CHEMICAL THERMODYNAMICS SETCoordinated by Michel Soustelle

This book is part of a set of books which offers advancedstudents successive characterization tool phases, the studyof all types of phase (liquid, gas and solid, pure or multi-component), process engineering, chemical andelectrochemical equilibria, and the properties of surfaces andphases of small sizes. Macroscopic and microscopic modelsare in turn covered with a constant correlation between thetwo scales. Particular attention has been given to the rigor ofmathematical developments.

This fourth volume of the set concerns the study of chemicalequilibria. The conditions of evolution of systems are firstdiscussed through the De Donder affinity method. Themodes of movement and stability under the action ofdisturbances are examined as well as the Gibbs and Duhemphase laws. The azeotropic transformations and thedescription of indifferent states complete this study.

The methods of experimental determination and thecalculation of different values associated with chemicalreactions (enthalpies, entropies, calorific capacities, free andconstant equilibrium enthalpies) lead to complex equilibriumcalculations and their graphical representations in variousforms: pole figures, generalized Ellingham diagrams, binary,tertiary and quaternary diagrams.

Michel Soustelle is a chemical engineer and EmeritusProfessor at Ecole des Mines de Saint-Etienne in France. Hetaught chemical kinetics from postgraduate to Master degreelevel while also carrying out research in this topic.

Z(7ib8e8-CBIGHD(www.iste.co.uk

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Michel S

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Chemical Equilib

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CHEMICAL ENGINEERING SERIES

CHEMICAL THERMODYNAMICS SET

Volume 4

Chemical Equilibria

Michel Soustelle

9781848218673-Case.qxp_Layout 1 17/09/2015 17:17 Page 1

Chemical Equilibria

Chemical Thermodynamics Set coordinated by

Michel Soustelle

Volume 4

Chemical Equilibria

Michel Soustelle

First published 2015 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA

www.iste.co.uk www.wiley.com

© ISTE Ltd 2015 The rights of Michel Soustelle to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2015951442 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-867-3

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Notations and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Chapter 1. Physico-Chemical Transformations and Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Characteristic parameters of physico-chemical transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1. Balance equation of a transformation . . . . . . . . . . . . . . . . . . 1 1.1.2. Values associated with a transformation . . . . . . . . . . . . . . . . 2 1.1.3. Standard values associated with a transformation . . . . . . . . . . . 3 1.1.4. Extent and rate of a transformation . . . . . . . . . . . . . . . . . . . 3

1.2. Entropy production during the course of a transformation in a closed system . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3. Affinity of a transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2. Affinity and characteristic functions . . . . . . . . . . . . . . . . . . . 6 1.3.3. Affinity and chemical potentials . . . . . . . . . . . . . . . . . . . . . 7 1.3.4. Affinity, reaction quotient and activities . . . . . . . . . . . . . . . . 8 1.3.5. Total differential of the affinity in variables Yl, Xm, ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.6. Derivatives of the affinity in relation to the extent and the chemical potentials . . . . . . . . . . . . . . . . . . . . 11

1.4. De Donder’s inequality – direction of the transformations and equilibrium conditions . . . . . . . . . . . . . . . . . . . 12 1.5. Heats of transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.1. Heat of transformation at constant pressure and temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

vi Chemical Equilibria

1.5.2. Heat of transformation at constant volume and temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.3. Variations in the heat of transformation at constant pressure with changing temperature – Kirchhoff relation . . . . . . . . . . . . . . . . . . . 17

1.6. Set of points representing the equilibrium states of a transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.7. Closed systems accommodating multiple reactions . . . . . . . . . . . . 19 1.8. Direction of evolution and equilibrium conditions in an open system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.9. Azeotropic transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 2. Properties of States of Physico-Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1. Laws of displacement of an equilibrium . . . . . . . . . . . . . . . . . . . 25 2.1.1. General form of the displacement laws . . . . . . . . . . . . . . . . . 25 2.1.2. Influence of a temperature disturbance . . . . . . . . . . . . . . . . . 26 2.1.3. Influence of a pressure disturbance . . . . . . . . . . . . . . . . . . . 27 2.1.4. Influence of the addition of a component . . . . . . . . . . . . . . . . 27 2.1.5. Influence of the addition of an inert component . . . . . . . . . . . . 32

2.2. Properties of all the equilibria in a system . . . . . . . . . . . . . . . . . . 33 2.2.1. Property of the set of balance equations of a system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.2. Linear combinations of balance equations . . . . . . . . . . . . . . . 36 2.2.3. Base of the vector space of the balance equations – Jouguet criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3. Phase laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.1. Reminder of Gibbs’ phase rule . . . . . . . . . . . . . . . . . . . . . . 41 2.3.2. Duhem’s phase rule in closed systems . . . . . . . . . . . . . . . . . 42 2.3.3. Comparison between the Gibbs variance and the Duhem variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4. Indifferent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.2. Condition of indifference of a state . . . . . . . . . . . . . . . . . . . 45 2.4.3. Set of indifferent points of equilibrium . . . . . . . . . . . . . . . . . 47 2.4.4. Gibbs–Konovalov theorem . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5. Thermodynamically-equivalent systems . . . . . . . . . . . . . . . . . . . 48 2.6. Stability of equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.6.1. De Donder’s general stability condition . . . . . . . . . . . . . . . . 49 2.6.2. Stability of a system with unilateral variations. . . . . . . . . . . . . 49 2.6.3. Stability of a system with bilateral variations . . . . . . . . . . . . . 51

Contents vii

2.6.4. Conditions of bilateral stability expressed in terms of chemical potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Chapter 3. Molecular Chemical Equilibria . . . . . . . . . . . . . . . . . . 55

3.1. Law of mass action – equilibrium constants . . . . . . . . . . . . . . . . 55 3.1.1. Expression of the law of mass action . . . . . . . . . . . . . . . . . . 55 3.1.2. Different forms of the law of mass action . . . . . . . . . . . . . . . 57 3.1.3. Use of solution models and application of the law of mass action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.1.4. Systems composed of a set of equilibria . . . . . . . . . . . . . . . . 63 3.1.5. Unit of the equilibrium constants . . . . . . . . . . . . . . . . . . . . 64 3.1.6. Variations of the equilibrium constants with temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.1.7. Influence of the choice of reference pressure on the equilibrium constant . . . . . . . . . . . . . . . . . . . . . . 66 3.1.8. Dissociative dissolution of a gas in a solid . . . . . . . . . . . . . . . 67

3.2. Graphical representations of equilibria – pole diagrams . . . . . . . . . 69 3.2.1. Principle of the pole diagram . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.2. Influence of a temperature change on a pole diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.3. Pole diagrams of two reactions in the same family . . . . . . . . . . 72

3.3. Representation of the evolution of an equilibrium with the temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3.1. Diagram in van ’t Hoff coordinates . . . . . . . . . . . . . . . . . . . 73 3.3.2. Ellingham diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4. Binary diagrams for chemical equilibrium . . . . . . . . . . . . . . . . . 88 3.5. Ternary diagrams of chemical equilibria . . . . . . . . . . . . . . . . . . . 89

3.5.1. Mode of representation . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5.2. Molar fractions at equilibrium and initial composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.5.3. Iso-Q curves in perfect solutions . . . . . . . . . . . . . . . . . . . . . 95 3.5.4. Iso-composition curves in perfect solutions . . . . . . . . . . . . . . 98

3.6. Quaternary diagrams of chemical equilibria . . . . . . . . . . . . . . . . 100

Chapter 4. Determination of the Values Associated with Reactions – Equilibrium Calculations . . . . . . . . . . . . . . . . . . 105

4.1. Reminders of a few thermodynamic relations . . . . . . . . . . . . . . . 105 4.2. Enthalpies of reaction – thermochemistry . . . . . . . . . . . . . . . . . . 110

4.2.1. Experimental determination of the reaction enthalpies by calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2.2. Calculation of the standard enthalpy at another temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

viii Chemical Equilibria

4.2.3. Influence of the pressure on the reaction enthalpies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2.4. Determination of the reaction enthalpies by calculation on the basis of other thermodynamic data . . . . . . . . . . . . 112 4.2.5. Enthalpies of formation . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2.6. Enthalpies of combustion . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.7. Dissociation energy, bond energy and enthalpies of formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3. Reaction entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.3.1. Planck’s hypothesis – calculation of the calorimetric entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.3.2. Spectroscopic determination of entropies – absolute entropies . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.3.3. The third law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.4. Specific heat capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.4.1. Calorimetric measurements of the specific heat capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.4.2. Spectral measurements of the specific heat capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.5. Experimental determination of the equilibrium constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.6. Calculation of the equilibrium constants on the basis of other thermodynamic data . . . . . . . . . . . . . . 136

4.6.1. Calculation of the equilibrium constants – method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.6.2. Calculation of the equilibrium constants – method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.6.3. Calculation of the equilibrium constants – method 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.6.4. Calculation of the equilibrium constants – method 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.6.5. Calculation of the equilibrium constants – method 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.7. Determination of the equilibrium constants on the basis of spectral data and statistical thermodynamics . . . . . . . . . . . 140 4.8. Thermodynamic tables and databanks . . . . . . . . . . . . . . . . . . . . 140 4.9. Estimation of thermodynamic data . . . . . . . . . . . . . . . . . . . . . . 142

4.9.1. Method of evaluating the energies of dissociation by spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.9.2. Group contribution methods . . . . . . . . . . . . . . . . . . . . . . . 143

4.10. Thermodynamic calculations for complex systems . . . . . . . . . . . . 146 4.10.1. Definition of the system . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.10.2. Output mode – graphical representation . . . . . . . . . . . . . . . . 147

Contents ix

4.10.3. Calculation method based on the equilibrium constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.10.4. Method of minimization of the Gibbs energy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Preface

This book – an in-depth examination of chemical thermodynamics – is written for an audience of engineering undergraduates and Masters students in the disciplines of chemistry, physical chemistry, process engineering, materials, etc., and doctoral candidates in those disciplines. It will also be useful for researchers at fundamental- or applied-research labs dealing with issues in thermodynamics during the course of their work.

These audiences will, during their undergraduate degree, have received a grounding in general thermodynamics and chemical thermodynamics, which all science students are normally taught, and will therefore be familiar with the fundamentals, such as the principles and the basic functions of thermodynamics, and the handling of phase and chemical equilibrium states, essentially in an ideal medium, usually for fluid phases, in the absence of electrical fields and independently of any surface effects.

This set of books, which is positioned somewhere between an introduction to the subject and a research paper, offers a detailed examination of chemical thermodynamics that is necessary in the various disciplines relating to chemical or material sciences. It lays the groundwork necessary for students to go and read specialized publications in their different areas. It constitutes a series of reference books that touch on all of the concepts and methods. It discusses both scales of modeling: microscopic (by statistical thermodynamics) and macroscopic, and illustrates the link between them at every step. These models are then used in the study of solid, liquid and gaseous phases, either of pure substances or comprising several components.

xii Chemical Equilibria

The various volumes of the set will deal with the following topics:

– phase modeling tools: application to gases;

– modeling of liquid phases;

– modeling of solid phases;

– chemical equilibrium states;

– phase transformations;

– electrolytes and electrochemical thermodynamics;

– thermodynamics of surfaces, capillary systems and phases of small dimensions.

Appendices in each volume give an introduction to the general methods used in the text, and offer additional mathematical tools and some data.

This series owes a great deal to the feedback, comments and questions from all my students are the Ecole nationale supérieure des mines (engineering school) in Saint Etienne who have “endured” my lecturing in thermodynamics for many years. I am very grateful to them, and also thank them for their stimulating attitude. This work is also the fruit of numerous discussions with colleagues who teach thermodynamics in the largest establishments – particularly in the context of the group “Thermodic”, founded by Marc Onillion. My thanks go to all of them for their contributions and conviviality.

This fourth instalment in the series is devoted to the study of chemical equilibria.

Chapter 1 describes transformations and chemical equilibria using Donder’s affinity method. Equilibrium conditions are examined in enclosed media, where one or more equilibrium states are present, and in open systems. The chapter closes with a general look at azeotropic transformations.

Chapter 2 is a general study of the properties of physical and chemical equilibria. Thus, we examine the laws of displacement of equilibria under the influence of various disturbances, and the Gibbs and Duhem phase laws. Following a general study of indifferent states, the chapter goes on to analyze the conditions of stability of equilibria.

Preface xiii

Chapter 3 discusses the different aspects of the law of mass action and the equilibrium constants associated therewith. A number of graphical representations used in the study of chemical equilibria are presented. In turn, we examine pole diagrams and equilibrium representations with temperature, with the generalization of Ellingham diagrams. The chapter ends with a presentation of binary, ternary and quaternary diagrams of chemical equilibria.

The fourth and final chapter in this volume is given over to the determination, both experimentally and by computation, of the values of the parameters associated with chemical reactions. Thermochemistry for enthalpy, the determination of the entropies, specific heat capacities and Gibbs energy values ultimately lead to the determination of the equilibrium constants. Analysis of the different thermodynamic tables and methods for estimating unknown values enable us to proceed to the practical application and finally computation of the equilibria by the equilibrium constant method and minimization of Gibbs energy.

Michel SOUSTELLE Saint-Vallier,

September 2015

Notations and Symbols

A : area of a surface or an interface. A: affinity. A, B,: components of a mixture. C: concentration. Ci: molar concentration (or molarity) of component i. CV, CP: specific heat capacity at constant volume and constant pressure, respectively. c: number of independent components. deS: entropy exchange with the outside environment. diS: internal entropy production. dω: elementary volume. E: energy of the system. Eb: balance equation.

E : mean total energy of an element in the canonical ensemble.

EC: total energy of the canonical ensemble. EI: potential energy due to interactions. Ej: energy of an element j of the canonical ensemble. E0: standard emf of an electrochemical cell. Ep: set of variables with p intensive variables chosen to define a

system. F: Helmholtz energy.

xvi Chemical Equilibria

mixmF : molar excess Helmholtz energy. xsiF : partial molar excess Helmholtz energy of component i. mixiF : partial molar mixing Helmholtz energy of component i.

iF : free energy, partial molar Helmholtz energy of component i. Fm: molar Helmholtz energy. fi: fugacity of the component i in a gaseous mixture.

0if : molar Helmholtz energy of pure component i.

f0 or 0if : fugacity of a pure gas i.

F: Faraday (unit). xsmG : excess Gibbs energy.

G~ : electrochemical Gibbs energy.

xsiG : partial excess molar Gibbs energy of component i.

G, iG , [G]: Gibbs energy, partial molar Gibbs energy of i, generalized Gibbs energy.

mG : molar Gibbs energy. mixmG : molar Gibbs energy of mixing.

g: degeneracy coefficient or multiplicity coefficient or statistical weight.

0ig : molar Gibbs energy of pure component i.

gi: coefficient of multiplicity of state i. g*: molar Gibbs energy of gas i at pressure of 1 atmosphere in a mixture.

0TH : standard molar enthalpy of formation at temperature T.

H , iH : enthalpy, partial molar enthalpy of i.

H: Hamiltonian. H: magnetic field.

xsmH : molar excess enthalpy. mixmH : molar mixing enthalpy.

Notations and Symbols xvii

xsiH : partial excess molar enthalpy of component i. mixiH : partial molar mixing enthalpy of component i.

h: Planck’s constant. 0ih : molar enthalpy of pure component i.

I, I1, I2, I3: moments of inertia. II: integral of configuration of the canonical partition function of translation.

iJ : partial molar value of J relative to component i. mixiJ : mixing value of J relative to component i. mixiJ : partial molar mixing value of J relative to component i. *iJ : value of J relative to component i in a perfect solution. *iJ : partial molar value of J relative to component i in a perfect

solution. 0ij : value of J for the pure component i in the same state of

segregation. j: rotational quantum number.

( )pjiK E, : thermodynamic coefficient associated with the set of variables Ep. Xj is its definition variable and Yi its definition function.

)Tr(iK : constant of change of equilibrium for phase transition Tr for

component i. )(cK : equilibrium constant relative to concentrations.

)( fK : equilibrium constant relative to fugacity values.

)(PK : equilibrium constant relative to partial pressure values.

K: equilibrium constant. kB: Boltzmann’s constant. Lt: latent heat accompanying transformation t. M: molar mass. m: total mass.

xviii Chemical Equilibria

mi: mass of component i. N: number of components of a solution or a mixture of gases or involved in a reaction or number of molecules of a collection. Na: Avogadro’s number. NA: number of molecules of component A. NC: number of elements in the canonical ensemble. ni: number of objects i in the system with energy εi or number of moles of component i. n: translational quantum number or total number of moles in a solution or a mixture. n(α): total number of moles in a phase α. P: pressure of a gas.

)subl(iP : sublimating vapor pressure of component i.

)vap(iP 0iP : saturating vapor pressure of component i.

Pc: critical pressure. Pi: partial pressure of component i. p: number of external physical variables. Q: heat involved. QP: heat of transformation at constant pressure. Q(P): reaction quotient in terms of partial pressures. Q(I): reaction quotient of a transformation in reference state (I). QV: transformation heat at constant volume. ℜ : reaction rate. R: perfect gas constant.

Ar : radius of the ionic atmosphere.

r0: minimum energy gap between two molecules. mixmS : molar mixing entropy. xsmS : partial excess molar entropy of component i. mixmS& : partial mixing molar entropy of component i.

S: oversaturation of a solution. Si, iS : entropy or partial molar entropy of i.

Notations and Symbols xix

S% : electrochemical entropy. xsmS : excess molar entropy. 0is : molar entropy of pure component i.

T: temperature mixcT : critical temperature of the mixture.

Tc: critical temperature. TF: Fermi temperature. Ti(Eb): boiling point of pure i. Ti(F): fusion (melting) point of pure i. T(sub): sublimation temperature. T(vap): vaporization temperature. T(Δα): phase-change temperature.

xsmU : excess molar internal energy. mixmU : mixing molar internal energy. xsiU : excess partial molar internal energy of component i. mixiU : partial mixing molar internal energy of component i.

U, iU : internal energy, partial molar internal energy of i.

U~ : internal electrochemical energy.

Um: molar internal energy. UR: crosslink internal energy.

0iu : molar internal energy of pure component i.

V, iV : volume, partial molar volume of i.

Vc: critical volume. VG: Gibbs variance. Vm: molar volume. vD: Duhem variance.

0iv : molar volume of pure component i.

v: quantum vibration number. vm: molecular volume.

xx Chemical Equilibria

vM: molar volume of solid at melting point. wi: mass fraction of component i. wij: energy of exchange between atoms i and j.

( )αkx : molar fraction of component k in phase α.

x, y, z: coordinates of a point in space. xi: molar fraction of component i in a solution. : mean value of y. Yi and Xi: conjugal intensive and extensive values. yi: molar fraction of component i in a gaseous phase. ZAB: molecular partition function of interaction between molecules. ZC: canonical partition function. ZC(A): canonical partition function of component A. ZC(I): canonical partition function of interaction. ZC(t): canonical partition function of translation. z: molecular partition function. ze: electron molecular partition function or electrovalence of ion i. zint: contribution of internal motions to the molecular partition function. zn: molecular partition function of nuclei. zpf: molecular partition function of a perfect gas. zr: rotational molecular partition function. zt: translational molecular partition function. zt(pf): translational molecular partition function of a perfect gas. zv: vibrational molecular partition function. α: Lagrange multiplier relating to the number of objects in a

collection. β: Lagrange multiplier relating to the energy of the objects in a collection.

( )PΓ E : characteristic function with the set PE as canonical variables.