THERMODYNAMICSAND AN INTRODUCTION TOTHERMOSTATISTICSSECOND EDITION

HERBERT B. CALLENUniversitv of Pennsvlvania

JOHN WILEY & SONSNew York Chichester Brisbane Toronto Singapore

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Copyright @ 1985, by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part ofthis work beyond that permitted by Sections107 and 108 of the 1976 United States CopyrightAct without the permission of the copyrightowner is unlawful. Requests for permissionor further information should be addressed tothe Permissions Department, John Wiley & Sons.

Lihrary of Congress Cataloging in Publication Datu

Callen, Herbert B.Thermodynamics and an Introduction to

Thermostatistics.

Rev. ed of: Thermodynamics.1960.Bibliography: p. 485Includes index.1. Thermodynamics. 2. Statistical Mechanics.

I. Callen, Herbert B. Thermodynamics. II. Title.III. Title: Thermostatistics.QC311.C25 1985 536', .7 85-6387ISBN 0-471-86256-8

Printed and bound in the United States of America

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PREFACE

Twenty-five years after writing the first edition of Thermodynamics I amgratified that the book is now the thermodynamic reference most fre-quently cited in physics research literature, and that the postulationalformulation which it introduced is now widely accepted. Neverthelessseveral considerations prompt this new edition and extension.

First, thermodynamics advanced dramatically in the 60s and 70s, pri-marily in the area of critical phenomena. Although those advances arelargely beyond the scope of this book, I have attempted to at leastdescribe the nature of the problem and to introduce the critical exponentsand scaling functions that characterize the non-analytic behavior of ther-modynamic functions at a second-order phase transition. This account isdescriptive and simple. It replaces the relatively complicated theory ofsecond-order transitions that, in the view of many students, was the mostdifficult section of the first edition.

Second, I have attempted to improve the pedagogical attributes of thebook for use in courses from the junior undergraduate to the first yeargraduate level, for physicists, engineering scientists and chemists. Thispurpose has been aided by a large number of helpful suggestions fromstudents and instructors. Many explanations are simplified, and numerousexamples are solved explicitly. The number of problems has been ex-panded, and partial or complete answers are given for many.

Third, an introduction to the principles of statistical mechanics hasbeen added. Here the spirit of the first edition has been maintained; theemphasis is on the underlying simplicity of principles and on the centraltrain of logic rather than on a multiplicity of applications. For thispurpose, and to make the text accessible to advanced undergraduates, Ihave avoided explicit non-commutivity problems in quantum mechanics.All that is required is familiarity with the fact that quantum mechanicspredicts discrete energy levels in finite systems. However, the formulationis designed so that the more advanced student will properly interpret thetheory in the non-commutative case.

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Fourth, I have long been pnzzled by certain conceptual problems lyingat the foundations of thermodynamics, and this has led me to an interpre-tation of the "meaning" of thermodynamics. In the final chapter-an"interpretive postlude" to the main body of the text-I develop the thesisthat thermostatistics has its roots in the symmetries of the fundamentallaws of physics rather than in the quantitative content of those laws. Thediscussion is qualitative and descriptive, seeking to establish an intuitiveframework and to encourage the student to see science as a coherentstructure in which thermodynamics has a natural and fundamental role.

Although both statistical mechanics and thermodynamics are includedin this new edition, I have attempted neither to separate them completelynor to meld them into the undifferentiated form now popular under therubric of " thermal physics." I believe that each of these extreme options ismisdirected. To divorce thermodynamics completely from its statisticalmechanical base is to rob thermodynamics of its fundamental physicalorigins. Without an insight into statistical mechanics a scientist remainsrooted in the macroscopic empiricism of the nineteenth century, cut offfrom contemporary developments and from an integrated view of science.Conversely, the amalgamation of thermodynamics and statistical me-chanics into an undifferentiated " thermal physics" tends to eclipse ther-modynamics. The fundamentality and profundity of statistical mechanicsare treacherously seductive; " thermal physics" courses almost perforcegive short shrift to macroscopic operational principles.* Furthermore theamalgamation of thermodynamics and statistical mechanics runs counterto the "principle of theoretical economy"; the principle that predictionsshould be drawn from the most general and least detailed assumptionspossible. Models, endemic to statistical mechanics, should be eschewedwhenever the general methods of macroscopic thermodynamics are suffi-cient. Such a habit of mind is hardly encouraged by an organization of thesubjects in which thermodynamics is little more than a subordinate clause.

The balancing of the two distinct components of the thermal sciences iscarried out in this book by introducing the subject at the macroscopiclevel, by formulating thermodynamics so that its macroscopic postulatesare precisely and clearly the theorems of statistical mechanics, and byfrequent explanatory allusions to the interrelationships of the two compo-nents. Nevertheless, at the option of the instructor, the chapters onstatistical mechanics can be interleaved with those on thermodvnamics ina sequence to be described. But even in that integrated option the basicmacroscopic structure of thermodynamics is established before statisticalreasoning is introduced. Such a separation and sequencing of the subjects

*The American Physical Society Committee on Applications of Physics reported lBulletin of thelPS, Vol 22 #10, 1233 (I9'1L)l that a survey of industrial research leaders designated thermody-namics above all other subjects as requiring increased emphasis in the undergraduate curriculum Thatemphasis subsequently has decreaserl

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preserves and emphasizes the hierarchical structure of science, organizingphysics into coherent units with clear and easily remembered interrela-tionships. Similarly, classical mechanics is best understood as a self-contained postulatory structure, only later to be validated as a limitingcase of quantum mechanics.

Two primary curricular options are listed in the "menu" following. Inone option the chapters are followed in sequence (Column A alone, orfollowed by all or part of column B). In the "integrated" option the menuis followed from top to bottom. Chapter 15 is a short and elementarystatistical interpretation of entropy; it can be inserted immediately afterChapter 1, Chapter 4, or Chapter 7.

The chapters listed below the first dotted line are freely flexible withrespect to sequence, or to inclusion or omission. To balance the concreteand particular against more esoteric sections, instructors may choose toinsert parts of Chapter 13 (Properties of Materials) at various stages, or toinsert the Postlude (Chapter 21, Symmetry and Conceptual Foundations)at any point in the course.

The minimal course, for junior year undergraduates, would involve thefirst seven chapters, with Chapter 15 and 16 optionally included as timepernuts.

P hiladelphia, P ennsyluania Herbert B. Callen

Preface

l. Postulates

15.

2. Conditions of Equilibrium

3. Formal Relations and Sample Sys-tems

4. Reversible Processes; Engines

Legendre Transformations

Extremum Principles in LegendreRepresentation

Maxwell Relations

Stability

First-Order Phase Transitions

Statistical Mechanics in EntropyRepresentation

Canonical Formalism

Generalized Canonical Formula-tion

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10. Critical Phenomena

11. Nernst

12. Summary of Principles

13. Properties of Materials

14. Irreversible Thermodynamics

Quantum Fluids

Fluctuations

Variational Properties and MeanField Theory

18.

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21. Postlude: Symmetry and the Conceptual Foundations of Thermodynamics

CONTENTS

PART IGENERAL PRINCIPLES OFCLAS SICAL THERMODYNAMICS

lntroduction The Nature of Thermodvnamics and the BasisThermostatistics

I THE PROBLEM AND THE POSTULATES1.1 The Temporal Nature of Macroscopic Measurements1.2 The Spatial Nature of Macroscopic Measurements1.3 The Composition of Thermodynamic Systems1.4 The Internal Energy1.5 Thermodynamic Equilibrium1.6 Walls and Constraints1.7 Measurability of the Energy1.8 Quantitative Definition of Heat-Units1.9 The Basic Problem of Thermodynamicsl.i0 The Entropy Maximum Postulates

2 THE CONDITIONS OF EQUILIBRIUM2.1 IntensiveParameters2.2 Equations of State2.3 Entropic Intensive Parameters2.4 Thermal Equilibrium-Temperature2.5 Agreement with Intuitive Concept of Temperature2.6 Temperature Units2.7 MechanicalEquilibrium2.8 Equilibrium with Respect to Matter Flow2.9 ChemicalEquilibrium

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3 SOME FORMAL RELATIONSHIPS,AND SAMPLE SYSTEMS

3.1 The Euler Equation3.2 The Gibbs-Duhem Relation3.3 Summary of Formal Structure3.4 The Simple Ideal Gas and Multicomponent

Simple Ideal Gases3.5 The "Ideal van der Waals Fluid"3.6 ElectromagneticRadiation3.7 The "Rubber Band"3.8 Unconstrainable Variables; Magnetic Systems3.9 Molar Heat Capacity and Other Derivatives

4 REVERSIBLE PROCESSES AND THEMAXIMUM WORK THEOREM

4.1 Possible and Impossible Processes4.2 Quasi-Static and Reversible Processes4.3 Relaxation Times and Irreversibility4.4 Heat Flow: Coupled Systems and Reversal of processes4.5 The Maximum Work Theorem4.6 Coefficients of Engine, Refrigerator, and

Heat Pump Performance4.7 The Carnot Cycle4.8 Measurability of the Temperature and of the Entropy4.9 Other Criteria of Engine Performance; Power Output and

"Endoreversible Engines"4.10 Other Cyclic Processes

5 ALTERNATIVE FORMULATIONSAND LEGENDRE TRANSFORMATIONS

5.1 The Energy Minimum Principle5.2 LegendreTransformations5.3 ThermodynamicPotentials5.4 Generalized Massieu Functions

6 THE EXTREMUM PRINCIPLE IN THELEGENDRE TRANSFORMED REPRESENTATIONS

6.1 The Minimum Principles for the Potentials-6.2 The Helmholtz Potential6.3 The Enthalpy; The Joule-Thomson or "Throttling" process6.4 The Gibbs Potential; Chemical Reactions6.5 Other Potentials6.6 Compilations of Empirical Data; The Enthalpy of Formation6.7 The Maximum Principles for the Massieu Functions

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Contents

7 MAXWELL RELATIONS7.1 The Maxwell Relations7.2 A Thermodynamic Mnemonic Diagram7.3 A Procedure for the Reduction of Derivatives in

Single-Component Systems

7.4 Some Simple Applications'7.5

Generalizations: Magnetic Systems

I STABILITY OF THERMODYNAMIC SYSTEMS8.1 Intrinsic Stability of Thermodynarnic Systems8.2 Stability Conditions for Thermodynamics Potentials8.3 Physical Consequences of Stability8.4 Le Chatelier's Principle; The Qualitative Effect

of Fluctuations8.5 The Le Chatelier-Braun Principle

9 FIRST-ORDER PHASE TRANSITIONS9.1 First-Order Phase Transitions in Single-Component Systems9.2 The Discontinuity in the Entropy-Latent Heat9.3 The Slope of Coexistence Curves; the Clapeyron Equation9.4 Unstable Isotherms and First-Order Phase Transitions9.5 General Attributes of First-Order Phase Transitions9-6 First-Order Phase Transitions in Multicomponent

Systems-Gibbs Phase Rule9.7 Phase Diagrams for Binary Systems

TO CRITICAL PHENOMENAlO.1 Thermodynamics in the Neighborhood of the Critical PointlO.2 Divergence and Stability10.3 Order Parameters and Critical Exponents10.4 Classical Theory in the Critical Region; Landau Theory10.5 Roots of the Critical Point Problem10.6 Scaling and Universality

II THE NERNST POSTULATEll.1 Nernst's Postulate, and the Principle of Thomsen

and Bertholotll.2 Heat Capacities and Other Derivatives at Low Temperatureslf .3 The "Unattainability" of Zero Temperature

T2 SUMMARY OF PRINCIPLESFOR GENERAL SYSTEMS

l2.I General Systems12.2 The Postulates

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12.3 The Intensive Parameters72.4 LegendreTransforms72.5 Maxwell Relations72.6 Stability and Phase TransitionsI2.7 CriticalPhenomenaI2.8 Properties at Zero Temperature

13 PROPERTTES OF MAT4RTALSf 3.1 The General Ideal Gas73.2 Chemical Reactions in ldeal Gases73.3 Small Deviations from "Ideality"-The Virial Expansion73.4 The "Law of Corresponding States" for Gases13.5 Dilute Solutions: Osmotic Pressure and Vapor pressure13.6 Solid Systems

14 IRREVERSIBLE THERMODYNAMICSI4.I General RemarksI4.2 Affinities and Fluxes74.3 Purely-Resistive and Linear Systems14.4 The Theoretical Basis of the Onsager Reciprocity14.5 ThermoelectricEffects14.6 TheConductivitiesI4.7 The Seebeck Effect and the Thermoelectric power14.8 The Peltier Effect14.9 The Thomsen Effect

PART IISTATISTICAL MECHANICS

15 STATISTICAL MECIIANICS IN THEENTROPY REPRESENTATION:THE MICROCANONICAL FORMALISM

15.1 Physical Significance of the Entropy for Closed Systems75.2 The Einstein Model of a Crystalline Solid15.3 The Two-State System75.4 A Polymer Model-The Rubber Band Revisited15.5 Counting Techniques and their Circumvention;

High Dimensionality

16 THE CANONICAL FORMALISM; STATISTICALMECHANICS IN HELMHOLTZ REPRESENTATION

16.1 The Probability Distribution76.2 Additive Energies and Factorizability of the partition Sum

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16.3 Internal Modes in a Gas16.4 Probabilities in Factorizable Systems16.5 Statistical Mechanics of Small Systems: Ensembles16.6 Density of States and Density-of-Orbital States16.7 The Debye Model of Non-metallic Crystals16.8 Electromagnetic Radiation16.9 The Classical Density of Statesf6.10 The Classical Ideal Gas16.11 High Temperature Properties-The Equipartition Theorem

17 BNTROPY AND DISORDER; GENERALIZEDCANONICAL FORMULATIONS

l7.I Entropy as a Measure of Disorder17.2 Distributions of Maximal Disorder17.3 The Grand Canonical Formalism

18 QUANTUM FLUIDS18.1 Quantum Particles; A "Fermion Pre-Gas Model"18.2 The Ideal Fermi Fluid18.3 The Classical Limit and the Quantum Criteria18.4 The Strong Quantum Regime; Electrons in a Metal18.5 The Ideal Bose Fluid18.6 Non-Conserved Ideal Bose Fluids; Electromagnetic

Radiation Revisited18.7 Bose Condensation

T9 FLUCTUATIONS19.1 The Probability Distribution of Fluctuations19.2 Moments and The Energy Fluctuations19.3 General Moments and Correlation Moments

20 VARIATIONAL PROPERTIES, PERTURBATIONEXPANSIONS, AND MEAN FIELD THEORY

20.1 The Bogoliubov Variational Theorem20.2 Mean Field Theory20.3 Mean Field Theory in Generalized Representation;

the Binary Alloy

PART IIIFOUNDATIONS

21 POSTLUDE: SYMMBTRY AND THE CONCEPTUALFOUNDATIONS OF THERMOSTATISTICS

21.7 Statistics

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21.2 Symmetry21.3 Noether's Theorem2I.4 Energy, Momentum and Angular Momentum; the Generalized

"First Law" of Thermodynamics21.5 Broken Symmetry and Goldstone's Theorem21.6 Other Broken Symmetry Coordinates-Electric and

Magnetic Moments27.7 Mole Numbers and Gauge Symmetry21.8 Time Reversal, the Equal Probability of Microstates,

and the Entropy Principle21.9 Symmetry and Completeness

APPENDIX ASOME RELATIONS IT{VOLVINGPARTIAL DERIVATIVESA.1 Partial Derivatives4.2 Taylor's ExpansionA.3 DifferentialsA.4 Composite FunctionsA.5 Implicit Functions

APPENDIX BMAGNETIC SYSTEMS

GENERAL REFERENCES

INDEX

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479

485

487