Chemical Kinetics and Transport

Chemical Kinetics and Transport

Peter C. Jordan Brandeis University

Waltham, MasSllchusetts

Plenum Press . New York and London

Library of Congress Cataloging in Publication Data

Jordan, Peter C Chemical kinetics and transport.

Includes bibliographical references and index. 1. Chemical reaction, Rate of. 2. Transport theory. I. Title.

QD501.J75731979 541'.39 ISBN-13: 978-1-4615-9100-9 e-ISBN-13: 978-1-4615-9098-9 DOT: 10.1007/978-1-4615-9098-9

First Printing - March 1979 Second Printing - January 1980 Third Printing - September 1981

© 1979 Plenum Press, New York

Softcover reprint of the hardcover 1 st edition 1979

A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013

All righ ts reserved

78-20999

No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming,

recording, or otherwise, withou t written permission from the Publisher

Preface

This book began as a program of self-education. While teaching under­graduate physical chemistry, I became progressively more dissatisfied with my approach to chemical kinetics. The solution to my problem was to write a detailed set of lecture notes which covered more material, in greater depth, than could be presented in undergraduate physical chemistry. These notes are the foundation upon which this book is built.

My background led me to view chemical kinetics as closely related to transport phenomena. While the relationship of these topics is well known, it is often ignored, except for brief discussions of irreversible thermody­namics. In fact, the physics underlying such apparently dissimilar processes as reaction and energy transfer is not so very different. The intermolecular potential is to transport what the potential-energy surface is to reactivity.

Instead of beginning the sections devoted to chemical kinetics with a discussion of various theories, I have chosen to treat phenomenology and mechanism first. In this way the essential unity of kinetic arguments, whether applied to gas-phase or solution-phase reaction, can be emphasized. Theories of rate constants and of chemical dynamics are treated last, so that their strengths and weaknesses may be more clearly highlighted.

The book is designed for students in their senior year or first year of graduate school. A year of undergraduate physical chemistry is essential preparation. While further exposure to chemical thermodynamics, statistical thermodynamics, or molecular spectroscopy is an asset, it is not necessary.

As a result of my perspective, the choice of topics and organization of the material differs from that of other kinetics texts. The introductory chapter, on equilibrium kinetic theory, is mainly review. The next two chapters, treating transport in both gas and solution, introduce many

v

vi Preface

chemically useful concepts. Chapter 4 focuses on phenomenology-the experimental methods used to determine a rate law, the meaning of rate laws and rate constants. The next three chapters are devoted to mech­anisms of thermal and photochemical reaction, first in systems that are stable to perturbation and then in those for which explosion or oscillation is possible. Much of this material is in the standard repertory. However, the sections devoted to heterogeneous catalysis, infrared laser photo­chemistry, and oscillating chemical reactions incorporate the results of recent, occasionally speculative, research. Chapter 8 treats single-collision chemistry-first, to relate the reaction cross section and the rate constant) and then, to demonstrate how mechanistic information can be deduced from scattering data.

The first of the theoretical chapters (Chapter 9) treats approaches to the calculation of thermal rate constants. The material is familiar-activated complex theory, RRKM theory of unimolecular reaction, Debye theory of diffusion-limited reaction-and emphasizes how much information can be correlated on the basis of quite limited models. In the final chaptpr, the dynamics of single-collision chemistry is analyzed within a highly simplified framework; the model, based on classical mechanics, collinear collision geometries, and naive potential-energy surfaces, illuminates many of the features that account for chemical reactivity.

I have tried to present both the arguments and the derivatioflS."so that they can be readily followed. However, some steps have been intentionally omitted. I believe that unless a student works through the material indi­vidually, the subject matter cannot be mastered. Examples have been chosen to illustrate general principles, not to be exhaustive. The problems, which are an indispensable adjunct to each chapter, provide further illustrations.

While the arrangement of the material is not traditional, each chapter, and many of the sections, are, to a great degree, self-contained. Topics may, if the instructor so desires, be presented in any order.

I am indebted to many individuals for their assistance. Foremost among these are my colleagues, Michael Henchman and Kenneth Kustin, who clarified my ideas and suggested many of their own; without their help this book would never have been written. The reviewers, Lawrence J. Parkhurst and Robert E. Wyatt, made many helpful and perceptive sug­gestions. Students at Brandeis University have rendered considerable assis­tance, especially in pointing out ambiguous or unclear passages. Finally I am grateful to Evangeline P. Goodwin for her rapid and accurate typing and to Virginia G. Steel for her excellent draftsmanship.

Peter C. Jordan

Contents

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xi

1. Kinetic Theory of Gases-Equilibrium

1.1. Introduction 1 1.2. The Maxwell-Boltzmann Distribution of Velocities 2 1.3. Determination of Pressure 6 1.4. Properties of the Maxwell Distribution 9 1.5. The Molecular Flux and Effusion of Gases . 13 Problems 14 General References 15

2. Kinetic Theory of Gases-Transport

2.1. Introduction . . . . 17 2.2. Mean Free Path . . . . . . . 18 2.3. Collision Frequency. . . . . . 20 2.4. Macroscopic Equations of Transport 24 2.5. Solution of the Transport Equations 26 2.6. Kinetic Theory of Transport and Postulate of Local Equilibrium 29 2.7. Simplified Transport Theory: Dilute Hard-Sphere Gas. 30 2.8. Comparison with Experiment ........... 35 Appendix. Coordinate Transformation by Jacobian Method. 44 Problems . . . . . 45 General References 47

3. Electrolytic Conduction and Diffusion

3.1. Introduction . . . . . . . . . . 49 50 3.2. Ionic Conduction in Solution-a Review.

vii

viii

3.3. Nernst-Einstein Relation 3.4. Electrolyte Diffusion 3.5. Stokes' Law and the Microscopic Interpretation of AD

3.6. The Mobility of H+ and OH- in Water 3.7. The Concentration Dependence of A 3.8. The Wien Effects Problems ..... General References

Contents

53 56 57 61 62 65 67 69

4. Determination of Rate Laws

4.1. Introduction . . . . . . 71 4.2. Stoichiometry, Rate Law, and Mechanism . . . . . . . 72 4.3. Elementary Rate Laws and the Principle of Mass Action. 73 4.4. Reaction Order and Molecularity . . . . . . . . 77 4.5. The Principle of Detailed Balance ....... 78 4.6. Experimental Methods for Determining Rate Laws 80 4.7. Integrated Rate Laws-First Order 85 4.8. Isolation Methods . . . . . . . 87 4.9. Relaxation Methods ...... 89 4.1 o. Determination of Rate Law from Data-Two Examples 93 4.11. Temperature Dependence of Rate Constants 98 Appendix. Linear Least-Squares Analysis 101 Problems . . . . . 103 General References 11 0

5. Stationary State Mechanisms

5.1. Introduction . . . . . . 5.2. Consecutive Reactions-First Order. . . 5.3. Consecutive Reactions-Arbitrary Order. 5.4. Unimolecular Decomposition-Gases . . 5.5. Radical Recombination-Gases 5.6. Complex Mechanisms-Gaseous Chain Reactions. 5.7. Simple Mechanisms-Solution . 5.8. Complex Mechanisms-Solution

113 114 120 124 127 131 135 139

5.9. Homogeneous Catalysis 142 5.10. Enzyme Catalysis 145 5.11. Heterogeneous Catalysis 147 Appendix A. Matrix Solution to the Coupled Rate Equation (5.2) 153 Appendix B. Approximations to "+ and ,,_ 156 Problems . . . . . 157 General References 163

Contents

6. Photochemistry

6.1. Introduction 6.2. Measurement of Intensity 6.3. Spectroscopic Review 6.4. Primary Processes 6.5. Secondary Processes 6.6. Chemiluminescence. 6.7. Orbital Symmetry Correlations: Woodward-Hoffman Rules 6.8. Laser Activation . . . . . . . . . 6.9. Laser-Controlled Isotope Separation Problems ..... General References

7. N onstationary State Mechanisms

7.1. Introduction . . . . . . . . 7.2. Thermal Explosion . . . .. 7.3. Population Explosion-Autocatalysis 7.4. The Lotka Problem: Chemical Oscillations. 7.5. The Belousov-Zhabotinskii Reaction ... 7.6. Other Oscillating Systems . . . . . . . . 7.7. Multiple Stationary States: The NO.-N.04 Reaction Problems ..... General References

8. Single-Collision Chemistry

8.1. Introduction . . . . . 8.2. Reaction Cross Section: Relation to the Rate Constant 8.3. Scattering Measurements-Molecular Beams . 8.4. Reaction Cross Section: Hard-Sphere Model . . 8.5. Reaction Cross Section: Ion-Molecule Systems 8.6. Reaction Cross Section: Atom-Molecule Systems 8.7. Product Distribution Analysis . . . . . . . . 8.8. Direct, Forward-Peaked-The Ar+ + D. System 8.9. Recoil-The D + Cl. System ..... . 8.10. Collision Complexes-The 0 + Br. System 8.11. State Selection Experiments Problems ..... General References

ix

165 166 167 169 178 184 185 188 192 195 199

201 202 205 209 214 220 226 229 232

233 234 237 240 244 248 252 254 257 259 261 263 267

x Contents

9. Theories of Reaction Rates

9.1. Introduction . . . . . 269 9.2. Potential-Energy Surfaces . 270 9.3. Activated Complex Theory. 276 9.4. Model Calculations of k.(T) 281 9.5. The Kinetic Isotope Effect . 287 9.6. Unimolecular Reaction-General Features 291 9.7. Unimolecular Reaction-RRKM Theory 293 9.8. Critique of RRKM Theory-Isomerization of Methyl Isocyanide 299 9.9. Thermodynamic Analogy 302 9.10. Gas-Phase Reactions 304 9.11. Solution Reactions . . . 306 9.12. Primary Salt Effect . . . 307 9.13. Diffusion-Limited Kinetics-Debye Theory 309 Appendix A. Statistical Thermodynamics . . . . 316 Appen4ix B. Evaluation of koo in RRKM Theory 318 Problems . . . . . 319 General References 323

10. A Dynamical Model for Chemical Reaction

10.1. Introduction . . . . . . . . . . . 325 10.2. Kinetic Energy of a Triatomic System . . 326 10.3. Model for Inelastic Collision . . . . . . 328 lOA. Effects Due to More Complex Potential-Energy Surfaces . 333 10.5. Model for Thermoneutral Chemical Reaction 335 10.6. Exoergic Reactions . . . 339 10.7. Endoergic Reactions 342 10.8. Noncollinear Geometries 345 10.9. Realistic Scattering Calculations 348 Problems . . . . . 351 General References 353

Physical Constants 355

Index . . . . . . 357

Notation

Those symbols which are only used once in the text do not appear in this listing. Familiar mathematical functions are also not included, nor are quantities that are simply intermediates in a derivation. Capital letters are listed before lowercase letters. Greek letters are listed at the end.

A Arrhenius A-factor, reaction affinity, energy ac­ceptor in excitation transfer reaction

ABC Product of principal moments of inertia of a nonlinear molecule

Ci Normality of the ith component Cv Constant volume heat capacity D Energy donor in excitation transfer reaction D Dissociation energy for dynamical model of

Chapter 10 Di Diffusion coefficient of the ith component E Energy

Ea Arrhenius activation energy E Electric field F Faraday's number

FA(C) = FA(u, v, w) Three-dimensional velocity distribution function for molecules of species A

FA(C) Speed distribution function for molecules of species A

L1Go+ Gibbs free energy of activation G{J,(u) x component of molecular flux as a function of

x component of velocity

xi

xii Notation

iJHo+ Enthalpy of activation J Ionic strength, moment of inertia of a linear

molecule h Intensity of a beam of molecules of species A

J(v) Normalized fluorescence intensity at frequency v J(J.., x) Position-dependent light intensity at wavelength J.. JA(x) Position-dependent intensity of a beam of mole­

cules of species A J Rotational quantum number

Ji Diffusive flux of the ith component J q Heat flux

K(T), K(T, p), K(T, p, I) Equilibrium constants K+ Equilibrium constant for activation Km Michaelis constant M Total mass

MA Molecular weight of species A No Avogadro's number NA Number of particles of species .A Nt Hypothetical number of activated complexes

N( E,,) Density of states for the ath degree(s) of freedom of an activated molecule with energy E~ in the specified degree(s) of freedom

P x Momentum flux in x direction P( E) Probability density for states of energy E

Q Enthalpy release per mole of reaction QA Molar partition function for species A Q+ Reduced molar partition function for the ac­

tivated complex Q(E, i,j), Q(E), Q(E) Reaction cross sections as functions of molecular

states i, j, collision energy E, or molar collision energy E

82Q( E, e, 1> )/8e 81> Differential cross section for molecules with collision energy E scattering into solid angle e, 1>

R Gas constant, total reaction rate Rj (Rr) Rate of forward (reverse) reaction

Si The ith singlet state of an electronic manifold iJ So t Entropy of activation

T Absolute temperature Ti The ith triplet state of an electronic manifold

V A Potential energy of a molecule of species A

Notation xiii

V Volume, limiting velocity of an enzyme-catalyzed reaction

Vcoll Collision volume of crossed molecular beams L1 vot Volume of activation X, Y Relative coordinates for collinear motion in the

dynamical model of Chapter 10 Yi Symbol denoting the ith reactant

[Yd Concentration of the ith reactant ZA Total collision frequency for molecules of spe­

cies A ai Activity of the ith component at Hypothetical activity of the activated complex b Impact parameter C Molecular speed

Co Velocity of light Ci Concentration of the ith component Ci Velocity of the ith particle

CM Velocity of the center of mass tv Constant volume heat capacity per molecule

d (dAB) Hard-sphere diameter (for an A-B collision) e Electronic charge, energy per mole

erf(x) Error function fA (c) Three-dimensional velocity distribution for mole­

cules of species A as a function of speed f Frictional force on a particle

g~ Degeneracy factor for the ath degree(s) of freedom

h Planck's constant ji Current density attributable to the ith ionic

species k Boltzmann's constant, rate constant for reaction

ken), k±n Rate constants for reaction kd Rate constant for irreversible decomposition

kdiff (kdiss) Rate constant for association (dissociation) in a diffusion-limited reaction

koo (ko) Limiting rate constant at high (low) pressure in unimolecular decomposition

kef:) Energy-dependent rate constant for reaction me Electron mass mi Mass of the ith particle

xiv Notation

ni Number density of the ith component n+ Hypothetical number density of the activated

complex P Pressure

Pi Partial pressure of the ith component p(x) Position-dependent scattering probability for

molecules in a beam pCb, E, i, j), pCb, E) Reaction probability as functions of molecular

states i, j, collision energy E, and impact param­eter b

qA Molecular partition function for species A q+ Reduced molecular partition function for the

activated complex qrx Partition function for the ath independent

degree(s) of freedom r Distance in spherical polar coordinates

r rx Rate of the ath step in a reaction sequence , i Position of the ith particle

'M Position of the center of mass t Time u x component of velocity, mobility in an electric

field u(r) Intermolecular potential function

v y component of velocity, volume per mole, reaction velocity in an enzyme-catalyzed reac­tion, vibrational quantum number

v Drift velocity in an external field w z component of velocity

x, y Position components for the equivalent particle in the dyIiamical model of Chapter 10

x, y, z Position components in Cartesian coordinates Zi Charge of the ith ionic species

ZA(C) Speed-dependent collision frequency per mole­

rex) L1

cule of species A Gamma function Reaction exo- or endoergicity for dynamical model of Chapter 10 Equivalent conductance of the ith electrolyte (at infinite dilution)

'l' Time- and position-dependent wave function

Notation xv

ai(A) Molar absorption coefficient of the ith com­ponent at wavelength A

fJ IjkT, angle characterizing mass ratio in dynamical model of Chapter 10

Y Thermal coupling constant for dissipation of heat to surroundings

Yi Activity coefficient of the ith component y+ Hypothetical activity coefficient for the activated

complex Y(A) Light attenuation coefficient per unit length at

wavelength A € Energy per molecule, dielectric constant, energy

parameter in Lennard-Jones potential €+ Classical barrier height per molecule in activated

complex theory €.j(v) Molar extinction coefficient for the ith com­

ponent at frequency v 1] Shear viscosity () Polar angular velocity III spherical polar co­

ordinates, polar angle in spherical polar co­ordinates, angle of deflection, angle charac­terizing energy partitioning in dynamical model of Chapter 10

()i Fractional surface coverage attributable to ad­sorption of the ith component

" Thermal conductivity, transmission coefficient "i Decay constant for the ith relaxation process A Wavelength, decay constant for an independent

kinetic process AA Mean free path of species A

Ai (AiO) Equivalent conductance of the ith ionic species (at infinite dilution)

fli Chemical potential of the ith component fli Electrochemical potential of the ith component

flij Reduced mass for relative motion of particles i and j

v Frequency v A Number of atoms of type A in a compound Vi Stoichiometric coefficient of the ith reactant,

vibrational frequency of the ith normal mode

XVI Notation

v+ Frequency defined by curvature at the peak of the reaction profile

~(t) Time-dependent progress variable ~(E) Depletion factor in the theory of unimolecular

reaction ei Mass density of the ith component

ei(E) Density of quantum levels in electronic state i at energy E

eiu) One-dimensional velocity distribution a Specific conductivity, hard-sphere diameter for

a collision, length parameter in Lennard-Jones potential, symmetry number of a molecule

r(A) Intrinsic lifetime of a photoexcited state cp Azimuthal angular velocity in spherical polar

coordinates, azimuthal angle in spherical polar coordinates, angle of deflection

CP;. Quantum yield at wavelength A CPI (cpQ) Fluorescence (phosphorescence) quantum effi­

ciency X Polar angle in spherical polar coordinates 1p Electric potential, position-dependent wave func­

tion w Oscillation frequency of a kinetic process, azi­

muthal angle in spherical polar coordinates