Statistical Thermodynamics For Beginners

496 Pages·2017·3.351 MB·English

by  Howard D Stidham

Statistical
Thermodynamics
for Beginners
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Statistical
Thermodynamics
for Beginners
Howard D Stidham
University of Massachusetts Amherst, USA
World Scientific
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Stidham, Howard D., author.
Title: Statistical thermodynamics for beginners / Howard D. Stidham
(University of Massachusetts Amherst, USA).
Description: Hackensack, NJ : World Scientific, [2016] | Includes bibliographical references.
Identifiers: LCCN 2016040516| ISBN 9789813149922 (hardcover ; alk. paper) |
ISBN 9813149922 (hardcover ; alk. paper) | ISBN 9789813149939 (pbk. ; alk. paper) |
ISBN 9813149930 (pbk. ; alk. paper)
Subjects: LCSH: Statistical thermodynamics.
Classification: LCC QC311.5 .S485 2016 | DDC 536/.7015195--dc23
LC record available at https://lccn.loc.gov/2016040516
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
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Preface
This book is intended to serve as a relatively thorough introduc-
tion to statistical mechanics for interested students who can regard
themselves as beginnersinthearea.Itisassumedthatthereaderhas
an abiding interest in mathematics and physics, and has had intro-
ductory courses in advanced calculus, thermodynamics, and quan-
tum mechanics. The book is based on a series of notes developed
over approximately 50 years of teaching the subject to first and sec-
ond year graduate students and to some fortunate undergraduates.
There is much more material than can be reasonably presented in
one semester, developed bit by bit over the years by the artifice of
reserving the latter part of the course for coverage of material not
included in previous years.
Thebookisorganizedbychapteronly,buthasanunstatedstruc-
ture. The first 5 chapters are concerned with the fundamentals of
statistical mechanics. The next 4 chapters are concerned with calcu-
lation of thermodynamic properties from spectroscopic data. This is
followed by three chapters on polymers of increasing difficulty. The
chapter on excluded volume is presented near the research level. The
remaining four chapters cover quantum statistics, adsorption, heat
capacityofcrystals,andimperfectgasesandliquids,notinthisorder.
Although thebook is intended as a textbook, thereare relatively few
problems at the ends of the chapters. Some chapters have no prob-
lemsatall.Thereasonforthisisthatgraduatestudentsareexpected
v
vi Statistical Thermodynamics for Beginners
to do research, generally working in another field to which statistical
mechanics is not directly applicable, and do not have much time to
solve routine problems. In addition, there are many very good prob-
lemsinotherbooksonstatisticalthermodynamics.McQuarrie’sbook
is an especially rich source of endless problems, should an instructor
feel motivated to augment the text. The problems given are largely
intended to emphasize some point in the text.
A close reading of the book will reveal the great debt owed
T. L. Hill, W. H. Stockmayer, Marshall Fixman and E. A. DiMarzio.
In particular, the lectures given by Marshall at Yale in 1965 resulted
in my notes on which I based several developments in the text,
notably the Gaussian probability for the infinite polymer chain,
the transfer method in the helix-coil transition, and the argument
for ensemble equivalence. It is probably impossible to surpass Hill’s
Introduction to Statistical Thermodynamics, and this book does not
try. But Hill’s book is out of print, and is no longer useful as a text-
book since it is of undependable availability. There are a number of
textbooks presently available, each with its own strengths and weak-
nesses. This book is a unique addition, containing some material not
easily available elsewhere.
Any serious student of statistical mechanics will sooner or later
run into a need to consult some of the classic works in the area. A
brief list of some of the prominent books follows. In particular, the
book by Mayer and Mayer has some surprising inclusions that are
not often seen in modern applications, but deserve inspection. In his
lectures on statistical mechanics at Yale in the 1960’s, Lars Onsager
always came to lecture carrying a copy of Mayer and Mayer, which
he never consulted, but which always occupied a prominent place on
the lecture table. It is especially recommended.
The list of books that should be available to a practicing statis-
tical mechanician includes several of a mathematical nature. Those
from my personal collection are:
M. Abramowitz and I. A. Stegun, Nat’l Bur. Stds. AMS 55, 1964,
th
5 Printing.
J. H. Hildebrand, “Advanced Calculus for Engineers,” Prectice-Hall,
New York, 1949.
Preface vii
L. B. W. Jolley, “Summation of Series,” Dover reprint.
E. T. Whittaker, “Analytical Dynamics,: Dover reprint.
E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis,”
th
4 Ed., Cambridge, 1927.
In addition, there are a number of books on statistical mechanics
that are useful. These include, but are not limited to, the following:
W.Band,“Introduction toQuantumStatistics,” VanNostrand,New
York, 1955.
M. Born and K. Huang, “Dynamical Theory of Crystal Lattices,”
Oxford, 1954.
D. Chandler, “Introduction to Modern Statistical Mechanics,”
Oxford, 1987.
N.Davidson,“StatisticalMechanics,”McGraw-Hill,NewYork,1962.
P. J. Flory, “Principles of Polymer Chemistry,” Cornell, Ithaca NY,
1953.
R. H. Fowler and E. A. Guggenheim, “Statistical Thermodynamics,”
Cambridge, 1939.
J. W. Gibbs,“The Collected Works of J. Willard Gibbs,” Vols. I and
II, Yale, 1948.
D.terHaar,“ElementsofStatisticalMechanics,”Constable,London,
1954.
nd
D.ter Haar, “Elements of Thermostatistics,” 2 Ed.,Holt, Rinehart
and Winston, New York, 1966.
T. L. Hill, “Statistical Mechanics,” McGraw-Hill, New York, 1956.
T. L. Hill, “Thermodynamics of Small Systems,” Benjamin, New
York, 1963.
T. L. Hill, “Introduction to Statistical Thermodynamics,” Addison-
Wesley, Reading, MA, 1960.
T. L. Hill, “Cooperativity Theory in Biochemistry,” Springer Verlag,
New York, 1985.
J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, “Molecular Theory
of Gases and Liquids,” Wiley, New York, 1954.
L. D. Landau and E. M. Lifshitz, “Statistical Physics,” Pergamon,
London, 1958.
viii Statistical Thermodynamics for Beginners
J. E. Mayer and M. G. Mayer, “Statistical Mechanics,” Wiley, New
York, 1940.
D. A. McQuarrie, “Statistical Mechanics,” Harper and Row, New
York, 1976.
D. A. McQuarrie, “Statistical Thermodynamics,” Harper and Row,
New York, 1973.
L. K. Nash, “Elements of Statistical Thermodynamics,” Addison-
Wesley, Reading MA, 1968.
G. S. Rushbrooke, “Introduction to Statistical Mechanics,” Oxford,
1949.
E. Schrodinger, “Statistical Mechanics,”Cambridge, 1960.
J. C. Slater, “Introduction to Chemical Physics,” McGraw-Hill, New
York, 1939.
nd
M. Toda, R. Kubo, N. Saito, “Statistical Physics,” 2 Ed., Springer
Verlag, Berlin, 1983, 1992.
R. C. Tolman, “Principles of Statistical Mechanics,” Oxford, 1938.
G. H. Wannier, “Elements of Solid State Theory,” Cambridge, 1959.
One important area not covered in the first edition of this book is
phase transitions. The interested reader may find it useful to consult
good textbooks of physical chemistry for the groundwork and then
graduate to such books as the following:
H. E. Stanley, “Introduction to Phase Transitions and Critical Phe-
nomena,”Oxford, 1971.
J.M.Yoemans,“StatisticalMechanicsofPhaseTransitions,”Claren-
don Press, Oxford, 1992.
“New Kinds of Phase Transitions,” V. V. Brazhkin, S. V. Buldyrev,
V. N. Ryzhov, H. E. Stanley, Eds., NATO Science Series,
Kluwer, Dordrecht/Boston, London, 2002.
Amherst, Massachusetts
May 2016
Contents
Preface v
Chapter 1. Review of Classical Mechanics
and Thermodynamics 1
Chapter 2. The Basis of Statistical Mechanics 19
Chapter 3. Steepest Descent 47
Chapter 4. Gaussian Distribution, Ensemble
Equivalence and Fluctuations 61
Chapter 5. A Variational Principle, Phase Space,
Classical Averaging and Ω 77
Chapter 6. Gases 91
Chapter 7. Homonuclear Diatomic Molecules 113
Chapter 8. Polyatomic Molecules 137
ix
x Statistical Thermodynamics for Beginners
Chapter 9. Residual Entropy and Chemical Equilibrium 165
Chapter 10. Polymers 183
Chapter 11. More Polymers 213
Chapter 12. Excluded Volume of Polymer Chains 239
Chapter 13. Imperfect Gases and Liquids 305
Chapter 14. Bose-Einstein, Fermi-Dirac and
Boltzmann Statistics, Quantum Gas 341
Chapter 15. Adsorption 371
Chapter 16. Heat Capacities of Solids 431
Index 481