattice fluid of monomeric segments associating to form dimer molecules, but otherwise athermal, then the system compressibility factor, Z, can be expressed a s l

Z = 2Zseg + Zbond (7.24)

where Zseg is the compressibility factor of the fluid of nonattracting monomers, and Zbond is

the correction for chemical bonding, which can be expressed via Qres, the noncombinatorial entropy term in the quasichemical approximation:

L ,z i,z } 1 Zbond "- 2 f In( 1 - f ) + f In f + - - - - f In f - (7.25) z z z z z

Using Equation (7.25) the compressibility factor has been calculated for the system of dimers and compared with computer simulations, (Figure 7.1). The Camahan-Starling expression has been used for Zseg , because, as we have already pointed out, the lattice-gas expressions would

dramatically underestimate the compressibility factor for the fluid of monomers. As can be seen from Figure 7.1, the quasichemical approximation gives a reasonable correction for the entropy loss due to chemical bonding, particularly at high densities. Another argument strongly supporting the usefulness of the quasichemical approach is that it gives, as we will show later, quite reasonable predictions of structural properties of associated fluids.

266

The first attempt to incorporate the quasichemical routine into a lattice-gas EOS of a chain-molecule fluid was made by Panayiotou and Vera (21); somewhat earlier the quasichemical approximation was employed in a cell-lattice EOS taking into account effects of molecular sizes (13), and in a lattice-gas theory of simple fluids composed of spherical molecules (71). Later, quite a few modifications of lattice-gas quasichemical EOS's have been proposed for practical applications (23-25,44,45). To the best of our knowledge one of the most general versions of a quasichemical lattice-gas EOS has been developed by Victorov and Smirnova (23,72). This equation, denoted hereafter as the HM (Hole Model), combines the Guggenheim-Staverman repulsion term, Equation (7.15), with a quasichemical residual contribution, which has a simple form:

eres V * _

R T lnX o (7.26)

Here X o is the solution of the set of quasichemical equations (7.22) for holes. Under various conditions this EOS reduces to the well-known lattice gas EOS's considered above, e.g., for a completely random r-mer fluid (corresponding formally to the case of infinite coordination number) the HM transforms into the LF EOS by Sanchez and Lacombe (19). If the interaction parameters COst for the groups of specific kinds are kept constant, regardless of the types of molecules these functional groups belong to, then the group-contribution EOS follows. This possesses enhanced predictive abilities. The HM has been applied to a great variety of fluids and fluid mixtures (23,67-69,72-75). In the next section we shall discuss the application of the various quasilattice and related EOS's in more detail.

The solution of the nonlinear set of quasichemical equations (7.22) cannot be performed analytically, and hence the quasichemical EOS's are doomed to be implicit in their general form. However, a special technique has been proposed (76) which expresses the solution of the equations as a series expansion, thus leading to explicit EOS's (25,44,45,77). Such an expanded version of the HM has been developed (44,45), which is well suited for practical purposes. The first explicit version of a quasichemical lattice-gas EOS is apparently due to Kumar et aL (25). This equation is written in a rather convenient way and it has been extensively used for practical applications (25,78,79). The original EOS of Kumar et al. mainly differs from other quasichemical equations proposed later in that it assumes the complete randomness of the distribution of the vacant lattice sites.

Besides the quasichemical versions of lattice-gas EOS's there is the 'chemical theory' EOS (80), which combines the Sanchez-Lacombe LF EOS (19) with association equilibrium constants. This theory has been used for modeling of phase equilibria in alkane-alcohol mixtures and gave good results (81), though they are somewhat inferior to those obtained by the HM (67).

An intermediate between a purely 'chemical' and a purely 'quasichemical' (i.e., related to the Guggenheim original approach (7) developed specifically for lattice models) version of an EOS has been proposed by Panayiotou and Sanchez (22) (Hydrogen Bonding Lattice Fluid, LFHB EOS). As in the chemical theories the thermodynamic functions are separated into physical and chemical parts. The original version of LF (22), or SAFT (82), is employed to describe the physical interaction between the molecules, while for the chemical part statistical mechanical arguments are used, rather than phenomenological empirical values of association constants. This statistical-mechanical approach (originally due to Veytsman (83)) considers the probabilities of hydrogen-bond formation between acceptors and donors as if they applied to

267

independent events. A quasichemical-type expression is obtained for the number of hydrogen bonds in a fluid (22) [a set of quadratic equations, similar to Equations (7.22) for the contact pairs]. The theory gives good results for associated fluids at supercritical and subcritical conditions (22,82,84,85); however, it is difficult to say whether it presents any improvement over quasichemical models, since no direct comparison has been made. However, the weak point of the LFHB and similar EOS (85) is the arbitrariness in the separation of the 'physical' and 'chemical' parts of the interactions, a flaw present intrinsically in any 'chemical-theory' EOS.

One of the possible routs to account for the effects of nonrandomness is provided by the local composition concept, which found a most extensive application in constructing GE- models of liquid solutions. This concept has recently been employed in the frame of lattice-gas EOS's (46,86,87). The relation of the concept with the quasichemical approximation to nonrandomness is well established in the literature.

The analytic relation between the 'chemical theories' and the quasichemical lattice model has been discussed in its general form (88). Early consideration was carried out for two particular types of association (89): (I) the formation of mixed AB associates, and (11) the consecutive association of monomers A to form i-mers A i (i=2,3,...). Let us denote the specific contact sites participating in strong interactions by O and H. If the consecutive association (type II) takes place in an A+B binary mixture, both specific contact sites belong to the A molecule. To model the association of type I assume that A molecules have one O-contact site each, and B molecules have one H-contact site each. The numbers of different molecular species can be expressed via the law of mass action and also can be obtained from the number of contact pairs (the latter being estimated in the quasichemical approximation). For an association of type I, the number of associates is simply equal to the number of contact pairs NOH. For type H, the most probable set of the numbers of different molecular species A i , NAi, is determined by the maximum number of ways this set can be generated on a lattice at given values of NOH, NA, and NB. Using the Guggenheim lattice statistics for the mixture of chain molecules A i (and the Lagrange multipliers method), one finds the following distribution for i- mers:

NA i = NAfl2(I_ fl)i-1 (i = 1,2,...) (7.27)

where/3 = (N A - NOH ) / N A .

The nonideality of the mixture was taken into account in the Guggenheim athermal and nonathermal approximations using the law of mass action (via the chemical potentials of molecular species). It was found that the quasichemical approach and the law of mass action give similar monomer and associates concentrations, the association constant, K, and the interchange energy for the specific interaction being related approximately as:

K = exp(- c0 OH/RT) (7.28)

with

268

K . _

X~ (xA , QA ' + xB, QB ' + x ~ Q ~ )

XA 1XB l oo

XA, (xBIQBI + ~ XA, QAi ) i=1

XA 1 XAi. !

(model I)

(model II)

(7.29)

Here x~ is the mole fraction of molecular species a, and Q~ is the number of contact sites for

molecular species c~ (a=A1, B 1, etc.) The difference between the two approaches is due to the fact that in the quasichemical

approach we deal with the numbers of pairs of different types and in the theories of association equilibria with the numbers of molecular species (monomers, dimers, etc.). Although, as has been shown, the approaches give similar results for the same association model, in general they differ in their capability to describe different types of association. The most serious restriction of contact-site models is their inability to describe the dependence of interaction parameters on the structure of associates (thus, the energy of the O-H bond is supposed to be the same for associates AB and AB 2, for A 2 and A3, A4, etc.). Some types of association (e.g. formation of only trimers or only tetramers) cannot be described by contact site models, yet theories of association equilibria do not have any difficulties in this respect. But there are situations when the contact-site models work well, whereas the approach using the law of mass action can hardly be suitable (e.g., systems where a network of strong bonds is formed).

An alternative approach for associated fluids, the SAFT (33,58,59), is today the most theoretically rigorous and it also became the most popular one. It employs the contact-site model and in this respect is similar to the quasichemical approaches considered above. In its essence, however, the SAFT is a perturbation method (33,60), giving the correction for the Helmholtz energy of a fluid of nonbonded segments (monomers) due to strong short-range attractions between specific sites (in the limiting case, of infinitely strong attractions due to the chemical bonding between segments). We merely give a very short outline of the theory, because Chapter 12 of this book discusses it in every detail. The difference, Abona, between the Helmholtz energy of the associating mixture with multiple associating sites and the Helmholtz energy of the reference fluid of non-associating components is given by (33):

AR°n; "-~i XiI~A~illnSiA--S--~2 I"t-~l (7.30)

where x/ is the mole fraction of the component i, n i is the number of associating sites on the

molecule of component i, the second sum is over all such sites on the molecule. Xf is the

fraction of molecules of component i not bonded at site A. These quantities can be found as the solution of the nonlinear equations:

xA II+ i L j B ~ j tj

269

(7.31)

Here p is the total number density and A AB is defined by:

AB f ref A0 = Jgu (12)f7 AB (12)d(12) (7.32)

f/jAB =exp - k T ) - I is the Mayer f-function, g~ef is the reference-fluid pair-correlation

function, and the integration is performed over all the orientations and separations of molecules 1 and 2. Thus knowledge of the pair distribution function of the reference fluid is sufficient to calculate the properties of the associated mixture. Often the distribution function for a hard sphere fluid is used together with a simplified expression for A~. (90):

AB .__ 4~gref (o')gAB exp - 1 A 0. (7.33)

where VAB is the association volume parameter (the integral that measures the volume

available for bonding to sites A and B), gref(o.) is the contact value of the pair distribution

function, CAB is the association potential (the square-well depth characterizing the specific AB

attraction). The EOS is obtained from the above equations by a standard routine. For instance, in the case of hard spheres with infinitely strong attractions, which make them stick into hard flexible chains, one can write:

Zchai n = ZHS + Zbond (7 .34 )

where ZHS is the compressibility factor of a reference fluid of free hard-sphere segments, and

Zbond is the contribution due to bonding, which corresponds to Equation (7.30) and infinitely

large CAB. From this the well-known SAFT EOS for athermal flexible r-mers (33,90), which is in excellent agreement with Monte-Carlo simulation results, follows. In a more general case of associating chain molecules interacting as well via dispersive attraction forces the compressibility factor consists of all these contributions:

Ztota 1 = Zchai n + Zasso c + Zdisp (7.35)

where Zasso c is, again, calculated with the aid of Equation (7.30), and the contribution of the dispersive interactions is accounted for by standard perturbation terms (91). The SAFT EOS has been applied to a great variety of pure fluids (58) and fluid mixtures (59,92,93). The generalization of this equation for the case of chains built by spheres of dissimilar sizes is

270

available (94). A special recurrent procedure to correct the EOS for long chains and to modify it for polymers has been proposed (95). Definitely, in the last few years the SAFT has become the most intensively used and rapidly developing method which is taking over quasilattice EOS's in the field of thermodynamic modeling of complex fluids for practical purposes. However, the SAFT is also based on certain approximations, some of them being similar to those of the quasichemical contact-site quasilattice approach. One of the most important simplifications of both methods is that the particular location of specific sites within a molecule is not reflected: thus, isomers can not be naturally distinguished. Nevertheless the effect of the location of contact sites can be very important. For instance, a two dimensional lattice system of squares, each having a couple of specific (attracting) edges, shows different types of orientational ordering, depending on whether the specific edges are adjacent or opposite (96). None of the existing molecular models, however, takes such an effect into account.

The relation between all three approaches for associating systems: the 'chemical theory' EOS's, the SAFT, and the quasichemical lattice models has recently been discussed (88). The analysis was performed using the Flory equal-reactivity principle (the assumption that the probability of bond formation between any two sites is independent on the actual shape and size of the particular clusters on which these sites lie). It was shown that the equations describing association within all three approaches are quite similar. The same conclusion was made earlier (32) for several particular association schemes. The calculation showed that all three approaches lead to quite similar results for association, the difference being mainly due to different physical-interaction terms in the EOS's. The fraction of nonbonded molecules is expressed by similar formulae, and the numerical evaluation for associating pure and mixed hard-sphere fluids gives similar results for all three EOS's and agrees quite well with computer simulation (32).

7.5 RESULTS OF THERMODYNAMIC MODELING BY THE QUASILATTICE EOS

The quasilattice EOS's have been extensively applied for modeling phase equilibria and calculating thermodynamic properties for molecular fluids. Liquid-gas (20,23,25,44- 46,73,80,81,85,86,97-100,102-104), gas-gas (103,106), liquid-liquid (20,101,103), solid-liquid (25,97) equilibria and equilibria with more than two coexisting phases (69,101) have been studied. Supercritical fluids and the solubility of solids in supercritical solvents were considered as well (77,84,98-100,104,105,107). The performance of the lattice gas EOS's has been compared with that of nonlattice equations, e.g., cubic EOS's (38,74,75,105), chemical theory EOS's (32,67,68) and the SAFT (32,68,69,82).

Various chemical classes of pure compounds and different types of fluid mixtures have been described by lattice gas EOS's. To the most extensive studies belong those reported in references (44-46,73,78,79,97-100,102,107). The EOS's served to model many systems of practical importance. Apart from systems containing polymers (20,25,86,87,98,102,108) there are mixtures of typical components for the petroleum industry (46,98,99,101,103), solutions with near critical solvents (69,98,99), and fluids containing associating compounds (46,67,73,81,82,85,97). Special attention has been paid to aqueous - organic mixtures (68,98,101).

Among the different versions of the lattice-gas quasichemical equations the HM (23) and the EOS (45,46,98) are, perhaps, the most extensively developed for practical applications. They have been thoroughly tested and their parameters have been determined for a variety of

271

substances. Parameters are reported for 211 pure components (44) and for approximately 40 functional groups (46,73,97-99), which can be built up into even more substances representing saturated and unsaturated hydrocarbons, aromatic compounds, alkanols, ethers, acetic acid, water, carbon dioxide, hydrogen sulfide, etc. The LF model has also originally been applied for many systems (38), however, as it is not capable of handling systems with strongly polar components properly, it has been overtaken by the extended versions of quasilattice EOS's. The KK lattice-gas EOS gives an accurate description of various substances in both the supercritical and the subcritical region (with 5-6 parameters for a pure component alone). This EOS is easy to use, and it has given really impressive results for diverse fluids [systems under extremely high pressures (100), supercritical fluids (100,107), polymers(102), water-containing systems(101,109), and mixtures with electrolytes (109)]. A systematic study has been made to obtain correlations for two parameters of this EOS with molecular volumes and surface area derived using Bondi's group-contribution method (102). However the empirical basis for the entropic correction parameter of this EOS is obvious, which leaves some doubts concerning the KK EOS predictive abilities. This is particularly true in the case of multicomponent systems. For these the HM is perhaps the most thoroughly tested among the quasilattice EOS's. Their performance for phase equilibria modeling will be illustrated mainly by the results obtained with the aid of the HM.

Table 7.1 and Figures 7.2, 7.14 contain the results for selected pure components; Figures 7.3-7.13, 7.15, 7.16 give examples of calculated phase diagrams for mixtures. In Table 7.1 Ap and AP are the absolute average deviations between the calculated and experimental values for saturated liquid density and saturated vapor pressure. The components are chosen to span quite different molecular characteristics: from nonpolar low boiling substances to strongly polar ones having small or long chain molecules. Comparison with other EOS's is given. These include the SAFT, the APACT (17) [one of the most highly developed versions of the chemical-theory EOS, which reduces for unpolar compounds to the PSCT (110)], and a lattice-fluid EOS, LFAS

Table 7.1 Description of selected pure components by the HM EOS, a group-contribution quasichemical model and several other EOS's.

Substance T(K) A9, % AP, % Model*) CO 2 216-298 1.7 0.3 HM (68)

216-298 0.8 3.0 APACT (68) 218-288 0.86 2.8 SAFT

Methane 91-188 5.4 1.5 HM (75) 91-180 0.81 0.62 APACT (111) 92-180 0.35 1.4 SAFT

Ethane 150-300 5.1 1.6 HM (75) 150-305 0.62 1.4 APACT (112) 160-300 1.6 1.8 SAFT

Propane 193-367 2.7 1.1 HM (75) 190-360 1.8 2.1 SAFT

Butane 213-423 2.2 1.6 HM 213-423 2.8 0.63 APACT 220-420 2.6 2.3 SAFT

Pentane 253-470 3.1 2.4 HM

272

253-470 3.3 0.60 APACT 200-430 1.0 4.4 LFAS 233-450 3.0 1.9 SAFT

Hexane 273-507 2.3 3.0 HM 273-503 2.1 0.44 APACT 220-430 0.42 4.1 LFAS 243-433 3.5 2.3 SAFT

Heptane 273-537 3.6 2.3 HM 273-537 2.1 1.1 APACT 240-470 0.16 4.0 LFAS 273-523 3.4 1.8 SAFT

Decane 273-433 0.5 4.1 HM (74) 278-525 1.4 0.89 APACT (111) 313-573 3.5 2.2 SAFT

Water 273-646.9 2.3 1.7 HM 319-643 1.1 0.96 HM

273-646.9 3.5 3.7 APACT 319-643 3.3 3.2 APACT 283-613 3.2 1.3 SAFT

473(single phase region) 46. HM 473(single phase region) 25 APACT

Methanol 270-502 1.0 1.3 HM (68) 270-502 1.5 3.2 APACT (68) 273-487 0.88 0.83 SAFT

Ethanol 253-513 0.85 0.19 HM 253-513 1.9 1.4 APACT 290-450 4.2 2.5 LFAS 302-483 0.86 0.83 SAFT

1-Propanol 293-536 1.6 2.3 HM 273-513 1.2 1.2 APACT (68) 290-460 2.3 1.2 LFAS 293-493 1.2 0.16 SAFT

1-Butanol 373-562 0.56 1.5 HM 293-476 1.0 0.5 APACT (68) 320-500 1.6 7.0 LFAS 313-493 0.23 1.0 SAFT

1-Dodecanol 393-513 0.9 3.8 HM (68) 393-513 0.4 2.9 APACT (68)

*) Unless otherwise stated the data are taken from the following references" HM and APACT (67), SAFT (58), LFAS (81). For CO 2 and alkanes the APACT EOS reduces to PSCT (110).

(80). The results for some of these EOS's were taken from original works in which, unfortunately, the considered temperature intervals were not the same, and, accordingly, they differ somewhat for different EOS's in Table 7.1. As far as the HM is concerned, very wide temperature intervals are taken extending almost over the whole liquid-state domain up to temperatures very close to the critical. The description of the single-phase states is illustrated for water by an isotherm at 473 K between 25 and 1000 bar. As can be seen from Table 7.1, the accuracy of the description of pure components with the aid of the HM is quite satisfactory for

273

practical purposes. Both the HM EOS and the SAFT EOS are density implicit and require the sets of nonlinear (quadratic) equations [Equations (7.22) and (7.31), respectively] to be solved in the course of calculations. In this respect the explicit expanded version of the model (45, 98), the local composition lattice-gas EOS (46,86,87), and the APACT EOS are somewhat easier to apply. The HM and several other lattice-gas models (46,86,98) are group-contribution approaches and as is the case with any group-contribution method the specification of groups and their parameter estimation is a difficult task. Once the parameters have been estimated, however, the models can be recommended for use, especially when there are no experimental data available for a given system.

A characteristic feature of the lattice-gas EOS's is their lower accuracy for the volatile nonpolar substances such as propane, ethane, methane or nitrogen. In this seemingly simpler case, the lattice gas EOS usually gives worse results than for longer-chain compounds (Table 7.1) and is often inferior to the PSCT, SAFT and modem versions of cubic EOS's, which can be made impressively accurate by means of an empirical technique (113). For longer chain alkanes the quality of description by the HM improves somewhat, though it is still not as good as that given by the PSCT, SAFT or by purely empirical correlations (113). The situation changes for strongly associated fluids. In this case the HM becomes quite accurate and competes well with other physically meaningful EOS's. It should be noted also, that even though the HM somewhat overshoots the critical temperature (as do many other classical EOS's), it gives a fairly nice shape to the coexistence curve for strongly polar substances (Figure 7.2) and works reasonably well at temperatures only several degrees below the critical point [e.g. for pure ethanol the deviation of the predicted critical temperature from the experimental one is about 8 K (67)]. The results become somewhat worse in the case of nonpolar components, particularly low boiling ones.

One of the shortcomings of the HM, as has been already mentioned is that it is density implicit. As one can infer from reference (44,98), the expansion of quasichemical terms, leading to an explicit EOS, basically does not spoil the quality of the description of pure components, even in the case of strongly polar ones (Figure 7.2). The local composition versions of the lattice gas EOS's (46,86,87) are also explicit. We note that these latter can provide certain improvement over HM in the description of low boiling substances (46).

For mixtures composed of light nonpolar compounds simple cubic EOS's often work better than the HM, particularly in the neighborhood of the critical states, Figure 7.3. The lattice-gas EOS's become more useful for the mixtures containing a long-chain component (Figures 7.4, 7.5). The capabilities of quasichemical EOS's are, again, unveiled to the full extent for systems with associating components, for which they give very good predictions of phase equilibria over a wide range of temperature and pressure, and are often superior to other EOS's. The examples of liquid-gas equilibrium diagrams for such systems are given in Figures 7.6-7.10. Reasonable results are obtained up to extremely high pressures reflecting the existence of the double homogeneous point and of the gas-gas equilibrium (Figure 7.10). The description of liquid-liquid equilibrium at normal pressures is quite good (Figures 7.11, 7.12). Multiphase equilibria can be modeled as well by the quasichemical EOS's (Figure 7.13). Serious problems were encountered however, in predicting the dependence of liquid-liquid equilibrium upon pressure: the dependence given by the HM EOS was far too weak (69). The performance of the lattice gas EOS in calculating caloric properties of pure fluids and mixtures is a field not very

274

1 O0 I I

co I, E o

0 3 v

> , ,

or) c" (1) n

0.80

0.60

0.40

0.20

000 *

250

] 350 450 550

Temperature (K)

Figure 7.2 Liquid-vapor saturation line for pure ethanol calculated by lattice-gas EOS's. Solid curve: the HM. Dashed curve: the expanded version of the quasichemical lattice-gas model (44). Dotted line: the random hole model by Kumar et al. (79), given after reference 44. Points: experimental data (128).

13..

v

(1)

o') o')

13_

_ i i i i i i i i i

J _ . , ~ 8 K

j ~ z ~

• / *

2 ~ " * ' ~ * f 253 K

,~~-" ,..~ ~ , _ _ _ _ _ - - -

n a - r - - 7 - - r , , , , , , 0 0.2 0.4 0.6 0.8 1 . 0

Mole fraction of 002

Figure 7.3 The results of the liquid-gas equilibria calculations for propane + carbon dioxide mixture at temperatures below and above the critical point of carbon dioxide. Solid curves: the HM (75). Dashed curves: the Peng-Robinson EOS (75). Points: experimental data (126,127).

t~ n

v

03

CO O0 03

13..

15

10

0 0.0 1.0

1 I I I

I 1 I I

0.2 0.4 0.6 0.8

Mole fraction of CO 2

275

Figure 7.4 Carbon dioxide + decane liquid-vapor coexistence curves predicted by the KK EOS. Points: experimental data (139). From reference 101.

352 K

0.0 0.2 0.4 0.6 0.8 1.0

Mole fraction of H2S

Figure 7.5 Description of the vapor-liquid equilibria in the hydrogen sulfide + heptane system by means of the expanded version of the quasichemical EOS by Kumar et al. (curves). Points: experimental data (141). From reference 140.

276

13_ v

(1) x,_,

x . . .

13..

0.20

0.15

0.10

0.05

I ' I ~ I ~ I '

363 K

343 K

323 K

0 . 0 0 1 , I ~ I , I t I , I

0.0 0.2 0.4 0.6 0.8 1.0

Mole fraction of ethanol

Figure 7.6 Description of the vapor-liquid equilibria in the ethanol + water system by means of the expanded version of the quasichemical EOS by Kumar et al. (curves). Points: experimental data (142). From reference 140.

I I I I

6.5 T-523 K -

5.5

4.5

~ 3 . 5 ) . T=473 K -

~- 2.5

1.5

0.00 ' '

0 1 Mole fraction of acetone

Figure 7.7 Vapor - liquid equilibrium for acetone + water system calculated by the HM (73) (curves). Points: experimental data (129).

3

2

© 2

1

0 0.0

l I I I I 1 473 K

448 K

0.2 0.4 0.6 0.8 1.0 Mole fraction of hexane

277

Figure 7.8 Prediction of the vapor-liquid equilibrium for hexane + ethanol by the HM EOS in its group-contribution version (solid curves) and by the APACT EOS (dashed curves) (67). Points: experimental data (130).

cl

v

©

--3 o3 if)

cl

80

60

40

20

~ i l v J I I i ~ i ~ l l l l 1

i i i I i i i 1 [ i i i i I I i 1 ] i 1 i i i i i i

1 10 100 1000 Hexanol Concentration (10 -3 g'c m-3)o

Figure 7.9 The vapor-liquid equilibria of the binary hexanol + sulfur hexafluoride at 362.8K predicted by the HBLF EOS without any adjustable mixture parameters (curves). Points: experimental data (137). From reference 84.

278

400 . . . . . . . . .

1 2 3 k ~43 2 1

300

• 200

lOO

O0 0.2 0.4 0.6 0.8 1.0

Mole fract ion of CO 2

Figure 7.10 Phase diagram for water + carbon dioxide system at elevated temperatures: 473K (1), 533 K (2), 538 K (3), 539 K (4), 541 K (5), 573 K (6). Solid curves: calculations by the HM EOS (103). Dashed curves: experimental data (131).

340

335 -

330 -

¢ 325 -

.~ 3 2 0 - 0

E 315 - © 1-

310 -

305 -

3OO 0.4

nm

• • •

t I I I I I /

0.5 0.6 0.7 0.8 0.9 1.0

Mole fract ion of acet ic acid

Figure 7.11 Liquid-liquid equilibrium in the acetic acid + dodecane binary mixture calculated (curve) with the aid of the group-contribution quasichemical EOS by High and Danner (24) and experimental data (points). From reference 138.

MeOH

a)

279

n-C 7 H20 C6OH

b)

n-C 7 H20 04E1

c)

H20 n-Clo

Figure 7.12 Liquid miscibility gaps in temary aqueous mixtures predicted by the HM EOS (solid lines) (69). Points and dashed lines: experimental data. (a), (b): water + n-heptane+ methanol and water+n-heptane+hexanol mixtures, respectively, at T = 298 K, P = 0.1 MPa. Composition is given in mole fractions, the experimental data are after reference 132. (c): water + n-decane + butoxyethanol mixture at T - 313 K, P - 20 MPa. Composition is given in weight fractions, the experimental data are after 133.

280

Ace tone

10 - --~ ,,

8 - EL.

d.)

00

(~ 6 - n

4 - C O 2 H20

Figure 7.13 The phase diagram for the carbon dioxide + acetone + water system at 313 K. Experimental data (134,135) are represented by solid tie lines with stars at their ends. Solid curves and dashed lines are the HM predictions (69). The triangles in the lower pressure cross-sections of the pressure-composition prism correspond to three-phase regions.

well explored, however some examples of quite reasonable results can be found in the literature (23,97,136).

The structural properties of associated fluids are reflected quite well by quasichemical quasilattice methods, in particular, this is true for the fraction of non-bonded molecules (32,67- 69), when compared with computer simulation for model systems and spectroscopic measurements. Figure 7.14 is an example of such a prediction for saturated liquid ethanol. In Figure 7.15 a comparison is given with computer simulation results on binary mixtures of Lennard-Jones molecules having conical specific sites (70). Figure 7.16 depicts the results for the carbon dioxide + hexanol mixture. The fractions of nonbonded molecules calculated by the HM agree better with the experimental data than those given by the SAFT and APACT.

For binary and temary mixtures many examples of phase-equilibrium calculations with other quasilattice EOS's are available in the literature (24,25,39,45,46,78-81,84-87,98-101,104- 107,136), therefore a general comparison can be made which shows basically the same trends for the EOS's with the effects of nonrandomness taken into account.

Thus the obvious merit of the lattice gas EOS's is their capability in describing more complex mixtures. This might be very important for practical applications of the EOS's. For instance, the quasilattice EOS's are not superior to the cubic ones for mixtures of typical oil and gas components. However, giving similar accuracy to these systems, the quasilattice EOS's

281

permit modeling of aqueous mixtures of these components, which essentially enlarges their range of applicability in comparison with the cubic EOS's. Another attractive feature of the quasilattice EOS's is that typically they give quite reasonable predictions over a wide range of temperature and pressure. This is due to the fact that these EOS's are related to a molecular

1.0 ' I ~ 1 ' I

c- O .m O t~ , x._

0 . 5 - 4,

0 / / / , c-- j ~ . . / . O -

_ J / ~ /- ......i - - - ' ' /

0.0 , ~ % - - - i " , , 2()0 300 400 500

Temperature (K)

Figure 7.14 The fraction of monomer molecules in saturated liquid ethanol. Spectroscopic data (125) and prediction by quasichemical quasilattice theories: the HM EOS (solid) and the LFHB theory (84) (dashed). APACT (67): dotted.

model, and the success of their application is gained not merely by fitting parameters for specific ranges of conditions, as is the case with some more empirical methods (113). The basic validity of this simple molecular model is confirmed by the good predictions it gives for fractions of nonbonded molecules in a fluid. Very important for practical usage is the group- contribution approach which is easy to implement for the quasilattice-contact-site EOS's. Group-contribution equations, such as the HM, or those proposed by High and Danner (24,138), Yoo et al. (98), Mattedi et al. (46) become rather convenient for mixtures formed by a very large number of homologues (103), in particular in combination with the continuous thermodynamics treatment (114).

It seems that the results given by quasilattice EOS's for fluids composed by nonpolar chain-like molecules can be improved. For these systems the most promising are, perhaps, the quasilattice continuous versions of the GFD family of the EOS's, as can be concluded from comparison with computer simulation results (27-32,47). However a better approximation applied separately for the repulsion or for the residual contribution does not necessarily guarantee better overall performance of the EOS. Thus a version of a quasichemical lattice gas EOS combined with the SAFT repulsion part did not show significant improvement over the original HM in the description of phase equilibria (74). Similar results have been obtained by trying several repulsion parts with a quasilattice residual term for a number of real fluids (115): the best description has been obtained with a lattice gas repulsion contribution.

282

As has been pointed out by several authors (74,104,116), a problem is encountered when trying to apply an EOS to phase equilibrium and to single phase (supercritical) states. It has been shown that it is very difficult to describe simultaneously the whole phase diagram of a pure substance unless two different sets of the model parameters are used for the sub- and super-critical states, the parameter-versus-temperature plots show an obvious discontinuity around the critical point (74,104). This peculiar feature does not seem to be an exceptional property of quasilattice models, it is more likely to be connected with some basic limitations of classical EOS's in reproducing the whole phase diagram including both the single-phase and the two-phase states.

a) b) 1.0

cl

E 0.8 o o

N.,.-

o c 0.6 0

o ~J X.-- 4-- 0.4 ©

E 0 ,-- 0 . 2 - o

i i •

0 . 0 I I I I

0.0 0.2 0.4 0.6 0.8 1.0 Mole fraction of comp. 1

E 0.8 O o

N--.

O c 0.6 0

o ~J

~- 0 4 ~ °

©

E 0 c 0.2 0

0.0 0.0 0.2 0.4 0.6 0.8 1.0

Mole fraction of comp. 1

Figure 7.15 Monomer fractions for component 1 in coexisting vapor (upper curves) and liquid (lower curves) phases for model binary systems with self-association (a) and cross-association (b). Points: computer-simulation data (70). Curves: prediction by the HM EOS (69) using the interaction energies between specific and nonspecific sites after reference (70), see below.

(a): U b°na = 8Ull , U22 - - 1.2Ull, f,2 = (UllU22) 0"5, T*=kT/Ull=I.1. (b): U b°nd = 10U, UI~ - U22 = U12 = U, T*=I.0.

In the both cases r 1 = r 2 = 1.

7.6 CONCLUSIONS

Basically it can be concluded that the quasilattice EOS's play an important role in the modeling of phase equilibria in fluid systems and that they are especially useful for studies of complex systems with components differing in size, polarity, and shape, and for modeling associated mixtures. The advanced versions of quasilattice models (22-25,44-46,98,107) can be successfully applied for the description of such systems and for the prediction of the behavior of multicomponent mixtures over a wide range of conditions and can be listed among the best modem EOS's in this area. Group-contribution formulations of quasilattice EOS's provide

283

remarkably good predictive abilities for systems containing various chain-molecule homologues. However, the quasilattice approach is not superior to the other modem EOS's for systems composed of small nonpolar and slightly polar molecules, in which case new modifications of the empirical cubic EOS's (113), or EOS's having more theoretical grounds (e.g. PSCT (110), SAFT(33,59)) give somewhat better results.

17

n 13 v

©

a..

' " " it I • : \\\ I I / " /; •

, d . ,\H II l ~ '/ / . p J I I • / //

• I I I

i I

II 1

5 -

, ,

0.0 0.22 0.4 0.6 0.8 1.0 Hexanol monomer fraction

Figure 7.16 Comparison of the experimental and calculated (68) fractions of hexanol monomeric molecules in vapor and liquid phases for carbon dioxide + hexanol mixture at 402K. Solid curves: the HM EOS. Dashed curves: the SAFT EOS. Dotted curves: the APACT EOS. Points: the experimental data (137).

Certainly the role of the lattice EOS's and their further development are influenced by the impressive results obtained for model systems by means of techniques which are more rigorous than the quasilattice approach. First of all the description of fluids of flexible r-mers should be addressed, for which the traditional Flory-Huggins-Miller-Guggenheim approximations presently give way to new equations showing excellent agreement with the behavior of the fluids studied by computer simulation. Such a family of quasilattice EOS's based on the empirical generalization to a continuous case is under development (30,31,47).

The consideration of strong directional interactions with the aid of the contact-site model in the quasichemical approximation still remains among the attractive features of the lattice models. A general mathematical form for equations for various types of association, the possibility of considering systems with spatial networks of associates, and satisfactory accuracy are the stimulating benefits of the quasichemical approach. It has somewhat better capabilities than methods based on the law of mass action and is simpler compared to SAFT. Of interest are the attempts to combine the quasichemical approach (for the description of strong attractions) with rather accurate expressions available for the contribution of repulsive and dispersive interactions into the EOS's (117,74,115).

284

One of the general trends of the last several years is that more rigorous methods, such as SAFT (33,91), are used very intensively in practical applications (59,92,95) somewhat diminishing the interest in the lattice EOS's in their traditional domain. The applications of quasilattice models shifts to other areas, such as modeling of nonuniform molecular systems, the structure of interfacial regions, micelles, microemulsions, and phase equilibria in such systems. As has already been mentioned above, a fluid of r-mers can be studied applying a formalism for a fluid of spherical segments with strongly associating contact sites to mimic the chemical bonds of the r-mers. A kind of sphere-and-bond formalism has been applied within the framework of a lattice model to study the concentration and orientation profiles of solutions near a flat surface (66). The way to extend the formalism for the case of a lattice-gas fluid with curved interface is quite clear and we are about to carry out the modeling of such systems. A lattice model can be an effective tool to apply the sphere-and-bond formalism in the study of local ordering in amphiphilic chain fluids (118,119). From the criterion of the system stability with respect to fluctuations of segmental concentrations the microphase separation can be revealed on the phase diagram (119). Another new application of the quasilattice EOS's is their combination with the methods of continuous thermodynamics to describe polydisperse polymer systems (120).

As concerns the further development of EOS's for bulk fluid phases, the description of the behavior of fluids in the near-critical region appears to be of major importance. A general shortcoming of the existing quasilattice versions of EOS's, is that nearly all of them have been derived in a mean-field approximation. This assumption leads to an analytic EOS incapable of giving the correct asymptotic behavior of a fluid in the near critical region, and is typically present in most of the EOS's derived for molecular fluids. It is likely, that the quasilattice approach will play an essential role in the search for nonanalytic EOS's. Thus, a practical nonanalytic approach, the renormalization-group theory, has been, to a great extent, developed for lattice models (121). Another possibility for describing correctly the asymptotic critical behavior is provided by decorated lattice models (122). As these are isomorphic to the Ising model, one can apply the exact results known for the latter. An alternative approach for practical applications is an empirical search for correlations, which are in good agreement with computer-simulation data for non-athermal lattice systems. A correlation of this kind recently yielded a very good description of the whole liquid-liquid coexistence curve (123) including the upper and the lower consolution points. The results are much more accurate than those obtained with the aid of the LCT or quasichemical approximation. The correlation, designed for modeling of liquid-liquid equilibrium in a non-compressible two-component lattice fluid, may work as well for liquid-gas equilibrium in a one-component lattice gas. The extension to the multicomponent case is ambiguous, though it is a challenge to solve this problem.

Acknowledgments

The authors are indebted to the Russian Foundation for Basic Research for financial support (projects 96-03-33-992a and 96-15-97399).

285

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Russian) 68, 203 (1995). 115. S. Mattedi, F.W. Tavares, and M. Castier, Fluid Phase Equilib. 99, 87 (1994). 116. J.M. Nitsche, Fluid Phase Equilib. 78, 157 (1992). 117. I. Farkas, S. Kemeny, E. Thury, and A.Deak, 13 th IUPA C Conference on Chemical

Thermodynamics, Clermont-Ferrand, France, Abstr. 69 (1994). 118. D.R. Kuespert and M.D. Donohue, Langmuir9, 431 (1993). 119. D.R. Kuespert and M.D. Donohue, J. Chem. Phys. 9, 9782 (1993). 120. Y. Hu, X. Ying, D.T. Wu, and J.M. Prausnitz, Fluid Phase Equilib, 98, 113 (1994). 121. J.S. Walker and C.A.Vause, J. Chem. Phys. 79, 2660 (1983). 122. G.C. Nielsen and J.M.H. Levelt Sengers, J. Phys. Chem. 91, 4087 (1987). 123. S.M. Lambert, D.S. Soane, and J.M. Prausnitz, Fluid Phase Equilib. 83, 59 (1993). 124. D.J. Tildesley and W.B. Streett, Mol. Phys. 41, 85 (1980). 125. W.A.P. Luck, Disc. Faraday Soc. 43, 115 (1967). 126. H.H. Reamer, B.H. Sage, and W.N. Lacey, Ind. Eng. Chem. 43, 2515 (1951). 127. W.W. Akers, R.S. Kelley, and T.G. Lipcomb, Ind. Eng. Chem. 46, 2535 (1954). 128. J. Timmermans, Physico-Chemical Constants of Pure Organic Compounds, Vol. 2, Elsevier,

New York, (1965). 129. J. Griswold and S.Y. Wong, Chem. Eng. Progr. Symp. Ser. 48, 18 (1952). 130. W. Poot, Th.W. de Loos, and F. Kohler, to be published. 131. K. Todheide and E.U. Frank, Z Phys. Chem., N.F. 37, 387 (1963).

288

132. T.M. Leteher, S. Wooten, B. Shuttleworth, and C. Heward, J. Chem. Thermodyn. 18, 1037 (1986).

133. C.L. Sassen, Th.W. de Loos, and J. de Swaan Arons, J. Phys. Chem. 95, 10760 (1991). 134. A.Z. Panagiotopoulos and R.C. Reid, ASC. Symp. Ser. 329, 115 (1987). 135. P. Traub and K. Stephan, Chem. Eng. Sei. 45, 751 (1990). 136. J.W. Kang, J.Y. Kim, K.-P. Yoo, and C.S. Lee, Fluid Phase Equilib. 150-151, 199 (1998). 137. D. Nickel and G.M. Schneider, J. Chem. Thermodyn. 21,293 (1989). 138. W. Wang, D.A. Tree, and M.S. High, FluidPhase Equilib. 114, 47 (1996). 139. H.H. Reamer and B.H. Sage, J. Chem. Eng. Data, 8, 508 (1963). 140. S.K. Kumar, U.W. Suter, and R.C. Reid, Ind. Eng. Chem. Res. 26, 2532 (1987). 141. H.-J. Ng, H. Kalra, D.B. Robinson, and H. Kubota, J. Chem. Eng. Data 25, 51 (1980). 142. P.C. Pemberton and C.J. Mash, J. Chem. Thermodyn. 10, 867 (1978)

l~quations oJ ~'tate Jor 1~ luwls and P luid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 289

8 THE CORRESPONDING-STATES PRINCIPLE

James F. Ely and Isabel M. F. Marrucho

Chemical Engineering and Petroleum Refining Department Colorado School of Mines Golden, CO 80401-1887, U.S.A.

8.1 Introduction 8.2 Theoretical Considerations 8.3 Determination of Shape Factors

8.3.1 Other Reference Fluids 8.3.2 Exact Shape Factors 8.3.3 Shape Factors from Generalized Equations of State

8.4 Mixtures 8.4.1 Van der Waals One-Fluid Theory 8.4.2 Mixture Corresponding-States Equations

8.5 Applications of the Extended Corresponding-States Theory 8.6 Conclusions References

290

8.1 INTRODUCTION

During the past 25 years there have been great advances in the theory of dense fluids and in the application of these theories to complex molecular systems. These advances which include both integral-equation and statistical-mechanical perturbation theory have been brought about primarily by the advent of faster, cheaper computers. As pointed out by Kreglewski (1), however, these results seem hopelessly complex for a chemical engineer who is generally interested in simple, practical solutions. With this complexity in mind, the most powerful tool available today Oust as 25 years ago) for making highly accurate, yet mathematically simple, predictions of the thermophysical properties of fluids and fluid mixtures is the corresponding-states principle. The power of the corresponding-states principle is that it allows the prediction of fluid properties with a minimum amount of information for the system of interest, given a detailed knowledge of few reference systems. The principle is well founded in molecular theory but certainly is not new. Its fundamentals and applications to pure-fluids and mixtures have been reviewed in almost all of the recently published thermodynamic and statistical mechanics books (2-5). As discussed by Leland and Chappelear (6) in their review of the corresponding-states principle 25 years ago, the basic concept of corresponding states is to apply dimensional analysis to the configurational portion of the statistical-mechanical partition function. The end result of this analysis is the expression of residual thermodynamic properties in terms of dimensionless groups. On an empirical basis, the corresponding-states principle was originally proposed by van der Waals who observed that the reduced form of his equation of state could be written for all fluids. All modem generalized engineering equations of state are examples of applications of this principle.

The original, two-parameter corresponding-states principle leads to an equation of state which expresses the residual compressibility factor in terms of a universal function of the dimensionless temperature and molar volume (or density):

Z r -~ P--~V-I = F(V*,T*) (8.1) RT

where p is the pressure, R is the gas constant and V* and 2"* are the dimensionless volume and temperature, respectively. Starting from a molecular basis, V* would be identical to V/N~ where ~ is the intermolecular-potential distance parameter and T* would be given by kT/c where k is Boltzmann's constant and c is the intermolecular-potential well depth. If one invokes the stability criteria for a pure-fluid critical point, the dimensionless volume and temperature would be given by V/Vc and T/Tc, respectively, where the subscript c denotes a value at the critical point. We note that this two-parameter corresponding-states principle can be applied to any polynomial equation of state which has a liquid-vapor critical point (6). Since Equation (8.1) says that all substances obey the same reduced equations of state we can make a slight transformation of this result to directly relate the properties of one fluid to another. For two fluids j and 0 which obey the simple corresponding-states principle we can write from Equation (8.1)

Z~ (V*, T*) = Zo(V*,T*)= F(V*,T*) (8.2)

o r

zs( ,rj)= Zo(ro,r o) (8.3) where Vo and 7'o are related to their corresponding values in the j-fluid by

V o = Vj ~hi and T o = Tj /f j (8.4)

291

where hj =VjC IV oc =o-j3/0-0 3 and fs. = Tf/T0 ~ = e j / c 0 . The quantities £ and hj are known as

equivalent-substance reducing ratios. Experimental evidence has shown that the two-parameter corresponding-states equation is

obeyed only by the heavy noble gases (Ar, Kr and Xe) and nearly spherical molecules such as methane, nitrogen and oxygen. In order to extend the corresponding-states theory to a larger spectrum of fluids, additional characterization parameters have been introduced into the basic two-parameter corresponding-states equation (8.1). Two main approaches have been followed in this parameterization. The first is to introduce the additional characterization parameters and then perform a multi-parameter first-order Taylor series expansion of the compressibility factor about the parameters. Mathematically,

• --" "-I- ~ i ( 8 . 5 ) Z(V*,T*,/~I,/~2,...,/~n) Z(V*,T*) {;~. :0} i : l {~:0}

where the {Li}are the characterization parameters. The derivatives appearing in this equation are typically evaluated by making finite-difference approximations using reference fluids that differ in the parameter of interest. A good example is the use of a single additional characterization parameter is the Pitzer acentric factor co (7,8). In this case

(c3Z~ = Z(T*, V*,co = col)-Z(T*, V*,co = 0) (8.6)

t,&o) co =0 col

and the corresponding-states model becomes

z ( r * , v* , co = co, ) - z ( r * , v* , co = o) Z(T*,V*,co) = Z(T*,V*,co = O)+

(01 co (8.7)

Examples of this approach and its generalizations include those of Pitzer (7-10), Lee and Kesler (11-13), Teja and co-workers (14-19) and, more recently, Johnson and Rowley (20,21) and Malanowski and Anderko (22). Both of the latter two references present concise overviews of the different techniques used in corresponding states, giving special attention to the development of four-parameter models which are capable of describing fluids that exhibit large deviations from the simple two-parameter corresponding-states principle due to size/shape and polarity/association effects. In addition, Reid, Prausnitz and Poling (23) summarize a large number of corresponding-states-like correlations that use this approach.

The second approach is to extend the simple two-parameter corresponding-states principle at its molecular origin. This is accomplished by making the intermolecular potential parameters functions of the additional characterization parameters {2;} and the thermodynamic state, e.g., the temperature T. This can be justified theoretically on the basis of results obtained by performing angle averaging on a non-spherical model potential. The net result of this substitution is a corresponding-states model that has the same mathematical form as the simple two-parameter model, but the definitions of the dimensionless volume and temperature are more complex. In particular the dimensionless volume becomes

V V V* = = (8.8)

No- 3 (T, {2 i }) Vc(P(T, {2 i })

and the dimensionless temperature takes the form

292

k T T T* = = (8.9)

E(T, {~'i }) LO( T' {/~'i })

The quantities 8 and q) which appear in these equations are referred to as shape factors and we refer to this method as the extended corresponding-states theory (ECST). Several review papers have been published that focus on this approach, the most extensive of which are those of Leland and Chappelear (6), Rowlinson and Watson (24) and Mentzer et al. (25). Applications of this method are presented in Section 8.5.

Thus far we have only introduced the pure-fluid corresponding-states principle which, as mentioned above, has a rigorous basis in molecular theory. The extension of this theory to mixtures cannot, however, be made without further approximation and the problem of rigorous, yet tractable, prediction of mixture properties remains unsolved. These approximations take the form of mixing rules which are a topic of another chapter in this volume. We will only discuss mixing rules from an illustrative basis to show problems that can arise in the implementation of a corresponding-states model. In that regard, we will focus our mixture discussions on the one-fluid theories~primarily the van der Waals one-fluid theory proposed by Leland et al. (26, 27) The essence of this model is that the properties of a mixture are first equated to those of a hypothetical pure-fluid whose properties are then evaluated via the pure-fluid corresponding-states principle. The one-fluid theory is capable of providing highly accurate results, especially when the extended pure-fluid corresponding- states formalism is used.

In the remainder of this review we have chosen to focus on the corresponding-states nature of generalized engineering equations of state and the more general shape-factor-based extended corresponding-states principle. We start by reviewing the basic molecular theory of pure-fluid corresponding states and then describe in some detail the extended corresponding- states principle. Some time will be spent discussing the methods of calculating and/or predicting shape factors. Part of this discussion will be spent examining common engineering equations to extract the dependence of their shape factors on temperature, volume and other characterization parameters. We will also try to illustrate the dependence of the 'experimental' shape factors on temperature, volume and for example, dipole moment by studying highly accurate equations of state for both nonpolar (e.g., hydrocarbons) and polar materials (e.g., refrigerants and water). Finally we discuss the implementation of the extended corresponding- states principle for mixture calculations and demonstrate some of the successes and difficulties encountered in the application of the model.

8.2 THEORETICAL CONSIDERATIONS

The direct calculation of thermophysical properties from statistical mechanics involves not only extraordinary mathematical complexity, but also detailed knowledge of the interactions between the molecules. Although considerable progress has been made in developing molecularly based predictive methods, most of the results have been obtained using more or less drastic approximations and are computationally complex. Also, most of these correlations are developed for specific types of fluids in certain regions of the phase diagram. Examples include the statistical-mechanical perturbation theories and integral-equation theories. The corresponding-states theory provides an alternative route to calculate thermophysical properties since it uses experimentally measured properties of one or more reference fluids to represent the solution of the configurational part of the partition function. In this section we briefly review the molecular basis of the corresponding-states theory.

293

As a detailed derivation can be found in the literature (2,4,7), only the basic assumptions of simple corresponding states are stated here: 1. The partition function can be factorized into a density-independent intra-molecular

contribution and a density-dependent configurational contribution. 2. The configurational contribution can be treated using classical statistical mechanics. 3. The intermolecular pair potential may be written as a product of an energy parameter and a

universal function f, which depends only on the distance between the molecules, r, e.g., u (r)= ~f(r/cr) where 6 is the potential well depth and o-is the collision diameter.

The first assumption is generally valid at low densities and even at high densities for simple fluids but it does not necessarily apply to polyatomic fluids or associating molecules. The second assumption excludes fluids that exhibit quantum behavior. The third assumption is the most restrictive since it excludes all non-spherically symmetric molecules.

Given these assumptions the corresponding-states principle may be easily derived from scaling arguments applied to the residual canonical-ensemble partition function, Qr = Q/Qideal. Using standard notation (28)

1 _ O "3N Qr(N,V,T) = ~--~-- f... fe-UN(ru)/kTdr N vN f"" fe-UN(r*N)/kTdr *N - e F(V*'T*) ' (8.10)

where UN is the configurational energy. Since the Helmholtz energy A is given by - kT In Q(N, V, T), this equation reduces to

i r

= -F(V" r ' ) (8.11) 6"T*

Applying this relationship to two conformal fluids (fluids which obey assumptions 1-3 given above) results in our basic working relation in corresponding-states theory, namely that if two fluids are conformal, their dimensionless residual Helmholtz energies are identical when evaluated at equivalent conditions

(Vj, L)= U A; (Vo, to)= L.4; / (8.12)

where we have used the definition of the equivalent-substance reducing ratios j~ and hj presented in the previous section. Relations between other thermodynamic properties of the two conformal fluids can be obtained by straightforward differentiation. For example, for the pressure one finds

£ f ,

(vj /hj, / f j) Pj (Vj, Tj) = ~-j Po (Vo, To) = - - Po (8.13)

In order to extend the simple molecular corresponding-states principle to non-spherical fluids, two approaches are possible. The first simply amounts to introducing models for the non- spherical interactions into the intermolecular potential. For example, the intermolecular potential between two axially symmetric molecules whose electrostatic interactions can be represented as point dipoles and quadrupoles can be modeled as (4)

2 02 H(F, 01,02,~12) -- ocL (F/ 0") "~- - ~ fg (01,02,~I2) -[- --7 fo (01'02'~12)

r r (8.14)

294

where 01, 02 and 012 are angles describing the relative orientation of the two molecules and ¢t and ® are the dipole and quadrupole moments, respectively. When this potential is used in the evaluation of the configurational energy and the residual partition function is made dimensionless one finds

A r - F(V*,T*;p*2,® .2) (8.15)

sT*

and the corresponding-states working equation becomes

-ol/ r fl'lj'Oj)= fJf~ ( V ° ' T ° ' [ d ° ' -- hj ' f j f~ hj' f jh~J (8.16)

From a theoretical view point this approach is perfectly viable. However, from a practical point of view neither theory nor experiment have provided quantitative details about how the equation of state depends on the multipole moments. Thus Equation (8.16) cannot be used directly in any corresponding-states predictions.

The second approach to extending the molecular corresponding-states principle to non- spherical molecules was suggested by the work on angle-averaged potentials by Rushbrooke (29), Pople (30, 31) and Cook and Rowlinson (32). For example, if the spherical portion of the potential of an axial dipolar molecule can be represented by the Lennard-Jones (12-6) model

(8.17)

and the dipole can be modeled as a point dipole as in Equation(8.14), Boltzmann averaging of the potential yields an effective spherical Lennard-Jones potential for which the parameters are temperature dependent and are given by / 4)2 / ),J6

c(T) = s o 1 + and o-(T) = 1 + (8.18) lZkTsocro 6 °'o lZkTc0cr06

Allowing the potential parameters to be temperature dependent retains the form of the simple, two-parameter corresponding-states relationship except that the equivalent-substance reducing ratios become functions of temperature. In particular,

A)(Vj, Tj ) = f j Aro(Vo, To)=f j Aro(Vj /hj(T),Tj / f j(T) (8.19)

The temperature dependence of the equivalent-substance reducing ratios somewhat complicates the thermodynamics of the thermal properties calculated from this model. For example, the internal energy of the fluid j is related to that of the reference fluid 0 by the equation

[?~= 1 - ~ , O T j J ~ o - ~ - ~ ) Z o (8.20)

295

In applying this formalism one must have knowledge of the reference fluid properties (as denoted by a 0 subscript) and the equivalent-substance reducing ratios. These reducing ratios are typically expressed in terms of the molecular shape factors 0 and (p that are defined as

c V.~ - - - _ _ 2 2 - - _ _ 2 2 Tj 0(Tj,ps,Oj,.. .) and hs J qg(Tj,ps,Os,...) (8.21)

fJ To c V o

8.3 DETERMINATION OF SHAPE FACTORS

The theoretical basis for the molecular shape factors was derived in the previous section. That analysis, which led to temperature-dependent shape factors represents an idealized case where the non-spherical potential parameters may be incorporated with the spherical parameters through angle averaging. Although that approach is correct in certain circumstances, it is of limited practical use since the intermolecular potential function for real fluids is not known precisely. Hence, one is forced to use macroscopic thermodynamic measurements to determine the shape factors and then try to develop a generalized correlation for them which depends on known molecular parameters. We shall refer to the shape factors determined from experimental data as the apparent or exact shape factors and their generalized correlation as the correlated shape factors. The shape factors are weak functions of temperature and, in principle, density and can be visualized as distorting scales that force the two fluids to conformality. Although there is no direct theoretical evidence for the density dependence of the shape factors, mathematical solutions for exact shape factors found by equating the dimensionless residual compressibility factor and Helmholtz energy of two pure- fluids exhibit a weak density dependence.

The first attempt to find exact shape factors is due to Leach (33), who equated the residual compressibility factor and fugacity coefficient of two fluids, viz.

z) (vj, rj)= z;(V / h,, r /f j)

Oj , ) = Oo / , / £ ) (8.22)

and solved for the equivalent-substance reducing ratios, fj and hj. In these equations the superscript r denotes a residual (real minus ideal) property evaluated at the temperature and molar volume of the system. For example, ~b r =f/pRT wheref is the fugacity of the fluid. At low density these equations become identical and the apparent shape factors were found by simultaneously solving corresponding-states relationships for the second and third virial coefficients

Bj(Ts )= hj Bo(Tj / f j ) (8.23)

Cs(Ts )= hJ Co(rs / f s)

As emphasized by Leland and Chappelear (6), the shape factors determined from the solutions to Equation (8.22) depend on both density and temperature and, as such, cannot be related to any sort of intermolecular pair potential. More recently, Massih and Mansoori have discussed the statistical mechanical basis of the shape factors (34).

Using methane as reference and a large number of pure normal hydrocarbons from C1-C15, Leach (35) obtained solutions to these systems of equations and empirically fitted the results in

296

terms of the acentric factor and the critical parameters. The set of correlated shape factors that were obtained is given by:

0 = l + ( c o j - c o 0) a, - a21n T ] + a 3 -

(8.24)

Z°~/1 + (coj -(.00)[j~ 1 (V; - ~2 )lnT; -]- J~3 (Vf - J~4 )]} q~ = Z~. t J

where 0 and cp are the shape factors, co the acentric factor, T* is the temperature,/I* the volume and Z ~ is the critical compressibility factor. The subscripts j and 0 indicate the fluid of interest and the reference fluid (methane), respectively. The values of the parameters reported by Leach et al. (33) for the 0 shape factor were al = 0.0892, a2 = 0.8493, a3 = 0.3063 and a4 = 0.4506 with the cp parameters being fll = - 0.9462, f12 = 0.7663,/33 = 0.3903 and/34 = 1.0177. When Vj.* >2.0 its value is set equal to 2.0; when Vj*< 0.5 its value is set equal to 0.5. These limits correspond to the virial region and dense liquid regions, respectively, where the apparent shape factors are independent of density. For other values of Vj-* between these limits, the shape factors were found to be density dependent. Note that if the apparent shape factors are defined according to Equation (8.24), two parameters in addition to T: and Y~ are introduced,

co s and Z~, giving rise to a four-parameter corresponding-states model.

In 1981 Ely and Hanley (36) developed an extended corresponding-states theory for the viscosity of hydrocarbon mixtures. In conjunction with that work they developed a wide- range reference-fluid equation of state for methane and a new set of correlation parameters for the apparent shape factors. The functional form of that correlation was the same as determined by Leach et al., Equation (8.24), but the parameters were somewhat different owing to the expanded range of temperature. The values of the parameters reported by Ely and Hanley for the 0 shape factor were al = 0.090569, a2 = 0.862762, a3 = 0.316636 and a4 = 0.465684 with the ~o parameters being/31 =-0.93281,/32 = 0.754639, ,83 = 0.394901 and/34 = 1.023545.

8.3.1 Other Reference Fluids

A disadvantage of the original Leach shape-factor approach is that the reference fluid is fixed as being methane. This introduces errors in the method when using it to calculate properties of fluids which have very different properties from methane or when the reduced temperature of the fluid of interest (target fluid) is less than the triple-point temperature of methane. Ely and Hanley attempted to overcome this latter problem by developing an equation of state for methane which had a fluid region extrapolated down to 40 K (T* = 0.21). Another solution to this problem was adopted Leach (35) and later by Rowlinson (24)who showed that it is possible to convert shape factors relative to one reference fluid into shape factors relative to another reference fluid, thereby allowing two different reference fluids to be used in the corresponding-states calculations. For example, Leach et aL (33) used methane and pentane in their original extended corresponding-states model. In developing the transformation equations we assume that we know the shape factors or the equivalent- substance reducing ratios of fluids i and j relative to some reference fluid which we shall denote as 0. The two fluids are at an equivalent corresponding state w h e n / ] / f o = I ] / ~ o = To and/I,. / hio = Vj. / hjo = Vo. Note that we have added a second subscript to make the reference fluid clear. Rewriting these equations we obtain relationships between the state parameters of

297

the i fluid with respect to fluid j, e.g., Ti =f0 Tj/fj0 and V,. = hio Vj/hjo. Finally, if we define the equivalent-substance reducing ratios for i relative to j as f j = f0 /fj0 and h,j = hio / hjo we obtain the same functional mathematical relationships between the state points of the i and j fluids as we started with for the i and j fluids relative to the reference 0. The difference, however, is thatfj, and h a involve the state points of both fluids, rather than just the state point of fluid i. The shape factors which are associated with fj. and h a were called relative shape factors by Leach et al. and are given mathematically by

Oi°(Ti*'Wi*) - Oj°(Ti-~*'Wi*) (8.25) 00" (Ti" V/" ) -- Ojo(T; , V;) - OJ° (--~ Oj°Tii " - (pj°Vi"

and

e,o(~*, v,*) _ ejo(~*, v,*) (P ° ( Ti * ' Vi * ) = (P j o ( T ; , V ; ) - (/3 J 0 I O j ° Ti * (/3 j ° Vii * i 0 ' ( / 9 io

(8.26)

These equations are non-linear and must be solved numerically. In 1987 Younglove and Ely (37) reported a wide-range equation of state for propane

based on the functional form of the 32-term modified Benedict-Webb-Rubin equation (MBWR-32) which was proposed by Jacobsen and Stewart (38) for nitrogen. The advantage of that equation of state was that its range of fluid states included the triple point of propane which is 85 K (e.g., the reduced triple point is 0.22). This development eliminated the need to use two reference fluids or to use an artificially extrapolated reference fluid as in the work of Ely and Hanley (36). To avoid having to use the shape-factor transformation formulas given above, Ely re-determined the apparent shape factors relative to the propane reference. In determining the apparent shape factors, a slightly different method than that used by Leach et al. was incorporated. In particular, at subcritical conditions the procedure suggested by Cullick and Ely (39) was used. In this procedure the vapor pressures and saturated-liquid densities of the target and reference fluids are equated

P~t(Tj) = pSat (Tj / L ' ) f j / h j (8.27)

p~at(Tj)- p~at(Tj/ f j ) /h j

and solved simultaneously for J) and hj.. At supercritical conditions the virial method given in Equation (8.23) was used. This procedure, unlike that used in previous studies, generates apparent shape factors which only depend on temperature. Their correlation gave the following results:

0 = 1 + (coj- COo)[0.05202-0.74981 lnTj* 1

(,o = Z--~ {1 +(coj- COo)[-0.14359 + 0.28215 lnT;l } (8.28)

298

More recently, Marrucho and Ely (40) developed a saturation boundary based method to evaluate the shape factors that is easily transferable between reference fluids. The method is based on the Frost-Kalkwarf vapor-pressure equation (41) and the Rackett equation (42) for saturated-liquid densities. Using these relations and Equations (8.27) one finds for the 0 shape factor

1-C o + 2(1- Tj*)2/V ln(Zj /Zo ) - ~ * + AC* lnTj* + B; /Tf 0 = (8.29)

1-C o + Bo/Tf and for

(Zj)(1-Tj *)2/7 (8.30) (~-'(Zo)(l-Z;/O)2/7

In deriving these equations we have assumed that 0 is close to one and therefore In 0 _=_ 0 - 1 and have defined AB* B ) - B* o and AC*= C ) - C O and neglected the D* term in the Frost-

Kalkwarf equation:

lnp"~at=B* -1 +C*lnr*+D' t - -~- -1 (8.31)

Since the reference fluid parameters appear explicitly in the shape-factor expressions, this formulation is easily transferable between reference fluids. In the supercritical region, Marrucho and Ely proposed a method of calculating the shape factors assuming that

(pj = Z o /Z~ and that isochores were nearly linear, resulting in an expression for the 0 shape

factor

o=lh°(p'-T~Y')+(h°y'-y°)TjlTf'po -yoTo To (8.32)

The superscript cr indicates the isochore which intersects the reference fluid saturation

boundary at Po = pho, ho = ZoPo/Z~p~ and y - (c~ / cT/')p, y c for the target fluid may be

obtained from the Frost-Kalkwarf equation as

Y~i = a P ~i ( C~i - B~i - 2 D* ) /(1- D* ) T~ (8.33)

As discussed in the next section, Mollerup (43) analyzed common cubic equations of state, such as the Soave-Redlich-Kwong (44) and Peng-Robinson (45) equations in terms of separating them into a shape-factor correlation and an equation for the pure reference fluid. The shape factors obtained from the cubic equations of state have the advantage of being similar to the ones developed by the more complex correlation schemes outlined here, but are simpler to use, since they are density independent and can be used with any reference fluid.

8.3.2 Exact Shape Factors From the late 1970's to the early 1980's, an increasing number of high-accuracy, analytic,

wide-range equations of state started to appear in the literature. The availability of these highly accurate equations of state allows one to 'exactly' (although numerically) solve the two-parameter corresponding-states relationship for the apparent molecular shape factors. The possibility of density dependence in the shape factors does, however, complicate the

299

thermodynamic description of the target fluid in terms of the reference fluid and the resulting solution for the shape factors themselves. The basic equation from the scaling of the partition function remains the same,

a~ (Vj, Tj) = a; (V 0 , T 0) = a~ (Vj / hi, Tj / fj ) (8.34)

where we have introduced a more compact notation which denotes a dimensionless residual property by a lower case letter with a superscript r to emphasize that the property represents the difference between the real and ideal values evaluated at the same volume and temperature. For example, in the equation above a r - - (A(V,T) - A idea! g a s ( g , T ) ) / RT. Using the thermodynamic relationship

r r

da r = - z - - - d v - U - - d r (8.35) V T

and the chain rule, we find the following thermodynamic relations between the target and reference fluids:

r r r

zj : Z o [ 1 - H v ] - u o F v

u~ = U o [ 1 - F r ] - z r H r (8.36) r r r r

sj = s o - z o H r - u o F r

The enthalpy and Gibbs energy can be constructed from their thermodynamic definitions and the relations given above. In the corresponding-states relations summarized in Equation (8.36), dimensionless derivatives of the equivalent-substance reducing ratios must be known. These derivatives are defined as

T ( a f j / and Fv= V [0 f j ] Fv = - f f ~ , O T ) v f-f ~,~-V-) r (8.37)

with similar expressions for the derivatives of h i . Given the thermodynamic relations summarized above, it is not possible to solve for the equivalent-substance reducing ratios without making some other assumption, i.e., the set of equations given in Equation (8.36) are under-determined since a knowledge of both the values and derivatives of J) and hj is required.

r r The simplest assumption is to choose the solution for which zj = z0, which requires the

relationship zroHv = - uroFv (8.38)

Figures 8.1-8.6 illustrate the results of the apparent-shape-factor calculations obtained using this technique with propane as a reference. Figure 8.1 shows the shape factors for methane obtained using the MBWR-32 equation of state of Younglove and Ely (37). Methane differs from propane in both size and molecular shape but both are non-polar. The shape factors reflect this in their weak temperature and volume dependencies. Figure 8.2 shows the same results obtained using the newer methane equation developed by Friend et aL (46). The results are similar with the differences in the shape factors being only a fraction of one per cent. We conclude that the volume dependence of the shape factors, although weak, is real and not due to artifacts of the equations of state used in the calculation.

300 1.08

1.04

9- 1.00

0.96

0.92

' I i I ' I ' I i

p*=2.7 C H 4 -

! -

p*=O.6

I I, l I I I t I I,,

1.12 ' I i I ' I i I i

CH 4

1.08

1.04

1.00

0.96

0.92 I 0.40 0.80 1.20 1.60 2.00 2.40

Figure 8.1 0 and ¢ shape factors for methane relative to a MBWR-32 propane reference using the MBWR-32 equation of state for methane (37).

Figures 8.3-8.5 show the apparent-shape-factors of carbon dioxide, 1,1,1,2-tetra- fluoroethane (R134a) and water relative to propane. Carbon dioxide (47) was chosen for illustration because of its large quadrupole moment which would lead to an inter-molecular potential which is substantially different from that of propane. In this case, the shape factors are nearly independent of volume and weak functions of temperature. R134a has a large dipole moment and exhibits a stronger volume dependence in its shape factors relative to propane than is observed with CO2. Finally water is highly polar and associating and exhibits stronger temperature and volume dependence than that observed in the other fluids. These

1.04

1.00

9- 0.96

0.92

0.88

' I ' I ' I ' I l

- p*=2.? C H,,,, _

_ _i

p*=l.

9*=0.6

p* =0.3

I , I i I , I ,

i I ' I * I i 1 '

301

1.12 i

C H 4

1.08

1.04

o

1.00

0.96

0.92 , , I 0.40 0.80 1.20 1.60 2.00 2.40

, T

Figure 8.2 0 and rp shape factors for methane relative to a MBWR-32 propane reference using the Schmidt-Wagner of state for methane (78).

figures illustrate that the extended corresponding-states approach is extremely powerful in that it can be used to make any fluid, regardless of intermolecular potential, conformal to a selected reference fluid. One would hope that by studying the shape factors of various families of fluids (e.g., refrigerants, alcohols, etc.), relative to a fixed reference fluid, behavioral trends could be identified and correlated with known molecular parameters. Figure 8.6 illustrates this type of relationship for the shape factors of various polar compounds as a function of acentric factor.

302 1.02

1.00

0.98

0.96

0.94

0.92

1.04

1.02 - -

1 . 0 0

a i ' I ' I '

_ • C O 2 -

1.7

p*=2

L - -

0 . 7

_ m

_

=0.3 _

i 1 I I t I ,

' I ' I ' I l

C O 2

p*=2.5

~ * = O . 3 ~ 0.7 ~

0.98 I I I I I ! I ...... 0.40 0.80 1.20 1.60 2.00

T

Figure 8.3 0 and (p shape factors for carbon dioxide relative to a MBWR-32 propane reference using the Schmidt-Wagner of state for carbon dioxide (47).

8.3.3 Shape Factors from Generalized Equations of State

Thus far we have discussed the determination of shape factors from equations of state which have a high degree of accuracy in representing the properties of the pure fluids. Mollerup (43) observed that, although this procedure offers a very high accuracy in the determination of the shape factors, it can be very time consuming in terms of evaluating equations of state. This led him to examine 'simple' generalized equations of state which are commonly used in engineering calculations. Examples include the Redlich-Kwong Soave (RKS) equation (44), the Peng-Robinson (PR) equation (45)and others. In our analysis we have also included the Carnahan-Starling de Santis (CSD) equation (48) because it has a temperature-dependent volume parameter and a more complex volume dependence. Although we do not normally think of these modified van der Waals types of equations of state in terms

303

1.30

1.20

9- 1.10

1.00

0.90

1.12

1.08

1.04

1.00

0 . 9 6

' I i I ' I

R134a

• = .

m

_ p * = 0 . 3

p * = 3 . 0

I I I I I I

' I ' I ' I

* = 3 . 0

p * = 0 . 3

R134a -

0 9 2 I I ~ I t I i 0 4 0 0 8 0 120 160 2 0 0

T

Figure 8.4 0and q~ shape factors for R134a relative to a MBWR-32 propane reference using the MBWR-32 equation of state for R134a (123).

of shape factors, they all contain a prescription for their determination. To illustrate this, consider the RKS equation given below

RT a(T*, co) p = - - (8.39)

v - b v(v +b)

where V is the molar volume, b is a volume parameter given by E2bRTZ/p c which is independent of temperature in this model and a is a parameter which is given by

304

1.30

1.20

1.10

9-

1.00

0.90

0.80 1.12

1.08

1.04

1.00

0.96

I ' I 1 I ' I ' l

J _- ~ H20

- p * = 1.8

2.2 _ p * - 3 . o 2 .

. ' I ' I ' I i I ' ]

p* = I .~

- .

I I w I i I i I t 0.40 0.60 0.80 1.00 1.20 1.40

T*

Figure 8.5 8 and q~ shape factors for water relative to a MBWR-32 propane reference using the Saul- Wagner equation of state for water (92).

where

a(T*,co) ~'~a (RTC)2 = a(T*, co) (8.40) pC

(8.41)

305

0.0004

0.0002

' I ' I ' I '

N2 © • CH4

0 2 0

0.0000

-0.0002

C2H 4

R12 [ ] ACO 2 R l 1 [ ]

• [ ] R22 C3H 6

C4H10 •

[ ] R32

R125 i- 1 R134a I-']

R123 [ ] A [ ] H20 _

R124

C5H12 •

- o . o o o 4 , I , I , I , - -0.20 -0.10 0.00 0.10 0.20

6o--o30

Figure 8.6 Comparison of the slope of the temperature dependent term in the 0 shape factor (see Equation (8.28)) as a function of acentric factor difference for various fluids.

The parameters -Qa and -62b are universal for this equation of state and m(co) is a simple quadratic function of the acentric factor. The residual compressibility factor and dimensionless residual Helmholtz energy are given by

and

z~ (Vj, Tj.) = bj a(T~, coj) (8.42) vj - R S (vj + )

Aj In - In RTj Vj bjRTj ( Vj

(8.43)

An examination of the corresponding-states relations, Equations (8.29) and (8.31), show that if the equivalent-substance reducing ratios are given by

hi= bj and f j = ajb° (8.44) bo a0b0

the mathematical corresponding-states principle is obeyed. For the shape factors, these

relations imply that cpj - Z o / Z~ and

_ a(T],coj) _ { l + m(coj) + Im(COo) - m ( c o j ) ] ~ } 2 (8.45)

Oj - a(To,, COo ) - 1 + m(COo)

306

where we have used the fact that T0* = Tj.*/0j in deriving the second relationship. As an example of a more complex modified van der Waals-type model, we have also

made shape factor calculations with the generalized Carnahan-Starling-De Santis (GCSD) equation which in dimensionless residual form is given by

4 yj + 2 y 2 4a(T])yj r . _ _ _ _ (8.46)

zj (1- yj)3 RTjb(T~)(1-4yj) where y is the packing fraction b(T*)/4 V and the parameters a(T*) and b(T*) are given by

a(T*) = ~")a ( R r c ) 2 elal(1-T*)+a2(1-T*)2] p~

(8.47) RT c

[1 + b a (1 - T*) + b z (1 - T*) 2 ] 7 Again the parameters if2 a and -Oh are universal for this equation of state with values of 0.461883 and 0.104999, respectively. At temperatures above the critical temperature, bl and b2 are set equal to zero. In this case (or any case in which the volume parameter has temperature dependence), we find a set of coupled equations for the shape factors which cannot be solved in closed form. For the GCSD equation

e[a'(1-Tf')+a2(1-T;)2] / 1 + b a (1- To* ) + b 2 (1-To) 2 ]

Oj = e [a,(1-To)+a20-To)z ] \ 1 + b, (1 - Tj) + b E (1 - T ; ) 2 ) (8.48)

~pj = Z__~ ( I + bl(1- T;) + bz(1- T;)Z i Z~ 1+ ba (1- T0*) + b2 (1- T0*) 2

where To* = Tj /0 j . In Figure 8.7 we have compared the Peng-Robinson shape-factor relations, Equation (8.45) and the CSD relations, Equation (8.48) for methane, relative to a propane reference, with the shape factors calculated using the MBWR-32 equations described above. This figure also shows results obtained using the generalized shape-factor correlation, Equation (8.24). These comparisons show a relatively good agreement at sub-critical conditions but fairly substantial differences in the supercritical region. The RKS model predicts a q~ shape factor which has a constant value of 0.965, as compared to 'exact' values which range between 0.92 and 1.02, as shown in Figure 8.7.

8.4 MIXTURES

In order to apply the simple corresponding-states theory to mixtures of conformal molecules, approximations must be made concerning the microscopic interactions and resulting structure of a mixture. A basic problem arises in this extension because the configurational energy in a mixture is a function of the position of the molecules and the species of the molecule located at these positions. This should be contrasted to the pure-fluid case where the molecules are indistinguishable and the energy of the system is only a function of molecular positions. Thus, for a mixture, the scaling arguments that led to Equation (8.11) for pure fluids, do not apply, even if the intermolecular potentials for all the mixture components are conformal.

I ' I ' I '

A

& A A , ~ A A S " " ' " " "

"" 0.6

0.3

0 . 8 8 ~ I I , I l I 0 .4 0 .8 1.2 1 .s 2 . 0

,

7

{ i

C H 4 - -

I

2 .4

1.12 | 1 ' ' t ' 1" '

t 1 08

1.04 O

1.00

0.92 ~ 1 , I , 1

307

0.4 0.8 1.2 1.6 2.0 2.4

T*

Figure 8.7 Methane shape factors calculated via several equations of state and correlations.- "exact" values calculated with MBWR-32 equations of state; © Cullick and Ely saturation boundary method; A Generalized correlation, Equation 8.28; * Carnahan-Starling-DeSantis equation and II Peng- Robinson equation of state.

The earliest theoretically based attempt to deal with this problem was to average the configurational energy of a mixture over all possible random assignments of species to a given position (49). This random mixing concept leads to an effective, hypothetical, pure-fluid potential (equivalent-substance potential) which can be used in the formalism developed for pure-fluids. Consideration of the explicit form of the conformal potential (e.g., Lennard-Jones potential) leads to mixing rules for the equivalent-substance reducing ratios of the hypothetical pure-fluid, i.e.,

and

where the subscript x denotes the hypothetical pure-fluid and for example fj =e 0/So where ij denotes the interaction of an 0" pair in the mixture. The term {x•} indicates the explicit dependence upon composition. For example, for the Lennard-Jones 12-6 potential

308

N N N N

fxh2 = Z Z xixjfiJ h2 and fxh4x = ~' ~_~ xixjfijh; ( 8 . 5 0 ) i=1 j=l i=1 j=l

Even though the random-mixing theory played an important role in the development of mixture theories, its predictions are not reliable due to its unrealistic physical basis (random assignment of molecules). However, the one-fluid concept, i.e., that the properties of a mixture can somehow be equated to those of a hypothetical pure-fluid whose properties can be evaluated from corresponding-states, has persisted and forms the basis for what are currently the most accurate corresponding-states models for mixtures.

8.4.1 Van der Waals One-Fluid Theory

The most successful corresponding-states theory for mixtures is called the van der Waals one-fluid theory. This theory was developed on a molecular basis by Leland and co-workers (26, 27, 50) and follows from an expansion of the properties of a system about those of a hard sphere system. A hard-sphere system is one whose molecules only have repulsive intermolecular potentials with no attractive contributions. The starting equation for the development of the van der Waals one-fluid (VDW-1) theory is a rigorous statistical- mechanical result for the equation of state of a mixture of pairwise-additive, spherically symmetric molecules.

Z = I _ 2 Z p N N 3kT Z Z XiX j ~u~ (r)g~i (r; p 'T' { xk } ; { ek } ' { crk } )r2dr (8.51)

i=1 j=l

In this equation Z is the compressibility factor pV/RT, go is the radial distribution function which gives the probability of finding a molecule of type i at a distance r from a central molecule of type j, u o. is the intermolecular potential whose parameters are c,j and or, 7, the prime denotes differentiation with respect to distance r, k is Boltzmann's constant and p is the number density. In the development of the VDW-1 theory the intermolecular potential is assumed to be composed of a hard-sphere term plus a long-range attraction, i.e.,

HS uij(r)=u o. + ruF*(r/cr~) (8.52)

where HS foO r < c r

U~/ = ~ 0 r > O " (8.53)

F(r) is a long-range attraction contribution to the potential, such as C6/r 6. Before one can proceed, assumptions concerning the radial distribution functions of mixture pairs must be made. In the development of this model, Leland proposed the mean-density approximation. This approximation amotmts to saying that the radial distribution function of the/j" pair is identical to that of a pure-fluid evaluated at the reduced conditions of the pair and a mean number density. Mathematically

gu (r; p,T, {xk } ; {sk }, {crk } ) = go (r / crij; p'Cr3x,kT / c~i ) (8.54)

where the subscript 0 denotes a pure-fluid distribution function. The next step is accomplished using the expansion techniques developed by Kirkwood et al. (51) for the distribution function of a real fluid in terms of that of a hard-sphere fluid, namely

309

. ' 3 : (r/o.O.,iOO.x).lt_Z[kr) go(r /Cro,PCrx,kT/s#) gHoS . ' 3 ~ij k~ln(iO'O.3 ) ( 8 . 5 5 )

n=l

where the Tn are complicated integrals over the hard sphere radial distribution function. Substituting these results into Equation (8.51) we find a temperature expansion for the mixture compressibility factor

r 2 ~ P £ Z Z X i X j £0" 3 ' 3 Zm~x - 3k r .=o i=1 j=l [ kT ) O'~iL~ln p O'x (8.56)

The analogous result for a hypothetical pure-fluid at the same reduced number density is n

2a-p S-, ( a',j "~ o.3T (p,O.x3) (8.57) zr = 3 k V ,zTd=o ~.-k-f ) o ,

Subtracting, one finds

r r n 3 - = (p x) (s.ss) Zx Zmix ~ =0 i=l j=l xixJ~io'iJ

Thus, we have at our disposal an infinite set of terms (coefficients of T ~) from which we can choose two for the determination of the potential parameters of the hypothetical pure-fluid. In the van der Waals one-fluid model the first two members of the series are chosen, giving

N N N N 3 3 O~xO" x : Z ZXiXjoC(jO'~ and O-x : Z Z x i x j o -3 (8.59)

i=0 j=0 i=0 j=0

Dividing through by the reference fluid parameters we obtain N N N N

fxhx = Z Zxixjf i jhij and h x = Z Zxix jho " i=0 j : 0 i=0 j=0

(8.60)

These are exactly the relations that van der Waals assumed in the generalization of his equation of state to mixtures. Within the mean-density approximation we see that these mixing rules are correct to order T 2 in the compressibility factor; terms that have higher-order temperature dependence in the reference fluid are not correctly mapped by the van der Waals mixing rules.

8.4.2 Mixture Corresponding-States Relations

The working equations for the mixture extended corresponding-states theory are exactly the same as in the case of a pure-fluid, Equation(8.36). The expressions for the derivatives of the equivalent-substance reducing ratios in terms of the component ratios are, however, somewhat complex. In particular, application of the formulas given above to mixtures requires derivatives offx and hx with respect to temperature, density and composition. An inspection of the mixing rules and the definitions of the equivalent-substance reducing ratios shows that the arguments of the shape factors are the effective temperatures and densities of components in the mixture. These, in fact, do not correspond to the temperature and density of the mixture unless the shape factors are identically unity. Thus, in a mixture, the arguments of the shape factors are themselves functions offx and hx. The dependence of these is nominally given by

310

Tj. =Tx~/fx and Vj. = Vxhj/hx. Differentiating these relations with respect to Tx one obtains two equations:

Fj(Tx)- Fj(Tj)[1 + Fj (Tx)- F x (Tx) ] + F s (Vj)[Hj(Tx)- g x (Tx) ] (8.61)

Hj (rx) = Hj (Tj)[I + ~(rx)-Fx(rx)]+ Hj(Vj)[Hj(rx)-Hx(rx)] Differentiation of the mixing rules with respect to temperature, Tx, yields

Jxnx----1N g I ] Fx(rx)+ Hx(r~)=-W-i-~_~_xixjfjh U F~(Tx)+ q--cH,(Tx) (8.62) i=1 j=l qo

and I N

Hx(Tx) = --~x .~l xixjhij q---Li Hi(Tx) (8.63) "= qo"

where qi = hi lz and qi = (qi + qy2. Simultaneous solution of Equations (8.55) for ~.(Tz) and Hj(Tx) and substitution into Equations (8.56) and (8.57) and subsequent solution for Fx(T~ and Hx(Tx) yields Fx(Tx) = 1 - $7/R and Hx(T~ = $6/R. Similar procedures yield for the volume derivatives

Fx(Vx) =($2+$4-$7) and Hx(Vx) = (S6 + S1- S3 + Dx) (8.64) Dx Dx

In order to perform phase-equilibrium calculations, fugacity coefficients can be calculated from the thermodynamic relationship

ln ~i -" g; nt- U;Fx(l'li)nt" zoHx(n i ) - ln Z (8.65)

and the necessary composition derivatives of the equivalent-substance reducing ratios are given by

Fx(nk) =S}k)S7-'S}k)(S2+S4) and Hx(nk) =S~k)(SI+S3)-S}k)S6 (8.66) D D

where k denotes component k in the mixture and Dx = (S1 + $3) $7 -- ($2 + $4) $6. The definitions of the sums Sm that appear in these results are given in Table 8.1.

8.5 APPLICATIONS OF THE E X T E N D E D C O R R E S P O N D I N G - S T A T E S T H E O R Y

Historically, applications of the extended corresponding-states theory have been limited by the scarcity of high-accuracy thermodynamic data and wide-range equations of state. As this data situation has improved over the past twenty years, refinements have been made in how the model is applied. In its original form, the extended corresponding-states model proposed by Leland et al. (6) incorporated a combined reference fluid of methane in the majority of PVT space and n-pentane in the low-temperature, high-density region. The shape factors used in this model were described in Section 3. Leland and coworkers originally applied this model to predicting vapor-liquid equilibrium in non-polar mixtures with a reported accuracy of about ten percent in the equilibrium K-values. Later, Fisher and Leland applied the model to predicting enthalpies, compressibility factors and fugacities in systems that did not deviate greatly from ideality (52).

311

Table 8.1 Sums required for evaluation of mixture corresponding-states properties

where

1 N N

Sl =~xhx ~"~Zxgxjfijh~i[1 Hi(Vi)]ni i=l j=l

1 U N

82 = ~xhx ZEXixjfijhijfi(Vi)Di i=1 j=l

1 N N

S 3 : ~xhxZEXixjfijhi j q----L-ini(Ti)Oi i=1 j=l qo

1 N N

84 =~xhxZZxixjfijh~ i qi [ I _ F i ( T / ) I D i i=1 j=l

~ ) _ 2x~ --gy )..xgLhik J x~x i=1

I N N S 6 ---h~xZZXixjho. qini(Ti)Oi

i=1 j=l qu 1 N U

84 =---~x ZZxixjhO'i=l j=l ~ijqi [1-- F/(T/)]Di

S~k) 2x~ N = 2-hx Zxihiki=l

D ; 1 = [1- H,. (V/)] [1 - Fi (T/) ] - Fi (V/)H/(T/)

Starting in 1969, Rowlinson and coworkers published a series of papers applying the extended corresponding-states model to a series of fluids and properties. In their application they used the Leach shape factors, but an extended methane equation of state that incorporated the low temperature data of Vennix (53,54) was used as the reference fluid. In the first paper, Watson and Rowlinson (55) predicted bubble-point temperatures and vapor compositions of argon+nitrogen+oxygen temary mixtures with satisfactory accuracy. Gunning and Rowlinson (56) calculated compression factors, enthalpies, Joule-Thomson coefficients and VLE for various systems and concluded that the extended corresponding-states principle had the advantage of requiting relatively little starting information and could be successfully applied to a wide variety of properties and fluids. Its primary disadvantage was pointed out to be a high degree of numerical complexity. In 1972, Teja and Rowlinson (57) applied the method to the prediction of critical and azeotropic states finding quantitative agreement for mixtures with one liquid phase and at least qualitative agreement in systems that had multiple liquid phases. Teja and Kropholler (58) and Teja (59) extended this study in 1975 by investigating azeotropie behavior in the mixture critical region. In the second study azeotrope formation and saturated-liquid densities in the CO2+ethane system were predicted with excellent results. In 1976, Teja and Rice (60) measured densities of benzene + alkane mixtures and compared their measurements to extended corresponding-states predictions using the Leach shape factors and Bender's methane equation of state (61) as the reference fluid. The average absolute percentage differences found between the measurements and predictions were less than two percent for alkanes from hexane to hexadecane. Teja (14) also applied the extended corresponding-states method to mixtures containing polar components such as ammonia and hydrogen sulfide.

In 1974 Goodwin published a very high-accuracy, wide-range equation of state for methane (62) which was capable of providing complete thermodynamic data from the triple point to 70 MPa and 500 K. This reference-fluid equation of state, along with the Leach shape factors, was used in a series of studies of liquefied-natural-gas (LNG) properties by Mollerup and co-workers. In the first study (63) Mollemp and Rowlinson found that it was possible to reproduce LNG densities to within 0.2%, even down to reduced temperatures of 0.3. In 1975, Mollemp (64) continued his study of LNG properties and reported results for phase equilibria, densities and enthalpies in both the critical- and normal-fluid regions. The method was also

312

applied to natural gas, liquefied petroleum gas (LPG) and related mixtures in a 1978 investigation (65, 66). Mixtures studied included methane through pentane and common inorganics such as N2, CO, CO2 and H2S. The paper reported density predictions to within ±0.2%, dew- and bubble-point errors "not exceeding those of good experimental data" and errors in liquid-phase enthalpies which were less than +2 kJ/kg.

During the mid-1970s, researchers at the National Institute of Standards and Technology (NIST) at Boulder, Colorado undertook a series of projects with the objectives of measuring and predicting the properties of LNG and related mixtures. One result from this project was an extended corresponding-states model for LNG densities developed by McCarty (67). That implementation used a 32-term, modified Benedict-Webb-Rubin equation of state for methane as the reference fluid and shape factors which had the same functional form as those proposed by Leach, but which had been re-fit to LNG density data. The model reproduced the available LNG density data to within +0.1%. Eaton et al. (68) used McCarty's methane equation to predict critical lines and VLE in methane+ethane mixtures. Another part of the NIST study focused on the development of predictive extended corresponding-states models for transport properties (36, 69-74). That work has been reviewed recently (75) and will not be included here. As mentioned in Section 8.4, however, the transport property work produced another reference-fluid equation of state for methane that was extrapolated to 40 K so as to avoid problems with the relatively high triple point of methane. That equation was later used by Romig and Hanley (76) as the reference-fluid equation to predict Type III phase equilibria in nitrogen+ethane mixtures.

In addition to the studies mentioned here, Mentzer et al. (77) summarized extended corresponding-states results for phase equilibriumwespecially for systems containing hydrogen and common inorganics. They found accurate pure-fluid predictions for non-polar compounds up to about C7. For mixtures they found a strong dependence on the binary interaction parameters but once those parameters were optimized for phase equilibrium, they could be used to represent a variety of properties accurately, without further optimization.

An overriding conclusion from all of these studies is that, while extended corresponding- states calculations are very accurate for systems of similar molecules, the predictions tend to decrease in accuracy as the system of interest deviates in size and shape from methane. More generally, as the components in a mixture become more dissimilar both in size and polarity, there is a marked decrease in the accuracy of the predictions. With this situation in mind, a research program was initiated by Ely and co-workers in the mid-1980's to investigate the possibility of improving the extended corresponding-states model in three areas: 1) Development of more realistic reference fluids, 2) Development of better shape-factor generalizations and 3) Development of improved mixing rules. In a sense, the most important part of this work has been the development of a base of high-accuracy equations of state that can be used as reference fluids or which can be used to generate 'exact' shape factors as described in Section 8.3. The equations have been reported by Younglove and Ely (37) and Ely and co-workers (46,47,78-82). In addition, during the same time frame, high-accuracy equations have been reported by Jacobsen, Pennoncello and Beyerlein and coworkers at the University of Idaho (38,83-90), Wagner and co-workers at Bochum (91-95) and deReuck and co-workers at Imperial College (96-98). Other work includes the studies of Haar and Gallagher on ammonia (99), Haar, Gallagher, and Kell on water (100) and equations summarized by Younglove (101). Most of the recent work in this area has focused on the development of high-accuracy equations of state for alternate refrigerants (102-107).

Examples of applications that have incorporated these new equations are diverse and are only briefly summarized here. Parrish (108, 109) performed measurements of ethane+propane

313

and propane+butane mixtures and compared those measurements to extended corresponding- states calculations using the 32-term MBWR equations for methane, ethane, propane and butane. Ely and coworkers have applied the extended corresponding-states principle with exact shape factors to a series of carbon-dioxide-rich mixtures including (0.98 CO2 + 0.02 N2) (110), (0.98 CO2 + 0.02 CH4) (111), (0.96 CO2 + 0.02 CH4 + 0.02 N2) (80), (0.99 CO2 q- 0.01 C2H6) (112), (0.96 CO2 + 0.02 CH4 + 0.01 N2 + 0.01 CzH6) (80) and (0.90 CO2 + 0.10 CH4) (80). The average absolute percent deviation in density prediction for all of these systems, which include approximately 500 data points, was 0.29% with a root-mean-square percent error of 0.43%. Figure 8.8 illustrates some of those results. Friend and Ely (113) also investigated the methane+ethane system over a wide range of conditions.

' ° I , , , t , r , 1 , 01 - co,, ,,, -!

0

0.0

-1.0

-2.0 0.00 10.00 20.00 30.00 40.00

X

CL -_.._ 0 m (13 r J

(3_ v 0 C ) . , e -

2.0

1.0

0.0

-i .0

-2.0 0.00

' I ' I ' I '

C02+ C2H 6

0 0 0

P ° l , I , I , lo.oo 20.00 30.00 40.00

2.0 f ~ t l t I t ] '

C02+ CH 4 1.0

©

o.o o c ~ o o o o

-1.0 L ~ I

-2.0 I u ~ l ~ , ~ l ~ , ~ 0.00 10.00 20.00 30.00

Pressure, MPa 40.00

Figure 8.8 Comparison of extended corresponding states predicted and measured densities of carbon dioxide rich mixtures (80).

314

2 . 0 0 / ' " ' ' ' ' " I ' ' ' ' ' ' " I ' ' ' ' ' ' " I

0.00 / ( ~ ~ - ~ ' ~ ~ ~ ' ~ - - ~ > ~ -

©© o

l , , , , J , , l , , i , l l , , l i I J , , , i , l - 2 . 0 0

2 . 0 0

0.00

- 2 . 0 0

I I I I I I ] ' r B l a n k e

I I I I I I I I

' ' ' ' ' " ' i ' ' ' ' ' " ' i ' ' ' ' ' " ' I ' ' ' ' ' " ' Kosolov

, , , , , , , , I , , , , , , , , I ©

I I I I l l l l i ,, I I I I I I I I

E 0 . m

G) r-~ > ,

¢/} ¢..

r-t

2 . 0 0

0 . 0 0

- 2 . 0 0

I I I I I I I i I

I I I 1 I I l i d I

' ' ' ' ' " ' I ' ' " ' ' ' " ' I ' ' " ' ' " ' I © Michels

, , , , , , , I , , , , , , , , I , , , , , , , , I

2 . 0 0

0.00

' ' ' ''"'I ' ' ' ''"'I , , , ,a,,,I ' ' ' ''"

Romberg

- 2 . 0 0 I I , I l l l l l I I I l l t i t l t , I I l l l J l

0.01 0 . 1 0 1 .00 1 0 . 0 0 Pressure, MPa

I I I I I I I I

1 0 0 . 0 0

Figure 8.9 Comparison of ECST calculated and experimental densities for air.

Air and air component mixtures (e.g., nitrogen, oxygen, argon and carbon dioxide) have been studied by Ely (114) and Clarke et al. (115,116). This system is especially well suited to the van der Waals one-fluid theory since its components are very similar in size. Figure 8.9 compares some typical predictions of ECST for the density of air.

In 1987, Gallagher et al. (117,118) used the extended corresponding-states formulation to generate tables of thermodynamics properties and phase equilibria for an isobutane + isopentane mixture for use as a working fluid in a geothermal power cycle. They concluded that predictions of the single-phase mixture pVT surface and the dew-bubble curve were generally better than 0.5% (usually 0.2%) except in the extended critical region. More

315

recently Levelt Sengers and co-workers (119) have applied the extended corresponding-states model to aqueous systems. In one study (120) a model was developed for carbon dioxide in water at near-critical and supercritical conditions. This study found that the model could be used reliably for the prediction of thermodynamic properties and phase boundaries at the water-rich end in a large region around the critical point of water. Figure 8.10 compares the predicted and experimental critical line, while Figure 8.11 shows the apparent molar volumes and excess volumes. The agreement in all cases is very satisfactory. A second study investigated the nitrogen+water system (121). For this system it was found that the Henry's constants near the pure-water vapor-pressure curve were well represented, but the mixture critical line and phase boundaries could not be accurately reproduced. An interesting observation from this study was that the phase behavior of generalized corresponding-states models has not been systematically explored and that such an exploration seems urgently needed. In a more recent study Gallagher et al. (122) used the extended corresponding states theory to predict critical lines in aqueous mixtures.

As a final application of the extended corresponding-states theory, Huber and Ely (123) developed a predictive corresponding-states model for refrigerant mixtures using R134a as the reference fluid and determining shape factors by the saturation-boundary method described in section 3. Generally the results were very good with deviations of the predictions being less than 1.0% for these highly polar mixtures..

3 0 0 1 .... [ 1 1 1

a_ 200

(b ::::3 o9

G) ,.- 1 0 0 cl

tO I

, A I !

!

,., 6 i

/ £x

_ 1 1 1 . . . . . . . t

0 0.1 0.2 0.3 0.4

X C 0 2

Figure 8.10 Comparison of calculated and experimental critical line for the carbon dioxide+water system (120).

316

0 E

03

E

:> 49-

2.0

1.5

1.0

0.5

I I 1 1 1 1 l

- p = 28 M P a

_ 0.0015 _

y - / / / ~ . - ~ - . ~ M Pa -

0 .1 1 1 ,1, 1, 1 I I

640 660 680 700 720

Temperature / K

Figure 8.11 Comparison of calculated and experimental apparent molar volumes in carbon dioxide+water mixtures along three supercritical isobars (120).

8.6 CONCLUSIONS

The extended corresponding-states theory incorporating shape factors can predict thermodynamic properties of mixtures very accurately, especially if the exact or saturation- boundary methods are used. Due to the higher level of complexity of this method, recent applications have been limited to studies where very high accuracy is of importance. Advances in the development of high-accuracy equations of state for a wide variety of substances (e.g., refrigerants and other polar compounds such as water and ammonia) has enabled researchers to extend the ECST methodology to a wide variety of systems and this trend will continue in the future.

Previous studies have shown that the approach is not very suited for the prediction of excess properties especially if there are substantial differences in molecular size. Since the pure correspondence is exact, this failure can be attributed to a failure of the van der Waals one-fluid mixing rules to correctly map the composition dependence of higher-order temperature terms in the reference-fluid equation of state. Efforts are Underway to develop

317

Acknowledgements J. F. Ely was supported by the U. S. Department of Energy, Basic Energy Sciences under grant number FG03-95ER41568 and I. M. F. Marrucho was supported by JINCT Programa Ciencia BD\1534\91-RM.

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M.L. Huber and H.J.M. Hanley, The Corresponding-States Principle: Dense Fluids, in Transport properties of fluids: Their correlation, prediction and estimation, J. Millat, J.H. Dymond, and C.A. Nieto de Castro, eds., Cambridge University Press (1995), p. 283. K.D. Romig, Jr. and H.J.M. Hanley, Cryogenics 26, 33 (1986). R.A. Mentzer, R.A. Greenkom, and K.C. Chao, Ind. Eng. Chem. Process Res. Dev. 20, 240 (1981). D.G. Friend, J.F. Ely, and H. Ingham, J. Phys. Chem. Ref Data 20, 275 (1991). M.L. Huber and J.F. Ely, Rev. Int. Froid 15, 393 (1992). J.A. Howley, J.W. Magee, and J.F. Ely, Gas Processors Association, Tulsa, OK, Research Report 136 (1993). K.B. Shubert and J.F. Ely, Int. J. Thermophys. 16, 101 (1995). J. Ratanapisit and J.F. Ely, Int. J. Thermophys. 20, 1721 (1999). R.T. Jacobsen, R.B. Stewart, and A.F. Myers, Adv. Cryog. Eng. 55, (1973). R.T. Jacobsen, R.B. Stewart, and M. Jahangiri, Int. J. Thermophys. 7, 503 (1986). M. Jahangiri, R.T. Jacobsen, R.B. Stewart, and R.D. McCarty, J. Phys. Chem. Ref Data 15, 593 (1986). R.T. Jacobsen, S.G. Penoncello, and S.W. Beyerlein, Fluid Phase Equilib. 79, 113 (1992). R.T. Jacobsen, S.G. Penoncello, and E.W. Lemmon, Fluid Phase Equilib. 80, 45 (1992). A. Kamei, S.W. Beyerlein, and E.W. Lemmon, Fluid Phase Equilib. 80, 71 (1992). S.G. Penoncello, R.T. Jacobsen, and E.W. Lemmon, Fluid Phase Equil. 80, 57 (1992). S.G. Penoncello, R.T. Jacobsen, and A.R.H. Goodwin, Int. J. Thermophys. 16, 519 (1995). R. Schmidt and W. Wagner, FluidPhase Equilib. 19, 175 (1985). A. Saul and W. Wagner, J. Phys. Chem. Ref Data 18, 1537 (1989). U. Setzmann and W. Wagner, J. Phys. Chem. Ref Data 20, 1051 (1991). R. Span and W. Wagner, J. Phys. Chem. Ref Data 25, 1509 (1996). R. Span, E.W. Lemmon, and W. Wagner, Int. J. Thermophys. 19, 1121 (1998). S. Angus, B. Armstrong, and K.M. De Reuck, International Thermodynamic Tables of the Fluid State-7 Propylene, Pergamon, Oxford (1980). K.M. De Reuck, International Thermodynamic Tables of the Fluid State-ll Fluorine, Blackwell, Oxford (1990). K.M. De Reuck, International Thermodynamic Tables of the Fluid State-12 Methanol, Blackwell, Oxford (1993). L. Haar and J.S. Gallagher, J. Phys. Chem. Ref Data 7, 635 (1978). L. Haar, J.S. Gallagher, and G.S. Kell, NBS/NRC Steam Tables. Thermodynamic and Transport Properties and Computer Programs for Vapor and Liquid States of Water in SI units, Washington, DC (1984), B.A. Younglove, J. Phys. Chem. Ref Data (Supplement) 11, 1 (1982). R. Tillner-Roth and H.D. Baehr, J. Phys. Chem. Ref Data 23, 657 (1994). R. Tillner-Roth, Int. J. Thermophys. 16, 91 (1995). R. Tillner-Roth, J. Phys. Chem. Ref Data 26, 1273 (1997). S i . Outcal.t and M.O. McLinden, Int. J. Thermophys. 16, 79 (1995). S.L. Outcalt and M.O. McLinden, J. Phys. Chem. Ref Data 25, 605 (1996). S.L. Outcalt and M.O. McLinden, Int. J. Thermophys. 18, 1445 (1997). W.R. Parrish, Fluid Phase Equilib. 23, 193 (1985). W.R. Parrish, Fluid Phase Equilib. 25, 65 (1986). J.F. Ely, W.M. Haynes, and B.C. Bain, J. Chem. Thermodyn. 21,879 (1989). J.W. Magee and J.F. Ely, Int. J. Thermophys. 9, 547 (1988). G.J. Sherman, J.W. Magee, and J.F. Ely, Int. J. Thermophys. 10, 47 (1989). D.G. Friend and J.F. Ely, Fluid Phase Equilib. 79, 77 (1992). J.F. Ely, Adv. Cryo. Eng. 35, 1511 (1990). W.P. Clarke, R.T. Jacobsen, and S.W. Beyerlein, Fluid Phase Equilib. 88, (1993). W.P. Clarke, R.T. Jacobsen, and S.W. Beyerlein, Int. J. Thermophys. 15, 1289 (1994).

320

117. J.S. Gallagher, Int. J. Thermophys. 7, 923 (1986). 118. J.S. Gallagher, D. Linsky, G. Morrison, and J.M.H. Levelt Sengers, Technical Note 1234,

National Institute of Standards and Technology, Gaithersburg, MD (1987). 119. J.M.H. Levelt Sengers and J.S. Gallagher, J. Phys. Chem. 94, 7913 (1990). 120. J.S. Gallagher, R. Crovetto, J.M.H. Levelt Sengers, J. Phys. Chem. Ref. Data 22, 431 (1993). 121. J.S. Gallagjer, J.M.H. Levelt Sengers, I.M. Abdulagatov, J.T.R. Waston, and A. Fenghour,

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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 321

9 MIXING AND COMBINING RULES

Stanley I. Sandier and Hasan Orbey*

Center for Molecular and Engineering Thermodynamics Department of Chemical Engineering University of Delaware Newark, DE 19716, U.S.A.

9.1 Mixing Rules for Cubic and Related Equations of State 9.1.1 The Van der Waals One-Fluid Model 9.1.2 Non-Quadratic Combining Rules for Van der Waals One-Fluid Model 9.1.3 Mixing Rules that Combine an Equation of State with an Activity-Coefficient

Model 9.1.3.1 Huron-Vidal Model 9.1.3.2 Wong-Sandler Model 9.1.3.3 Combination of Excess-Energy Models and Equations of State at Low or

Zero Pressure 9.1.4 Commentary on Mixing Rules for Cubic Equations of State

9.2 Mixing Rules for the Virial Family of Equations of State 9.2.1 The Van der Waals One-Fluid Model 9.2.2 Generalized Extended Virial Equations of State and Their Mixing Rules

9.3 Mixing Rules Based Upon Theory and Computer Simulation 9.4 Conclusions References

*Deceased

322

Equations of state (EOS) are powerful tools in chemical engineering practice since they can be used to correlate and/or predict the thermodynamic properties and phase behavior of pure fluids and mixtures over large ranges of temperature and pressure. However, since there is no exact statistical-mechanical solution relating the properties of dense fluids to their intermolecular potentials, or detailed information available on intermolecular potential functions, all equations of state are, at least partially, empirical in nature. The equations in common use can be classified as being of the van der Waals family of cubic equations, of the extended virial family of equations, or equations based more closely on results from statistical mechanics and computer simulations. Such equations are discussed elsewhere in this book. Here we are concerned with mixing and combining rules that extend the use of equations of state developed for pure fluids to mixtures. There have been many such models proposed in recent years, and we have had to pick and choose among them. We have decided to discuss only those which havestood the test of time, or are reasonably new and appear to be important advances.

One of the exact results we do have from statistical mechanics is the virial equation of state

Pv B C Z = ~ = 1 + - - +--T+... (9.1)

RT v v

where Z is the compressibility factor, P is the pressure, R is the gas constant, T is the temperature, v is the molar volume, and B and C are second and third virial coefficients. The virial coefficients are related to the intermolecular potential between molecules, and for pure fluids they are functions of temperature only. Also, for mixtures the only composition dependence of the virial coefficients is given by

B = ~ ~_~ xixjBu(T ) (9.2) i j

C = Z Z Z xix, x~ cgik (T), etc. (9.3) i j k

Since Equations (9.2) and (9.3) are exact results, these equations can be considered low- density boundary conditions that should be satisfied for mixtures by other equations of state when expanded into the virial form.

At present, there is no exact high-density-mixture boundary condition for mixture equations of state. However, there is the observation that, at liquid densities, empirical activity- coefficient models, such as those of van Laar, Wilson, NRTL, UNIQUAC, etc. provide a good representation of the excess, or nonideal, part of the energy of mixing. Therefore, another boundary condition that could be imposed is

excess free energy of mixing excess free energy of mixing calculated from an equation = calculated from an activity- of state coefficient model

(9.4)

323

In what follows, we will consider mixing rules for each of the classes of equations of state mentioned above. As we shall see, Equation (9.2) has frequently been used in the formulation of mixing rules for equations of state, Equation (9.3) is used as the basis for mixing rules with extended virial equations of state, and Equation (9.4) has been used to develop equation-of- state mixing rules in recent years.

9.1 MIXING RULES FOR CUBIC AND RELATED EQUATIONS OF STATE

The van der Waals equation

R T a p _ _ m (9.5)

v - b V 2

was the first equation of state that could describe, at least qualitatively, both the vapor and liquid phases, and therefore phase transitions. In this equation the parameter b is a volume- exclusion parameter accounting for the fact that molecules cannot overlap, and the parameter a results from the attraction between molecules. The van der Waals equation has been used as the basis for many other, more accurate equations of state that are also cubic in volume, such as the Peng-Robinson equation (1)

R T a ( T ) P = ~ - (9.6)

v - b v (v - b) + b(v + b)

which, because of its volume dependence and the fact that the attraction parameter has been made a function of temperature, provides a more accurate representation of fluid behavior and is widely used in the chemical industry, as is the almost equivalent Soave-Redlich-Kwong (2) equation. The important question that arises is how does one extend these equations (and others) to mixtures.

9.1.1 The Van der Waals One-Fluid Mixing Model

One method of generalizing a pure-fluid equation of state to mixtures is the one-fluid model proposed by van der Waals. The underlying assumption of the van der Waals one-fluid model is that the same equation of state used for pure fluids can be used for mixtures, if a satisfactory way can be found to obtain the EOS parameters for mixtures. The common method for doing this is based on expanding the equation of state in powers of (b/v); for the van der Waals equation one obtains

a

R T n=0 R T v

To satisfy the boundary condition of Equation (9.2), the composition dependence of a two- parameter cubic equation of state of the van der Waals family must conform to the relation

324

B ( T , x ) ~'~'~xixjBij(T) b a ~i ~j ( -~T! . . . . XiX j b - (9.8) i j RT " " ~i

It should also be noted that

C(T, x) = ~ ~ ~ xixjx kCUk (T) = b 2 (9.9) i j k

It is not possible to set a composition dependence of the b parameter to simultaneously satisfy both Equations (9.2) and (9.3).

Until recently, the most common way of choosing mixture parameters was to use the so- called van der Waals one-fluid mixing rules to satisfy only Equation (9.8) with the mixing rules:

a = ~ ~ x i x j a O. (9.10) i j

b = Z Z xixjbij (9.11) i j

The following combining rules are frequently used to obtain the cross coefficients a,j and b O. from the corresponding pure-component parameters:

a o. = 4aiiajj (1 -- k o. ) (9.12)

l (bii + bjj )O-~ij ) (9.13)

where k O. and 6ij are the binary-interaction parameters obtained by fitting equation of state predictions to measured phase-equilibrium and volumetric data. Generally 60 • is set to zero, in which case Equation (9.11) reduces to

b = Z xibi (9.14) i

where bi is the volume parameter for component i (subscript ii and i are used interchangeably in this chapter for pure component parameters depending upon the number of summations in an equation). When fitting vapor-liquid equilibrium (VLE) data, it is found that k o. is approximately zero for relatively ideal mixtures, such as alkane mixtures, whereas for other mixtures, it is not only nonzero but will also change in value with temperature.

There is only a qualitative justification for Equations (9.12) and (9.13). The a parameter is related to attractive forces and, from intermolecular potential theory (3), the parameter in the attractive part of the intermolecular potential for a mixed interaction is given by a relation like Equation (9.12). Similarly, the excluded-volume parameter b would be given by Equation (9.13) if the molecules were hard spheres. However, in general, there is no direct relation between the attractive part of the intermolecular potential and the a parameter in a cubic EOS, and real molecules are not hard spheres. Further, it could be argued that even for the case of

325

hard spheres it is really the collision diameters that are additive, so that if (Yii is the hard-sphere diameter for species i, then

1 cr 1/3 )3 t~i = -2( ii "q- I~jj) which implies b,7 = (bil/3 n t- bjj (9.14b)

since the volume is proportional to the third power of the diameter. Conformal solution theory and other molecular-based theories have led to other mixing and/or combining rules for intermolecular-potential parameters and, by analogy, for equation-of-state parameters. But as there is no direct relationship between these two parameter sets, except for the van der Waals equation, there is only a tenuous basis for such mixing rules, and little evidence that they are superior to the van der Waals one-fluid rules.

The shortcoming of the van der Waals one-fluid mixing rules is that they are only applicable to mixtures which exhibit relatively moderate solution nonidealities as seen in Figures 9.1 and 9.2. In Figure 9.1, the Peng-Robinson equation of state, as modified by Stryjek and Vera (4), is shown to represent the vapor-liquid equilibrium behavior of the CO2 + n- butane system (5) very accurately with a single binary-interaction parameter; but, as seen in Figure 9.2, the VLE data for the acetone + water binary system at 200°C (6) cannot be adequately correlated with a single binary parameter and the van der Waals mixing rule. Why the van der Waals one-fluid mixing rule cannot describe highly nonideal mixtures can be understood by starting with the relation between the molar excess Gibbs energy of mixing, G ex, and fugacity coefficients obtained from an equation of state:

G ex =RTIlnqg-~xilnq)il,. (9.15)

where qo and q)i are the fugacity coefficients of the solution and of the pure component i calculated, using the mixture and pure component parameters, respectively, from the equation

l n q g = l n ( f ( ~ P ) ) 1 i(RT p) = --~ .,.--~- - d v - In Z + (Z - 1)

oo

(9.16)

For the van der Waals cubic EOS this leads to the following expression for the excess Gibbs energy of a binary mixture"

G eX

RT ~ = l n e bii --~FI _ XlX2 a11b22 _ a22b11 2

i=' (RT_b) RT(xlb11+x2b22) bll b22

2XlX2~falla220-k12) Xlbll + x2b22

(9.17)

326

This equation shows that the excess Gibbs energy, computed from a van der Waals-family cubic EOS and one-fluid mixing rules, contains three contributions. The first, which is the Flory free-volume term, comes from the hard-core repulsion terms and is completely entropic in nature. The second term is very similar to the excess Gibbs-energy term in regular-solution theory, and the third term is similar to a term that appears in augmented regular-solution theory (7). As a result the combination of a cubic EOS with the van der Waals mixing rules can only represent those mixtures that have a moderate degree of solution nonideality describable by augmented regular-solution theory. Many mixtures of industrial interest have a much greater

13.

v (D L . "--1 fa9 t o

13_

4000

3000

2000

1 000

0 0.0 0.2

8000 Ill ' ' ' ' ' ' ' ' e31 0.93 K • •

7000 • 344.26 K • •

6000 •377.59 K • • A /

5000 -

I I 1 ~ I ,

0.4 0.6 0.8 1.0

Mole Fraction of Carbon Dioxide

Figure 9.1 Correlation of carbon dioxide + n-butane vapor-liquid equilibrium data of reference 5 (points) with the modified Peng-Robinson equation of state [see reference 4] and with the van der Waals one-fluid mixing rules with a single binary-interaction parameter.

degree of solution nonideality and therefore require different mixing rules. One misconception that has occurred in the use of the van der Waals one-fluid mixing rule

is that setting the binary-interaction parameter k12 equal to zero results in an ideal solution. Equation (9.17) clearly shows that this is not the case since, even with k12=0, G ~x is not equal to zero; indeed none of the three terms in the equation vanish. Also, as shown in Figure 9.2, setting k12=0 for a mixture of very different components such as acetone and water not only does not produce an ideal solution, but can lead to unrealistic phase-behavior predictions. Therefore, a value of zero is not always the best choice of a default guess for k12 in the absence of experimental data.

327

35

30 '-" ",, ",

m"'c,l,'a / f ~ . .- ..: :.. 25

13" I . " " . '

I ,.. , . , "

20 I . . - . "

15 t l 0.0 0.2 014 016 0.8 1.0

Mole Fraction Acetone

Figure 9.2 Correlation of acetone + water vapor-liquid equilibrium data of reference 6 (points) at 200°C with the modified Peng-Robinson equation of state. The solid lines are correlations with the Huron-Vidal mixing rule, the long and short dashed lines are correlations with the van der Waals one- fluid mixing rule with a single binary interaction parameter fit to experimental data, and set to zero, respectively. The dotted lines result from the Wong-Sandler mixing rule with all model parameters set to zero (see last paragraph of Section 9.1.3.2 for explanation).

9.1.2 Non-Quadratic Combining Rules for Van der Waals One-Fluid Model

One empirical approach to eliminate the shortcomings of the van der Waals one-fluid model for a cubic EOS has been to provide additional composition dependence by adding parameters to the combining rule for the a parameter, generally leaving the b-parameter rule unchanged. Some examples are the combining rules of Panagiotopoulos and Reid (8)

k o. = X o - ( % - X j i ) x i (9.18)

Adachi and Sugie (9)

k O. = Kii + l~i(x i - X j ) (9.19)

Sandoval et aL (10),

= x xi + x;,x, + o s(xo + )O- x , - x, ) (9.20)

328

and Schwartzentruber and Renon (11,12)

k U = K U + l U m O x i - mjixj (X i "]- Xj ) (9.21) mijx i + mjix j

where in the last equation KO= Kj'i, lj" i = lij, mji-- 1-m O" and Kii = lii = 0. It should be pointed out that these combining rules do not satisfy any of the boundary conditions given by Equations (9.2) to (9.4). By an appropriate choice of parameters, these combining rules reduce to one another, and to Equation (9.12), and they have been shown to provide good correlations of vapor-liquid equilibrium data for complex binary mixtures, including highly nonideal systems that previously could be correlated only with activity-coefficient models. An example is given in Figure 9.3, where it is seen that the VLE for the ethanol + heptane system at 60°C (5) can be correlated with the two-parameter Adachi-Sugie combining rule much better than with the single-parameter van der Waals mixing rule.

The relationship between an activity coefficient and the fugacity coefficients from an EOS is

r i = [q)i (T, P, x)/q)i (T, P)] (9.22)

where ~ i ( T , P , x ) i s the fugacity coefficient of species/in a mixture and ¢Pi(T,P)is the pure-

component fugacity coefficient, both obtained from the EOS. Consequently, the two interaction Parameters per binary system, in the class of combining rules being considered here, can be related to activity coefficients over the whole composition range or to the values at a specified composition, such as the two infinite-dilution activity coefficients of a mixture, (12,13,14) thus eliminating the need for VLE data over the whole composition range. In this latter case one writes Equation (9.22) at the two infinite-dilution limits, obtaining two equations for the two parameters in Equations (9.18) to (9.21), in terms of the pure-component EOS parameters and infinite-dilution activity coefficients. In this case it is possible to predict the phase behavior of some highly nonideal systems successfully using only infinite-dilution activity-coefficient information.

However, there are several problems associated with these multiparameter combining rules that limit their use. One is that, due to added composition dependence in the a parameter, this group of combining rules fails to satisfy the theoretical quadratic composition dependence of the second virial coefficient given in Equation (9.2). This fact is usually ignored on the grounds that pure-component second virial coefficients calculated from a cubic EOS are in poor agreement with experimental data. However, the equation for the fugacity coefficient of a species in a mixture,

lnq~i = ~ v ~ o [ V - ~,v,u~ j d V - l n Z (9.23)

where Ni is the mole number of species i in the mixture and V is the total volume, shows that the evaluation of the fugacity coefficient from the equation of state involves an integral of a

329

70

60

n 5 0

~ 40 ,,," . . . .

3 0 1/'/ ~ / / / Two Paramete r Mixing Rule 7 / f vdW Mixing Rule

J J i I i I i 1 L 20 0.0 0.2 0.4 0.6 0.8 1.0

Mole Fraction of Ethanol

Figure 9.3 Correlation of ethanol + n-heptane vapor-liquid equilibrium data of reference 5 (points) at 60°C with the modified Peng-Robinson equation of state. The solid lines are correlations with the two parameter Adachi-Sugie (9) mixing rule and the dashed lines are correlations with van der Waals one- fluid mixing rule with a single binary-interaction parameter.

term that includes a partial derivative with respect to composition from zero density to the density of interest. Therefore, an error in the composition dependence of the equation of state at low densities will affect the fugacity coefficient calculated from Equation (9.23) at all densities. It should be stressed that the correct composition dependence of the mixture second virial coefficient and accurate values of pure-component second virial coefficients are two different problems, and both will affect the computed fugacity coefficient. Models that have the correct second-virial composition dependence eliminate one of these sources of error.

A second difficulty associated with mixing rules in this category, as pointed out by Michelsen and Kistenmacher (15), is that they do not result in the correct treatment of multicomponent mixtures containing two or more very similar components. To see this, consider a mixture of three components, and for the sake of demonstration allow two of these components, such as 2 and 3, to become identical. (This is not a trivial thought experiment, as there are multicomponent mixtures of industrial interest in which two or more components have very similar EOS parameters, for example, the isomers 1-methyl-naphthalene and 2- methyl-naphthalene.) In this case, to be internally consistent, the mixing rule for the parameter a should reduce to that for a binary mixture with the composition (x 2+xs) for the new

component 2+3. The one-fluid mixing/combining rule of van der Waals satisfies this test, whereas the group of combining rules being considered here does not.

330

Yet another problem with Equations (9.18) to (9.21) is the so-called dilution effect, which refers to the fact that as the number of components in a mixture increases, mole fractions of individual components in the system become smaller. This leads to smaller contributions from the terms with the higher-order composition dependence and the added parameter(s), thus effectively reducing the mixing rule to the quadratic one-fluid mixing rule of van der Waals as the number of components in the mixture increases.

9.1.3 Mixing Rules that Combine an Equation of State with an Activity-Coefficient Model

Many mixtures of interest in the chemical industry exhibit strong nonideality, greater nonideality than results from regular-solution theory, and have traditionally been described by activity-coefficient (or excess Gibbs energy) models for the liquid phase, and an equation of state model for the vapor phase. However, there are numerous problems with the activity- coefficient description. For example, there are difficulties in defining standard states especially for supercritical components, the parameters in these models are very temperature dependent, and since a different model is used for the vapor and liquid phases critical phenomena are not predicted. Also other thermodynamic properties (densities, enthalpies, entropies, etc.) cannot be obtained from the same model. Therefore, there has been interest in mixture equation-of- state models that are capable of describing greater degrees of nonideality than is possible with the van der Waals one-fluid model.

9.1.3.1 The Huron-Vidal Model

Vidal (16) and later Huron and Vidal (17) proposed the first successful combination of equations of state and activity-coefficient models by using Equation (9.4) in the following form

G ; ( r , v = Gaos(r,P = ®,x) (9.24)

where G~X(T,P=oo, x) and G~Xos(T,P=oo, x)are the excess Gibbs energies at infinite pressure

(i.e., at liquid-like densities) calculated from an activity-coefficient model and the equation of state, respectively. Since GeX=AeX+pv ex, A ex in the liquid state is almost independent of pressure, and at infinite pressure from an equation of state, vi=bi, in order to use Equation (9.24) it is necessary that

V ex =V-ZXiV i =b -Zx ib i = 0 (9.25) i i

That is, to use Equation (9.24), Equation (9.14) must also be used. This provides the two equations necessary to determine the two EOS parameters. The resulting mixing rule for the a parameter, then, is

a : (9.26)

v

- I f u q

1 3 .

, , , , ,

~ ' 250c '

1 000

4 ....

2 -

loo

0.0 0.2 0.4 0.6 0.8 1.0 Mole Fraction of Acetone

331

Figure 9.4 Correlation of acetone + water vapor-liquid equilibrium data of reference 6 (points) with the modified Peng-Robinson equation of state. The solid lines are correlations with the Huron-Vidal mixing rule and the NRTL excess energy model. Note that different parameters are used for each isotherm.

where C is a numerical constant that depends on the particular EOS used, and G ex is an excess Gibbs energy of mixing expression appropriate for the mixture of interest.

This mixing model has been successful for describing the VLE of highly nonideal systems. An example is shown in Figure 9.4 for the VLE of the acetone + water system (6) in which the three-parameter NRTL activity-coefficient model was used for G e×. However, each of the isotherms in this figure were calculated with a different set of parameters (i.e. three parameters have been fit to each isotherm). Also, none of the parameter sets are the same as those reported in the DECHEMA Chemistry Data Series (5) where this same activity- coefficient model is used to correlate the data directly.

However, there are several theoretical and computational difficulties associated with the Huron-Vidal mixing rule. For example, this mixing rule has not been successful in describing nonpolar hydrocarbon mixtures. Also this mixing rule does not satisfy the quadratic composition dependence required by the boundary condition of Equation (9.2). Further, as already mentioned, even though the Huron-Vidal approach allows the use of G ~x models with an EOS, the parameters are not the same as those obtained when correlating data directly with the activity-coefficient model. Consequently, one cannot use parameter tables (for example, the DECHEMA Chemistry Data Series) developed for excess Gibbs-energy models at low pressure with this equation of state model. The reason for this is that the excess Gibbs energy of mixing, both from experiment and as calculated with an equation of state, is very pressure dependent as seen in Figure 9.5 for the methanol + benzene system at 373 K. Therefore, the

332

0.45

0.35

0.25

0.15 v

0.05

-0.05 0.0 0.2 0.4 0.6 0.8 1.0

Mole Fraction of Methanol

Figure 9.5 The excess Gibbs and Helmholtz energies of mixing for the methanol + benzene mixture at 1 bar and 1000 bar calculated from the modified Peng-Robinson equation of state and the Wong- Sandier mixing rule.

excess Gibbs energies at infinite pressure and at the pressure at which experimental data were obtained can be very different. Finally, the G ex parameters in this model are very temperature dependent, so that while the model can be useful for correlation, it has little extrapolative or predictive capability.

9.1.3.2 Wong-Sandler Model

Wong and Sandier (18) have developed a mixing rule that combines an equation of state with an excess Gibbs energy model but produces the desired EOS behavior at both low and high densities without being density dependent, uses existing G ex tables, allows extrapolation over wide ranges of temperature and pressure, and provides a conceptually simple method of accurately extending LYNIFAC or other low-pressure VLE prediction methods to high temperature and pressure. This mixing rule is based on several observations. First, while Equations (9.10) and (9.11) are sufficient conditions to ensure the proper composition dependence of the second virial coefficient, they are not unique. In particular, the van der Waals one-fluid mixing rules of Equations (9.10) and (9.11) place constraints on the two functions a and b to satisfy the single relation

333

a~/1 = b a (9.27) B(xi'T)= Z Z x i x j B o ( T ) - - Z Z x i x j b°"---~J -- R--T

The original version of the Wong-Sandler mixing rule uses the last equality above as one of the restrictions on the parameters, together with the combining rule

- ~ ) l Ilbii - aii l (bjj ajj l l (1-kij ) (9.28) b- o = -~ RTJ + - RTJ

which introduces a second-virial-coefficient binary-interaction parameter ko.. Note that Equation (9.27) does not provide relations for the parameters a and b separately, but only for the sum b-a/RT, so that an additional equation is needed. Also, by using Equation (9.27) as one of the relations to determine the equation of state parameters, the proper composition dependence of the second virial coefficient is assured, regardless of which additional equation is used.

The second equation in their mixing rule is based on the observation that the excess Helmholtz energy of mixing is relatively insensitive to pressure as can be seen in Figure 9.5. Consequently to an excellent approximation:

G ex (T,P = l bar, x )= A ex ( T , P = l bar, x )= A ex (T, high pressure, x) (9.29)

The first of these equalities follows from the fact that G e X = A ex + Pv ex, and that the P v ex term is very small at low pressures. The second of these equalities is a result of the essential pressure independence of A ex at liquid densities. Therefore, the second equation for the a and b parameters comes from Equation (9.4) in the form of

ox = = ox v x) = A, °x(v, low P, x) = G, (v, low P, x) AEos(T,P m,x) A ~ ( , P = m , ex (9.30)

where subscript 7 refers to a property obtained from an excess-energy model. Combining Equations (9.27) and (9.30) gives the following mixing rules

b a -- R---~ = Z Z x i x j b - (9.31) i j U

and

G•X a a i . . . . ~i Xi ~ (9.32) CRT bRT • beRT

where the cross term in Equation (9.31) is obtained from Equation (9.28), and any excess Gibbs-energy model may be used in Equation (9.32).

This EOS/activity-coefficient-model mixing rule combination satisfies the low-density boundary condition that the second virial coefficient is quadratic in composition, and the high- density condition of producing an excess Gibbs energy like that produced by currently used

334

activity-coefficient models, while the mixing rule itself is independent of density. This model provides a more correct alternative to earlier ad-hoc density-dependent mixing rules (19,20,21) that violate the one-fluid model in that the equations of state for pure fluids and mixtures have different density dependencies. This mixing rule has been successful in a number of ways. First, when combined with any cubic EOS that gives the correct saturation pressures for organic chemicals and an appropriate activity-coefficient model for the G ex term, it has been shown to lead to excellent correlations of vapor-liquid, liquid-liquid and vapor.liquid-liquid equilibria. Indeed, these correlations are generally comparable to those obtained when the same activity-coefficient models are used directly (5). Consequently, this new mixing rule extends the range of application of equations of state to mixtures that previously could be correlated only with activity-coefficient models.

Second, since low-pressure G ex information has been used in developing this mixing rule, Wong et aL (22) found that activity-coefficient parameters reported in databanks, such as the DECHEMA Data Series (5), could be used directly and with good accuracy in the new mixing rule without the need of refitting experimental data. Further, they found that, as a result of the inherent temperature dependence in equations of state, the parameters in the excess-Gibbs- energy model used in the EOS mixing rule were much less temperature dependent than when the same excess-energy model was used directly. In fact, for many mixtures the Gibbs-energy- model parameters in the EOS could be taken to be independent of temperature, thereby allowing extrapolation of phase behavior over wide ranges of temperature and pressure. This is shown in Figure 9.6 for the system ethanol + water (23), where activity-coefficient-model

13.

(9

otj

13.

1 0000

1 000

6 5 ~ 4 0.0

350 C

325 C 300 C

275C

250 C

200 C

150C

0.2 0.4 0.6 0.8 1.0

Mole Fraction of Ethanol

Figure 9.6 Predictions of the vapor-liquid equilibrium for the ethanol + water system with the modified Peng-Robinson equation of state and the Wong-Sandler mixing rule with temperature- independent parameters (22). The experimental data shown as points are from reference 23.

335

parameters reported in the DECHEMA Chemistry Data Series (5) at 4 bar are seen to lead to excellent predictions up to 200 bars. A third observation is that, since this mixing rule uses information at low pressure, it can also be used to make predictions at high pressure based on low-pressure prediction techniques, such as UNIFAC and other group contribution methods (24). This is demonstrated in Figure 9.7 for the n-pentane + ethanol (25) binary system, where the UN/FAC predictions at 20°C are shown as dashed lines, and the solid line shows the predictions using the Wong-Sandler mixing rule fitted to UNIFAC predictions. The remaining solid lines are the extrapolations of this model to higher temperatures and pressures with temperature-independent UNIQUAC parameters. The predictions at temperatures of 373 K and 423 K are seen to be in excellent agreement with the experimental data.

2

lOOO

"~ lOO n

I,..

o l 01 (1)

13. 10 -~.t~~ .... ~ ... . -~ ... . L ~ 1 _ ............ _

i : 1 0.0 0.2 0.4 0.6 0.8 1.0

Mole Fraction of Pentane

Figure 9.7 Prediction of the vapor-liquid equilibrium for the n-pentane + ethanol system with the modified Peng-Robinson equation of state and the Wong-Sandler mixing rule with parameters obtained from low-pressure predictions of the UNIFAC model (24). The dashed lines are predictions based on the UNIFAC activity-coefficient model at 293 K, and the points are experimental data (25).

A desirable characteristic of an excess-energy-based mixing rule is that it converges smoothly to the conventional van der Waals one-fluid mixing rules for some values of its parameters. This is useful because in multicomponent mixtures it may occur that only some of the binary pairs form highly nonideal mixtures requiring mixing rules such as those described in this section, while other binary pairs in the same mixture can be adequately described by use of Equations (9.10) and (9.11). Orbey and Sandler (26) have recently shown that, by a slight reformulation of the Wong-Sandler mixing rule, convergence to the van der Waals one-fluid mixing rules can be accomplished. They did this by retaining the mixing rule of Equations (9.31) and (9.32), but replacing the combining rule of Equation (9.28) by

336

(b---~--f) l (b i+bj ) -4Ri ;J (1- ku) (9.33)

This introduces the binary interaction parameter in a manner similar to that in Equation (9.12) of the van der Waals one-fluid mixing rule. Next the following modified form of the NRTL equation was used for the excess-Gibbs-energy term:

with

a ex I~jxjaji'~ji 1 ---~~ixi L ~kXk-~k i (9.34)

G i i = bj exp(-cz'rji) (9.35)

where bj is the volume parameter in the equation of state for species j. This modified NRTL form was suggested earlier by Huron and Vidal (17) for use in their model.

For binary mixtures there are four dimensionless parameters in the above equations, a, 1:12, ~:21 and k12. This mixing rule can be used as a four-parameter model to correlate the behavior of complex mixtures, or in a number of other ways that have fewer adjustable parameters. For example, for selected values of a and k12, one can solve Equation (9.34) in the infinite-dilution limit for v21 as

bl exp(_ot,rl2 ) ~'21 = In ~/12 --~'12 b22 (9.36)

where Y12 is the infinite-dilution activity coefficient of species 1 in 2. A similar relation for "~12

is obtained by index rotation. In fact, Orbey and Sandler (26) found that setting a=0.1 and k12=0 worked quite well for several nonideal mixtures of organic chemicals, so that one could use the model to predict the complete phase behavior over large ranges of temperature and pressure from infinite-dilution activity coefficients at a single temperature. This model could also be used in a completely predictive fashion, in the absence of any experimental data, with infinite-dilution activity coefficients obtained from the UNIFAC model. Various forms of this reformulated Wong-Sandler mixing rule have been shown to be successful for highly nonideal binary mixtures. An example is shown in Figure 9.8 for the 2-propanol + water binary mixture. Also, by setting a=0 and solving for v21 from

C [2 a~la2 ( l_k12)_a l l vzl =-R-f L bl + b 2 -~ (9.37)

('~12 is obtained by index rotation) one recovers the van der Waals one-fluid mixing rule with a single binary-interaction parameter k12 that can be used for the slightly nonideal binary pairs in

a multicomponent mixture. It should be noted that Equation (9.37) is not unique, and other expressions that lead to the van der Waals mixing rules are possible (26).

337

1 2500 ,, ," " . / .

1 0000

7500 , , ,T , 7 25o c

. . . . . : ..........~ ._ _. . : ..: ._.--.-:..~ - - ' 2 ~ ~ - r: - O - i ~ -

5000

200 C

v

f f l

13_

2500

150C

0 J I ~ L i ~ i ! r

0.00 0.20 0.40 0.60 0.80 1.00

Mole Fraction of 2-Propanol

Figure 9.8 Predictions of the vapor-liquid equilibrium for the 2-propanol + water system with the reformulated Wong-Sandler mixing rule and the modified Peng-Robinson equation of state. The solid lines are predictions made with parameters determined from infinite-dilution activity coefficients predicted with the UNIFAC model, the dashed lines are predictions with parameters from the correlation of the lowest temperature isotherm, and the dotted lines are correlations with parameters obtained by correlating each isotherm separately (26). The experimental data (points) are from reference 23.

The correlative capabilities of the Wong-Sandler mixing rule have also been investigated for systems for which no equation-of-state model worked very well. One example is correlation of the vapor-liquid equilibrium of polymer-solvent systems which is important for the design of solvent devolatilization processes. Mixtures of ethylene and long-chain alkanes can be considered as prototypes for low-molecular-weight polymers in supercritical solvents. Such systems were recently studied by Orbey and Sandler (27) by coupling the Wong-Sandler mixing rule with the Flory-Huggins model for the excess Gibbs energy. A three-parameter correlation of data for the ethylene + n-tetracontane system (28) to the critical point of that mixture by this approach is shown in Figure 9.9.

338

A

v (1) L_

(!.1

13.

45000

40000

35000

30000

25000

20000

1 5000

10000

5000

\ \

\, \

, \

'm \

[] 423.15 K

[] 373.15 K

0 i I

0.0 0.2 1.0

", \ "~. \ ~

". ~

. . . . . . . ~-- ' i~

0.4 0.6 0.8

Mole Fraction C4oH82

Figure 9.9 Partial pressure of ethylene in n-tetracontane. The lines are the result of a three-parameter correlation obtained with the Peng-Robinson equation of state and the Wong-Sandler mixing rule (27), and the experimental data (points) are from reference 28.

This mixing rule has also been applied to hydrogen + hydrocarbon mixtures (29). However, in this case there can be a problem, depending on the temperature, if the conventional temperature dependence is used for the parameter a of the equation of state. This arises from the fact that the temperature-dependent a term in cubic equations of state is usually obtained by fitting vapor pressure and other subcritical data, and consequently is poorly defined at high reduced temperatures. As discussed in Chapter 4, several different temperature functions have been proposed. Some of these temperature functions, including the ones originally proposed for the Peng-Robinson and Soave-Redlich-Kwong equations, lead to the following problem in the Wong-Sandler mixing rule with hydrogen due to its low critical temperature and with other asymmetric mixtures containing very light components. The b parameter in this mixing model is, from Equations (9.31) and (9.32)

b= iai X 1 93,, 1- ~.x, biRT+-C~

339

The denominator of this equation contains three terms. The excess Gibbs-energy term can be negative or positive, and should vanish at high temperatures. For simplicity this term is neglected in the discussion here. In this case, unless the (a/biR7) term is larger than unity for all the components of the mixture, there will be a composition of the (liquid or gas) mixture at which the denominator becomes zero or even changes sign from negative to positive. The first of these is theoretically not allowed because of the discontinuity that results in the value of b of the mixture, the second case is not allowed because it leads to meaningless negative values for the b parameter. To avoid this, it is necessary at all temperatures to have

a i bg - ~ < 0 (9.39)

for all the components of the mixture, which ensures that the (ai/biRT) term is always larger than unity. For cubic equations of state this term can be written in a generalized form; for example, for the Peng-Robinson equation of state

a = O.45724 R2Tc2 a(T) Pc

b = 0.0778 RTc P~

a _ _ (0.45724) a(T)

bRT 0.0778 Tr =5"87712( a(T)-~ )

(9.40)

where subscript c denotes the critical property, Tr=T/Tc, and a(T) is the temperature- dependent generalized function specific to each EOS. Thus the requirement of Equation (9.39) becomes a(T)>_Tr/5.87712 for the Peng-Robinson equation of state. For most cubic equations of state at reduced temperatures below 2 this requirement is met but, depending upon the temperature dependence of a(7), may not be satisfied at higher reduced temperatures. A simple solution for mixtures with one or more highly supercritical components is to use the equation a(T) =Tr/M for the highly supercritical component, with M being an EOS-dependent constant, for example 5.87712 for the Peng-Robinson EOS. This removes the singularity, and does not cause any problems or inaccuracies in the calculation of phase behavior or thermodynamic properties.

One final observation, setting G ex equal to zero in the Wong-Sandler mixing rule does lead to the prediction of almost ideal solution behavior, as shown in Figure 9.2 for the acetone + water system. The predictions in this figure are to be compared with the unrealistic results of setting k12 =0 in the van der Waals one-fluid model, also shown in that figure. A conclusion that can be drawn from this figure is that, while using a default value of G ex equal to zero in the Wong-Sandler mixing rule may not lead to accurate predictions, it will not result in unphysical predictions that could cause difficulties in process simulations.

340

9.1.3.3 Combination o f Excess-Energy Models and Equations o f State at Low or Zero Pressure

The problem with the Huron-Vidal model, which results from the pressure dependence of the excess Gibbs energy of mixing, has led a number of modifications based on the idea of equating the excess Gibbs energies of mixing from an equation of state and from an activity- coefficient model at low (or zero) pressures, as originally suggested by Mollerup (30). For example, in developing the UNIWAALS equation, Gupte et al. (31) used the combination of Equations (9.4) and (9.14) at the experimental pressure. However, this approach has the disadvantages of not satisfying the second-virial-coefficient boundary condition, and having to solve the volumetric equation of state for the liquid density three times for each step in the iterative solution of a phase-behavior calculation for a binary mixture, first for each of the two pure components, and then for the mixture.

Another complication that arises with this method and, in fact, with most finite-pressure combinations of equations of state and activity-coefficient models, is that there may not be a liquid-density solution for one or more of the pure components at the temperature and pressure of the mixture. This either restricts the method to mixtures in which all the pure components are liquids at the temperature and pressure of interest, or requires some sort of extrapolation technique. Further, there is an essential inconsistency in this method in that it produces mixture-equation of state parameters that are functions of pressure (or density). Consequently, when applying the same equation of state to both the pure fluids and their mixture, the volumetric dependence of the equation of state for the mixture is different from that for the pure fluids due to the implicit pressure dependence of the mixture parameters. Gani et aL (32) eliminated some of these problems, however, the computational aspects of the resulting model remain too complex for routine use.

To eliminate, or at least simplify, the problem of requiting multiple solutions of the equation of state, most other modifications of the Huron-Vidal mixing rule have made the connection between G~ x and G~o s at a fixed pressure, usually at zero pressure. Before

analyzing these models, several general comments need to be made. The starting point for combining excess-energy and equation of state models is, for the case of the excess Gibbs energy of mixing,

G~Xos (T, P, x ) = ln~m~ (r, P, x)- Z xiln~i(T'P) RT i

= ( Z n ~ x ( T ' P ' x ) - Z x i Z i ( T ' P ) ) - ( i

1 , ~ ,p,x) Z,,~, - 1 X-~ r Z i - 1 _ ~

Vrrfix i oo V i

(9.41)

and equivalently, for the excess Helmholtz energy of mixing,

AEos(T,P,x) _ Z xiln R T i 2 i - - ~ F i J Vrrfix

v(r.e,) Zi _ 1 1 av=-Ex I 1 i ~ Vi

(9.42)

341

From the conventional definition of an excess-property change on mixing, it is necessary that the pure components and the mixture be in the same state of aggregation. It is obvious from these equations that the excess Gibbs or Helmholtz free energies of mixing computed from an equation of state is a function of pressure, whereas activity-coefficient models are independent of pressure or density. Therefore, the equality between G ex (or A ex) from an equation of state and from an activity-coefficient model can be made at only a single pressure. Models that combine equations of state and activity-coefficient models can be categorized into two groups: those that make this link at infinite pressure, such as the models discussed in the previous section, and those that make this connection at low or zero pressure. Here we are concerned with the latter group.

For van der Wails-type cubic equations, Equations (9.41) and (9.42) take the form

1 GEOS = Zrrflx - E x iZ i - E xiln -~ x,.ln + RT i • . •

vi

(barmRT - ai

(9.43)

and

Fl-bn~x A~°Xs =-~ ix i l n (~ i ) - ~ x i l n | V~x +l arfflx IC('Vnflx)-~Xi( " ~,.,,)C('v~) R T • i [ l _ bi b ~ R T i OilfLI ) "

[_ vi

(9.44)

Here C(v) is a molar volume-dependent function specific to the EOS chosen; for the Peng-

equation C ( v ) = l / ( 2 ~ / 2 ) l n { ( v + ( 1 - 4 ~ ) b ) / ( v + ( l + ~ ) b ) } . For later reference we Robinson

note that, in the limit of infinite pressure, vi--'-~bi and Vmix----~brnix [so that ( V m i x = b m i x ) = C ( v i = b i ) = C * , and for the Peng-Robinson equation C *= [In (~/2 -1)]/~/2 = -0.62323].

The modified Huron-Vidal mixing rule of Michelsen (33,34) is the most used of the category of mixing rules discussed in this section. Its basis is to use Equation (9.43) at P=0 to obtain

x I° ll il i llai GEos(P=0) _~ixi l n v ~ - b ~ ( an~x C(v ° ) - ~ i x i C(v?) R---T = v~ ° -ff ii - Zi x i ln + ~ b = R T rfftx . b i R Z

(9.45) which, to look more like the Michelsen expression, can be rewritten as

I il = + xiln (9.46) i R T •

342

with

( = / a q { a } = - l n v (T ,P O) _ 1 + C(v °) b

(9.47)

In Equations (9.46) and (9.47), the notation q {a} is being used to indicate that q is a function of a = a/bRT. These equations use one of the degrees of freedom in choosing the mixture- equation of state parameters. The second equation that is used to define the mixing rule completely is Equation (9.14). However, it should be noted that since it is no longer necessary to be concerned with the divergence of the excess Gibbs energy at infinite pressure, it is not necessary to impose Equation (9.14); other choices, such as the last equality of Equation (9.27), could be used as well.

A problem that arises with this and other versions of the zero-pressure mixing rules is what to do at temperatures when there is no liquid root to the equation of state for some of the pure components or the mixture. Michelsen (33) eliminated this problem by choosing a cut-off value of a, G(cutoffi for which a liquid root exists, and for smaller values of a using the linear extrapolation

q(a) = qo + q,a (9.48a)

It was then suggested that Equation (9.48a) be used for all values of a, resulting in Equation (9.46) being replaced by

1 IG~X(T'P = O,x) I~" rrfix - - 7 . X i a i +

L 7" • q l RT + Zxiln( bnf~x l]

i Cbi)J (9.49a)

The combination of Equations (9.14), (9.46) and (9.49a) is referred to as the MHV1 model. Later, for better accuracy, a quadratic extrapolation for q(a) was proposed

q(a ) = qo + ql a + q2 a 2 for a < a~.~ff (9.48b)

with the constants q0, ql and q2 chosen to insure continuity of the function q(a) and its derivatives. This gives the following quadratic equation to be solved for amix=amix/(bmixRT) •

ql an~ - ~ X i a i + q2 a ~ -ZXgag = + Zxgln (9.49b) • i RT •

which, together with Equations (9.14) and (9.48b), is the MHV2 model (35). An alternate derivation of the MHV1 model is to start with Equation (9.45) and assume

(v ° )= C(v°)= q, which immediately gives that v~°/bn ~ - vi°/bg - constant, so that C ~ ,

Equation (9.49a). From this derivation we see that an assumption inherent in the MHV1 model is that the ratio of the zero-pressure liquid molar volume to the close packing parameter b is the same for the mixture and for each of the pure components.

343

In spite of the fact that the MHV1 and MHV2 models do not satisfy the second-virial- coefficient boundary condition, and include an extrapolation procedure, they have been found to give quite reasonable correlations and predictions of experimental data for highly nonideal systems. However, the MHV2 model is less accurate than the Wong-Sandler mixing rule for extrapolations over large temperature ranges, and when using binary data to make ternary and multicomponent predictions (36).

Recently Tochigi et al. (37), using the observation we have discussed, proposed a modified Huron-Vidal mixing rule consistent with the second-virial-coefficient boundary condition by combining Equations (9.27), (9.45) or (9.46), and (9.48b). In their implementation they have eliminated the binary interaction parameter in Equation (9.27) leading to

an~ x I ~ i ai I{G~"(T,x)~i ( ~ ' - / } ] R T = bn~ Xi biRT +--ql ~+RT • xiln (9.50a)

and

ixi( 0b,

_ _ _ + xiln 1 Z Xi hiNT ql R T i

This mixing rule model has been tested for several binary and three temary systems and shown to be approximately equivalent in accuracy to the MHV 1 model.

Boukouvalas et al. (38) proposed a new mixing rule by forming the following linear combination of the Huron-Vidal and Michelsen models

a ~ 1 - ~ G~X + xiln + x i b-~-R r + -ql ' R r q l " " "

a i b i R T

(9.51)

In deriving this mixing rule the authors have ignored the pressure dependence of the excess Gibbs energy of mixing (which is why activity-coefficient parameters had to be recorrelated when using the Huron-Vidal model) and have assumed that G~, of the Huron-Vidal model

e)c which is evaluated at infinite pressure, and G r of the Michelsen model, which is evaluated at

zero pressure, are identical. They have then shown that using the UNIFAC model in their mixing rule, with an appropriate choice of the additional parameter ~ (0.36 for the original UNIFAC model, and in the range from 0.65 to 0.75 for the modified UNIFAC model), led to reasonable predictions for many systems.

An earlier activity-coefficient-based equation of state mixing rule is that of Heidemann and Kokal (39). Their zero-pressure based mixing rule is, like Michelsen's, based on Equation (9.45) and the two are identical at temperatures below which there is a zero-pessure solution for the liquid volume. At higher temperatures for which there is no zero pressure solution for the liquid volume, they obtain an apparent zero-pressure liquid volume by solving (following their notation)

344

3 a i ai ~i = bi = 1+ ~ RT +6 biRT P=O

(9.52)

where 13 and 6 are equation of state-dependent parameters chosen to insure continuity of ~ and its temperature derivative at temperatures at which a zero-pressure-volume solution can no longer be obtained. Heidemann and Kokal also set the standard-state pressure by demanding that

e v ) = ~ xiZ i (9.53) Zmix -- - ~ mix i

They then have ~mix, amix=(amix/bmixRT), and bmix as unknowns; these are obtained by an iterative process using the set of nonlinear equations that includes Equations (9.45), (9.52) and (9.53), and the equation of state.

A simple model similar to the zero-pressure models discussed above has recently been proposed by Orbey and Sandier (40). This has an easy-to-understand basis, results in equations that are almost identical to other models of this class in use, and leads to predictions of vapor- liquid equilibria over a broad range of temperatures with temperature-independent parameters, and is comparable in quality to other models in this class. The basis is to assume that for all fluids there is a universal parameter, u, that relates the liquid molar volumes to their close- packed hard-core volumes by v=ub, where u is positive in value and larger than unity at any given temperature. This simple relation is used for both the mixtures and for the pure liquids. In this case Equation (9.44) can be rewritten as

x lu xl E I I (ail A~os = -~xi ln ~ + b~xR T) Zi xi biRT R T • ub i - ~i xiln 1-u C(ubna x ) _ C ( u b i)

/ _ • bn~xRT) C(ubnax) ~ixi( ai biRT) C(ubi)

(9.54)

Further, u is unity at both infinite pressure and at very low temperatures; for simplicity, as an approximation, setting u=l at all conditions (so C(b)=C *) the following equation results:

A~x°SRT = - ,~. xiln ( ~ / + . C* ( b~xRTana x ~xi_.~l. ai (9.55)

The relation given by Equation (9.55) uses only one degree of freedom in determining the a and b parameters for the mixture equation of state, so that this equation can be coupled with either Equations (9.11), (9.14) or (9.27) to obtain a new mixing rule. The first alternative, Equations (9.11) and (9.55), leads to an algebraically simple mixing rule that is very similar to the MHV1 model, but with the ql replaced with C*, and, like that model, does not satisfy the second-virial-coefficient composition dependence. For the Peng-Robinson equation of state, ql = -0.53 (34), while C *= -0.62323 as discussed earlier. However, this small difference is

345

3 7 5 0 ~

3250

2750

2250

1750

1250 0.00 0.20 0.40 0.60 0.80 1.00

3750

3250

~.~ 2750

2250

1750

1250 , ~ , ~ r , ~ , 0.00 0.20 0.40 0.60 0.80 1.00

3750 . , , , ,

.--.. 3125 n

2500 .-n 01 ol

n 1875

1250 0.00 0.20 0.40 0.60 0.80 1.00

Mole Fraction Acetone

Figure 9.10 Comparison of various predictive models for the vapor-liquid equilibria of the acetone + water binary system at 200°C with the modified Peng-Robinson equation of state with the experimental data (points) of reference 6. (a) The solid lines are the results of Equations (9.14) and (9.55), and the short and long dashed lines are from MHV1 and MHV2 models, respectively. (b) The results of the Equations (9.14) and (9.55) with C*- -0.623 (solid lines), -0.52 (dashed lines) and -0.68 (dotted lines). (c) The results of Equations (9.14) and (9.55) (solid lines) and the models of Wong and Sandler (dash-dot lines) and Boukouvalas et al. (dotted lines).

significant when extrapolating or predicting vapor-liquid equilibria. An example is shown in Figure 9.10 in which the VLE of the acetone + water binary mixture is predicted at 200°C based on a van Laar-excess-Gibbs-energy model with parameters obtained at 20°C which were reported in the tables of the DECHEMA Chemistry Data Series (5). This figure includes predictions based on the MHV1, MHV2, Wong-Sandler, Boukouvalas et al. models, and the zero-pressure model discussed above with various values of the parameter C*. The most

346

accurate predictions are obtained with the Wong-Sandler model. The results of the model discussed immediately above and the Boukouvalas et al. model are comparable, though the former is better in the midconcentration range. Of the zero-pressure models, the simple one proposed above is clearly superior in extending-low pressure information to higher temperatures and pressures. Another mixing rule can be developed by combining Equations (9.27) and (9.55); this model satisfies the second-virial-coefficient boundary condition, and results in a mixing rule that is slightly more complex than the one above, and very similar to the model of Tochigi et al. (37), again with ql replaced with C*. This model has not yet been tested.

9.1.4 Commentary on Mixing Rules for Cubic Equations of State

For almost a century, the van der Waals one-fluid mixing rules were the only ones used to apply cubic equations of state, developed for pure fluids, to mixtures. Initially, the only innovation was the use of one or more binary interaction parameters. Even with that change, cubic equations of state led to reasonable correlations and predictions only for mixtures with small excess Gibbs energies of mixing, such as hydrocarbons, and hydrocarbons with inorganic gases. After a number of unsuccessful attempts at extending the range of equations of state by introducing ideas such as density-dependent mixing rules and other ad hoc procedures, theoretically-based mixing rules have now emerged that allow equations of state to describe mixtures with large excess Gibbs energies of mixing. These mixing rules also allow extrapolations and predictions over large ranges of temperature and pressure. However, each of these mixing and combining rules for cubic equations of state has some limitations, as has been discussed in this section, so that this area is one that continues to deserve further research.

9.2 MIXING RULES FOR THE VIRIAL FAMILY OF EQUATIONS OF STATE

9.2.1 The Van der Waals One-Fluid Model

The virial equation of state of Equation (9.1) represents the volumetric behavior of real gases as an infinite power series in inverse molar volume about the ideal gas state. This equation is a MacLaurin series in that each term is a correction to the sum of the preceding terms. Following the suggestion by Kamerlingh Onnes (41) B is called the second virial coefficient, C is the third virial coefficient, and so on. From statistical mechanics, these coefficients are related to the forces between molecules, and for a pure fluid these virial coefficients are functions of temperature only. Also, the composition dependence of each of the virial coefficients in a mixture is known from statistical mechanics, and is given by Equations (9.2) and (9.3). Since the virial equation is only applicable to gases, it is of limited use; however, the known composition dependence of the virial coefficients provides the basis for the mixing rule for this family of equations of state.

Augmented forms of the virial EOS that include terms to represent the infinite number of terms in the series have been proposed for use in accurate PvT and phase behavior calculations. Among these is the equation of Benedict, Webb, and Rubin, (BWR) (42)

347

Bo R T act Z = I + + + + ~

2 v v RTv s R T 3 v2 (1 v ) exp(- v ) (9.56)

developed to fit the P v T data of methane, ethane, propane, and n-butane, and later extended to include eight addditional hydrocarbons up to n-heptane (43). In Equation (9.56), a, b, ¢, A, B,

C, a and Y are the equation-of-state parameters. This equation is useful for the high-accuracy calculation of density and derived properties, such as enthalpy, fugacity, vapor pressure, and latent heat of vaporization. The BWR equation is a closed form of the virial EOS as it contains an exponential term that can be expanded into an infinite series in reciprocal molar volume representing all the remaining terms in the series. This exponential term makes a large contribution to the equation of state at high density and in the critical region. Expanding the BWR equation in virial form, we obtain the following

/ Ao co// R,a / Bo R T R T 3 b - I +

Z = I + + . . . 3 _ yc + (9.57) 2 ,124 . . . . v v R T 3

The most common method of extending the BWR equation to mixtures is based on the van der Waals one-fluid idea that the mixture and the pure fluids should satisfy the same equation of state, and on insuring the correct composition dependence of as many virial coefficients as possible by using the equations

Bo : ~ Xi Boi i=l

A o - - x i ( A o i ) 2 , i=l

C o -- x i ( C o i ) ~ i=l

a:E x a) l 13 E n 1

i=l i=l

(9.58)

as originally proposed by Benedict, Webb, and Rubin (42). These combining rules, that do not contain any adjustable parameters, are satisfactory for light hydrocarbon mixtures, but not for mixtures containing nonhydrocarbons or even heavy hydrocarbons. Stotler and Benedict (44) first proposed the use of a single interaction parameter m, 7 (which is similar to the interaction parameter used with cubic equations of state) in the mixing rule for Ao in the BWR equation

1

Ao = Z Z x i x j ( A o i A o j ) ~ mu (9.59)

348

This mixing rule was later used by Orye (45) for hydrocarbon mixtures, and by Nohka et al. (46) for refrigerant mixtures with nitrogen, argon, methane, etc.

A shortcoming of extended virial equations of state in their application to mixtures in the manner described above arises from the fact that, given the uncertainty in experimental data, the many pure-component equation of state parameters cannot be uniquely determined. Consequently, many sets of constants are possible depending on the source of the experimental data and the weighting factors used in the regression. In fact, significantly different sets of constants have been reported by different investigators for the same pure fluids as discussed below. Although all these sets of the constants are, in general, satisfactory for the representation of a pure fluid, they can lead to quite different results for mixtures. This is especially true when mixing rules such as those above are used for a mixture containing components whose equation of state constants differ considerably in magnitude. This problem can be alleviated by using generalized forms for the equation of state parameters, as will be discussed in the next section.

Many modifications have been proposed for improving the overall accuracy of the BWR equation by increasing the number of terms, and thus increasing the number of constants. One of the first such modifications was by Strobridge (47):

P = RTp + [C1RT + C2 + C3 + C4 "1- C5 ~ p2 p3 p4 T F F ) nt-(c6ez-I-C7) nt- C8 Z at- C15 P

I(~2 9 (C12 C13 C14)]p3 Clo Cll) + + + exp(-?' 0 2) + + T 3 + Z 4 ) T 2 T 3 T 4 P 9.

(9.60)

where p is molar density and the Ci are equation of state constants. This 16-constant equation set the pattern for many of the later modifications, including those by Bender (48), (20 constants), Morsy (49), (10 constants), Starling (50), (11 constants), Jacobsen and Stewart (51), (32 constants), Lee and Kesler (52), (12 constants), Nishiumi and Saito (53), (15 constants), Schmidt and Wagner (54), (32 constants), and the AGA Natural Gas Equation by Starling (55), (53 terms). Each of these equations uses mixing rules much like those of Equations (9.58). The increased number of terms and constants generally improves the accuracy of the equation for pure fluids, but for the reasons discussed above, may not result in increased accuracy when applied to mixtures. A more detailed description of equations of state of this class can be found elsewhere in this book, Chapters 10 and 12 and elsewhere (56).

9.2.2 Generalized Extended Virial Equations of State and Their Mixing Rules

One method of avoiding the problem of nonuniqueness of the parameters of the extended virial equation, based on the principle of corresponding states, is to change from temperature and density as the variables with fluid-specific parameters to reduced properties and parameters that are either universal constants or that are generalized in terms of some fluid property, such as the acentric factor c0. Joffe (57) was the first to generalize the parameters in the BWR equation of state by following the suggestion of Kamerlingh Onnes (41) and expressing the reduced pressure (Pr = P/Pc) as a function of reduced temperature (Tr = T/Tc) and ideal reduced volume (Pcv/RTc) with eight universal constants. In this way the equation becomes a two-parameter (Pc and ire) corresponding-states method that is not very accurate. Later Opfell et aL (58) put the BWR equation in corresponding-states form using the reduced

349

pressure, reduced temperature and acentric factor by making its constants, except for ?', linear functions of the acentric factor. Similarly, Edmister et aL (59) made seven of the eight constants quadratic functions of the acentric factor to improve accuracy, keeping the product of a and a constant. A more extensive generalization by Starling and Han (60) used the 11- constant form of Starling's modified BWR equation

P=oRT+ BoRT-Ao T~+-~ ~ + bRT-a-

/ 3 c p (1+ + a a + 106 + ---~ ?' p2)exp(_~, p2) (9.61)

but made the ten constants linear functions of the acentric factor, and Eo an exponential function of the acentric factor, and used the critical volume in the generalization. The dimensionless constants in reduced form are

P ci Aoi Pci Boi = A~ + B~ (-Di - - A2 + B2 ( D i

RTci

P ci Coi

RT~i

2 -- A3 + B 3 f o i Pci~/i = A4 + B 4 f o i

2 2 Pci ai

P c i b i = As + B 5 0 ) i , - A6 + B 6 f o i RTci

2 3 Pci Ci

[£)ciai = A7 + B7 o) i , - As + B8 (-Oi Rr~i

2 3 Pci Ci

P c i a i = A7 + B7 (oi , - A8 --t- B8 (_Di RT~i

2 2 P c i D o i P c i d i _

- - A 9 + 9 9 (_Di, ~ - A l o + B l o o i RT4i RTZi

Pci Eoi

RT~i - All + BllCoiexp(-3.Scoi) (9.62)

Starling and Han (60) used the original BWR mixing rules, Equations (9.58), except for introducing a binary interaction parameter, as follows

i=l j=l

Co- xixj(CoiCo )l'2 l- 3 i=l j=l

350

[ $1/2 )4 Do--~~xixj~DoiDoj) (1-ko. i=l j=l

I d = ~ 1 / 3 Xi ai

i=l

n

Eo=~Zxix j (EoiEoj) l /2(1-ko. ) s (9.63) i=l j=I

The approach discussed above for the generalization of extended virial equations is to write the equation in reduced form and then make the constants functions of the acentric factor. In this approach, even though the EOS constants are linear functions of the acentric factor, the calculated thermodynamic properties are not. It is not possible, with this type of generalization, to produce an equation of state in which the molar volume (or the compressibility factor) and other derived properties are linear functions of the acentric factor, even though such behavior is experimentally observed.

Pitzer and co-workers (61-63) and Curl and Pitzer (64) developed correlations for the compressibility factor (and thus other thermodynamic properties) as a linear function of the acentric factor at given reduced pressure and reduced temperature in the following form

Z ( T r , Pr, fO) = Z (°) (Tr, Pr) + 09 Z (1) (Tr, Pr) (9.64)

Here Z ~°) is the compressibility factor of a simple fluid (or=0) and Z 0) is a departure function, both of which are only functions of reduced pressure and reduced temperature. Lee and Kesler (52) expanded on this idea and made it useable with computers by proposing

Z ( T r , Pr, O) = Z (°) (Tr, Pr) + CO Z (r) (Tr, e r ) -- Z (°) (Tr, Pr)

O(r) (9.65)

where Z ~°) and Z (r) a re the compressibility factors for a simple reference fluid (noble gas, co=0) and a second reference fluid (essentially n-octane for which co (r) is 0.3978) at the same reduced conditions. Lee and Kesler used the following reduced form of the modified BWR equation of state to represent both Z ~°) and z~r):

z = ( P r v r J = I + B + C +D s c43 2 (fl +vY~)exp("vY2) (9.66) \ T r J Vr Vr Vr Trvr

where

_ vPc Vr m u

RTc

B=bl b2 b3 b4

Tr T 2 T~

C = cl - ¢---'L2 "q - ~

T r

and

C3

r~

D = dl + d__22 Tr

They also suggested the following mixing rules for the effective critical parameters

TC = L ~ ~ x i x j v c i j T c o . 12c i=l j=l

n

3" Vc z~, Xi X j VcO" i=l j=l

and n

(3) = Z X i O) i i=l

with the following combining rules

/" 1/3-- 1/3"~ 3

Zci RTc i Yci :

Pc i

Tco = ~Tci Tcj

351

(9.67)

(9.68)

Zci "-- 0.2905 - 0.085 O)i (9.69)

P C =

(0.2905 - 0.085co) RTc

Vc

Note that the mixing rules here for the generalized, extended virial equations are of a different nature than the ones considered previously in this section. While the previous mixing rules combined the values of the equation of state parameters of the pure fluids to obtain the mixture parameters, the mixing rules here use values of the critical temperature, critical pressure, and acentric factor of the pure components to obtain effective values for these physical properties of the mixture. To apply the Lee-Kesler equation to vapor-liquid equilibria, Joffe (65) proposed adding an interaction parameter to Equation (9.68), which Ploecker et al. (66) and Oellrich et aL (67) also did, as well as modifying the mixing rule for Tc as follows

T~ = -iT-i x i x j vco Tcij Vc i=1 j=l

352

Zcij -- 4 Zci Zcj k ij (9.70)

Other, similar mixing and combining rules have been proposed, and it is difficult to establish that one set is clearly superior to the others.

The two classes of mixing rules discussed here (one for the equation of state parameters and the other for the effective critical properties) are the only ones used for the family of extended virial equations of state. Unfortunately, these mixing and combining rules are only suitable for mixtures of hydrocarbons, and for hydrocarbons with inorganic gases. Perhaps future work based on using a different mixing rule for each term of the extended virial equations will allow this family of equations of state to be used for more nonideal mixtures than is currently the case.

9.3 MIXING RULES BASED UPON THEORY AND COMPUTER SIMULATION

One important contribution to the theory of mixing rules comes from the exact results for the composition dependence of the virial coefficients that we have already used in the formulation of boundary conditions. However, from theory and computer simulation, we also have other information that can be used in developing mixing and combining rules. For the case of a fluid of hard spheres (without attractive forces), we now know that an accurate equation of state is the Carnahan-Starling expression (68):

p= RT [ l +rl +rl2 - rl31 v (1-77 )3 (9.71)

where 77 = roper316 with cr being the hard-sphere diameter and p the density. The comparable expression for a mixture of hard spheres, due to Mansoori et al. (69), is

I 3 3] RT 6 77 30102 3772 7/37/2 p = ~ _ _ o + ~ + ~ + ~ (9.72) V 7C Ll--r/3 (1_//3) 2 (1_//3) 3 (1_//3)3 J

where r/j : (~/6)~-"~ DiO'/. Note that, while the combining rules for each of the r/i are different i

(due to the different exponent on or), the mixture equation has the same overall density dependence as the pure fluid equation, and reduces to it in the limit of all components becoming identical. Also, no binary-interaction parameters are used, as is typical in the repulsive or hard-sphere part of an equation of state.

To match the results of molecular dynamics computer simulations for square-well molecules, Alder et al. (70) used a double-power-series expansion in reduced inverse temperature and volume for the attractive part of an equation of state. Modifications of their attractive term have been used in a number of equations of state, such as the augmented and perturbed-hard-sphere equations, the perturbed-hard-chain equation, and the BACK equations of state (71). The general form of this attractive term is

353

F ttr--- ~'~'~ Anm(~TI n \T,l(v°lm (9.73) n m

where the Amn a r e universal constants, Vo is the close-packed volume and u is an effective well depth, both of which are taken to be depend upon temperature in using these equations to describe real fluids. In this case the mixing rules commonly used are

Y o - -ZZXiXjVo,U and u : Z Z x i x j u u i j i j

(9.74)

with the combining rules

O,i ll v°ii 1J3+ v° 13132 andu= uiiu 0k t (9.75)

Consequently, in all the equations that include the double power-series expansion above (or its variations), such as the family of perturbed-hard-chain equations, the equation of state parameters are related to molecular rather than critical properties, and the mixing and combining rules are quadratic in composition for the attractive term, and based on hard-sphere theory for. the repulsive term. Of course, the Carnahan-Starling expression is only correct for hard spheres, and most molecules are not spherical. Therefore, other equations of the same general form have been proposed for molecules of various geometrical shapes. For example, there is the equation of Boublik (72) for hard convex bodies

Pv _ Z _ 1 + (3a - 2)~ + (3a2- 3a + 1)~ 2 - a 2 ~ 3 (9.76) R T (1-~) 3

where a is the surface integral of the radius of curvature divided by three times the molecular volume, and

_ zr ~t~ Vo (9.77) 6 v

where Vo is the hard-core volume. However, there is no obvious or theoretically based mixing rule to extend this equation (and related equations by others) to mixtures of nonspherical molecules.

The last type of equation of state that we will consider is the Statistical Associating-Fluid Theory (SAFT) of Chapman et aL (73) and Huang and Radosz (74). In the SAFT model, a molecule is considered to consist of a collection of segments, and the Helmholtz energy is written as:

A = A ig + A seg + A cha~ + A ass°c (9.78)

354

which is a sum of contributions from the ideal gas, the intermolecular interactions between segments, the formation of chains from segments, and association due to hydrogen bonding and donor-acceptor interactions. The sum of the Camahan-Starling and Alder et aL expressions for interactions among spheres is used for the segment term (Aseg), the first-order perturbation expression of Wertheim (75-79) is used to account for association (A aSs°c) and, based on the work of Chapman (80), also for chain formation (Achain). The segment Helmholtz energy per mole of molecules, A seg , is

A seg = r A seg= r (Ao ~ + Ao disp) (9.79)

where r is the number of segments per chain, and the subscript o refers to a property of a single segment. The segment properties are further assumed to be a combination of a hard sphere term represented by the Helmholtz energy form of the Camahan-Starling expression and a dispersion term given by the expression of Alder et al. with constants as modified by Chen and Kreglewski (80). The contribution to the Helmholtz energy for chain formation comes from an expression due to Chapman (80) and Chapman et al. (73) based on Wertheim's first order perturbation theory (75-79)

A~in

R T

2 - r r / ] - ( 1 - r ) I n 2 ( 1 - r r / ) 3

The association term from Wertheim's theory, is given by

(9.80)

- ~-' In X A - + - - ( 9 . 8 1 ) R T 2

where the sum is over all association sites, M is the number of association sites on each molecule and X A is the mole fraction of molecules which are not bonded at site A. The application of this equation of state requires the replacement of the Carnahan-Starling term with the expression above due to Mansoori et aL, and more complicated chain and association terms given by the authors. We will not reproduce these mixture-model expressions here. However, the important point to note in the SAFT equation, and other equations of this type, is that each of the terms has its own theoretically-based mixing rule that is different from the mixing rules for other terms in the same equation.

9,4 CONCLUSIONS

Equations of state have been used for the accurate description of the P-v-T behavior and thermodynamic properties of fluids for decades. It is mixing and combining rules that allow an equation of state developed for pure fluids to be used for mixtures. Historically, there have been many examples in which an equation of state could be used to accurately describe the properties of two or more pure fluids, but not of their mixtures. This is clearly a failure of the mixing and/or combining rules used. For almost a century, only the simplest mixing rules were widely used; the classical van der Waals one-fluid rules for cubic equations of state, and mixing rules based on the virial equation for more complicated equations of state

355

In recent years there has been a great deal of research on mixing rules and some real advances. Perhaps the most important has been the combination of equation of state and free energy models. These mixing rules, discussed in this chapter, have resulted in simple cubic equations of state providing accurate descriptions of the phase behavior of highly nonideal mixtures over large ranges of temperature and pressure with temperature-independent parameters. Further, these new mixing rules can incorporate mixture group contribution methods, such as UNIFAC, allowing for a simple extension of these models to high temperatures and high pressures with the existing parameter tables. New mixing rules have also been developed for other classes of equations of state. In this area the progress has been slow and less noteworthy. One interesting feature that has emerged, largely from molecular theory, is the use of different mixing rules for the attractive and repulsive contributions to the equation of state.

Much remains to be done in the area of mixing rules. In fact, the formulation of each new equation of state brings with it the question of the mixing rules to be used.

REFERENCES

1. D.Y. Peng and D.B. Robinson, Ind. Eng. Chem. Fund. 15, 59 (1976). 2. G. Soave, Chem. Eng. Sci. 27, 1197 (1972). 3. J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird, Molecular Theory of Gases and Liquids, Sec. 1.3

and Ch. 13, Wiley, New York (1954). 4. R. Stryjek and J.H. Vera, Can. J. Chem. Eng. 64, 820 (1986). 5. J. Gmehling and U. Onken, Vapor-Liquid Equilibrium Data Compilation, DECHEMA Chemistry

Data Series, DECHEMA, Frankfurt am Main (1977). 6. J. Griswold and S.Y. Wong, Chem. Eng. Symp. Set. 48, 18 (1952). 7. J.H. Hildebrand, J.M. Prausnitz, and R.L. Scott, Regular and Related Solutions, Van Nostrand

Reinhold, New York (1970) 8. A.Z. Panagiotopoulos and R.C. Reid, ACS Symp. Ser. 300, 571 (1986). 9. Y. Adachi and H. Sugie, Fluid Phase Equilib. 23, 103 (1986).

10. R. Sandoval, G. Wilczek-Vera, and J.H. Vera, Fluid Phase Equilib. 52, 119 (1989). 11. J. Schwartzentruber and H. Renon, Fluid Phase Equilib. 52, 127 (1989). 12. J. Schwartzentruber and H. Renon, Ind. Eng. Chem. Res. 28, 1049 (1989). 13. C. Torres-Marchal, A.L. Cantalino, and R.M. DeBrito, Fluid Phase Equilib. 52, 111 (1989). 14. K.A. Pividal, C. Sterner, S.I. Sandier, and H. Orbey, Fluid Phase Equilib. 72, 227 (1992). 15. M.L. Michelsen and H. Kistenmacher, Fluid Phase Equilib. 58, 229 (1990). 16 J. Vidal, Chem. Eng. Sci. 33, 787 (1978). 17. M.-J. Huron and J. Vidal, Fluid Phase Equilib. 3, 255 (1979). 18. D.S.H. Wong and S.I. Sandler, AIChE J. 38, 671 (1992). 19. S. Michel, H.H. Hooper, and J.M. Prausnitz, Fluid Phase Equilib. 45, 173 (1989). 20. T.W. Copeman and P.M. Mathias, ACS Symp. Set. 300, 352 (1986). 21. S.I. Sandler, K.-H. Lee, and H. Kim, ACS Symp. Set. 300, 180 (1986). 22. D.S.H. Wong, H. Orbey, and S.I. Sandier, Ind. Eng. Chem. Res. 31, 2033 (1992). 23. F. Barr-David and B.F. Dodge, J. Chem. Eng. Data 4,107 (1959). 24. H. Orbey, S.I. Sandler, and D.S.H. Wong, Fluid Phase Equilib. 85, 41 (1993). 25. S.W. Campbell, R.A. Wilsak, and G. Thodos, J. Chem. Thermodyn. 19, 449 (1987). 26. H. Orbey and S.I. Sandler, AIChEJ. 41,683 (1995). 27. H. Orbey and S.I. Sandler, AIChEJ. 40, 1203 (1994). 28. T.W. de Loos, W. Poot, and R.N. Lichtenthaler, Bet. Bunsenges. Phys. Chem. 88, 855 (1984).

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29. H. Huang, S.I. Sandler, and H. Orbey, Fluid Phase Equilib. 96, 143 (1994). 30. J. Mollerup, Fluid Phase Equilib. 25, 323 (1986). 31. P.A. Gupte, P. Rasmussen, and A. Fredenslund, Ind. Eng. Chem. Fund. 25, 636 (1986). 32. R. Gani, N. Tzouvaras, P. Rasmussen, and A. Fredenslund, Fluid Phase Equilib. 47, 133 (1989). 33. M.L. Michelsen, Fluid Phase Equilib. 60, 47 (1990). 34. M.L. Michelsen, Fluid Phase Equilib. 60, 213 (1990). 35. S. Dahl and M.L. Michelsen, AIChE J. 36, 1829 (1990). 36. H. Huang and S.I. Sandier, Ind. Eng. Chem. Res. 32, 1498 (1993). 37. K. Tochigi, P. Kolar, T. Izumi, and K. Kojima, Fluid Phase Equilib. 96, 215 (1994). 38. C. Boukouvalas, N. Spiliotis, P. Coutsikos, and N. Tzouvaras, Fluid Phase Equilib. 92, 75 (1994). 39. R.A. Heidemann and S.L. Kokal, Fluid Phase Equilib. 56, 17 (1990). 40. H. Orbey and S.I. Sandier, Fluid Phase Equilib. 111, 53 (1995). 41. H.K. Kamerlingh Onnes, Archives Neerlandaises 6, 874 (1901). 42. M. Benedict, G.B. Webb, and L.C. Rubin, J. Chem. Phys. 8, 334 (1940). 43. M. Benedict, G.B. Webb, and L.C. Rubin, Chem. Eng. Prog. 47, 449 (1951). 44. H.H. Stotler and M. Benedict, Chem. Eng. Prog. Symp. Ser. 49, 25 (1953). 45. R.V. Orye, lnd.Eng.Chem. Process Design Dev. 8, 579 (1969). 46. J. Nohka, S. Eiji, Y. Arai, and S. Saito, J. Chem. Eng. Japan 6, 10 (1973). 47. T.R. Strobridge, NBS Technical Note No. 129 (1962). 48. E. Bender, Proceedings on the 5th Symposium on Thermophysical Properties, ASME,

New York (1970), p. 227. 49. T.E. Morsy, J. Chem. Eng. Data 15, 256 (1970). 50. K.E. Starling, Hydrocarbon Process. 50, 101 (March 1971). 51. R.T. Jacobsen and R.B. Stewart, J. Phys. Chem. Ref Data, 2, 275 (1973). 52. B.I. Lee and M.G. Kesler, AIChEJ. 21,510 (1975). 53. H. Nishiumi, J. Chem. Eng. Japan 13, 178 (1980). 54. R. Schmidt and W. Wagner, Fluid Phase Equilib. 19, 175 (1985). 55. K.E. Starling, Revision of AGA No.8 Equation, Gas Research Institute, Chicago (1991). 56. S.I. Sandler, H. Orbey, and B.I. Lee, Models for Thermodynamic and Phase Equilibria

Calculations, Ch. 2, S.I. Sandler, ed., Marcel Dekker, New York (1994). 57. J. Joffe, Chem. Eng. Prog. 45, 160 (1949). 58. J.B. Opfell, B.H. Sage, and K.S. Pitzer, Ind. Eng. Chem. 48, 2069 (1956). 59. W.C. Edmister, J. Vairogs, and A.J. Klekers, AIChE J. 14, 479 (1968). 60. K.E. Starling and M.S. Han, Hydrocarbon Process. 51, 129 (March 1972). 61. K.S. Pitzer, J. Am. Chem. Soc. 77, 3427 (1955). 62. K.S. Pitzer and R.F. Curl, Jr, Ind. Eng. Chem. 49, 2369 (1957). 63. K.S. Pitzer, D.Z. Lippmann, R.F. Curl, Jr., and C.M. Huggins, J. Am. Chem. Soc. 77, 3433 (1955). 64. R.F. Curl, Jr. and K.S. Pitzer, Ind. Eng. Chem. 50, 265 (1958). 65. J. Joffe, Ind. Eng. Chem. Fund. 15, 298 (1976). 66. U. Ploecker, H. Knapp, and J.M. Prausnitz, Ind. Eng. Chem. Process Design Dev. 20, 127

(1978). 67. L. Oellrich, U. Ploecker, J.M. Prausnitz, and H. Knapp, Int. Chem. Eng. 21, 1 (1981). 68. N.F. Carnahan and K.E. Starling, AIChEJ. 18, 1184 (1972). 69. G.A. Mansoori, ACS Symp. Ser. 300, 314 (1986). 70. B.J. Alder, D.A. Young, and M.A. Mark, J. Chem. Phys. 56, 3013 (1972). 71. S.S. Chen and A. Kreglewski, Ber. Bunsenges. Phys. Chem. 81, 1049 (1977). 72. T. Boublik, J. Chem. Phys. 63, 4084 (1975). 73. W.G. Chapman, K.E. Gubbins, G. Jackson, and M. Radosz, Ind. Eng. Chem. Res. 29,

1709 (1990). 74. S.H. Huang and M. Radosz, Ind. Eng. Chem. Res. 29, 2284 (1990).

75. M.S. Wertheim, J. Stat. Phys. 35, 19 (1984). 76. M.S. Wertheim, J. Stat. Phys. 35, 35 (1984). 77. M.S. Wertheim, J. Star. Phys. 42, 459 (1986). 78. M.S. Wertheim, J. Stat. Phys. 42, 477 (1986). 79. M.S. Wertheim, J. Chem. Phys. 85, 2929 (1986). 80. W.G. Chapman, J. Phys. Chem. 93, 4299 (1990).

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Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 359

10 MIXTURES OF DISSIMILAR MOLECULES

Enrico Matteoli*, Esam Z. Hamad** and G. Ali Mansoori

Department of Chemical Engineering University of Illinois 810 S. Clinton Street, Chicago, IL 60607, U.S.A.

10.1 Introduction 10.2 Background 10.3 Grand Canonical Ensemble 10.4 Analytic Theory of Dissimilar Mixtures

10.4.1 Analytic Solutions for Low to Moderate Pressures 10.5 Binary Mixtures 10.6 Test of C, 7 Closure

10.6.1 Hard-Sphere Mixtures 10.6.2 Lennard-Jones Mixtures 10.6.3 Real Mixtures

10.7 Vapor-Liquid Equilibria 10.8 Liquid-Liquid Equilibria 10.9 Summary References

* Permanent Address: Instituto di Chimica Quantistica ed Energetica Molecolare, C.N.R., Via Risorgimento 35, 56100 Pisa, Italy ** Permanent Address: Chemical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

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10.1 INTRODUCTION

The behavior of mixtures of dissimilar molecules is modeled using a statistical- mechanical solution theory based on the grand canonical ensemble. The strength of this solution theory is its applicability to dissimilar mixtures consisting of species for which, generally, intermolecular potential-energy functions are not accurately known and conventional statistical-mechanical theories of mixtures have failed to predict their behavior. Other models of mixtures of dissimilar molecules are dealt with in other chapters of this book.

The solution theory reported in this chapter is based upon the exact relations between the direct correlation-function integrals of mixtures. These relations indicate that, for example, for a binary mixture, only one of the three direct correlation-function integrals is independent. These relations are combined with an appropriate closure expression for the cross-direct correlation-function integrals to derive analytic expressions for partial and total molar properties, activity coefficients, and equation of state of dissimilar mixtures. The results reported here indicate the possibility of introduction of simple closure expressions relating unlike and like-interaction direct correlation integrals. This approach is shown to be applicable for vapor-liquid and liquid-liquid equilibria calculations of dissimilar mixtures.

One major advantage of this solution theory is the fact that it makes it possible to produce analytic expressions for total and partial molar properties of dissimilar mixtures at finite concentrations. Also, because of independence of this solution theory from the nature of intermolecular-interaction potentials in solutions, it is applicable to mixtures consisting of complex molecules with or without intermolecular-polar and association forces.

10.2 BACKGROUND

Accuracy in prediction of properties of mixtures is one of the major concems in scientific research and engineering calculations (1-3). The traditional molecular theories of mixtures usually start with the consideration of intermolecular potential-energy functions and their applications in a certain statistical-mechanical ensemble (4-8). In the case when the intermolecular potential-energy functions of species of the mixture are not well defined, and this is almost always the case for real fluids, we have to resort to other means of prediction.

One of the interesting problems in statistical mechanics is the prediction of mixture properties from those of the pure components comprising the mixture. There exists a number of techniques to do this. The conformal-solution theory (9-12) has offered a practical and rather accurate theoretical approach to this problem when the intermolecular potential-energy functions of species of a mixture conform to a certain mathematical form. For mixtures of dissimilar molecules for which the intermolecular potential-energy functions are not conformal, the conformal-solution theory may be quite successful. Another statistical- mechanical route to predict behavior of mixtures is the statistical-mechanical solution theory presented here.

The origin of the present statistical-mechanical theory of mixtures goes back to the equation of state derived from the grand canonical ensemble (13-18). The theory introduced here provides us with a strong tool for developing analytic expressions for total and partial thermodynamic properties of multi-component mixtures at finite concentrations as well as solutions at infinite dilution (18-21). It has also been applied to phase equilibria (vapor-liquid and liquid-liquid equilibria) of mixtures with success. In this chapter the details of this theory will be presented and it will be applied to property prediction of a number of dissimilar mixtures of practical interest.

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10.3 GRAND CANONICAL ENSEMBLE

In the grand canonical ensemble, where the temperature T, the volume V, and the chemical potential ~t, of each component in the system are fixed, the partition function, E is (22-24):

E (T, V,t) = Z ... Z Q(T, V,N) e g u / k r (10.1) N1 Nc

where c indicates the number of components, Q is the canonical partition function, ~t=~tl,g2,...,~tc, Ni is the number of molecules of component i, N = N1, Nz, ..., Nc,

c and the scalar product ~t . N = Z ~tkNk .

k=l

The radial-pair distribution function, go, in the grand canonical ensemble is defined as

g i j ( r )={ lV2 / N i N j l Z Z ... Z zNm!)eg 'N/kr l . . . Ie -e /krdr3 . . .drN}/=-T,V,~t ) Ni>i N j> j m=l

(10.2)

where ~ is the interaction potential, r is the vector of coordinates, k is Boltzmann's constant and rc denotes the product over all components.

A relation between the isothermal compressibility, Kr = -(1/V)(OV/aP)r, and g(r) can be derived from the grand-canonical ensemble (22,24)

c c p k r • v = [ B I / p Z Z x ix j IBI0 (10.3)

i = l j = l

where P is the number density and IBI,j denotes the cofactor of the element B o in the c x c matrix B and [B[ is the determinant of B. The elements, Bo., of the matrix are:

Bij = Pxi[6ij + pxjGij ] ( 1 0 . 4 )

where 6, 7 is the Kronecker delta, and G o. is called the radial-distribution-function integral, or fluctuation integral, or the Kirkwood-Buff integral. It is defined by the following expression, where r indicates the distance between molecules i and j:

~3

G 0" = I [gij(r) - 1] 4~r2dr (10.5) 0

In addition to the mixture isothermal compressibility, the Kirkwood-Buff solution theory gives the partial molar volumes and the derivatives of the chemical potential with respect to mole fractions in terms of the fluctuation integrals (24-26):

kT[O < N i >/O~tj]V,T,~ti¢ j = < N i >< N j > Gij./V + 6ij < N i > (10.6)

362

where <AT,.> is the average number of particles of type i in the grand canonical ensemble. Provided information is available about the G O • integrals, Equation (10.6) can be used to

calculate chemical potentials of components of a mixture. Also the information about G o. can provide us with the means of calculating other properties of a mixture such as the isothermal

compressibility ~:r, as given by Equation (10.3), and the partial molar volumes V i as given by

the following expression:

- - C C C

PVi = E x j IBI/j / E E XjXk IBljk (10.7) j=l j=lk=l

where B, [BI/j and [Blik have been defined previously. The above equations have been studied extensively for the case of infinitely dilute

solutions (27-31). The major difficulty in utilizing the fluctuation theory in mixture-property calculations at finite concentrations is the lack of knowledge about the fluctuation integrals, Go.'s. For solutions with finite compositions, expansions of the G/fs in powers of concentration are available (25,26). However, the coefficients in these expansions are given in terms of third, fourth and higher order correlation functions. Since little is known about correlation functions of order higher than two, the expansions are currently of limited use.

Theoretical calculation of G o • integrals requires knowledge about the radial-distribution functions, go. The radial-distribution functions can generally be calculated using the theory of intermolecular potential-energy functions in the context of a partition function. However, for complex mixtures, such as mixtures consisting of asymmetric, highly polar, and associating molecules the intermolecular potential-energy functions are not very well known. Also the existing techniques of calculating radial-distribution functions from the knowledge of intermolecular potential-energy functions require extensive numerical calculations (4-6,23).

Pearson and Rushbrooke (32) reformulated the fluctuation theory expressions, Equations (10.3), (10.6) and (10.7), in terms of direct correlation-function integrals, Co., as defined by

C O. = ~ co(r )dr . (10.8) v

In this equation c0.(r ) is the direct correlation function, which is related to the radial- distribution function by the Ornstein-Zernike relation

C

ctj(rl2) = h/j(rl2) - E Pk~ Cik(r13)hjk(r23)dr3; hij(r) = go(r) - 1; Pk = PXk k=l

(10.9)

where h O. is the total correlation function and xk is the mole fraction of component k. In terms of the direct correlation-ftmction integrals, the isothermal compressibility, the partial molar volume, and the chemical potential become (32,33)

C C

( p k T r . T ) -1 = 1 - p Y'. ~ xixjCi j (10.10) i=lj=l

363

C C C

oV~ = ( 1 - 9 Z x jCo) / (1 -P Z Z XjXk Cjk) j=l j=lk=l

(10.11)

(kT)-l x i (1 - xj )(c3btic3xj ) = Sijxi { l + pC/jp ~] xk ( Cik + Cjk ) k=l

C C C

+102 ~ 2 X k X m ( C i k C j m - C i j C k m ) } / { 1 - P 2 2 XkXm Ckm} k=lm=l k=lm-1

(10.12)

According to Equations (10.10)-(10.12), the fluctuation theory expressions are reformulated with respect to another molecular function, c u rather than go. It should be pointed out that the following relations hold between the fluctuation integrals, G,7's, and the direct correlation- function integrals, Co.'s, (32,33)

G = C + GXC (10.13)

where G is a c x c matrix with elements pC u and X is a diagonal matrix with elements X O. = < Ni/N>.

In solving the above equations for isothermal compressibility, partial molar volumes and chemical potentials in a mixture, knowledge about the Co.'s and their interrelationships is necessary. In the theory described below, a method is introduced by which one can derive analytic expressions for the C•'s.

10.4 ANALYTIC THEORY OF DISSIMILAR MIXTURES

To derive the relations among the C, 7 integrals, we use the mathematical fact that the mixed second partial derivatives of a function with respect to two variables are equal at all points where the derivatives are continuous. Let us consider the mole fractions x;, total pressure P, and the absolute temperature T as the independent variables in a mixture in one phase. Then, for the chemical potential, ~i = ~ti(T,P, xl,x2, .. , Xc-1) , of component i in a mixture one has (19):

O[O~i/Oxj]/OP = O[Ogi/OP]Ox j = [OVi/Oxj]p, T i,j = 1,.., c (10.14)

O[ Oktilc3x j ]lOxk = c3[ OpilOx k ]c3xj i,j,k= 1 .... c and j ~: k (10.15)

In the above equations, the partial molar volume, V i and agi/Oxj are related to the direct

correlation-function integrals by Equations (10.11) and (10.12). By inserting Equations (10.11) and (10.12) into Equations (10.14) and (10.15), we shall derive a number of expressions between the Co.'s of the mixture. In a c-component mixture there exist c(c+1)/2 direct correlation-function integrals which are considered as unknowns in Equations (10.14) and (10.15). One should note that there are c(c- 1) relations of the type of Equation (10.14) and c(c- 1)(c-2)/2 relations of the type of Equation (10.15). Of all these equations only [c(c+1)/2]-1 of them are independent. Since there are c(c+1)/2 unknowns, one additional relation will be needed to determine all the unknowns regardless of the number of components involved. The

364

resulting equations for the direct correlation-function integrals consist of a set of non-linear partial differential equations. The general analytic solution of the above set of equations is not presently available. In what follows we report the analytic solution of these equations for mixtures at low to moderate pressures.

10.4.1 Analytic Solutions for Low to Moderate Pressures

For liquids at low to moderate pressures, it may be assumed that the activity coefficient is not dependent on pressure. In mathematical form, indicating with ~tio the standard chemical potential and using the definition of activity coefficient, f , of component i in a mixture,

ln(xij~) = (~t i - ~tio ) /kT (10.16)

this means

[ Olnf t./OP ]xi, T = [ a ( ~ t i - ~tio )/Oe ]xi , T / kT = (V i - v i ) / k T = 0

If V i = vi, Equation (10.11) can be written in the following form:

C C C

p ~., xjCij = 1 -v iP (1 -p Z Z XjXk Cjk) j=l j=lk=l

i = l , . . . , c (10.17)

Given that

9(1 - 9 Z Z x i x j C i j ) = (kTr.T) -1 (10.18)

Equation (10.17) then becomes

C

Z xjpCij = I - v i ( k T K T ) -1 j=l

i = l , . . . , c (10.19)

where the mixture isothermal compressibility is now given by

K T = -Zx i (Ov i /O P) /Zx i v i (10.20)

The set of Equations (10.15) does not simplify upon making the assumption of low pressure. To overcome this difficulty and to provide the one additional relation needed, the following closure relation is assumed to be valid for all the cross-direct-correlation-function integrals (C u, i¢:j)(19,20):

C i j = t ~ j i C i i + t ~ i j C j j i,j = 1,...,C and i ~: j (10.21)

365

where cxj; and ct 0. are binary interaction parameters ((Zji~-(Zij" which, at low to moderate pressures, are functions of temperature only. In earlier publications (18,19) we demonstrated that this form of the C U closure expression is quite satisfactory in representing the cross-direct- correlation-function integrals of various non-polar, polar, and hydrogen-bonding binary mixtures. Substituting Equation (10.21) in Equation (10.19) gives the following set of linear equations:

C

( Z x jo t j l )PCl l + x2c~12PC22 + ... + XcOt, lcPCcc = 1 - Vl/(kTK T) j=l

C

Xl~21oC11 + ( Z xjotj2)PC22 +... + XcOt2cPCcc = 1 - v2 / (kTKT) j=l

C

Xl°tClPC11 + X2@c2PC22 +"" + ( Z x ja jc )OCcc = 1 - Vc/(kTKT) j=l

(10.22)

where ff, ii=l. The above relations represent a set of c linear questions for the unknowns 9Cii. To solve for pCii, it is more convenient to use matrix notation:

A. C = b (10.23)

where A is a matrix with elements Aii = ~_.,xjffji, A• = xj%. and b and C are vectors with elements b~=l-vi/kTKr and C~=gCii, respectively. The solution of Equation (10.23) is:

¢

PQck = Z [A lik bi/lA[ (10.24) i=1

where [AIik is the cofactor of the element Aik in the determinant [A[.

By using the direct correlation-function integrals calculated above one can derive the expression for the activity coefficient of a species in a multi-component mixture in the following form

N(O lnfk/ONk ) = 1- gCkk + vk2lkTZ xi(Ovi/OP)T ] -1 (10.25)

As an application, this theory, which we repeat is valid only at low to moderate pressures when Equation (10.21) is valid, will be used for derivation of activity coefficients and vapor- liquid and liquid-liquid equilibria calculations of binary mixtures.

366

10.5 BINARY MIXTURES

For a two-component mixture, Equations (10.14) and (10.15) reduce to

(02}.tl [ aXlaP) T = (OVi----/C3Xl)T,p (10.26)

(02kt2 [ OXlOP)r = (OV2---/aXl )T,P (10.27)

Then; by substituting Equations (10.11) and (10.12) in Equations (10.26) and (10.27), we get the following two relations between Cll, C22, and C12:

2 kTO([1- xlPC11 - x2PC22 + XlX292(CllC22 -C12 )]/[1- ~Zxix jpCi j]} /OP

=XlO{[ 1 - xlpC 11 - x2pC12 ]/[t9(1 - Z Z xixjpCi j)] }/Or 1 (10.28)

and

kTO{[1- xlPC11 - x2PC22 + XlX202(CllC22 - C122)]/[1 - ~,XxixjpCij l}/OP =x20{ [ 1 - xzPC22 - XlP C12 ]/[p( 1 - Z ~ xix jpC O" )] }/Ox 2

(10.29)

These two equations constitute two independent expressions relating Cll, C22 and C12 in a binary mixture. In principle, Equations (10.28) and (10.29), together with Equation (10.21) relating C~2 to Cll and C22, can be used to solve for the three quantities C~ 1, C22 and C12 with the following results:

pC 1 ! = W(1 - o,12v2/v 1)x22 + Wx 1 (Ctl2X 1 + x 2)vl/v 2 + (1 - ~12)x2 + o;12x 1

ct21x22 + XlX 2 + Ctl2Xl 2 (10.30)

pC22 = W(1-ct21Vl/V2)Xl2 + Wx2(ct21x2 + Xl)V2/Vl + (1 - ~21)Xl + ~21x2 (10.31) 2 2

ct21x 2 + XlX 2 + ~12Xl

pC12 _ :W(ctl2xl2 + ct21XlX2Vl/V 2 + ~12XlX2V2/Vl + ~21x22) + ~12Xl + ct21x 2

ct21x22 + XlX 2 + ~12Xl 2 (10.32)

where

W = - [kT(xlK ~/v 2 + X2KT2/v 1 ]-1 (10.33)

and where KT~ and KT2 are the isothermal compressibilities of the pure components 1 and 2,

respectively. With the availability of pure-fluid thermodynamic data and binary interaction parameters ix21 and ~12, Equations (10.30) - (10.33) can be used for calculation of mixture direct-correlation-function integrals.

367

Given the direct-correlation-function integrals, activity coefficients can be obtained by substituting the Cu's in Equation (10.25), which for binary mixtures can also be written as

lnJ] = -~2 (pC12 - W - 1 ) d x 2 / x 1 (10.34)

Substituting for pC12 from Equation (10.32) and performing the integration, an analytical expression for activity coefficient results

In./] = -Llxln[x 1 + L2/L1)x2]+ (1/2)(Lfr - 1)ln[a/c~12 x2 + ~i 1 - 2)x 2 + 1]

+ (1/2)[(2cq 2L 2 - L 1)1: 4-1 - 2x12 ]/qln[(14- x2/r-)/(1 + x2/r+)] (10.35)

with

L 1 = kTKT1/v 2, L 2 = kTr.T2/v 1, a = tx 21 4- Ctl2 - 1, q = 1 - 4tx 21~12)1/2, r + = ( 1 / 2 ) ( 1 - 2 t x 1 2 + q)/a,

"r, = (~21Vl/V2 + ~lZV2/Vl - 1)/(~21 L2 + ~12L~ - L1L2)

(10.36)

The expression for ln~ can be obtained by interchanging subscripts 1 and 2 in Equation (10.35) and in the expression for r -+. The quantities a, q and "~ remain unchanged under this operation.

To test the accuracy of the closure given by Equation (10.21), we apply it to various binary liquid-mixture calculations, Then the activity-coefficient expression, Equation (10.35), is used for vapor-liquid and liquid-liquid equilibria calculations of a number of binary dissimilar mixtures.

10.6 TEST OF Cls CLOSURE

In order to demonstrate the accuracy of the C• closure, given by Equation (10.21), we apply it first to two model-fluid mixtures (hard'sphere and Lennard-Jones mixtures). Then it will be applied to a variety of real-fluid binary mixtures consisting of dissimilar and complex molecules.

1 0 . 6 . 1 H a r d - S p h e r e M i x t u r e s

The validity of Equation (10.21) can be tested by comparing the fluctuation integrals or direct correlation-function integrals calculated by the present theory with their exact values. For this purpose we first consider a model-fluid mixture (hard-sphere mixture) for which accurate direct correlation-function integrals are already available. The Percus-Yevick theory (34) gives quite accurate expressions for the direct correlation-function integrals of a hard- sphere mixture. In Table 10.1 the C12/Cll values calculated by the Percus-Yevick theory are compared with the result of geometric mean closure, C12=(CllC22) v2, and the arithmetic mean (a12=c~21=1/2) closure, Equation (10.21), for a binary equimolar mixture of hard spheres with ~1~/cr22=1.5. Table 10.1 shows clearly that both geometric- and arithmetic-mean closures are accurate enough for a hard-sphere mixture. However, the geometric-mean closure is more

368

accurate at high densities, while the weighted arithmetic-mean closure is more accurate at low densities.

Table 10.1 Values of C12/Cll for an equimolar hard-sphere mixture (Oll/O22--'1.5) calculated according to different closure equations and compared with the Percus-Yevick theory (19).

(~/6)p(~223 P-Y C12=(CllC22) V2 % D e v . C12--(I,12Cl1-1-(z21C22 % Dev. 0.05 0.5265 0.5047 4.1 0.5243 0.4 0.10 0.4806 0.4677 2.7 0.4772 0.7 0.20 0.4066 0.4034 0.8 0.4039 0.7 0.30 0.3530 0.3526 0.1 0.3537 -0.2 0.40 0.3141 0.3142 0.02 0.3202 -1.9 0.50 0.2844 0.2844 0.0 0.2969 -4.4

10.6.2 Lennard-Jones Mixtures

The hard-sphere intermolecular potential-energy function is purely repulsive, to test the proposed closure models for mixtures of molecules with intermolecular potentials which possess both attractive and repulsive parts, the Lennard-Jones-potential model is considered here. The chemical potential of a mixture of Lennard-Jones fluids at infinite dilution is predicted here and it is compared with available computer-simulation results. The chemical- potential predictions are compared with the infinite-dilution simulation data (35) in Table 10.2 According to this table for molecules with the same intermolecular energy parameters ~12/~;22--1, the weighted arithmetic-mean closure shows better agreement with the simulation data at cr12/~22>1. Geometric-mean closure shows better agreement with the simulation data at t~12/t~22<l.

Table 10.2 The chemical potential at infinite dilution of Lennard-Jones mixtures (19) for kT/e.,22=l.2, OCY223=0.7.

((~12/(~22)3 ~12/E22 ktoolr/k T Simulated C12=(CllC22) l& C12--c£12Cl1-]-(z21C22 Data (35)

0.3 1.0 -1.30 -1.75 +4.14 0.5 1.0 -1.63 -1.80 -1.29

0.75 1.0 -1.83 -1.86 -1.92 1.0 1.0 -1.93 -1.99 -1.99 1.5 1.0 -2.00 -2.26 -1.65 2.0 1.0 -2.48 -2.58 -1.52

10.6.3 Real Mixtures

Data on fluctuation integrals and direct correlation-function-integrals are available for a variety of binary mixtures (20,37). Such data can be used for the test of closure expressions. The experimental C• integrals for real mixtures are calculated from experimental G/2 integrals (5,22) by inverting Equation (10.13). In Table 10.3 the results of this fit are reported for 22 different binary mixtures; for eight of these mixtures, the calculated C12 values are compared with the experimental data in Figures 10.1 - 10.4.

The validity of the C12 closure, Equation (10.21), can be tested for real mixtures by plotting C12/Cll data versus C22/Cll data and considering the linearity of the resulting curve.

369

P C i j

-20

- 80

- 1 4 0

- 2 0 0

~ ~ ' . . . . . . • , . . . ~ ...'._ • . ' . . ' . . . ~ . . ' . . .

0 Q Q ° 0 0 Q Q ip Q

ID Q

la •

Q • o °

0 0

0 ®

o

V÷PROPRNOL •

® o

t 1 l I 1 I I I 1

. I .Z .3 .4 .5 .6 .2 .0 .9 1

pCij

10

-40

-90

- 1 4 0

- 1 9 0

_

® ®

o 0

-- IP -

0

0

o

o

¥+RCETONE

! 1 I 1 I I 1 i I

O

0 .1 . 2 . 3 . $ . 5 . 6 . 7 .8 . 9 )

Figure 10.1 The variation of 9C O. with composition for water(W) (1) + propanol (2) and for water (1) + acetone (2) at 25°C. The points represent experimental data (18,20) for pC12 (triangle), pCll (square), and pC22 (circle). The solid curve represent the PC12 prediction by the present theory using weighted- arithmetic-mean C12 closure.

Figures 10.5 and 10.6 show the plot of C12/Cll v e r s u s C22/Cll for four different binary mixtures. The linearity of these plots suggests that the weighted arithmetic-mean closure, Equation (10.21), is quite sufficient for real mixtures. As a further test of the validity of the weighted arithmetic-mean closure for other real mixtures the deviations of C12 data from the C12 value calculated using Equation (10.21) are reported in Table 10.3. The maximum deviation for all mixtures in Table 10.3 is about 22%. This large error happens where the C U goes to zero and crosses the horizontal axes. Since some of the experimental data have uncertainties up to 30-40% (37), one concludes that the weighted arithmetic-mean closure is a good closure approximation for real mixtures.

370 0 o o o l w o m o o o o e

pCl i -10o ® ®

-200 l ® ® ° e m e e e

V + D H S O • • ® • ® ® ® ® ® • ®

0 .1 .Z .5 .4 .5 ,6 ,7 .S .9

Xl

I _ooo®e e -110 I . . . . . . . . . . e~mm

| S O ~

0 .1 .Z .3 .4, .$ .6 .7 .6 .9

X]

Figure 10.2 The variation of pC# with composition for water (W) (1) + dimethylsulfoxide (DMSO) (2) and for water (1) + aminoethanol (2) at 25°C. The points represent experimental data (18,20) for pC]2 (triangle), pC]] (square), and PC22 (circle). The solid curve represents the pC]2 prediction by the present theory using weighted-arithmetic-mean C]2 closure.

| ..... , i . . . . . ~, 1 | i • ~ -" ®o m

-Z6 TCN+TItF @ o D o ® O O ® o ° ° e ° ° ~

o ® o o ° ° P C i j - 3 0

-54

- 3 1 t • •

-4.2 I i I I I t I I

0 .1 .Z .3 .4 .S .6 .Y .O .9

Xl

., ~ " m 2 ® e , _ , , , r, ,

°e@°°°eee@eee@®e e -36 - TCH+DI(]XRNE

e ® e pCi j -4,o i

-44, w m • •

-48 • • • • • • - II 8 i

-SZ • 8 • . . ! ! | f I l ... f f . f

0 .I .Z .3 .¢ .5" .6 .7 .0 .9

XI

Figure 10.3 The variation of pC U with composition for tctrachloromcthanc (TCM) (1) + tetrahydrofuran (THF) (2) and for TCM (1) + dioxane (2) at 25°C. The points represent experimental data (18,20) for pC~2 (triangle), pC]] (square), and pC22 (circle). The solid curve represents the pC12 prediction by the present theory using weighted-arithmetic-mean CI~_ closure.

P C t I

-26-'/2 • e 1 e ~ | e ® ee e e l ! ® el e|® 1 | |

-30

-34

- 3 1 1

I I I I I I I I I •

0 .1 .Z .3 .4 . 5 .6 . 7 .8 .9

371

-3O I I m I I i i I

-40 ||mm'mm | m| mm mm-

p C i j

-50

-60 e • e

0 .l . 2 .3 .4 .5 .6 .7 .a .9

Figure 10.4 The variation of oct/with composition for cyclohexane (CY-HEX) (1) + ethylether (2) and for CY-HEX (1) + propylether (2) at 25°C. The points represent experimental data (18,20) for pC12 (triangle), pCll (square), and pC22 (circle). The solid curve represents the pC12 prediction by the present theory using weighted-arithmetic-mean C12 closure.

"I

o -

-0.4

=. ...... i-"i¶" -=

I I 1 i i J . -o .z o.o o.z 0.4 o.8 o.e 1.o

Czz/C~

Figure 10.5 The test of the lmearity of C12/C~1 v e r s u s C22/Cll for water (1) + methanol (2) (circles) and water (1) + amionoethanol (2) (squares) binary mixtures. The curves represent the weighted arithmetic- mean closure, Equation (10.21), compared with the experimental data at different compositions (19,20).

372

• is"

:t / ' J

i I " i i t I I

4.0 8.0 Iz.0 18.o zo.o z4.o zs.o Czz/Ctl

Figure 10.6 The test of the linearity of C12/Cll versus C22/C1~ for tetrachloromethane (1) + ethanol (2) (circles) and tetrachloromethane (1) + n-butanol (2) (squares) binary mixtures. The curves represent the weighted arithmetic-mean closure, Equation (10.21), compared with the experimental data at different compositions (19,20).

1 0 . 7 V A P O R - L I Q U I D E Q U I L I B R I A

To test the possibility of vapor-liquid-equilibrium (VLE) prediction by the activity- coefficient expression, Equation (10.35), a number of complex systems are chosen at low pressures. The vapor phase is assumed to be an ideal gas and the vapor pressure of the pure component, Pi ° is represented by the Antoine equation. Two kinds of pure-liquid property data are needed in the activity-coefficient equation: they are the specific volume and the isothermal-compressibility values. The VLE data are fitted to the activity-coefficient expression, by minimizing the following objective function

S = 2 ( P - Pexp) 2/Sp + 2 ( y - Yexp) 2/Sy (10.37)

where P is the calculated total vapor pressure, y is the calculated vapor molar fraction of one component, and Sp and Sy are the values of the sums of squares of P and y deviations, respectively, that result when Y~(P-Pexp) 2 and Z(y-Yexp) 2 are minimized individually. Equation (10.37) allows optimization of Equation (10.35) simultaneously with respect to both experimental vapor-pressure, Pexp, and vapor-phase mole fraction, Yexp, data. The results of the calculations are shown in Table 10.4.

373

Table 10.3 Deviations of real-fluid direct correlation integrals from the relation C12--o(,12Cl1+Ot,21C22 . Mixture Temperature S Max Dev xl at

[C °] xl00 [%] Max Dev. water+methanol 0 2.1 7.7 0.0

. . . . 25 1.9 7.5 0.0

. . . . 60 2.2 -2.6 0.4 water+ethanol 25 14.5 22 0.0

. . . . 50 3.2 4.1 0.1

. . . . 90 4.1 6.7 0.1 water+propan-l-ol 25 2.7 7.4 0.0 water+tert-butanol 25 1.5 -4.6 1.0 water+acetonitrile 30 2.3 -11 1.0 water+acetone 25 1.9 -4.4 1.0 water+dimethyl sul foxide 25 5.9 -8.7 0.0 water+tetrahydrofuran 25 4.3 -8.8 1.0 water+ 1,4-dioxane 25 4.0 -5.9 1.0 water+2-aminoe than o 1 25 6.9 -6.2 1.0 CC14-methanol 25 0.32 -2.4 0.0 CC14+ethanol 25 0.28 -2.1 0.0 CC14+propanol 25 0.12 - 1.3 0.0 CC14+n-butanol 25 0.12 - 1.2 0.0 CC14+tetrahydrofuran 25 0.0004 .005 0.9 CC14+l,4-dioxane 25 0.00003 0.003 0.0 CY-hexane+diethylether 25 0.014 -0.4 1.0 CY-hexane+dipropylether 25 0.030 -0.6 0.0 CY-hexane+dibutylether 25 0.002 0.2 0.0 CY-hexane+ethylbutylether 25 0.020 -0.5 0.0 CY-hexane+dimethoxymethane 25 0.24 -2.1 1.0 CY-hexane+diethoxymethane 25 0.0005 -0.02 1.0 CY-hexane+diethoxyethane 25 0.19 - 1.8 0.0 CY-hexane+ 25 0.16 1.0 1.0

diethyleneglycol dimethylether

In this table S=E { ( C 1 2 , p r e d - - C12,exp} 2 and %Dev.=100 { C 1 2 , p r e d - - C12 ,exp ) /C12 , exp} .

374

In this table comparisons are also made with the results of the Wilson correlation (36) which is selected because it gave the best over-all fit to the large number of mixtures studied in VLE calculations. Table 10.4 also includes parameters a21 and c~12, and optimized values of Sp and Sy. Figures 10.7 - 10.9 show the vapor-liquid-equilibrium composition (x-y)- diagrams for the systems methanol + water, 1-propanol+water and ethanol+benzene.

Table 10.4 Parameters of the present activity coefficient model, Equation (10.35), and comparisons with the Wilson correlation (18,19).

Equation (10.35) Wilson Mixture T/°C (z21 C~,.12 .... 102Sp 102Sy 10~2Se ..... 102Sy Methanol+Water 60 0.3869 0.3562 2.0 0.10 1.0 0.07 Ethanol+Water 40 0.2900 0.3561 1.4 0.09 1.1 0.06 Ethanol+Water 55 0.2920 0.3799 0.7 0.14 0.4 0.13 Ethanol+Water 70 0.2942 0.4048 1.6 0.11 0.6 0.13 Propan- 1-ol+Water 30 0.2392 0.2722 0.07 0.46 0.09 0.64 Ethyl Acetate+Ethanol 40 0.5105 0.2989 0.03 0.04 0.02 0.04 Ethanol+Benzene 25 0.1028 0.6393 0.009 0.008 0.004 0.01 Ethanol+Benzene 40 0 . 0 9 7 4 1 0.6458 0.2 0.07 0.1 0.05 Ethanol+Benzene 55 0.09688 0.6498 0.9 0.13 0.5 0.13 Acetone+Chloroform 25 0.2907 0.6404 0.04 0.24 0.04 0.24

According to Table 10.4 and Figures 10.7 - 10.9, it can be concluded that the present theory can predict the vapor-liquid equilibria of the systems studied as well as the Wilson equation (36).

o

t,,-

o .

t o

,q,

o

0.0 0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 .0

X I

Figure 10.7 The vapor-liquid cquilibrium composition (x-y) diagram for methanol (1) + water (2) at 60°C. The squares are the experimental data, the solid curve is the present theory and the dashed curve is the Wilson correlation; diagonal is Raoult's-law line.

w

C~

c::;

c6

tt~

(:5

cu

O I I I , , I I l I l I ....

0 . 0 0.1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1.0

375

Figure 10.8 The vapor-liquid equilibrium composition (x-y) diagram for 1-propanol (1) + water (2) at 30°C. The squares are the experimental data, the solid curve is the present theory and the dashed curve is the Wilson correlation; diagonal is Raoult's-law line.

O.

C~

D-

c~

C~

c~ c~

c~ 0

c~

0 6 .i ..... i I I , I I I I i

0 . 0 011 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 .7

X I

0.8 0.9 1.0

Figure 10.9 The vapor-liquid equilibrium composition (x-y) diagram for ethanol (1) + benzene (2) at 55°C. The squares are the experimental data, the solid curve is the present theory and the dashed curve is the Wilson correlation; diagonal is Raoult's-law line.

376

10.8 LIQUID-LIQUID EQUILIBRIA

Phase splitting in liquid mixtures may occur when two phases have a lower total Gibbs energy than one phase at constant temperature and pressure. A necessary condition for stability of a two-component homogeneous mixture is (2,3,38,39):

dkti/dxi > 0 i = 1,2 (10.38)

In terms of the activity coefficients, Equation (10.38) becomes:

d(lnfi)/dx i + 1/x i > 0 (10.39)

Substituting for Inf. from Equation (10.35) gives

(~12Xl + (1,21x2)(LlX 1 + L2x2) - XlX2(Cl,21Vl/V2 + ~12v2/v 1 - 1) > 0 (10.40)

provided parameters 0~21 and (El2 are positive. For liquid-liquid equilibria (LLE) the left-hand side of the inequality given by Equation (10.40) should be negative. Considering the magnitudes of the different quantities one concludes that the present activity-coefficient model is capable of predicting LLE. The Wilson correlation (36), on the other hand, is known to be unable to predict LLE, because it always satisfies inequality (10.40).

The present activity-coefficient theory is tested versus experimental LLE data. For the variation of the isothermal compressibilities with temperature, which is needed in this calculation, a three-parameter equation is used (40). Temperature and pure-component volume dependence of parameters ~21 and ~12 for a number of organic binary systems exhibiting LLE are represented by the following expressions (3,4)

ct21 = (v2/vl)exp(t~o21/RT) (10.41)

Ctl2 = ~o12(Vl/V2) (10.42)

where ao21 and ao12 are constants (independent of temperature and composition). Their functional form was suggested by examining the behavior of the tx21 and tXl2 values obtained for all mixtures shown in Table 10.3.

The experimental LLE data are fitted to the activity-coefficient equation with Equations (10.41) and (10.42) for ~21 and Ctl2, and by taking as objective function, S, the deviation from unity of the ratio of the activities in the two liquid phases:

S = Z{1-Xl(1)fl(1)/[Xl(2)fl(2)]} 2 + 2{ 1-x2(1)f2(1)/[x2(2)f2(2)]}2 (10.43)

where superscripts (1) and (2) are for phases 1 and 2, respectively. Table 10.5 shows values of parameters tXo21 and Oto12 and optimum S for a number of binary systems. The variation with the compositions of two LLE systems (methanol+heptane and formic acid+benzene) with temperature is shown in Figures 10.10 and 10.11. According to these figures the agreement with the data is satisfactory. The largest deviation of theory from experimental data occurs near the upper critical solution temperature.

o. o

o. g

Overall, the present theory is capable of formulating analytic expressions for activity coefficients in mixtures, for this purpose it is necessary to define closure expressions for the cross-direct-correlation-function integrals. Application of the activity-coefficient expressions resulting from this theory for vapor-liquid and liquid-liquid equilibria calculations has been as successful as the other activity-coefficient expressions available. To make this theory predictive, relations which correlate the ot o. values to temperature, pressure and molecular structure and properties are to be developed; work aiming to this goal is in progress.

Table 10.5 LLE parameters of the present activity-coefficient model for different binary mixtures (18,19). Mixture Oto21 ~o12 103S Methanol+Heptane - 1070.9 1.0243 0.12 Methanol+Cyclohexane -980.29 1.0191 0.21 Formic Acid+Benzene -748.15 0.97983 1.7 Phenol+Octane -1077.0 1.0031 7.3 Methanol+Carbondisulfide -797.20 1.0186 0.45 Nitrobenzene+Hexane -975.97 1.0027 2.0 1,3-Dihydroxybenzene+Benzene -937.02 1.0032 5.5

In this table, S is defined by Equation (10.43).

).0 0.9 1.0

[

I I I I I ,

0.I O.Z, 0.3 0.4 0.5 0.6 0.7 0.8

x !

377

Figure 10.10 The liquid composition x versus temperature (T-x) diagram for liquid-liquid equilibri-um of methanol (1) + heptane (2). The squares are the experimental data and the solid curve is the present theory.

378

0

i

0.0

' ° o ~o

I ol 0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8

X I

Figure I0.11 The liquid composition versus temperature (T-x) diagram for liquid-liquid equilibrium of formic acid (1) + benzene (2). The squares are the experimental data and the solid curve is the present theory.

10.9 SUMMARY

Equations (10.14) and (10.15), which are the basis of the present theory, are mathematically exact. The choice of the closure expression, Equation (10.21), relating the cross-direct-correlation integral to the other two integrals in a binary mixture allows us to solve these equations simply and analytically using only the pure-fluid thermodynamic data. It is demonstrated here that simple geometric- and arithmetic-mean closures are needed for simple model fluids. It is also demonstrated that a weighted-arithmetic-mean closure expression is sufficient to represent the relation between the cross-direct-correlation integral to the other two correlation integrals of binary mixtures. Success of this theory in its application to a variety of mixtures comprised of components with polar and associating intermolecular potential energies is indicative of its promise as a strong technique for prediction of properties of complex dissimilar mixtures of practical interest.

The present theory has allowed us to perform calculations of properties and phase equilibria of mixtures. There are several points to be noted about the advantages of the present techniques compared with other existing thermodynamic calculation methods for mixtures: (i) The present theory allows us to perform thermodynamic calculations for the whole range of mixture compositions and not just at the infinite-dilution and high-concentration limits as has been the case for most of the fluctuation-theory techniques. (ii) The present theory is applicable for mixtures consisting of dissimilar species with large differences in molecular size, shape, and energetics. It is specifically useful for polar and associating molecular fluids for which, generally, no accurate intermolecular-potential-energy functions are available. (iii)

379

With the application of the weighted-arithmetic-mean closure for the cross-direct-correlation- function integrals, Equation (10.21), it has become possible to derive analytic expressions for activity coefficients in complex mixtures consisting the dissimilar molecules.

The resulting activity-coefficient expressions allow us to perform vapor-liquid and liquid- liquid equilibria computations for such mixtures. The accuracy of these calculations are is good as the best available phase-equilibria computational techniques for mixtures. (iv) What makes the relations among the direct-correlation-function integrals (or fluctuation integrals) introduced here particularly interesting is the fact that only one closure relation is needed to determine all the binary-mixture properties. Detailed studies are under way in order to develop analytic expressions for total and partial molar properties of multicomponent systems using the present theory.

Acknowledgement

This research is supported by U.S. National Science Foundation Grant CTR-9108595.

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Equilib. 26, 1 (1986); R.G. Rubio, M. Diaz Pena, and D. Patterson, Advances in Thermodynamics 2, 241 (1990).

18. E. Matteoli and G.A. Mansoori, eds., Advances in Thermodynamics 2, Taylor & Francis, New York (1990).

19. E.Z. Hamad and G.A. Mansoori, Fluid Phase Equilib. 54, 1 (1989); Z. Phys. Chem. N.F. 166, 63 (1990); J. Phys. Chem. 94, 3148 (1990); Advances in Thermodynamics 2, 95 (1990); Fluid Phase Equilib. 85, 141 (1993).

20. L. Lepori, E. Matteoli, E.Z. Hamad, and G.A. Mansoori, Advances in Thermodynamics 2, 95 (1990); Z. Phys. Chem. N.F. 162, 27 (1989).

21. E.Z. Hamad, U.A. E1-Nafaty, and G.A. Mansoori, Fluid Phase Equilib. 79, 21 (1992).

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22. T.L. Hill, Statistical Mechanics, McGraw-Hill, New York (1956). 23. D.A. McQuarrie, Statistical Mechanics, Harper and Row, New York (1975). 24. J.G. Kirkwood and F.P. Buff, J. Chem. Phys. 19, 774 (1951). 25. F.P. Buff and R. Brout, J. Chem. Phys. 23, 458 (1955). 26. F.P. Buff and R.M. Schindler, J. Chem. Phys. 29, 1075 (1958). 27. A. Ben-Naim, Water and Aqueous Solutions, Plenum, New York (1974); Hydrophobic

Interactions, Plenum, New York (1980); Advances in Thermodynamics 2, 211 (1990). 28. H.D. Cochran, L.L. Lee, and D.M. Pfund, Fluid Phase Equilib. 34, 219 (1987); Advances in

Thermodynamics 2, 69 (1990). 29. P.G. Debenedetti and S.K. Kumar, AIChEJ. 34, 1211 (1988). 30. A.A. Chialvo, Ph.D.Dissertation, Clemson University (1988); Advances in Thermodynamics 2,

131 (1990). 31. Y.J. Kwon and G.A. Mansoori, Jr. Supercritical Fluids 6, 173 (1993). 32. F.J. Pearson and G.S. Rushbrooke, Proc. Soc. Edinburgh, Sec. A. A64, 305 (1957). 33. J.P. O'Connell, Molecular Physics 20, 27 (1971); Fluid Phase Equilib. 6, 21 (1981); Advances in

Thermodynamics 2, 45 (1990). 34. J.K. Percus and G.J. Yevick, Phys. Rev. 110, 1 (1957). 35. K.S. Shing, Ph. D. Thesis, Comell University (1982); K.S. Shing, K.E. Gubbins, and K. Lucas,

Mol. Phys. 65, 1235 (1988). 36. G.M. Wilson, J. Am. Chem. Soc. 86, 127 (1964). 37. E. Matteoli and L. Lepori, J. Chem. Phys. 80, 2856 (1984); J. Phys. Chem. 92, 6997 (1988). 38. H.C. Van Ness and M.M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions,

McGraw-Hill, New York (1982). 39. I.F. Radzyminski and W.B. Whiting, Fluid Phase Equilib. 34, 101 (1987). 40. W. Brostow and P. Maynadier, High Temp. Sci. 11, 7 (1979).

Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All rights reserved 381

11 C R I T I C A L R E G I O N

M. A. Anisimov a and J. V. Sengers a'b

a Institute for Physical Science and Technology and Department of Chemical Engineering University of Maryland, College Park, MD 20742, U.S.A. bphysical and Chemical Properties Division National Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A.

11.1 Introduction 11.2 Critical Region in the Van der Waals Theory 11.3 Asymptotic Scaled Equation of State 11.4 Corrections to the Asymptotic Scaled Equation of State 11.5 Crossover between Scaling and Classical Asymptotic Critical Behavior 11.6 Crossover between Scaling and Classical Nonasymptotic Critical Behavior in the Extended

Critical Region 11.7 Global Crossover Equation of State for One-Component Fluids 11.8 Isomorphic Scaled Equation of State for Near-Critical Binary Fluids 11.9 Renormalization of Critical Exponents 11.10 Isomorphic Description of Fluid-Fluid Phase Equilibria 11.11 Crossover between Vapor-Liquid and Liquid-Liquid Critical Phenomena 11.12 Summary and Outlook References

382

11.1 INTRODUCTION

The purpose of this chapter is to review the current status of equations of state for fluids and fluid mixtures in the critical region. Ionic and chemically reacting fluids near critical points are specifically considered in another chapter of this volume (1). Critical phenomena in fluids have been the subject of many theoretical and experimental studies during the past thirty years. The most striking result of these studies has been the discovery of critical-point universality: the microscopic structure of fluids becomes unimportant in the vicinity of a critical point (2-4). Moreover, the principle of critical-point universality has been extended to near-critical binary mixtures (isomorphism approach) (5-9). These discoveries make it possible to develop universal equations of state for fluids in the critical region and enable us to look into the problem of formulating global equations of state for dense fluids and fluid mixtures from an entirely new point of view.

There is a well known statement of Landau and Lifshitz which reads "Unlike solids and gases, liquids do not allow a general calculation of their thermodynamic quantities or even their temperature dependence. The reason for this is the presence of strong interactions between the molecules of the liquid without having at the same time the smallness of the vibrations which makes the thermal motions of solids so simple" (10). For many years the ideal-gas model was the only universal basis for developing equations of state for dense fluids. The effects of interactions between the molecules in dense gases were then treated as corrections in the framework of the virial expansion. Even the equation of state of Van der Waals, which implies vapor-liquid condensation and the existence of a critical point, has not changed the situation in principle. As also discussed by Landau and Lifshitz, the Van der Waals equation can be inferred from the truncated virial expansion with a specific temperature dependence of the second virial coefficient (10). As a consequence, the Van der Waals equation is an illegal extrapolation of the virial expansion to the dense-fluid region. It is not surprising, therefore, that the Van der Waals equation fails to provide a quantitative description of fluids in the critical region.

As an alternative approach, attempts have been made to describe the thermodynamic properties of dense fluids in terms of a perturbation expansion around a universal equation of state for dense hard spheres (11). The problem is that the hard-sphere limit is not approached in practice at any temperature and density. Various versions of a rigorous approach to the theory of dense fluids have been developed based on integral equations for multiple-correlation functions (12). This approach cannot solve the problem of a description of real fluids even far away from the critical point and even when the intermolecular potential is known, because the equations cannot be closed. The lack of a good theory for dense fluids has led to a large number of empirical equations of state. As a rule, these equations of state avoid the critical region, having failed in a correct description of the critical anomalies consistently. If one succeeds in describing the pressure in the vicinity of the critical point using an equation of state in a polynomial form, for example, calculation of the specific heat capacity from such an equation of state may produce large errors (13). This discouraging situation changed 25-30 years ago with the discovery of the principle of universality of the thermodynamic behavior of condensed matter near critical points and second-order phase transitions. This principle of universality goes so far that diverse systems, such as one- and multicomponent fluids near their critical points, magnetic materials near the Curie point, binary alloys near the order-disorder transition, etc., can all be described by the same type of scaled equation of state (14). Applicability of critical-point universality goes far beyond

383

applicability of corresponding states. The principle of critical-point universality finds its physical origin in the phenomenon that

long-range fluctuations of the order parameter (density in one-component fluids or/and concentration in fluid mixtures) dominate in the critical region and that the range of these fluctuations becomes much larger than any other microscopic scale. The spatial extent of these critical fluctuations is determined by a correlation length ~', which diverges at the critical point. As a consequence, the behavior of the thermodynamic properties becomes singular at the critical point. The mathematical nature of the asymptotic, singular, critical behavior is now well understood: it can be characterized by scaling laws with universal critical exponents.

When a fluid is not exactly at the critical point, the correlation length ~' will be finite. Never- theless, one must consider the possible effects of critical fluctuations at all temperatures and densities where the correlation length is significantly larger than the average intermolecular distance. This is, in fact, a very wide range of temperatures and densities, namely at all temperatures and densities where any critical enhancement of the compressibility is observed (15- 17). We refer to this region as the extended or global critical region (18). It has become possible to extend the principle of critical-point universality to develop equations of state for the entire global critical region. Furthermore, methods which modify a classical equation of state by incorporating the nonclassical critical behavior have also been suggested recently (19-24). These methods use a transformation to convert a classical equation of state, that is valid far away from the critical region, into an equation of state that also correctly describes the near-critical thermodynamic behavior.

This chapter is organized as follows. We first consider the critical behavior of a Van der Waals fluid. Although actual fluids do not satisfy Van der Waals critical behavior, the Van der Waals equation remains conceptually very important as a universal equation of state in the classical theory that neglects density fluctuations. We shall then consider a universal scaled equation of state that is valid in the asymptotic vicinity of the critical point. To extend the scaled equation of state to a larger region around the critical point crossover between two kinds of critical behavior, namely, between asymptotic singular behavior and classical (Van der Waals like) critical behavior, is to be considered. The next issue to be discussed is crossover between such an extended critical equation of state and a classical equation of state that describes noncritical (regular) behavior far away from the critical region. Such a global crossover equation of state bridges the gap between the two universal regimes in the fluid state, namely the universal asymptotic regime near the critical point and the universal ideal-gas limit. Finally, we shall discuss the approach to formulating scaled equations of state for binary fluid mixtures, including those that exhibit a continuous transformation from vapor-liquid to liquid-liquid critical phenomena.

11.2 CRITICAL REGION IN THE VAN DER WAALS THEORY

From a microscopic viewpoint the classical theory of the critical point corresponds to a mean- field approximation which neglects local inhomogeneities (fluctuations) of density (25). In this approximation a complex intermolecular interaction is replaced by an average effective field, having an equal effect on each molecule. A typical mean-field model is the Van der Waals equation of state

384

p _ R T _ a _ R T p _ a P 2 (11 1) v - b V 2 1 - b p

where P is the pressure, T the temperature, v the molar volume, p the molar density, and where a and b are constants associated with the attractive and repulsive intermolecular forces, respectively.

The Van der Waals equation can be represented in the universal dimensionless form:

p , _ 8T'p* _ 3p,2 (11.2) 3-p*

o r

p , _- 8(AT* + 1)(Ap* + 1) _ 3(Ap* + 1)2 (11.3) 2 - @ *

where

T* = AT* + 1 = T/Tc, P * = A P * + 1 = P/P~, p* = Ap* + 1 = p / p c . (11.4)

The critical temperature T c , the critical pressure Pc, and the critical molar density Pc are related to the constants a and b:

T c = 8a /27bR, Pc = a / 2 7 b 2 , Pc = 1/3b, z c = Pc/PcRTc = 3/8 (11.5)

where R is the molar gas constant. In the vicinity of the critical point, that is defined by the conditions

(OP/OR)T = 0, (0 2P/OR 2 )r = 0 (11.6)

the Van der Waals equation can be expanded in an analytical power series (10, 26):

P*= 1 + ~ AT* + a o AT*AR* + (1/3 [)u o (Ap*) 3 + ... (11.7)

with

% = 4 , a 0 = 6 , u 0 = 9 (11.8)

Equation (11.7) corresponds to the Landau expansion (10) of the density of the Helmholtz energy A, i.e., the Helmholtz energy per unit of volume V, taken in the dimensionless form:

A* = A/PcV= AA* (AT*, Aft) + Ao*(T* ,/5)

where

AA* (AT*, &C)=(1/2)aoAT*(AI; ~ + (1/4!)Uo (&o*) 4 +...

and

Ao*(T*, P* ) = Pg0*( T• ) -P0* (T*)

385

(11.9)

(11.10)

(11.11)

Here go(T*) = Pcg(P = Pc,T*)/P , the dimensionless molar Gibbs energy (chemical potential) g taken along the critical isochore, and P0 *= 1 + PlAT * + ... are analytic functions of temperature. The function Po*(T*) is the pressure along the critical isochore withpl = a0 = (T]Pc)(OP/OT)v at the critical point. We also note that another appropriate way to make the Helmholtz-energy density and pressure dimensionless is to divide them by poRTe which differs from Pc by a factor ofzc (18).

Along the liquid-vapor coexistence curve we have asymptotically

AP icxc = + [-(6ao/uo)AT*] 1/2 (11.12)

where plus and minus correspond to the liquid and vapor branches of the coexistence curve, re- spectively. The higher-order odd terms in Equation (11.10), proportional to AT *(Aft )3 and

A * (Aft)5, yield asymmetric contributions to P cxc responsible for the linear dependence of the coexistence-curve diameter on temperature (27). Specifically, the coefficient of the cubic term is zero in the Van der Waals equation of state.

From Equation (11.10) one can see that the inverse dimensionless susceptibility (Z*) -1 = ( p * ) - I (Op,/o#) r = (Og*/Ot~)r will asymptotically vary as

aoAT* + (1/2)Uo(Ap,)2 (AT*> 0)

Q~*)-I __ { (11.13)

2a 0 AT* (AT*< 0, A o* = AP*cxc)

The condition (2"*)-l = 0 determines the stability limit (spinodal)"

A * *] 1/2 Ap* = P sp = + [-(2ao/uo)AT (11.14)

At p = Pc the isochoric molar heat capacity C v exhibits a finite jump upon crossing the critical temperature (10,27):

lim [ Cv(T<_T~)- Cv(T>__Tc) ] =z c (3a02/u0) R (11.15)

386

r -L

The isobaric molar heat capacity Cp can be calculated in accordance with the thermodynamic relation

Cp : C V + (T/p 2) (Op/OT)2 (Op/Op)r (11.16)

which asymptotically behaves as

%9/0 2 (a 0 A T* )-1 (AT *> 0, p* = 1)

Ce/z~R = { %2[(912)Uo1 -'/3 (AP*) -2/3 ( A T ' - 0) (11.17) O(0 4/3 [(912)Uo] -1/3 (A T* )-2/3 (AP* = 0)

Equations (11.7), (11.9)-(11.17) are valid for any classical (analytic) equation of state in the critical region; however, the values of the coefficients %, a 0, and u 0 will be different from the Van der Waals values given by Equation (11.8) and will not be universal. A remarkable feature and advantage of the dimensionless form of the Van der Waals equation of state, as for any other equation of state which obeys a law of corresponding states, is universality. However, its fundamental disadvantage, like that of any other classical equation of state, is its failure to describe the actual behavior of fluids in the critical region.

11.3 ASYMPTOTIC SCALED EQUATION OF STATE

It has been known since the beginning of the century, as shown in a review by Levelt Sengers (28), that the vapor-liquid coexistence curve follows a cubic power law rather than the quadratic law implied by Equation (11.12). Furthermore, in the sixties it was discovered that the isochoric heat capacity diverges at the critical point (29,30). It was also found that the inverse compressibility does not obey Equation (11.13) and is not linear in AT*, as was predicted by the classical theory, but rather follows a nonanalytic power law with an exponent larger than unity (31). At the present time the nonanalytic singular behavior of the thermodynamic properties of fluids near their critical points is a well established fact (2-4,16,26,27,32-36). The nonclassical critical behavior of the thermodynamic properties is a consequence of the long-range fluctuations of an order parameter. In one-component fluids the order parameter is associated with density. The spatial extent (correlation length) of the density fluctuations diverges at the critical point and becomes larger than any molecular scale in the critical region. As a result, the physical properties in the critical region depend only on such fundamental features as the symmetry of the order parameter and the spatial dimension rather than on the microscopic structure of matter.

According to statistical thermodynamics (10) the density fluctuations of a one-component fluid diverge at the critical point because the isothermal compressibility diverges. As a result of these fluctuations, the local density will actually be a function of the position r . In the classical theory such a spatial dependence leads to the presence of a gradient term-- (Vp') 2 in the local Helmholtz-energy density. Hence, the classical local Helmholtz-energy density AAc~ will actually have an expansion of the form

387

AAc~[AT*,Ap*(r)] = (1/2)aoAT*[(Ap*(r)] 2 + (1/4!)Uo[(Ap*(r)] 4 + (1/2)Co(VP*) 2 (11.18)

where c o is an additional system-dependent coefficient which determines the amplitude of the correlation length of the density fluctuations. In the classical theory the correlation length is given by (10,18)

4 = (coz*) ~ (11.19)

In the classical theory the correlation lenght (diverges as

4' = ~o±lZXT*l -~ (11.20)

with

- - +

~o = (Co/ao) ~ (11.21 a)

at Ap* = 0 above the critical temperature and with

fo- - (Co/2ao) ~ (11.2 lb)

at Ap* = Ap*cx ° below the critical temperature. The resulting fluctuation contribution to the susceptibility 2"* has been evaluated by Vaks et

al. (37) and that to the specific heat capacity by Levanyuk (38) for second-order phase transitions in solids. In our notation the results for fluids along the critical isochore in the one-phase region can be written in the form (18)

2'* = (aoAT*) -1 {1 + [UoVo/8ZCZca2(fo+) 31 (AT*) -~} (11.22)

where v 0 is the average molecular volume, and

m

A C v / R = [Vo/167C(~o)3](AT*) -~ (11.23)

A comparison of the fluctuation corrections with the main parts of the thermodynamic properties, namely with unity in Equation (11.22) for the susceptibility and with the classical jump 3aoZ/uo given by Equation (11.15) for the heat capacity, yields a condition for the range of validity of the

388

classical theory (Ginzburg criterion (39)). Hence, it follows from Equations (11.22) and (11.23) that the fluctuations are negligible and that the classical theory is valid only when

AT" > > N C (11.24)

The parameter N~, called the Ginzburg number (32), is defined by

No= no(u~/a4z 2) [v0/(f0+)3] 2 (11.25)

where n o is a number which is not system-dependent but which depends on the choice of the physical property considered. For example (18), if the susceptibility is taken as a measure of the distance from the critical point, it follows from Equation (11.22) that n o = 1/64 rc 2 .

To estimate the magnitude of the Ginzburg number for fluids we need to know the relationship between the coefficients in the mesoscopic expression (11.18) for the local Helmholtz-energy density and the molecular parameters of fluids. Little information concerning this relationship is currently available, so that it is difficult to evaluate the Ginzburg number a priori. If we adopt the values a 0 = 6 and u 0 = 9 for a Van der Waals fluid in accordance with Equation (11.8) and assume that

V 0 = g~(~O )3

then N~ = 0.01 (18). Surprisingly, such a rough approximation gives a correct order-of-magnitude estimation of N~ for simple fluids, as will be discussed in Section 11.5. Thus classical equations of state cannot be valid unless AT* >> 0.01, which corresponds for fluids like carbon dioxide to T - T c >> 3 K. If the condition (11.24) is not satisfied, the effects of critical fluctuations cannot be reduced to the small corrections given by Equations (11.22) and (11.23). Instead, the fluctuations yield a major contribution to the thermodynamic properties.

The renormalization group theory (RG) has been developed to deal with the effect of critical fluctuations (4,33-35,40-43). To characterize the critical behavior, systems are classified into universality classes depending on the nature of the order parameter. Fluids possessing a scalar order parameter belong to the universality class of the 3-dimensional Ising model (equivalent to the 3-dimensional lattice-gas model). It means that the properties of fluids in the critical region obey the same universal power laws with the same universal exponents and that the equation of state of fluids has the same form as that of the 3-dimensional Ising model (2,4,14,33).

To specify the thermodynamic behavior of a fluid near the critical point the pressure is decomposed as

P = p s(hl,h2) + pr(T,g) (11.26)

Here pr is a regular backgrotmd term, which is an analytic function of the variables T and g, and where ps(h 1, h2) is a singular function of two scaling fields, namely a strong field h 1 and a weak field h2:

389

h 1 = Ag* (11.27)

h 2 = A T* (11.28)

where Ag* = g* -g¢* with an arbitrary value of the chemical potential g¢* at the critical point. The corresponding scaling densities qk 1 (order parameter) and ~ conjugate to the scaling

fields h 1 and h 2 a r e

~b 1 = A,o* (11.29)

-- As* (11.30)

where As* = T c (s - sc)/P¢, with s = S/Vthe entropy per unit volume and s¢ its critical value which is arbitrary unless its absolute value is defined by the third law of thermodynamics. The regular part of the pressure is

p r, = p r/p c = Ag* - A 0 : Po (11.31)

with the same analytic function Po = 1 + PlAT * + ..., as defined in Section 11.2. The asymptotic nonanalytic critical behavior of the thermodynamic properties can be described

in terms of scaling laws initially proposed by Widom (44). The singular part of the pressure satisfies a scaling law of the form

ps , = p s / p c = h2]h2l-~fo(Zl) (11.32)

with

z, = h , / Ih21 ~+~ (11.33)

and where f0 (z 1 ) is a universal scaling function. The indices a = 0.110 + 0.003, fl = 0.3255 + 0.0020, and y= 1.239 + 0.002 are universal critical exponents (35,36,45-49) interrelated by

a + 2fl + 7 "= 2 (11.34)

The strong susceptibility 2"1 = (c32ps*/Oh?)h = (c3(Pl/C3h2)h, which is proportional to the isothermal susceptibility (Op/Og)v, exhibits a st}ong singularit~ at the critical point. Specifically, in the one-phase region along the critical isochore h i = Ag*= 0 and the susceptibility diverges as

2', = 2'* = Fo(AT*)-r (11.35)

We note that the isothermal susceptibility has no regular part a s (02pr*/Og 2)T "-- 0.

390

The weak susceptibility X2 : (02ps* /h Z)h, = (O~2/Oh2)h is proportional to the critical enhance- ment of the isochoric specific heat capacity and exhibits a'weak singularity:

z~ : rcp~(aG) /ec : A d ( a r * ) - " (11.36)

The order parameter in zero field h~ = 0 varies along the two branches of the phase boundary as

(/91 : mpcxc : 4- BolAT*I" (11.37)

In addition we define a cross susceptibility

X12 : )(21 : (O(,°2/C3hl)h2 : (O(,o1/Oh2)h, : h~-lLSfo/(Z1) - (fl+~)zlfo//(z1)] (11.38)

where fo/(zl)=dfo/dzland f~/(Zl)=d2fo/dz 2. The cross susceptibility is proportional to the temperature derivative of the density along the coexistence curve. For h 1 = 0 and h 2 > 0 the order parameter is zero and the cross susceptibility vanishes. In Equations (11.35) - (11.37) A0 ÷ , B 0 , and Fo + are system-dependent critical amplitudes interrelated by the universal amplitude ratio aAo+Fo+/Bo 2 = 0.05814-0.001 (47), so that only two critical amplitudes are independent. The critical amplitudes are defined by the asymptotic behavior of the scaling functionflz) in zero field h 1 =0

A0 + = (2-c0(1-cr)f0 (0), F0 + --f0" (0), B0 =f0 '(0) (11.39)

The amplitudes of the scaling susceptibilities can also be introduced below T c as A o- and Fo-. The ratios

Ao +/Ao-= 0.523+0.009, Fo + / F o- = 4.954-0.15 (11.40)

are universal (47). The sign of the order parameter ~1 is defined by the sign of hi = Ag*, the sign of the second scaling density ~ is defined by the sign of h 2 = AT*:

- z~* - [Ao-/(l-a)] (AT *)I AT* ]-~ (11.41)

Hence, the molar heat capacity at constant total density, proportional to the weak susceptibility )(,2 = (O~o2/Ohz)h,, remains always positive. We also note that in the thermodynamic relation

G = rv(O ~l~/Or ~-)~- r (o ~g/Or ~)~ (11.42)

the derivative (3 2p/OT 2 )V is assumed to be responsible for the divergence of C v at the critical point (27), and the temperature dependence of the pressure along the critical isochore contains a nonanalytic term --- Ao-+(AT') 2 I A7' ~ . The molar isobaric heat capacity Ce and the isobaric thermal expansivity (1/V)(OV/OT)e, being proportional to the isothermal compressibility, are strongly

391

divergent at the critical point. The thermodynamic speed of sound w in accordance with the thermodynamic relation

w ~ = ( o P / @ ) ~ = ( c d G ) ( o P / @ ) ~ (11.43)

vanishes at the critical point as Cv -1 . The correlation length diverges along the critical isochore in the one-phase region as

= ~o + (AT*) -v (11.44)

where the universal critical exponent v = 0.630+0.001 (27,32,36,47) is related to the exponent a by the hyperscaling relation (14)

v = (2 - a)/d (11.45)

where d is the dimension of space. The amplitude ~0 + is related to the critical amplitude A 0 +of the weak susceptibility )(2 by a universal relation (two-scale factor universality) (32,36,47)

ctAo +(~o +)3 (pc/RTc)NA __ 0.0188 + 0.0015, (11.46)

where NA is Avogadro's number. The connection between the correlation length and strong susceptibility, which in the classical theory is given by Equation (11.19), can now be written above the critical temperature as

= ~0 + (2"1/F0+) v/v (11.47)

Strictly speaking, Equation (11.47) is only valid at p = Pc, but is often used as an approximation for p ~ Pc also (27).

The scaled equation of state can also be expressed in terms of T and p which are more practical variables than T and g . For this purpose one needs to transform the field-dependent potential P(T,g) into the Helmholtz energy A(T, p):

A(T, p)/V = - P(T,g) + pg (11.48)

Just as the pressure, the Helmholtz-energy density can be decomposed into a singular part and a regular part

A* = A/P c V=A s*+A r* (11.49)

where the regular part A r* = A 0 (T* p*)= * * • , P go (7 ' ) -P0 (7 ' ) has the same form as the analytical background given by Equation (11.11) in the classical theory. However, the singular part A s* is not equal to the classical critical enhancement hA *, given by Equation (11.10), but is now a nonanalytic function:

392

A" *(h2,~b~)= _ p s . + ~b 1 h~ = h22 I hz I -'~ ¢Oo(Zz) (11.50)

where

zz = q~l/lh21 ~ (11.51)

and where ~Oo(Z2) is another scaling function related tOfo(z~). The scaling field h 1 = ( 0 A s*/o~O1)h2 has the form

hi = ~1 I~l ]8-1 ho(z2-1) (11.52)

with a new scaling function h o ( Z2 -1 ) and a new universal critical exponent 6 = 4.80 + 0.02 (36), which is related to fl and 7' by

fl (8 -1 ) = 7' (11.53)

Along the critical isotherm h 2 = 0 and

h 1 = D O ~b~ I~b~ 18-1 (11.54)

with a critical amplitude Do = h 0 (0) which is related to To + and B 0 by To + D O B0 ~-~ = 1.65 + 0.1 (36). In zero-ordering field (h 1 = 0) below T c Equation (11.51) yields z 2 = z 0 = B 0.

The scaling functions f0 (z0, ¢Po (z2) and h 0 ( z2 -1 ) are universal and, in principle, can be calculated with some approximations from the RG theory (50,51). Hence, the singular parts of all thermodynamic properties of a fluid in the asymptotic vicinity of the critical point can be obtained provided one possesses the critical parameters, any two system-dependent scaling critical amplitudes, and the regular amplitude @ proportional to the slope of the saturation-pressure curve at the critical point. The absolute value of go and its temperature dependence in linear approximation do not affect any measurable physical quantity as they depend on the arbitrary choice of zero points of energy and entropy. Therefore, in linear approximation one may take go =go* = 0 (8).

Expressions for the derivatives of the scaling functions f0 (Zl) and ~o o (zz), as given by various approximations of the RG theory, are very cumbersome and inconvenient for practical use (27,33,52), in particular, if one needs to pass from thermal to caloric properties. In fact, an explicit closed form of the scaling functions has never been formulated for the 3-dimensional Ising model by the RG theory. Moreover, the change from positive to negative values of AT* involves a mathematical difficulty of passing across the singular critical point (33). Therefore, a phenomenological parametric representation of a scaled equation of state which operates with a positive variable characterizing a distance from the critical point has become popular (52-60). Such a representation has appeared to be consistent with the asymptotic RG theory (61,62) and is simple for experimental use and verification (27,63,64). The representation requires a transformation to parametric variables r and 0, where r measures the distance from the critical point and 0 a location on a contour of constant r. A transformation commonly used is

CP

h, = a r~50(1 - 0 2) (11.55)

h 2 = rO(1 - b 2 0 2) (11.56)

where a is a system-dependent constant and b is a universal constant. The conditions 0 =0 and 0 = ±1 correspond to h~ = 0 : 0 =0 on the critical isochore in the one-phase region and 0 = ± 1 on the liquid and vapor branches of the coexistence boundary as illustrated in Figure 11.1. We note that the thermodynamic quantities are not singular on bypassing the critical point along the contour r = constant and, hence, are analytic functions of O.

i

O=-l/b " ~

393

h2

. , (r,o) I

f /

/ /

t I o=+l/b

Figure 11.1. The variables r and 0 in the parametric equation of state.

The simplest approximation of a parametric equation of state, known as the l inear m o d e l (55), assumes a linear dependence of the order parameter on 0 :

394

~b 1 = k r~O (11.57)

where k is the second system-dependent coefficient in addition to a in Equation (11.55). In the linear model the universal constant in Equation (11.56) is ot~en defined as

b 2 = ( y - 2 f l ) / y ( 1 - 2 f l ) (11.58)

Explicit expressions for a number of thermodynamic properties and for the universal amplitude ratios implied by the linear model can be found elsewhere (27,52,64). Strictly speaking, the linear model as used by many investigators does not include a fluctuation-induced analytic contribution to the singular part of the thermodynamic potential (18). This problem can be remedied, as recently done by Anisimov et al. (65) for the fluid-fluid critical point in chiral systems, by generalizing the expression ofP s* to

, 2 202)2 _ p s (r,O) = akr2-a[ f (O) - 02(1 - 0z)] + (ao/2Uo)r2(1 - b (11.59)

where the last term is the fluctuation-induced analytic background. The second scaling density q~2 is obtained by differentiation of Equation (11.59), so that:

~02 = a k r 1-as(O) - ( a 2 / u o ) r ( 1 - b 202) (11.60)

Using the definitions of the susceptibilities in Equations (11.35) and (11.36) one finds

ZI = ( k / a ) r -rCl(0 ) (11.61)

Z2 : a k r - % 2 ( O ) - a2/Uo (11.62)

in agreement with the RG calculations. In Equations (11.59) - (11.62)J(0), s(0), and ci(0 ) are known functions of 0 (27,64) and B is the universal critical exponent, which can be found from relation (11.53).

With the expression for b 2 given by Equation (11.58) the linear model is referred to as the "restricted" linear model (27). Substitution of the values of the critical exponents into Equation (11.58) yields for the restricted linear model

b 2 = 1.35944 (11.63)

The restricted-linear-model expressions for the amplitudes and the amplitude ratios of the asymptotic power laws become simple as given in Table 11.1 (60)

395

Table 11.1 Amplitudes of critical power laws and universal amplitude ratios in the restricted linear model.

Asymptotic critical amplitudes

Ao _ aky(7 - 1) /-'o = k/a B o = k/(b 2_ 1)13 2b2a

Universal amplitude ratios

Ao/A o = [(1-2fl)/O'- 1)]2(b 2_ 1)~ Fo/F o = (1-2fl)(b 2_ 1)/2(y- 1)

The RG theory indicates that the linear model is not exact, but it is correct up to second order in e = 4 - d (61-63,66). Higher-order approximations such as the restricted cubic model (55) or the quintic model (67) have also been considered. In the restricted cubic model

~b 1 = kr~O(1 +cO 2) (11.64)

with c -- 0.055 as an additional constant. Explicit expressions for most common thermodynamic properties derived for the restricted cubic model can be found in the literature (64).

The linear model has been used to describe the thermodynamic properties of a number of fluids in the critical region (52,63,68). It has been found that this simple model yields a satisfactory representation of the equation of state only in a very narrow range of temperatures AT * << 0.01 and densities A,o* << +0.1.

An important test of the quality of the asymptotic parametric equation of state is to compare the values of the asymptotic amplitude ratios with those obtained for the 3-dimensional Ising

• + - + -

model. As seen from Table 11.2, the ratlosA 0/A o and Fo/F o of the restricted linear model are in + + 2 agreement with the Ising values within the uncertainties of the theoretical values, while aA 0 Fo/B o

is almost 3% lower than the best Ising value (60). Fisher et al. (59) have shown that it is impossible to satisfy the Ising values for all amplitude ratios either in the linear or the cubic parametric model. They suggested an "extended cubic model" which contains additional universal parameters to be adjusted so that all Ising amplitudes are satisfied.

In real-fluid samples a phase transition always takes place in the presence of various disturbing factors, such as temperature and pressure gradients and impurities (32). Near the critical point, due to the large compressibility, even very small perturbations may lead to a dramatic response. Among the typical disturbing factors, the gravitational field is most dangerous as it cannot be eliminated or even reduced in ground-based experiments. Since in the presence of gravity the density of a fluid is a function of height, all thermodynamic properties will vary with the height. Because experimental probes sample the fluid over a finite height, there are practical limits to the resolution obtained in near-critical fluid experiments (27,36,64,69,70). These

396

gravitational limitations can be reduced by performing experiments on near-critical fluids in space (64). The most recent measurements of the isochoric heat capacity of sulfur hexafluoride up to 1 ~tK from the critical point, performed on a Shuttle mission, have confirmed the predictions of the asymptotic RG theory (71).

Table 11.2 Universal amplitude ratios.

Restricted linear model (60) Ising model (47)

Ao/A o 0.525 0.523+0.009

1-o/F o 4.86 4.95+0.15

+ + 2 aA o Fo/B o 0.056 0.0581+0.0010

+ 8 - I FoDoB o 1.68 1.67+0.10

11.4 CORRECTIONS TO THE ASYMPTOTIC SCALED EQUATION OF STATE

It turns out that the scaling laws with theoretical exponent values are only valid at temperatures within 10 -3 or even 10 -4 of the critical temperature (72) and that corrections to asymptotic scaling are required if accurate data are to be represented in a larger range. The RG theory predicts the presence o~,con2~uen, ~ingularities, in the dependence of the thermodynamic properties on h 2 of the order h 2 , h 2 , h 2 and so on (73,74), where (48,75) A s = 0.50 - 0.54 is a new independent universal correction-to-scaling exponent. Instead of the asymptotic scaling-law expressions, given by Equations (11.32), (11.50) and (11.52), one now obtains expansions which are often referred to as Wegner expansions (73)"

p,s = t~/p~ = h 2 ih~F,~ fo (z~) [1 + h2A~f~ (z) + ...] (11.65)

A,S(h2 ,~bl ) = _p*s + ~ hi = hz2 [h z F= .~ (z z ) [1 + hzA'fl (z2) + ... ]

hi = ~ll~l~-'[h0 ( z ( ' ) + I~ ~ " ]~1 ( Z ; 1 ) + "'" ]

(11.66)

(11.67)

With the first corrections to scaling, the power laws (11.35) and (11.36) for the susceptibility along the critical isochore and the power law (11.37) for the order parameter become

/7if 1 - " ~ * - - ~ 0 + (AT*)-Y [1 + L + (AT*)A, + . . .1 (11.68)

397

Z2 = T ~ p c A C v / P c = Ao + (AT*) -°< [1 + A, + (AT*) A, + ...] (11.69)

q~ : AT~x~ : -+B0 IAT*I ~ [1 + B, + IAT*I A~ + ...] (11.70)

The system-dependent amplitudes F ( , A( , and B~ + are interrelated by two universal amplitude ratios A ~ + / ~ + = 0.9 + 0.1 and B 1 + / F ~+ = 0.8 + 0.2 (45,46,75-77), so that only one of these amplitudes is independent. Deviations from the asymptotic laws observed experimentally in the range of AT * - 10 -3 - 10 -2 can be roughly absorbed by the first Wegner correction. The Wegner corrections have a universal nature: they are present in the thermodynamic properties of anysystem that exhibit scaling behavior near a second-order phase transition. In addition, in fluids there are other nonasymptotic corrections. The simple Ising model that describes a second- order phase transition in magnetic systems, and, consequently, the lattice gas that models the vapor-liquid phase transition, has full symmetry with respect to the sign of the scaling field h 1. Real fluids, however, possess the symmetry of the lattice gas only asymptotically, so that one should account for vapor-liquid asymmetry. As a consequence of the asymmetry, in fluids the scaling fields actually are linear combinations of the chemical potential and temperature (8,33):

h 1 = a 1Ag + a 2 AT (11.71)

h z = b~ A T + b z Ag (11.72)

In one-component fluids the scaling fields can be normalized by taking (8)

a 1 = pc~Pc , b 1 = 1/T~ (11.73)

Since go* and (Og ~/OT*!h:O = - a 2 / a 1 depend on zero energy and zero entropy, the coefficient a 2 in Equation (11.71) is arbitrary and for simplicity may be taken as zero. As the path h 1 = 0 in first approximation coincides with the critical isochore, the difference

S c - a z / a 1 = pc -1 (OP/OT)c = z c R a o (11.74)

is well defined at the critical point. The coefficient b 2 in Equation (11.72) is often referred to as the mixing parameter; it has an important physical consequence, since it causes a nonclassical behavior of the coexistence-curve diameter. The scaling densities ~b i and ~ generally appear to be linear combinations of the density and entropy (8) which, after adopting a 2 -- 0 , become

~b~ = Ap" - b 2 AS* (11.75)

= As" (11.76)

Along the critical isochore

As' = [A0+/(1-a)IAT* I AT'I- ~ (11.77)

398

and A,o" along the two branches of the phase boundary varies as

Ap*cxc = ¢~ + b2 ¢2 = +Bo IAT*I ~ [1 + B~ IAT*I A-" + ...] + B a AT* IAT*I -= (11.78)

with

Ba = b2 [Ao-I(1-a)](Pc /Pc) (11.79)

Here the subscript "a" indicates the term associated with vapor-liquid asymmetry. Once a critical value of the entropy or, equivalently, of the coefficient a 2 is adopted, the mixing parameter becomes well defined and can be detected experimentally from the coefficient of the singular diameter of the coexistence curve (78):

(PL + Po) / 2Pc = 1 + B~ AT* IAT *1 -= (11.80)

where PL and PG denote the density along the liquid and vapor branches, respectively. The expansion for the density difference along the coexistence curve is given by:

COL-PG )/2p¢ BolAT*I u [1 + B llAT*[ < "- -I- . . . ] (11.81)

In addition, the path of zero field h~ = 0 no longers coincide with the critical isochore: below T c this path coincides with the singular diameter and, above T c it is defined by

Ap*= b 2 [Ao+/(1-a)]( AT *)' -'* (11.82)

as shown in Figure 11.2. As a result, along the critical isochore in the one-phase region the order parameter and the chemical potential vary with AT* as

¢~ = q~ = As*= b2 [Ao + / ( l -a ) ] (AT*)'-~ (11.83)

Ag* = h 1 = (f0+)-l( AT *) v ~1 = [b2Ao + / ( l - a ) F0 + ] (AT*) 1-a+r (11.84)

Hence, the third derivative of the chemical potential, (O 3g lOT 3 )p at P=Pc above T c diverges at the critical point as (AT*) -~+ v-2. In the classical theory a = 0 and Y = 1, and the effect of the liquid- gas asymmetry is reduced to analytic corrections like the rectilinear diameter of the coexistence curve and linear temperature dependence of the molar heat capacity background. We emphasize that accounting for the mixing of the field variables does not affect the asymptotic critical scaling- law behavior which remains the same as in the symmetric lattice-gas model. The only effect of the mixing is the appearance ofnonasymptotic corrections. However, as we shall show in Section 11.8, in binary mixtures the mixing of field variables has more significant consequences, changing in some cases the asymptotic critical behavior itself. We also note that the two different nonasymptotic contributions, namely the first Wegner correction and the asymmetric term can be separated in experiments with vapor and liquid densities at coexistence. This cannot be done

399

T I

i

/i hi = O-- -~ l i

CP

coexis tence curve To , / i i i i i i i i i i i i i i i

Pc P

Figure 11.2 The path h 1 = 0 above T c and the coexistence-curve diameter below T c after account has been made for vapor-liquid asymmetry.

easily for the heat capacity and compressibility which along the critical isochore above T c now satisfy

Tc Pc ACv/Pc = A0 + (AT*)-~ [ 1 + A1 + (AT*)& + ...] + Aa+(AT * ) v- 2a_ Bc r (11.85)

2"* = F0 + (AT*)-r[1 + ~ + (AT*)A + ...] +/--a+(AT * )-~,

with (7)

A~+Fo + = C+Ao + = (1 - a) 2 Ba 2

(11.86)

(11.87)

400

In Equation (11.85) Bcr is the fluctuation-induced analytical part of the weak susceptibility (18). One can see that the terms associated with the vapor-liquid asymmetry are less important in the vicinity of the critical point than the first Wegner corrections (7 '- 2 a = 2As) and, hence, less detectable. We also note that for consistency one should include higher-than-linear terms in the analytic expansions for the regular (background) functions go* (T*) and P0* (T*). In particular, the terms of order (AT*) 2 give a background to the isochoric molar heat capacity. A contribution of the asymmetric term -- AT*(Ap* )3 in the local Helmholtz-energy expansion (11.18) gives a linear term - AT* in Equation (11.78) in addition to the nonanalytic term - [AT* ]l - ,. Moreover, the asymmetric term - (Aft)5 leads to additional nonanalytic terms in expressions for the thermodynamic properties with a new universal critical exponent A a -- 1.32 (79,80). We note, however, that these terms are of higher order than the second Wegner corrections since 2A s < A a.

Another way to extend the range of validity of the scaled equation of state is to use modified physical variables. In recent years it has become evident that in dealing with nonasymptotic critical behavior in the extended critical region it is more advantageous to use the inverse temperature 1 / T rather than the temperature T, the ratio P / T rather than the pressure P and the ratio g / T r a t h e r than the actual chemical potential g. Theoretical arguments for adoption of these alternative physical-field variables have been presented (74,81,82). The dimensionless field- dependent potential and reduced variables are then defined as

lY = P T¢/TP¢ , g = gPc T~/TP¢ , T = - T~ /T , p = f f = p/p~ (11.88)

with

dT' = - ~ d T + i f dg (11.89)

where t7 - (U/VPc) is a reduced energy density (U is the internal energy). The corresponding dimensionless Helmholtz-energy density becomes , 4 - A T c / V T P c l The reduced field- dependent thermodynamic potential P is the same function of the scaling fields h i and h 2 as P*, but the scaling fields now are defined as

hi = a l (g -gc ) + a2(T-Tc) = al Ag~ + a2 A~ (11.90)

and

h2 = /~IAT + /~2Aff (11.91)

while the scaling densities, with the adoption of al = /~1 -- T/T¢ and a2 -- 0, are

~01 = Ap* - /~2At2, ~02 = At? (11.92)

401

In the near vicinity of the critical point the difference between the two sets of variables becomes unimportant and it disappears asymptotically.

The inclusion of the first Wegner correction, of the vapor-liquid asymmetry, of higher-order terms in the expansion of the regular part, and adoption of the modified variables have made it possible to extend the range of applicability of scaled equations of state. The parametric model, revised and extended in such a way, yields an accurate representation of the various thermodynamic properties in a range of temperatures and densities bounded by (36,52,60,83-87)

- 0.006 _< AT* _< + 0.06, - 0.3 _< Ap* _< + 0.3 (11.93)

We note that although the range of applicability of revised and extended scaling is larger than that of simple asymptotic scaling, it is still very limited, especially below T c. On the other hand, empirical introduction of apparent (effective) critical exponents usually improves the practical representation of thermodynamic properties to AT * > 0.1. The effective exponents are slightly different from the theoretical universal values and usually slightly shifted toward the mean-field values. Moreover, they depend on the range of temperatures and densities of the fits and may vary from fluid to fluid. In principle including higher-order terms in the Wegner expansion might be helpful to extend the scaling procedure. However, there is evidence that the Wegner series has a poor convergence, so that the range of applicability of a truncated Wegner series is rather restricted (87). Hence, in order to extend the scaling approach to a practical range of temperatures and densities it is necessary to develop an alternate procedure that somehow takes a summation of the Wegner expansion into account.

11.5 CROSSOVER BETWEEN SCALING AND CLASSICAL ASYMPTOTIC CRITICAL B E H A V I O R

Summation of the Wegner expansion, in principle, should yield a crossover from the asymptotic scaling behavior to the classical (mean-field) critical behavior (4). Theoretical approaches to deal with this crossover problem have been considered by many investigators (45,46,66,88-107). Most of these approaches are based on the RG theory of critical fluctuations. Although an exact solution of this problem for 3-dimensional Ising-like systems is still not available, the main physical features of the crossover phenomenon, which reflect a competition between universality and nonuniversality, have been clarified. We shall first consider a simple crossover problem, namely the crossover between the asymptotic singular behavior and the asymptotic analytic (mean-field) critical behavior within the critical domain. We shall refer to this problem as the asymptotic crossover. This crossover problem has been investigated in considerable detail in the RG theory (66,79,92). Explicit expressions of the crossover susceptibilities in zero field have been obtained by Belyakov and Kiselev on the basis of the e- expansion in first order e (96). An alternative solution has been obtained by Bagnuls and Bervillier from a field-theoretical approach applied to the 3-dimensional space (45,46,93). Another solution of the asymptotic crossover problem, referred to by Tang et aL (76) as crossover model II, can be obtained as an approximation to a more general approach based on a so-called match-point method of the RG theory (89) as developed by Nicoll et al. (66,79,92) and further implemented by Albright et al. (95) and Chen and co-workers (100,101). A detailed comparison between these three solutions has been made in reference 18. It has been shown that the asymp-

402

totic crossover is determined by a single temperature scale t x, as first introduced by Kouvel and Fisher (108). Therefore, the crossover behavior of the Helmholtz-energy density may be made universal by simply rescaling the distance from the critical point. The asymptotic crossover behavior implies that the susceptibility close to the critical temperature will diverge along the critical isochore according to a power law with the Ising critical exponent 7/= 1.24, while for larger values of AT* it will vary according to the power law with the classical exponent 7" =1. Similarly, the molar heat capacity crosses over from a power law with a =0.11 for small AT* to a non- diverging heat capacity with a = 0 for larger AT*. One can define an effective susceptibility exponent Yeff and an effective heat-capacity exponent aef f as (108)

~ e f f = - AT*(0 ln;~/OAT*)h - o , 1-

a e f f : - AT*(O l n A C v / c 3 A T * ) h l : o (11.94)

where A C v is the singular critical part of C v without the fluctuation-induced analytic background term Bcr (see Equation (11.85)). Other effective critical exponents can be introduced in a similar fashion. In the asymptotic crossover theory the effective exponents are universal functions of the rescaled distance from the critical temperature A T * / t x .

The explicit forms for the crossover free-energy density will depend on the different approximations adopted in solving the RG theory equation. The solution based on the match-point method proposed by Nicoll and coworkers (66,79,92) goes beyond the results obtained by earlier investigators in (45,46,94,96,106,107) in that it is not restricted to systems in zero field h 1 and explicitly includes so-called c u t o f f e f f e c t s (92). In the infinite cutoff approximation, which neglects the discrete microscopic structure of fluids, all approaches mentioned above give the same crossover behavior in zero field.

We begin with rewriting the asymptotic Landau expansion (11.10) for the classical local Helmholtz-energy density (11.18) in rescaled form

AAcl = ( 1 / 2 ) t M 2 + ( 1 / 4 ! ) u * u A M 4 + (1/2)(VM) 2 (11.95)

with

t = c t AL M = c Aft, u *fi-A = Uo/C; (11.96)

where

C t - ao(Vo)2/3/Co = ( ~ : o ) - 2 ( V o ) 2/3 , 1/2 -1/3 2

c = c o v o , c t cp = a 0 (11.97)

and where the coefficient u* is the universal coupling constant of the RG fixed point for 3- dimensional Ising-like systems, u*-- 0.472 (76); A = qD VO 1/3 is a dimensionless wave number with qD the actual cutoffwave number, characterizing a discrete structure of matter with spacing qD -~.

403

As a measure of a distance from the critical point we consider a parameter, x, related to the inverse correlation length in such a way that in the classical limit to- ~-J Vo 1/3 so that A/x = ~-1 v01/3 = qD ~ which in the classical theory is proportional to the square root of the inverse susceptibility 2" and has the form

2 Kcl = (OA,4cllOM2)t - t + ( l l 2 ) u * ~ A M 2 (11.98)

In the asymptotic critical limit tc = C,q ~ - 1 V01/3 with Cq "" 0.7. In the asymptotic crossover theory the two scale factors c t and cp play the role of the two system-dependent amplitudes, while u- and A are generally independent crossover parameters. As a result of the long-range fluctuations, the Helmholtz-energy density is renormalized upon approaching the critical point in such a way that

t -, t T $ t -1/2 , M -, M D 112 U TM (11.99)

and by the addition of a kernel term ( - ½ i l K ) w h i c h replaces the gradient-square term in the

classical expression (11.95) for AA:

A,~ = (1/2) t M z T D +(1/4!) u*OA M4D z U - (1/2) PK (11.100)

The corresponding expression for K 2 becomes

= [OAAIO (AID 1/2 )2 ] , = t T + (I/2)u*OAMZDU (11.101)

The rescaling functions can be calculated from the coefficients in the RG equation as evaluated by Schloms and Dohm (97). For practical applications one needs to represent these functions by closed-form approximants; a number of possible approximants have been analyzed by Tang et al.

(76). The choice made by Chen et al. (101) and referred to as crossover model II in reference 76, is

T= y(Zv-1)/as, D = y(v-zv)/as, U = y~/a,, K = ( v / a g z A ) ( Y - " / a ' - 1) (11.102)

where Y is a crossover function such that

1 - (1 - U-)Y = ( u A / x ) Y v/As (11.103)

Upon substitution of Equations (11.102) and (11.103) into Equation (11.100) one obtains a crossover equation for the Helmholtz-energy density AA*. In the asymptotic critical limit A/K-, (3O9

Y = (x/u/ l ) As/v -, 0 (11.104)

404

and one recovers from Equation (11.100) the critical behavior in the form given by Equation (11.66) with values for the critical ratios of the amplitudes of the critical power laws as derived

• ÷ - + 6 - 1

by Tang et aL (76). Specifically, Fo/F o and FoDoB o are reproduced with their theoretical accuracy, Ao/A o within 5% and AoFo/B:o within 10% (cf, Table 11.2). Near the critical point the crossover susceptibility reduces to the Wegner expansion (11.68) with

F o = goN~-l/ctc: and F 1 = glNGA(1-u --) (11.105)

where go ~ 1.99 and gl ~0.104 are universal coefficients (18). The system-dependent parameter

N~ = no(ffA)2/ct (11.106)

with the universal number n o ~ 0.0314, is equivalent to the Ginzburg number defined by Equation (11.25). Indeed, it follows from Equation (11.97) that

(u.ffA)2/c = Uo2 2"Vo 'aoCo3 u~v2/a 4 (~0+)6 (11.107)

The mean-field behavior is recovered for ffA/x << 1 and controlled by the gaussian fixed point at which uA - 0. As u is inversely proportional toA (Equation (11.96)), in the infinite-cutoff approximation u = 0, so that the product uA remains finite and vanishes at the gaussian fixed point even when A -. ~. In this approximation the crossover scale t x = N~. If the cutoff parameter A is finite, u is not zero for real fluids and the crossover scale should be redefined as (18)

t x = /G/(1-U-) 1/As (11.108)

We refer to this redefined crossover scale as the effective Ginzburg number. In the asymptotic crossover theory t x is a single crossover parameter which plays two roles simultaneously: it controls the convergence of the Wegner series, which has the form (1 + gl (AT */tx) A" + ...), and indicates the range of validity of the classical theory. One can recover from the mean-field limit the root-square corrections to the classical behavior of the susceptibility and heat capacity in the scaled form (tx/AT*) v2 similar to Equations (11.22) and (11.23) (18). In simple fluids the cutoff parameter A is of order unity (the characteristic microscopic scale is of the order of a molecular size). This is why in simple fluids the crossover to the classical regime is never completed within the critical domain where ( i s large, unless there is a special reason for very small 6. The crossover to mean-field behavior in the asymptotic crossover model occurs only if 6 <1. If 6 _> 1, the crossover scale is not defined by this theoretical model. For 6 = 1 all the correction-to-scaling terms in the Wegner expansion disappear (the effective Ginzburg number becomes infinite) and the critical exponents within the entire critical domain are equal to their asymptotic (Ising) values. For 6 >1 the effective critical exponents y and a monotonically increase in this asymptotic crossover model with increase of AT *, as the correction-to-scaling amplitudes of the Wegner expansion are negative, and the crossover to the classical regime never happens either.

405

The crossover Helmholtz-energy density defined by Equations (11.100) - (11.103) allows one to calculate all thermodynamic properties along different thermodynamic paths. Note that Equation (11.103) defines the crossover function Y implicitly. This complicates the fitting procedure. To describe the crossover of the compressibility and heat capacity along the critical isochore explicit crossover expressions can be obtained from an e-expansion in first-order approximation (96,106,107). In particular, the crossover function becomes:

y - l = + 1-u (11.109)

When e = 4 - d = 1, Equation (11.109) becomes equivalent to the expression for the crossover function obtained by Belyakov et aL (109).

However, while the results obtained in first order e are useful to illustrate several physical aspects of the crossover phenomena, they will not be quantitatively correct in 3-dimensional systems. A simple form of the crossover susceptibilities Z~ and Z2 along the critical isochore (in zero field) in the one-phase region was proposed by Belyakov and Kiselev (96) as phenomenological approximants to their crossover solution obtained from a first order c- expansion:

aT*/t×=(l+Xoyfi Adv)(y-,)/A, ~ l - 1 +(1+A5 ,~dv yV/A, ] (11.110)

Zz/Bcr = A C v / R B c r = (1+ ZoZ1A'/v ) ~/As-1 (11.111)

where/'Ill --- a0 tx .~fl, the rescaled susceptibility and 2"0 = 2.333, an empirical constant. The two amplitudes, namely, a 0 and the so-called critical background Bcr = aoZ/uo, and the crossover scale tx are to be found by fitting to experimental data. This model does not recover the proper square- root-of-temperature corrections in the classical limit, as it was the s-expansion solution, whereas the match-point solution does. However, as further discussed in Section 11.6, in the classical limit far away from the critical point any asymptotic crossover model will be inadequate anyway and some phenomenological adjustments will, in practice, be necessary to provide the crossover to a regular behavior.

A comparison between the three solutions to the asymptotic crossover problem obtained by Anisimov et aL, Belyakov and Kiselev, and by Bagnuls and Bervillier is presented in Figure 11.3. All three solutions give a very similar crossover of the susceptibility and molar heat capacity along the critical isochore in the one-phase region. The universal crossover susceptibility in the form of Equation (11.110) was used by Meier et aL to fit experimental light scattering data in polymer blends (110,111) and by Seto et al. to describe small-angle neutron-scattering data in a three-component microemulsion system (112). The asymptotic crossover theory described in this section has been applied to susceptibility and heat-capacity data of a polymer blend (113).

406 0.12

0.10

0.08

0.06

0.04

0.02

0.00 1.24

1.20

1.16

112

1.08

1.04

,k X\

- \ \ -

I I I I I I I

~ ~ I I I I • "'- .~ ,~

\ \

- * ~ \ \ \ \

1.00 I I I i I I i

- 4 -:3 - 2 - 1 0 i 2 3 4

log (AT/NG)

F i g u r e 11.3 Effective weak and strong susceptibility exponents as a function of A f " / N a . Asterisks: Asymptotic crossover theory of Belyakov and Kiselev (96). Dashed curve: Asymptotic crossover theory of Bagnuls and Bervillier (45). Solid curve: Asymptotic crossover theory described in Section 11.5. From

reference 18.

11,6 CROSSOVER BETWEEN SCALING AND CLASSICAL NONASYMPTOTIC CRITICAL BEHAVIOR IN THE EXTENDED CRITICAL REGION

The crossover equation for the Helmholtz-energy density presented in the previous section accounts for a crossover between asymptotic singular critical behavior and asymptotic classical critical behavior within a critical domain (in practice for AT*<10 -2 and Ap *<I0 -1 ). However, it is inadequate to deal with the complete crossover behavior of fluids with short-range forces. The reason is that for such fluids the classical theory does not become valid until so far away from the

407

critical point that any asymptotic critical behavior has become inadequate. Hence, in order to deal with the extended critical region where the correlation length is still larger (but not very much larger) than the distance between molecules, one needs to consider the nonasymptotic aspects of the crossover behavior. Another important disadvantage of the asymptotic crossover model is that it is not applicable for a coupling constant 6 larger than unity and, hence, for systems with negative Wegner correction amplitudes (114,115). However, the possibility of negative corrections follows from field-theoretical RG approaches (93,116) and from the analysis of the asymptotic critical behavior of 3-dimensional Ising lattices (117). Moreover, negative first Wegner corrections have been reported for some aqueous and nonaqueous ionic solutions near the consolute critical point (118,119) whereas for simple fluids the corrections are usually positive. With a negative first Wegner correction term the crossover from Ising behavior to the mean-field behavior, becomes nonmonotonic and sharper than usual (114,115).

Two different approaches to the problem of nonasymptotic crossover have been pursued. The first is to try to implement the RG procedure for a fluid of molecules that interact with a given interaction potential. Parola and coworkers (103,120-122) and also Zhang and Badiali (123) have applied a RG procedure to fluids on the basis of the Ornstein-Zernike equation for the pair correlation function as reviewed by Meroni (102). White and coworkers (105,124,125) and subsequently Tang (126), Lue and Prausnitz (127) and Jiang and Prausnitz (128) have shown how one can implement a global renormalization procedure for fluids in the phase-space cell approx- imation originally proposed by Wilson (129). These approaches yield a good qualitative rep- resentation of the thermodynamic behavior of simple fluids over large ranges of temperatures and densities. While these approaches towards developing a theory for the global thermodynamic behavior of fluids and fluid mixtures are very promising, there are numbers of limitations. They do not lead to explicit expressions for equations of state; the RG procedure has to be implemented numerically. With the possible exception of the work of Zhang and Badiali (123), one does not recover the theoretically correct Ising exponent values exactly, since one obtains r/= (2 v- ~,)/v = 0 rather than theoretical value r/= 0.033.

The second approach is an extension of the crossover theory based on the Landau-Ginzburg- Wilson theory of critical phenomena outlined in the previous section and adopted by our research group (18,101,130-136). The advantage of this approach is that it yields a crossover equation of state for all Ising-like systems; hence, it should be applicable to all fluids and fluid mixtures with short-range forces regardless of the complexity of the intermolecular potential. This crossover approach is also applicable to systems with negative Wegner correction terms (114,115,137) and is theoretically supported by the exact solution of the 3-dimensional spherical model (138). The disadvantage, as in any scaling-based theory, is that it cannot yield specific system-dependent properties such as the critical parameters and the critical amplitudes without experimental information.

In order to deal with the full nonasymptotic effects of the critical fluctuations we need to incorporate three features in addition to those considered in the previous section. First, a rigorous analysis of the Crossover problem for the spherical model (92) has shown that the actual classical limit is not reached when A/x--1 as suggested by Equation (11.103), but when A/to-.0. This means that t¢ -~ o, ¢ corresponds to the spatial extent of the critical part of the correlation length of the critical fluctuations only and should disappear far away from the critical point, while the actual correlation length tends to the range of intermolecular forces. This feature can be incorporated by replacing A/to in Equation (11.103) for the crossover function with [ 1 + (A/to)2] v2.

408

Instead of Equation (11.103), the crossover function Yis now to be determined fi'om the condition (101)

1 - (l-u-') Y = u-[1 + (A/K)2] 1/2 ] ''v/As (11.112)

Secondly, we need to include higher-order terms in the Landau expansion (11.10). As the next approximation we consider

A A c l - - (1/2) t M 2 + (1/4!) u*t2A M 4 + (%5/5!) M '

+ (%6/6!) M 6 + (a14/4!) tM 4 + (a22/2!2!) t2M 2 (11.113)

As pointed out by Nicoll and Albright (66,139), the crossover behavior of the M 5 term can be accounted for by replacing M 5 by M 5 D 5/2 V U with

V = Y (2A~- ~)/2A, (1 1.1 1 4)

which supplements the transformation given by Equation (11.99) for t and for the even powers of M. In Equation (11.14) Aa(= 1.32) is a new correction-to-scaling exponent.

Third, while t = ctAT, the definition of the order parameter M in Section 11.4 can be generalized to account for vapor-liquid asymmetry according to (c f , Equation (11.96)):

M = cp~ 1 -dl A T = cp (A,5 - bzA tT) - d 1A iF (11.115)

Hence, the classical expansion (11.113) transforms to the renormalized Helmholtz-energy density expansion (18,101,130-136):

AA =(1/2) t M 2 7 D + (1/4!) u*OAMaD2U - ( 1 / 2 ) t2K + (aos/5!) MSDS/2 "/~ + (%6/6!) M 6 D3/2U 3/2+ (a14/4!) tM 4 D 2 U1/2 + (a22/2!2!) t 2 M 2 DU-v2 (11.116)

The corresponding equation for x 2 becomes (132)

K 2 = t7 + ½ u * O A M 2 D U +(aos/5!)MSD3/2VU +(ao6/6!)M 6 D 3/2U3/2 + (a14/4!) tM4D2U 1/2 + (a22/2!2!) t 2 M 2 DU-1/2 (11.117)

The total Helmholtz-energy density is given by A(T) - Z£4 +/fg0(~ - /50(~ (cf , Equation (11.9)) after substitution of Equation (11.116) into this equation, while the analytic functions g0 and 150 are represented in practice by truncated Taylor expansions in terms of powers of AiF (101).

For the inverse susceptibility in zero field in the one-phase region this crossover solution implies

409

X, -1 = c 2 t y (v-1)/a (1 + (u*/2co ) { 2 ( t c / A ) 2 [1+ (A/~c) z]

× [(1/co) + (1 -U-)Y/(1-(1 -u')Y)] - [(2 - e -~ )/(O]} -1) Jr- 0 (t 2 ) (11.118)

with x 2 = t7 + 0 (t z ) = t~zv-o/A + 0 (t 2 ) and co = As/v. Close to the critical temperature

(tc/A << 1, Y<<I) the behavior of the susceptibility is the same as given by the asymptotic crossover model. Far away from the critical point (to/A>> 1) Y-+ 1 and

Z1 1= cp 2 (t +St) + (1/2) c~ a22(t +St) 2 (11.119)

where

8 t = c t ( T c - T ~ m f ) / T = - (1/2) u*t0OA 2 (11.120)

with t o ~ 0.3752/(1+0235u-). Here, fit is the shift between the actual critical temperature T c and the mean-field critical temperature Tc rnf A more general consideration of the renormalization of the value of the mean-field critical temperature (140) shows that the numerical value of t o should be close to unity. The numerical difference could arise from the fact that M-dependence of the kernel term (1/2) ilK, as was implemented by Equation (11.116) into the crossover theory, is not rigorous. This problem is still under investigation.

We note an important conceptual difference between the asymptotic and nonasymptotic crossover models. While the former is characterized by a single crossover scale, the latter is controlled by two independent parameters 6 and A / c t 1/2 : 11 determines the sign of the first Wegner correction amplitude (76) and A / c t 1/2 controls the crossover to the mean-field regime which is recovered in the limit A/~¢ <<1. For 6 <<1 there is no qualitative difference between the nonasymptotic and asymptotic crossover behavior: the crossover is still monotonic and is, in practice, not completed within the critical domain where ~' ~ tc -1 is large and A is of order unity. This is a typical situation in simple fluids. As the regime A/t¢<<I is not accessible within the critical domain, for this case we can retain the concept of the effective Ginzburg number as a characteristic crossover scale t x, defined by Equation (11.108) in the asymptotic crossover model. In complex fluids, however, the parameter A is not necessarily associated with the actual microscopic cutoff, but rather may reflect another characteristic spacing on a scale larger than a molecular size (114,115,137). I fA is small enough ( i . e . , the characteristic spacing is large), the nonasymptotic behavior of the model of Nicoll and Bhattacharjee (92) implies that the crossover to the mean-field critical exponent 7/= 1 is still possible within the critical domain even for 6 _> 1. We note that the effective coupling constant OA may be small if A is small. In this case the crossover is not monotonic, since it is controlled by two independent crossover parameters: 6 >1 is responsible for a negative first Wegner correction amplitude and drives the effective susceptibility exponent upward with increase oft, and small A provides a decrease of the exponent downward to the mean-field value y = 1 even for relatively small x. An analysis of light-scatter- ing data shows that in several ionic solutions (114,141) and in polymer solutions (115) near the consolute critical point the effective susceptibility exponent definitely exhibits a sharp or even nonmonotonic crossover between Ising-like and classical values and that the theoretical model of Nicoll and Bhattacharjee (92) gives a good explanation of this phenomenon.

410

The version of the nonasymptotic crossover Helmholtz-energy density described above has been applied to represent the thermodynamic properties of a variety of fluids. It yields a good representation of the thermodynamic properties in a range of densities, temperatures and pressures indicated in Figures 11.4 and 11.5. As examples we show in Figures 11.4 and 11.5 the thermodynamic speed of sound of ethane and the isobaric molar heat capacity of methane at the critical density (133). The crossover model, based on the renormalized six-term Landau expansion, represents the thermodynamic properties in a broad range of temperature and density approximately bounded by 2" >-- 2.2. A representation of the nonasymptotic crossover in terms of parametric variables was developed by Luettmer Strathmann et al. (142).

7 0 0 // 6 0 0

rn 5 0 0

400

300

- 2 0 0

1 0 0 5 0 1 0 0 1 5 0 2{)0

p, k g . m -a

D - T = 1 9 0 . 5 7 K • - T = 1 9 3 . 0 6 K • - T = 2 0 0 . 1 6 K • - T = 2 1 0 . 1 6 K • - T = 2 2 3 . 1 6 K

• - T = 2 4 8 . 1 6 K < > - T = 2 7 3 . 1 5 K o - T = 2 9 8 . 1 4 K + - T = 3 2 3 . 1 4 K z x - T = 3 4 8 . 1 4 K ( > - T = 3 7 3 . 1 5 K

2 5 0

Figure 11.4 Sound velocity of ethane at various temperatures as a function of density. The symbols indicate experimental data and the curves values calculated from the crossover equation of state (133).

A phenomenological parametric crossover equation of state developed by Kiselev and coworkers (15,143-149) sometimes yields an even broader range of description, and is easier to use. In this approach one starts from a revised and extended parametric equation for the Helmholtz energy (150,151) and modifies each term so that they all become analytic far away from the critical point. For this purpose a simple crossover function R(q) is introduced as

R(q) = 1 + q2/(q o + q) (11.121)

Here q0 is an empirical constant, while the variable q is related to the parametric variable r by

q =gr . (11.122)

10-~ l a - P= I].2737 MPa | | , , - P=10.3421 MPa

"7 CH~ t:m

~ 3

(a)

190 210 230 250 270

T, K 80

60-

40-

20

0 180

e - P=5.0000 MPa o P'-5.5158 MPa ( b )

P=6 .8948 MPa

CH4

i~o z6o zlo z~o z~o z4o T, K

411

Figure 11.5 The isobaric specific heat capacity Cp of methane at various pressures as a function of temperature. The symbols represent experimental data and the curves values calculated with the crossover equation of state (133).

The coefficient g plays the role of an inverse crossover scale similar to that in the asymptotic crossover model. The critical part of the Helmholtz-energy density is represented as

A A = krZ-~ R ~(q) [ aY/o(O) + ~a C, rA'R-A'(q) ~(0) ] (11.123)

(with summation from i = 1 to i=5) and the parametric representation of the order parameter is generalized to

qk 1 = k r~R -~+~ (q)O, (11.124)

where k and a are the coefficients of the asymptotic linear model described in Section 11.3; c i are

empirical system-dependent coefficients; A, (A 1 = A s, A 2 = 2A 1, A 3 = A 4 = y + 13-1, A 5 = A a) are various correction-to-sharing exponents; Y0 (0 ) and T 1 ( 0 ) are universal asymptotic and first- Wegner-correction functions, ~2-5 (0) are empirical universal functions. In the critical limit q-.0, the expression reduces to that of the asymptotic, linear-model-based, scaled equation of state

412

(150,151). In the classical limit q-.oo, Equation (11.123) yields an expansion of the Helmholtz- energy density similar to that given by Equation (11.113) with a modified analytical background proportional to t 2 (143) (see also reference 15).

The crossover behavior of the effective exponents y~ff and aef f implied by the parametric crossover model of Kiselev et al. appears to deviate from that derived from the theoretical crossover solutions (18). Specifically, the crossover is too sharp and the classical limit is not approached monotonically. Alternatively, one could redefine the function R(q) as

R(q) = [1 + q2/(q o + q)]2 (11.125)

with

q : (gr) 2. (11.126)

This modification yields some improvement (18). The coefficients of the Wegner expansions (see Equations (11.68)-(11.70), are imposed as terms with adjustable coefficients in this model (c i in Equation (11.123)), while in the crossover Landau expansion they are recovered automatically upon expanding the crossover function Y. The parametric crossover model of Kiselev is useful as it is able to represent actual thermodynamic properties in a broad range of temperatures and densities (15).

11.7 GLOBAL CROSSOVER EQUATION OF STATE OF ONE-COMPONENT FLUIDS

While a nonasymptotic crossover for the Helmholtz-energy density based on a transformed Landau expansion is capable of yielding a good representation of the thermodynamic properties of supercritical fluids over a large range of temperatures and densities, it has a sizable number of system-dependent adjustable coefficients, namely the scaling-field constants ct, cp, b2, d 1 relating the Landau variables t and M to the physical variables A 7 ~ and A/~ (see Equation (11.115)), the crossover parameters 6 and A, the coefficients ao in the classical Landau expansion (11.113) and the coefficients in the Taylor expansions for the background regular functions g0(~ and /50(~. Nevertheless, its validity is still restricted to a domain around the critical point. For chemical- engineering applications one often prefers to use simple equations of state that are generalizations of the Van der Waals equation that should in principle be applicable everywhere depending on the quality of the equation. The question arises whether such closed-form equations can also be corrected for the effects of critical fluctuations by a transformation similar to the one specified by Equation (11.99). This question is also conceptually important. The mean-field critical region is well defined by the classical equation of state. The mean-field critical parameters and the constants of the classical equation of state are interrelated. A corresponding global equation of state would explicitly demonstrate how fluctuations change the classical critical behavior and how the classical critical point drifts to its actual position. Such equations should provide both the universal critical behavior and noncritical regular behavior up to the ideal-gas limit as well as the high-density limit.

To formulate the approach we consider three regions. The first is the region far away from the critical point not affected by fluctuations at all. The second is the asymptotic critical region where the physical properties manifest universal scaling behavior. The third region is an extended

413

critical region where the properties exhibit crossover between the universal scaling behavior and nonasymptotic classical behavior. We will call it the crossover region. In short, the algorithm to develop global equations of state should proceed as follows: 1. A proper classical equation of state, based on data obtained far away from the critical point, is chosen. The critical point defined by this equation is the mean-field (classical) critical point which is located at a temperature above the actual critical temperature. 2. The classical equation of state is expanded around the classical critical point in the form of the analytical 6-term Helmholtz-energy density expansion. The coefficients of the expansion are expressed through the constants of the equation of state. The values of the expansion coefficients will provide certain constraints for the crossover parameters. 3. The temperature and density are expressed through Air and Aft which are then renormalized by the nonasymptotic crossover transformations presented in Section 11.6. It should yield the universal asymptotic behavior of the Helmholtz energy close to the critical point and the location of the actual critical point affected by fluctuations. Kostrowicka Wyczalkowska et al. (21) have shown how this procedure can be used to incorporate the affects of the critical fluctuations into the equation of state of Van der Waals.

If there existed an exact classical equation of state which could perfectly describe experimental data of a fluid far away from the critical point, the mean-field critical point would be unique for the fluid. Then, in principle, only two additional system-dependent parameters, namely the crossover parameters O and A/c t, in addition to the coefficients of the classical equation of state, are needed to specify the global equation of state. Unfortunately, such an equation does not exist and different equations of state applied to the same fluid will give different locations of the mean- field critical point. Moreover, as mentioned in the previous section, the shift of the critical temperature derived from the nonasymptotic crossover model is not exact. This is why an additional adjustment of the shift might in practice be needed. Van Pelt et al. (17) have applied such an approach to the Camahan-Starling-DeSantis (CSD) equation of state, which was used by Morrison and McLinden to describe the thermodynamic properties of various alternative refrigerants (152). It was assumed that the critical density, as implied by the CSD equation, is not affected by fluctuations. A shift of the critical pressure corresponds to a shift of the critical temperature along the critical isochore. In Figure 11.6 we show the coexistence curve of R135a (17). The dotted curve represents the coexistence curve when the classical CSD equation is fitted to the experimental data (152). The solid curve represents the coexistence curve implied by the CSD equation after being transformed to account for critical fluctuations.

A similar but somewhat more empirical method for incorporating scaling laws with an analytical equation of state has been developed by Kiselev (22). This approach is based on a transformation of temperature and molar volume rather than temperature and density. This approach has been applied to the cubic Patel-Teja equation of state (22,23) and, more recently, to the SAFT equation of state (24,153).

It should be noted that the asymptotic scaling law behavior is characterized not only by universal critical exponents but also by universal amplitude ratios (cf., Table 11.2). It is not always clear to what extent empirical procedures to introduce the scaling laws into classical equations of state also incorporate good estimates for the amplitude ratios. Specifically, this appeared to be a serious problem in a procedure proposed earlier by Fox (154), as discussed by Tang et al. (76).

414

[.-,

400 , I , I , I i I i I , -

3 5 0

3O0

0 2 4 6 8 10 12

p / mol.L-1

Figure 11.6 Coexisting vapor-liquid densities of R134a as a function of temperature. The symbols indicate various experimental phase-boundary data, the dashed curves represent values calculated from the classical CSD equation, and the solid curve represents values calculated from the transformed CSD equation to include the effect of critical fluctuations (17). The diamond indicates the location of the classical (mean- field) critical point.

11.8 I S O M O R P H I C S C A L E D E Q U A T I O N OF STATE OF NEAR-CRITICAL BINARY FLUIDS

With the growth of supercritical technology and with the progress in the theory of critical phenomena during recent decades, more and more scientists and engineers are interested in the phase behavior and thermophysical properties of binary and multicomponent fluids in the critical region (155,156). Binary fluid mixtures, being a bridge between one-component fluids and complex multicomponent fluid systems, may themselves exhibit complex phase behavior and different kinds of critical phenomena (157-159).

415

Based on the geometry of the pressure (P) - temperature (T) projections of critical and three- phase lines, fluid phase diagrams can be classified into six major types as elucidated by Van Konynenburg and Scott (158) and further discussed in a recent report issued by IUPAC (160). In a mixture with Type I phase behavior the vapor-liquid critical line runs continuously from the critical pointof component 1 to the critical point of component 2. In mixtures with Type II phase behavior the vapor-liquid critical line is continuous as well, but an additional liquid-liquid critical line appears in the liquid region. A Type III phase diagram has two distinct vapor-liquid critical lines: one critical line starts at the critical point of the component with the higher critical temperature, but never approaches the critical point of the other component moving instead to higher pressures; the other critical line starts at the critical point of the component with the lower critical temperature and meets a three-phase line in a critical end point. Mixtures of water with carbon dioxide and with hydrocarbons are examples of systems exhibiting Type ffl behavior. In mixtures with Type IV or Type V phase behavior the critical locus between the critical points of the pure components is interrupted by a three-phase liquid-liquid-gas region terminated at either end by critical end points, namely an upper critical end point (UCEP) and a lower critical end point (LCEP). Such types of phase behavior can be observed in mixtures of hydrocarbons when the number of the carbon atoms differs considerably. While the methane+ethane system is a typical example of a mixture with Type I phase behavior, the methane+n-hexane system is an example of a system with Type V behavior. As shown schematically in Figure 11.7, in a system with Type V phase behavior one branch of the critical locus, which starts at the critical point CP 2

of the component with the higher critical temperature and ends at a LCEP, represents a continuous crossover between vapor-liquid critical phenomena and liquid-liquid critical phenomena. Another branch of the critical locus starts at the critical point CP~ of the component with the lower critical temperature (methane) and terminates at a UCEP. In the case of Type IV phase-behavior there is an additional liquid-liquid critical line at low temperatures which ends at another critical end point (159). Type VI phase behavior is characterized by a continuous critical locus between the critical point of the two pure components, but at lower temperatures there is a three-phase line which is bounded above and below by critical end points. Experimental studies of fluid phase equilibria in binary mixtures at high pressures show that there are continuous transitions between different types of phase diagrams that can be described qualitatively by a Van der Waals equation of state (158).

The concept of critical-point universality has been extended to binary fluid mixtures almost 30 years ago (5-7) provided one selects an appropriate set of thermodynamic variables associated with the scaling fields. This approach is often referred to as isomorphism of critical phenomena (7-9). The stability criterion for a binary mixture reads (157):

(O~A/Ox~)~,~( o~A/o V ~ ) ~,x - j A / O x O v)~] ~ >_ o (11.127)

where x is the mole fraction of component 2, designated as the solute. At either a vapor-liquid or a liquid-liquid critical point the determinant (11.127) should vanish. Depending on the choice of the density variable associated with the order parameter, the molar density or the Composition, one can rewrite Equation (11.127) in terms of two different response functions:

416

RCP

.° -.% o" •

°°

."'C P 1 / CP2

_1/

Figure 1!.7 Schematic phase diagram of a mixture exhibiting Type V phase behavior.

(7-9). To specify the appropriate set of thermodynamic variables one needs to analyze the stability criterion for a binary mixture (157):

- ( 0 P / 0 v)~ , , = - ( 0 e / 0 v) ~,x - [ ( @ / 0 v)~,x] ~/(op/Ox)~,~ ~ o

o r

(O#/Ox)p,r = (O#/OX)v,r + [(OP/O x) ~,v] 2 / (OP/O V) r,x -> 0

(11.128)

(11.129)

In these equations # = (OAIOx)v,r = ~ -#1 is the difference between the chemical potentials of the two components 1 and 2.

While stability criterion (11.127) is a natural one for "plait" (vapor-liquid critical) points as a generalization of condition (11.6) for a one-component critical point, (OP/OV)v,u cannot serve as a response function near the consolute point of an almost incompressible liquid mixture. Similarly, criterion (i 1.127)will fail in the one-component limit where (OP/OV)v, xvanishes. This is why two different sets of isomorphic variables are usually chosen, one for vapor-liquid critical phenomena and another one for liquid-liquid critical phenomena (32). The most general formulation of the isomorphism approach (8,9) encompasses all possible kinds of critical phenomena in binary fluids (vapor-liquid, liquid-liquid, and gas-gas). In binary mixtures the density of the thermodynamic potential, £1 = -PVper unit of volume, is a function of three field variables, namely the temperature T, the chemical potential of the solvent, #1, and the difference #, between the chemical potentials of the two components:

417

dP = sdT + Pdgl + 92dg (11.130)

where/92 = P x is the partial molar density of the solute. The application of the principle of critical-point universality to binary fluids implies that the thermodynamic behavior of near-critical mixtures is always characterized by only two scaling fields h 1 and h 2. Therefore, in binary fluids the density of the thermodynamic potential -P*, is still the same function of the scaling fields h 1 and h 2 as implied by Equation (11.32). However, now these scaling fields are, in general, linear combinations of three physical field variables, and AT= T- T c, Agl = g - glc and Ag - g - go:

h 1 = alA/.t 1 +a2AT+a3A, u (11.131)

h 2 =blAT+b2A, tt 1 +b3A, u (11.132)

where Tc,/~ lc, and ~t c are the critical values of T,/~ 1, and/4 respectively. The critical parameters To,/~ lc, and/~ c and the coefficients a i and bi depend parametrically on the position on the critical line, which can be specified by any of the three variables To,/~ lc, and ~t c.

The three physical densities, namely, the total molar density p, the partial molar density P2 and the entropy density s are linear combinations of the two scaling densities ~o 1 and ~o 2, such that

Ap=p-pc =pcRTc (alcPl +bzcP2) (11.133)

=X,O--Xc/3c =/92--P2c =PcRTc(a3(Pl +63(/92) (11.134)

=pc (a e (11.135)

where Pzc, xc, and s c are the critical values of/92, x, and s, respectively. In one-component fluids the zero-field condition (h i - 0) corresponds in first approximation

to the path along the critical isochore above the critical temperature and along the coexistence curve below the critical temperature at constant overall density (p = Pc). Generally, in binary fluids the zero-field condition can be realized along different paths, for example, along the critical isochore at constant ~ = ~c, along the critical isobar at constant overall composition, and along the critical isotherm at constant overall composition. The explicit expression for the weak scaling field h 2 at h 1 = 0 will depend on the specific choice of the thermodynamic path. For mixtures the scaling susceptibilities 2"1, 2"2, and Zl2, as well as the scaling densities q)l and q)2, are not directly measurable quantities. However, experimentally accessible second derivatives of the thermodynamic potential can be expressed in terms of 2"1, 2"2, and 2"12 by applying appropriate thermodynamic transformations. Two characteristic system-dependent parameters K 1 and K 2 and

418

which determine the difference between the one-component-like critical behavior and binary-fluid- like behavior, are defined as

K1 = -(a3/al-Xc) (11.136)

and

K2 = b2a3/al'b3 (11.137)

K 1 and K 2 are determined by the shape of the critical locus (8,9):

d x c K 0 (11.138) K I = R T c dl.t c

with

1 ] K0 - L~-~- ~ - _ (11.139)

PoRTe dxc h~ 0,/~=~c

and

1 d x c d T c

K 2 - Tc dt, t c d x c . (11.140)

The derivative dx c / dlzc cannot be measured directly. However, the so-called critical-line- condition (9) gives a realistic approximation of the connection between the chemical-potential difference and composition along the critical line: ¢te ---RTlnxc so that

d x ¢ ~ x¢(1- xc) R:r (11.141)

419

The applicability of the critical-line condition has been discussed by Rainwater and Friend (161) and Anisimov et al. (9). In the infinite dilute-solution limit (x---~0) the parameter K 0 is reduced to the so-called Krichevskii parameter (162) with

h 1 = 0,/.t =/.t c p=p¢ ,x---~ 0 (11.142)

In the pure-component limit K 1 and K 2 vanish and the scaled equation of state for binary fluids reduces to the one-component equation of state presented above. The parameter K1 vanishes also at the critical azeotrope, while K 2 vanishes at the critical-temperature extrema (8,9). Expressions for various thermodynamic properties of binary fluids derived from the isomorphism approach have been presented elsewhere (8,9,163,164).

11.9 RENORMALIZATION OF CRITICAL EXPONENTS

Experimental conditions may or may not correspond to the isomorphic paths for which at least one field variable must be kept constant. For example, if the thermodynamic properties as a function of temperature are studied at constant density, the chemical potential difference/z must be kept constant to make the path isomorphic. Instead of/z, in practice, the concentration x is usually kept constant, so that the experimental variables do not coincide with the theoretical ones.

As the connection between the theoretical (isomorphic) variables and the experimental variables (non-isomorphic) can be non-analytic, the apparent (experimentally observed) critical singularities can be different from those calculated for the isomorphic paths. This phenomenon, known as the renormalization of the critical exponents, was predicted by Fisher (165). As a simple example, we consider a dilute solution near the vapor-liquid critical point of the pure solvent. Consider an experiment at constant p = Pc and at constant composition in the one-phase region. The relation between the experimental temperature scale T - T c at x=xc and the theoretical temperature scale h2= T-T~ at/x=/z~ is given by

A T * ( x ) / r 2 = ( A T * ( / t ) / z 2 ) l - a { 1 + [ A T * ( / z ) / r 2 ] a} (11.143)

with

r-Vc(X-x c) AT*(x) = T~(x=Xc) (11.144)

and

420

AT*(~) = . (11.145)

The characteristic (renormalization crossover) temperature r 2 in Equation (11.143) is defined by

1/2 + 2 1/2

[x~(1-Xc) J (11.146)

Another characteristic temperature

/ ' 1 = [Fgxc(1-xc)K2] 1/~ rox

Xe(1-X c)

1/v

(11.147)

determines a range of distortion of the strong singularity in the isothermal compressibility and appearance of divergence in the osmotic susceptibility. Usually, with the exception of the critical azeotrope where K0=0, z" 2 << r~. At AT* >> r 2 :

AT*~) = AT*(x) (11.148)

and no changes in the apparent singularities are expected. However, at AT* << r 2 (asymptotically close to the critical point)

AT* ~)/I" 2 --- [ A T * (x)/'/" 2]1/(1 -a) (11.149)

As a result, the critical anomalies measured along the path p = Pc and x - x c asymptotically will exhibit critical exponents renormalized by a factor 1/(1-~). Note, that in the Van der Waals (classical) theory a = 0 and the connection between AT*(x ) and AT (/z) ~s analytic. As 1/(1-(~) > 1, the apparent asymptotic singularities of the quantities that diverge at the critical

point will be stronger than those in one-component fluids, whereas the isochoric molar heat capacity at constant composition Cv, ~ will approach a finite value. The crossover expression for Cv~ reads (163)

Cv, x A 0 Ocr[AT'(Iz)/~2] a C ~ -~ - + (11.150)

R v2[ AT* (,a)/v 2 + 1 ] [ AT* (~)/v 2 ]'~ + 1 R

421

where AT*(/z) is related to AT*(x) by Equation (11.149) and Bcr represents a fluctuation-induced modification of the regular background molar heat capacity cr~ In the linear-model parametric _ . Vrlf", equation of state Bet = ao2/Uo according to Equation (11.62). At AT > > r 2 (but still in the critical domain):

r Cv'~ - AoAT*(x) -'~- Bcr + Cv'x (11.151) R R

diverges just as in one-component fluids. At AT* << r 2

Cv~

R A° [1-( AT* (x) ) r2 -gcr mZ*(x)] ~ t T 2 + CV'XR (11.152)

which implies that Cv. x exhibits a finite cusp with an infinite slope. Note that the fluctuation- induced, analytic, background contribution Bcr to Cv, x vanishes at the critical point of a binary fluid.

In practice, in many systems the renormalization of the critical exponent a is not observed as the condition AT* << r 2 is often not satisfied in an experimentally accessible temperature range. At infinite dilution as well as upon approaching a temperature minimum on the critical locus the crossover temperature r 2 -~ 0 and the renormalization becomes unobservable. If the concentration dependence of Tc(x ) is steep enough, the renormalization of the critical exponents can be detected even in dilute solutions. In solutions of n-heptane in ethane the renormalization of the isochoric heat-capacity exponent has been clearly observed (166) for a n-heptane mole fraction x = 0.03 with the crossover temperature r 2 between 10 -2 and 10 -3 as shown in Figure 11.8. On the other hand, the isochoric heat capacity at constant composition of mixtures of carbon dioxide and ethane, in the vicinity of the temperature minimum on the critical line, shows the same asymptotic singular behavior as that of the pure components (166).

11.10 ISOMORPHIC DESCRIPTION OF FLUID-FLUID PHASE EQUILIBRIA

A comprehensive review of vapor-liquid equilibria of binary mixtures and their correlation in terms of a scaled equation of state has been presented by Rainwater (167). Building on a procedure proposed by Moldover and Gallagher (168), Rainwater and coworkers have correlated vapor-liquid equilibria data for many mixtures (169-178) including multicomponent mixtures (178-180) based on a scaled equation of state for mixtures with effective critical exponents (see also Section 11.4), referred to as the Leung-Griffiths model (181). Both the 6-term crossover Landau expansion discussed in Section 11.6 and the crossover equation of Kiselev have been extended to mixtures on the basis of the isomorphism principle (132-134,145-149). A crossover procedure has been applied by Kiselev and Friend (23) to cubic equations of state for mixtures. Although their approach maybe useful for chemical engineering practice, it is not consistent with

422

the isomorphism principle and, hence, will be inadequate for representing the critical behavior of mixtures in the near-vicinity of the critical point.

Here we present a simple approach to the description of near-critical, fluid-fluid equilibria which does not require an explicit form of the equation of state. The approach is based on the fact

206

Cv, J/(MOL. K) : (~) |

T e e !

fl :;

I I I i I I I 1 i I I i I 1 ~ 1 I i JO J8 z~5 5 T, ° C

Cv/T, J/(MOL-K 2)

$ e o

2 + + :'0..,, .,

o o + ++ ~° t l~

o o:2: 8 ~ o ~ o + + •

I i I t t - h - - 3 - 2 log AT*

(b)

Figure 11.8 (a) Influence of equilibrium impurities on the behavior of the heat capacity near the critical point (166). (b) Renormalization of the heat-capacity critical exponent for an ethane+n-heptane mixtures in the one-phase region. (1) pure ethane; (2) 0.01; (3) 0.03 mole fractions of n-heptane in ethane.

that in the two-phase region the ordering field hi=0. Therefore, the two scaling densities ¢1, and ¢2 which determine the coexisting densities and concentration according to Equations (11.133) and (11.134), depend only on the non-ordering field h 2.

For a description of the coexistence properties in a wider region, one should add to Equations (11.133) and (11.134) regular terms accounting for the vapor-liquid asymmetry, such as the linear

• 2 diameter proportional to h 2 and higher-order terms h 2 :

423

AR = (AP)cx c = Pc(al*q) I + b 2 (P2 + dllh21 + d, l lh21 z)

Ap2 = (AP2)cxc = A(px) = Pc(a3~01 + b3*~2x c + d2lhz[ + d221h212)

(11.153)

(11.154)

• • • •

where a, - a l R T c , b 2 - R T c , a c - a 3 R T c , b 3 = b 3 R T c , a n d d 1, dll, d 2,d22 are additional system-dependent coefficients. Hence, for the concentrations in the two-phase region

Xc 2 c x = (11.155)

1 + A p / p c

As the critical composition changes along the critical line, x¢ must be specified. For vapor-liquid equilibria of Type I binary fluids the chemical-potential difference/z can be used in a hidden field and in the first approximation

_ dx c x c = Xc(tt ) = x + A/l (11.156)

d/~ c

where ~- is an overall (experimentally assigned) composition. Further one can expand the chemical-potential difference A/z(/z~,T, P) in the vicinity of the critical point

T , P P t , P PI, T

AP (11.157)

with a constraint given by the two-phase equilibrium condition h 1 = 0:

a 1 A/t 1 + a 2 AT + a 3 Aft = 0 (11.158)

The cross-sections of the two-phase domain at constant x - x correspond to the equation

c) + 1 + @ *

= 0 (11.159)

X c Xc(~) = A~t (11.160)

Rrc

which, supplemented with Equations (11.156) and (11.157), gives P - T bubble-dew curves. The

424

cross-sections at constant T and P give the coexisting densities and concentrations along isobars and isotherms, respectively. In the limit of the pure solvent, when ~- -~ 0

a 3 --. O, b 3 --, O, (dx c Id/zc) x c l R T c (11.161)

and the P - x isothermal or T-x isobaric coexistence curves exhibit a so-called "bird beak" (182)" concentrations of both vapor and liquid phases depend on AP or AT linearly, and have the same tangent. Specifically, along the critical isotherm of the pure solvent in first approximation,

AP XL : XG : Xc(/Z) - (11.162) tO0

We note that neither coexisting density nor concentration is to be identified with the order parameter ~0~; both of them are combinations of the two scaling densities ~0~ and 092 and a regular background term. To describe the actual densities and concentrations at coexistence, one should choose an explicit form for the scaling densities ~01 and ~02 in a zero-ordering field. Depending on the range of h 2 to be described, either asymptotic (Equation (11.37)) or crossover (based on minimization of Equation (11.116)) models can be used.

This approach has been applied to describe P - T curves at constant concentrations (183), and to isothermal coexisting densities and concentrations (134) of methane-ethane mixtures. For the P - T cross-sections the asymptotic model for ~01 was used and for describing the isothermal coexistence properties the crossover model has been applied.

One can see from Figures 11.9 and 11.10 that this simple approach yields a good representation of experimental data. The same approach with incorporating the crossover model for calculating ~0~ and fP2 has been applied recently to one of the most challenging binary fluid systems, namely, aqueous solutions of sodium chloride (134), to describe densities and concentrations at vapor-liquid equilibrium, and isochoric specific heat in the one-phase region. The description shows a crossover from asymptotic, Ising-like, critical behavior to classical behavior. As the salt concentration increases, the region of Ising behavior decreases, although pure classical behavior is never attained. Anomalously low solubility of sodium chloride in the vapor phase of the solutions at very low concentrations requires an additional modification of the model to include possible association/dissociation effects.

11.11 CROSSOVER BETWEEN VAPOR-LIQUID AND LIQUID-LIQUID CRITICAL PHENOMENA

Next we consider the case that the critical locus is a continuous line which starts at the vapor- liquid critical point of the solvent and terminates in a consolute critical endpoint as indicated in Figure 11.7. Along the critical line the physical fields in Equations (11.131) and (11.132) exchange their roles: A/z plays the role of hidden field in dilute solutions, but it is the ordering field in incompressible liquid mixtures, while A/z 1, does the same in the opposite way. The variation of the coefficients a~ and a3 along the critical locus results in the transformation of the order parameter (/91 from density to concentration, while the variation of the coefficients b 2 and b 3

is responsible for the transformation of the origin of the nonasymptotic asymmetric corrections.

425

7 o - - 1

a - 2 ~ '~

(3 •

EL 3

2

1 18o l go z6o zlo zlo zgo 2~0 a6o

T, K

(a)

5 . 5

5.0

© EL

4.5

d L.

O3 O3 4 .0 (D k_

CL

3 . 5

3.0

0 - ,3 o -- 4 4 • -- 5

, , , J , , , l I , i . . . . . . . u ~)1 ' 0.01

m o e fract ion of ethane, R~

(b)

Figure 11.9 Pressure-temperature diagram (a)and pressure-concentration diagram (b) of vapor-liquid equilibrium in methane-ethane mixtures. The symbols represent experimental data and the solid curves represent values calculated on the basis of the isomorphism approach ((134) and (183), respectively). (a): (1) 0.075; (2) 0.15; (3) 0.3; (4) 0.5; (5) 0.7; (6) 0.85; (7) 0.95 mole fraction of ethane. The critical points are shown as asterisks. (b): (1) 0.00253; (2) 0.00932; (3) 0.0186; (4) 0.0251; (5) 0.0509 mole fraction of ethane. From reference 134.

Such an exchange occurs in both scaling fields h 1 and h 2 which are controlled by the variation of the coefficients a 1 and a 3 and of b 2 and b 3 along the critical line.

426 65

60

55

© 5O

O__ 45

L. D 35 CO 6O ~ 3O k..

D_ 25

2O

15

10

, - B i s c h o f f & R o s e n b a u e r ( 8 8 ) I~ R o s e n b a u e r & B i s c h p f f ( . 8 7 )

- K h a i b u l l i n & B o r i s o v ( 6 6 ) • - B i s c h o f f ( 9 1 ) o U r u s o v a ( 7 4 ) . . # O l a n d e r & L i a n d e r ( 5 0 ) x - c r i t i c a l p o i n t

, v l l V | l , 1 , v | , i , l , , | v , , , | l ~ , , , , i , 1 1 i , | , , 0 .01 0.1 I 10

m a s s f r a c t i o n o f N a C l , •

(a)

7 0 o - U . . . . . . ( 7 5 ) " " (b) A - K h a i b u l l i n & B o r i s o v ( 6 6 )

6 5 • - B i s c h o f f ( 9 1 ) 6 0 x - c r i t i c a l p o i n t I

0 5 5

.~ 5o 4 5

~_ ~o ~ 3s

~_ ~o 25

2O

15

10 0 2 4 0 4 4 0 6(~0 8 4 0 t 0 0 0 1 2 0 0

Density, kg .m -3

Figure 11.10 Vapor-liquid coexisting concentrations (a) and densities (b) of aqueous NaC1 solutions along critical isotherms from 380 °C to 500 °C (155). The symbols represent experimental data and the solid curves represent values calculated on the basis of the isomorphism approach. The dashed curves represent the critical locus. From reference 134.

The coefficient al = al /RT~ will vary continuously from unity in the one-component limit to zero in the incompressible liquid-mixture limit. On the other hand, the coefficient a 3 = a J R T c

will vary from zero to one. To make this variation explicit, it is convenient to express the coefficients a~ and a 3 in terms of polar coordinates (8,9,184,185):

a 1 - r ( x ) cosq9 (11.163)

a 3 : r ( x ) sin tp (11.164)

427

where the angle ~o varies from ~0 = 0 in the pure solvent limit (x-,0) to ~o = re/2 in the incompressible liquid-mixture limit. The radius r(x) can be approximated as r(x) = 1/RT c. Hence,

a 3

a 1

- x - K 1 = tan ~o (11.165)

The parameter K 1 changes from zero in the pure solvent to infinity in the incompressible liquid- mixture limit. In practice, as the critical line terminates at the critical endpoint, the coefficient a 1 does not reach zero and K 1 remains finite. The parameter K 2 changes from zero in the pure solvent limit to a finite value in the incompressible limit since the slope of the critical temperature line ( d T /dx) is finite. Both parameters K 1 and K2 can be found from the shape of the critical line. To determine K 1 one should have, in addition, a slope of the saturation pressure as a function of temperature or find K 1 from a fit to liquid-liquid equilibria data (163). Figure 11.11 illustrates the striking difference in the thermodynamic properties predicted for the two critical endpoints, upper (vapor-liquid) and lower (liquid-liquid), in mixtures. One can also notice in this figure the difference between the experimental path (p = Pc, x = constant) and the isomorphic path (p = Pc, /~ = constant) in the dilute solution of n-hexane in methane near the upper critical endpoint.

A complete scaled equation of state that would describe simultaneously liquid-vapor, liquid- liquid, and liquid-liquid-vapor equilibrium has not been developed yet. An interesting attempt to describe liquid-liquid-vapor equilibrium near the lower critical endpoint has recently been made by Rainwater (186).

11.12 SUMMARY AND OUTLOOK

Thermodynamic states of fluids and fluid mixtures near critical points are characterized by the presence of long-range fluctuations in the density and/or concentration. As a consequence, the thermodynamic properties of fluids and fluid mixtures satisfy universal scaling laws with universal critical exponents in the near-vicinity of critical points, provided one adopts appropriately chosen thermodynamic variables. The apparent difference between the behavior of actual thermodynamic properties observed experimentally in mixtures with various types of critical points is accounted for by a transformation from the actual experimental variables to the appropriate theoretical variables.

The universal scaling laws are valid in the near-vicinity of critical points where the range of the critical fluctuations is so large that any microscopic length scale can be effectively neglected. However, outside this near-critical region the range of the fluctuations is still appreciable. Therefore, to include the effects of critical fluctuations on the thermodynamic properties in the global phase diagram one needs to extend the theory of critical phenomena to include a crossover from the universal scaled behavior in the near-vicinity of critical points to the classical behavior

428

10 4

~-, 10 3

o ~...q

~ 0 2

©

.~ 101 ,z::: ©

, ~...q

10 0

10 "1 O.

\ . \

\ . \

\ \

\

\ . \ .

N.

" - , . pure CH 4

N \ ,

\ \

\

U C E P -. . . . . . . . . . . . . - \

o.1 1 lO

IT- Tel (K)

L C E P

Figure 11.11 Dimensionless isothermal compressibility RTc (OplOP)r,~ of the methane+n-hexane system near the critical end points. The solid curves correspond to the critical isobar at constant x near the lower critical endpoint (LCEP) and the critical isochore at constant x near the upper critical endpoint (UCEP). The dashed curve corresponds to the isomorphic isochoric path p =/A near the UCEP. The dashed-dotted curve represents the isothermal compressibility along the critical isochore of pure methane near the critical point. From reference 163.

of the thermodynamic properties far away from the critical point where the effects of critical fluctuations are negligibly small.

As discussed in Section 11.6, three major approaches have been developed to deal with crossover critical phenomena. The approach described in the present chapter is based on a renormalized Landau expansion (18,95,101,102). Other approaches are numerical approaches based on the hierarchical theory developed by Reatto, Parola, and Meroni (102-104,120-122) and the phase-space-cell approximation method developed by White and coworkers (105,124,125). Each of these methods has advantages and disadvantages. The advantage of the method presented

429

in Section 11.6 of this chapter is its universal character: it can be applied to various systems with different levels of complexity.

The present chapter deals mostly with simple fluids and fluid mixtures where the crossover phenomenon is, basically, governed by a single parameter, the Ginzburg number. However, in general, the crossover phenomenon will be governed by two independent parameters associated with two different length-scales, namely, a molecular length and an additional mesoscopic length characteristic for the system structure competing with the correlation length of the critical fluctuations. The different roles of these two parameters becomes significant in ionic solutions and in polymer solutions where a nonmonotonic crossover of the susceptibility from asymptotic Ising- like behavior to classical behavior has been observed (114,115,137,141). As the characteristic length-scale in complex fluid may be associated with an additional order parameter, a coupling between different order parameters and possible multicritical phenomena are to be considered (187-189). In particular, the crossover temperature in polymer solutions corresponds to the condition under which the correlation length is equal to the radius of gyration of polymer molecules. The study of crossover critical phenomena in complex fluids is a topic of active current interest (1). One may expect many interesting further developments on this problem in the near future.

Acknowledgments

The research is supported by the Division of Chemical Sciences of the Office of Basic Energy Sciences of the U.S. Department of Energy under Grant DE-FG02-95ER-14509. We are indebted to V. A. Agayan and A. A. Povodyrev for their assistance in preparing the manuscript and to J.C. Rainwater for a careful reading of the manuscript.

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