Phase Transformations

CHEMICAL ENGINEERING SERIES

CHEMICAL THERMODYNAMICS SET

Volume 5

PhaseTransformations

Michel Soustelle

Chemical Thermodynamics Set

coordinated by Michel Soustelle

Volume 5

Phase Transformations

Michel Soustelle

iSlE WILEY

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:

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© ISTE Ltd 2015

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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: 2016933308

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-868-0

Contents

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

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

Chapter 1. Phase Transformations of Pure Substances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Standard state: standard conditions of a transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Classification and general properties of phase transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1. First-order transformations and the Clapeyron relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2. Second-order transformations . . . . . . . . . . . . . . . . . . . . . . 7

1.3. Liquid–vapor transformations and equilibrium states . . . . . . . . . . . 16 1.3.1. Method of two equations of state, using the Clapeyron equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.2. Gibbs energy and fugacity method . . . . . . . . . . . . . . . . . . . . 18 1.3.3. Unique equation of state method . . . . . . . . . . . . . . . . . . . . . 19 1.3.4. The region of the critical point and spinodal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.5. Microscopic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.6. Liquid–vapor equilibrium in the presence of an inert gas . . . . . . 26

1.4. Solid–vapor transformations and equilibriums . . . . . . . . . . . . . . . 28 1.4.1. Macroscopic treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.2. Microscopic treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.5. Transformations and solid–liquid equilibria . . . . . . . . . . . . . . . . . 30 1.5.1. Macroscopic treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.5.2. Microscopic treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vi Phase Transformations

1.6. Diagram for the pure substance and properties of the triple point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.7. Allotropic and polymorphic varieties of a solid . . . . . . . . . . . . . . 35

1.7.1. Enantiotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.7.2. Monotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.7.3. Transition from enantiotropy to monotropy and vice versa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.8. Mesomorphic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Chapter 2. Properties of Equilibria Between Binary Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.1. Classification of equilibria between the phases of binary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2. General properties of two-phase binary systems . . . . . . . . . . . . . . 45

2.2.1. Equilibrium conditions for two-phase binary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2.2. Conditions of evolution of a two-phase binary system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3. Graphical representation of two-phase binary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.1. Gibbs energy graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.2. Phase diagram in the mono- and bi-phase zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3.3. Isobaric cooling curves . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.4. Isobaric representation of three-phase binary systems . . . . . . . . . . . 66 2.4.1. Gibbs energy curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.4.2. Isobaric phase diagram in tri-phase regions . . . . . . . . . . . . . . 68 2.4.3. Isobaric cooling curves with tri-phase zones . . . . . . . . . . . . . . 70

2.5. Isothermal phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.6. Composition/composition curves . . . . . . . . . . . . . . . . . . . . . . . 73 2.7. Activity of the components and consequences of Raoult’s and Henry’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 3. Equilibria Between Binary Condensed Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.1. Equilibria between phases of the same nature: liquid–liquid or solid–solid . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.1.1. Thermodynamics of demixing . . . . . . . . . . . . . . . . . . . . . . 76 3.1.2. Demixing in the case of low reciprocal solubilities . . . . . . . . . . 79 3.1.3. Demixing of strictly-regular solutions . . . . . . . . . . . . . . . . . . 81

3.2. Liquid–solid systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Contents vii

3.2.1. Thermodynamics of the equilibria between a liquid phase and a solid phase . . . . . . . . . . . . . . . . . . . . 86 3.2.2. Isobaric phase diagrams of equilibria between a solid and a liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.2.3. Solidus and liquidus in the vicinity of the pure substance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.3. Equilibria between two solids with two polymorphic varieties of the solid . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.4. Applications of solid–liquid equilibria . . . . . . . . . . . . . . . . . . . . 102

3.4.1. Solubility of a solid in a liquid: Schröder–Le Châtelier law . . . . . . . . . . . . . . . . . . . . . . . . 102 3.4.2. Determination of molar mass by cryometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.5. Membrane equilibria – osmotic pressure . . . . . . . . . . . . . . . . . . 106 3.5.1. Thermodynamics of osmotic pressure . . . . . . . . . . . . . . . . . . 107 3.5.2. Osmotic pressure of infinitely-dilute solutions: the Van ‘t the Hoff law . . . . . . . . . . . . . . . . . . . . . . . . 109 3.5.3. Application of osmotic pressure to the determination of the molar mass of polymers . . . . . . . . . . . . . . . 110 3.5.4. Osmotic pressure of strictly-regular solutions . . . . . . . . . . . . . 111 3.5.5. Osmotic pressure and the osmotic coefficient . . . . . . . . . . . . . 112

Chapter 4. Equilibria Between Binary Fluid Phases . . . . . . . . . . . 113

4.1. Thermodynamics of liquid–vapor equilibrium in a binary system . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2. Liquid–vapor equilibrium in perfect solutions far from the critical conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2.1. Partial pressures and total pressure of a perfect solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.2.2. Isothermal diagram of a perfect solution . . . . . . . . . . . . . . . . 119 4.2.3. Isobaric diagram of a perfect solution . . . . . . . . . . . . . . . . . . 120 4.2.4. Phase composition curve . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.3. Liquid–gas equilibria in ideal dilute solutions . . . . . . . . . . . . . . . 122 4.4. Diagrams of the liquid–vapor equilibria in real solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.4.1. Total miscibility in the liquid phase . . . . . . . . . . . . . . . . . . . 125 4.4.2. Partial miscibility in the liquid phase, heteroazeotropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.5. Thermodynamics of liquid–vapor azeotropy . . . . . . . . . . . . . . . . 129 4.5.1. Relation between the pressure of the azeotrope and the activity coefficients of the liquid phase at the azeotropic composition . . . . . . . . . . . . . . . . 129

viii Phase Transformations

4.5.2. Relation between the activity coefficient and the temperature of the azeotrope . . . . . . . . . . . . . . . . . . . . . . 130

4.6. Liquid–vapor equilibria and models of solutions . . . . . . . . . . . . . . 132 4.6.1. Liquid–vapor equilibria in strictly-regular solutions . . . . . . . . . 132 4.6.2. Liquid–vapor equilibrium in associated solutions . . . . . . . . . . . 137

4.7. Liquid–vapor equilibria in the critical region . . . . . . . . . . . . . . . . 140 4.8. Applications of liquid–vapor equilibria . . . . . . . . . . . . . . . . . . . 143

4.8.1. Solubility of a gas in a liquid . . . . . . . . . . . . . . . . . . . . . . . 143 4.8.2. Determination of molar masses by tonometry . . . . . . . . . . . . . 145 4.8.3. Determination of molar masses by ebulliometry . . . . . . . . . . . 146 4.8.4. Continuous rectification or fractional distillation . . . . . . . . . . . 149

Chapter 5. Equilibria Between Ternary Fluid Phases . . . . . . . . . . . 163

5.1. Representation of the composition of ternary systems . . . . . . . . . . . 163 5.1.1. Symmetrical representation of the Gibbs triangle . . . . . . . . . . . 163 5.1.2. Dissymmetrical representation of the right triangle . . . . . . . . . . 168

5.2. Representation of phase equilibria . . . . . . . . . . . . . . . . . . . . . . 169 5.2.1. Isothermal projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.2.2. Conjugate points and conodes . . . . . . . . . . . . . . . . . . . . . . 170 5.2.3. Isopleth sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.3. Equilibria in liquid phases with miscibility gaps . . . . . . . . . . . . . . 171 5.3.1. Representation of the miscibility gap . . . . . . . . . . . . . . . . . . 171 5.3.2. Sharing in liquid–liquid systems . . . . . . . . . . . . . . . . . . . . . 173 5.3.3. Application of sharing between two liquids to solvent extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.4. Liquid–vapor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.4.1. Isothermal and isopleth sections (boiling and dew) . . . . . . . . . 182 5.4.2. Distillation trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.4.3. Systems with two distillation fields . . . . . . . . . . . . . . . . . . . 186 5.4.4. Systems with three distillation fields . . . . . . . . . . . . . . . . . . 187

5.5. Examples of applications of ternary diagrams between fluid phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.5.1. Treatment of argentiferous lead . . . . . . . . . . . . . . . . . . . . . 187 5.5.2. Purity of oil products: aniline point . . . . . . . . . . . . . . . . . . . 188 5.5.3. Obtaining concentrated ethyl alcohol . . . . . . . . . . . . . . . . . . 189

Chapter 6. Equilibria Between Condensed Ternary Fluid Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.1. Solidification of a ternary system with total miscibility in the liquid state and in the solid state . . . . . . . . . . . . . . . 192 6.2. Solidification of a ternary system with no miscibility and with a ternary eutectic . . . . . . . . . . . . . . . . . . . . . . . 192

Contents ix

6.2.1. Invariant transformations of a liquid–solid ternary system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.2.2. Representations of the ternary system with no miscibility in the solid state . . . . . . . . . . . . . . . . . . . . . . . 194 6.2.3. Lowering of the melting point of a binary system by the addition of a component . . . . . . . . . . . . . . . . . 199 6.2.4. Slope at the ternary eutectic . . . . . . . . . . . . . . . . . . . . . . . . 202

6.3. Ternary systems with partial miscibilities in the solid state and ternary eutectic . . . . . . . . . . . . . . . . . . . . . . . 204 6.4. Solidification of ternary systems with definite compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

6.4.1. Ternary system with a binary definite compound binary with congruent melting . . . . . . . . . . . . . . . . . . . 208 6.4.2. Generalization to the case of a ternary compound and of multiple definite compounds . . . . . . . . . . . . . . . . 211 6.4.3. Definite compound with incongruent melting: quasi-peritectic transformation . . . . . . . . . . . . . . . . . . . . 213

6.5. A peritectic transformation in one binary system and total miscibility in the other two . . . . . . . . . . . . . . . . . . . 215 6.6. The ternary peritectic transformation . . . . . . . . . . . . . . . . . . . . . 217

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

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 and chemical thermodynamics which all science students are normally taught, and will therefore be familiar with the fundamentals. This may includes the principles and 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, 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 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 Phase Transformations

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 at the École nationale supérieure des mines (engineering school) in Saint Étienne 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 fifth volume is devoted to the study of transformations between phases of pure substances and of binary and ternary systems.

Chapter 1 examines equilibria between the phases of a pure substance. Following a presentation of the different types of transformations – particularly of the first and second order – we study, in turn, at the macroscopic and microscopic scale, liquid–vapor, liquid–solid and solid–vapor equilibria. Solid-solid equilibria are examined for monotropy and enantiotropy. Finally, mesomorphic states are described.

Chapter 2 is devoted to the study of the general properties of phase equilibria in binary systems. The isobaric and isothermal modes of representation are discussed, as are composition curves. The chapter closes with the representation

Preface xiii

of the activities of the components in two-phase systems, with the particular cases of Raoult’s and Henry’s laws being examined.

In Chapter 3, we study the equilibria of binary systems between condensed phases. The thermodynamics of demixing is studied, before being applied to strictly-regular solutions. Liquid–solid equilibria, with or without demixing, of one or both of the phases, define eutectics and peritectics, with the reactions corresponding to each of these. Solid–solid equilibria with polymorphic varieties are studied and compared to the previous equilibria, with the reactions at the eutectoid and peritectoid points. Two applications of liquid–solid equilibria are examined through the lens of the Schröder–Le Châtelier solubility lens and cryoscopy. Finally, the thermodynamics of equilibria of membranes is studied, with an application to strictly-regular solutions.

Chapter 4 is given over to equilibria of binary liquid–vapor phases. These systems are studied far from and near to the critical conditions. Applications for different types of solution are presented. The thermodynamics of azeotropy is examined in detail, and then applied to strictly-regular solutions. A certain number of applications of these equilibria are presented, such as tonometry, ebulliometry and continuous fractional distillation.

In Chapter 5, we look at ternary systems with modes of representation, before going on to study systems where no solid phase is present. The presence of miscibility gaps in liquid phases is applied to solvent extraction, and particularly to the treatment of argentiferous lead ores. Liquid–vapor equilibria lead us to distillation fields and to binary and ternary azeotropes. The chapter concludes with the application of the charts to alcohol purification and determination of the aniline point.

Finally, Chapter 6 describes ternary systems involving solid phases. The concepts of ternary solidus and liquidus, and the existence of binary and ternary eutectics, are analyzed. Definite compounds, binary or ternary and exhibiting congruent or incongruent melting are represented, and the different means of solidification, with one, two or three solid phases, are examined.

Michel SOUSTELLE February 2016

Notations and Symbols

{ }gas pure, { }{ }gas in a mixture, ( )liquid pure, ( )( )liquid in solution,

solid pure, solid in solution

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

:D dielectric constant of the medium

D(T/ΘD): Debye function

d: distance between two liquid molecules

diS: internal entropy production

E: energy of the system

E0: internal energy associated with a reaction at the temperature of 0 K

Eb: balance equation

Ep: set of variables with p intensive variables chosen to define a system

xvi Phase Transformations

F: Helmholtz energy

: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

F: faraday (unit)

fi: fugacity of component i in a gaseous mixture 0:if molar Helmholtz energy of pure component i

f0 or 0:if fugacity of a pure gas i

:xsmG excess 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: osmotic coefficient 0:ig molar Gibbs energy of pure component i

g(r): radial distribution function

g*: molar Gibbs energy of gas i at pressure of 1 atmosphere in a mixture

0:TH standard molar enthalpy of formation at temperature T

H , :iH enthalpy, partial molar enthalpy of i

H: Hamiltonian

Notations and Symbols xvii

:xsmH molar enthalpy

:mixmH enthalpy of mixing

:xsiH partial molar excess enthalpy of component i

:mixiH partial molar enthalpy of mixing of component i

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

:iJ partial molar value of J relative to component i

:mixiJ value of mixing 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

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

( ):TriK constant of equilibrium change for the phase transition Tr

for component i

KAX: solubility product of the solid AX ( ):iK αβ coefficient of sharing of the compound i between the α and

β phases

Kd: dissociation constant ( ):crK equilibrium constant relative to concentrations

( ):frK equilibrium constant relative to fugacity values

( ):PrK equilibrium constant relative to partial pressure values

xviii Phase Transformations

Kr, (I)rK , (II)

rK , (III):rK equilibrium constant or equilibrium constant in convention (I), in convention (II), in convention (III)

Ks: solubility product

kB: Boltzmann’s constant

Lt: latent heat accompanying the transformation t

M: molar mass

M: magnetic moment or Madelung’s constant

s:m mass of solute s in grams per kg of solvent

m: total mass

mi: mass of component i

N: number of components of a solution or a mixture of gases or involved in a reaction

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: total number of moles in a solution or a mixture

n(α): total number of moles in an α phase

<n>: average number of gaps neighboring a molecule in a liquid

:mixcP critical pressure of the mixture

P: pressure of a gas ( ):subl

iP sublimating vapor pressure of component i ( )vap

iP 0:iP saturating vapor pressure of component i

:mixrP relative pressure of the mixture

Pc: critical pressure

Notations and Symbols xix

Pi: partial pressure of component i

p: number of external physical variables

Q: heat involved

Qa: reaction quotient in terms of activities

QP: heat of transformation at constant pressure; reaction quotient in terms of partial pressures

Qr: reaction quotient of transformation r

QV: transformation heat at constant volume

ℜ : reaction rate

R: perfect gas constant

R: distillation reflux rate

:mixmS molar entropy of mixing

:xsiS partial excess molar entropy of component i

:mixiS partial molar entropy of mixing of component i

S: oversaturation of a solution

:iS entropy or partial molar entropy of i

:xsmS excess molar entropy 0:is molar entropy of pure component i

T: temperature

:mixcT critical temperature of mixture *:T second-order transition temperature

:mixrT relative temperature of the mixture

T(Az): boiling point of the azeotrope

Tc: critical temperature

TF: Fermi temperature

xx Phase Transformations

Ti(Boil): boiling point of pure i

Ti(F): fusion (melting) point of pure i

Ts: sublimation temperature

Tv: vaporization temperature

:xsmU excess molar internal energy

:mixmU molar internal energy of mixing

:xsiU excess partial molar internal energy of component i

:mixiU partial molar internal energy of mixing of component i

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

Um: molar 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 0:iv molar volume of pure component i

vm: molecular volume

vM: molar volume of solid at melting point

wi: mass fraction of component i ( ):kx molar fraction of component k in the phase

x, y, z: coordinates of a point in space

xi: molar fraction of component i in a solution

<y>: average value of y

Notations and Symbols xxi

Yi and Xi: intensive and extensive conjugate values

yi: molar fraction of component i in a gaseous phase

α: relative volatility

Γ: characteristic function

:γ activity coefficient of the component regardless of the reference state used

γ0: activity coefficient of a solvent

γi: activity coefficient of the species i (I):iγ activity coefficient of component i in the pure-substance

reference (II):iγ activity coefficient of component i in the infinitely-dilute-

solution reference (III):iγ activity coefficient of component i in the molar-solution

reference

γs: activity coefficient of a solute

Δr(A): value of A associated with transformation r

:jiε Wagner interaction coefficient

ΘD: Debye vibrational temperature

ΘE: Einstein vibrational temperature

μi: chemical potential of compoent i, (L)iμ , (G)

iμ : chemical potential of component i in the liquid and gaseous states, respectively

( ):k ρν algebraic stoichiometric number of component Ak in reaction ρ

ξ: reaction extent

ρ: density

xxii Phase Transformations

:Φ practical osmotic coefficient; expansion pressure

:iΦ coefficient of fugacity of component i in a gaseous mixture

φ: number of phases

:iφ coefficient of fugacity of gas i in a mixture

φ0 or 0:iφ coefficient of fugacity of a pure gas

:iχ calorimetric coefficient relative to variable xi

χT: coefficient of compressibility at temperature T

Π : osmotic pressure

1

Phase Transformations of Pure Substances

In this chapter we shall examine the transformations undergone by a definite pure compound, with no chemical alteration. These transformations belong to the category of phase transitions, or phase transformations. Hence, this chapter excludes the transformations of isomerization and decompositions which are accompanied by a chemical alteration – i.e. a modification of the molecule.

1.1. Standard state: standard conditions of a transformation

The “standard state” of a substance at temperature T is defined as the state of the pure substance at that temperature, at a pressure of 1 bar and as its stable state of aggregation in these conditions (solid, liquid or gas). If the substance is a gas, its behavior is perfect. If the pure substance is a crystalline solid, its stable state of aggregation determines the crystalline system. A transformation takes place in standard conditions if it occurs with the components in their standard state in the final state. This means that at the start of the reaction, the substance is in a non-standard state, which is unstable, and that it returns to its stable standard state at the end. Thus, only the state of aggregation may possibly have been modified by the transformation. This is indeed a phase transformation, in that the state of aggregation defines a phase.

Phase Transformations, First Edition. Michel Soustelle.© ISTE Ltd 2015. Published by ISTE Ltd and John Wiley & Sons, Inc.

2 Phase Transformations

1.2. Classification and general properties of phase transformations

Phase transitions are classified into different types. The advantage of classifying definite compound transformations depending on their order lies in the fact that a series of relations characterizes each order.

The first classification of these transitions is attributable to Ehrenfest, in which we say that a transformation is of order n when at least an nth derivative of the characteristic function in relation to its canonical variables undergoes a discontinuity for certain values of those variables, when the derivatives of order less than n are continuous. The most important are the first- and second-order derivatives which, by their discontinuities, respectively give us the first- and second-order transformations.

This classification has been abandoned, because it does not allow for the possibility of divergences other than discontinuities of derivatives of the Gibbs energy. However, numerous models allow for such divergences within the thermodynamic limit – i.e. when the system’s dimensions increase indefinitely. For example, a derivative such as the specific heat capacity (second derivative of the Gibbs energy in relation to temperature) is thought to be divergent during the ferromagnetic transition.

The current classification still distinguishes between first- and second-order transformations, but the definitions used are different.

First-order transformations are those which involve latent heat. In other words, they are transformations accompanied by an associated enthalpy value. During the course of these transformations the system absorbs or emits a certain fixed amount of energy, which is usually fairly large; as that energy cannot be transferred instantaneously between the system and the external environment, the transformations take place over extended periods of time, during which not all parts of the system undergo the transformation at the same moment.

During these transformations the systems are heterogeneous, meaning that at each moment of the transformation they involve the simultaneous presence of multiple phases, and therefore interfaces between those phases. Such is the case with numerous transitions between the solid, liquid and gaseous phases.

Phase Transformations of Pure Substances 3

T

H

P

V

T

G

T

S

T

V

Figure 1.1. Shape of the curves representative of the different values during a first-order transformation

Figure 1.1 shows the profiles of the modifications of first-order transformation functions; discontinuity of H (associated enthalpy), volume with temperature and pressure, entropy with temperature and continuity but with a change in the slope of the Gibbs energy. For simplicity’s sake, curved segments have been represented by linear segments which do not accurately reflect reality.

Second-order transformations are continuous-phase transitions: there is no latent heat associated with the transition, the system is homogeneous, the transition takes place within the phase at every point and no interfaces manifest themselves. These transformations are also known as continuous transformations. There are a number of substances which exhibit second-order transitions; let us cite the following examples:

– the transition between the two forms I and II of liquid helium;

– ferromagnetic substances whose transition point is the Curie temperature, at which they cease to manifest ferromagnetism;

– certain alloys which exhibit so-called order–disorder transitions1 in a fixed composition:

- superconductive substances at the point of disappearance of the property,

- certain crystals, such as ammonium salts which, at a low temperature, undergo what is known as a “lambda” transformation.

Unlike with the first-order phase-transition, in a second-order transition, the two states are not localized separately in space, and thus there is no interface: the two states constitute a single, unique phase.

1 These transitions were examined in Volume 3 of this set [SOU 15c].


T

H

T

S

T

G

T

V

P

V

T

χ

T

CP

T

α

Figure 1.2. Shape of the different functions in a second-order transformation

The transition between a superfluid, liquid and gas near to the critical point is a second-order phase-transition.

Figure 1.2 shows the profile of the variations of the different functions at the transition point. To simplify, the curves have been represented by linear segments. We see the continuity of entropy with temperature but with a change of slope. The same is true of enthalpy and volume. The curve of Gibbs energy takes place with no change in slope and the variations in expansion coefficient, isothermal compressibility and specific heat capacity at constant pressure exhibit discontinuities at the transition.

1.2.1. First-order transformations and the Clapeyron relation

According to the definition given by Ehrenfest, at least one first derivative of the characteristic function in relation to a variable undergoes a discontinuity, though the characteristic function itself (the zero-order derivative) remains constant. We first examine it in the case of a system with two physical variables: temperature and pressure. The characteristic function is the Gibbs energy, such that:

G ST

∂ = −∂

[1.1a]

and:

G VP

∂ =∂

[1.1b]


The first-order transformation is characterized by a discontinuity of entropy or volume at a given pressure and temperature, at which the system can exist in two states: 1 and 2. Such is the case, in particular, with phase changes of pure substances (melting, vaporization, sublimation, and polymorphic transformation of a solid).

Let us now establish the Clapeyron relation, which governs these first-order transformations.

Consider states 1 and 2, both stable simultaneously in the same thermodynamic conditions (i.e. same pressure and temperature). The Gibbs energy is constant, which is expressed by the following:

1 2G G= [1.2]

If the temperature and pressure are modified by an infinitesimal amount, the Gibbs energy takes on the new value G + dG, and the new continuity condition is written thus:

1 1 2 2d dG G G G+ = + [1.3]

which, in view of equation [1.2], gives us:

1 2d dG G= [1.4]

Relation [1.4] is simply the application of the general equilibrium condition to the balance equation of the phase change (stoichiometric coefficients equal to +1 and -1).

However, in state 1, the differential of the Gibbs energy is:

1 1 1d d dG S T V P= − + [1.5]

A similar expression can be written for state 2. The continuity condition [1.4] becomes:

1 1 2 2d d d dS T V P S T V P− + = − + [1.6]


For simplicity’s sake, let us set the differences:

2 1rV V V= − [1.7a]

and:

2 1r S S S= − [1.7b]

These are the changes in volume and entropy associated with the transformation 1 2.

By feeding these values back into expression [1.6], we find:

d d 0r rS T V P− = [1.8]

This relation is known as the Clapeyron equation. In particular, it shows that if one of the intensive values conjugal to one of the variables (e.g. the entropy) undergoes a discontinuity, the intensive value conjugal to the other variable (in this case the volume) also undergoes a discontinuity, and these two discontinuities are linked to one another by the Clapeyron equation.

Often, the term “latent heat L of the transformation between states 1 and 2” is used to denote the amount of heat involved in reversible conditions, and therefore:

rL T S= [1.9]

Hence, there is equivalence between the first order as understood by Ehrenfest and the first order in the new classification. Transformations which are accompanied by latent heat exhibit a discontinuity of entropy.

By applying the Clapeyron equation [1.8], we obtain:

dd

r

r r

SP LT V T V

= = [1.10]

The Clapeyron equation can be generalized to apply to a system defined by the set of p physical variables iZ , whose conjugal extensive variables are

iX . The characteristic function, then, is a function . By thinking about


that function in the same way as we did with the Gibbs energy, we find the result:

( )d 0ii r i ii

X Zε = [1.11]

iiε is a switch whose value is +1 if the value Zi is intensive and -1 if Zi is extensive.

Thus, we generalize the Clapeyron equation, which shows that if a first derivative of the characteristic function i undergoes a discontinuity, there is at least one other first derivative which also undergoes a discontinuity. The magnitudes of the different discontinuities are linked by equation [1.11].

1.2.2. Second-order transformations

Second-order transformations cannot result from the Clapeyron equations, because the first derivatives are continuous.

1.2.2.1. Ehrenfest equations

According to Ehrenfest’s definition, we are dealing with a second-order transformation if a second derivative of the characteristic function is discontinuous, with that function and its first derivative being continuous. According to the new definition, it is sufficient for entropy to be continuous (no latent heat).

Ehrenfest established the relations mirroring Clapeyron’s relations in second-order transformations.

In the case of chemical systems (choosing the variables –P and T as before) we write the continuity of the Gibbs energy between states 1 and 2, at equilibrium in the same conditions of pressure and temperature:

1 2G G= [1.12]

The continuity of the two first derivatives is expressed by the two relations:

1 2

G GP P

∂ ∂=∂ ∂

[1.13a]


and:

1 2

G GT T

∂ ∂=∂ ∂

[1.13b]

If we modify the Gibbs energy by an infinitesimal amount, the first derivative in relation to pressure assumes the value:

1 1

dG GP P

∂ ∂+∂ ∂

The new continuity condition becomes:

1 1 2 2

d dG G G GP P P P

∂ ∂ ∂ ∂+ = +∂ ∂ ∂ ∂

[1.14]

and from this we get the condition for the derivative in relation to pressure:

1 2

d d 0G GP P

∂ ∂− =∂ ∂

[1.15]

Similarly, we write the continuity of the derivative in relation to temperature:

1 2

d d 0G GT T

∂ ∂− =∂ ∂

[1.16]

By replacing the first derivatives with volume and entropy, equations [1.15] and [1.16] take the form:

1 2 1 2

d d d d d d 0G G V V V VP T P TP P P T P T

∂ ∂ ∂ ∂ ∂ ∂− = + − + =∂ ∂ ∂ ∂ ∂ ∂

[1.17]

1 2 1 2

d d d d d d 0G G S S S SP T P TT T P T P T

∂ ∂ ∂ ∂ ∂ ∂− = − − − − − =∂ ∂ ∂ ∂ ∂ ∂

[1.18]


However, for reasons of symmetry of the characteristic matrix, we have:

S VP T

∂ ∂= −∂ ∂

[1.19]

By substituting into relation [1.18], this continuity condition becomes:

1 2 1 2

d d d d d d 0G G V S V SP T P TT T T T T T

∂ ∂ ∂ ∂ ∂ ∂− = − − − =∂ ∂ ∂ ∂ ∂ ∂

[1.20]

Remember the meanings of the derivatives of volume and entropy in relation to pressure and temperature, which are respectively linked to the expansion coefficient β, the isothermal compressibility coefficient χ and the specific heat capacity at constant pressure CP:

V VT

β∂ =∂

[1.21a]

V VP

χ∂ = −∂

[1.21b]

pCST T

∂ =∂

[1.21c]

If we feed these definitions back into relations [1.17] and [1.20], the continuity conditions become:

1 1 2 2d d d d 0V P V T V P V Tχ α χ α− + + − = [1.22]

1 21 2d d d d 0P PC C

V P T V P PT T

α α− + + − = [1.23]

From relation [1.22], we deduce:

dd

r

r

PT

αχ

= [1.24]


From relation [1.23], we can extract:

dd

r P

r

CPT VT α

= [1.25]

By combining equations [1.24] and [1.25], we obtain:

( )2r r r PTV Cα χ= [1.26]

Equations [1.24], [1.25] and [1.26] constitute the Ehrenfest equations. They express the relations linking the discontinuities of the expansion coefficient, the compression coefficient and the specific heat capacity, with one another and with the derivative of the pressure in relation to temperature.

As was the case with the Clapeyron equation [1.11] the Ehrenfest equations can be generalized to apply to other pairs of variables Zi and iY . Such is the case, for example, with the ferro-paramagnetic transformation involving temperature/entropy and magnetic moment/magnetic field pairs.

1.2.2.2. Landau theory

Landau theory aims to offer a general description of second-order transformations, disregarding the peculiarities applying to any specific system. Indeed, the manifestation of order suggests that there is a certain degree of universal behavior.

Given that the transformation takes place without any spectacular results, it can be considered that the effects (discontinuity or indetermination of second derivatives of the Gibbs energy or another property) associated with this type of transformation depend essentially on a universal value known as the order parameter (or parameters), whose exact definition depends on the physical nature of the transition. Thus, the occurrence of the transformation manifests as the continuous evolution of that parameter.

1.2.2.2.1. Symmetry and order parameters

Phases before and after a transition often have different symmetries, although this is not always the case.


For instance, in the transition between a crystalline solid and a fluid (liquid or gas), the substance moves from a solid state to a liquid state. The solid state is stable at a low temperature, is ordered and less symmetrical, because the properties, whilst they are periodic, change from one point in the solid to another, and differ with direction (anisotropy). The liquid is stable at a higher temperature, less ordered but more symmetrical, with properties that are more homogeneous from one point to another. We say that the solid–fluid transition breaks the symmetry. The ferromagnetic–paramagnetic transition breaks the symmetry, because of the disappearance of magnetic domains containing aligned magnetic moments in the low-temperature state.

The breaking of symmetry is important in the behavior of a phase-transition. Landau noted that it was not possible to find a continuous and derivable function between states having different symmetry. This explains why it is not possible to have a critical point in a solid–fluid transformation.

Transformations which break symmetry are necessarily either first- or second-order. In general, the most symmetrical phase is stable at a high temperature.

The breaking of symmetry necessitates the involvement of additional variables to describe the state of the system. These are the (intensive) variables, which are the order parameters introduced by Landau. Table 1.1 gives a few examples of order parameters.

We can posit that for the stable phase at high temperature, less ordered and more symmetrical, the order parameter is null (s = 0), whereas it is different to zero in the stable phase at a lower temperature, which is more ordered and less symmetrical (s 0).

Transition Order parameter

Solid–fluid Fraction of liquid phase

Ferro–paramagnetic Magnetization M

Order–disorder transition in alloys Fraction of sites correctly occupied

Gas–liquid Difference between the densities of the liquid and the gas

Table 1.1. Examples of order parameters


NOTE 1.1.– It is possible to show that we cannot define order parameters if there is no breaking of symmetry, but Landau nevertheless introduced a pseudo-order parameter in these cases, as seen from the last row in Table 1.1 for the gas–liquid transition, which does not break symmetry.

1.2.2.2.2. Second-order transitions according to Landau

Consider a phase-transition with breaking of symmetry and therefore an order parameter. At a given temperature, the state of stable equilibrium corresponds to a value of the order parameter which renders the Gibbs energy G(s, T) minimal.

In the vicinity of the transition temperature T* at which the order parameter takes the values of 0, that parameter is small, and thus we can take the first terms of the MacLaurin expansion of the Gibbs energy in the form:

2 3 40 1 2 3 4( , ) ( ) ( ) ( ) ( ) ( ) ...G s T G T A T s A T s A T s A T s= + + + + + [1.27]

In this expression, 0 ( )G T is the Gibbs energy of the stable phase at high temperature:

– for T > T* ( )G T must be minimal for s = 0 and A2(T) > 0;

– for T < T*, ( )G T must be minimal for s 0 and A2(T) < 0.

The simplest choice fulfilling these conditions is:

*2 2( ) ( )A T a T T= − where a2 > 0 [1.28]

For a second-order transition, we can show that the coefficients of the odd-powered terms are null 1 3( ( ) ( ) 0)A T A T= = and that 4 ( )A T is a positive constant 4 4( ( ) 0)A T a= > , and therefore:

* 2 40 2 4( , ) ( ) ( )G s T G T a T T s a s= + − + [1.29]

The value of the order parameter is the solution to the equation which renders a zero value of the derivative of the function [1.29]:

* 32 4

( , ) 2 ( ) 4 0G s T a T T s a ss

∂ = − + =∂ [1.30]


– for T > T*, the only solution is s = 0;

– for T < T*, there are three solutions: a maximum at s = 0 and two minima for:

*0 2 4( ) / 2s s a T T a= ± = ± − [1.31]

In the vicinity of the temperature T*, the order parameter behaves like *( )T T− , so the curve s(T) has a vertical tangent at T = T* (see Figure 1.3).

1 T/T*

s1

0

Figure 1.3. Variation of the order parameter with changing temperature

Figure 1.4 illustrates the three modes of variation of G(s) depending on the temperature value.

G-G0

T>T*

s

G-G0

T=T*

s

G–G0 G–G0

T>T*

s

T=T*

s

G-G0

T<T*

s

s0 -s0

T<T*

–s0 s0

G–G0

s

Figure 1.4. Variations of the Gibbs energy with the order parameter

We can see that, although at a given pressure a first-order transformation occurs at a very specific temperature, a second-order transformation takes place in a temperature range limited to a so-called critical temperature, which is no longer a transition temperature.


Second order T*

s = 0

s = 1G

T T*

First order

G

T

Figure 1.5. Comparison of the variations in the Gibbs energy with temperature of the first- and second-order transformations

This can be seen clearly in Figure 1.5, which compares the Gibbs energies of a system undergoing a first-order transformation, with a rupture of the slope at the transition T0, and a system undergoing a second-order transformation in the temperature domain of variation of the order parameter between 1 and 0, and ending at the critical temperature T*.

If we examine the variation of specific heat capacity at the critical temperature during the second-order transition, taking account of relations [1.27] and [1.31], we obtain:

– in the vicinity of the critical temperature (slightly above it), s = 0 and:

*

20

2PT T

GC T

T ε= +

∂= −∂

[1.32]

– in the vicinity of the critical temperature (slightly below it), s 0 and:

*

2 *2022

42PT T

G TC T aT aε= −

∂= − +∂

[1.33]

which gives us the discontinuity of the specific heat capacity:

*22

42t PTC aa

= − [1.34]


1.2.2.2.3. Critical exponents

To represent the behavior of the physical properties in the vicinity of the critical temperature of a second-order transformation, we usually use power laws dependent on the temperature. These laws are practically the same regardless of the substance undergoing the transition in question. These variations are written as:

– for the order parameter:

*s T Tβ

∝ − [1.35]

– for the specific heat capacity:

*PC T T

δ−∝ − [1.36]

– for the isothermal compressibility:

*T Tγ

χ−

∝ − [1.37]

These exponents are known as the critical exponents. For example, the constant α is the critical exponent associated with the specific heat capacity. Because the transition has no latent heat, it is imperative that δ be less than 1, otherwise the enthalpy would no longer be continuous with the temperature.

NOTE 1.2.– By virtue of the Ehrenfest equation [1.26], a relationship exists between the critical exponents associated with the specific heat capacity, the expansion coefficient and the isothermal compressibility.

1.2.2.2.4. Limitation of Landau’s model

In actual fact, Landau’s model proves true only for a limited number of transitions. This limit is probably due to the fact that this theory is an approximation, which does not take into account fluctuations of the order parameter in the vicinity of the transition, where the value of that parameter is of a similar order of magnitude to its fluctuations. There is a more elaborate model, developed by Ginsburg and Landau, which incorporates terms of fluctuations in the development of the Gibbs energy given by relation [1.27].


1.3. Liquid–vapor transformations and equilibrium states

The liquid–vapor system of a pure substance A obeys the balance equation:

(A) = {A} [1R.1]

This balance equation highlights that the gas is considered to be pure in its phase.

Such a system has a Gibbs variance equal to 1, meaning that there is only one intensive variable which needs to be established in order to define the equilibrium. The gas pressure known as the saturating vapor pressure is a function of the temperature. It is a first-order transformation – i.e. it is accompanied by a latent heat of vaporization, and therefore discontinuities of the first derivatives of the Gibbs energy (volume, entropy and enthalpy). There are a number of different ways in which we can approach the study of a liquid–vapor phase-change and define the equilibrium conditions. Below we examine four such methods.

1.3.1. Method of two equations of state, using the Clapeyron equation

On the basis of the Clapeyron equation, supposing the gaseous phase to be pure and therefore having the saturating vapor pressure A

vapP , the equilibrium of the liquid-to-gas transition is written as:

{ } ( )( )0 0

A0 0 0

A A

dd

vapv v

v

h hPvT T T v v

= =−

[1.38]

To explain the equilibrium conditions, it is helpful to feed the equations of state for the liquid and the gas back into expression [1.38]. Very frequently, when the pressure is moderate, the molar volume of the liquid is negligible in comparison to that of the gas, so we can assume that

{ } ( )0 0A Av v>> , and use only the equation of state of the gas in relation [1.38],

using the SRK equation of state, for example, to represent the vapor phase. Obviously, it is not acceptable to neglect the molar volume of the liquid when conditions are quite far removed from the critical conditions.


The enthalpy of vaporization is positive, because vaporization is an endothermic phenomenon. As the molar volume of the gas is always greater than that of the liquid, the slope of the curve P(T) is positive.

We frequently encounter the perfect gas approximation as the equation of state of the gas, by writing:

{ }0A

A

Rvap

TvP

= [1.39]

which gives us the new form of the Clapeyron equation:

A2

A

d dR

vapv

vap

hPT

P T= [1.40]

By integrating between two temperatures T1 and T2, where the respective saturating vapor pressures are

( )1AvapP and

( )2AvapP , and ignoring the variations in

enthalpy with changing temperature, we obtain:

( )

( )

2

1

A

A 1 2

1 1lnR

vap

vvap

P hP T T

= −

[1.41]

Remembering that the boiling point is defined as the temperature at which the saturating vapor pressure is 1 bar, then if the reference pressure P0 is chosen as equal to 1 bar, we find:

0A

A0

1 1ln lnR

vapvap v

eb

hPP

P T T= = −

[1.42]

This relation uses standard enthalpy, because boiling takes place in standard conditions.

The curve representing the logarithm of the saturating vapor pressure as a function of the inverse of the temperature, then, is a straight line whose slope is equal to the opposite of the standard enthalpy of vaporization (positive) (Figure 1.6).


Figure 1.6. Variations of the logarithm of the saturating vapor pressure as a function of the inverse of the temperature

It must never be forgotten that this relation [1.42] is founded on the hypothesis that the gas’ behavior is perfect, which is a very significant approximation, because clearly, for example, in the presence of the liquid, the pressure in an enclosed volume does not depend on the quantity of gas, as suggested by the Boyle–Mariotte law. However, relation [1.42] is still accepted, because we are very far from the critical conditions.

1.3.2. Gibbs energy and fugacity method

We use the de Donder equilibrium condition. By making the affinity equal to zero, in view of the influence of pressure on the Gibbs energy of the liquid, we find:

{ } ( ) { } ( ) ( )0 0 * * 0 0A(A) AA A A A 0A R ln 0

vapvapf

g g g g T v P PP

= − + = − − + − + − = [1.43]

If we ignore the liquid volume term, the fugacity of saturation – i.e. the fugacity at equilibrium – becomes:

{ } ( )* 0 0A AA

0 exp expR R

vapv

g g gfP T T

−= − = −

[1.44]

Again, assuming usage conditions approximately corresponding to the perfect gas law, the fugacity is replaced by the pressure, which gives us the following equilibrium condition:

0A

0 expR

vapv gP

P T= −

[1.45]


By writing the Gibbs energy of vaporization as a function of the corresponding enthalpy and entropy, we obtain:

0 0 0A

0lnR R R

vapv v vg h sP

P T T= − = − +

[1.46]

If we compare the two equivalent expressions [1.46] and [1.42], we find:

0 0

Rv v

eb

h sT

= [1.47]

This relation [1.47] merely expresses the fact that the liquid and the vapor are in a state of equilibrium at the boiling point ( 0

eb 0g = ).

1.3.3. Unique equation of state method

We now choose a unique equation of state, applicable both to the liquid and the gas. For instance, if we choose a cubic equation in a volume, such as the van der Waals equation, we can show that with these equations, the isothermal curve P(v0), depending on the temperature domain, presents two or no positive extrema, and the first extremum is inevitably a minimum, because both parts of the isotherm of the liquid and the gas, in order to yield stable phases, must be decreasing ( )/ 0P V∂ ∂ < .

Let us trace such an isotherm at temperature T, which is represented by the curve a’abedd’ in Figure 1.7.

As soon as the two phases are present simultaneously, the Gibbs variance falls to 1. If the temperature is static, then so too is the pressure, and the curve needs to be replaced by a horizontal line ad. Thus, in the curve, we can distinguish five regions:

– between a’ and a: the material is completely liquid with the molar volume 0( )lv in a;

– between d and d’: the material is entirely gaseous with the molar volume 0( )gv in d;


– between a and b: the liquid phase is stable, but a curve of the Gibbs energy would show that the liquid is less stable than the gas; nonetheless, we can conserve liquid which is metastable;

v0(a)

v0

Branche de liquéfaction

v0(b) v0(c)

v0(e) v0(d)

K

P

Branche de vaporisation

Domaine supercritique

Liquide Vapeur Spinode

C

a

b

c e

d

d’

a’

T

T

Tc

Tc

A I J D

Supercritical domain

Liquefaction branch

Vaporization branch

Liquid Vapor Spinode

v0(a) v0(b) v0(c) v0(e) v0(d)v0

P

Tc

T

a’

a

b

ce

d

T

Tc

d’

C

A I J D

K

Figure 1.7. Clapeyron diagram of a pure substance, showing an isotherm and the critical isotherm

– between b and e: the phase is not stable because the diagonal thermodynamic coefficient /P V−∂ ∂ is negative;

– between e and d: the gaseous phase is stable, but a curve illustrating the Gibbs energy would show that the gas is less stable than the liquid; however, it remains gaseous until a germination facilitates the transformation; the gas is metastable.

On an isotherm with two extrema (see Figure 1.7), the two areas between the horizontal and each of the extrema are equal. Indeed, at thermodynamic equilibrium, the integral of the function P(v0) along the closed line abcdea must be null, because it represents the variation of the internal energy on a loop. However, the internal energy is a state function, and thus we must have:

{ } { } ( ) ( )

a

A A A Ad d d d 0c d c

a c d c

P V P V P V P V+ + + =

[1.48]


Hence:

( ) { }( ) { } ( )( )A A A Ad d 0a d

c c

P P V P P V− + − =

[1.49]

From this, we deduce:

( ) { }( ) { } ( )( )A A A Ad da c

c d

P P V P P V− = −

[1.50]

This demonstrates the equality of the areas. Thus, this property enables us to position the horizontal platform on an isotherm.

At every point K on the segment ad, the matter occupies the volume:

{ } { } ( ) ( ) ( ) ( ) { }( ) { }0 0 0 0 0

K A A A A A A A AV x v x v x v v v= + = − + [1.51]

The fractions { }Ax and ( )Ax respectively represent the fraction of gas and

the fraction of liquid at point K, with their sum being equal to 1.

From relation [1.51], we deduce:

( ){ }

( ) { }

0A

A 0 0A A

KV vx

v v

−=

−

Thus:

( )AKdxad

= [1.52]

This expression is the lever rule, which we can use on a diagram to evaluate the molar fraction of the liquid at K by the ratio between the lengths of the segments Kd and ad.

1.3.4. The region of the critical point and spinodal decomposition

If we alter the temperature, the point a follows a branch AC called the vaporization branch. Similarly, the point d follows a branch DC known as the liquefaction branch. These two branches intersect at point C, which is


an inflection point with a horizontal tangent to the isotherm obtained at temperature Tc (the critical temperature). This is the first isotherm with no extremum. At point C, the pressure is the critical pressure Pc. Above that point we are in the supercritical domain, there is no longer any equilibrium between the liquid and the vapor, and we have a gaseous-type phase, occupying all the available volume. Indeed, near to the critical point (slightly above it), there are conditions of temperature and pressure for which the transition between the gas and the liquid becomes a second-order transition, and the system has a milky appearance because of fluctuations in the density of the medium which interferes with the light. This phenomenon is known as critical opalescence. Thus, at the critical point, the isothermal compressibility coefficient χ becomes infinite.

If the temperature varies, the points which correspond to the extrema b and e each describe a curve. These curves intersect at the critical point and represent the curve along which the separation of the phases is inevitable, which is known as the spinode, because of the name of the transformation – spinodal – which, starting with a metastable liquid phase, leads to the mixing of a liquid phase and a stable gaseous phase, or vice versa.

1.3.5. Microscopic modeling

The goal of microscopic modeling is to find a priori values for the vapor pressure and the enthalpy of vaporization.

Take relation [1.43], but expressing the molar Gibbs energies of the liquid and the gas on the basis of expressions drawn from statistical thermodynamics, stemming from the models of the gas and the liquid.

Let us choose the perfect gas approximation. We can use statistical thermodynamics to calculate the Gibbs energy as a function of the partition function .pfZ This relation is written as:

{ } { } { } B BA A A k ln kpfG F PV T Z N T= + = − + [1.53]


Using the Stirling approximation with expression [1.53], we have the following for one mole (i.e. Na molecules):

{ }{ } { }

0BA 0 0

A A

R-R ln R ln R ln kpfz Tg T T T T

v v= + + [1.54]

In the perfect gas approximation, this Gibbs molar energy of the gas becomes:

{ }{ }

0BA 0

A

-R ln R ln R ln kpf vapA

zg T T P T T

v= + + [1.55]

The atomic partition function pfz is made up of the translation term (including the volume) and the terms of internal motions (vibrations, rotations, electronic motions).

With regard to the liquid, it is helpful to choose the best possible model –namely that of Eyring2, in all probability. This model is complex, and it is laborious to use directly. However, it should be noted2 that this model tends toward the smooth potential model2 on the side of the gas and towards the Mie model2 on the side of the solid. Thus, we choose the smooth potential model for the liquid–vapor transformation, as we shall choose the Mie model when we study melting (see section 1.4).

The Gibbs molar energy of the liquid, established using the smooth potential model, is:

( )( )

( )0 0A A0

A

-R ln Rpf fa

z vg N T T Pv

vε= − − + [1.56]

By equaling the molar Helmholtz energies of the gas [1.55] and the liquid [1.56], in conformity with relation [1.43], by ignoring the volume of the liquid, we obtain:

BA

B

kln lnk

vap

f

TP

T evε≅ − + [1.57]

2 See Chapter 1 of Volume 2 in this set of books [SOU 15b].


This gives us the saturating vapor pressure:

B BA

B

k kexpk

vap

f

T TP

ev Tε += − [1.58]

Based on expression [1.56], the enthalpy of liquid is:

( )( )

( )

0(A)0 2 0 0

a intA A

/ 3- -N R2

g Th T T Pv u

Tε

∂= = + + +

∂ [1.59]

For the perfect gas, we have:

{ }{ }( )0A0 2 0

intA

/ 5- R2

g Th T T u

T

∂= = +

∂ [1.60]

In that we can consider the internal molar contribution (rotations, vibrations and electronics of the molecules) to be identical for the liquid and the gas, the enthalpy of vaporization is:

{ } ( )0 0 0

aA A N Rvh h h Tε= − ≅ + [1.61]

By comparing expressions [1.58] and [1.61], we find:

0B

Ak exp

Rvap v

f

hTP

v T= − [1.62]

and by comparing relations [1.46] and [1.62], we can deduce the entropy of vaporization:

0 BkR lnvf

Ts

v= [1.63]

Thus, generally, the saturating vapor pressure is written in the form:

0

A expR

vap vhP aT

= − [1.64]


Note that for a large number of substances yielding non-associated liquids, the value of the coefficient a does not vary hugely from one substance to another. It seems that the average value of 2.3×104 is highly accurate for all these substances. Thus, we can write the saturating vapor pressure in the form:

04

A 2,3.10 expR

vap vhPT

≅ − [1.65]

The boiling point satisfies this approximation at the pressure of 1 bar, which enables us to write:

04ln 2,3.10 10

Rv

eb

hT

≅ ≅ [1.66]

or in the form:

0

10Rv

eb

hT

≅ [1.67]

This expression is the first form of Trouton’s rule.

In light of the equilibrium at the boiling point, we can also write:

00 10 Rv

veb

hs

T= ≅ [1.68]

This is the second form of Trouton’s rule, which stipulates that the entropy of vaporization at boiling of all the liquids is essentially the same and equal to 10 R, which is 83.2J/mole. Although this law has no theoretical basis and is founded on a common approximate evaluation of the parameter a (relation [1.66]), Table 1.2 (last column) shows that the rule follows fairly closely for many substances, with the exception of hydrogen and helium in the last two rows.


Substances Teb(K) Δvh0(kj/mole) Δvs0(J/mole.K)

Neon 27.2 1.7347 63.954

Nitrogen 77.5 1.5048 73.568

Argon 87.5 6.27 71.896

Oxygen 90.6 6.9388 76.494

Ether 307 27.0446 88.198

Carbon sulfide 319 27.1282 85.272

Chloroform 334 16.5946 86.944

Carbon tetrachloride 350 29.8452 85.272

Benzene 353 30.723 86.944

Methyl salicylate 497 45.98 92.796

Organic compounds (non-associated)

– 83 to 96

Methanol 337.7 35.0284 103.664

Formic acid 373.6 23.1572 61.864

Helium 4.29 0.09196 21.318

Hydrogen 20.4 0.89452 43.054

Table 1.2. Boiling point, enthalpy and entropy of vaporization at the boiling point of a number of gases (Fowler and Guggenheim, 1949)

NOTE 1.3.– Equation [1.62] enables for the measurement of the free volume vf of the liquid. We can compare it with the method described using thermomechanical coefficients (see section 1.3.3 in Volume 2: Modeling of liquid phases), returning to the example of chloroform, which yielded the value of 0.44 cm3/mole; this new method yields the value of around 1cm3/mole.

1.3.6. Liquid–vapor equilibrium in the presence of an inert gas

Up to now, we have only looked at vaporization in the presence of pure vapor of substance A. Now we consider the influence of the presence of an inert gas, mixed with the vapor of A, meaning that we are now examining the transformation:

(A) = {{A}} [1R.2]


The presence of the inert gas results in a pressure in the chamber which is the sum of pressure of the inert gas and the saturating vapor pressure of A – i.e.:

{ }{ } { }{ }ine AvapP P P= + [1.69]

The equilibrium is written by an expression equivalent to relation [1.43], which we write directly with the saturating vapor pressure and the total pressure P:

{ } ( ) { } ( ){ }{ } ( )A* * 0 0

(A)A A A A 0A R ln 0vapP

g g g g T v P PP

= − + = − + − + − = [1.70]

If we subtract relations [1.43] and [1.69] term by term, and pass to the exponential, we obtain a relation between the new vapor pressure of A in the mixture, { }{ }A

vapP and the vapor pressure of pure A { }AvapP :

{ }{ } { }{ }A0

(A)AA expR

vapvap vap

P PP P v

T

−= [1.71]

For small volumes of liquid we sometimes need to make do with the

limited expansion of the exponential, thus written:

{ }{ }

{ }

{ }A A0(A)

A R

vap vap vapA

vap

P P P Pv

P T

− −= [1.72]

Thus, the saturating vapor pressure is slightly modified in the presence of

an inert gas. To gain an idea of the importance of the correction, we choose the example of water, which gives us 372.78K at the pressure of 1 bar with a molar volume of the liquid of

2

0 5 3( ) 1.88 10 m /moleH Ov −= × . We choose the

presence of nitrogen at the pressure of 10 bars, and the total pressure becomes 11 barsP ≅ . The calculation gives us the following ratio:

{ }{ } { }

{ }

22

2

2

H OH O 0(H O)

H O

11 1 0.006372.8R

vap vap

vap

P Pv

P

− −= ≅

We can see that the correction is, in fact, extremely small and justifies the use of approximation [1.72].


1.4. Solid–vapor transformations and equilibriums

The solid–vapor transformation is known as sublimation. It can be represented in the pure gas phase by the following balance equation:

<A> = {A} [1R.3]

As with the liquid–vapor equilibrium (see section 1.2), these transformations can be studied at macroscopic and microscopic equilibrium.

1.4.1. Macroscopic treatment

The treatment based on the Clapeyron equation is absolutely parallel to that of the liquid–vapor equilibrium; equation [1.38] is transposed by writing the enthalpy of sublimation and the molar volume of the solid in the form:

( )

{ }( )0 0

s sA0 0 0s AA

dd

subl h hPT T v T v vΔ

< >

= =−

[1.73]

The enthalpy of sublimation is positive, as sublimation is an endothermic process. Because the molar volume of the gas is always greater than that of the solid, the slope of the curve P(T) is positive.

Using the same approximations as in section 1.2.1, the boiling point is replaced by the sublimation temperature at atmospheric pressure Ts. We find:

( ) 0sA

0

1 1lnR

subl

s

hPP T T

= − [1.74]

and:

( )( )

0sA

A2

dd RT

sublsublhP

PT

=

[1.75]


Similarly, equation [1.70] is replaced by:

{ } ( )* * 0 0AA AA 0R ln 0

sublPg g T v P P

P< > < >− + − + − = [1.76]

which, instead of relation [1.46], gives us:

0sA

0 expR

subl gPP T

= − [1.77]

and similarly, in the presence of an inert gas, with the same approximations as in section 1.2.5, in place of relation [1.72] we obtain:

{ }{ }( ) ( )

( )

( )AA 0 A

A R

subl subl subl

Asubl

P P P Pv

TP < >

− −= [1.78]

The corrections due to the presence of an inert gas are of the same order as the liquid, because the molar volumes of liquids and solids are of the same order of magnitude.

1.4.2. Microscopic treatment

For the Gibbs molar energy of the gas, with the same approximations, we use relation [1.55], written by separating the contribution of translation and the contributions of the internal movements of the molecule, in the form:

{ }( )

{ }

3/2B0

A B intA 3 0A

2 k-R ln R ln R ln k R ln

hsublm T

g T T P T T T zv

π= + + + [1.79]

For the solid, we again see the case of the molecular solid with a unique Einstein frequency (see volume 3 in this series (Soustelle, 2015)). By combining the internal contributions of the molecule and operating at


temperatures much higher than the Einstein temperature, we show that we can write, for a mole:

a 0 A BA a a int

A

N kln 3N ln N lnR vc

TZ z

Tε

ν< >

< >< >

= + +

[1.80]

However, the Gibbs molar energy of the pure solid is given by the general relation:

0 0A c A AR lng T Z Pv< > < > < >= − + [1.81]

so by using relation [1.80]:

0 0BA a 0 A int A

A

kN 3R ln R lnv

Tg T T z Pvε

ν< > < > < >< >

= − − − + [1.82]

By equaling the two expressions of the Gibbs energies of the gas [1.79] and the solid [1.82], we obtain:

( ){ }

3/2B 0a

A 0 A B A A3 0A

2 kNln ln 4ln k ln hR h

subl m TP T Pv

T vπ

ε ν< > < > < >= − + − + + [1.83]

Obviously, as with the previous cases, we can neglect the pressure term.

Expression [1.83] can be used a priori to calculate a sublimation pressure. The sublimation temperature, obtained for a pressure of 1 bar, cannot be explicitly stated, but can be approximated by computer calculation.

1.5. Transformations and solid–liquid equilibria

The solid–liquid transformation is known as melting. It can be represented by the following balance equation:

<A> = (A) [1R.4]

We shall also touch on the macroscopic and microscopic treatments.


1.5.1. Macroscopic treatment

The treatment on the basis of the Clapeyron equation is absolutely parallel to that of the liquid–vapor equation: equation [1.38] is transposed, by writing the enthalpy of fusion and the molar volumes of the solid and the liquid, in the form:

( )0 0

f f0 0 0

f (A) A

dd

h hPT T v T v v< >

= =−

[1.84]

The enthalpy of fusion is positive, because melting is endothermic. The slope of the curve P(T) is fairly significant, because the molar volumes of the solid and the liquid are fairly similar. It is usually positive, because the molar volume of the liquid is a little greater than that of the solid, with the exceptions of water, bismuth, arsenic and antimony, for which the slope is negative.

1.5.2. Microscopic treatment

We now choose the Mie model (see [SOU 15b]) for the liquid: that which approximates the solid. The Gibbs molar energy is then calculated using the relation:

0 0B(A) a 0(A) int(A) (A)

(A)

kN 3R ln R R lnTg T T T z Pv

hε

ν= − − − − + [1.85]

The Gibbs molar energy of the molecular solid is given by expression [1.82]. We apply the equation of the Gibbs molar energies, supposing that the internal movements of the molecule are the same in the solid and the liquid, and thus we can set:

int(A) int Az z < >≈ [1.86]

Because we have 0(A) 0 Aε ε < >< and (A) Aν ν < >< , then we will have the inequality 0(A) 0 Ag g < >< for low temperatures and the inequality 0(A) 0 Ag g < >> at higher temperatures. Thus for a certain temperature, the melting point, at


atmospheric pressure. In that we neglect the term ( )0 0A (A) / R fP v v T< > − , we

have the equality:

0 A 0(A) A

B (A)

3ln 1k fT

ε ε νν

< > < >−

= + [1.87]

Again, by neglecting the pressure term, equations [1.86] and [1.82] can be used to calculate the enthalpy of fusion:

( )0 0 0f (A) A a 0(A) 0 ANh h h ε ε< > < >= − = − [1.88]

By combining equalities [1.87] and [1.88], we find:

00f A

f(A)

3R ln Rf

hs

Tνν

< >= + = [1.89]

In general, the Einstein frequency of the solid is slightly greater than or similar to that of the liquid A (A)ν ν< > ≥ . Relation [1.89] shows that the entropy of fusion will not be overly different to R, and the melting point will practically be given by the ratio 0

f / Rh for practically all molecular substances.

NOTE 1.4.– By considering relation [1.89] and the definition of the Gibbs energy of fusion, we can write:

( )0 0f f 1

f

Tg T h

T= − [1.90]

1.6. Diagram for the pure substance and properties of the triple point

Figure 1.8(a) represents certain isotherms in the (P, V) diagram which extend across the whole of the domain, ranging from the molar volume of the solid to that of the gas. The solid lines represent the isotherms at three


temperatures T1, T2 and T whilst the dotted lines delimit the two-phase domains.

– at temperature T1< T, we have a two-phase platform of equilibrium of the gas with the solid;

– at temperature T 2> T, we see the melting and then vaporization of the liquid, with the two corresponding two-phase platforms;

– at temperature T, there is a single platform, which goes from the molar volume of the solid to that of the gas. This transition delimits the two-phase solid–gas domain. On that platform, the two biphasic solid–liquid and liquid–gas domains are also based. Thus, at that temperature and pressure of those platforms, there are three phases, and the Gibbs variance of the system becomes equal to zero.

Liquide

TTE

gaz

C

E

F

Ba

vv<A> v(A) v{A}

TT1

T2

C

Solide

Liquide

Liqu

ide

+ so

lide Liquide

+Gaz

Solide +Gaz

Gaz

P

b

P

solide

I

A

Liquid

Solid

Solid + Gas

Liquid + Gas Li

quid

+

Solid

Gas

a b

T2

T

T1

V<A> V(A) V{A}

V

P

P

A

Solid Liquid

E

C

E

F

TTE

Gas

Figure 1.8. a) Isotherms of the pure substance; b) pressure–temperature diagram

Figure 1.8(b) shows a representation of all three transformations in the P–T diagram, displaying the curves which correspond to each of the equilibria. The liquid–gas equilibrium curve IC finishes at point C, which is the critical point of coordinates. The solid–liquid equilibrium curve IB, which is almost vertical, is unlimited. The curve AI represents the solid–gas equilibrium. As we can see, it is possible to go around the point C starting in the gas domain and going to the liquid domain without crossing the curve IC.

At point I, the three phases are present; this is the point with zero variance, known as the triple point. This point is unique because if three phases are present, then the temperature and pressure are perfectly determined (null variance).


At the triple point I, as the enthalpy is a function of the state, it does not vary along the entire boundary of a closed circuit. Thus if we imagine a cycle around point I, we immediately find the following relation between the enthalpies of melting, sublimation and vaporization:

s f vh h h= + [1.91]

Consider a point E located at the extension of the sublimation curve, sufficiently close to I so that the enthalpies of sublimation and vaporization are the same as at point I. The point E defines a temperature TE.

Immediately, from Figure 1.8(b), we can read the following inequality at temperature TE between the pressures of vaporization and sublimation:

{ }( )( ) ( )( )vap subl

AAEE TT

P P<

[1.92]

However, in light of equation [1.91], we deduce that:

svh h<

[1.93]

The two inequalities [1.92] and [1.93] enable us to write:

( )( ) ( )( )vap sublA A2 2R RE E

v s

T TE E

h hP P

T TΔ Δ< [1.94]

By comparing with expressions [1.40] and [1.74], we find:

( ) ( )vap sublA Ad d

d dE ET T

P PT T

< [1.95]

Thus, at the triple point, the slope of the sublimation curve is greater than the slope of the vaporization curve.

The three curves are not necessarily limited to their point of intersection I; they can be stretched (as illustrated by the dotted lines in Figure 1.8(b)), but we shall show that on these extended curves, the equilibria in question are metastable.


Indeed, by combining the inequality [1.92] with relations [1.45] and [1.77], we can write:

sexp expR R

vg gT T

− > − [1.96]

so:

sv g g> [1.97]

In view of the definitions of v g and s g , inequality [1.97] gives us:

{ } ( ) { } AA A Ag g g g< >− > − [1.98]

Hence the affinity of the fusion reaction is written as:

( ) ( ) AA A AA 0g g< >< >→ = − < [1.99]

Thus according to de Donder’s inequality, at point E the solid state is unstable in comparison to the liquid state, and therefore the equilibrium of sublimation is metastable. This conclusion merely confirms the difference in slope between the curves of sublimation and vaporization, and means that the extension of the sublimation curve cannot pass beneath the liquid vaporization curve.

1.7. Allotropic and polymorphic varieties of a solid

We say that a substance has two allotropic varieties if it exists in the solid state in two different crystallized forms – e.g. sulfur α and sulfur β, or white phosphorus and black phosphorus. If the substance is not a simple substance but a compound one, we say that the solid exists in two polymorphic forms. Thus, ice exists in six polymorphic forms, meaning that ice can exist in six different crystalline forms. The transformation from one form, which we shall call phase α, into another variety (phase β) is known as the allotropic or polymorphic transition αβ, and will be represented by the balance equation [1R.5]:

<A>(α) = <A>(β) [1R.5]


1.7.1. Enantiotropy

We say that there is enantiotropy (from the Greek, meaning transformable in both directions) if, in the (P, T) diagram, there is a region of stability for each of the polymorphic (or allotropic) forms. Such a case – which is found, for instance, with the two allotropic varieties of sulfur – is represented in Figure 1.9(a).

Two clear solid domains appear, with transformation from one to the other, melting of the high-temperature form and sublimation of each of the two forms. The result is that the triple point I1, which would exist in the absence of the β form, is replaced by three new triple points:

– point I1 where the two solid phases and the vapor phase coexist;

– point I2 where the high-temperature solid phase β, liquid and vapor coexist;

– and potentially a triple point I4, which is not shown in the figure, in the upper part where the two solid phases and the liquid phase would coexist.

A triple point I3 can be materialized, and appears as a triple point where the vapor phase, liquid phase and low-temperature solid phase all coexist: thus three metastable phases at that point.

P

T

I1

I2 I3

I4

Liquid

<A>α <A>β

Gasα-gas β-gas

αβ β-liq

M N

Q

P R

P

T

<A>β

<A>α

Gas

Liquid

α-gasβ-gas

α-liqM QP R

a) b)

Figure 1.9. Polymorphic forms: a) enantiotropy; b) monotropy

At the pressure corresponding to that of point M, at the temperature of point H, TH, a mole of the phase α transforms into a mole of the β phase


when the amount of heat corresponding to the molar enthalpy of transformation at that pressure, ( ) ( )A A

h h h α< > < >= − , is injected into the

system. The heated β phase melts at the temperature TP of the point P, absorbing that amount of heat which corresponds to the enthalpy of fusion

( )( )

f ( )A Ah h h= − . The liquid phase then vaporizes.

A rapid increase in temperature at point M may prevent the allotropic transformation from α into β and lead to the direct melting of the α phase without passing through the β phase. This melting occurs at the temperature TQ of the point Q, which is lower than TP, but nevertheless the liquid phase thus obtained is metastable in relation to the liquid, up to the temperature TP. By knowing the enthalpies of fusion of the β phase and of the αβ transformation, and the corresponding equilibrium temperatures, we shall be able to calculate the melting point TQ of the metastable phase.

Let us express the difference between the Gibbs energies of fusion of the two solid phases at the same temperature. Using the definitions, we can write:

( ) ( )( ) ( )

f f A AA Ag g g g h T s− = − = − [1.100]

As Ah and As are, respectively, the enthalpy and entropy of the

αβ transformation at temperature TH, the two solid phases are in equilibrium, so their Gibbs energy of transformation is null: 0g = , and therefore:

hT

s= [1.101]

Consider TP, which is the melting point of the stable β phase for which we have ( )

f ( ) 0Pg T = . If we express the enthalpy of fusion of the α phase at the melting point of that phase, TQ, relation [1.100] becomes:

( )f Q Q( )g T h T s= − [1.102]


Also, by applying relation [1.90]:

( ) ( ) Pf Q f

Q

( ) 1 Tg T h

T= − [1.103]

Thus, by taking account of relations [1.102] and [1.103]:

( ) PQ f

Q

1 Th T s h

T− = − [1.104]

Additionally, at the triple point I3, we would have a relation similar to [1.91]:

( ) ( )f f Hh h T h= + [1.105]

By substituting back into expression [1.104] and taking account of equation [1.101], we obtain:

( )( )P Pf

H Q

1 1T Th h h

T T− = + − [1.106]

which can be written in the form:

( )( )

fQ P ( )

f /P H

h hT T

h T T h+

=+

[1.107]

This calculation contains the hypothesis of phase-change enthalpies that are independent of the temperature.

Ostwald’s law applies to metastable phase transformations. For example, a liquid, when supercooled at M, will solidify, first giving the metastable β phase and then the stable α phase.

Figure 1.10(a) shows the Gibbs molar energy over the course of the path MR from Figure 1.9(a).


For ionic crystals, it is theoretically possible to calculate the temperature of transition between two polymorphic varieties by writing that, at that temperature, the lattice energies of the two varieties computed are equal. Unfortunately, the calculations of lattice energy are too imprecise – particularly because of the repulsive term – for such a theoretical determination to be possible with an acceptable degree of accuracy.

TQ

TPTH TQ TH

g0(α) g0(β)

g0(liq)

G

T

G

T

g0(α)

g0(β)

g0(liq)

a) b)

Figure 1.10. Diagrams of Gibbs energy in the case of a) enantiotropy; b) monotropy

1.7.2. Monotropy

Monotropy (from the Greek, meaning change in only one direction) is the existence of two allotropic varieties, one of which has a stable domain. We can see that the diagram in Figure 1.9(b) is a normal diagram, but in dotted lines, we also see the appearance of a metastable β phase. As we follow path MR, the Gibbs molar energy is given by Figure 1.10(b), the enthalpy of the metastable β phase is always greater than that of the other phases. When cooled, the supercooled liquid will solidify, first forming the β phase and then the α phase (Ostwald’s law).

1.7.3. Transition from enantiotropy to monotropy and vice versa

It may happen that, at sufficiently high levels of pressure, the respective positions of curves of transformation TT’ and fusion I1F1 and I2F2 will become inverted. In that case, the curves intersect at the fourth triple point I4. At pressures higher than that corresponding to that point I4, the enantiotropic varieties become monotropic (Figure 1.11(a)) and the monotropic varieties become enantiotropic (Figure 1.11(b)). The first case applies to sulfur, while the second applies to carbon (stable graphite at


low pressure and stable diamond at high pressure), and to white phosphorus with black phosphorus.

I4 α

Liquid

β

P

T

I4

α

Liquid

βP

T

a) b)

Figure 1.11. Transition a) from enantiotropy to monotropy; b) from monotropy to enantiotropy

1.8. Mesomorphic states

The transition of a pure substance from the solid to the liquid state usually occurs quite suddenly when a very specific temperature is reached. However, there are some substances known as mesogenic substances, in which the phase change appears to happen in several steps. These steps demonstrate the existence of intermediate states, between the solid crystalline state and the liquid state, known as mesomorphic states. This phenomenon was first observed by F. Reinitzer in 1888; Reinitzer had noticed that cholesteryl hydrochloride melted at 145.5°C into a cloudy liquid, then turned into a colored liquid and finally into a clear, limpid fluid only at 178.5°C. He then remarked that the intermediate states with liquid appearance were anisotropic, and resulted in the existence of birefringence (double refraction). X-ray diffraction studies at small and large Bragg angles reveal that, beginning with the crystallized solid, we first see the formation of flat crystals, layers and leaves, which is known as the smectic state (Figure 1.12). As the temperature is increased, this state transforms into filamentary (whiskery) crystals, which is known as the nematic state (Figure 1.12), and by further increasing the temperature, we finally obtain the clear, isotropic liquid state.


Etat nématique Etat smectiqueSmectic state Nematic state

Figure 1.12. Illustration of the smectic and nematic states

The filaments or whiskers seen in the nematic state, which is the closest to the true liquid, are mobile in relation to one another and are disordered; their orientation is different from one elementary volume to another. That orientation is susceptible to a magnetic field, which gives rise to the current main usage of that state, known as liquid crystals, in electronic displays.

This progressive melting, which is a property encountered in many substances (mainly organic), is attributable to anisotropy of the forces of cohesion of the molecular crystals, which leads to the need for different temperatures to overcome these forces and break the bonds depending on the direction. Thus, at a certain temperature, the forces of cohesion in a direction Ox would be broken, leading to melting in that direction, and we would see the formation of the flat crystals characteristic of the smectic state. A further rise in temperature would break the cohesive forces in another direction Oy, leading to the formation of the whiskery crystals of the nematic state, and finally, a third temperature increase would overcome cohesion in the third direction Oz, producing the isotropic liquid. Thus, within a given temperature range, melting is replaced by three first-order transformations (Figure 1.13), which each occur at a very specific temperature, almost independently of the pressure.

NOTE 1.5.– We use the term thermotropic to speak of crystals which undergo mesomorphic melting under the influence of temperature, as opposed to lyotropic substances, which also exhibit mesomorphic states by progressive dissolution of the substance by gradual addition of appropriate solvents.


P

T T1 T2 T3

Crystalline state

Smectic state

Nematic state

Liquid state

Figure 1.13. Successions of thermotropic mesomorphic states

A de Gennes model, which generalizes the Landau model to first-order transitions, is applicable to the transformations of mesomorphic states. For example, the order parameter is defined in the nematic state by a function of the orientation of the filamentary crystals, whose value is zero in the liquid state and non-zero in the nematic state. In the case of 1st-order transformations, in order to account for the latent heat of transformation, the expansion [1.27] of the Gibbs energy as a function of the order parameter retains a term of an odd degree (the third-degree term), which is independent of temperature and negative.

2

Properties of Equilibria Between Binary Phases

A binary system is a system that contains two components, which may be in molecular, atomic or ionic form, which are not liable to react with one another.

2.1. Classification of equilibria between the phases of binary systems

A binary system, as defined above, must necessarily contain two independent components. In the context of thermodynamics, with two external intensive variables (pressure and temperature), the Gibbs variance of such a system will be given by the relation:

2 2 4Dv φ φ= + − = − [2.1]

As the variance must be null or positive, the number of phases satisfies the following inequality:

4φ ≤ [2.2]

If they are not single-phase, such systems can be characterized by the nature of the phases present, and then by the distribution of the two components between those phases. In two-phase systems, we distinguish between:

– equilibria involving only the condensed phases, which may be;



- equilibria between a solid and a liquid,

- equilibria between two liquids,

- equilibria between two solids;

– equilibria involving a gaseous phase, which may be;

- equilibria between a liquid and a gas,

- equilibria between a solid and a gas.

In this type of system, there are as many equilibria between independent phases as there are couples of independent phases. If α, β and γ denote three phases, for instance, we would have:

– an equilibrium for component 1A between the two phases α and β: 1A in the α phase and 1A in the β phase:

( ) ( )1 1A A= [2R.1]

– an equilibrium for component 1A between the two phases α and γ: 1A in the α phase and 1A in the γ phase:

( ) ( )1 1A A= [2R.2]

– an equilibrium for component 1A between the two phases β and γ: 1A in the β phase and 1A in the γ phase:

( ) ( )1 1A A= [2R.3]

It is easy to see that the third equilibrium is a linear combination of the first two. Hence, there are only two independent equilibria.

Similarly, for the compound 2A , we would also have two independent equilibria chosen from among the following three:

– an equilibrium for the component 2A between the two phases α and γ:

2A in the α phase and 2A in the γ phase:

( ) ( )2 2A A= [2R.4]

Properties of Equilibria Between Binary Phases 45

– an equilibrium for component 2A between the two phases α and γ: 2A in the α phase and 2A in the γ phase:

( ) ( )2 2A A= [2R.5]

– an equilibrium for component 2A between the two phases β and γ : 2A in the β phase and 2A in the γ phase:

( ) ( )2 2A A= [2R.6]

In a two-phase system, we would have the two equilibria [2R.1] and [2R.4].

2.2. General properties of two-phase binary systems

Two-phase binary systems are extremely numerous and important, and of differing natures. We shall now discuss the properties common to all these types of systems.

2.2.1. Equilibrium conditions for two-phase binary systems

Consider a system comprising two phases α and β, and two components A1 and A2. We let 1x denote the molar fraction of component 1A in the mixture. Thus, (1-x1) is the molar fraction of component 2A . In each of the phases, the molar fractions of the components from 1A are ( )

1x α and ( )1x β . The molar

fractions ( )11 x α− and ( )

11 x β− are those of the components from 2A . We shall use y and 1 y− to denote the molar fractions of each of the phases such that:

number of moles of phasetotal number of moles of the two phases

y = [2.3]

De Donder’s equilibrium condition, when applied to equilibria [2R.1] and [2R.4], gives us the equality of the chemical potentials between the components of the same component in the two phases:

( ) ( )1 1μ μ= [2.4]

( ) ( )2 2μ μ= [2.5]


We explicitly state the chemical potentials by choosing a convention for the activity coefficients; ( )0

1αμ and ( )0

1βμ represent the partial Gibbs molar

energies of substance 1A in the reference state, in the same state of segregation as the α phase and the β phase. Based on relation [2.4], we obtain:

( ) ( ) ( ) ( )0 01 1 1 1R ln R lnT a T aμ μ+ = + [2.6]

We deduce from this that at equilibrium, the activities of component A1 must fulfill the condition:

( )

( )

( ) ( )0 01 1 1

1

expR

aTa

μ μ−= − [2.7]

Similarly, using relation [2.5], we obtain the following for compound A2:

( )

( )

( ) ( )0 02 2 2

2

expR

aTa

μ μ−= − [2.8]

2.2.2. Conditions of evolution of a two-phase binary system

In order to define the conditions of evolution of the equilibria when a variable is modified, we state the equality of the variations in chemical potential for the reactions [2R.1] and [2R.4], which gives us:

( ) ( )1 1d dμ μ= [2.9]

( ) ( )2 2d dμ μ= [2.10]

These two equations enable us to define all possible states of equilibrium between the two phases when we alter an intensive variable such as the temperature or pressure, or if we add or remove a certain amount of a component of the mixture. Obviously, it is possible to expand these last two relations [2.9] and [2.10] in terms of activities or of partial molar enthalpies and entropies of the standard state, as we shall do when demonstrating the Gibbs–Konovalov theorem in section 2.3.2.2.3.


2.3. Graphical representation of two-phase binary systems

Two families of graphical representation are useful and easy to use for the equilibria of binary mixtures: Gibbs energy graphs and phase diagrams.

2.3.1. Gibbs energy graphs

On a Gibbs energy graph, the ordinate axis shows the Gibbs molar energy of a system – i.e. its Gibbs energy for one mole of mixture – and the abscissa axis shows its composition. The graph is plotted for a given temperature and pressure. For equilibria between phases, these diagrams are constructed either on the basis of models of solutions or using thermodynamic data drawn from experience when the phase is stable. For an α phase, the Gibbs molar energy is written as follows, using the pure compounds as states of reference (convention I):

( ) ( ) ( )( )0( ) 0( ) ( ) ( )1 2 1 1 1 1R ln R ln 1mG g g T x T xγ γ= + + + − [2.11]

and similarly for a second β phase.

We can distinguish two types of curves obtained from these relations: those which exhibit only an extremum (Figures 2.1 and 2.2(a)) and those which exhibit two inflection points and three extrema (Figure 2.2(b)).

In the first case, to represent the two phases, it is necessary to have two curves – one for each phase – obeying equations similar to [2.11]. These two curves may exhibit different respective positions in the diagram.

Figure 2.1. Relative arrangements of the Gibbs energy curves of two binary phases


In Figure 2.1(a), the concave parts of both curves G(α) and G(β) face downwards. Thus, for these two curves, the second derivative of the Gibbs energy in relation to the composition is negative, which means that, for all compositions, both phases are unstable in relation to the two pure substances.

In Figure 2.1(b), the concavity of curve G(α) faces upwards, so its second derivative is positive in all compositions, and it is stable in relation to the pure miscible substances in all proportions. The curve G(β) is downwardly concave, so its second derivative is negative, and the β phase is unstable in relation to the two pure substances and also in relation to the α phase. Thus, the stable system would be in the form of the α phase.

a) b)

S1

O

xsmG

C2

S2

1C' 2C'

C1

1 0A1

1x A2

A1A2

C1

1'C 2'C

C

C2 D

0 1

1μ

2μ

)α(G

)β(G

Gm

1x

Figure 2.2. Representation of two Gibbs energies of two phases at equilibrium

In Figure 2.1(c), both curves G(α) and G(β) are upwardly concave. Thus, for these two curves, the second derivative of the Gibbs energy in relation to the composition is positive, meaning that, for all compositions, the two phases are stable in relation to the two pure substances. However, the Gibbs energy is lower for the α phase, which is therefore stable in relation to the β phase. The latter, which is stable in relation to its pure components but unstable in relation to the α phase, is said to be metastable.

In Figure 2.2(a), the two curves G(α) and G(β) are both upwardly concave. Thus, for these two curves, the second derivative of the Gibbs energy in relation to the composition is positive, which means that, for all compositions, the two phases are stable in relation to the two pure


substances. However, these two curves intersect, and thus there are two points C1 and C2 that have a common tangent, which means that, at these points, the chemical potentials of component A1 represented by segment A2C is the same in both phases. Similarly, the segment A1D, which is the chemical potential of component A2, is also the same in these two phases. Thus, the phases which correspond to the compositions

1Cx (abscissa of point C1) and

2Cx (abscissa of point C2) are at equilibrium, because equations [2.4] and [2.5] are satisfied. Thus, we distinguish three domains of composition:

For overall compositions xM’ of a mixture of the two phases such that 1 2C M Cx x x< < (Figure 2.3(a)). The mixture includes the α phase at the

composition P'x and the β phase at a composition Q'x . This phase is such that the three points PMQ are aligned. Thus, we can prepare all possible mixtures of the phases.

a)

H P

M Q

C1

C2 G(α)

G(β)

Gm

x1C’1 C’2 Q’ P’ M’

C

Q

b)

C’1

G(α)

C2

C1

P M

G(β)

Gm x1

H

C’2Q’ M’ P’

Figure 2.3. Stability of systems formed of two phases

Let y represent the fraction of the α phase in the mixture; by finding the balance of component A1 in the whole of the mixture, we can write:

P ' Q ' M '(1 )yx y x x+ − = [2.12]

From this, we deduce:

Q' M '

Q ' P '

MQ M'Q'PQ P'Q'

x xy

x x−

= = =−

[2.13]


and:

P ' M '

Q' P '

PM1PQ

x xy

x x−− = =−

[2.14]

Let us now show that the Gibbs molar energy of the mixture is equal to the ordinate of the point M. For the mixture, we can write:

( )mix P Q1G yG y G= + − [2.15]

PG and QG respectively represent the Gibbs molar energies of the α and β phases of our mixture.

In addition, if we calculate the ordinate of the point M lying on the straight line PQ – i.e. such that:

P P 'G ax b= + [2.16a]

and:

Q Q'G ax b= + [2.16b]

we obtain:

M mixG G= [2.17]

Thus, the point M is representative of the mixture for its composition and its molar enthalpy.

The tangent common to the two curves is external to these curves, and is situated below them. Hence, every point belonging to the curve ( )G α or curve

( )G β is above the points C1 and C2. Thus, any point M between two points P and Q on these curves is also below the point H, of the same composition as at point M, and whose phases have the same compositions as at points C’1 and C’2. As the Gibbs energy at point M is less than that at point H, the mixture represented by point M is not stable in relation to the mixture represented by the point H. Thus, we conclude that for any overall composition between those of the points of contact of the common tangent,


the stable system is made up of that mixture of the two phases at equilibrium, in varying proportions depending on the composition at point H, and such that the fraction of α phase is given by:

1

2 1

H ' C'

C' C ' 0 2 1 2

H'C' HCC' C' C C

x xy

x x

−= = =

− [2.18]

The overall Gibbs energy of the system is represented by the points H of the segment C1C2, with the equation:

( ) ( )( ) ( ) ( )2 1

2 1 1

1 2

C CC 1 C C

T Tm

G GG G x x G

x x−

= − +−

[2.19]

We shall now examine the region where 1

0 M Cx x< < , where in Figure 2.3(b) the point M, representative of the global mixture, is situated to the right of the point C1.

To create the mixture, as before, we shall use a certain amount of α phases represented by a point P on the curve ( )G α and an amount of the β phase represented by a point Q on the curve ( )G β .

Of course, the abscissa value of the point M must be between that of P and that of Q. If it were not so, it would be impossible to create the mixture of phases to achieve the composition at M, and the pure α phase is the only one that is stable.

If M is between P and Q, then the point M must inevitably be higher than the point H, situated at the same abscissa value and on the curve ( )G (see Figure 2.3(b)), and thus the β phase is the only one which is stable.

Thus, if the molar fraction of the component A1 in the system, at M, is less than that of C1, the only stable phase is the pure α phase. The total Gibbs energy is given by the curve G(α), whose equation is:

( ) ( )( )0( ) ( )1 1 1R ln 1m mG G g T xα αα αγ= = + − [2.20]


For compositions xM such that 2C M 1x x< < , by the same reasoning as in

the previous region, we find the result that the β phase is the only stable one, and the Gibbs energy is given by the curve G(β), whose equation is:

( ) ( )0( ) ( )1 1 1R lnm mG G g T xγ= = + [2.21]

NOTE 2.1.– We can see, from Figure 2.3(b), that the extension of the common tangent C1C2 is situated below the point H, but it is not possible to prepare a mixture of the phases with the compositions represented by C1 and C2 and with the global composition xM, because the abscissa of the point M is not situated between those of points C1 and C2.

NOTE 2.2.– A figure like Figure 2.3(b) is sometimes constructed by representing the curve of the Gibbs energy of mixing of a phase, given, for example, for the α phase, by:

( ) ( ) ( )( ) 0( ) 0( ) ( ) ( )1 2 1 1 2 2- R (ln ln )mix

m mG G g g T x xγ γ= − = + [2.22]

This curve then passes through the points (0,0) and (1,0). In these conditions, the β phase, in order to respect the translation of origin, represents the function ( )transl

mG β :

( )

( ) ( )

( ) 0( ) 0( ) 0( ) 0( ) 0( ) 0( )1 2 1 2 1 2

( ) ( )1 1 2 2

- - -

R ln R ln

translm mG G g g g g g g

T x T xγ γ

= − = +

+ + [2.23]

In the second type of Gibbs energy graph, exemplified by Figure 2.2(b), the enthalpy curve has three extreme points. A single function correctly models the two phases and their coexistence. The curve presents two points of inflection, denoted by S1 and S2. We shall now show that this function contains the phase change. Indeed, in Figure 2(a), we distinguish five domains:

– the interval A2C1, which is presented in exactly the same way as the domain situated to the left of C1 in Figure 2.3(b), so the consequences are the same – i.e. in that interval, we have only one stable phase, which we call the α phase;


– the interval C1S1 exhibits the same characteristics as the interval C1C2 shown in Figure 2.2(a), and consequently we have a mixture of the α phase and β phase with the respective compositions

1Cx and 2C ;x

– the interval S1OS2 exhibits the peculiarity of having downward concavity, and thus the unique phase is not stable in relation to its components, but those components are not stable in relation to the mixture with the composition C1C2. Thus, in that interval, as in the previous case, we shall have a mixture of the same two phases α and β;

– the interval S2C2 is identical, at every point, to the interval C1S1, and thus we draw the same conclusion: stability of the same mixture of phases;

– finally, the interval C2A1 is, at every point, the counterpoint of the right-hand side of point C1 in Figure 2.3(b), so only the β phase will be stable.

In conclusion, on curves with two minima, we draw the same conclusions as in the case of the two curves with one minimum and secants: a unique phase for overall compositions less than

1Cx and greater than 2Cx . Between

the overall compositions 1Cx and

2Cx , we have a mixture of the two phases in variable proportions, with the compositions

1Cx and 2Cx .

2.3.2. Phase diagram in the mono- and bi-phase zones

The phase diagram is the basic tool for the use of the thermodynamic data of a binary system. As a function of either the temperature (isobaric diagrams) or of the pressure (isothermal diagrams), it gives us the nature and composition of the phases at equilibrium.

2.3.2.1. Construction of the isobaric phase diagram in the mono- or biphasic regions

The phase diagram is constructed with the same abscissa axis as the Gibbs energy graph – i.e. indicating both the overall composition of the mixture x1 and those of the two phases ( )

1x and ( )1x . We use the temperature

as an ordinate; the pressure remains constant.

Onto this system of axes we impose the sets of points C1 and C2 from the Gibbs energy graph (obtained at a temperature TM) when the temperature varies (see Figure 2.4).


Figure 2.4. Phase diagram and corresponding Gibbs energy curve

Thus, we obtain a first curve T(α), which is the set of points C1 when the temperature varies, and represents both the composition of the α phase at equilibrium between phases and the separation between the single-phase region (α phase) and the two-phase region. This region is further delimited by the curve T(β) with the set of points C2, which also represents the composition of the β phase at equilibrium. To the right of the curve T(β), the domain is again monophasic, so the β solution is the only stable phase.

The curves T(α) and T(β) are two conjugate curves of the diagram. The points C1 and C2 are known as the conjugate points.

NOTE 2.3.– The diagrams are sometimes represented with the phase compositions expressed in mass fraction (w). As the mass fractions and the molar fractions are not proportional, often the graduation in mass fraction is borne on an abscissa axis placed in the upper part of the diagram.

2.3.2.2. Properties of phase diagrams in regions with one or two phases

We shall now examine a few remarkable properties which are general to all phase diagrams.


2.3.2.2.1. The lever rule or law of chemical moments

As we have seen, the abscissa axis of the phase diagram is exactly the same as that of the Gibbs energy curve, so relation [2.18], giving the proportion of each phase making up a two-phase system represented by a point H in the diagram, is applicable. This is what is known as the lever rule or law of chemical moments.

2.3.2.2.2. Crossing a line in the diagram

As every non-horizontal line in the diagram is the set of points of a boundary C1 or C2 of the Gibbs energy curve, it follows that every non-horizontal line in the diagram separates a single-phase region from a two-phase region, and thus each time that we move around the diagram, we cross a non-horizontal line, the number of phases varies by one unit, and changes from one to two or from two to one.

2.3.2.2.3. Gibbs–Konovalov theorem

The Gibbs–Konovalov theorem governs the shape of the conjugate curves of the graph. When they meet, it is as stated below for isobaric and isothermal graphs.

THEOREM FOR ISOBARIC GRAPHS.– Any meeting between two conjugate lines of the isobaric phase diagram is a tangential meeting around an extremum (i.e. a maximum or minimum) of temperature (Figure 2.5).

THEOREM FOR ISOTHERMAL GRAPHS.– Any meeting between two conjugate lines of the isothermal phase diagram is a tangential meeting around an extremum (i.e. a maximum or minimum) of pressure.

We shall only demonstrate this theorem in the case of isobaric diagrams, but it is perfectly easy to perform a demonstration for isothermal diagrams, mirroring the one below in all aspects.

Figure 2.5. Contacts with maximum and minimum between two conjugate curves in the phase diagram


In describing the nullity of the variation of the chemical potential (extremum of the curve) for the α and β phases of a two-phase binary system containing the components A1 and A2, we obtain:

( ) ( ) ( ) ( )( ) ( )1 1 2 2d d d dm mV P S T x xμ μ− = + [2.24]

( ) ( ) ( ) ( )( ) ( )1 1 2 2d d d dm mV P S T x xμ μ− = + [2.25]

By subtracting relation [2.25] from relation [2.24], we find:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )1 1 1 1 2 2

2 2

d d d d d

d

m m m mV V P S S T x x x

x

μ μ μ

μ

− − − = − +

− [2.26]

By feeding equations [2.9] and [2.10] into expression [2.25], we obtain:

( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )

1 1 1 2 2 2

d d

d d

m m m mV V P S S T

x x x xμ μ

− − −

= − + − [2.27]

However, the sum of the molar fractions of a phase is 1. Hence:

( ) ( ) ( ) ( )2 2 1 11 1x x x x− = − − − [2.28]

By applying this equation to relation [2.27], we obtain:

( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( )1 1 1 2d d d dm m m mV V P S S T x x μ μ− − − = − − [2.29]

or:

( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( )1 1 1 2d d d dm m m mV V P S S T x x μ μ− − − = − − [2.30]

At constant pressure, because ( )1μ is a function only of temperature and

of ( )1x , we can always write:

( )( )

( )( )

( )1 1

1 11

d d dx TTx

μ μμ ∂ ∂= +∂∂

[2.31]


Similarly, for ( )2d μ we find:

( )( )

( )( )

( )2 2

2 11

d d dx TTx

μ μμ ∂ ∂= +∂∂

[2.32]

By substituting these last two equations back into expression [2.30] and making dP = 0, we find:

( ) ( ) ( )( )( ) ( )

( )

( ) ( )( )( )

( )

( )

( )

( ) ( ) 1 21 1

1

1 21 1

1 1

ddm m

TS S x xT T x

x xx x

μ μ

μ μ

∂ ∂− − − − − =∂ ∂

∂ ∂− −∂ ∂

[2.33]

We can demonstrate the following relations for the variations of the chemical potential with the molar fractions:

( )

( )

( )

( ) ( )+∂

∂=∂∂

11

1

1

1 1lnRxx

Tx

γμ [2.34]

and:

( )

( )

( )

( ) ( )

( )

( ) ( )2 2 2

1 1 2 1 1

ln ln1 1R R1

T Tx x x x xμ γ γ∂ ∂ ∂= − = −

∂ ∂ ∂ − [2.35]

By substituting the final two identities into formula [2.33], we find:

( ) ( ) ( )( )( ) ( )

( )

( ) ( )( )( )

( )

( )

( ) ( ) ( )

( ) ( ) 1 21 1

1

1 21 1

1 1 1 1

dd

ln ln 1 1R1

m mTS S x x

T T x

T x xx x x x

μ μ

γ γ

∂ ∂− − − − −∂ ∂

∂ ∂= − − + +∂ ∂ −

[2.36]


This expression can be simplified to:

( ) ( ) ( )( )( ) ( )

( )

( ) ( )( )( )

( )

( )

( ) ( ) ( )

( ) ( ) 1 21 1

1

1 21 1

1 1 1 1

dd

ln ln 1R1

m mTS S x x

T T x

T x xx x x x

μ μ

γ γ

∂ ∂− − − − −∂ ∂

∂ ∂= − − +∂ ∂ −

[2.37]

using the Gibbs–Duhem relation at constant pressure and temperature in the form:

( )

( )

( )

( )

( )

( )2 1 1

1 2 1

ln lnxx x xγ γ∂ ∂= −

∂ ∂ [2.38]

and by substituting this expression into relation [2.37], we are led to:

( ) ( ) ( )( )( ) ( )

( )

( ) ( )( )( )

( )

( )

( ) ( ) ( )

( ) ( ) 1 21 1

1

1 11 1

1 1 1 1

dd

ln 1R 11 1

m mTS S x x

T T x

xT x x

x x x x

μ μ

γ

∂ ∂− − − − −∂ ∂

∂= − + +∂ − −

[2.39]

On condition that ( )1x is different to 1, this expression can be simplified

to:

( ) ( ) ( )( )( ) ( )

( )

( ) ( )( )( )

( )

( ) ( )

( ) ( ) 1 21 1

1

1 1 1

1 1 1

dd

R ln 11

m mTS S x x

T T x

T x x

x x x

α

μ μ

γ

∂ ∂− − − − −∂ ∂

− ∂= −− ∂

[2.40]

If we examine expression [2.40] for abscissa values different to ( )1 0x = ,

on condition that the term in brackets on the right-hand side is not equal to zero, we can see that, on the curve giving the temperature as a function of


( )1x , if the temperature passes through an extremum ( ( )

1d / d 0T x = ) then the two phases have the same composition ( ) ( )

1 1x x= .

Consequently, if the two phases have the same composition, the temperature passes through an extremum.

Observe the bracket on the right-hand side of relation [2.40]. There is nothing to prevent this term from having a value of 0. What we can say is that for perfect solutions, this property is verified, because the derivative

( )

( )1

1

lnxγ∂

∂is zero.

Similarly, for a strictly-regular solution, the defining relation gives us:

( )

( )( )( )11

1

ln 2 1B xTx

γ∂ = − −∂

[2.41]

Only in an extremely special case may B have the value:

( ) ( )( )1 12 1T

Bx x

=−

[2.42]

which would render the bracket equal to zero. Even in this case, there is nothing to say that the compositions of the two phases cannot be equal.

In fact, there is no known example where the bracket on the right of relation [2.40] is equal to zero.

We could again apply a similar reasoning process for the β phase, in which case we would obtain the relation:

( ) ( ) ( )( )( ) ( )

( )

( ) ( )( )( )

( )

( ) ( )

( ) ( ) 1 21 1

1

1 1 1

1 1 1

dd

R ln 11

m mTS S x x

T T x

T x x

x x x

μ μ

γ

∂ ∂− − − − −∂ ∂

− ∂= −− ∂

[2.43]


Thus, the curve T(β) would have the same property as its conjugate, provided that ( )

1x β is different to 1. If the compositions are the same for the two curves, they come into contact, and as they both have a horizontal tangent at that point, they are tangents to one another.

Hence, the Gibbs–Konovalov theorem is demonstrated.

Such an extremum in the phase diagram is known as an azeotropic point. The term comes from a Greek word meaning passage without change.

2.3.2.2.4. The different types of azeotropic points

In practice, we find three types of azeotropic points, illustrated by Figures 2.6(a), 2.6(b) and 2.6(c).

Figure 2.6. Phase diagrams at the maximum; a) at the azeotropic point; b) for a definite compound; c) at the critical point

In the first case shown in Figure 2.6(a), if we descend vertically from the point of the maximum, we cross two simultaneous conjugate lines, passing from a mono-phase zone (β phase) to another mono-phase zone (α phase). Thus, we distinguish two cases:

– if, in the vicinity of that extremum, there is no phase which is never adjacent to one of the axes x1 = 0 or x1 = 1, we say that the mixture, at the composition corresponding to the extremum, is an azeotrope (see Figure 2.6(a));

– if, in the vicinity of a maximum, a new α phase manifests itself which does not have a domain of existence in zones adjacent to one of the axes x1 = 0 or x1 = 1 (the α phase is never in contact with the axes), then the mixture


will be considered a new pure definite compound formed of the two components A1 and A2, whose formula will be given by the proportions of the maximum. This is the case illustrated by Figure 2.6(b).

In the third scenario (Figure 2.6(c)), the maximum (or minimum) lies on the two conjugate curves which intersect and finish at that point, and we experience a vertical shift at M, moving from a single-phase zone (α or β phase which are identical) to a two-phase zone (mixture of α and β phases), then the point M is called a critical point, and the composition of the mixture at point M is the critical composition. The existence of this point entails a direct switch from the α phase to the β phase, above the maximum, without the appearance of an interface: this situation is absolutely similar to the critical point in the pure substance between the liquid and gaseous phases – hence its name. In this case, we see the phenomenon of demixing of the liquids.

At the points where the temperature passes through an extremum, the system is subjected to an extra condition which is the equality of composition of the phases, so there is a decrease by one unit in the number of degrees of freedom of system, which decreases from 1 to 0 at constant pressure, so that the system undergoes the transformation at temperature constant, as a pure substance would do. The difference between an azeotropic mixture and a definite compound is marked by the influence of pressure, if pressure is an equilibrium variable. The temperature of an azeotropic mixture varies with changing pressure, whilst the temperature of the maximum corresponding to a definite compound does not.

2.3.2.2.5. Property of the vertical axes at ( )1 0x = and ( )

1 1x = in the phase diagram

If we consider the two axes ( )1 0x = and ( )

1 1x = , they represent the two pure components A2 and A1 which, by definition, are single-phased. Two cases may be encountered:

– the region which is in contact with one of those axes is a single-phase region, so the corresponding axis does not belong to the diagram – it corresponds only to a specific composition of the phase characterizing that region, for which the other component has a molar fraction of 0;

– the region which is in contact with one of these axes is a two-phase region, so the corresponding axis is a line in the graph which separates the


monophasic domain (infinitely thin). In the first domain, we find only the pure corresponding component. In the second (two-phase) domain, we find a mixture of a phase comprising the corresponding component and another two-component phase.

NOTE 2.4.– In the latter case where the axis is a line in the graph, it can be extended to meet its conjugate line (see Figure 2.7(a)), but the meeting point will not be an extremum because the compositions ( )

1 0x α = and ( )1 1x α = have

been excluded from the domain of application of the Gibbs–Konovalov theorem (see the passage from relation [2.39] to relation [2.40] and its equivalent to establish [2.43]).

2.3.2.3. Particular configurations of a diagram in the regions with one or two phases

We shall now examine two particular cases where the conditions of miscibility of the phases are extreme.

2.3.2.3.1. One component is completely immiscible with the other

In Figure 2.7(a), the component A2 is partially miscible with component A1, which crystallizes in the β phase, whilst the component A1 is totally immiscible with component A2, which crystallizes in an α phase which is therefore pure.

Hence, the diagram is made up of two conjugate lines, one of which is part of the line x1 = 0.

2.3.2.3.2. The two components may be entirely miscible with one another

In Figure 2.7(b), both components are, in certain temperature ranges, completely miscible with one another, so the two phases are solutions. One – the α phase – is a solution of component A1 in component A2. The other – the β phase – is a solution of component A2 in component A1. In fact, there is no longer any reason to distinguish between the two α and β phases. This being the case, no vertical axis is a conjugate line of the diagram. Thus, if the two components are solids, the existence of such a case requires that the two pure solids crystallize in the same system. The solid solution then presents the same crystalline system as the two pure substances.


Figure 2.7. Diagram with a) zero miscibility of a component into the other; b) total miscibility between the two components

2.3.3. Isobaric cooling curves

The isobaric cooling curve, at constant pressure, gives us the temperature of a system over time as it cools.

Consider the isobaric phase diagram in Figure 2.8(a). It shows three domains – one biphasic domain containing the α and β phases, and the other two monophasic, on either side of the two conjugate curves C1 and C2, with one containing the α phase and the other the β phase.

Let us consider a mole of the β phase at composition x1,1, taken to the temperature T0 (Figure 2.8(a)). Allow it to cool naturally to the temperature TA, corresponding to the intersection A of the vertical of overall composition with the curve T(β), which gives the composition of the β phase at equilibrium.

Over a period of time dt of that cooling, by conduction (we shall ignore the other losses) the system loses an amount of heat proportional to the difference between its temperature T and that of the external medium Text, which is the quantity dQ given by:

( )dextdQ T T tλ= − [2.44]


a)

T(α)

T(β)

α

α+β

β

T0

TA

TB

TE

TD

A’

b)

T

x1

T

t

A

BE

D

B’

E’D’

O’1

O1 O’2 O2

x1,1 x1,2

Figure 2.8. Correspondence between phase diagram and cooling curves

During that same period, that amount of heat lost by the system leads to its cooling, such that:

dpdQ C T= [2.45]

pC is the average specific heat capacity of the β phase, which is

similar to the sum of the specific heat capacities of the two components of the system, A1 and A2, weighted by the composition x1,1.

The heat balance leads us to equalize the two expressions [2.44] and [2.45], and by integration, we find:

( )0 ext extexpp

tT T T TCλ= − − + [2.46]

This is the equation for the curve between points O’1 and A’1 in Figure 2.8(b).

Let us now consider the part of the cooling within the two-phase zone between points A and E in Figure 2.8(a). The transformation of the β phase into the α phase as the temperature decreases is an endothermic transformation. By virtue of the laws of equilibrium displacement, this enthalpy is positive. The heat balance, therefore, will contain a new term


which decreases the rate of cooling – hence the appearance of point A’ (Figure 2.8(b)), and the decrease the slope of the cooling curve. This new law will apply up until point E (Figure 2.8(a)). At that point, the whole of the β phase is transformed, we again have a domain comprising only one phase, and the cooling curve again obeys the law [2.46] of natural cooling of a phase.

The cooling experiment performed at another composition x1,2 will give us the curve O’2B’D’, shown in Figure 2.8(b), on which the points of rupture of the slope are shifted, in relation to the points B and D of phase change to the new composition.

Thus, the experimental plot of a cooling curve enables us to pinpoint the temperatures of separation of the domains in the phase diagram. Thus, we can see that by repeating the experiment for different compositions of the diagram, it is possible to plot that diagram.

NOTE 2.5.– In the case of a system with zero variance, such as phase change of a pure substance, the cooling curve exhibits a temperature plateau throughout the phase change, because the temperature of the system is fixed throughout that transformation.

In practice, it is very tricky to locate the points A’, E’, B’ and D’ in the curve shown in Figure 2.8(b). Hence, it is preferable to use differential thermal analysis to find these points. In order to do so, we take two identical crucibles. In one, we place the mixture under examination, and in the other, an “inert” substance as a reference – i.e. one which does not undergo any transformation in the temperature range in question, and whose specific heat capacity and heat conductivity are close to those of the mixture. Thus, frequently, we use aluminum oxide α for oxides, salts and other non-metallic substances, and a metal without phase change to study alloys. The two crucibles each have a thermocouple, which pinpoints their temperature and, by comparing them, enables us to directly measure the temperature difference between the substance (T) under study and the reference (Tref). The two crucibles are placed symmetrically in a kiln, raised to the higher temperature T0 and kept at that temperature long enough for all the transformations to take place. The crucibles are then removed from the kiln and allowed to cool naturally. By plotting the difference in temperature between the two samples as a function of the temperature of the sample studied, we see a baseline close to a plateau, so long as the system does not


undergo any modification, with the two crucibles following the same temperature curve.

At point A, as the temperature of the system sinks less quickly than the reference sample, a peak appears which expands and then disappears when the system reaches point E. The points at which the peak begins to deviate from the baseline mark the two temperatures TA and TE.

T-Tref

TTA TE

Figure 2.9. Detection of temperatures of transformations by differential thermal analysis

2.4. Isobaric representation of three-phase binary systems

Let us now consider binary systems presenting three phases at equilibrium. If a binary system is three-phased, its number of degrees of freedom (at constant pressure) becomes null. Thus, if three phases are at equilibrium, the temperature and composition of the phases are fixed.

2.4.1. Gibbs energy curve

Let T3 denote the temperature at which three phases – α, β and δ – coexist.

In order for the three phases to be at equilibrium, it is necessary and sufficient for the chemical potentials of each of the components to be the same in the three phases, meaning that we have:

( ) ( ) ( )1 1 1μ μ μ= = [2.47a]

and:

( ) ( ) ( )2 2 2μ μ μ= = [2.47b]


This means that on the Gibbs energy graph, the three curves representing each of the phases must, at temperature T3, exhibit a common tangent (Figure 2.10). This common tangent, by the abscissa values at the points of tangency H1, H2 and H3, defines the compositions ( )

1x , ( )1x and ( )

1x of the three phases at equilibrium at that temperature. Thus, we can show that, in the same way as for mixtures with two phases (see section 2.3.1), any mixture whose overall composition x1 lying between ( )

1x and ( )1x will

evolve spontaneously towards a mixture of these three phases, in varying proportions depending on the composition.

Figure 2.10. Representation of tri-phase zones in the Gibbs energy graph and the phase diagram of a binary system

The figure can no longer be used to calculate the proportion of each of the three phases, because with the three unknowns – the fractions of each phase y(α), y(β) and y(δ) – we have only two equations: the sum of the fractions of phase equal to 1 and the balance equation for component A1, which is insufficient.


2.4.2. Isobaric phase diagram in tri-phase regions

As with two-phase mixtures, we construct the isobaric phase diagram based on the curves of the Gibbs energy at the same temperature by preserving the same scales for the abscissas. As the three tangential curves can only exist at the temperature T3, this means that any mixture with the overall composition x1 will appear on the horizontal at T3 between the projections h1 and h3 of the points H1 and H3 which correspond to the α and β phases. The composition of the δ phase will be represented by a point h2 which is the projection of H2, lying somewhere between h1 and h3. For a composition less than ( )

1x α , only the α phase will be present, and for a composition greater than ( )

1x β , only the β phase will be present.

When the three phases are at equilibrium, they are in a state of two-by-two equilibrium, which means that the point h3, as shown in Figure 2.10 for instance, must also be on the equilibrium curve αδ and on the αβ curve. Thus, the points h1, h2 and h3 belong to two-phase equilibria curves.

α α+δ

α+β b)

Γ2

Γ1

Γ’1

α

α+β

α+δ

a) Γ1

Γ’1Γ2

Figure 2.11. Extension of the two-phase equilibria curves at the tri-phase point

The two-phase equilibria curves do not intersect in an arbitrary way at the tri-phase point. For example, in Figures 2.11(a) and 2.11(b), the curves Γ1, Γ

2 and Γ’1 (which is the extension of Γ1) present two possible configurations. However, the configuration shown in Figure 2.11b is not possible, because then the extension of Γ’1 would be in a single-phase domain, although it must always be in a two-phase domain in which the α phase is present – i.e. it must respect the configuration shown in curve Figure 2.11(a).

If we change the temperature – e.g. at a temperature T3+dT which is higher than T3, two scenarios may be found for the position on the Gibbs


energy graph of the minimum of the curve Gδ, in relation to the tangent common to the two curves Gα and Gβ.

In the case represented in Figures 2.12(a) and 2.12(b), on the Gibbs energy curve, the minimum relative to the intermediate δ phase is situated below the tangent common to the two curves of the α and β phases. Thus, there is the possibility of a common tangent between α and δ, so a mixture α + δ is stable between the abscissa values of the points of contact of that tangent, Hα and Hδ. Similarly, a tangent common to the curves G(δ) and G(β) leads to a zone of stability of the mixture δ + β between the abscissas of points H’δ and Hβ. It follows that all the points on the curve G(δ) situated between the abscissa values of points Hδ and H’δ have a Gibbs energy lower than that of those points, so between those two points the δ phase is the only stable one.

Thus, we obtain a phase diagram represented by Figure 2.12(b), which is known as a eutectic diagram.

Figure 2.12. Eutectic and peritectic tri-phase zone

In the case represented in Figures 2.12(c) and 2.12(d), on the Gibbs energy curve in the first figure, the minimum relative to the intermediate δ


phase is situated above the tangent common to the two curves of the α and β phases. Hence, between the points Hα and Hβ, only a mixture of the two α + β phases is stable; the δ phase cannot exist. Hence, we find the so-called peritectic diagram shown in Figure 2.12(d).

NOTE 2.6.– At a temperature T3-dT, which is slightly less than the temperature of the platform, we may legitimately wonder whether it is possible to have either of the two configurations given by Figures 2.12(a) and 2.12(c). It is easy to show that not all choices are possible: indeed if, in Figure 2.12(b), below the platform, we had a configuration identical to the upper part, the δ phase would constantly be between the α and β phases, so there would never be a tri-phase zone. Similarly, if in Figure 2.12(d) we find the mixture α + β below the platform, there would never be a δ phase and thus never a tri-phase zone. Hence, the eutectic configuration and the peritectic configuration defined in Figures 2.12(b) and 2.12(d) are the only ones possible.

2.4.3. Isobaric cooling curves with tri-phase zones

Figure 2.13(b) shows the two cooling curves which correspond to the compositions Xa and Xb in the phase diagram shown in Figure 2.13(a).

Figure 2.13. Cooling curves at different compositions intersecting the triphase zone

At the abscissa value Xa, up to temperature TB, we have cooling of only one phase. Between the temperatures TB and T3, we have cooling of a two-phase zone, as we saw in section 2.3.3. At the temperature T3 the three


phases are present and within the system, having no degree of freedom, the temperature remains constant. Thus, we have a platform whilst the δ phase transforms into the α and β phase depending on the reaction, which takes place from left to right on cooling:

= a + [2R.7]

This is what is known as a eutectic reaction.

Below T3, when the previous reaction is finished, we again see cooling of a two-phase zone.

If we place ourselves at abscissa Xb, which corresponds to that of point t2. Up to temperature T3, we have the conventional cooling of a single phase. Having reached point t2 the eutectic reaction takes place, so again we have the temperature platform; except that now, it is longer than in the previous case, because the amount of δ phase to break down is greater than before. When the δ phase is completely transformed, we see the cooling of a two-phase mixture.

In reality, because of the existence of the eutectic reaction, we find two origins at the α phase after the temperature T3: one part that existed before the reaction and one part due to the reaction. Similarly, if we perform cooling at a higher abscissa than that of point t2, the β phase below temperature T3 will have two origins: one part that existed above T3 and the other part from the eutectic reaction. This result explains why, very frequently, we find an artificial division of the region below the eutectic platform by a vertical line passing through t2. On the left, this vertical marks the stability of a mixture of a phase α and the eutectic (which, itself, is composed of a mixture α + β in proportions determined by the application of the lever rule between the abscissa values of the points t1, t2 and t3). To the right of this vertical, we find a mixture of the β phase and the same eutectic mixture. This distinction is justified by the morphological examination of the mixtures of phases below the eutectic point in the case of solid phases, to which we shall return.

If, instead of choosing a eutectic diagram, we had worked with a peritectic diagram, then in the same conditions we would have found a platform of temperature. In this instance, this platform would be due to a


peritectic reaction, which takes place from left to right as the temperature drops, and examining Figure 2.12(d) leads us to write:

a + = [2R.8]

We find these different types of diagrams essentially in the equilibria between a solid and a liquid, and between two solids.

2.5. Isothermal phase diagrams

If the temperature is now kept constant, we can use the same Gibbs energy curves to plot the points of tangency common when pressure is varied and we obtain an isothermal phase diagram in which pressure is on the ordinate axis and, as before, composition is on the abscissa axis.

These isothermal diagrams are used only in liquid–vapor systems, because pressure has practically no influence on condensed-phase systems.

Figure 2.14. Obtaining a composition curve from an isothermal diagram


Stemming from the same points on the Gibbs energy curves, the diagrams of isothermal binary phases have the same properties and the same configurations as the isobaric diagrams. We have seen that they respect the Gibbs–Konovalov theorem for the extrema of pressure. If the isobaric diagram includes an azeotropic mixture, the isothermal diagram has one too; yet nonetheless, we shall see that for a maximum isobaric azeotrope, there is a corresponding minimum isothermal azeotrope.

The fact that liquid phases are less often immiscible than solid phases renders universally-eutectic diagrams much rarer. Also, as gases are always miscible with one another, it is not possible to find peritectic points, so Figure 2.12(d) is not possible in vaporization diagrams.

2.6. Composition/composition curves

Sometimes we use composition curves, which show the composition of a phase ( ( )

1x ) as a function of the composition of the other ( ( )1x ) at a given

pressure or temperature. We can easily obtain these curves on the basis of an isobaric or isothermal phase diagram. We can cite the example of the plot of an isobaric composition curve.

2.7. Activity of the components and consequences of Raoult’s and Henry’s laws

Figure 2.15 shows the profile of the activities ( )ia of each of the

components in each of the phases as a function of the compositions. The curve ( )

1 shows the activity of component A1 in the α phase; the curve ( )2α gives the activity of component A2 in the α phase; the curve

( )1 illustrates the activity of the component A1 in the β phase; the curve ( )2 gives the activity of component A2 in the β phase. At the center of the

figure, we see the two-phase zone in which there is no phase with a composition corresponding to that zone.

The ordinates of points M1(α) and M1

(β) fulfill the condition of equality (equation [2.3]) of the chemical potentials. Similarly, the points M2

(α) and


M2(β) fulfill the condition of equality [2.4] of the chemical potentials. Except

in very specific cases, the points M1(α) and M1

(β) have different ordinates, and the same is true for the points M2

(α) and M2(β).

For a phase – say for example the β phase – if the solution is sufficiently dilute, then we can apply Raoult’s law for component A1, which plays the role of the majority solvent, with Henry’s law for component A2, which plays the role of the minority solute. Thus, we find the Raoult and Henry lines, respectively represented for each component in each phase.

Figure 2.15. Activity of the components in the phases

Having examined the general properties of equilibria between phases in systems with two components (binary systems), we shall now devote Chapters 3 and 4, respectively, to the features of equilibria between the condensed phases and to liquid–vapor equilibria.

3

Equilibria Between Binary Condensed Phases

When looking at the equilibria between condensed phases, we can usually discount the influence of pressure – at least within that parameter’s normal range of values. In fact, the role played by pressure only becomes apparent when it is extremely high, and in the case of membrane equilibrium (see section 3.5). Thus, when the pressure is normal, chemical systems can be viewed as having only temperature as an intensive variable. All data will generally be evaluated at the standard pressure of 1 bar.

If the system contains two phases, there is now only one degree of freedom. This means that if the temperature is set and remains constant, then the molar fractions of the components in each of the phases are also fixed. If the system is three-phased (triphasic), there are no longer any degrees of freedom – there is only a state of equilibrium in which the temperature and the different molar fractions are fixed.

We distinguish:

– equilibria between condensed phases of the same nature: liquid–liquid or solid–solid equilibria;

– equilibria between a liquid and a solid.

The first of these lead to the phenomenon of demixing. The second lead to phase diagrams, which are of considerable practical importance – particularly in materials science.



3.1. Equilibria between phases of the same nature: liquid–liquid or solid–solid

Two solids or two liquids at equilibrium are said to be miscible if the mixture contains only one phase, or immiscible if the mixture comprises two phases.

In the case of liquids, thermodynamics experts do not draw a distinction in terms of structure, considering them all to be in the same state of segregation. In the case of solids, the two phases generally differ in terms of their crystalline system.

We see that a solution with multiple components undergoes demixing if it spontaneously separates into two phases – β and α – which are also solutions of the same components, but of different compositions. In general, if the solution is liquid, the two solutions α and β obtained after demixing are also liquids. If the solution α is a solid solution, the solutions α and β obtained after demixing are also solid solutions. As demixing only involves condensed phases, pressure has practically no influence upon it, so all data are quoted at the pressure of 1 bar.

3.1.1. Thermodynamics of demixing

The phenomenon of demixing, separating two phases which differ only in terms of their composition, can only take place if the curve of the Gibbs energy has – at least within a certain temperature range – three extrema, similar to the shape of the curve shown in Figure 2.2(b). This configuration facilitates the existence of a tangent common to two points of the same curve. These two points of tangency give us the compositions of the two conjugate points of the phase diagram, at the temperature at which the Gibbs energy curve is obtained.

In general, miscibility increases with temperature, and at a temperature T1 which is higher than temperature T2 on the curve with two minima, the Gibbs energy curve now has only one minimum (Figure 3.1). This means that at a higher temperature there is only one phase, and thus the two conjugate lines intersect, giving an azeotropic point, which is a critical point (see section 2.3.2.2.4). Thus, for a maximum critical temperature, it will be

Equilibria Between Binary Condensed Phases 77

the miscibility gap, with a closed top as shown in Figure 3.2(a). We can therefore switch from absolutely any point in the domain of the solid solution α to the domain of solution β by increasing the temperature and adding the component A1. Such cases arise, for instance, with mixtures of water and dimethylamine.

Figure 3.1. Gibbs energy of a solution undergoing demixing

More rarely, in other cases miscibility increases as temperature decreases. This is called retrograde solubility. Here, the temperature T1 is lower than T2, the demixing domain is present above a critical demixing temperature, and we obtain a miscibility gap at a minimum temperature such as that represented by Figure 3.2(b). This may be encountered, for instance, with mixtures of water and piperidine.

Finally, it sometimes happens that the system exhibits both a higher and a lower critical temperature of demixing, so that the two-phase domain is closed; we say that we have a demixing gap or window. This is the case represented in Figure 3.2(c). This may be encountered, for instance, with water–nicotine mixtures.


Figure 3.2. Demixing zones; a) at maximum critical temperature; b) at minimum critical temperature; c) demixing loop

It is worth noting that the existence of critical temperatures of demixing is more common with liquids than with solids. Very often, in the case of solids, the domain of immiscibility is limited to higher temperatures by fusion (melting). Referring to Figure 2.12(b), at a temperature lower than the three-phase zone we do indeed have a miscibility gap that is not limited by a critical temperature, but by melting, represented by the horizontal line t1t3.

To calculate the two compositions of the phases at equilibrium, ( )1x and

( )2 ,x we need only apply relations [2.7] and [2.8] for each component.

In the case of liquids, for two solutions which differ only by composition but are made up of the same two species, those species have the same Gibbs molar energy in the pure state in both phases:

0( ) 0( )1 1g g= [3.1a]

and: 0( ) 0( )2 2g g= [3.1b]

Thus, relation [2.7] gives us:

( ) ( ) ( ) ( )1 1 1 1x xγ γ= [3.2a]

and:

( ) ( )( ) ( ) ( ) ( )1 2 1 21 1x xγ γ− = − [3.2b]


By writing the activity coefficients as a function of the compositions, we obtain a system of two equations in terms of ( )

1x and ( )1 ,x whose solutions,

if they exist ( 10 1x< < ), give the abscissa of the conjugate points of the phase diagram. The biphasic zone obtained between the two conjugate lines when the temperature varies is called the miscibility gap.

NOTE 3.1.– In this particular case of equilibrium between two immiscible liquid phases, the points M1

(α) and M1(β) in Figure 2.15 have the same

ordinate value. The same is true for the points M2(α) and M2

(β).

In the case of solids, the difference between the single-phase domains is often due to a difference in the crystalline systems, so we can no longer apply equation [3.1]; the two phases are no longer in the same state of aggregation and we must resort to the general expression [2.7]. Furthermore, in this case, as the component A2 crystallizes in the α phase in the pure state, the Gibbs energy 0( )

2g to be used in the calculations is that of the component A2, which crystallizes in the system of the β phase. However, this crystallization is often impossible, and it follows that 0( )

2g is a hypothetical value. The same is true on the other side of the diagram for 0( )

1g .

There are cases, far from a hypothetical critical point, where the miscibility is zero. Then, the single-phase zones α and β are reduced to the axes, and the two-phase zone is in fact a mixture of component A1 and component A2, created with each of the components initially being pure in their phase.

3.1.2. Demixing in the case of low reciprocal solubilities

Far from a critical point of demixing, the reciprocal solubilities of the components may sometimes be so low that we can apply Raoult’s and Henry’s laws (see section 2.7) to them.

Regarding component A2, we can consider A2 to be the solvent and can apply Raoult’s law to it; hence A1 is the solute to which we can apply Henry’s law, so that using Henry’s coefficient iHK , we have:

( )2 1γ = [3.3a]


and:

( )( )1 1HKγ = [3.3b]

Conversely, component A1 is the solvent and we apply Raoult’s law to it, whereas A2 is the solute and justifiable by Henry’s law, thus:

( )1 1γ = [3.4a]

and:

( )( )2 2 HKγ = [3.4b]

By feeding these activity coefficient values into expressions [2.7] and [2.8], choosing the pure-substance reference I (pure substances) (in this case, we generally cannot use the simplified forms [3.1] and [3.2]), we obtain the two equilibrium conditions:

– for A1:

( ) ( ) ( )( )0( ) 0( )

1 1 11 1 1

1

expRTH

g g xK x x

K−= = [3.5]

– and for A2:

( ) ( ) ( )( )( ) ( )( )0( ) 0( )2 12 2

1 2 12

11 1 exp

RT KH

H

K xg gx K x

−−− = − = [3.6]

Thus, we obtain the following for the composition of the β phase:

( ) ( )( )( ) ( )

1 1 2 2( )1

1 1 2 2

H H

H H

K K K Kx

K K K K

−=

− [3.7]

and for the composition of the α phase:

( )( )( ) ( )2 2( )

11 1 2 2

H

H H

K Kx

K K K K

−=

− [3.8]


Remember that temperature obviously acts on constants K1 and K2, but also on Henry’s constants. Indeed, it must be remembered that this Henry’s constant obeys relation1:

a 0,Nexp

Rs

sHKTε

= − [3.9]

In this last relation, the sign of the interaction ε0s may be positive or negative. This tells us that we may have normal solubilities which increase with temperature, and retrograde solubilities which decrease as the temperature increases.

3.1.3. Demixing of strictly-regular solutions

We know that in a strictly-regular solution, the activity coefficient of a component obeys the relation2:

2 2a 121 2 2

NlnR

wBx x

T Tγ = = [3.10]

By virtue of these expressions, the activity coefficients of the two components of a strictly-regular solution are given, for an α phase, by:

( )( ) ( )( )2 2( ) a 121 1 1

Nln 1 1R

wB x xT T

γ = − = − [3.11]

( )( ) ( )( )2 2( ) a 122 1 1

NlnR

wB x xT T

γ = = [3.12]

The expressions are identical for the β phase.

As the strictly-regular solutions are symmetrical in relation to the compositions of the two components, we must strictly have:

( ) ( )1 11 x x− = [3.13]

1 See [SOU 15b], Volume 2 of this set of books, Chapter 3, relation [3.45]. 2 See [SOU 15b], Volume 2 of this set of books, Chapter 2, relation [2.59] and Chapter 3, relation [3.37].


Since we are dealing with liquid phases we can apply the simplification of relations [3.2a] and [3.2b], and by virtue of relation [3.13] we obtain:

( ) ( ) ( ) ( )1 1 1 1x xγ γ= [3.14a]

and: ( ) ( ) ( ) ( )1 2 1 2x xγ γ= [3.14b]

If two points on the Gibbs energy curve as a function of the composition have a common tangent, by virtue of the rule of symmetry it is horizontal. In these conditions, to find the compositions of the two phases at equilibrium upon demixing, rather than using expressions [3.14a] and [3.14b], it is better to cancel out the first derivative of the Gibbs energy of mixing in relation to the composition.

Recalling that3 the Gibbs energy of mixing of a strictly-regular solution is:

mixm

1 1 1 1 1 1(1 ) ln (1 )ln(1 )R

G B x x x x x xT T

= − + + − − [3.15]

Figure 3.3. Gibbs energy of mixing of a strictly-regular solution

3 See [SOU 15b], Volume 2 of this set of books, Chapter 2, relation [2.68].


The first derivative in relation to 1x , therefore, is:

mixm

11

1 1

dR (1 2 ) ln

d 1

GxBT x

x T x= − +

−

[3.16]

It is not easy to make this derivative equal to zero analytically, so we shall circumvent the problem by looking for the conditions which produce two points of inflection on the curve. In order to do so, we shall zero the second derivative, which is:

mix2 m

21 1 1

d 1R 2 0d (1 )

GBT

x T x x= − + =

− [3.17]

This derivative becomes 0 for the values:

( )1

21 1

2

TBx α

+ −=

[3.18a]

and:

( )1

21 1

2

TBx β

− −= [3.18b]

The term underneath the radical must be positive. From this, we deduce the following conclusions:

– if / 2B T > , there are two points of non-horizontal inflection (Figure 3.3), given by relations [3.18a] and [3.18b]. These points are the two particular points S1 and S2 which correspond to two compositions that are called the spinodal compositions (e.g. for B/T = 2.7 the compositions of those points are 0.255 and 0.745). The presence of these two points of inflection implies the existence of three extrema (Figure 3.3), whose value is


0.5 because of symmetry for the maximum, and two minima on either side, so that the symmetry is preserved;

– if / 2B T = , the second derivative only becomes 0 for x = 0.5, without changing the sign, so this is a minimum (point K in Figure 3.3). This curve corresponds to the critical temperature of demixing;

– if / 2B T < , there is no longer any value which cancels out the second derivative (Figure 3.3), so there is no longer an inflection point, and we are left with only the minimum for x = 0.5; there is no longer demixing in the solution.

We have thus demonstrated the possibility of immiscibility at low temperatures (large B/T ratio). Consequently, we shall have a miscibility gap at a maximum critical point similar to that shown in Figure 3.2(a), but symmetrical.

It is not possible to give an account of a miscibility gap at a minimum critical point, as in Figure 3.2(b), as given by the strictly-regular solution model. However, it is possible to give an account of such a miscibility gap at a minimum critical temperature by using the solution model, for which the activity coefficient of a component is given by:

( ) ( )2 6(I)1 1 1ln ' 1 ' 1B x C xγ = − + − [3.19]

The same relation is applied to second component. If we apply this expression to each of the phases with B’=3 and C’ = 4, we obtain a minimum critical temperature.

3.2. Liquid–solid systems

In the case of liquid–solid systems, we can envisage the presence of two or three phases. Indeed, in the presence of four phases, the Gibbs variance would fall to 0 and if the pressure has no influence, or is fixed, the number of degrees of freedom becomes negative, which is impossible.

In the two-phase system, one is liquid and the other solid, and we may find both components in the two phases with the two equilibria between the components of the same nature as equilibria [2R.1] and [2R.4].


Alternatively, it may be that the two components are in the liquid phase and only one is in the solid phase – we then speak of solubility of the solid in the liquid, and the system then accommodates only one equilibrium, such as [2R.1].

In systems involving three phases, the number of degrees of freedom drops to 0 if the influence of pressure is negligible, which means that there is no longer a free variable: if temperature is fixed, then so too are the compositions of the phases. We encounter two cases:

– the case of two liquid phases which are not miscible, denoted by “liq 1” and “liq 2”, and a solid phase. In such a case, we have the equilibria:

( ) ( )liq1 liq21 1A A= [3R.1]

( ) ( )liq1 liq22 2A A= [3R.2]

( ) ( )liq1 sol1 1A A= [3R.3]

( ) ( )liq1 sol2 2A A= [3R.4]

( ) ( )liq2 sol1 1A A= [3R.5]

( ) ( )liq2 sol2 2A A= [3R.6]

Yet it is easy to show that these six equilibria are not independent, and that balance equation [3R.5] is the difference between balance equations [3R.1] and [3R.3]. Similarly, balance equation [3R.6] is the difference between balance equations [3R.2] and [3R.4]:

– the case of two immiscible solid and a single liquid phase is strictly symmetrical;

– the case of two immiscible liquid phases and a single, pure solid phase, containing only one component – e.g. A1. Thus, we have for example the following equilibria:

( ) ( )liq2 sol1 1A A= [3R.7]

( ) ( )liq2 liq11 1A A= [3R.8]

( ) ( )liq2 liq12 2A A= [3R.9]

( ) ( )liq1 sol1 1A A= [3R.10]


Yet of these equilibria, only three are independent. We can see that the balance equation [3R.10] is the difference between the balance equations [3R.7] and [3R.8].

3.2.1. Thermodynamics of the equilibria between a liquid phase and a solid phase

Equilibria between a solid and a liquid in binary systems are obviously related to the general issue of equilibria between the phases of binary systems, as discussed in Chapter 2. However, as only the condensed phases are involved, pressure only influences these equilibria at very high values and not in the usual range of pressure. The number of degrees of freedom also shrinks by one. The graphical representations, therefore, will be isobaric phase diagrams usually obtained at the normal pressure of 1 bar.

Using relations [2.7] and [2.8], the equality of the chemical potentials for each of the components between the two phases at a given temperature results in the two ratios of the activities:

( )

( )( )

LLS1

1S1

( )aK T

a= [3.20a]

and: ( )

( )( )

LLS2

2S2

( )aK T

a= [3.20b]

Constants ( )LS1K and ( )LS

2K are defined as functions of the Gibbs energies of the reference states which depend only on temperature, in accordance with:

( )0(L) 0(S)

LS 1 11 exp

RK

Tμ μ−= − [3.21a]

and:

( )0(L) 0(S)

LS 2 22 ) exp

RK

Tμ μ−= − [3.21b]


By derivation of expressions [3.21a] and [3.21b] and application of Helmholtz’s law, and introduction of the variation in partial molar enthalpy of the reference states upon melting, we obtain:

( )LS 0(L) 0(S) 01 1 1 f 1

2 2

d lnd R RK H H HT T T

−= = [3.22a]

and:

( )LS 0(L) 0(S) 02 2 2 f 2

2 2

d lnd R RK H H HT T T

−= = [3.22b]

These expressions adapt to the reference state chosen.

If we choose the pure-substance reference (I), we obtain the relations:

( )

( )( )

(L)(I)LS (II)1 1

1S(S)(I)1 1

( )Lx

K Tx

γγ

= [3.23a]

and:

( )

( )( )

L(L)(I)2 1 LS (I)

2S(S)(I)2 1

1( )

1

xK T

x

γ

γ

−=

− [3.23b]

The difference between the Gibbs energies of the states of reference becomes the difference between the Gibbs energies of the pure substances – i.e. the enthalpy of melting of the pure substances, which gives us the new form of relations [3.22a] and [3.22b]:

( )0(L) 0(S) 0

LS 1 1 f 11 exp exp

R Rg g g

KT T−= − = − [3.24a]

and:

( )0(L) 0(S) 0

LS 2 2 f 22 ) exp exp

R Rg g g

KT T−= − = − [3.24b]


The difference between the enthalpies of the reference states becomes the enthalpy of fusion of the pure substances, so relations [3.22a] and [3.22b] are reformulated:

( )LS 0(L) 0(S) 01 1 1 f 1

2 2

d lnd R RK h h hT T T

−= = [3.25a]

and:

( )LS 0(L) 0(S) 02 2 2 f 2

2 2

d lnd R RK h h hT T T

−= = [3.25b]

These relations involve enthalpies of fusion 0f 2h or latent heats of fusion.

These enthalpies are positive, as the rise in temperature encourages fusion.

For example, these relations can easily be applied to the case of perfect solutions, for which the activity coefficients in reference (I) are equal to 1. Thus, we obtain:

( )

( )( )

LLS (I)1

1S1

( )xK T

x= [3.26a]

and:

( )

( )

( )L

1 LS (I)2

S1

1( )

1

xK T

x

−=

− [3.26b]

The influence of temperature on the equilibrium constants always obeys relations [3.25a] and [3.26b], which depend not on the model of the solution but solely on the reference state chosen.

If we choose reference state (II), which is an infinitely-dilute solution, we then need to choose a solvent – A1, for example, and A2 becomes the solute.

For the solvent A1, we know that the convention is the same as in convention (I), so the isothermal expression will be of the same form as relation [3.23a].


For the solute A2, the reference is the infinitely-dilute state, and the isothermal equilibrium law takes the form:

( )

( )( )

L(L)(II)LS (II)2 2

2S(S)(II)2 2

( )xK T

xγγ

= [3.27]

NOTE 3.2.– The definition of Henry’s constant is given, as we recall, by the relation4:

(I) 0

(II)ln lnR

s s siH

s

gK

Tγ μγ

∞ −= = [3.28]

In light of this relation and of relation [3.23b], expression [3.27] can be written as:

( )

( )

( ) ( )

(L)(I)L(L)(II) 2

2 2 (L)(I) (S)LS (II) LS (I)2 2

2 2(S)(I) (L)S(S)(II) 2 2

2 2 (S)(I)2

( ) ( )H

H

xK

K T K TKx

γγγγγγ

= = [3.29]

With regard to the influence of temperature on the fusion constants, for the solvent relations [3.24a] and [3.25a] are applicable. For the solute, obviously, we would have:

( )(L) (S)

LS (II) 2 22 exp

RK

Tμ μ∞ ∞−= − [3.30]

(L)2μ∞ and (S)

2μ∞ respectively denote the chemical potential at infinite dilution of the solute in the solution in the liquid state, and in the solution in the solid state. The application of the derivation simply gives us:

( )

2

)L(2

02f

2

)L(2D

(II)LS2

RRdlnd

THh

TH

TK

∞∞+== [3.31]

4 See [SOU 15a], Volume 1 of this set of books, Chapter 3, relation [3.33].


(L)2D H∞

is the partial molar enthalpy of dissolution of the solute at

infinite dilution, 0f 2h is the enthalpy of fusion and

(L)2H∞

is the partial molar enthalpy of the liquid solute at infinite dilution.

3.2.2. Isobaric phase diagrams of equilibria between a solid and a liquid

We shall now examine isobaric phase diagrams with the different forms described in sections 2.3.2 and 2.4.2, depending on the number of phases present – i.e. depending on the miscibilities of the components in each of the phases. The diagram will always include two conjugate curves:

– that giving the composition of the solid and also the temperature of fusion beginning at a given concentration, which is known as the solidus;

– that giving the composition of the liquid and also the temperature of solidification beginning at a given composition, which is the liquidus.

In the case of melting diagrams, it must be borne in mind that, as they involve the emergence of solid phases, the diagrams are thermodynamic diagrams theoretically obtained at equilibrium. In reality, it is possible to obtain other phases than those shown by a diagram. Indeed, as the temperature changes, solidification begins to take place, with the process of germination. Similarly, the composition of solid phases gradually becomes homogeneous, by the process of diffusion. These two processes are relatively slow, and their rates are very heavily influenced by temperature. For example, at low temperature, diffusion is extremely slow, and over the course of cooling we may obtain superposed solid phases of the same nature but differing compositions, as the composition of the solid has not had time to homogenize.

We shall now examine the various types of diagrams encountered in the case of equilibrium between a liquid and a solid.

3.2.2.1. Miscible components in all proportions in the two phases

The simplest diagrams are those which represent total miscibility of the components, both in the solid state and in the liquid state. There are three possible forms of these diagrams: a single spindle diagram (Figure 3.4(a)), maximum diagram (Figure 3.4(b)) and minimum diagram (Figure 3.4(c)).


Figure 3.4. Diagrams of melting with complete miscibility in the solid and liquid phases

In the latter two cases, the extremum corresponds to an azeotropic composition, but as we are dealing with condensed phases, the melting point of that azeotrope of melting does not vary with pressure; hence, there is no longer any difference between that azeotrope and a definite compound formed between the two components, and whose composition xC corresponds to the abscissa of the extremum C (see section 3.2.2.5).

The diagram behaves like two half-diagrams with a single spindle corresponding to two systems A2-C and A1-C, adjacent to one another. The only nuance lies in the existence of a horizontal tangent to the composition of the intermediate compound. This tangent is attributable to the application of the Gibbs–Konovalov theorem (see section 2.3.2.2.3) and to the fact that the molar fractions at that point are neither 0 nor 1.

3.2.2.2. Equilibria between a solid and a liquid with demixing of the solid phase

Total miscibility in the solid state, though, is fairly exceptional, and we are usually faced with systems that exhibit zero or partial miscibility in the solid state, which is why we see the possibility of three-phase mixtures (see section 2.4.2).

For example, Figure 3.5 shows the fusion diagram where two solids are totally immiscible, with the two liquids being entirely miscible. This configuration reveals a eutectic point E. At this point, we see an invariant equilibrium which is written in the direction of cooling:

LiqE = Solid A1 + Solid A2 [3R.11]


Figure 3.5. Melting diagram with eutectic point

In the diagram, the solid zone has been artificially divided into two zones by the vertical dotted line passing through point E. This vertical is not a curve of the diagram, because on both sides we see the presence of a mixture of crystals of the two pure substances A1 and A2, just like the eutectic mixture. However, in terms of texture, the mixture corresponding to the eutectic composition is formed of crystals of A1 and A2 whose dimensions are very different to those of the other crystals: they are generally much finer, and in the case of metals, we can clearly distinguish this mixture from the other crystals by metallographic examination.

With newly-improved examination techniques, it is possible to show that ultimately, complete immiscibility is extremely rare – particularly in the case of mixtures of metals or oxides, and that usually there is a small degree of solubility of each of the elements in one another, which results in the presence of two single-phase zones of solid solutions in the vicinity of the pure substances.

Figure 3.6 gives an example of such a diagram with a eutectic point, which is formed of a mixture of the two solid solutions α and β.

The invariant reaction at the eutectic point corresponds to equilibrium (written in the direction of cooling):

LiqE = αA + βB [3R.12]


αA and βB respectively denote the solid solution rich in component A2 with its composition at point A and the solid solution rich in component A1 with its composition at point B.

Figure 3.6. Eutectic point with partial miscibility in the solid state

As before, the vertical line drawn through E does not form part of the diagram – it simply enables us to distinguish the eutectic mixture, which exhibits the same microstructures as those mentioned above (very fine grains).

Usually, in addition to the liquidus and the solidus, the diagram also includes the limit of the miscibility gap in the solid state; that gap is ended at high temperatures by melting. In the case shown in the figure, the solubility curves in the solid state show normal solubilities (which increase with temperature), as opposed to retrograde solubilities, which are occasionally encountered and would be expressed by slopes with the opposite sign.

Figure 3.7 shows a fusion diagram with the presence of a peritectic point. The mixture of solutions α and β which is deposited below the point P exhibits the same microstructural characteristics as a eutectic solution (very fine grains). At point P, upon melting, the peritectic reaction gives us the second solid solution on the basis of the liquid whose composition corresponds to point T and the first solid solution with the composition of point B, which is written as follows in the direction of cooling:

βB + LiqT = αP [3R.13]


Figure 3.7. Peritectic diagram with partial miscibility in the solid state

Liq1 +βΒ

i

Liq1 Liq2+βΒ

Liq1+ αΑ

A B

Liq1 + Liq2

A1 A2

E αA+βΒ

βΒ

αΑ

Figure 3.8. Eutectic diagram with miscibility gap in the liquid state

3.2.2.3. Equilibria between a solid and a liquid with demixing of the liquid phase

Sometimes, the above diagrams become more complicated – e.g. with the emergence of a miscibility gap in the liquid phase (see Figure 3.8), which reveals two three-phase points A and B, with the invariant equilibrium:

βD + Liq1A = Liq2B [3R.14]


We are then led to consider two liquid phases L1 and L2, with differing compositions, which become one above the critical point K.

3.2.2.4. Three-phase reactions in liquid–solid systems

In fact, given the possibility of having three-phase mixtures, we find different types of invariant reactions which, depending on the reaction and the phases present, are known by different names. Table 3.1 lists the invariant reactions found in the direction of melting.

Obviously, we see the eutectic reaction at points E in Figures 3.5, 3.6 and 3.8, and the peritectic reaction from point P in Figure 3.7.

Table 3.1. Invariant equilibria in a three-phase solid–liquid medium

3.2.2.5. Systems with formations of definite compounds

Definite compounds can also form in cases of non-total miscibility in the solid state. We then distinguish two cases.

If the intermediate compound keeps its composition when it melts, as is the case in Figure 3.9, we say that we have a definite compound with congruent melting. The full diagram is the juxtaposition of two diagrams – one pertaining to the system A2C and the other to the system A1C. We can see that the two “half-diagrams” in Figure 3.9 resemble the diagram in Figure 3.6, with two eutectic points. Whilst eutectic E1 would be a mixture


of the solid solutions α and γ (of A2 in C), the eutectic E2 is a mixture of the solid solutions β and γ of A1 in C. It is not possible to distinguish the two sides of the vertical passing through C, which is not a curve belonging to the diagram. The whole zone, in fact, corresponds to non-stoichiometry of the intermediate compound. Of course, by application of the Gibbs–Konovalov theorem, we find that the tangents to the point C are horizontal.

If the intermediate compound decomposes before melting (as shown in Figure 3.10), we say that we are dealing with a definite compound with incongruent melting.

Figure 3.9. Melting diagram showing a definite compound with congruent melting

The branch of the liquidus corresponding to the deposit of intermediate compound C is then limited at point T, above which only A1 (or a solid solution β) can be deposited. However, it is possible, by extrapolation, to determine the melting point of the intermediate compound C. Here, point T is called a transition point. The vertical line passing through C is a part of the diagram if there is no deviation from stoichiometry of C; otherwise it is replaced by a limit of the domain of a solid solution γ. The diagram can then no longer be distinguished from a diagram with a peritectic point as shown in Figure 3.7, except for the fact that the composition of one of the crystal boundaries corresponds exactly to that of C – i.e. with a generally simple ratio A2/A1 between the molar fractions.


Figure 3.10. Melting diagram showing intermediate compound with incongruent melting

3.2.3. Solidus and liquidus in the vicinity of the pure substance

We shall now give an approximation of the equation for the liquidus in the vicinity of a pure component; in this instance, we shall choose A2. The evolution of the liquidus obeys the differential equation:

(L) (S)2 2d dμ μ= [3.32]

This equation, expanded as a function of the variable of composition x1 and of the temperature, becomes:

(L) (L) (S) (S)(L) (S)2 2 2 21 1(L) (S)

1 1

d d d dx T x Tx T x Tμ μ μ μ∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂ [3.33]

Thus, by expressing the chemical potentials as a function of the activities, and using the Gibbs–Duhem relation:

( )(L) (S) (L) (S)(L) (S)2 22 21 1(L) (S)

1 1

ln lnR d R d da aT x T x S S T

x x∂ ∂− = −

∂ ∂ [3.34]

However, at the point corresponding to the pure substance A2, we have (L) (S)

1 1 0x x= = , so the temperature is the melting point of A2:

2( )FT T= [3.35]


The activity of the liquid solution uses the pure liquid component A2 as a reference, and for the solid solution the pure solid component A2 is taken as a reference. In these conditions, in view of Raoult’s law in solutions, the activity of A2 is equal to its molar fraction, which is 1-x1, and its partial molar entropy is the entropy of pure A2 in the same state of condensation. The relation then comes to the limit x2 = 1, so x1 = 0:

1

(L) (S)01 1

2( ) f 20

d d-Rd dF

x

x xT s

T T=

− = [3.36]

At the melting point, the variation in the Gibbs energy of fusion is zero, and hence the entropy of fusion is linked to the enthalpy of fusion by the following relation:

00 f 2

f 22( )F

hs

T= [3.37]

Relation [3.36] can also be written in the form:

1

2 (L) (S)2( ) 1 1

0f 2 0

R d d- 1d d

F

x

T x xh T T =

− = [3.38]

If the solubility of A1 in A2 in the solid state is very low, we have:

( ) ( )1 1

(L) (S)1 10 0

d / d d / dx x

T x T x= =

<< [3.39]

so:

( ) ( )1 1

(L) (S)1 10 0

d / d d / dx x

x T x T= =

>>

By an initial approximation, relation [3.38] can be reduced to:

2

22( )

0 (L)f 2 1 1

R dd )

F F

x

T Th x

=

− = [3.40]

NOTE 3.3.– The slope of the liquidus is more strongly negative when approximation [3.39] is more fully respected – that is to say, the solid is all the more immiscible when the solubility of that solid is zero.


In the case that approximation [3.39] is no longer true, we use mL to represent the slope of the liquidus at the origin, and mS for the slope of the solidus at the same point. We have:

( )2

(S)S 1 1

d / dF xm T x

== [3.41a]

and:

( )2

(L)L 1 1

d / dF xm T x

== [3.41b]

For simplicity’s sake, we posit that:

2( )*0

f 2

R0FT

mh

= > [3.42]

Relation [3.38] then assumes one of two forms:

*S

L *S

m mm

m m=

− [3.43a]

or:

*L

S *L

m mm

m m=

− [3.43b]

If Sm is negative, the difference between *Sm m− is positive, because *m

is positive, and thus in view of relation [3.43a], the slope Lm is also negative.

Similarly, relation [3.43a] tells us that if Lm is negative, then Sm is also negative.

Hence, if the slope at the origin of the solidus is negative (Figure 3.12(a)), then that of the liquidus is also negative, and vice versa. This corresponds to what is known as cryoscopic reduction in the melting point of the solvent by the addition of the solute. This being the case, the diagram would contain a eutectic point.


A particular practical application of cryoscopic reduction is seen in the use of salt on roads to help melt ice and snow.

Figure 3.11. Relative positions of the solidus and liquidus in the vicinity of the pure substance;

a) with positive slopes; b) with negative slopes

By virtue of the same reasoning, where we showed that the direct theorem and its reciprocal were both true, the contrary theorem holds true as well, so a positive slope of the liquidus leads to a positive slope of the solidus (Figure 3.11(b)).

3.3. Equilibria between two solids with two polymorphic varieties of the solid

When one of the components or an intermediate compound, in the solid state, exhibits two allotropic varieties, this leads to equilibria between the solid phases.

The resulting diagrams are very similar to the liquid–solid diagrams and also exhibit points with three phases, which are the site of invariant reactions. The eutectic transformation takes the name of eutectoid, and the peritectic reaction becomes peritectoid.

Table 3.2 gives the main diagrams and the invariant reactions on cooling, associated with and linked to the transformations in the solid state.

Solid iron presents two allotropic varieties. Figure 3.12 gives the iron part of the iron–graphite diagram. In fact, there are two iron–carbon diagrams,


which are fairly similar to one another: one pertains to carbon in graphite form, and the other to the iron–cementite system. Cementite is iron carbide C3Fe, which is, in fact, metastable and therefore gives rise to metastable equilibria. In Figure 3.12, we see the existence of an invariant eutectoid reaction where, at 740°C, the solid solution γ, at the eutectoid concentration, transforms upon cooling into the solid solution α and solid graphite carbon. The diagram also presents a peritectic point P.

Table 3.2. Invariant reactions in a three-phase solid–solid system

Figure 3.12. Partial diagram for iron–graphite, on the side of iron


3.4. Applications of solid–liquid equilibria

We shall now examine two applications of liquid–solid diagrams: one is linked to the solubility of a solid in a liquid; the other is cryometry.

3.4.1. Solubility of a solid in a liquid: Schröder–Le Châtelier law

We speak of solubility of a solid when we have the system of equilibrium of a pure solid substance and of the liquid solution in a solvent.

3.4.1.1. Thermodynamics of solubility

We shall distinguish the two components of a system as solvent A0 and a solute As. Look again at Figure 3.5. The left-hand branch of the liquidus describes the equilibrium between the pure solid A2, which will be our solvent A0, and the liquid solution. We can apply relations [3.20a] and [3.20b] with an activity of the solvent in the solid state equal to 1. We obtain the equation for this branch of the liquidus, so that, if T0(F) is the temperature of fusion of the pure solvent, in our case:

( )0( )

0(L) f 0

0 2ln 1 dRF

T

s T

hx T

Tγ − = [3.44]

Integration of this equation, using the hypothesis of enthalpy of fusion which practically does not vary with temperature, gives us:

( )0

(L) f 00

0( )

1 1ln 1Rs

F

hx

T Tγ − = − [3.45]

If the solution is perfect, the activity coefficient of the solvent is 1, and relation [3.45] becomes:

( )0

(L) f 0

0( )

1 1ln 1RF

sF

hx

T T− = − [3.46]

Relation [3.46] constitutes the Schröder–Le Châtelier law, which is a good approximation to express the solubility of solids in liquids.


Similarly, the left-hand branch of the liquidus in Figure 3.5 describes the equilibrium between the liquid solution and the pure solid component As. With the same hypothesis of temperature-independent enthalpy of fusion, we obtain the relation parallel to equation [3.45], which gives the equation for this branch of the liquidus:

0(L) f

( )

1 1lnR

ss s

s F

hx

T Tγ = − [3.47]

At the eutectic point, the two branches intersect, so the point E obeys both relations [3.45] and [3.47]. By solving the system composed of these two equations, we obtain the coordinates of the eutectic point E.

Relations [3.45] and [3.47] can be used to determine the activity coefficients.5

3.4.1.2. Curves of solubility of salts in water

The solubility of salts in water usually obeys phase diagrams with no solid solutions; indeed, salts are rarely miscible with ice or with one another. On the other hand, it is not uncommon for several hydrates to exist in the solid state, which are thus definite compounds, intermediate between water and the anhydrous salt. These hydrates may exhibit congruent or incongruent melting. Figure 3.13(a) shows the example of the diagram for the water–FeCl3 system, which leads to four hydrates, all with congruent melting: FeCl3, 6H2O, FeCl3, 7/2H2O, FeCl3, 5/2H2O and FeCl3, 2H2O. Given the immiscibility of solids with one another, in such a diagram the solidus is composed purely of the horizontal and vertical curves, which are fairly easy to place. Hence, most of the time we do not represent the solidus, instead contenting ourselves with representing the branches of the liquidus. In addition, as it is often interesting to consider the solubility aspect, we modify the coordinates of the axes by reporting the temperature as the abscissa value and the ordinate values showing the concentration of salt in moles per liter or grams per liter, to obtain the solubility curve. As only the liquidus is represented, we find curves comparable to that shown in Figure 3.13(b), plotted in the case of sodium sulfate in water, which presents a definite compound: decahydrate. Along the branches, we display the nature of the deposited solid. In the figure, we also see that decahydrate exhibits 5 See [SOU 15b], Volume 2 of this set of books, Chapter 5, section 5.3.1.


incongruent melting and that the anhydrate has a reciprocal solubility, meaning that this solubility decreases as temperature increases. In general, the bottom of the curve, near to the origin, is constituted by the branch of deposition of ice up to the eutectic point.

Figure 3.13. Salt–water system; a) phase diagram of the water–FeCl3 system; b) solubility curve of sodium sulfate

3.4.2. Determination of molar mass by cryometry

Cryometry is the measurement of the reduction in melting point of a solvent in the presence of one or more solutes. This measure is used, in particular, for determining the molar mass of the solute and measuring the activity of the solvent in a solution.

As with the previous application, we draw the distinction between the solvent A0 and one or more solutes As.

Let 0( )FT denote the reduction in melting point of the solvent, given by the difference:

0( ) 0( )F FT T T= − [3.48]

As the temperatures T and 0( )FT are very close, equation [3.46] is written thus:

( )0

0( )(L) f 02

0( )

ln 1R

Fs

F

Thx

T− = [3.49]


If the molar fraction of the solute is low, we can replace the logarithm with the first term in its Maclaurin series expansion, and write:

00( )(L) f 02

0( )RF

sF

Thx

T≅ [3.50]

By introducing the mass of solute per kilogram of solvent sm , defined by the relation:

(L)

0

1000 s ss

M xM

=m [3.51]

relation [3.50] is then written as:

20( ) 0

0( ) 0f 0

R1000

FF s

s

T MT

h M= m [3.52]

We define the cryoscopic constant of the solvent, which depends solely on that solvent, by:

20( )

0( ) 00f 0

R1000

Fcryo

TK M

h= [3.53]

For example, in the case of water, the calculated value of the cryoscopic constant is 1860kg.mole-1.

By combining relations [3.52] and [3.53], we deduce the molar mass of the solute from the measurement of the cryoscopic reduction, using the expression:

0( )0( )

ss cryo

F

M KT

= m [3.54]

This is Raoult’s law of cryometry. We see that, in fact, it is a limit law, valid at a concentration of zero (so that the activity coefficient is 1). In order


to use this relation, we measure the reduction in the melting point for different amounts of the solution, and then plot the curve giving the reduction 0( )FT as a function of the amount sm of solution. The value of

0( )FT to be used in relation [3.54] is then obtained by extrapolation of the

curve for 0sm = .

If the solution is not hugely dilute and contains multiple solutes, it is easy to deduce, from relation [3.45], the activity coefficient of the solvent in the form:

0f 0 ( ) (L)

0 120( )

ln ln 1R

F

sF

h Tx

Tγ ≅ − − − [3.55]

This relation can be used to measure the activity coefficient of the solvent at the temperature of fusion of the solution.6

3.5. Membrane equilibria – osmotic pressure

The phenomenon of osmosis is the selective passage of a solvent, with the exclusion of the solute, from a solution across a so-called semi-permeable membrane. We begin with a solution composed of a solvent, such as water, placed at atmospheric pressure P0 and a solute with large molecules, polymers or sugars, and place an empty capillary tube, closed at the bottom with a semi-permeable membrane, in the solution.

If we allow the system to evolve, we see that at equilibrium, only the solvent is passed through the membrane and fills the capillary up to a certain height h. This means that a pressure difference is established (Figure 3.14), represented by the height h, between the pure solvent at pressure P and the solution in the tank at constant pressure P0. This pressure is generally the normal pressure of the laboratory. The difference in pressure 0P P− thus noted is called the osmotic pressure, denoted by Π.

6 See [SOU 15b], Volume 2 of this set of books, Chapter 5, section 5.3.2.


Figure 3.14. Demonstration and measurement of osmotic pressure

The origin of the phenomenon of osmosis is to be found in the selective solubility of the solvent in the membrane, whereas the solute is insoluble. The solvent can therefore diffuse into this membrane, as there is a difference in the chemical potential of the solvent between the two compartments. It is possible to use various natural membranes, such as animal tissue, or artificial ones, such as parchment paper, cellulose hydrates, etc.

3.5.1. Thermodynamics of osmotic pressure

In order to explain the thermodynamics of osmosis, we state that diffusion ceases when the chemical potentials are the same on both sides of the membrane, and thus, at equilibrium, there is equality of the chemical potentials of the solvent between the solution at pressure P0 ( 0 0 0( , , ))P T xμ and the pure solvent at pressure P ( 0 ( , )g P T ), so:

00 0 0( , , ) ( , )P T x g P Tμ = [3.56]

Although we are dealing with condensed solutions, to explain osmotic pressure thermodynamic study must take account of the variations in chemical potential with pressure. The chemical potential of the pure solvent varies with pressure according to the relation:

00

gv

P∂ =∂

[3.57]


0v is the molar volume of the solvent at a given pressure. Strictly speaking, that molar volume varies with pressure, but it can be treated the same way as the molar volume of the pure solvent at normal pressure 0

0v , which depends itself very little on the pressure. The chemical potential of the pure solvent at pressure P is therefore written:

( )0

0 00 0 0 0 0 0 0 0( , ) ( , ) d ( , )

P

Pg P T g P T v P g P T v P P== + = + − [3.58]

The chemical potential of the solvent in the solution at pressure P0 will be:

0 (I)0 0 0 0 0 0 0( , , ) ( , ) R lnP T x g P T T xμ γ= + [3.59]

The relation at equilibrium will be as follows, as we substitute expressions [3.58] and [3.59] into expression [3.56]:

( )0 (I) 00 0 0 0 0 0 0 0( , ) R ln ( , )g P T T x g P T v P Pγ+ = + − [3.60]

By definition of the osmotic pressure, we have:

0P PΠ = − [3.61]

Relation [3.60] is simplified to:

( )0 (I)0 0 0 0R ln 0v P P T xγ− = = [3.62]

In view of definition [3.61], this gives us the following expression of the osmotic pressure:

(I)0 00

0

R lnT xv

Π γ= − [3.63]

This expression is valid no matter what the nature of the solution. The order of magnitude of the osmotic pressure observed is 607 hPa for a concentration of sugar in water of 0.0292 mol/l, at a temperature of 0°C.


3.5.2. Osmotic pressure of infinitely-dilute solutions: the Van ’t Hoff law

In the case of dilute ideal solutions, the activity coefficient of the solvent is 0 1γ = , which gives us the osmotic pressure:

000

R lnTx

vΠ = − [3.64]

This relation also gives us the osmotic pressure that a perfect solution would have, for which the activity coefficient of the solvent is also 1.

In the case of a single solute whose molar fraction is xs, we can write:

( )00

R ln 1 sT x

vΠ = − − [3.65]

As the solution is very dilute, we can content ourselves with the first term in the Maclaurin expansion of the logarithm, which gives us:

00

Rs

T xv

Π = [3.66]

As this solution is very dilute, it is easy to show that the molarity of the solute (Cs) is simply linked to its molar fraction by:

00

ss

xC

v≅ [3.67]

When substituted back into relation [3.66], this gives us the following for the osmotic pressure of a dilute solution:

R sTCΠ ≅ [3.68]

or, by bringing in the quantity of the solute (number of moles ns), we obtain:

00 R sv TnΠ ≅ [3.69]


This law, which is completely analogous to the perfect gas law, is called Van ’t Hoff’s law for osmotic pressure. Note that this law requires not only that the solution be perfect, but it must also be very dilute. This double requirement means it is impossible to define the perfect solution on the basis of this relation.

If the solution contains multiple solutes at the molar fraction xs, relation [3.64] is written thus:

00

R ln 1 ss

T xv

Π = − − [3.70]

With the same reasoning, for very dilute solutions, we find the expression of the osmotic pressure:

R ss

T cΠ ≅ [3.71]

and Van ’t Hoff’s law becomes:

00 R s

s

v T nΠ ≅ [3.72]

We can see, therefore, that the solution behaves like a mixture of solutions with a single solute, each with an osmotic pressure sΠ . Thus, we have:

ss

Π Π≅ [3.73]

Note the similarity with the equation of state for the perfect gas and with the behavior of the mixture of perfect gases.

3.5.3. Application of osmotic pressure to the determination of the molar mass of polymers

Van ’t Hoff’s law is used for the determination of molar masses – particularly for macromolecules, for which the cryometric method (see section 3.4.2) gives too small a difference between the melting points of the solution and the solvent. For this purpose, we introduce the mass


concentration 0C of the solvent (mass per unit volume of the solution or of the solvent, if it is sufficiently dilute). If Ms is the molar mass of the solute, this mass concentration is linked to the molarity by:

0s

s

CM

=C [3.74]

By combining this with expression [3.68], we obtain:

0R

sTM

Π= C [3.75]

We can see that knowing the osmotic pressure and the mass concentration leads to the molar mass of the solute. This law is written in the form:

0

R

s

TM

Π =C

[3.76]

In practice, we experimentally measure the ratio 0/ CΠ for different values of the mass concentration, and generally obtain a curve whose equation is similar to the form:

00

R

s

TB

MΠ = + CC

[3.77]

This curve is extrapolated to a mass concentration of zero, which is the point of validity of relation [3.76], and thus we can use this last relation to calculate the molar mass of the solute.

3.5.4. Osmotic pressure of strictly-regular solutions

We shall now examine what becomes of the Van ’ t Hoff law in the case of a strictly-regular solution.

Using relation [3.10b] of strictly-regular solutions for the solvent, we have:

(I) 20ln sBxγ = [3.78]


By introducing that activity coefficient into the relation [3.63], the osmotic pressure of a strictly-regular solution is:

2

0 0

RR ss

B xT xv v

Π = − [3.79]

In light of relation [3.65], the osmotic pressure becomes:

0R R ss

T B v CCΠ = − [3.80]

We can see that it is possible to characterize a strictly-regular solution if, by plotting Π/Cs as a function of the concentration, we obtain a straight line. The slope of the line can be used to calculate the coefficient B.

3.5.5. Osmotic pressure and the osmotic coefficient

If we compare relations [3.63] and [3.64], which respectively give the osmotic pressure of a real solution and a perfect dilute solution, we see that the ratio of the osmotic pressures at the same temperature and the same concentration can be written:

(I) (I)real 0 0 0

perfect 0 0

ln ln1ln ln

xx x

Π γ γΠ

= = + [3.81]

We can see that this ratio defines the gap between the real solution and perfection. This ratio defines the osmotic coefficient (g):

(I)0

0

lnln

gx

γ= [3.82]

This ratio is sometimes used to characterize a real solution. The ratio [3.81] explains the origin of the name of this coefficient. The osmotic coefficient is not entirely equivalent to an activity coefficient in characterizing a real solution. Indeed, the reference solution which we have used to obtain relation [3.81] is not only perfect but also dilute. This partly explains why this osmotic coefficient is so rarely used.

4

Equilibria Between Binary Fluid Phases

Liquid–vapor equilibria (LVE) are of great practical importance for the design and optimization of unit operations used widely in the chemical, petroleum and cosmetics industries. It is also these equilibria which are most commonly used to validate and calibrate models of solutions.

4.1. Thermodynamics of liquid–vapor equilibrium in a binary system

Consider two components present in a liquid phase at equilibrium with a vapor phase. For each of the two components, we express the chemical potential in the liquid phase in the form:

(L) 0(L) (L)R lni i i iT xμ μ γ= + [4.1]

0(L)iμ is the chemical potential of the component i in the reference state of

the liquid phase.

For the gaseous state, the chemical potential is written as:

(G) *(G) R ln ii ig T fμ = + [4.2]

*(G)ig is the chemical potential of the pure component i in the gaseous

state, which depends solely on the temperature; fi is its fugacity at the total pressure P chosen.



If to find the fugacity of the gas i in the mixture we adopt Lewis’ law, which states that the fugacity of a gas in a mixture is the product of the fugacity of the pure gas at the same pressure by its molar fraction in the mixture (remember1 that this is equivalent to considering a perfect solution of imperfect gases), then the chemical potential becomes:

(G) *(G) (G)0R ln ii i ig T x Pμ φ= + [4.3]

Using the equilibrium condition [2.4] or [2.5] for each component, we obtain:

(G) *(G) 0(L)0(LV)

(L) expR

i i i ii

i i

x P gK

Tx

φ μγ

−= − = [4.4]

(LV)iK is the vaporization constant of component i, which depends only

on the temperature, the pressure and the reference state chosen to express the chemical potentials in the liquid phase. In order to study the influence of temperature, at constant pressure, we can derive relation [4.4] in the form:

( ) ( )0(L) *(G)

(LV)/ /

d lnd R

i i

i

T g TK T TT T

μ∂ ∂−

∂ ∂= [4.5]

For the gaseous phase and for the liquid, if we use Helmholtz’s laws of variations of the Gibbs energies and the chemical potentials with temperature, relation [4.5] becomes:

0(L)(G) 0(G)d lnd R

i i iK h HT T

−= [4.6]

0(G)ih is the molar enthalpy of the pure component i in the gaseous state

and 0(L)iH is the partial molar enthalpy of the component i in the liquid in

the reference state.

1 See [SOU 15a], Volume 1 in this set of books, Chapter 8, relation [8.20].

Equilibria Between Binary Fluid Phases 115

If we know the variations of the enthalpies with temperature, we can integrate relation [4.6]. In the hypothetical situation that the enthalpies do not in practice depend on the temperature, we obtain:

0(L)0(G)(L )

0

1 1lnR

iV ii

i

h HK

T T−≈ − [4.7]

With regard to the influence of pressure, we derive relation [4.4], remembering that the term *(G)

ig is independent of pressure, so:

0(L)

(LV)d lnd R

i

iK PP T

μ∂∂=

[4.8]

Also, the chemical potential varies with pressure in accordance with:

0(L)0(L)iiV

Pμ∂

=∂

[4.9]

Using relation [4.9], relation [4.8] becomes:

0(L)(LV)d lnd R

iiK VP T

= [4.10]

0(L)iV is the partial molar volume of the component i in the reference

state. By integrating relation [4.10], we obtain the vaporization constant at temperature T and pressure P. As the variations in partial molar volumes of the liquids with pressure are extremely slight, we can integrate relation [4.10] and suppose they are constant, thus:

( )0

0(L) 00(L)(LV) 1ln d

R Ri

Pii

i iP

V P PK V P

T T

−= = [4.11]

We shall now consider the expression of relations [4.6] and [4.7] for each of the two references of the state liquid: reference (I) for which we choose


the pure substance and reference (II) for which we choose the infinitely-dilute state.

In the pure-substance reference (I) for component i, the partial molar enthalpy of the reference state is the molar enthalpy of the pure liquid component i:

0 0(L)i iH h= [4.12]

The difference 0(G) 0(L)-i ih h thus represents the molar latent heat of

vaporization of the pure component i, 0v ih , at the chosen pressure P:

0(G) 0(L)0v i i ih h h= −

[4.13]

Hence, the vaporization constant of i is identical to its saturating vapor pressure given by relation [1.42]. In this chapter, it will be denoted by the symbol 0

iP . If the reference pressure is taken to be 1 bar, which shall remain implicit in the rest of this chapter, then we shall have:

00

( )

1 1lnR

v ii

i Eb

HP

TT≈ −

[4.14]

( )i EbT is the boiling point of pure i at the pressure P.

Relation [4.4] then becomes:

(G) *(G) 0(L)0(LV)(I) 0

(I) (L) expR

i i i iii

i i

x P g gK P

Tx

φγ

−= − = = [4.15]

If we choose reference (II), the infinitely-dilute substance i, the partial molar enthalpy of the reference state for the solute is the partial molar enthalpy of component i in an infinitely-dilute liquid solution:

(L)0(L)

iiH H∞

= [4.16]


Relation [4.7] becomes:

(L)0(G)(LV)(II) 1 1ln

Ri i

ii

h HK

TT

∞

∞−

≈ − [4.17]

iT ∞ is the vaporization point of pure i at pressure P, at infinite dilution.

Relation [4.4] becomes:

( ) ( )

0 (G) *(G) (L)(LV)(II)

II II( ) (L)exp

Ri i i i i

iLi i i i

f x P gK

Tx xφ μ

γ γ

∞−= = − = [4.18]

It is easy to show, as we did for liquid–solid equilibria (see section 3.2.1), that between the two constants ( )LV (II)

iK and 0iP , there exists the following

simple relation:

0)L((LV)(I)i

)L((LV)(II)iiHiHi PKKKK == [4.19]

(L)iHK is Henry’s constant for the component i in the liquid phase.

We can apply these general laws to different families of solutions, starting with perfect solutions.

4.2. Liquid–vapor equilibrium in perfect solutions far from the critical conditions

Far from the critical conditions, we assume that the fugacity coefficient of each gas is equal to one. Because the liquid solution is perfect, the activity coefficients of both components in convention (I) are also equal to one. For each of the components, relation [4.14] becomes:

(G) (L)01 11 1x P P P x= = [4.20a]

and:

( ) ( )(G) 0 (L)1 2 2 11 1x P P P x− = = − [4.20b]


NOTE 4.1.– By finding the ratio between the two relations [4.20], we see that we have:

( ) ( )(G) (G) 0 (L) 0 (L)1 1 1 1 1 1

(G) 0 0 (L)L(G) ( )2 2 2 21 11 1

x x P x P xx P P xx x

= = =− −

[4.21]

The ratios between the molar fractions of the two components in the two phases are proportional to one another, and the coefficient of proportionality is the ratio between the saturating vapor pressures.

4.2.1. Partial pressures and total pressure of a perfect solution

We wish to plot the evolution of the total pressure and the partial pressures of each substance as a function of the composition of the liquid phase.

From relations [4.20], we can immediately deduce the partial pressures of the two components in the gaseous phase:

(L)01 1 1P P x= [4.22a]

and:

( )0 (L)2 2 11P P x= − [4.22b]

Figure 4.1 shows the partial pressure curves for each component as a function of the composition of the liquid phase. These curves are straight lines. The total pressure can immediately be deduced from the sum:

( )0 0 (L) 01 2 1 2 1 2P P P P P x P= + = − + [4.23]

The total pressure curve, therefore, is also a straight line.


10A2pur

A1pur

)L(1x

P

01P

02P

P1P2

P

Figure 4.1. Total pressure and partial pressures in liquid–gas equilibrium of a perfect binary solution

4.2.2. Isothermal diagram of a perfect solution

From equations [4.20], we deduce the composition of the liquid phase and that of the gaseous phase, which take the forms:

0(L) 21 0 0

1 2

P Px

P P−=−

[4.24a]

and: 0 0

(G) 1 11 0 0

1 2

P P Px

P P P−=

− [4.24b]

Figure 4.2. a) Isothermal and b) isobaric phase diagram of an equilibrium between a vapor and a perfect solution

If we plot the curves representative of these two functions, we obtain the isothermal phase diagram shown in Figure 4.2(a). The upper curve is the same


straight line as in Figure 4.1, and represents the total pressure as a function of the composition of the liquid phase. The lower curve gives that same pressure as a function of the composition of the gaseous phase at equilibrium.

The top part, above the upper curve, is the domain of stability of the single-phase liquid mixture. The bottom part, below the lower curve, is the domain of stability of the single-phase gaseous mixture. The part between the two curves is the domain of stability of the mixture in two phases: gas and liquid. The case chosen in Figure 4.2(a) is that of a compound A2 which is less volatile than compound A1; at the same temperature (e.g. that shown in Figure 4.2(a)), the vapor pressure of the compound A2 is less than that of compound A1.

4.2.3. Isobaric diagram of a perfect solution

By feeding back functions of the type of [4.14] into the two equations [4.24], we obtain representative functions of the lines of the isobaric diagram (Figure 4.2(b)):

02

01(L)

1 0 01 2

0 01 1

1 1expR

1 1 1 1exp expR R

v

v v

hP

TTx

h hT TT T

− −

=

− − −

[4.25a]

and:

01

1(Eb)(G)1 0 0

1 2

1(Eb) 2(Eb)

01

1(Eb)

1 1expR

1 1 1 1exp expR R

1 1expR

v

v v

v

hT T

xh h

T T T T

hP

T T

P

−

=

− − −

− −

[4.25b]

We can see, even at this stage, that these functions are highly complex even for a perfect solution, and we note that for non-perfect solutions, the corresponding expressions will rapidly become extremely complicated.


In Figure 4.2(b), the temperatures 1(Eb)T and 2(Eb)T are respectively the boiling points of pure compounds A1 and A2 at the pressure chosen to plot the diagram. As the compound A2 is the less volatile, its boiling point is higher than that of compound A1.

The single-phase liquid phase is stable at low temperatures; the gaseous phase is stable at higher temperatures. The two curves delimit the closed domain of stability of the mixture of the two liquid and gas phases. Each of the phases is a homogeneous mixture of the two compounds A1 and A2. The upper curve is called the dew-point curve and the lower one is the boiling curve or the vaporization curve.

4.2.4. Phase composition curve

In isothermal and isobaric conditions, we wish to plot the curve giving the composition of the gaseous phase as a function of that of the liquid phase at equilibrium.

By eliminating the pressure term between relations [4.24], we can deduce the equation for the composition curve:

( )(L)0

1(G) 11 L( )0 0 0

2 1 2 1

P xx

P P P x=

+ − [4.26]

This curve is a branch of a hyperbola, represented in Figure 4.3.

Figure 4.3. Curve of composition of liquid–gas equilibrium in the case of a perfect solution


4.3. Liquid–gas equilibria in ideal dilute solutions

A so-called ideal dilute solution means that there is a difference between a solvent which we shall call A0, and a solute which is denoted by As. In order to study the liquid–vapor equilibrium of ideal dilute solutions, we choose reference (II) – the infinitely-dilute solution. In this reference, the solvent has an activity coefficient of 1, which is the same as in the pure-substance reference (I). Therefore, we can use relation [4.21a]:

(G) 0 (L)0 0 0 0x P P P x= = [4.27]

The solute has an activity coefficient which is also equal to 1, but in reference (II), and therefore we can apply relation [4.18] to it, in the form:

( )(LV) II(G) (L)s s s sx P P K x= = [4.28]

By combining relations [4.27] and [4.28], we deduce the composition of the gaseous phase at temperature T:

( )

( )

(LG) II(L)0 (LG) II 0

0

s

s

K Px

K P−=−

[4.29a]

And that of the liquid phase at equilibrium:

( )

( )

(LG) II0(G) 00 (LG) II 0

0

s

s

P K Px

P K P

−=

− [4.29b]

NOTE 4.2.– By comparing relations [4.29a] and [4.29b], it is easy to deduce the relation between the compositions of the two phases:

(G) 0 (L)0 0 0

(G) (L)0 01 1sH

x P xx K x

=− −

[4.30]


Figure 4.4. Vapor pressures of the components of a binary mixture in a dilute solution

From expressions [4.29a] and [4.29b], we deduce the curve of pressure as a function of the compositions of the mixture and the vapor pressure curves for each of the components as a function of the composition of the liquid phase. These curves are straight lines, but of course, they are limited for low solute concentrations, and therefore pertain only to the left-hand side of the diagram in Figure 4.4.

The partial pressure curve for the solvent intersects the composition axis at the saturating vapor pressure of the solvent, whilst the vapor pressure curve of the solute would intersect the axis xs = 1 at a pressure equal to the equilibrium constant of vaporization in convention (II).

As a function of temperature, by applying relation [4.14] for the solvent we obtain:

00

00(Eb)

1 1lnR

v HP

T T= − [4.31]

Similarly, we apply relation [4.17] for the solute, which yields:

(L)0(G)

(Eb)

1 1lnR

sssH

s

h HK

T T

∞−

= − [4.32]


NOTE 4.3.– We can write:

( )(L) (L)0(G) 0(G) 0(L) 0(L) 0ds ss s s s v s sh H h h H h h H

∞ ∞ ∞− = − − − = + [4.33]

0v sh is the enthalpy of vaporization of the solute in the pure state, and

d sH ∞ represents the enthalpy of passage of a molecule of solute from the pure state to the infinitely dilute state, which is the heat of dissolution of the solute at infinite dilution.

We can also write relation [4.30], including the molar fraction of the solute, in the form:

( )0 (L)0 0 1 sP P x= − [4.34]

We can define the reduction in vapor pressure of the solvent by the difference:

00 0 0= −P P PΔ [4.35]

Thus, for the relative reduction in vapor pressure of the solvent, using relation [4.34], we obtain:

(L)00

0s

s

Px

P

Δ= [4.36a]

If only one solute is present, this law becomes:

(L)00

0s

Px

P

Δ= [4.36b]

This is Raoult’s law of tonometry, which states that the relative reduction in the vapor pressure of the solvent of a binary solution is equal to the molar fraction of the solute. We can see that this law is valid only for ideal solutions and far from the critical conditions, to allow the fugacity of the gas to be assimilated to its pressure.


For example, in the case of a solution of naphthalene in benzene, we obtain the following results; for a molar fraction of naphthalene of 0.043, the relative reduction in the vapor pressure of benzene is 0.044, whereas for a molar fraction of 0.917, the reduction is 0.891.

This law can be used to experimentally determine the molar mass of the solute (see section 4.8.2).

4.4. Diagrams of the liquid–vapor equilibria in real solutions

The equilibria involving real solutions include activity coefficients which depend on the composition and temperature, and fugacity coefficients, which depend on the temperature and the total pressure. As soon as the solutions can no longer be considered to be perfect, the idea of immiscibility in the liquid phase may be encountered. Thus, we distinguish two families of equilibria depending on whether or not the components are miscible in all proportions in the liquid phase.

4.4.1. Total miscibility in the liquid phase

When the components are totally miscible in the liquid state, real solutions are set apart from perfect solutions or ideal solutions by the fact that the intervention of the activity coefficients will result either in tendencies toward hetero-attraction between the different species (negative deviation and activity coefficients less than 1), or else in a tendency toward heterorepulsion between the different species (positive deviation and activity coefficients greater than 1). Hence, none of the curves is a straight line.

4.4.1.1. Isobaric diagrams

We distinguish two families of isobars when the liquids are totally miscible:

– the spindle isobar (Figure 4.5(a)), which resembles the curve encountered for perfect solutions, but the upper curve is no longer a straight line. A spindle such as this is the result of the existence of a single shared tangent between the Gibbs energy curves of the gaseous phase and the liquid phase – i.e. of only one point of intersection between these two curves;


– isobars presenting an extremum – i.e. a gas–liquid azeotrope, which may be maximal, if its boiling point is higher than the boiling points of the two pure substances (Figure 4.5(b)) at the same total pressure, or minimal in the opposite situation (Figure 4.5(c)). Such an extremum is the result of the existence of two common tangents between the Gibbs energy curves, meaning that there are two points of intersection between these two curves.

Of course, in accordance with the Gibbs–Konovalov theorem (see section 2.3.2.2.3), at the composition of the azeotrope, the tangent is common to the two curves and it is a horizontal.

Figure 4.5. Isobaric diagram of vaporization with complete miscibility of the liquid

4.4.1.2. Isothermal diagrams

Like isobaric diagrams, isothermal diagrams also exhibit the two families:

– spindle diagrams (Figure 4.6(a));

– diagram with an extremum, with the azeotrope (Figures 18.6(a) and (b)) being maximal or minimal.

If the azeotrope is a minimum on the isotherm, this means that its vapor pressure at a given temperature is less than that of the pure substances; the azeotropic mixture is less volatile, so this corresponds to a maximum of the azeotrope on the isobaric diagram, and similarly a maximum azeotrope on the isothermal diagram corresponds to a minimal azeotrope on the isobaric diagram.

As the system is divariant, in view of the extra condition of equality of the molar fractions in the two phases, unlike the case of the pure substance, the boiling point of the azeotrope is a function of the pressure.


Figure 4.6. Isothermal diagram of vaporization with complete miscibility of the liquid

4.4.1.3. Partial pressures and total pressure

Figures 4.7 show the partial pressure of each component and the total pressure as a function of the compositions. Also, in these figures, we have included the straight lines pertaining to perfect solutions as illustrated in Figure 4.1.

Figure 4.7. Partial pressures and total pressure for real solutions; a) with positive deviation; b) with negative deviation

If we examine the curve of the total pressure, we see that it is higher than the pressure above a perfect solution in the case of positive deviation – i.e. with activity coefficients greater than 1 (Figure 4.7(a)) – although the total pressure is less than the pressure above a perfect solution in the case of


negative deviation, with activity coefficients of less than 1 (Figure 4.7(b)). Furthermore, we can see from these figures that the more imperfect the solution (that is, the greater the difference between the activity coefficients and 1), the more the presence of an extremum is favored – a maximum in the first case, and a minimum in the second).

4.4.2. Partial miscibility in the liquid phase, heteroazeotropes

The presence of two liquid phases leads to the existence of a point where three phases are present – i.e. of a eutectic or peritectic zone (see section 2.4.2).

In an isobaric diagram, peritectic presence is impossible, because this would imply the existence of two gaseous phases, which is impossible. Hence, we shall have eutectic-type diagrams (Figure 4.8(a)). In the case of liquid–vapor equilibria, we call these eutectics heteroazeotropes; the name derives from the fact that at point E, the boiling point remains constant, as it is for a normal azeotropic mixture. Nonetheless, for the same reasons, the isothermal diagram presents only a peritectic zone; never a eutectic zone. The liquids L1 and L2 are respectively a solution of A2 in the liquid A1 and a solution of A1 in the liquid A2.

Figure 4.8. Isobaric liquid–vapor phase diagrams with heteroazeotrope

Figure 4.8(b) shows the example of the case where two liquids are totally immiscible, and the liquid is composed of a mixture of the two phases L1 and L2. However, in this case, each liquid contains only a single pure component A1 or A2. An example of such a diagram is that of the binary water–


isobutene system. In the case of total immiscibility, at a given temperature, the vapor pressures of the components are not modified with changing composition, because each liquid remains of pure phase, and the total pressure is constant as the sum of the two saturating vapor pressures.

4.5. Thermodynamics of liquid–vapor azeotropy

We suppose that a binary system leads to a liquid–vapor azeotrope. At a given temperature, the pressure of the azeotrope is fixed; similarly at a chosen temperature, the pressure of the azeotrope is determined.

4.5.1. Relation between the pressure of the azeotrope and the activity coefficients of the liquid phase at the azeotropic composition

For the composition of the azeotrope, the two phases are equititer, and therefore:

(G) (L)1( ) 1(Az)Azx x= [4.37]

We accept that gases are far from the critical conditions, which means that we can set the fugacity coefficients at the value of 1. For component A1, we can write the equilibrium condition [4.14] in the form:

(G) (L) (L)(I)011 1 1x P P x γ= [4.38]

For the composition of the azeotrope, at a set temperature, and in view of equation [4.37], we obtain:

(L)(I)1(Az) 0

1

P

Pγ = [4.39a]

(L)(I)1(Az)γ is the activity coefficient of component A1 in the liquid phase at

equilibrium and at the composition of the azeotrope.


Similarly, for the second component A2, we obtain:

(L)(I)2(Az) 0

2

P

Pγ = [4.39b]

Thus, by knowing the equilibrium pressure in the azeotrope P , we are able to find the two activity coefficients in this composition: (L)(I)

1(Az)γ and (L)(I)2(Az)γ . These relations can be exploited to determine the activity coefficients

in the azeotropic mixture.

Based on relations [4.39a] and [4.39b], we can see that if the pressure at equilibrium P is greater than each of the two saturating vapor pressures of the two components 0

1P and 02P (azeotrope at maximum pressure and

minimum temperature), then then two activity coefficients at the composition of the azeotrope ( )

1(Az)Lγ and ( )

2(Az)Lγ are greater than 1, and hence

the solution exhibits positive deviation. Conversely, if the pressure at equilibrium P is lower than each of the two saturating vapor pressures of two components 0

1P and 02P (azeotrope with minimum), then the two activity

coefficients at the composition of the azeotrope (L)(I)1(Az)γ and (L)(I)

2(Az)γ are less

than 1, and hence the solution exhibits negative deviation.

4.5.2. Relation between the activity coefficient and the temperature of the azeotrope

By applying relation [1.42], which gives the relation between the vapor pressure of a pure liquid and the temperature, we immediately find the activity coefficient at the azeotropic composition of each of the components as a function of the temperature of the azeotrope and its boiling point:

−Δ

==)()E(1

01

01

)()I)(L()Az(1

11R

lnlnAzb

vAz

TTh

P

Pγ [4.40a]

−Δ

==)()Eb(2

02

02

)()I)(L()Az(2

11R

lnlnAz

vAz

TTh

P

Pγ [4.40b]


By finding the ratio, member by member, between the two equations [4.40a] and [4.40b], we obtain:

(Az)00

(Az ) 1(E ) 2(Eb)210(Az) 1 (Az) 2(Eb) 1(Eb)

02

ln

ln

z bv

v

PT T ThP

P h T T TP

−=

− [4.41]

This equation enables us to calculate the temperature of the azeotrope, when we know the pressure and the saturating vapor pressures of the two components A1 and A2.

(Az) (Az)0 01 20 0

1 2(Az) 1(Eb) 2(Eb)

(Az ) (Az)0 01 1(Eb) 2 2(Eb)0 0

1 2

ln ln

ln ln

v v

zv v

P Ph h

P PT T T

P Ph T h T

P P

−=

− [4.42]

It is possible to give a simpler expression of this equation by applying the Trouton approximation [1.67], which gives an invariant ratio between the molar latent heat of vaporization and the boiling point, enabling us to write:

0 02 1

2(Eb) 1(Eb)

v vh hT T

≈

[4.43]

Equation [4.40] then becomes:

2(Eb) 1(Eb)0 01 2

(Az)

0 01 2

ln ln

ln ln

P PT TP P

TP P

P P

−=

− [4.44]

Hence, it is possible to calculate the temperature of the azeotrope at the chosen pressure P.


4.6. Liquid–vapor equilibria and models of solutions

The general laws of liquid–vapor equilibria can, of course, be applied to absolutely any model of a solution, provided that the model in question expresses the activity coefficients as a function of the composition of the liquid. Most of the time, the expressions obtained for the vapor pressures or the expressions of the dew-point curve and boiling curve are so complex that it is only possible to calculate them using a computer. Various software packages are available, based on models of solutions.

Liquid–vapor equilibria usually give rise to sophisticated solution models. It is these equilibria which are generally used for determining the parameters of these models in binary mixtures. For instance, we can cite the determination of the parameters of binary systems for the UNIQUAC model2, determination of the contributions of groups in the different variants of the UNIFAC model3 or the use of the PSRK4 and VTPR5 models.

4.6.1. Liquid–vapor equilibria in strictly-regular solutions

The strictly-regular solution model is able to represent the presence of azeotropes with maximum or minimum vaporization. In addition, as we know that the strictly-regular solution also represents, in certain conditions, demixing in the liquid state (see section 3.1.3), it is interesting to look at how this immiscibility combines with the liquid–vapor equilibrium.

4.6.1.1. Azeotropy of strictly-regular solutions

We shall now examine the properties and the conditions of occurrence of these azeotropes.

To begin with, we suppose that such an azeotrope does exist. We then establish the conditions for its existence.

If the azeotrope exists, we can apply the general relations we have just established (see section 4.2), in combination with the expressions given for the activity coefficients of a strictly-regular solution. 2 See [SOU 15b], Volume 2 in this set of books, Chapter 3. 3 See [SOU 15a], Volume 1 in this set of books, Chapter 8, relation [8.20] 4 See [SOU 15a], Volume 1 in this set of books, Chapter 8. 5 See [SOU 15b], Volume 2 in this set of books, Chapter 3.


We know6 that the activities of the components of the strictly-regular solution are:

( )21 exp 1Ba x x

T= − [4.45a]

( ) ( )22 1 exp Ba x x

T= − [4.45b]

4.6.1.1.1. Relation between temperature and composition of the azeotrope

By combining one of relations [4.45] and relation [4.38b], we immediately obtain the following for one of the components:

( )02(L)(I) (L) 2

2(Az) 1(Az)(Az) (Az) 2(Eb)

1 1lnRvhB

xT T T

γ− = = − [4.46a]

Similarly for the other component:

( )02(L)(I) (L) 1

1(Az) 1( Az)(Az) (Az) 1(Eb)

1 1ln 1RvhB

xT T T

Δγ− = − = − [4.46b]

By dividing relations [4.46a] and [4.46b], member by member, we obtain:

2(L) 01( Az) 2(Eb) (Az) 1(Eb)2

(L) 01(Eb) (Az) 2(Eb)11( Az)1

v

v

x T T ThT T Thx

Δ −=

−− [4.47]

By applying the Trouton approximation [1.67], as before, we obtain:

(L)1( Az) 2(Eb) (Az)

(L)1(Eb) (Az)1( Az)1

x T T

T Tx

−≈

−− [4.48]

6 See [SOU 15b], Volume 2 in this set of books, Chapter 2, section 2.5.7.


This relation enables us to calculate the composition of the azeotrope using just the boiling points of the two components and that of the azeotrope (Az)T .

Similarly, by subtracting relation [4.47] from relation [4.46a] term by term, we obtain, in light of relation [4.45a]:

( ) ( )02 2(L) (L) 2

1( Az) 1( Az) 0(Az) 1

1 ln PB x xT P

− − = [4.49]

which gives us a new relation between the composition of the azeotrope and its boiling point, and this time the saturating vapor pressures of 1 and 2:

0(Az)(L) 2

1( Az) 01

11 ln2 2

T Px

B P− = + [4.50]

4.6.1.1.2. Relation between composition and pressure in the azeotrope

If we now consider transformations at a given constant temperature T, we can write, by applying the law of liquid–vapor equilibrium [4.14] to the case of the strictly-regular solution (relations [4.45a] and [4.45b]) at the composition of the azeotrope:

( )2 (Az)(L)1( Az) 0

2ln

PBx

T P= [4.51a]

and:

( )2 ( )(L)1( Az) 0

11 ln AzPB

xT P

− = [4.51b]

By dividing relations [4.51a] and [4.51b], term by term, we obtain the composition of the azeotrope as a function of its pressure and the saturating vapor pressures of the two pure components:

(Az)2(L) 0

1( Az) 2(L)

(Az)1( Az)0

1

ln

1 ln

Px P

PxP

=−

[4.52]


This relation allows us to calculate the composition of the azeotrope when we know its pressure at a given temperature.

4.6.1.1.3. Relation between the pressure and temperature of the azeotrope

By combining the two relations [4.50] and [4.51b], we obtain:

20(Az) (Az)2

0 0(Az) 1 1

1 ln ln2 2

T PPBT B P P

+ =

[4.53]

which gives us the relation between the pressure and the boiling point of the azeotrope:

20(Az)0 2

(Az) 2 01

ln ln ln4 4

T PBP PT B P

= + +

[4.54]

4.6.1.1.4. Condition for the existence of the azeotrope

Until now, we have assumed that there was an azeotropic mixture for the strictly-regular solution. We shall now discuss the conditions that are necessary for the azeotrope to exist. In order for this to happen, the molar fractions calculated on the basis of the azeotropic properties must be between 0 and 1:

(L)1( Az)0 1x< < [4.55]

By applying relation [4.50], this condition is expressed by:

0(Az) 2

01

10 ln 12 2

T PB P

< + < [4.56]

This inequality can also be written in the form:

0(Az) 2

01

1 ln 1T P

B P− < < [4.57]


Two cases are therefore possible:

– if B < 0, 0 0 0(Az) 2 2 1 and P P P P< < , then we have the possibility of

negative azeotropy, i.e. a minimum on the isothermal boiling curves;

– if B > 0, 0 0 0(Az) 2 2 1 and P P P P> > , we then see the possibility of positive

azeotropy, i.e. a maximum on the isothermal boiling curves.

Thus, the strictly-regular solution model is the simplest model to represent a liquid–vapor azeotrope; indeed, it is obvious that perfect solutions never exhibit an azeotropic mixture.

4.6.1.2. Liquid–vapor equilibrium and demixing of the liquid

Based on relations [4.45a] and [4.45b], the partial vapor pressures above the solution will thus be:

( )2110

1

exp 1P Ba x xP T

= = − [4.58a]

and:

( ) ( )2220

21 exp

P Ba x x

TP= = − [4.58b]

By deriving these functions in relation to the composition, we find that these curves pass through extrema whose coordinates would be the same values of ( )

1x α and ( )1x β as given by relations [3.18a] and [3.18b]. We can

therefore go back to the same discussion (Figure 4.9).

Figure 4.9. Vapor-pressure curves and intersections with the miscibility gap


– if / 2B T > , we have two extrema. However, we can see that between two points I and J, for the same level of activity – i.e. at the same vapor pressure – there are three possible corresponding mixtures for the solution, which runs counter to the phase rule, because the variance would drop to zero, though it is at least 1 because we had to fix the temperature in order to obtain these values. Thus, the curve must be replaced by the horizontal IJ, passing through the point of symmetry M x = 0.5 (giving us a variance of 1), which defines two phases. The compositions of those phases are respectively equal to that of the phases at equilibrium of each of the demixings. This platform is the result of the intersection of the vapor-pressure and miscibility-gap curves;

– if / 2B T = , the derivative takes a value of 0 only for x = 0.5 (point K in Figure 4.9). That point corresponds to the critical temperature, which limits the miscibility gap;

– if / 2B T < , there is no longer any value which yields a 0 value of the derivative, and thus no longer any point of inflection. Hence, there is no longer an intersection between the miscibility gap and the vapor pressure curve.

4.6.2. Liquid–vapor equilibrium in associated solutions

We shall now examine an associated solution in which two components A and B, in order to account for the non-ideality of the solution, could be replaced by three components A, B and the dimer A2, which is the result of the association of two molecules7 of A.

In order to study the liquid–gas equilibrium in such a solution, we write that the liquid phase thus contains all three species, whilst the gaseous phase contains only two: A and B.

We can model the liquid phase using two descriptions:

– either we consider the liquid phase to be a non-perfect solution of An moles of A and Bn moles de B (initial quantities), with the respective molar

fractions Ax and Bx given by:

7 See [SOU 15b], Volume 2 in this set of books, Chapter 2, section 2.5.


AA

A A

nx

n n=

+ [4.59a]

and:

BB A

A A1

nx x

n n= = −

+ [4.59b]

The chemical potentials will be Aμ and Bμ and the activity coefficients

Aγ and Bγ . This is the first description of the solution; – or we consider the solution to be a perfect solution of A'n moles of the

monomer A, 2A'n moles of the dimer A2 and B'n moles of B with the molar

fractions:

2

AA

A A

''' ' 'B

nx

n n n=

+ + [4.60a]

2

2

2

AA

A A

''

' ' 'B

nx

n n n=

+ + [4.60b]

and:

2

BB

A A

''' ' 'B

nx

n n n=

+ + [4.60c]

We can then show that the activity coefficients (I)Aγ and (I)

Bγ in the first description obey the equations:

(I) AA

A

'xx

γ = [4.61a]


and:

(I) BB

B

'xx

γ = [4.61b]

Thus, given that the system is at equilibrium, we can write the following for the component A, in light of relation [4.61a]:

(G) (L ) (L )0 0A A A AA A A'Px P P x P xγ= = = [4.62a]

Similarly for component B, on the basis of relation [4.61b], we obtain:

(G) 0 (L ) 0 (L )B B B B B A B'Px P P x P xγ= = = [4.62b]

For example, for component A, we obtain the expression of its partial pressure in the form:

( ) ( )( )

(L ) A B0A A A

A B

1 )1 / 2

n nP P x

n nα

α+ −

=− +

[4.63]

α, which is the extent of the association reaction at equilibrium, can be calculated using the relations:

( ) ( )[ ]2A A

A B2 22A A

' / 2(1 / 2)

' 1

x nK n n

x n

α αα

= = − +−

[4.64]

( )

( )( )Δ

−=T

Th

z

zK

r

m

m

R

)(exp

0

2A

0

A0

2 [4.65]

In this last relation, ( )2

0Amz and ( )( )0

Amz are the molecular partition

coefficients of the species A and A2, and 0( )rh T is the fundamental energy level of vibration of the molecule A2.


4.7. Liquid–vapor equilibria in the critical region

Figure 4.10. Liquid–vapor diagram in the vicinity of the critical point

If we reach temperatures higher than the critical temperature of the most volatile component, the loop formed by the dew-point curve and evaporation curve does not extend to the axis corresponding to the pure volatile component. Figure 4.10 shows a network of isothermal boiling curves and dew-point curves for a binary system in which the critical point of a compound is reached (A1). The dew-point curves (illustrated by solid lines) and boiling curves (dotted) intersect at a critical point, where they are maximal and tangential to one another. For a given pressure, there is not necessarily a condensation point – for example, at the point M at the temperature T3 in Figure 4.10. At high temperatures, the mixture is homogeneous at any level of pressure. The maximum pressure on the isobaric curve is known as the cricondenbar; the maximum of temperature on isothermal curves is the cricondentherm.

If we represent the isologue curves, i.e. the curves for equititers in the pressure–temperature diagram, we can distinguish (Figure 4.11) the two curves of volatilization of the pure components, each one limited to the critical point. On the left, limited at K2, we see the curve pertaining to compound A2. On the right, we have the curve relating to compound A1, finishing at its critical point K1. Between these two curves, the isologue


curves take the shape of a loop with a maximum. Two isologue curves are represented in Figure 4.11, for two different compositions.

Figure 4.11. Isologue curves in the vicinity of the critical region

On an isologue, we distinguish three points of different natures. The point C corresponds to the critical point of the solution (temperature of the curve which yields a maximum at the chosen composition in the isothermal diagram shown in Figure 4.10). The point E corresponds to the maximum of the isologue (horizontal tangent Tjb) and the point D gives the maximum temperature at which two phases may be at equilibrium at the corresponding composition (vertical tangent Tjc).

The set of critical points C constitutes the critical line K1K2. It is tangential to the isologue at point C (shared tangent Tja). Any point M, situated between the vaporization curves of the pure substances and the critical line, is at the intersection of two lines: one is a line of composition of a liquid phase and the other a composition of the vapor phase. These two lines are not conjugate lines, except at an azeotropic point, because then the gaseous phase and the liquid phase at equilibrium have the same composition. The internal domain between the two curves of vaporization of the pure components and the critical curve is the zone where the two phases coexist.


If we compress a mixture with a given composition at a temperature lower than that of point C, the compression of the gaseous mixture will successively give us a heterogeneous mixture and then the liquid at the highest point of the loop.

TF TD

P

T

F’

K1 K2

E

A2

Tjb

D

F

G1

G2 G3

G4

Figure 4.12. Change of state without the appearance of an interface in the isologue diagram for the liquid–vapor equilibrium

For a temperature between that of point C and that of point D – e.g. at the temperature TF of the point F in Figure 4.12 – when the pressure increases, the gaseous mixture yields its first drop of liquid as it crosses the right-hand part of the isologue at F’. As the pressure continues to rise, the volume of liquid also increases, before diminishing, and finally the last drop disappears at F. This rather rare phenomenon of vaporization by compression is known as retrograde vaporization.

For temperatures higher than the maximum temperature of point D (TD) (Figure 4.12), the mixture will no longer liquefy through isothermal compression. If, though, whilst raising the pressure above point E (path G1G2), we cool it below the temperature of C (path G2G3) and then relieve the pressure (path G3G4), we find that it is again in the liquid state without ever having seen the appearance of an interface.


4.8. Applications of liquid–vapor equilibria

There are numerous applications for liquid–vapor equilibria. Some of them are crucial in industry, such as distillation for the chemical and petrochemical industries; others are useful in the laboratory, such as ebulliometry; others still are present in nature, such as the solubility of gases.

4.8.1. Solubility of a gas in a liquid

We speak of solubility of a gas in a solvent when the solvent’s vapor pressure is much lower than that of the solute, which means that there is practically no vapor of the solvent in the gaseous phase. We can apply the general laws of liquid–vapor equilibria to the solute alone, with ( )

0 0Gx = . Thus, we shall have the expression (because the gas is far from the critical conditions):

( )I (L )0i i i iP P xγ= [4.66]

From this, we deduce the molar fraction of the gas in the liquid:

( ) ( )

0

(L )I I0

expRv

iii

i i i

gPP Tx

P γ γ= = [4.67]

By developing the Gibbs molar energy of vaporization, as a function of the partial pressure and the temperature, we obtain:

( )0 0I(L )ln ln ln

R Rv v

i ih s

x PT

γ= + − − [4.68]

Generally speaking, the activity coefficient of the solute in the solution varies with temperature, even though the enthalpies and entropies of vaporization can be considered constant.

If we apply these results to perfect solutions, we make ( ) 1Iiγ = , and

relation [4.67] gives:

(L )0i

ii

Px

P= [4.69]


Thus, the solubility of a gas yielding a perfect solution is proportional to its partial pressure, and is independent of the nature of the solute.

As a function of temperature, relation [4.69] is written as:

0 0(L )ln ln

R Rv v

ih s

x PT

= + − [4.70]

As the enthalpy of vaporization of the pure substance 0vh is positive,

solubility decreases as the temperature increases. The heat of dissolution is exactly the same as the heat of liquefaction.

By introducing the mass of solute per kilogram of solvent, defined by the relation:

(L )

0

1000 i ii

M xM

=m [4.71]

and by combining with relation [4.68], we obtain:

00

1000 i ii

i

M PM P

=m [4.72]

Thus, the mass of gas dissolved is proportional to its partial pressure (this

is Henry’s old law of solubility of gases), but is not independent of the solvent (its molar mass comes into play).

If we adopt the hypothesis of low solubility, it is preferable to switch to convention (II), and the law becomes:

( )(L )

II(LV)i

ii i

Px

K γ= [4.73]

If the solution is dilute and ideal, then by introducing Henry’s constant, we can write ( )II (L )

i iHKγ = , and the solubility is then written as follows, by introducing the definition [4.71]:

0 0(LV) (L ) 01000 1000

i ii

i i ii iH

PM PM

M K M K P= =m

[4.74]


The solubility remains proportional to the partial pressure, the proportionality coefficient is (LV)1 / iK , and this solubility depends on the solvent.

NOTE 4.4.– If we express the solubilities on the basis of the molar fraction (L ) ,ix they become independent of the solvent.

In the case of an arbitrary solution, we apply relations [4.67] and [4.68], and we see that if the solution exhibits positive deviation ( (I) 1iγ > ), the solubility is less than in the case of perfect solutions. On the contrary, for solutions with negative deviations, the solubility is greater than in the ideal solution.

The solubility may vary in one direction or the other with changing temperature, depending on the variations of the activity coefficients. When the solubility increases with temperature, we speak of retrograde solubility.

For example, for a strictly-regular solution, the application of relations [4.67] and [4.45a] and [4.45b] yields:

( )(L ) 0 0

2(L )

lnln

R R1

i v i v ii

i

x H S BP

T Tx= + − −

− [4.75]

If the constant B is positive, the left-hand term in the above expression can vary in either direction with changing temperature depending on the relative values of B and the enthalpy of vaporization. Conversely, if the constant B is negative, the left-hand term decreases if temperature rises.

NOTE 4.5.– We have just examined the solubility of gases in liquids. The same methodology could and would be used to examine the phenomena of solubility of gases in solids.

4.8.2. Determination of molar masses by tonometry

Tonometry is the determination of the molar mass of a solute by measuring the relative reduction in vapor pressure of the solvent.


To begin with, with a single solute, equation [4.36b] is written as:

(L )00

0s

Px

P= [4.76]

We can introduce the mass of solute ms, expressed in grams per kilogram of solvent, defined by relation [4.71]. By feeding this definition back into relation [4.76] we obtain:

0

000

1000 PPMmM s

s Δ= [4.77]

We know that this last relation is only true in an infinitely-dilute medium. Thus, in order to determine the molar mass of a substance, we prepare multiple dilute solutions of that substance in a known solvent. These solutions differ from one another only in terms of their mass concentrations. For each of them, we measure the reduction in vapor pressure and in each case calculate a value of the molar mass of the solute by applying relation [4.77]. The true value of the molar mass of the substance under study is obtained by extrapolation to the origin of the curve, giving the values of sM thus calculated as a function of the mass concentration sm .

4.8.3. Determination of molar masses by ebulliometry

Ebulliometry is the use of the elevation of the boiling point of a solvent A0 containing a non-volatile solute As to measure the molar mass of that solute.

As the vapor pressure of the solute is much lower than that of the solvent, we can consider that the gaseous phase is pure, containing only the vapor of the solvent. The equilibrium is expressed solely by the equality of the chemical potentials of the solvent in both phases:

( )( )0I L*(G) 0(L ) (L )

0 0 0R lng g T xγ= + [4.78]


This expression can be written in the form:

( )( )0

*(G) 0(L )I L (L) 0 0

0lnR

g gx

Tγ

−= [4.79]

Using Helmholtz’s relation of the variation of the Gibbs energies with temperature, for volatilization, we have:

(L)(I) (L) 00 0 0

2ln

Rvx h

T T

γ∂= −

∂ [4.80]

By integrating, supposing the enthalpy of vaporization to be independent of the temperature, between the boiling point of the pure solvent 0( )EbT and its boiling point T above the solution, we obtain:

0(L)(I) (L) 00 0

0(Eb)

1 1lnRvh

xT T

γ = − − [4.81]

We posit that the difference of the boiling point of the solvent is between that of the solution and that of the pure solvent:

(Eb) 0(Eb)T T T= − [4.82]

In general, the differences are not great, so we can make the approximation:

20(Eb) 0(Eb)TT T≅ [4.83]

In the case of a perfect solution, the activity coefficient of the solvent is equal to 1, and therefore we obtain:

00 (Eb)(L)

0 20(E )

lnRv

b

h Tx

T≅ − [4.84]

If the solution contains several solutes, we can write the molar fraction of the solvent:

(L) (L)0 1 s

s

x x= − [4.85]


When these expressions are substituted back into expression [4.84], the difference in the boiling point of the solvent becomes:

( )20 Eb (L)

(Eb) 00

R ( )s

sv

TT x

h= [4.86]

As the enthalpy of vaporization is negative, the temperature difference is a reduction of the boiling point. This law is known as Raoult’s law of ebulliometry.

If the solution contains only one solvent, Raoult’s law of ebulliometry is written as:

( )20(Eb) (L)

( ) 00

REb s

v

TT x

h= [4.87]

With the molar masses being expressed in grams, if we let sm denote the mass of solute in grams per kilogram of solvent, we can write:

(L) 0

1000s

ss

M mx

M= [4.88]

This yields the relation giving the molar mass of the solute as a function of the difference in boiling points:

( ) 20 0(Eb)

0( )0

R

1000s

sEbv

M T mM

Th= [4.89]

We define the solvent’s ebullioscopy constant by the relation: 2

0 0(Eb)eb 0

0

R

v

M TK

h= [4.90]

For example, for water, ebK (calculated) = 514 K.g.mole-1. Thus, we apply the law in the form:

)Eb(eb 1000 T

mKM s

s Δ= [4.91]


In fact, as we have seen, this is a limit law. To determine the molar mass Ms in the case of solutions that are not infinitely-dilute, we measure the difference in temperature for different compositions – i.e. different values of ms – and plot the curve represented by the right-hand side of equation [4.91] as a function of sm . The curve obtained is extrapolated to the origin. The limit value obtained gives us the molar mass of the solute.

We cannot fail to draw the parallel between cryometry and ebulliometry (see section 3.4.2), which are, in fact, in both cases, the exploitation of the variation of the temperature at which the phase of the solvent changes in view of the presence of the solute.

NOTE 4.6.– If the solution is not very dilute, expression [4.84] is replaced by:

00 (Eb)(L)(I) (L)

0 20(Eb)

ln ln 1Rv

ss

h Tx

Tγ ≅ − − − [4.92]

With this relation, we are able to measure the activity coefficient of the solvent at the solution’s boiling point.

4.8.4. Continuous rectification or fractional distillation

Fractional distillation, also known as rectification, is a method for separating out the components of a homogeneous liquid mixture using the isobaric diagram of liquid–vapor equilibrium therein. It is performed in installations called distillation columns.

4.8.4.1. Insufficiency of simple distillation

Simple distillation consists of raising the mixture to boiling point in a still and condensing the vapor at the top of the still.

Consider a binary liquid, formed of two components A1 and A2, each characterized by its boiling point in the pure state T1(Boil) and T2(Boil). We suppose that component A1 has the lower boiling point, and say therefore that A1 is the more volatile or lighter component. Thus by contrast, A2 is the less volatile component and is said to be heavier. We wish to separate the two components out of the mixture by raising its temperature. The isobaric liquid–vapor diagram has the appearance of a single spindle, as shown in


Figure 3.5(a). To simplify the notations, in this section devoted to fractional distillation, x shall represent the composition of a liquid phase and y that of a gaseous phase. Thus, in relation to our previous system of notation, we have the equivalencies:

(L)1x x= [4.93a]

and:

(G)1x y= [4.93b]

If we raise the liquid mixture to its boiling point, it produces a vapor whose composition is y and a liquid whose composition is x; these two values are linked by relation [4.25].

Input x0

Distillate xD

Reflux

Heavy products

nth tray

Tray 0

T1(Boil)

T2(Boil)

T0(Boil)

Figure 4.13. Diagram of the distillation column

If we simply raise the temperature, the vapor becomes richer in terms of A1 whilst the liquid becomes richer in A2. However, at no time do we manage to separate the two components: at most, we can hope to enrich the vaporized part by raising the temperature of the system and condensing the vapor at that composition. Even theoretically, simple distillation is incapable of separating the two components from one another.

4.8.4.2. Rectification in the case of a single-spindle binary system

To overcome this difficulty we can employ fractional distillation, which consists of recovering the condensed vapor and enriching it in a second column at a higher temperature, repeating this operation a great many times,


which requires very large facilities that are not viable. Hence, it is preferable to carry out continuous fractional distillation, known as rectification.

4.8.4.2.1. Feasibility of separation by fractional distillation

For this purpose we use a vertical fractionating column (see Figure 4.13), filled with a solid inert material, which allows the flow of liquids and the rising of a gas and encourages contact between the gas and the liquid as much as possible. We place that column in an appropriate heating system, so that in normal thermal conditions the temperature in the column is continually decreasing as we go further up it, such that the bottom of the column is at the boiling point T2(Boil) of the less volatile compound, and the head (top) of the column is at the boiling temperature T1(Boil) of the more volatile.

We introduce the initial liquid solution into the level of column which is at the boiling point of the mixture T0(Boil). We call this point of introduction “tray 0”. The liquid thus begins to boil in this tray, giving off a vapor with the content y0, which rises up the column, and a liquid with composition x0, which descends by gravitation.

The rising vapors reach ever-colder areas, reaching eventually temperatures below their dew point, so that a portion of the vapor condenses and begins to descend. As it rises towards the top of the column, the vapor becomes increasingly rich in component A1, and conversely, as we descend the column, we find a mixture which is increasingly rich in component A2. The final result is that at internal thermodynamic equilibrium, at each level, the column is filled with a liquid–vapor mixture such that the two phases are at equilibrium at the temperature of the level in question. Hence, at least in part, the separation of the two components will be achieved by this operation.

4.8.4.2.2. Total reflux operation; theoretical trays

In the situation described above, no material comes out of the column and we say that the column operates with total reflux; the reflux of a column is defined as the ratio of the amount of material reinjected into the column from the top to the amount of material extracted from the top of the column. The reflux is total when all of the material is recycled in the column and the extracted quantity is zero. Obviously, this is a theoretical


operation with zero production, but it presents a certain advantage in studying the way in which the column works. This is the state of our column at thermodynamic equilibrium. We shall focus on this for a time, though later on we shall come back to more realistic conditions of operation, with partial reflux and therefore a certain amount of production.

In order to model the column, we replace the infinity of points at different temperatures with horizontal slices, dividing it up. To define a slice, we refer to the phase diagram in Figure 4.14; we find the tray “0” there, where we find the liquid introduced at its boiling point at composition x0 and the vapor of composition y0 in equilibrium. We define the following slice, situated in the diagram below tray “0”, which finishes with “tray N = 1”, by writing that the composition of the liquid in this new tray is equal to the composition of the vapor in tray “0”, expressed by:

1 0x y= [4.94]

This operation defines the point L1, from which we derive the tray N = 1 which defines the point V1 of composition x1.

Figure 4.14. Division of the distillation column into slices (fractions)

We proceed, step by step, at each step generalizing the law of passage [4.90] in the form:

1i ix y −= [4.95]


We reach the bottom of the spindle on the platform N = pt-1, as near as possible to the boiling point of A1. Thus, the number pt is called the number of theoretical trays in the column for the distillation of our mixture. The term “tray” has been chosen because, originally, distillation columns actually were made up of real trays such as those represented in Figure 4.15, which allow the liquid to descend and the vapor to rise; this device ensures the leaching of the liquid by the vapor.

Figure 4.15. Diagram of the tray in the distillation column using real trays

The operation which we have carried out around the bottom of the spindle can also be carried out at the upper part of tray “0”, up to the level closest to the boiling point of the component A2. Thus, we define a number of theoretical trays in the column, (pt)c, which is the sum of the number of theoretical trays pt added below the inlet point and the number of new trays added above the inlet point. This maximum number, which in fact varies little with the nature of the liquid being processed, is a good overall characteristic of our column irrespective of the composition of the mixture injected.

Figure 4.13 shows the theoretical trays thus defined.

4.8.4.2.3. The Fenske equation

We shall now consider the calculation of a number of theoretical trays in a column, adopting the conditions of total reflux by the Fenske method. In order to do so, we consider that the liquid mixture exhibits perfect behavior, whose thermodynamic properties are those described in section 4.2.


We define the volatility α, relative to the most volatile component A1, in relation to the least volatile component A2, at a given temperature, by the ratio of their saturating vapor pressures at that temperature, such that:

010

2

P

Pα = [4.96]

This ratio, which is greater than 1, is a function of temperature only. Using relation [1.42], it is easy to show that we have:

0 02 1expR

v vH HT

α −= [4.97]

However, by applying the Trouton approximation in the form [4.43], we see that the numerator of the exponential is negative, and therefore the relative volatility increases when la temperature increases.

Equation [4.30] which, at a given temperature, tells us the composition of the vapor phase as a function of that of the liquid phase at equilibrium in the tray i, is written as follows, introducing the relative volatility and our new simplified notations:

( )1 1i

ii

xy

x

α

α=

+ − [4.98]

This function is represented by the branch of a hyperbola shown in Figure 4.3, with a parameter α. To ensure better separation, we can see that it is good to have as high a value of this parameter as possible.

Relation [4.98] can also be written in the following symmetrical form:

1 1i i

i i

xyy x

α=− −

[4.99]

Let us first examine the situation in tray zero: relation [4.99] assumes the form:

0 0

0 01 1xy

y xα=

− − [4.100]


In tray N = 1, the same relation gives us:

1 1

1 11 1

xy

y x

α=

− − [4.101]

However, we apply the definition of the heights of the trays [4.95], which is then:

1 0x y= [4.102]

Thus, we continue to define each tray until the ptth (N = pt-1), which then

gives us:

1 1N N

N N

xyy x

α=

− − [4.103]

still with the law of generation [4.95], which is written:

1N Nx y −= [4.104]

Hence, we can finally deduce the tray N, which obeys the series of relations:

21 2

1 2...

1 1 1 1N N N N

N N N N

xy y y

y y yxα α α− −

− −= = = =

− − − − [4.105]

because in the previous tray, we must have:

1 2

1 21 1N N

N N

y yy y

α− −

− −

=− −

[4.106]


and thus, from tray to tray, we obtain the law which links the composition of the vapor in tray “0” to the composition of the liquid in the last tray:

0

01 1NN

N

y x

y xα=

− − [4.107]

We can easily deduce an expression of the relative volatility:

1 0

1 0

11

N N

N

y x

y xα −

−

−=

− [4.108]

and thus, by moving to logarithms, we obtain the Fenske equation:

( )( )

0

0

1ln

1ln

N

N

y yx y

Nα

−−

= [4.109]

This law thus enables us to calculate the number of theoretical trays (pt = N+1) in a column between two compositions. If, as an initial composition, we choose that of tray “0”, we obtain the number of theoretical trays attached to the mixture. If we choose tray “0” to be that which is closest to the pure liquid A2, we obtain the total number of theoretical trays in the column.

NOTE 4.7.– If we examine the trays that we have defined on the basis of Figure 4.14, we see that the number thereof is not remotely similar to that calculated by Fenske’s method. Indeed, in this calculation, we only used one value of α – that which corresponds to the temperature T0(Boil) of tray “0” – which is not the case with the trays defined in Figure 4.14. Thus, if we choose Fenske’s method, we obtain an approximate value of the number of theoretical trays.

4.8.4.2.4. McCabe and Thiele diagram

McCabe and Thiele devised a method for graphically determining the number of theoretical trays, using the diagram shown in Figure 4.16.


Figure 4.16. McCabe and Thiele diagram

This diagram is plotted on a system of axes: the composition of the vapor phase on the ordinate axis, and composition of the liquid phase on the abscissa axis. This system of axes shows, firstly, the conditions of isothermal equilibrium, given by the branch of the hyperbola shown in Figure 4.3, and whose equation is given by [4.97], and secondly the first bisector, defined by the equation:

y x= [4.110]

Each point on the equilibrium curve has a corresponding level of the isobaric phase diagram in Figure 4.14 – in other words, we have a triplet of values: the temperature at which the equilibrium in question occurs, the composition of the vapor phase and the composition of the liquid phase corresponding to that equilibrium. Thus, the McCabe–Thiele diagram is isobaric, but does not have the drawback of being isothermal, as Fenske’s calculation is.

On the equilibrium curve, we show the point E0, which corresponds to the first equilibrium of our input composition (x0, y0 and T0(Boil)). E0 represents tray “0”. The horizontal line starting at E0 intersects the first bisector in B1 at the point x1 = y0. This horizontal generates the composition of the liquid phase of the second tray. The vertical at B1 intersects the equilibrium curve at E1 – the point at which the equilibrium characterized by the triplet x1, y1 and T1(Boil) is achieved (i.e. which represents tray N =1). By continuing in this manner, we successively construct the points B2, E2, B3, E3, …BN-1, EN-1, BN, EN. This last point represents the last tray in the column. This tray is fixed by the composition we wish to obtain at the output of the column.


McCabe’s method exhibits the advantage, over Fenske’s method of calculation, of not requiring the solution to be perfect. Indeed, the equilibrium curve may very well be the real curve, plotted on the basis of the set of triplets characterizing the real isobaric diagram in Figure 4.14 instead of the theoretical branch of a hyperbola relative to perfect solutions.

In comparison to the plot of the theoretical trays obtained from the phase diagram in Figure 4.14, the plot obtained in Figure 4.16 is more accurate, because the two scales of ordinates and abscissas are identical, whereas the scale of the temperature in Figure 4.14 is a logarithmic scale inverse in relation to the titers.

4.8.4.2.5. Partial reflux operation. Sorel’s equation

Up until now, we have worked with total reflux, leading to zero production. We are now going to determine a number of “real” trays, characterized by a value of the reflux rate.

Let Gn denote the number of moles of gas deriving from the ptth tray with

the content Ny , Ln the number of moles of liquid dropping down from the tray N+1 whose composition is 1Nx + , and D the number of moles of distillate separated at the content Dx .

The total material balance at the top of the column is written as:

G Ln n D= + [4.111]

The particular balance pertaining to the component A1 at the top of the column would be:

1G N L N Dn y n x Dx+= + [4.112]

We define the reflux rate as being the ratio of the quantity of liquid refluxed to the quantity let out – i.e.:

LnR

D= [4.113]


By combining the last three equations, we deduce:

11 1D

N NxRy x

R R+= ++ +

[4.114]

This is Soret’s equation.

The first thing to note is that the Soret equation replaces relation [4.95] for the generation of the last tray.

NOTE 4.8.– If the reflux rate tends toward infinity, we return to using relation [4.95].

We are going to use the Soret equation to deduce a graphical method adapted from McCabe and Thiele’s approach, to calculate an approximate value for the number of “real” trays.

Figure 4.17. McCabe–Thiele corrected by the Soret line

In order to do so, on the McCabe diagram (Figure 4.17) we draw a straight line starting at the point S with the coordinates:

1S N

N

y yx x +

==

which intersects the ordinate axis at the point:

11

Nxy z

R+= =

+ [4.115]


The equation for this line is:

11 1

NxRy xR R

+= ++ +

[4.116]

By comparing expression [4.116] with Soret’s equation [4.114], we can see that it is the same equation, provided we assimilate xD and xN+1. Thus, this straight line replaces the bisector in the McCabe–Thiele diagram, because it gives us the equation for generation of successive slices of the column. We call it the Soret line. The new construction is thus identical to that of Figure 4.14, but points B1, B2, etc., must now be taken to be on the Soret line. There are two noteworthy consequences:

– the number of trays increases as a function of the reflux rate R, because by increasing R, we bring the Soret line closer to the equilibrium curve and thus decrease the height of the slices;

– there is a limit to the content of the input solution used to produce a mixture with the content xN+1; this limit can be read from Figure 4.17. The point of intersection of the Soret line and the ordinate axis (y = z) must be situated below the ordinate y0 which is equal to the composition of the input vapor.

When the columns were actually composed of trays, the “practical” number of trays was still higher than the “real” number, because our construction supposes that the phase equilibria are perfectly achieved on each tray. In practice, though, this is not the case. This means that the so-called equilibrium curve is shifted towards the straight line, which brings the Soret line even closer, and therefore decreases the height of the slices and increases the number of trays.

4.8.4.3. Fractional distillation in the presence of an azeotrope

Let us now consider the case where the isobaric phase diagram contains an azeotrope of vaporization. By definition, this azeotrope gives off a vapor of the same composition as its liquid phase, so it boils at a constant temperature, emitting the same vapor, and thus behaves like a pure substance in the sense that it blocks distillation. It is as though the diagram in fact contains two spindles, and we shall need to distinguish four cases:

– if the azeotrope is a maximum, as in the case of a mixture of water and nitric acid, and if the input composition is richer in water than the azeotrope,


the simple construction in Figure 4.14 demonstrates that we obtain pure water at the top of the column and the azeotropic mixture at the bottom;

– if the azeotrope is still a maximum and the input composition is less rich in water than the azeotrope, the construction shows that we obtain pure nitric acid at the top of the column and the azeotropic mixture at the bottom;

– if the azeotrope is a minimum, as in the case of the mixture of water and alcohol, and if the input composition is richer in water than the azeotrope, the simple Figure 4.14 shows that we obtain pure water at the bottom of the column and the azeotropic mixture at the top;

– if the azeotrope is a minimum and the input composition is less rich in water than the azeotrope, a simple construction such as Figure 4.14 shows that we obtain pure alcohol at the bottom of the column and the azeotropic mixture at the top;

In conclusion, for mixtures exhibiting a maximum, only the excess component in relation to the composition of the azeotrope can be obtained practically pure in the condensate, but we separate out only a part of that, because another part goes to the azeotropic state. For mixtures exhibiting a minimum, only the excess component in relation to the azeotrope can be obtained practically pure in the vapor state, but a fraction of this component will go to contribute to the azeotropic mixture in the distillate.

We say that the azeotrope blocks the distillation.

NOTE 4.9.– Unlike with the pure substance, the boiling point of an azeotrope depends on pressure, so it is possible to change this temperature by modifying the pressure. We exploit this observation to “break” the azeotrope. In order to do so, after an initial distillation during which the azeotrope is collected, it is distilled again at a pressure such that the input composition has flipped to the other side of the extremum at the new pressure, so distillation can continue at the new pressure and we can obtain the second component in the pure state, at least for a fraction.

5

Equilibria Between Ternary Fluid Phases

When we move from the binary to the ternary system, in terms of the characterization of the phases, the addition of a component results in the addition of a molar fraction. Thus, we have two independent molar fractions, because the sum of the three molar fractions in a phase is equal to 1.

5.1. Representation of the composition of ternary systems

Two representations are used for ternary systems: one symmetrical, because the three components play symmetrical roles within the system; the other dissymmetrical, because one of the three components plays a more important role than the other two – generally this is the role of the solvent, with the other two components being considered to be solutes.

5.1.1. Symmetrical representation of the Gibbs triangle

The composition of a mixture M, with fractions x1, x2 and x3 =1 – x1 – x2, is represented in an equilateral triangle, whose three vertices represent each of the pure components A1, A2 and A3 (Figure 5.1(a)).

The side A3A1 is graduated from 0 to 1 in the direction from A3 toward A1, and the molar fraction x1 of component A1 at point M, projected onto the side A1A3 in parallel to A3A2 in the direction of the arrow A3A3, is such that:

( ) 3 11

3 1

A aMA A

x = [5.1]



In Figure 5.1(a), we can read: 1(M) 0.70x = .

The side A1A2 is graduated from 0 to 1 in the direction from A1 toward A2, and the molar fraction x2 of component A2 at point M, projected onto the side A1A2 in parallel to A1A3 in the direction of the arrow x2, is such that:

( ) 1 22

1 2

A aMA A

x =

[5.2]

In Figure 5.1(a), we can read: 2 (M) 0.30x = .

Figure 5.1. Symmetrical representations of the composition of a ternary system

The side A2A3 is graduated from 0 to 1 in the direction from A2 toward A3, and the molar fraction x3 of component A3 at point M, projected onto the side A3A2 in parallel to A1A2 in the direction of the arrow x3, is such that:

( ) 2 33

2 3

A aMA A

x =

[5.3]

In Figure 5.1(a), we can read: 3 (M) 0,37x = .

It is easy to show that in the equilateral triangle, we have A1a2 + A2a3 + A3a1 = A3A1 = 1.

Equilibria Between Ternary Fluid Phases 165

One of the consequences of this representation is that all the mixtures including the same proportion of a component (A1, say) are represented by a parallel a1y to the opposite side A2A3, because the distance from that side must remain constant.

A further consequence is that if we add or remove increasing amounts of a component A1 to or from a ternary mixture, the ratio of the molar fractions of the other two does not vary: that is, the ratio Ma2/Ma3 remains constant, and the figurative point of the mixture shifts along a straight line passing through the figurative vertex of the component added or removed. By adding more of a component we move closer to its corresponding vertex; conversely, we move away from the vertex by removing the substance.

In Figure 5.1(a), starting from point M, the addition of component A1 causes movement along the line A1M towards A1. At the point m on this line and the side A3A2, there is no longer any component A1; the side A3A2 represents the binary mixture formed by the two components A3 and A2.

Thus, each of the sides represents one of the binary systems that are subsets of the ternary.

A segment such as A1m is a so-called quasi-binary section of the diagram. The ratio between the molar fractions of the other components is constant within that segment. In Figure 5.1(a), along the straight line A1m, we have:

( )( )

22

3 3

M.

Mxx

Constx x

= = [5.4]

Consider two phases, with different compositions, represented by the points P and Q (Figure 5.2(a)). We wish to obtain the position of the point M, representing the mixture of these two phases, which are supposed to be miscible. The composition at point M is linked to the phase composition of the mixture. We respectively use f(P) and f(Q) (f(P) + f(Q) =1) to denote the phase fractions of the initial two mixtures – i.e. the ratios of the numbers of moles of components in each of the phases to the total number of moles in


the system. The material balance in terms of component A1 enables us to write that the molar fraction x1 of the resulting mixture (point M) is such that:

( ) ( ) [ ] ( ) [ ]1 1 1 1 1 1M (P) P 1 (P) Q (P) (Q) (P) (Q)x f x f x x x f x= + − = − + [5.5]

This expression can also be written in the form:

1 1

1 1

(Q) (M) MQ(P)(Q) (P) PQ

x xf

x x−= =−

[5.6]

Similarly, the balance for component A3 gives us:

3 3

3 3

(Q) (M) MQ(P)(Q) (P) PQ

x xf

x x−= =−

[5.7]

Thus, the representative point after mixing is represented by the molar barycenter of the initial solutions, represented by the points P and Q. This is none other than the lever rule or chemical moment rule, which we have already encountered in the case of binary systems (section 2.3.2.2.1).

Figure 5.2. Representation: a) of a mixture of two solutions; b) of three solutions


We now consider the mixture S formed by the mixing of three solutions P, Q and R (Figure 5.2(b)). The three initial phase fractions are now f(P), f(Q) and f(R) (f(P) + f(Q) + f(R) =1). We write the balances for all three components in turn:

– balance for A1:

( ) ( ) ( ) ( )1 1 1 1S (P) P (Q) Q (R) Rx f x f x f x= + + [5.8]

or:

( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1S R (P) P R (Q) Q Rx x f x x f x x= + − + − [5.9]

– balance for A2:

( ) ( ) ( ) ( )2 2 2 2S (P) P (Q) Q (R) Rx f x f x f x= + + [5.10]

or:

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2S R (P) P R (Q) Q Rx x f x x f x x= + − + − [5.11]

– balance for A3:

( ) ( ) ( )3 2 1S 1 S Sx x x= − − [5.12]

Thus, the overall composition corresponds to the point S situated at the intersection of the segments RR’, PP’ and QQ’, such that:

( ) SR 'RRR '

f = [5.13a]

( )( )P R'QQ PR '

ff

= [5.13b]

and:

( ) ( ) ( ) RSP Q 1 RRR '

f f f+ = − =

[5.13c]


A variant of the symmetrical representation is shown in Figure 5.1(b), where the height of the triangle is 1 and the coordinates are obtained by projection of the point M perpendicularly to the sides.

5.1.2. Dissymmetrical representation of the right triangle

When one of the components plays a particularly important role – e.g. the role of a solvent – in comparison to the other two, it is preferable to represent the composition of the mixtures using Cartesian coordinates. Such is the case for aqueous saline solutions, in which water plays the role of the solvent. It is also the case with solvent extraction, where we distinguish between the initial solute and solvent, on the one hand, and the final solvent (or diluent) on the other. This time, it is the solute which plays the important role.

The component in question occupies the origin of the axes; the molar fractions of the other two are reported on the two perpendicular axes. The composition of a mixture represented by point M (Figure 5.3) is given by:

( ) 0 21

0 1

A aMA A

x = [5.14a]

( ) 0 12

0

A aMA A2

x = [5.14b]

The isosceles right triangle obtained by joining points A1 and A2 acquires the same meaning and the same property as the former equilateral triangle, from which it is derived by deformation.

Because A0a2 is the molar fraction of A0, a2A1 represents the fraction of (A2+A0). However, we have:

A2d = a2A1 = a2M + Md [5.15]

As a2M represents the molar fraction of A2, Md indeed represents the molar fraction of A0.


Similarly, we can show that the chemical moments rule applies in this triangle just as it does in the equilateral triangle. Relations [5.7] and [5.13] apply in the isosceles right triangle.

Figure 5.3. Dissymmetrical representation of the composition of a ternary system, in Cartesian coordinates

5.2. Representation of phase equilibria

The lines representing phase equilibria in binary systems are replaced by layers in the space for ternary systems. The representation of these layers in space rapidly becomes complicated, and it is difficult to read and interpret the diagram, so these layers are replaced by their planar (2D) projections.

5.2.1. Isothermal projections

Equilibrium layers are constructed in a right prism with the equilateral triangle of composition (see Figure 5.4) or a rectangle as a base. On the side, the temperature is reported vertically. When pressure influences the equilibria, the construction is isobaric at a definite pressure.

The right prism is intersected by horizontal planes, each characterized by its temperature. The isothermal sections are projected onto the base triangle. Thus, a point M is projected to m, accompanied by the value of its temperature.

M

a1

a0

a2

A1

A0 A2

1

1

d

x1

x2


(T)

M

m P

T

T

D

d

10

1 0

Figure 5.4. Isobaric projections in the horizontal plane and isoplethic projections in the plane P

5.2.2. Conjugate points and conodes

In the triangle of composition, the projection of the points of the layers at a given temperature will reveal, at that temperature, biphasic zones delimited by lines of the diagram (Figure 5.5).

In such zones, segments connect two conjugate points in the diagram. These segments are known as conodes or conodal lines.

At each endpoint C and D of a conode, the points give the compositions of the two phases which are at equilibrium at point M, if we apply the lever rule.

If we wish to see the influence of temperature, we project several isothermal sections onto the triangle of composition.

M C D

A1

A3 A2

x2

x3

x1

1 0

0 1

1 0

Figure 5.5. Isothermal projection of a biphasic zone and its conodes


5.2.3. Isopleth sections

A second section is used in ternary diagrams. This time, the right prism is divided by a plane P which contains a quasi-binary section of the diagram. This means that, in that plane, the ratio between the molar fractions of two components remains constant. For example, in Figure 5.4, we examine the section through the lens of the plane DdP, passing through m. The point P is characterized by the composition A1-A3 (0.60). Such a section is known as an isopleth section.

5.3. Equilibria in liquid phases with miscibility gaps

In the condensed phase, equilibria of miscibility in liquid phases will not depend on the pressure. The only variables will be the temperature and the composition of the phases, represented by the molar fractions of the two components in that phase.

5.3.1. Representation of the miscibility gap

The miscibility gap develops with a surface in the right prism used for the representation. The projections onto the triangle of composition of the sections of that surface by horizontal planes at different temperatures give us curves, such as those in Figure 5.6, representing the isotherms of the miscibility gap in the acetone–phenol–water system. These isothermal sections may be closed, such as the curve obtained at the temperature of 87 K, or open, pressed to one side of the triangle, like the curves at 68 and 36 K. These curves exhibit critical points K, K’, K1, which are the projections of the isothermal critical points, and a critical line, which is the set of all the projections of the critical points. We can see, in the chosen example, that the gap expands at low temperatures, revealing a higher ternary critical point in the vicinity of the temperature of 92 K.

We can also give an isopleth section in the plane defined by points A and M in Figure 5.6. This section, represented in Figure 5.7, resembles a miscibility diagram of a binary system, apart from the fact that one of the extremities of the composition axis represents not a pure substance but the binary water–phenol mixture corresponding to the composition at point M.


Figure 5.6. Isothermal sections of the liquid miscibility gap in the phenol-water-acetone system

We see the appearance of the pseudo-binary triple point K3 at the temperature of 87 K. Below 65 K, the gap is composed of a mixture of two liquid phases, one of which has exactly the composition of the pseudo-binary system water–phenol at the point M.

Figure 5.7. Isopleth section in the plane AM from Figure 5.6


5.3.2. Sharing in liquid–liquid systems

When a solute is soluble in two solvents which are immiscible or have very low miscibility, it is divided between the two solvents; we then say that it is shared between the two solvents. We distinguish two cases depending on whether or not the shared substance has the same constitution (the same formula) in the two phases.

5.3.2.1. The shared substance has the same constitution in the two solvents

In this case two immiscible liquid components are shared, which are the α and β phases. The component Ai is shared between the two phases, identically in both phases. The distribution between the two phases respects an equilibrium, which is written:

Ai (α phase) = Ai (β phase)

This equilibrium results in the equality of the chemical potentials of the component between the two phases:

( ) ( )i iμ μ= [5.16]

This is expressed by the following relation between the activity coefficients:

( )( )

( )i

ii

aK

a= [5.17]

The equilibrium constant ( ) ,iK which depends on the temperature, is called the partition coefficient of component Ai between the α and β phases.

If the solution is sufficiently dilute, the relation is applicable between the concentrations with the constant ( )(III)

iK , which is therefore the partition coefficient in reference (III) – infinitely-dilute solution – or at molar concentration, expressed by the concentrations.

( )( )(III)

( )i

ii

cK

c= [5.18]


This relation constitutes the partition law of a component between two phases.

In order to express the partition coefficient, we shall think in terms of the vapor–solution equilibrium. Indeed, according to the principle of superposition of the equilibria, if the α and β phases are at equilibrium with one another, we can write that they would be at equilibrium with the same vapor phase. Relation [5.17] can then be applied for both phases, as follows:

( )LV( )i

ii

fK

a= [5.19a]

and:

( )LV( )i

ii

fK

a= [5.19b]

By feeding back the values of the activities, found by the last two expressions, into relation [5.17], we obtain:

( )

( )( )

LV( )

( ) LVi i

ii i

a KK

a K= = [5.20]

Thus, the partition coefficient is the ratio between the vaporization constants of the component in the two solutions.

The fugacity of the gases remains the same in both cases, and if the molar fractions in the liquid phases tend toward zero, the activity coefficients tend toward 1. Thus, we have:

( )( )

( )

( )0lim

Li

ii

xi

xK

x→= [5.21]

To determine the partition coefficient, we first plot (Figure 5.8(a)) the ratio between the molar fractions ( ) ( )/i ix xβ α as a function of the molar fraction in one of the solutions ( )

ix α for different values a1, a2, … aj of the composition ( )

ix β .


The curves obtained are extrapolated for ( ) 0ix = , so we have b1, b2, … bj for the values obtained. We then plot (Figure 5.8(b)) these points as a function of the other composition ( )

ix . By extrapolation of the curve obtained to zero, we then obtain the partition coefficient.

( )βix

( ) ( )αβii xx /

( )αix

( )4axi =β

( )3axi =β

( )2axi =β

( )1axi =β

4a 3a 2a 1a

( ) ( )αβii xx /

( ) ( )LVαLVβ / ii KK

1b2b3b4b

1b2b

3b4b

a) b)

Figure 5.8. Experimental determination of the partition coefficient

NOTE 5.1.– If the component Ai is a solid, the concentrations of Ai in each of the two phases is then the solubility of the solid in each of the phases, and if those solubility values are low, then relation [5.18] is applicable to the solubilities in the form:

( )( )

( )i

cii

sK

s= [5.22]

Hence, the partition coefficient is equal to the ratio between the solubilities of the component in each of the phases.

5.3.2.2. The shared substance does not have the same constitution in the two solvents

Suppose that the component Z is shared between the α phase, in which it is in the form ( )Z , and the β phase, in which it is dimerized in the form

( )2Z .

However, there is always a very small amount of monomer in the β phase at equilibrium with the dimer. This equilibrium is written:

( ) ( )2Z 2Z= [5.23]


Only the monomeric form is shared between the two phases. Thus, the equilibrium of sharing is written as:

( ) ( )Z Z= [5.24]

The equilibrium between the monomer and the dimer means that, in a sufficiently-dilute solution, we can write:

( ) 2

( )2

[Z ]

ZdK= [5.25]

The equilibrium of sharing of the species Z between the two phases gives us:

( )

Z( )2

[Z ]

ZK= [5.26]

Let ( )0[Z ] denote the total concentration of Z in the β phase. The law of

conservation of the elements means we have:

( ) ( ) ( ) ( )0 2 2[Z ] [Z ] 2[Z ] 2[Z ]= + ≅ [5.27]

supposing the dimer to be greatly in the majority in the β phase. Equation [5.25] means we can write:

( ) ( ) 1/2 ( )2 0[Z ] [Z ] [Z ]

2d

d

KK= = [5.28]

In light of equation [5.26], we obtain:

( )2 Z

( )

[Z ][Z ] d

KK

= [5.29]

This relation can also be written:

( )2 0 Z( )

[Z ] 2[Z ] d

KK

= [5.30]


Relation [5.29] is the new form of the partition coefficient when the component is in the form of a dimer in one phase and a monomer in the other.

Of course, in each case this relation, which we have established for sharing between a monomer and a dimer, needs to be adapted to the molecular shape of the solute in each of the solvents.

5.3.3. Application of sharing between two liquids to solvent extraction

Solvent extraction consists of extracting a solute from a solution by the addition of a second solvent which is immiscible with the first, and in which the solute is more soluble. The two phases, which are immiscible, are then separated, usually by decantation.

5.3.3.1. Discontinuous extractions

In discontinuous extraction, the initial solution is brought into contact with the second solvent, known as the diluent. The two phases are separated. The operation is then repeated using the remnants of the initial solution and a new injection of fresh second solvent. The operation can be repeated a great number of times.

Let ( )0n denote the initial amount of solute in the initial solution

(α phase). After the first extraction operation, ( )1n moles of solute remain in

the initial solvent.

Let us respectively use l and v to denote the volumes of the initial phase and of the second solvent used. During the first extraction, ( ) ( )

1n n− moles of solvent pass into the second solvent. By application of relation [5.16], assuming the solutions to be sufficiently dilute and, to simplify the notation, using K to denote the partition coefficient of the solute between the two solvents in convention (III), we can write:

( ) ( )0 1

( )1

n nl K

nv

−

= [5.31]


From this, we deduce the residual amount of solute in the first solvent after the first operation:

( )( ) 01

n vn

Kl v=

+ [5.32]

After a second extraction, performed with the same volumes on the residual amount of material, we can show in the same way that the amount of solute remaining in the initial solvent is:

2( ) ( )2 0

vn n

Kl v=

+ [5.33]

Thus, after the pth extraction the amount of solute remaining in the initial solvent will be:

( ) ( )0

p

pvn n

Kl v=

+ [5.34]

In practice, we want the ratio ( ) ( )0/pn n to be as small as possible for a

given volume of extractant product (i.e. the product pl remains constant). It is easy to show that it is preferable for p to be large and l small. In other words, it is more effective to repeat the operation a large number of times with a small amount of extractant each time than to perform a single extraction with a much greater volume of extractant.

NOTE 5.2.– Dishwashing is actually an excellent example of extraction using a solvent (in this case, water). It is more effective to wash the plates several times using a little water each time than to wash them once with a huge amount of water. This finding is put into practice in dishwashers.

5.3.3.2. Multi-stage counterflow extraction

In the opposite direction, across the different stages, we run the final solution or the extract, which becomes richer in solute, the initial solution or raffinate, which is depleted by successive extractions.

Figure 5.9 illustrates the principle of such a counterflow system. The initial solution enters from the left in the first stage, with an amount L0 moles


at the composition x0; the solution which flows out of that first stage and into the second contains an amount L1 moles and has a composition x1; and so it continues until we reach stage N, which produces the residue with the amount LN moles at composition xN.

Figure 5.9. Diagram of a multi-stage counterflow extraction

Conversely, the new solvent enters from the right at stage N, with an amount L’N+1 moles at the composition xN+1, which can actually be zero. On exiting the system, from right to left at stage 1, we find the extract with a quantity of L’1 moles, at the composition y1.

Let us express the different material balances.

The overall balance has the form:

0 1 1' 'N NL L L L++ = + [5.35]

The overall solute balance is:

0 0 1 1 1 1' 'N N N NL x L y L x L y+ ++ = + [5.36]

The overall balance at stage j is:

1 1' 'j j i iL L L L− ++ = + [5.37]

The balance of solute at stage j is:

1 1 1 1' 'j j j j i j i jL x L y L x L y− − + ++ = + [5.38]

Between stages 1 and j, we can write an overall balance of solute in the form:

1 0 1 1 0 1 1' 'j j j jL x L y L x L y− + ++ = + [5.39]

from which we deduce:

0 01 0 1

1 1' 'j jL L

y x x yL L+ = − + [5.40]

L0, x0

L’1, y1

L1, x1

L’2, y2

L2, x2

L’3, y3

Stage 1 Stage 2 Stage N LN-1, xN-1

L’N, yN

LN, xN

L’N+1, yN+1


This expression gives the composition on output from the stage j as a function of the inlet composition, and we know the input and output compositions for stage 1. This gives us the equation for a straight line, known as the operational line, which has the slope L0/L’1.

For the whole of the installation with N stages, we have:

0 01 0 1

1 1' 'N NL L

y x x yL L+ = − + [5.41]

From this calculation, we deduce a first method for determining the theoretical number of stages in the extraction installation, which is McCabe’s method.

On the x-y axis system (Figure 5.10) we plot the equilibrium curve or sharing curve, which is only a straight line in the case of dilute ideal solutions or perfect solutions. We then plot the operational line given by relation [5.40]; in order to do so, we plot the point with coordinates (x0, y1) and point (xN, yN+1). We then plot the theoretical stages, as done by McCabe’s method, found in binary distillation (see section 4.8.4.2.4).

NOTE 5.3.– Very frequently, Figure 5.10 is plotted by choosing the mass fractions instead of the molar fractions. In addition, the method requires that the two solvents be practically completely immiscible.

Figure 5.10. Determination of the theoretical number of stages by the McCabe method


There is a second method that can be used, on the basis of the phase diagram, to obtain the theoretical number of stages. In order to do so, note that relation [5.37] can alternatively be written in the form:

1 1' 'j i i jL L L L D− +− = − = [5.42]

This equality at each stage shows that the difference D between the flow of material in two stages is constant. Thus, all the operative lines relating to the different stages all pass through the same point N.

This immediately gives us a method to graphically determine the theoretical number of stages:

– we plot the lines L0L’1 and LNL’N+1 (see Figure 5.11). These straight lines intersect with one another at N;

– knowing L’1, we can determine L1 (the conjugate point in the equilibrium curves);

– we plot the straight line L1N, which intersects the binodal curve at L’2;

– knowing L’2, we determine L2 and so on, until we reach a value near to LN.

The theoretical number of stages, then, is represented by the number of operational lines plotted on the graph.

Figure 5.11. Graphical determination of the theoretical number of stages of a multi-stage liquid–liquid extraction


NOTE 5.4.– The number of stages calculated is, obviously, a theoretical number of stages which is always less than the true number, because the assumption is made that perfect equilibrium is achieved at each stage, which is clearly not the case in practice.

5.4. Liquid–vapor systems

In liquid–vapor systems, we have four variables which have the potential to influence the equilibria: pressure, temperature and the two variables of composition. Liquid–vapor ternary systems are usually studied at a given pressure, often the standard pressure of 1 bar. Hence, the diagrams plotted are isobaric.

5.4.1. Isothermal and isopleth sections (boiling and dew)

In a right prism with the triangle of composition for its base surface and with temperature for its side, liquid–vapor equilibria result in the existence of two curved surfaces or layers:

– the dew layer, which gives the composition of the vapor phase and the dew point;

– the boiling layer, which gives the composition of the liquid phase at equilibrium, and the temperature at which the mixtures begin to boil.

Between these two layers, the two liquid and vapor phases coexist, whereas above the dew layer only the vapor phase exists, and below the boiling layer only the liquid phase exists.

Thus, we represent equilibria by two types of sections:

– isothermal sections projected onto the triangle of composition;

– isopleth sections in the plane of a pseudobinary.

Figure 5.12(a) shows an example of isothermal sections. In it, we see for three temperatures – 130, 111 and 90°C – firstly the projections of the sections of the dew layer (curves R130, R111 and R95), and secondly the projections of the sections of the boiling layer (curves E130, E111 and E95). In addition, in this figure a number of conodes are represented at 110°C which join the conjugate points L and V between the dew curve and boiling curve at that temperature.


Figure 5.12(b) shows the isopleth section corresponding to the pseudobinary system represented by the point M in Figure 5.12(a). As this section contains the pure component A1, we find the same boiling point as of this component at the chosen pressure. On the other hand, the endpoint of the axis of composition represents not a pure substance, but rather a mixture M, and thus the dew and boiling curves are not at the same temperature at that point.

Figure 5.12. Isothermal a) and isopleth sections A1M b) of the dew and boiling layers

In Figures 5.12, the liquids must be completely miscible. In the liquid state, it may be that a miscibility gap is superposed on the liquid–vapor diagram, as in the example of Figure 5.13. This gap, at the temperature of 88°C for example, gives us two conjugate points L1 and L2 from which extend the two branches of the boiling curve, L2L’ and L1L. In addition, the dew curve R 88 gives the corresponding conjugate points.

Figure 5.13. Liquid–gas isothermal section with a zone of liquid demixing


Figures 5.12(a) and 5.12(b) correspond to the dew layer and boiling layer, which have no points in common except for the pure components. In certain cases these layers may come into contact, leading to the formation of azeotropic mixtures. Thus, we distinguish different cases:

– there is a binary azeotrope which exists for one of the binary mixtures. For example, there is an azeotrope of composition Az2-3 in the binary mixture A2-A3. This azeotrope is expressed in the ternary diagram by a peak line (or minimum line) A1Az2-3 (Figure 5.14(a));

– there are two binary azeotropes. In the ternary diagram, this is shown by a peak line (or minimum line) which joins these two azeotropes;

– there are three binary azeotropes. This is expressed in the diagram by three peak (or minimum) lines, which intersect at a triple point where we find a ternary azeotrope (Figure 5.14(b)); this would be the highest or lowest point of the azeotropic lines.

A1 A1

a) Az2-3

A3 A2

x2

x3

x1

1 0

0 1

1 0

b) Az2-3

Az2-1

Az1-3

Az

A3 A2

x2

x3

x1

1 0

01

1 0

Figure 5.14. Systems with a binary azeotrope a) and a ternary azeotrope b)

5.4.2. Distillation trajectories

Figure 5.15(a) shows the mixture L1 which boils at a temperature T1, giving a vapor V1 of which we collect a small amount. The figurative point of the liquid residue defines the straight line L1V1 as we move further away from V1; the boiling point rises, and we reach L2, which belongs to the isotherm of boiling T2. The corresponding vapor has the composition V2 and a dew point at T2. If we take a new amount of vapor away, the representative


point will define the straight line L2V2 heading away from V2, ending up at L3. Continuing in this manner, little by little, with the distances L1L2, L2L3, L3L4, … tending progressively toward zero, the figurative point will define a trajectory, delimiting the polygon L1L2L3L4…. Thus, point by point, we construct the curve of evolution of the liquid during the distillation process. This curve is known as the vaporization trajectory or boiling trajectory. At each point L, the tangent to the boiling trajectory is a conjugate line issuing from this point, leading to the representative point V of the vapor, which separates at the same temperature.

At each point of the ternary diagram, there is a boiling trajectory which passes through, and only one, because distillation takes a unique path. The origin of the curves (Figure 5.15(b)) is the vertex representing the most volatile A1, and it terminates at the vertex corresponding to the less volatile component A2.

Clearly, we can plot dew trajectories or liquefaction trajectories which will have the same shape, but directed in the opposite direction, running from the less volatile component toward the most volatile, which liquefies last. At each point V, the tangent is a conjugate line issuing from that point and ending at the representative point of the liquid L, which separates at the same temperature.

a)

L2

L3

V1 V2

V3

L4

A1

A3A2

x2

x3

x1

1 0

0 1

1 0

b) A1

A3A2

x2

x3

x1

1 0

0 1

1 0

Figure 5.15. Principle and graph of the distillation paths

Hence, by distillation, we can fully collect each of the pure components in decreasing order of volatility.


5.4.3. Systems with two distillation fields

If the liquid–vapor diagram shows a peak line (or minimum line) ending at a binary azeotrope (Figure 5.14(a)), this line cannot be crossed by distillation, because it marks the point at which the vapor and the liquid have the same composition.

Suppose the three components are, in decreasing order of volatility, A1, A3 and A2. The distillation trajectories (Figure 5.16) cannot cross that line, and the ternary diagram is then divided into two regions: there are two distillation fields. The distillation trajectories then finish either at component A3 or at component A2, depending on whether the initial mixture is to the left or to the right of the boundary. By distillation of a mixture located to the left, we shall successively obtain A1, the azeotrope Az2-3 and A3; it is impossible to obtain A2 by distillation of such a mixture. In the case of a mixture situated on the right, distillation successively gives us A1, the azeotrope Az2-3

and A2; we can never obtain A3 by distilling such a mixture.

We also obtain a system with two distillation fields if the system contains two azeotropes. The two fields are separated by the line connecting the two azeotropes. Depending on whether these azeotropes are maximal or minimal, distillation in each of these fields will yield different results; but it is not possible by this method to obtain a single, pure component, even incompletely, because we would obtain the two azeotropes.

Az2-3

A1

A3 A2

x2

x3

x1

1 0

0 1

1 0

Figure 5.16. System with two distillation fields


5.4.4. Systems with three distillation fields

Here we find ourselves in the situation illustrated by Figure 5.14(b). The triangle includes three lines which cannot be crossed by distillation, and is made up of three distillation fields, each of which passes through one of the vertices. Depending on whether the azeotropes are maximal or minimal, and depending on the starting point, we shall obtain – at best and incompletely – only one component in the pure state by distillation: that which is contained in the field in question.

5.5. Examples of applications of ternary diagrams between fluid phases

Numerous physical-chemical operations in industry rely on the properties of a ternary diagram between fluid phases. We shall examine three here.

5.5.1. Treatment of argentiferous lead

Silver is often found alongside lead in lead ores, and obviously it is desirable to be able to extract it. The binary mixtures of the metals (lead–silver and zinc–silver) are miscible, in the liquid state, in all proportions. The mixture of lead and zinc is only partially miscible. On melting, two liquid layers appear: one which is rich in zinc, and the other rich in lead.

M

M’

Q N

R

Pb Zn

Ag

x2

x3

x1

1 0

0 1

1 0

Figure 5.17. Separation of silver in the ternary system Pb-Ag-Zn


The partition coefficient of the silver between these two layers is significantly in favor of the zinc-rich layer. This gives rise to the Parkes process, which is used to extract silver from argentiferous lead.

In this process, to separate the silver from the lead, the mixture is melted at 650°C and zinc is added to it. Thus, the composition moves along the M-Zn line shown in Figure 5.17. This curve intersects the ternary miscibility limit at 650°C. Zinc is added until we reach the point Q within the biphasic zone, such that the corresponding conode RN is, in R, as close as possible to the binary mixture Ag-Zn, and in N, as close as possible to the binary mixture Pb-Zn. In Q, there is separation into two phases:

– one in R, rich in silver and zinc;

– the other in N, rich in lead.

Thus, almost all of the silver passes into the zinc, forming the less dense layer which solidifies first upon cooling, so it is easy to separate. To extract the rest of the silver from the lead, at point N we pass through a stream of superheated steam, which oxidizes the zinc to form zinc oxide, and thus the composition moves along the line Zn-N to the point M’ which gives lead that is practically free of silver.

5.5.2. Purity of oil products: aniline point

Consider a mixture of an aliphatic hydrocarbon and a benzenic hydrocarbon. We wish to find the proportion of the benzenic hydrocarbon in the mixture. In order to do so, we measure what is known as the aniline point.

Figure 5.18 shows the ternary diagram of three substances:

– A1 is the aromatic hydrocarbon;

– A2 is the aliphatic hydrocarbon;

– A3 is aniline.

The miscibility gap is plotted at three temperatures, in decreasing order: T1, T2 and T3.


N

M

T1T2

T3

A1

A3 A2

x2

x3

x1

1 0

0 1

1 0

Figure 5.18. Determination of the aniline point of a mixture of hydrocarbons

If we consider the binary carbide (BC) of aliphatic and aniline hydrocarbons, at the composition of the point M, if the temperature is greater than T2, the mixture is homogeneous; on cooling, two phases emerge and the first sign of cloudiness appears at temperature T3.

If we add the aromatic hydrocarbon (A1) to the mixture M, the representing point moves along the straight line MA1 to point N, which is closer to A1 when there is more of the aromatic hydrocarbon present. At N, for example, on cooling, the cloudiness begins to appear at temperature T1. Thus, by measuring that temperature at which the cloudiness manifests itself, we are able to find the proportion of aromatic hydrocarbon in a known mixture of aliphatic hydrocarbon and aniline. This method is used to dose the proportion of aromatic hydrocarbon in a mixture of aliphatic and aromatic hydrocarbons. The temperature T1 is known as the aniline point of the mixture.

5.5.3. Obtaining concentrated ethyl alcohol

Given the existence of a binary azeotrope, it is not possible by distillation to obtain mixtures of water and ethyl alcohol (rich in water) which have an alcohol content higher than 96.5%. In order to obtain higher alcohol contents, and especially pure alcohol, we carry out a ternary distillation.


Figure 5.19. Ternary azeotrope of the alcohol–water–benzene mixture

To the water–alcohol mixture, represented by point R in Figure 5.19, we add a certain amount of benzene. The three couples – water–alcohol, water–benzene and alcohol–benzene – form binary azeotropes, respectively represented by points I, J and K in Figure 5.19. The mixture of the three substances forms a ternary azeotrope (point M). The distribution lines of the distillation fields, therefore, begin at I, J and K and intersect at M (boiling at 64.9°C, 18.5% alcohol and 75.4% benzene).

If we start at point R, the addition of benzene leads to the straight line R-benzene terminating at the representative point. We stop adding the substance when the representative point is located between points S and Z; in this case, we are in the distillation field of the alcohol. By distillation, water is brought into the ternary azeotrope (M) and then the excess benzene passes, along with another portion of the alcohol, into the corresponding binary (Point K: boiling point 68.25°C and 32.4% alcohol). Finally, we obtain pure alcohol at the end of the curve. Of course, under no circumstances must the representative point of the mixture surpass the point Z, because then we enter into the distillation field of benzene, and the alcohol will be completely drawn into the ternary azeotrope and into the binary system K (alcohol–benzene).

6

Equilibria Between Condensed Ternary Fluid Phases

Liquid–solid and solid–solid ternary systems only involve condensed phases, hence the influence of pressure is negligible in such systems and all data are collected at normal pressure.

The variance of such systems is 4-φ. Thus, there can be a maximum of four phases at equilibrium.

We shall use the representation of isothermal projections on the triangle of composition, as we did in the previous chapter (see section 5.1).

For two phases at equilibrium the variance is 2, which means we need to know the temperature and a composition variable; the other composition variable will be a function of the previous two. Thus, these equilibria are represented by a surface and the projections of isothermal sections onto the triangle by a line.

For three phases at equilibrium the variance becomes 1, and thus the temperature is sufficient to determine the equilibrium. Hence these equilibria will be represented by a line, and each projection of an isothermal section shall be represented by a point.

For four phases at equilibrium the variance becomes zero, and thus equilibrium would be represented by a point in space and by a point in a single isothermal section.



There is a wide variety of possible diagrams, depending on the properties of the binary systems included in the ternary diagram, the miscibilities of the solid and liquid phases, the existence or otherwise of definite compounds with congruent or incongruent melting, etc. We shall examine the most common cases.

6.1. Solidification of a ternary system with total miscibility in the liquid state and in the solid state

The case of solidification of the ternary system with total miscibility both in the solid phase and in the liquid phase is very similar to the case of the equilibrium in liquid–vapor ternary systems with miscibility in the liquid phase, as studied in section 5.4. The boiling and dew layers are respectively replaced by the solidus and liquidus layers. The former gives the composition of the solid phase and the temperature at which melting begins or solidification finishes; the latter gives the composition of the liquid phase and the temperature at which melting finishes or solidification begins. Thus, we can again use representations similar to those given by Figures 5.12(a) and 5.12(b), with the liquidus replacing the dew curve and the solidus replacing the boiling curve.

There is no reason not to consider the existence of binary and ternary azeotropes, which in this case will be heteroazeotropes, leading to representations similar to those given in Figures 5.14(a) and 5.14(b).

The boiling and liquefaction trajectories, defined for liquid–vapor systems (see section 5.4.2), are now replaced by solidification trajectories and melting trajectories. Each point in the triangle of composition will contain a solidification trajectory. In the case of azeotropes, we shall be led, as in Figure 5.16, to multiple solidification fields.

6.2. Solidification of a ternary system with no miscibility and with a ternary eutectic

We now suppose that the liquids are miscible in all proportions, but that the three solids A1, A2 and A3 are completely immiscible with one another.

Equilibria Between Condensed Ternary Fluid Phases 193

6.2.1. Invariant transformations of a liquid–solid ternary system

We have seen that if the system contains four phases, its variance is zero. Let us examine the case of a liquid phase L and three solid phases α, β and γ. The different transformations associated with the possible invariant equilibria with these four phases are, in the direction of solidification:

– ternary eutectic transformations, which we write in the form:

L [6R.1]

Figure 6.1(a) illustrates this kind of transformation;

– ternary peritectic transformations, which we write in the form:

L [6R.2]

Figure 6.1(b) illustrates such a transformation.

In ternary systems there is a new invariant transformation, for which there is no equivalent in binary systems. This is the transformation [6R.3] represented by Figure 6.1(c), called the quasi-peritectic transformation.

L + α β + γ [6R.3]

a)

L

α

γ β L

α

βγ

b)

L

α

β γ

c)

Figure 6.1. Diagrammatic representations of the invariant transformations on the solidification of a ternary system;

a) ternary eutectic; b) ternary peritectic; c) quasi-peritectic


6.2.2. Representations of the ternary system with no miscibility in the solid state

As is the case with all ternary systems, there are three possible modes of representation to define the system:

– the complete representation in three-dimensional space of the isobaric ternary diagram;

– the planar representation of the projection, onto the ternary diagram, of an isothermal section of the above three-dimensional diagram;

– the planar representation of an isopleth section of the three-dimensional diagram.

We shall mainly use the first two representations.

6.2.2.1. The isobaric three-dimensional representation

Figure 6.1(a) shows the spatial representation of the triangular-based prism containing the different elements of the diagram.

)2A(FT

A1

T1

T

A2A3 e1

e2

e3

e

E2

E3

E1

E

M

N

Q

a)

)3A(FT

)1A(FTe2

e1

e3

e

T= )1A(FT

n

mq

Lq+A1+A2

Ln+A2+A3

L+A1

L+A3 L+A2

b)

Lm+A1+A3

1 0

A1

A3 A2

x2

x3

x1

10

0 1

1 0

Figure 6.2. Solidification diagram with no miscibility in the solid state with a ternary eutectic (adapted from [DES 10]). For a color version of the

figure, see www.iste.co.uk/soustelle/transformations.zip


There we can see each of the three binary systems A1A2, A2A3 and A1A3. These systems all display the conventional form of a binary system with no miscibility in the solid phase and with total miscibility in the liquid phase (see Figure 3.5). We can distinguish the binary eutectic points (E1, E2 and E3), each of which are the site of an equilibrium transformation, which we could represent by a reaction (on cooling) – e.g. for the eutectic E1 of the binary A2A3:

LE1 A2 + A3 [6R.4]

The three layers of the liquidus respectively run through the melting points of the three pure substances

1(A )FT , 2(A )FT and

3(A )FT , intersecting in pairs, along the lines issuing from a binary eutectic, and converge at a point E.

Along these lines E1E, E2E and E3E, the system is univariant. Thus, in order to define it, we simply need to establish one variable – e.g. the temperature – so the composition of the liquid phase is established. Along these lines, we can represent the equilibrium transformations by the typical system of symbols (on cooling) which gives the three univariant transformations:

Liquid (E1E) A2 + A3 [6R.5]



This means that any liquid whose composition is on the vertical of one of the points on one of these lines, on cooling, experiences the onset of solidification at a certain temperature, and this temperature evolves during the course of the precipitation. We can see that this is completely different from what happens for the eutectic of a binary system, which is an invariant point. The lines E1E, E2E and E3E are called the univariant lines of the system.

The point E, which is the meeting point of the three univariant lines, has perfectly-defined coordinates ( (E)FT , 1(E)x , 2(E)x ) because the variance of the system is zero there. Its temperature is lower than that of the three binary eutectics E1, E2 and E3. This point represents the ternary eutectic. The liquid


with an initial composition at the vertical of that point begins to solidify at temperature (E)FT at equilibrium with the three pure substances A1, A2 and A3, with each solid in its own phase. The equilibrium equation is thus written as follows, in accordance with the reaction [6R.1]:

L A1 + A2 + A3 [6R.8]

The temperature does not evolve as solidification continues.

6.2.2.2. Projection of an isothermal section

We cut the right prism shown in Figure 6.2(a) with a plane at temperature 1(A )FT such that:

2 1 3 1(E ) (A ) (E ) (E )F F F FT T T T< < <

The three layers are intersected along a section MNQ. We project that section onto the triangle of composition. In this triangle, we also represent the projection of the three univariant lines. Thus, we obtain the isothermal planar projection represented in Figure 6.2(b). The points M, N and Q are projected to m, n and q, and the points E1, E2, E3 and E are respectively projected to e1, e2, e3 and e.

On that planar projection, we distinguish three-phase zones:

– the zone of the triangle A1A3m, which contains the solids A1 and A3 and the liquid with composition Lm;

– the zone of the triangle A2A3n, which contains the solids A2 and A3 and the liquid with composition Ln;

– the zone of the triangle A1A2q, which contains the solids A1 and A2 and the liquid with composition Lq.

We also have two-phase zones:

– the zone of the curvilinear triangle A1mq, which contains the solid A1 and the liquid L;

– the zone of the curvilinear triangle A3mn, which contains the solid A3 and the liquid L;


– the zone of the curvilinear triangle A2qn, which contains the solid A2 and the liquid L;

– the single-phase zone of the curvilinear triangle mnq, where the liquid phase is alone.

In each biphasic zone, we have represented a few conodes, which connect a point representing a pure substance (solidus) and points of the liquidus. (A1-mq, A2-nq and A3-nm). With these conodes, we are able to determine the composition of the liquid L at equilibrium with the solid at each of the points in the two-phase domain.

Figure 6.3(a) gives an illustration of several isothermal projections of the bismuth-tin-lead ternary system, and the projection of the univariant lines. The isothermal projections of the liquidus intersect on the univariant lines, and we can see that the temperature of the ternary eutectic is lower than that of binary eutectics.

6.2.2.3. Cooling and solidification of a given liquid

Consider Figure 6.3(b), which shows the univariant lines of a ternary system. The arrows at the end of these lines indicate the direction of decreasing temperature. Consider an initial mixture whose composition corresponds to the point M, for an initial temperature higher than the temperature of the liquidus in M: (M)FT . We cool this mixture. Once we

reach the temperature (M)FT the first crystals appear: in this case, crystals of A2. The composition of the liquid becomes poorer in A2 and shifts along MA2 in the direction of the point L. According to the law of moments, at L, the amount of A2 deposited will be given by the ratio ML/A2L.

At any point J, at temperature (J)FT we can express the phase fractions of the different phases formed. With the overall composition at M being the barycenter between the three phases J, A1 and A2, the fraction of liquid remaining is given by the ratio MK/JK. The point K, which is aligned with M and J, also gives us the fraction of A1 in relation to the total solid. Thus, we should have:

2

2

(K)A 2(K)

A 1

KAKA

ff

= [6.1a]


and:

2 1

(K) (K)A A

MJJK

f f+ = [6.1b]

At a temperature lower than (L)FT , there is precipitation of the binary eutectic along the line e3e. As the eutectic line evolves we reach the temperature of the ternary eutectic (E)FT , where cooling triggers the precipitation of that eutectic until the liquid phase disappears entirely.

a) b)

e

e1

e2

e3

K M

L

I

J

A1

A3 A2

x2

x3

x1

1 0

0 1

1 0

Pb

nSiB

x2

x3

x1

1 0

0 1

1 0

E1

E2

E3

200 175 150

200

250

300

200 150

125 E

Figure 6.3. a) Isotherms and univariant lines of the bismuth-tin-lead system; b) univariant lines and trajectory of solidification of the liquid M. For a color

version of the figure, see www.iste.co.uk/soustelle/transformations.zip

6.2.2.4. Solidification trajectory

In the same way as for the distillation trajectories seen in Chapter 5 (section 5.4.2), starting at a point M we can define the solidification trajectory on the basis of the tangent to the projections of the liquidus. The triangle of composition is divided into three zones by the univariant lines. In each of these zones, we have a solidification field. All the solidification trajectories starting at any given point M lead towards the corresponding monovariant line (that which intersects the extension of the straight line joining point M to the vertex of the triangle contained within the field in question); then, the solidification trajectory follows that univariant line and finishes at the ternary eutectic.


6.2.3. Lowering of the melting point of a binary system by the addition of a component

Let us consider a binary eutectic E3 in the binary system formed by the two pure substances A1 and A2. Suppose that we have a pure component A3 which is completely immiscible with A1 and A2 in the solid state, but which is entirely miscible in the liquid state. Consider a point M on the univariant line E3E (E is the ternary eutectic) (see Figure 6.4).

1AFT (

E3

)( 2AFT

1A

2A3A

3A

3xT d/d

E

Figure 6.4. Lowering of the melting point of a binary eutectic by the addition of a third component (adapted from [DES 10]). For a color version of the figure, see www.iste.co.uk/soustelle/transformations.zip

We shall employ reasoning similar to that which we used to study the slopes of the liquidus of a binary (see section 3.2.3). The solid–liquid equilibrium for each of the components A1 and A2 enables us to write:

(liq) (sol)1 1μ μ= [6.2]

(liq) (sol)2 2μ μ= [6.3]

The chemical potentials of the two components in the liquid phase (liq) (liq)1 2 and μ μ depend on the three variables: T, x1 and x2. We add a small

amount of a third component A3. Then, the new equilibrium condition imposes the following for the component A1:

(liq) (sol)1 1d dμ μ= [6.4]


and for component A2:

(liq) (sol)2 2d dμ μ= [6.5]

Let us write the differential for each of the chemical potentials. For that of A1 in the liquid phase, we have:

(liq) (liq) (liq)(liq) 1 1 11 1 2

1 2

d d d dx x Tx x T

μ μ μμ ∂ ∂ ∂= + +∂ ∂ ∂

[6.6]

For that of A1 in the solid phase, as A1 is pure the chemical potential will be the Gibbs molar energy of the pure substance A1, which is a function of temperature only, and thus:

(sol) *(sol)(sol) 1 11d d dg

T TT T

μμ ∂ ∂= =∂ ∂

[6.7]

Similarly, for the component A2 we shall have for the liquid phase:

(liq) (liq) (liq)(liq) 2 2 22 1 2

1 2

d d d dx x Tx x T

μ μ μμ ∂ ∂ ∂= + +∂ ∂ ∂

[6.8]

and for the solid phase:

(sol) 0(sol)(sol) 2 22d d dg

T TT T

μμ ∂ ∂= =∂ ∂

[6.9]

Furthermore, we know that the partial derivatives of a chemical potential and of a Gibbs energy in relation to temperature are such that:

(li )(liq)

qi

iST

μ∂= −

∂ [6.10a]

and:

*(sol)0(sol)ii

gs

T∂

= −∂

[6.10b]


By substituting expressions [6.6] and [6.7] back into relation [6.4], we find the following, in light of relations [6.10a] and [6.10b]:

( )(liq) (liq) (liq) 0(sol)1 111 2 1

1 2

d d dx x S s Tx x

μ μ∂ ∂+ = −∂ ∂

[6.11]

Similarly, on the basis of relations [6.5], [6.8], [6.9] and [6.10], we find:

( )(liq) (liq) (liq) 0(sol)2 221 2 2

1 2

d d dx x S s Tx x

μ μ∂ ∂+ = −∂ ∂

[6.12]

By multiplying equation [6.11] by x1 and equation [6.12] by x2, and adding the equations term by term, in light of the Gibbs–Duhem relation which is written as follows at a constant temperature:

1 1 2 2 3 3d d d 0x x xμ μ μ+ + = [6.13]

we obtain:

( ) ( )(liq) (liq)(liq) 0(sol) 0(sol)1 23 3 1 1 2 2 dx d x S s x S s Tμ− = − + − [6.14]

In addition, the chemical potential of the component A3 in the liquid can be written thus, if we choose the pure-substance reference:

(liq) 0(liq)3 3 3 3R ln R lng T T xμ γ= + + [6.15]

At constant temperature, the activity coefficient γ3 is a function of the two composition variables x1 and x2, and therefore by differentiating the function [6.15] we obtain:

(liq) 3 3 33 1 2

3 1 2

d ln lnd R d dxT x x

x x xγ γμ ∂ ∂= + +

∂ ∂ [6.16]

By substituting the result back into equation [6.14] we find:

( ) ( )3 3 31 2

31 2

(liq) (liq)0(sol) 0(sol)1 21 1 2 2

d ln lnd dR Rd d d

x x xT Tx

T x T x T

x S s x S s

γ γ∂ ∂− − + +∂ ∂

= − + − [6.17]


If the molar fraction x3 of component A3 is made to tend toward zero, then equation [6.17] is reduced to:

3

33

(E )

3 f (E )0

Rdd

F

x

TTx S

→

= [6.18]

3f (E )S is the entropy of melting of the binary eutectic E3, given by:

( ) ( )3

(liq) (liq)0(sol) 0(sol)1 2f (E ) 1 1 2 2S x S s x S s= − + − [6.19]

and 3(E )FT is its melting point.

However, at the equilibrium of melting we can write that the Gibbs energy of melting is zero, and therefore that:

3 3 3f (E ) (E ) f (E )FH T S= [6.20]

Hence, relation [6.18] becomes:

3

33

2(E )

3 f (E )0

Rdd

F

x

TTx H

→

= (with 3f (E ) 0H > ) [6.21]

Thus, we can see that adding component A3 to the mixture of the two pure substances A1 and A2 lowers the melting point of the binary eutectic of components A1 and A2, because of the miscibility of the liquids. This result is similar to the cryoscopic law obtained for binary mixtures.

6.2.4. Slope at the ternary eutectic

We are still considering a ternary system formed of the three components A1, A2 and A3, exhibiting a ternary eutectic and no miscibility whatsoever in the solid state, but complete miscibility in the liquid state.

As we did in section 6.2.3, let us express the slope along the univariant line E3E, using expressions [6.11] and [6.12] but choosing x1 and x3 as


composition variables. At equilibrium, along the length of the univariant line E3E, we can write:

By analogy with relation [6.11]:

( )(liq) (liq) (liq) 0(sol)1 1 11 1

1 3 3 3

d dd d

x TS sx x x x

μ μ∂ ∂+ = −∂ ∂

[6.22]

and by analogy with relation [6.12]:

( )(liq) (liq) (liq) 0(sol)2 1 22 2

1 3 3 3

d dd d

x TS sx x x x

μ μ∂ ∂+ = −∂ ∂

[6.23]

By eliminating dx1/dx3 between equations [6.22] and [6.23], along the line E3E, we have:

( ) ( )3

3 1,2(liq) (liq)(liq) (liq)0(sol) 0(sol)2 13 1 21 21 1

dd

line E E

x HTx S s S s

x xμ μ

=∂ ∂− − −

∂ ∂

[6.24]

Let us define M1,2 by the relation:

2 2 2

1,2 2 21 2 1 2

G G GMx x x x

Δ Δ Δ∂ ∂ ∂= −∂ ∂ ∂ ∂

[6.25a]

and 1,2D by the relation:

( ) ( )(liq) (liq)(liq) (liq)0(sol) 0(sol)2 11 21,2 1 2

1 1

D S s S sx x

μ μ∂ ∂= − − −∂ ∂

[6.25b]

M1,2 is a determinant associated with the ternary liquid along the univariant line E3E issuing from the binary eutectic E3 of the system A1-A2. ΔG is the Gibbs molar energy of mixing the ternary liquid.

By permuting the two concentration variables so that one of them is relative to the added component, the same type of relation as equation [6.24] can be obtained along the other two univariant lines, respectively issuing from the binary eutectic E1 relative to the binary mixture A2-A3 and from the binary


eutectic E2 relative to the binary mixture A3-A1. These lines converge toward the ternary eutectic E:

1

1 2,3

1 2,3

dd

line E E

x MTx D

= [6.26]

2

2 1,3

2 1,3

dd

line E E

x MTx D

= [6.27]

Relations [6.24], [6.25] and [6.26] are, in particular, applicable at the temperature and composition of the ternary eutectic E. In this case, we have:

M1,2 = M1,3 = M2,3 = M

By eliminating the value M, which becomes common to relations [6.24], [6.25a], [6.25b] and [6.26], we are led to relations between the slopes of the tangents to the univariant lines at the melting point and at the composition of the ternary eutectic. These relations are in the form:

2 2

1 1,3

1 2 2,3 2

d dd d

line E E line E E

x DT Tx x D x

= [6.28a]

2 2

2 2,3

2 3 1,3 3

d dd d

line E E line E E

x DT Tx x D x

= [6.28b]

2 2

1

1 3 3

d dd d

line E E line E E

xT Tx x x

= [6.28c]

Relation [6.28c] can be deduced directly from expressions [6.28a] and [6.28b].

6.3. Ternary systems with partial miscibilities in the solid state and ternary eutectic

We now envisage the case of a ternary system exhibiting total miscibility in the liquid state, partial miscibility in the solid state between the three


components and a ternary eutectic which results from three binary eutectics of the couples of components – i.e. each component can, in the solid state, dissolve a small amount of each of the other two components.

The liquidus (Figure 6.5) is exactly the same as that in the previous case – that of a ternary system which exhibits no miscibility in the solid state and three binary eutectics (see section 6.2.2.1): in other words, it is made up of three layers, each passing through the melting point of a pure component and intersecting in pairs along the three lines of univariant transformations issuing from a binary eutectic, converging on the ternary eutectic.

A1 A2

A3

)1A(FT)( 2AFT

)( 3AFT

T

α

β

γ

E2

E1

E3

E

N

R

Q

P

Figure 6.5. Ternary system with partial miscibility in the solid phase and eutectic (adapted for [DES 10]). For a color version of the figure,

see www.iste.co.uk/soustelle/transformations.zip

Line E2E, for example, gives us the composition of the liquid phase during the univariant transformation:

Liquid α + γ [6R.9]

The univariant line RP gives, over the course of that transformation, the composition of the ternary solid solution α issuing from R.


The solidus (Figure 6.5), which shows us the composition of the solid solution at equilibrium with the liquid phase, is a layer like that which is based on points

1(A )FT , R, P and Q for the α phase. The ternary eutectic solidification is represented by points such as P. It is only with overall compositions situated within the horizontal triangle passing through the ternary eutectic E (and the point P, notably) that solidification finishes at the composition of the ternary eutectic.

The isothermal sections have different appearances depending on the temperature chosen.

Figure 6.6(a) shows the isothermal section obtained for a temperature lower than the melting point of the ternary eutectic

1(E )FT . In it, we can see the three single-phase solid zones in the vicinity of the vertices of the triangle of composition, the three two-phase zones containing two solids whose conodes obviously depend on the limits of solubility, and finally the three-phase solid zone at the center. This section contains no liquid zone, because the section does not intersect any layer of the solidus and liquidus.

b) a)

A1

A2A3

α

βγ

αα + βα + γ

β + γ

α +γ + β

e1 S A2

A1

A3

α

βγ

liq

α + βα + γ

β + γ

liq +α + γ liq +α + β

Liq +γ + β

liq+γ

liq+β

liq+α

e2

e3

e1

Figure 6.6. Projections of isothermal sections of the diagram in Figure 6.5(a); a) temperature lower than the ternary eutectic; b) temperature between the ternary

eutectic and the binary eutectic with a lower melting point. For a color version of the figure, see www.iste.co.uk/soustelle/transformations.zip

Figure 6.6(b) shows the isothermal section obtained with a temperature between the melting point of the ternary eutectic (E)FT and the lowest

melting point of a binary eutectic 2(E )FT (see Figure 6.5). This section crosses


the liquidus around the ternary eutectic, which gives us the single-phase liquid zone around the projection of that point. With conodes depending on that line and on the boundary of a zone with a solid solution, we find a two-phase zone with the liquid and a solid phase – say, α. The two-phase solid zone has conodes which depend on the boundaries of single-phase zones α and β, for example, and finally zones with three phases (liquid + two solid phases) are situated in triangles bounded by each of the corresponding phases (liquid + α + β, for instance).

Figure 6.6(b) also shows the projections onto the triangle of composition of the three univariant transformation lines which link each of the three binary eutectics (points e1, e2 and e3) to the ternary eutectic.

The evolutions in temperature can best be tracked by way of isopleth sections.

)3E(FT

a) M

M’

A1 N

)(NFT)1A(FT

T

liqliq + α

liq + β

liq + α + ββ

α

α + β +γ

b)

A1 S

liq

)1A(FT

)S(FT

liq + α

liq + α+ β

liq + β

α

α + β

β

Figure 6.7. Isopleth sections; a) along the vertical plane passing through A1N; b) along the vertical plane passing through A1S. For a color version of the

figure, see www.iste.co.uk/soustelle/transformations.zip

Figure 6.7(a) represents the isopleth section as the vertical plane containing the line A1N in Figure 6.5. The cooling of a liquid mixture represented by point M allows for the deposition of the ternary eutectic, with a temperature which remains constant during that solidification at point M’.

Figure 6.7(b) represents the isopleth section as the vertical plane containing the line A1S in Figure 6.6(a). This isopleth, unlike the last one, never crosses the horizontal triangle containing the melting point of the ternary eutectic (and


point P from Figure 6.5). It follows from this that this eutectic is never deposited, and thus no zone with three solid phases appears in this section.

6.4. Solidification of ternary systems with definite compounds

Like binary systems (see section 3.2.2.5), ternary systems can yield definite compounds, which are combinations between components.

In the context of ternary systems, such compounds may be one of two types:

– compounds involving only two components, which are known as binary definite compounds;

– compounds which are combinations of all three components of the system, known as ternary definite compounds.

In addition, as happens in the case of binary systems, these compounds may exhibit:

– congruent melting, meaning that they melt without decomposing;

– incongruent melting, being unstable – these compounds break down before they melt.

The combinations of these different elements lead to numerous varieties of diagrams. In order to illustrate the presence of definite compounds, we shall discuss the case of a single binary definite compound with congruent melting, which we generalize by triangulation of the ternary diagram, and finally touch upon the case of definite compounds with incongruent melting.

6.4.1. Ternary system with a binary definite compound binary with congruent melting

Thus, we shall consider a ternary system with components A1, A2 and A3 in the case of the existence of a binary definite compound C with congruent melting between components A1 and A2. So as not to render the diagram overly complex, we shall restrict our discussion to the case where the solids are completely immiscible with one another, but the liquids are completely miscible. The extension to the case of partial miscibility of the solids will become easy to extrapolate, using the concepts introduced in section 6.6.3.


NOTE.– Even if the diagram does not show a homogeneous zone of solid solution around the composition of the definite compound C, we must remember that these compounds always exhibit a slight deviation from stoichiometry.

)2A(FT

)A( 3FT)1A(FT

A1

A3

A2

C

c

PE2 E1

E’3 E3

E’ E

e2

e1

e’1 e1

e’e

Figure 6.8. Ternary system with a binary definite compound with congruent melting (adapted from [DES 10]). For a color version

of the figure, see www.iste.co.uk/soustelle/transformations.zip

Suppose that each binary contains a binary eutectic, so we have four binary eutectics – respectively:

– E1 for the binary A2A3;

– E2 for the binary A3A2;

– E3 for the binary CA2;

– E’3 for the binary A1C.

These binary eutectics yield two ternary eutectics:

– E’ for the ternary A1CA3;

– E for the ternary A2CA3.

Figure 6.8 shows a spatial representation of the diagram. We can see that, in fact, it is the juxtaposition of two ternary diagrams of the same nature as encountered in section 6.2.2.1 (see Figure 6.2(a)):

– the diagram of the ternary system A1A3C;

– the diagram of the ternary system A2CA3.


We find three layers of liquidus passing through the melting points of the components

1(A )FT , 2(A )FT and

3(A )FT , in addition to which we have the layer of liquidus E’PEE3E’3, which defines the melting point and the composition of equilibrium between the liquid and the intermediate compound C.

Also in Figure 6.8, we see the projection onto the triangle of compositions of the following univariant transformation lines:

– e1e, which is the projection of E1E;

– e2e’, which is the projection of E2E’;

– e3e, which is the projection of E3E;

– e’3e, which is the projection of E’3E;

– e’e, which is the projection of E’E.

Figure 6.8 also shows the isopleth section defined in the vertical plane passing through the points A3 and C. This section shows a classic eutectic at point P. It is possible to demonstrate that the point P is a maximum of the univariant curve E’PE. The composition corresponding to the point P gives a eutectic crystallization of crystals of A3 and of C.

For any composition other than that which corresponds to point R, cooling gives a ternary eutectic solidification at E’ if the initial composition is within the triangle A1eA3, or a ternary eutectic solidification at E if the initial composition is within the triangle A3eA2.

The isothermal sections obviously depend on the temperatures chosen. Here we have chosen to illustrate two.

Figure 6.9(a) represents the projection onto the ternary diagram of the isothermal section obtained for a temperature Ta such that:

( )F RT <Ta< 3(A )FT

The single-phase liquid is clustered in the zone LALBLCLDLE only. We see the appearance of four biphasic zones with their conodes and two triphasic zones in the triangles A1LBC and CLCA2.


Figure 6.9(b) represents the projection of the isotherm at temperature Tb such that:

Tb< ( )F RT

We can see two domains of stability of the single-phase liquid, six biphasic domains and their conodes and six triphasic domains in triangles.

A2

A1

A2 A3

C

e1

e2

e’3

e3 e

e’

L1L2

L+D+A3

L+A1+C

L+A2+C

L+A1+A3

L+A2+A3

LL+C

L+A1

L+A3

L+A2

L

b)

A3

A1

e1

C

e’3

e

LA

e2

e3

e’

LB

LC

LD LE

L+A1+C

L+A2+C

L+A1

L+C

L+A2L+A3

L

a)

Figure 6.9. Isothermal sections of a ternary system with a binary definite compound displaying congruent melting. For a color version of the figure,

see www.iste.co.uk/soustelle/transformations.zip

It can be remarked that the two liquids which correspond to the compositions of the two points L1 and L2 apparently have the same composition, because they are both at equilibrium with the solid C. This is obviously false, because if we consider the above remark, each point L1 and L2 is at equilibrium with a non-stoichiometric solid C, which does not exhibit the same deviation in both cases, and therefore the liquids at L1 and L2 do not, in fact, have the same composition.

6.4.2. Generalization to the case of a ternary compound and of multiple definite compounds

We noted in the previous case (section 6.4.1) that the diagram of a ternary system with a definite compound of a binary could be reduced to the juxtaposition of two elementary ternary diagrams, ultimately defined by a triangulation of the ternary diagram with the construction of the two triangles A1CA2 and A2CA3 (Figures 6.9). We shall generalize this property to cases where several binary compounds appear, or indeed in the presence of a ternary definite compound. Figure 6.10(a) shows a triangulation in the


case of the presence of two binary definite compounds C1 and C2. In fact, in this case, there are two possibilities for triangulation:

– the first defining the triangles A1C1C2, A3C1C2 and A2C2A3;

– the second using the triangles A1C1C2, A2C1A3 and A2C1C2.

The distinction between these two possibilities is demonstrated by an experiment where we allow a liquid, whose composition is represented by the point M within the triangle C1KC2, to cool. Two possibilities may be encountered:

– either the solid A2 never crystallizes (or a solid solution of the same structure as A2 in the case of partial miscibility), so the point M falls within the triangle C1C2A3, and it is the first triangulation which is correct;

– or the solid A3 never crystallizes (or a solid solution of the same structure as A3 in case of partial miscibility), so the point M falls within the triangle A2C1C2, and it is the second triangulation which is correct.

a)

C1

C2

K

M

A3 A2

A1 b)

C

A3 A2

A1

Figure 6.10. Triangulation of the ternary diagrams for: a) two definite compounds on two different binaries; b) a ternary definite compound

Figure 6.10(b) shows the triangulation in the case of formation of a ternary definite compound represented by point C. The three elementary triangles will then be A1CA3, A1CA2 and A2CA3. In this figure, we have also represented the projections of the univariant transformations lines issuing from the three binary eutectics converging toward a ternary eutectic.

The same method could be used to generalize the triangulation to apply to any given scenario.


Figure 6.11. Ternary with a binary compound C with incongruent melting; a) layers of the liquidus; b) projection of the univariant transformation lines and

solidification lines. For a color version of the figure, see www.iste.co.uk/soustelle/transformations.zip

6.4.3. Definite compound with incongruent melting: quasi-peritectic transformation

We shall illustrate the concept of a binary definite compound with incongruent melting by examining a ternary system exhibiting such a compound on the binary A2-A3. We suppose the solids to be completely immiscible.

Figure 6.11(a) gives the representation of the liquidus in space. In it, we can distinguish four layers:

– 1(A )FT E2E’EE3, which leads to the deposition of the solid A1 through

cooling;

– E’1E’EE1, which leads to the deposition of the solid C through cooling;

– E1EE3 2(A )FT , which leads to the deposition of the solid A2 through cooling;

– 3(A )FT E2E’E’1, which leads to the deposition of the solid A2 through

cooling.

These layers intersect along five univariant transformation lines; E2E’, E’1E’, E1E, E3E and E’E.


a) b)

A3 N’

M’

J’K’

j’m’ k’

)3A(FT )N(FT

A3 N

JK

M

jmk

)3A(FT )N(FT

Figure 6.12. Isopleth sections with binary definite compound exhibiting incongruent melting; a) in the vertical plane A3N’; b) in the vertical plane A3N

On the triangular planar projection (Figure 6.11(b)), we see the two triangles A3A1C and A1A2C, resulting from triangulation, as in the case of a compound with congruent melting (Figure 6.9), but the layer of deposition of the compound A1 spills over into the triangle CA1A3, because the univariant transformation line E’1E’, which replaces the eutectic, is not situated within the triangle A3A1C.

To trace a trajectory of solidification of liquids whose compositions correspond to points m and m’ (Figure 6.11(b)), it is preferable to examine the isopleth sections by a vertical plane passing through A3m or A3m’.

The liquid whose composition corresponds to M’ deposits the solid A3, as shown by the isopleth section in Figure 6.12(a). Therefore, the projection of the solidification trajectory onto the triangle of composition follows the line m’N’, moving further away from A3 until we reach the line e’e at k’ in Figure 6.11(b) – i.e. point K’ in Figure 6.12(a). At that point, we see the following transformation:

Liquid + A3 C [6R.10]

However, the point E’ also belongs to the crystallization curve E’E1, and hence to equilibrium with the crystals of A1. The four phases – the liquid and the three solids A1, A3 and C – are at equilibrium at temperature (E ')FT , so the liquid solidifies by an invariant transformation:

Liquid E’ + A3 A1 + C [6R.11]


This transformation is called a quasi-peritectic transformation. This type of solidification is valid for any liquid M’ whose projection m’ is within the quadrilateral A3Ce’A1.

The liquid whose composition is at point M, but which is situated in the triangle CA1A2, also deposits the solid A3, as is illustrated by Figure 6.11(b). The projection of the trajectory therefore follows the line A3N, moving away from A3 until it too touches the line e’e at point k – i.e. point K from Figure 6.12(b), with invariant quasi-peritectic solidification. The point m is also inside the quadrilateral A3Ce’A1.

Then, the quasi-peritectic transformations finish with an excess of liquid, and we see the solidification of a binary eutectic with solidification of crystals A1 and C. In both cases, the projections of the solidification trajectories follow the line e’e to the point e, which corresponds to the point E of precipitation of the ternary eutectic, via the transformation:

Liquid A1 + C + A2 [6R.12]

The points situated outside the quadrilateral A3Ce’A1 follow a solidification path conforming to the triangle to which they belong.

6.5. A peritectic transformation in one binary system and total miscibility in the other two

We shall now consider a ternary system characterized by the following properties of the binary systems:

– total miscibility in the solid state for the two components A1 and A2;

– total miscibility in the solid state for the two components A1 and A3;

– existence of an invariant transformation due to the peritectic point P in the binary system A2A3.

In light of the dual total miscibility in the solid state of the binary systems A1-A2 and A1-A3, the different solid phases involved in the ternary system will all have the same crystallographic structure, and differ only by their compositions.


Figure 6.13(a) shows a spatial representation of the diagram of solidifica-tion of such a ternary. We find the single spindles of solidification in the two binaries with fully-miscible solids.

A1

A2 A3

PD

Q

Rβ β’

K

a)T A1

A2 A3

β

b)

Liq + β

β + β’

Liq + β+β’

Liq +β’

β’p

r

k

d

qLiq

Figure 6.13. Ternary diagram with a peritectic on a binary; a) spatial representation; b) isothermal section at a temperature with intersection with the peritectic line

(adapted from [DES 10]). For a color version of the figure, see www.iste.co.uk/soustelle/transformations.zip

The peritectic point P of the binary A2-A3 propagates in the ternary system along a line interrupted at Q. At that point, the liquid is at equilibrium with the solid solution β, whose composition is given by that of the point K, which is the minimum of the monovariant transformation line RKD. This curve defines the composition of the solution formed by the peritectic transformation, which can be written as follows, upon cooling:

Liquid + β β’ [6R.13]

Figure 6.13(b) shows the example of the projection of an isothermal section obtained at a temperature T between the melting point at P and that at Q: ( )F PT < T < ( )F QT . This figure also gives the nature of the phases in each domain with examples of conodes in the biphasic domains and the projections of the univariant transformation lines pq (projection of PQ) and rkd (projection of RKD).


6.6. The ternary peritectic transformation

Let us now consider the solidification of a ternary system whose binary subsystems possess the following properties:

– the two binary systems A1-A2 and A1-A3 exhibit a peritectic transformation;

– the third binary system, A2-A3, exhibits a eutectic transformation;

– the three components yield three solid solutions: α for that of the structure of A1, β for that of the structure of A2 and γ for that of the structure of A3.

A1

A2 A3

E

P2

a)

β0

β

β1

βs

α1

α0

α2

γ1

γ0

γ

γs

T

P1

P

α

A1

A2 A3

b)

β0

β

β1 βs αs

α0

α2

α1

γ1

γ0

γs

eγ

p1

p2

p

d

Figure 6.14. Diagram of a system with a ternary peritectic; a) spatial representation; b) projections of the univariant transformation lines (adapted from [DES 10]). For a

color version of the figure, see www.iste.co.uk/soustelle/transformations.zip

Figure 6.14(a) gives the spatial representation of the solidification diagram for this ternary system. The ternary peritectic transformation is given, in the direction of cooling, by the reaction:

Liquid + βs + γs αs [6R.14]

The triangle Pβsγs and the point αs define the compositions of the solid phases which are involved in this invariant peritectic ternary transformation.


The univariant transformation lines αsα1 and αsα2 define the evolution of the composition of the α phase during the transformation. The same is true for the lines γsγ1 and γsγ2, representing the evolution of the composition of the γ phase.

Figure 6.14(b) represents the projection, in the triangle of composition, of the univariant transformation lines of the system.

During the solidification of the liquid mixture, the peritectic ternary transformation can only take place if the liquid’s initial composition is represented by a point whose projection on the triangle of composition is within the triangle γsβsp.

For the cooling of mixtures whose representative point is projected within the triangle αsβsγs, solidification is followed by an evolution of the compo-sition of the solid phase along one of the univariant transformation lines αsα0, βsβ0 and γsγ0.

For compositions situated within the triangle pγsαs, the invariant transformation is followed by the peritectic transformation:

Liquid + γ α [6R.15]

For the end of solidification of the compositions within the triangle pγsαs, we distinguish:

– compositions entering into the triangle pαsd, for which solidification finishes with the solidification of the γ phase by divariant transformation between γ and the remaining liquid;

– the triangle dγsαs, in which solidification finishes when the liquid phase is entirely exhausted.

The same observations can be made within the triangle pβsαs, which is also divided into two triangles.

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Index

A, B, C

allotropic transformation, 35 variety, 35

aniline point, 188 argentiferous lead, 188 associated solutions, 137 atomic partition function, 23 azeotrope, 60

of vaporization, 160 azeotropic point, 60 azeotropy

negative, 136 positive, 136

binary azeotrope, 184 definite compound, 208, 211, 213 eutectic, 198, 206, 209 system, 43

birefringence, 40 boiling

curve, 121 layer, 182, 192 point, 17 trajectory, 185

boundary, 186 cementite, 101

chemical moment, 55, 166, 169 Clapeyron

equation, 6, 7, 10, 16, 17, 28, 31 relation, 5

congruent melting, 95, 208 conjugate points, 170 conodal line, 170 conode, 170 continuous transformation, 3 cooling curve, 63 cricondenbar, 140 cricondentherm, 140 critical

exponents, 15 line, 141 opalescence, 22 point, 22, 61, 95, 140 pressure, 22 temperature, 22 temperature of demixing, 84

cryometry, 104 cryoscopic

constant, 105 law, 202 reduction, 99

cubic equation, 19 Curie temperature, 3



D, E, F

de Donder (equilibrium condition), 18

de Gennes, 42 definite compound, 91, 95 demixing, 81, 91, 94, 136

loop, 78 dew

layer, 182 trajectory, 185

dew-point curve, 121 diamond, 40 differential thermal analysis, 65 discontinuous extraction, 177 distillation

column, 149 field, 186, 190

ebulliometry, 146 ebullioscopy constant, 148 Ehrenfest, 7

classification, 2 equations, 10

Einstein frequency, 32 temperature, 30

enantiotropy, 36, 39 enthalpy of

fusion, 31, 32, 34, 37 sublimation, 28, 34 vaporization, 34

equilibrium condition, 45 eutectic

point, 91, 103 reaction, 71, 95

diagram, 69, 73, 128

eutectoid, 100 extract, 178 Fenske equation, 153 ferromagnetism, 3 ferro-paramagnetic

transformation, 11

fractional distillation, 149 fusion constant, 89 gas–liquid azeotrope, 126

G, H, I

Gibbs triangle, 163 variance, 43

Gibbs energy curve, 66 graph, 47 of fusion, 32 of mixing, 52 of mixing, 82 of vaporization, 19

Gibbs–Konovalov theorem, 55, 73 Ginsburg and Landau model, 15 graphite, 39 heat of dissolution, 124 helium I – helium II transition, 3 Henry’s

coefficient, 79 constant, 81, 89

Henry’s law, 74, 79 (solubility of gas), 144

hetero- attraction, 125 azeotrope, 128 repulsion, 125

hydrates, 103 incongruent melting, 96, 208, 213 inert gas, 26 infinitely-dilute solution, 109 invariant reaction, 95, 100 iron–graphite, 100 isologue curve, 140 isopleth section, 171, 182, 194, 207 isothermal

compressibility, 15 phase diagram, 72 section, 169, 171, 182

Index 223

L, M, N

lambda transformation, 3 Landau

model, 42 theory, 10

latent heat, 2, 6 of vaporization, 16

lattice energy, 39 lever

law, 55 rule, 21, 166, 170

Lewis’ law, 114 liquefaction

branch, 21 trajectory, 185

liquid crystals, 41 liquid–liquid extraction, 177 liquid–vapor equilibrium, 16, 28 liquidus, 90, 93, 102, 192 mass

concentration, 111 fraction, 54

maximum critical temperature, 76 McCabe’s method, 180 McCabe–Thiele diagram, 156 melting trajectory, 192 mesogenic, 40 mesomorphic states, 40 Mie model, 23 minimum critical temperature, 78 miscibility gap, 84, 137, 171, 183 model, 23 molar

enthalpy, 114 latent heat of vaporization, 116 mass of the solute, 125 mass, 110, 145, 146

monotropy, 39 multi-stage counterflow extraction,

178 nematic state, 40, 41, 42

number of “real” trays 159 theoretical trays in the column,153

O, P, Q

operational line, 180 order parameter, 10, 15 order–disorder transition, 3, 11 osmosis, 106 osmotic

coefficient, 112 pressure, 106

Ostwald’s law, 38, 39 Parkes process, 188 partial reflux, 158 partition law, 174 partition coefficient, 173 perfect solution, 59, 109 peritectic, 93, 128

reaction, 72, 93 transformation, 215 type, 70, 71, 94

peritectoid, 100 phase fraction, 45, 197 polymorphic transition, 35 positive azeotrope, 136 PSRK model, 132 pure substance diagram, 32 quasi-

binary, 165, 171 peritectic transformation, 215

R, S, T

raffinate, 178 Raoult’s law, 74, 79

of cryometry, 105 of ebulliometry, 148 of tonometry, 124

real solution, 125


rectification, 149 reduction in vapor pressure of the

solvent, 124 reference state, 114 reflux rate, 158 relative volatility, 154 retrograde solubility, 77 Schröder–Le Châtelier law,102 semi-permeable membrane, 106 simple distillation, 150 smectic state, 40, 41 smooth potential, 23 solid-liquid equilibrium, 30 solidification trajectory, 192 solidus, 90, 93, 192 solubility, 175

curve, 93, 103 of a gas, 143 of a solid, 102 of salts, 103

solutions negative deviation, 145 positive deviation, 145

Sorel’s equation, 158 Soret line,160 specific heat capacity, 15 spinodal composition, 83 state of segregation, 46 strictly-regular solution, 59, 81, 111,

132, 145 sublimation, 30, 34 superconductive, 3 supercritical domain, 22 transformation,

first-order, 2, 4, 41 second-order, 2, 3, 7

ternary azeotrope, 184, 190 definite compound, 208, 211 distillation, 189 eutectic, 193, 198, 202, 206, 209 peritectic transformation, 193, 217

theoretical number of stages, 180 trays, 153

thermotropic, 41 three-phase binary system, 66 tonometry, 145 total reflux, 151 triangulation, 212, 214 triple point, 32 Trouton approximation, 131 Trouton’s rule, 25 two-phase

domain, 33 region, 54

U, V

UNIFAC, 132 UNIQUAC, 132 univariant

line, 195 transformation, 195, 207

van ’t Hoff’s law, 110 van der Waals, 19 vaporization, 16, 24, 26, 34, 116

branch, 21 constant, 114 curve, 121 trajectory, 185

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2015 SOUSTELLE Michel Chemical Thermodynamics Set

Volume I - Phase Modeling Tools

Volume 2 - Modeling of Liquid Phases

Volume 3 - Thermodynamic Modeling of Solid Phases

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CHEMICAL THERMODYNAMICS SETCoordinated by Michel Soustelle

This book is part of a set of books which offers advanced studentssuccessive characterization tool phases, the study of all types ofphase (liquid, gas and solid, pure or multi-component), processengineering, chemical and electrochemical equilibria and theproperties of surfaces and phases of small sizes. Macroscopic andmicroscopic models are in turn covered with a constant correlationbetween the two scales. Particular attention has been given to therigor of mathematical developments.

This fifth volume is devoted to the study of transformations andequilibria between phases. First- and second-order pure phasetransformations are presented in detail, just as with themacroscopic and microscopic approaches of phase equilibria.

In the presentation of binary systems, the thermodynamics ofazeotropy and demixing are discussed in detail and applied tostrictly-regular solutions. Eutectic and peritectic points areexamined, as well as the reactions that go with them. The study ofternary systems then introduces the concepts of ternaryazeotropes and eutectics. For each type of solid-liquid system, theinterventions of definite compounds with or without congruentmelting are taken into account. The particular properties of thedifferent notable points of a diagram are also demonstrated.

Michel Soustelle is a chemical engineer and Emeritus Professor atEcole des Mines de Saint-Etienne in France. He taught chemicalkinetics from postgraduate to Master degree level while alsocarrying out research in this topic.

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

Phase Transformations

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