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THE PRINCIPLES OF

CHEMICAL EQUILIBRIUM

WITH APPLICATIONS IN CHEMISTRY

AND CHEMICAL ENGINEERING

BY

KENNETH DENBIGH, F.RS., '" ..

FOURTH EDITION

' CAMBRIDGE . UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge CB2 1RP, United Kingdom

CAMBRIDGE UNIVERSITY PRESS

The Edinbu1·gh Building, Cambridge CB2 2RU, United Kingdom

40 West 20th Street, New York, NY 10011-4211, USA

10 Stamford Road, Oaldeigh, Melboume 3166, Australia

©Cambridge University Press 1966, 1971, 1981

This book is in copyright. Subject to st-atutory exception

and to the proYisions of relevant collective licensing agreements,

no reproduction of any part may take place without.

the written permission of Cambridge University Press.

First published 1955

Reprinted 1957, 1961, 1964

Second edition 1966

Third edition 1971

Reprinted 1973, 1978

Fourth edition 1981

Reprinted 1986, 1987, 1989, 1992, 1993, 1997

Britiah Library cataloguing in publiration data

Denbigh, Kenneth George

The principles of chemical equilib1ium -4th edition

1. Thermodynami~.\1

2. Chemi~.alreactiollS

I. Title

541'.369 QD504 8()-40925

ISBN 0 521 23682 7 hardback

ISBN 0 521 28150 4 paperback

Transferred to digital printing 2002

PREFACE TO THE FIRST EDITION

My aim baa been to write a. book on the general theory of chemical

equilibrium, including ita statistical d~velopment, and displa.ying

its numerous pra.ctica.l applioa.tions, in the laboratory and industry,

by means of problems. It is hoped that the book may be equa.lly

useful to students in their final years of either a chemistry or

a. chemica.l engineering degree.

Thermodynamics is a subject which needs to be studied not once.

but several times over at advancing levels. In the first round, usually

taken in the first or second year of the degree, a. good deal of attention

is given to calorimetry, before going forward to the second la.w. In

the second or third roun~ch as I a.m concerned with in this book

-it is assumed that the student is a.lready very familiar with the

concepts of temperature and heat, but it is useful once again to go

over the basis of the first and second laws, this time in a more logioa.l

sequence.

The student's confidence, and his ability to apply thermodynamics

in novel situations, oa.n be greatly developed if he works a. consider￾able number of problems which are both theoretioa.l and numerical

in character. Thermodynamics is a quantitative subject and it can be

mastered, not by the memorizing of proofs, but only by detailed and

quantitative applioa.tion to specific problems. The student is therefore

advised not to a.iin at committing anything to memory. The three or

four basic equations which embody the 'laws', together with a. few

defining relations, soon become familiar, and all the remainder can

be obtained from these as required.

The problems at the end of each chapter have been graded from

the very easy to those to which the student may need to return

several times before the method of solution occurs to him. At the

end of the book some notes are given on the more difficult problems,

together with numerioa.l answers.

Questions marked C.U.C.E. are from the qualifying and final

examinations for the Cambridge University Chemioa.l Engineering

degree, and publioa.tion is by permission. The symbols which occur

in these questions are not always quite the same as in the text, but

their meaning is made clear.

In order to keep the size of the book within bounds, the thermo￾dynamics of interfaces has not been included. The disOUB8ion of gal￾vanic cells and the activity coefficients of electrolytes is a.Iso rather

brief.

iv Preface

Part I conta.ins the ba.sis of thermodynamics developed on tradi￾tional lines, involving the Ca.mot cycle. Pa.rt II conta.ins the ma.in

development in the field of chemica.! equilibria., and the methods

a.dopted here have been much influenced by Guggenheim's books, to

which I am greatly indebted. Pa.rt III contains a. short introduction

to sta.tistica.l mechanics along the lines of the Gibbs ensemble and

the methods used by R. C. Tolman in his Principle8 of StatiBtical

M ec.1uJnic8.

It is a. great plea.sure to acknowledge my gratitude to a. number of

friends. In particular, my best thanks are due to Dr Peter Gray,

Professor N. R. Amundson, Dr J. F. Davidson and Dr R. G. H. Watson,

for helpful criticism and suggestions, and to Professor T. R. C. Fox,

for stimulating and friendly discussions on thermodynamics over

several years. Finally I wish to express my appreciation of the good

work of the Cambridge University Press, and my thanks to Messrs

Jonathan and Philip Denbigh, for help with the proof correcting, and

to my wife for help in many other ways.

CAMBRIDGE

Ocloba- 1954

K.G.D.

PREFACE TO THE FOURTH EDITION

My work for this edition has been mainly a. revising of the text in the

light of recent contributions to the literature. Many new references

have been added, and there are also certain changes of emphasis.

The difficulties in the way of establishing chemical thermo￾dynamics in a fully rigorous manner have been described afresh by

:Munster in his GlassicoJ, Tkerrrwdynamic8 (1970). As he has said, the 1

'laws' do not constitute a complete set of axioms, especially in the

case of systems having variable composition.

As regards entropy, one way of dealing with these difficulties is

simply to postulate its existence, rather than seeking to prove it.

However this method seems to me not sufficiently satisfying for the

student. Far better, in my view, to put forward the classica.l

arguments as well a.s they can be put, and to develop simultaneously

the statistica.l interpretation of the second law, so a.s to create a

linkage of thermodynamics with the rest of physics and chemistry.

This leaves my previous scheme for Chapter 1 essentially

unchanged. But I have become better aware than previously,

especially from Popper, that there is a certain hazard in using the

statistical argument, even at the elementary level of the present

volume. If the argument is put forward in terms of 'lack of informa￾tion' about micro-states, this may well create the impression,

Preface v

although quite unwarrantably, that thermodynamics contains very

subjective elements. Some of my re-writing has been intended to

correct that impression.

Although 'the information theory approach' is very helpful,

especially in an heuristic sense, I believe it has also somewhat

obscured the central issue, relating to the second law, of how

irreversible phenomena can ever occur. The fact that thermodynamic

systems are incompletely specified is only part of the story, although

an important part. One has also to ask questions about de facto

initial conditions, and how they can arise. These questions can only be

answered, in my view, by referring to the pervasive irreversibility

within the total environment.

Apart from these points concerning Chapters 1 and 11, various

footnotes have been added and improvements have been made to

Chapter 14, and to the section in Chapter 6 which deals with lambda

transitions.

In earlier editions I expressed my indebtedness to Professors

Guggenheim, Peter Gray and John Row Iinson for suggesting various

improvements to the text. I should now like to express my gratitude

to Professors J. A. Campbell, T. W. Weber and N. Agmon for

providing me with substantial lists of errors and misprints. Many

other correspondents have sent very helpful remarks, and to these

also I offer my best thanks.

Aprill980 K. G. D.

CONTENTS

Preface to the First Edition

Preface to the Fourth Edition

List of Symbols

V aluu of Physical Constants

PART I: THE PRINCIPLES OF

THERMODYNAMICS

Chapter 1: First and Second Laws

1·1 Introduction

vii

page iii

iv

xvii

xxi

3

1·2 Thermodynamic systems 5

1·3 Thermodynamic variables 6

1·4 Temperature and the zeroth law 9

1·5 Work 14

1·6 Internal energy and the first law 15

1·7 Heat 18

1·8 Expression of the first law for an infinitesimal process 19

1·9 Adiabatically impossible processes 21

1·10 Natural and reversible processes

1·11 Systematic treatment of the second law

1·12 Final statement of the second law

1·13 A ct:iterion of equilibriwn. Reversible processes

1·14 Maximwn work

1·15 The fundamental equation for a closed system

1·16 Swnmary of the basic laws

1·17 Natural processes as mixing p:r:ocesses

1·18 The molecular interpretation of the second law

Problems

23

25

39

40

43

45

46

48

56

60

viii Contents

Okapter 2: Auxiliary Functions and Conditions of Equilibrium

2·1 The functions H, A and G 11age 63

2·2 Properties of the enthalpy

2·3 Properties of the Helmholtz free energy

2·4 Properties of the Gibbs function

2•5a Availability

2·5b Digression on the useful work of chemical reaction

~·6 The fundamental equations for a closed system in terms of

H, A and G

2·7 The chemical potential

2·8 Criteria of equilibrium in terms of extensive properties

2·9 Criteria of equilibrium in terms of intensive properties

2·10 Mathematical relations between the various functions of .

state

2·11 Measurable quantities in thermodynamics

2·12 Calculation of changes in the thermodynamic functions

over ranges of temperature and pressure

2·13 Molar and partial molar quantities

2·14 Calculation of partial molar quantities from experimental

data

Problem8

PART II: REACTION AND PHASE

EQUILIBRIA

Okapter 3: Thermodynamics of Gases

3·1 Models

3·2 The single perfect gas

3•3 The perfect gas mixture

3·4 Imperfect gases

3·5 The Joule-Thomson effect

63

66

67

70

72

76

76

82

85

89

94

98

99

104

106

Ill

Ill

ll4

119

120

Contents ix

3•6 The fugacity of a. single imperfect ga.s page 122

3·7 Fuga.cities in a.n imperfect ga.s mixture 125

3·8 Temperature coefficient of the fuga.city a.nd sta.nda.rd

chemical potential 127

3·9 Ideal ga.seous solutions a.nd the Lewis and Ra.nda.ll rule 128

Problem8 130

Chapter 4: Equilibria. of Reactions Involving Gases

4·1 Introduction 133

4·2 The stoichiometry of chemical rea.ction 133

4·3 Preliminary discussion on reaction equilibrium 135

4·4 Concise discuBBion on reaction equilibrium 139

4·5 The equilibrium constant for a. ga.s rea.ction 140

4·6 The temperature dependence of the equilibrium constant 143

4·7 Other forms of equilibrium constant for perfect ga.s

mixtures 146

4·8 Free energies a.nd entha.lpies of formation from the

elements 148

4·9 Some examples 149

4·10 Free energies of formation of non-ga.seous substances or

from non-ga.seous elements 153

HI Preliminary discussion on rea.ction equilibria. involving

ga.ses together with immiscible liquids a.nd solids 156

4-12 Concise discussion on rea.ction equilibria. involving ga.ses

together with immiscible liquids a.nd solids 159

4-13 Example on the roa.sting of galena. 161

4·14 Mea.surement of the free energy of rea.ction by use of

ga.lva.nic cells 163

4-15 Alternative discUBBion of the ga.lva.nic cell 167

4·16 Number of independent rea.ctions 169

4·17 Conditions of equilibrium for several independent

rea.ctions 172

4·18 General rem&rks on simultaneous reactions

4•19 General remarks on maximum attainable yield

Problems

Chapter 5: Phase Rule

5·1 Introduction

5·2 The phase rule for non-reactive components

5·3 The phase rule for ,rea.ctive components

5·4 Additional restrictions

5·5 Example of the application of the phase rule

5·6 Alternative approach

5·7 Two examples from the zinc smelting industry

Problems

Chapter 6: Phase Equilibria. in Single Component'Systems

page 173

175

177

182

184

187

188

188

191

191

104

6·1 Introduction 196

6·2 The Clausius-Clapeyron equation 197

6·3 The enthalpyofvaporizationtanditstemperature coefficient 200

6·4 Integration of the Clausius-Clapeyron equation 202

6·5 The effect of a second gas on the vapour pressure of~

liquid or solid 203

6·6 Lambda transitions 207

Problems 213

Chapter 7: General Properties of Solutions a.nd the Gibbs￾Duhem Equation

7·1 The Gibbs-Duhem equation 215

7·2 Pressure-temperature relations 216

7·3 Partial pressure-composition relations 221

7·4 The empirical partial pressure curves of binary solutions 222

7·5 Application of the Gibbs-Duhem equation to the partial

pressure curves 232

Contents

7·6 Application of the Gibbs-Duhem equation to the total

pressure curve

7·7 The Gibbs-Duhem equation in relation to Ra.oult's and

xi

page 235

Henry's laws 236

7•8 The Gibbs--Duhem equation in relation to the Margules

and van La.a.r equations 240

Probkma

Chapter 8: Ideal Solutions

8·1 Molecular aspects of solutions

8·2 Definition of the ideal solution

8·3 Ra.oult's and Henry's laws

8·4 Imperfect vapour phase

8·5 The mixing properties of ideal solutions

8·6 The dependence of vapour-solution equilibria on

temperature and pressure

8•7 Nernst's law

8•8 Equilibrium between an ideal solution and a pure

crystalline component

8·9 Depression of the freezing-point

8•10 Elevation of the boiling-point

8·11 The osmotic pressure of a.n ideal solution

8·12 The ideal solubility of gases in liquids

8·13 The ideal solubility of solids in liquids

Problems

Chapter 9: Non-Ideal Solutions

9·1 Conventions for the activity coefficient on the mole

242

244

249

249

252

252

255

256

257

260

261

262

264

266

267

fraction scale 270

9·2 The activity coefficient in relation to Ra.oult's and Henry's

laws 271

9·3 The use of molality and concentration scales 274

9·4 Convention for the activity-coefficient on the molality

scale 276

9·5 The effect of temperature and pressure 278

xii Contents

9·6 The determination of activity coefficients page 281

9·7 The Gibbs-Duhem equation applied to activity coefficients 284

9·8 The calculation of the activity coefficient of the solute 284

9·9 Excess functions of non-ideal solutions 285

9·10 The activity

9·11 The osmotic coefficient

Problem&

0/w,pter 10: Reaction Equilibrium in Solution. Electrolytes

287

288

288

10·1 Reaction equilibrium in solution 292

10·2 Free energy of formation in solution. Convention

concerning hydrates 295

10·3 Equilibrium constants expressed on the molality and

volume concentration scales 298

10·4 Temperature and preBBure dependence of the

equilibrium constant 299

10·5 Ratio of an equilibrium constant in the gas phase and

in solution 301

10·6 Notation for electrolytes :102

10·7 Lack of significance of certain quantities 303

10·8 DiBBocia.tion equilibrium and the chemical potential of

the electrolyte 304

10·9 Activity coefficients 305

1 0·1 0 Phase equilibrium of an electrolyte. Solubility product 307

10·11 Equilibrium constant for ionic reactions 309

1 0·12 Magnitude of activity coefficients of charged and

uncharged species 310

1 0·13 Free energy of dissociation 312

10·14 The hydrogen ion convention and the free energies and

entha.lpies of formation of individual ions 314

10·15 Activity coefficients and free energies as measured by

the use of the galvanic cell 316

10·16 Activity coefficients by use of the Gibbs-Duhem

equation

10·17 Partial preBBure of a. volatile electrolyte

10·18 Limiting behaviour at high dilution

ProblerrUJ

322

324

:125

327

Contents

PART III: THERMODYNAMICS IN RELATION

TO THE EXISTENCE OF MOLECULES

Chapter II: Statistical Analogues of Entropy and Free Energy

11·1 Thermodynamics and molecular reality page 333

11•2 The quantum states of macroscopic systems 333

11·3 Quantum states, energy states and thermodynamic states 334

11·4 Fluctuations 335

11·5 Averaging and the statistical postulate 336

11·6 Accessibility 337

11·7 The equilibrium state 338

11·8 Statistical methods 339

11·9 The ensemble and the averaging process 340

11·10 Statistical analogues of the entropy and Helmholtz free

energy 345

11·11 Comparison of statistical analogues with thermodynamic

functions 350

1 1·12 Thermal and configure,tional entropy 353

11·13 Appendix I. Origin of the canonical distribution 356

11·14 Appendix II. Entropy analogues 359

Problem 360

Chapter I2: Partition Function of a Perfect Gas

12·1 Distinguishable states of a. gas and the molecular

partition function

12·2 SchrOdinger's equation

12·3 Separability of the wave equation

12·4 Factorization of the molecular partition function

12·5 The translational partition function

12·6 The internal partition function

12·7 Thermodynamic properties of the perfect gas .

12·8 The Maxwell-Boltzmann distribution

361

365

367

3'71

372

376

377

383

xiv Contents

12•9 Dist1ibution over translational and internal states

12·1 0 Number of translational states of a. given energy

12·11 The Maxwell velocity distribution

12·12 Principle of equipa.rtition

12·13 Appendix. Some definite integrals

Problems

Chapter 13: Perfect CeystaJ& a.nd the Third Law

13·1 Nornnalco-ordinates

13·2 The Schrodinger equation for the crystal

13•3 The energy levels of the ha.rnnonic oscillator

13•4 The partition function

13·5 The Ma.xwell-Boltzma.nn distribution

13·6 The high temperature approximation

13·7 The Einstein approximation

13·8 The Debye approximation

13·9 Comparison with experiment

13·10 Vapour pressure at high temperature

13·11 The third law-preliminary

13•12 Statement of the third law

13·13 Tests and applications of the third law

Problema

Chapter 14: Configurational Energy a.nd Entropy

14·1 Introduction

14·2 Example 1: the lattice model of mixtures

14·3 Example 2: the Langmuir isotherm

Chapter 15: Chemical Equilibrium in Relation to Chemical

Kinetics

15·1 Introduction

15·2 Kinetic speci~

page 386

387

390

392

394

396

397

400

401

402

405

406

408

409

411

414

416

421

424

427

429

432

436

439

440

Contents XV

15•3 Variables detennining reaction rate page 441

15·4 Forward and backward processes 442

15•5 Thermodynamic restrictions on the form of the kinetic

equations 444

15·6 The temperature coefficient in relation to thermodynamic

quantities 449

15·7 Transition-state theory 450

15·8 The equilibrium assumption 453

15·9 The reaction rate 455

Appendix. Answers to Problems and Comments 460

IM..ex 487