CALLEN THERMODYNAMICS AND AN INTRODUCTION TO THERMOSTATISTICS SECOND EDITION HERBERT B. CALLEN University o f Pennsylvania JO H N WILEY & SO N S New York Chichester Brisbane Toronto Singapore Copynght < 1985, by John Wiley & Sons, Inc. AH rights reserved Published simultaneously in Canada Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108 of the 1976 United States Copynght Act without the permission of the copynght owner is unlawful Requests for permission or further inform ation should be addressed to the Permissions Departm ent, John Wiley & Sons / ibrary of Congress Cataloging in Publication Data: Callen, H erbert B Therm odynamics and an Introduction to Therm ostatistics Rev ed of. Thermodynamics 1960 Bibliography p 485 Includes index.
1 Thermodynamics 2 Statistical Mechanics I Callen, H erbert B Thermodynamics II Title III Title Thermostatistics Q C 311C25 1985 536'7 85-6387 Printed in the Republic o f Singapore 10 9 8 Sara .and to Jill, Jed, Zachary and Jessica PREFACE Twenty-five years after writing the first edition o f Thermodynamics I am gratified that the book is now the thermodynamic reference most fre quently cited in physics research literature, and that the postulational formulation which it introduced is now widely accepted. Nevertheless several 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 are largely beyond the scope of this book, I have attempted to at least describe the nature of the problem and to introduce the critical exponents and scaling functions that characterize the non-analytic behavior of ther modynamic functions at a second-order phase transition.
This account is descriptive and simple. It replaces the relatively complicated theory of second-order transitions that, in the view of many students, was the most difficult section of the first edition. Second, I have attempted to improve the pedagogical attributes of the book for use in courses from the junior undergraduate to the first year graduate level, for physicists, engineering scientists and chemists. This purpose has been aided by a large number of helpful suggestions from students and instructors.
Many explanations are simplified, and numerous examples 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 has been added. Here the spirit of the first edition has been maintained; the emphasis is on the underlying simplicity of principles and on the central train o f logic rather than on a multiplicity of applications.
For this purpose, and to make the text accessible to advanced undergraduates, I have avoided explicit non-commutivity problems in quantum mechanics. All that is required is familiarity with the fact that quantum mechanics predicts discrete energy levels in finite systems. However, the formulation is designed so that the more advanced student will properly interpret the theory in the non-commutative case. till Preface Fourth, I have long been puzzled by certain conceptual problems lying at 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 thesis that thermostatistics has its roots in the symmetries of the fundamental laws o f physics rather than in the quantitative content of those laws. The discussion is qualitative and descriptive, seeking to establish an intuitive framework and to encourage the student to see science as a coherent structure in which thermodynamics has a natural and fundamental role. Although both statistical mechanics and thermodynamics are included in this new edition, I have attempted neither to separate them completely nor to meld them into the undifferentiated form now popular under the rubric of “ thermal physics.” I believe that each of these extreme options is misdirected. To divorce thermodynamics completely from its statistical mechanical base is to rob thermodynamics of its fundamental physical origins.
Without an insight into statistical mechanics a scientist remains rooted in the macroscopic empiricism of the nineteenth century, cut off from 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 mechanics are treacherously seductive; “ thermal physics” courses almost perforce give short shrift to macroscopic operational principles.* Furthermore the amalgamation of thermodynamics and statistical mechanics runs counter to the “ principle of theoretical economy”; the principle that predictions should be drawn from the most general and least detailed assumptions possible. Models, endemic to statistical mechanics, should be eschewed whenever the general methods of macroscopic thermodynamics are suffi cient.
Such a habit of mind is hardly encouraged by an organization of the subjects in which thermodynamics is little more than a subordinate clause. The balancing of the two distinct components of the thermal sciences is carried out in this book by introducing the subject at the macroscopic level, by formulating thermodynamics so that its macroscopic postulates are precisely and clearly the theorems of statistical mechanics, and by frequent explanatory allusions to the interrelationships of the two compo nents. Nevertheless, at the option of the instructor, the chapters on statistical mechanics can be interleaved with those on thermodynamics in a sequence to be described. But even in that integrated option the basic macroscopic structure of thermodynamics is established before statistical reasoning is introduced.
Such a separation and sequencing of the subjects *The American Physical Society Committee on Applications of Physics reported [ Bulletin o f the A P S , Vol 22 # 1 0 , 1233 (1971)] that a survey of industrial research leaders designated thermody namics above all other subjects as requiring increased emphasis in the undergraduate curriculum. That em phasis subsequently has decreased Preface ix preserves and emphasizes the hierarchical structure of science, organizing physics 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 limiting case of quantum mechanics. Two primary curricular options are listed in the “ menu” following.
In one option the chapters are followed in sequence (Column A alone, or, followed by all or part of column B). In the “ integrated” option the menu is followed from top to bottom. Chapter 15 is a short and elementary statistical interpretation o f entropy; it can be inserted immediately after Chapter 1, Chapter 4, or Chapter 7. The chapters listed below the first dotted line are freely flexible with respect to sequence, or to inclusion or omission.
To balance the concrete and particular against more esoteric sections, instructors may choose to insert parts of Chapter 13 (Properties of Materials) at various stages, or to insert the Postlude (Chapter 21, Symmetry and Conceptual Foundations) at any point in the course. The minimal course, for junior year undergraduates, would involve the first seven chapters, with Chapter 15 and 16 optionally included as time permits. Philadelphia, Pennsylvania Herbert B. Callen Preface to the Fourth Printing In the issuance of this fourth printing of the second edition, the publisher has graciously given me the opportunity to correct various misprints and “minor” errors.
I am painfully aware that no error, numerical or textual, is truly minor to the student reader. Accordingly, I am deeply grateful both to the numerous read ers who have called errors to my attention, and to the charitable forbearance of the publisher in permitting their correction in this printing. November, 1987 Herbert Callen X Preface 1. Conditions of Equilibrium 3.
Formal Relations and Sample Sys tems 4. Reversible Processes; Engines 15. Statistical Mechanics in Entropy Representation 5. Extremum Principles in Legendre Representation 7.
Generalized Canonical Formula tion 8. First-Order Phase Transitions 10. Summary of Principles 20. Variational Properties and Mean Field Theory 13.
Properties of Materials 14. Postlude: Symmetry and the Conceptual Foundations of Thermodynamics CONTENTS PARTI GENERAL PRINCIPLES OF CLASSICAL THERMODYNAMICS l Introduction The Nature of Thermodynamics and the Basis of Thermostatistics 2 1 THE PROBLEM AND THE POSTULATES 5 1.1 The Temporal Nature of Macroscopic Measurements 5 1.2 The Spatial Nature of Macroscopic Measurements 6 1.3 The Composition of Thermodynamic Systems 9 1.4 The Internal Energy 11 1.6 Walls and Constraints 15 1.7 Measurability of the Energy 16 1.8 Quantitative Definition of Heat—Units 18 1.9 The Basic Problem of Thermodynamics 25 1.10 The Entropy Maximum Postulates 27 2 THE CONDITIO NS OF EQUILIBRIUM 35 2.2 Equations of State 37 2.3 Entropic Intensive Parameters 40 2.4 Thermal Equilibrium—Temperature 43 2.5 Agreement with Intuitive Concept of Temperature 45 2.8 Equilibrium with Respect to Matter Flow 54 2.9 Chemical Equilibrium 56 xi xii Contents 3 SO M E FORMAL RELATIONSHIPS, AND SAM PLE SYSTEM S 59 3.1 The Euler Equation 59 3.2 The Gibbs-Duhem Relation 60 3.3 Summary of Formal Structure 63 3.4 The Simple Ideal Gas and Multicomponent Simple Ideal Gases 66 3.5 The “Ideal van der Waals Fluid” 74 3.7 The “ Rubber Band” 80 3.8 Unconstrainable Variables; Magnetic Systems 81 3.9 Molar Heat Capacity and Other Derivatives 84 4 REVERSIBLE PROCESSES AND THE M AXIM UM WORK THEOREM 91 4.1 Possible and Impossible Processes 91 4.2 Quasi-Static and Reversible Processes 95 4.3 Relaxation Times and Irreversibility 99 4.4 Heat Flow: Coupled Systems and Reversal of Processes 101 4.5 The Maximum Work Theorem 103 4.6 Coefficients of Engine, Refrigerator, and Heat Pump Performance 113 4.7 The Carnot Cycle 118 4.8 Measurability of the Temperature and of the Entropy 123 4.9 Other Criteria of Engine Performance; Power Output and “Endoreversible Engines” 125 4.10 Other Cyclic Processes 128 5 ALTERNATIVE FORMULATIONS AND LEGENDRE TRANSFORMATIONS 131 5.1 The Energy Minimum Principle 131 5.4 Generalized Massieu Functions 151 6 THE EXTREMUM PRINCIPLE IN THE LEGENDRE TRANSFORMED REPRESENTATIONS 153 6.1 The Minimum Principles for the Potentials 153 6.2 The Helmholtz Potential 157 6.3 The Enthalpy; The Joule-Thomson or “Throttling”Process 160 6.4 The Gibbs Potential; Chemical Reactions 167 6.6 Compilations of Empirical Data; The Enthalpy ofFormation 173 6.7 The Maximum Principles for the Massieu Functions 179 Contents xiii I MAXWELL RELATIONS 181 7.1 The Maxwell Relations 181 7.2 A Thermodynamic Mnemonic Diagram 183 7.3 A Procedure for the Reduction of Derivatives in Single-Component Systems 186 7.4 Some Simple Applications 190 7.5 Generalizations: Magnetic Systems 199 8 STABILITY OF THERMODYNAMIC SYSTEM S 203 8.1 Intrinsic Stability of Thermodynamic Systems 203 8.2 Stability Conditions for Thermodynamics Potentials 207 8.3 Physical Consequences of Stability 209 8.4 Le Chatelier’s Principle; The Qualitative Effect of Fluctuations 210 8.5 The Le Chatelier-Braun Principle 212 9 FIRST-ORDER PHASE TRANSITIONS 215 9.1 First-Order Phase Transitions in Single-Component Systems 215 9.2 The Discontinuity in the Entropy-Latent Heat 222 9.3 The Slope of Coexistence Curves; the Clapeyron Equation 228 9.4 Unstable Isotherms and First-Order Phase Transitions 233 9.5 General Attributes of First-Order Phase Transitions 243 9.6 First-Order Phase Transitions in Multicomponent Systems—Gibbs Phase Rule 245 9.7 Phase Diagrams for Binary Systems 248 10 CRITICAL PHENOMENA 255 10.1 Thermodynamics in the Neighborhood of the Critical Point 255 10.2 Divergence and Stability 261 10.3 Order Parameters and Critical Exponents 263 10.4 Classical Theory in the Critical Region; Landau Theory 265 10.5 Roots of the Critical Point Problem 270 10.6 Scaling and Universality 272 II THE N E R N ST POSTULATE 277 11.1 Nernst’s Postulate, and the Principle of Thomsen and Bertholot 277 11.2 Heat Capacities and Other Derivatives at Low Temperatures 280 11.3 The “ Unattainability” of Zero Temperature 281 12 SUM M ARY OF PRINCIPLES FOR GENERAL SYSTEM S 283 12.2 The Postulates 283 X IV Contents 12.3 The Intensive Parameters 284 12.6 Stability and Phase Transitions 286 12.8 Properties at Zero Temperature 287 13 PROPERTIES OF MATERIALS 289 13.1 The General Ideal Gas 289 13.2 Chemical Reactions in Ideal Gases 292 13.3 Small Deviations from “Ideality”—The Virial Expansion 297 13.4 The “ Law of Corresponding States” for Gases 299 13.