An Introduction to Statistical Thermodynamics

Terrell L. Hill

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486130903
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

"A large number of exercises of a broad range of difficulty make this book even more useful…a good addition to the literature on thermodynamics at the undergraduate level." — Philosophical Magazine
Although written on an introductory level, this wide-ranging text provides extensive coverage of topics of current interest in equilibrium statistical mechanics. Indeed, certain traditional topics are given somewhat condensed treatment to allow room for a survey of more recent advances.
The book is divided into four major sections. Part I deals with the principles of quantum statistical mechanics and includes discussions of energy levels, states and eigenfunctions, degeneracy and other topics. Part II examines systems composed of independent molecules or of other independent subsystems. Topics range from ideal monatomic gas and monatomic crystals to polyatomic gas and configuration of polymer molecules and rubber elasticity. An examination of systems of interacting molecules comprises the nine chapters in Part Ill, reviewing such subjects as lattice statistics, imperfect gases and dilute liquid solutions. Part IV covers quantum statistics and includes sections on Fermi-Dirac and Bose-Einstein statistics, photon gas and free-volume theories of quantum liquids.
Each chapter includes problems varying in difficulty — ranging from simple numerical exercises to small-scale "research" propositions. In addition, supplementary reading lists for each chapter invite students to pursue the subject at a more advanced level. Readers are assumed to have studied thermodynamics, calculus, elementary differential equations and elementary quantum mechanics.
Because of the flexibility of the chapter arrangements, this book especially lends itself to use in a one-or two-semester graduate course in chemistry, a one-semester senior or graduate course in physics or an introductory course in statistical mechanics.

Part I

Principles of Quantum Statistical Mechanics

CHAPTER 1

STATISTICAL-MECHANICAL ENSEMBLES AND THERMODYNAMICS

1-1 Introduction.

The object of thermodynamics is to derive mathematical relations which connect different experimental properties of macroscopic systems in equilibrium—systems containing many molecules, of the order of, say, 10²⁰ or more. However useful, these interconnections of thermodynamics give us no information at all concerning the interpretation or explanation, on a molecular level, of the observed experimental properties. For example, from thermodynamics we know that experimental values of the two heat capacities C_p and C_v for a given system must be interrelated by an exact and well-known equation, but thermodynamics is unable to furnish any explanation of why particular experimental values of either C_p or C_v, taken separately, should be observed. Such an explanation falls rather within the province of statistical mechanics or statistical thermodynamics, terms which we regard in this book as synonymous. That is, the object of statistical mechanics is to provide the molecular theory or interpretation of equilibrium properties of macroscopic systems. Thus the fields covered by statistical mechanics and thermodynamics coincide. Whenever the question “why?” is raised in thermodynamics—why, for example, a given equilibrium constant, Henry’s law constant, equation of state, etc., is observed—we are presented with a problem in statistical mechanics.

Although thermodynamics itself does not provide a molecular picture of nature, this is not always a disadvantage. Thus there are many complicated systems for which a molecular theory is not yet possible; but regardless of complications on the molecular level, thermodynamics can still be applied to such systems with confidence and exactness.

In recent years both thermodynamics and statistical mechanics have been extended somewhat into the nonequilibrium domain. However, the subject is new and changing, and the foundations are still a little shaky; hence we omit this area from our consideration. An exception is the theory of absolute reaction rates, which we discuss in Chapter 11. This approximate theory is based on a quasi-equilibrium approach which makes it possible to include the theory within the framework of equilibrium statistical mechanics.

Aside from the postulates of statistical mechanics themselves, to be introduced in the next section, the foundation on which our subject is based is quantum mechanics. If we seek a molecular interpretation of the properties of a system containing many molecules, as a starting point we must certainly be provided with knowledge of the properties of the individual molecules making up the system and of the nature of the interactions between these molecules. This is information which can in principle be furnished by quantum mechanics but which in practice is usually obtained from experiments based on the behavior of individual molecules (e.g., spectroscopy), pairs of molecules (e.g., the second virial coefficient of an imperfect gas), etc.

Although quantum mechanics is prerequisite to statistical mechanics, fortunately a reasonably satisfactory version of statistical mechanics can be presented without using any quantum-mechanical concepts other than those of quantum-mechanical states, energy levels, and intermolecular forces. Only in Part IV of the book is it necessary to go beyond this stage.

Another very helpful simplification is that the classical limit of quantum mechanics can be used, without appreciable error, in most problems involving significant intermolecular interactions. Problems of thi...

Table of contents

• DOVER BOOKS ON PHYSICS
• Title Page
• Copyright Page
• Dedication
• PREFACE
• Table of Contents
• Part I - Principles of Quantum Statistical Mechanics
• Part II - Systems Composed of Independent Molecules or Subsystems
• Part III - Systems of Interacting Molecules
• Part IV - Quantum Statistics
• APPENDIX I - NATURAL CONSTANTS
• APPENDIX II - MAXIMUM-TERM METHOD
• APPENDIX III - METHOD OF UNDETERMINED MULTIPLIERS
• APPENDIX IV - THE LENNARD-JONES POTENTIAL
• APPENDIX V - NORMAL COORDINATE ANALYSIS IN A SPECIAL CASE
• APPENDIX VI - VIBRATIONAL FREQUENCY DISTRIBUTION IN A SOLID CONTINUUM
• APPENDIX VII - GENERALIZED COORDINATES
• INDEX
• A CATALOG OF SELECTED DOVER BOOKS IN SCIENCE AND MATHEMATICS