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ERRATA On p. 22, the exponent in equation (4) should be divided by the number 2—lost in the foregoing derivation by a careless approximation. On p. 87 line 4 slightly underestimates the total number of rotational quantum states. Copyright Copyright © 1965, 1966, 1974 by Leonard K. Nash All rights reserved. Bibliographical Note This Dover edition, first published in 2006, is an unabridged republication of the 1974 second edition of the work originally published in 1965 by Addison-Wesley Publishing Company, Inc., Reading, Massachusetts. International Standard Book Number: 9780486137469 Manufactured in the United States by Courier Corporation 44978502 www.doverpublications.com Preface Macroscopic thermodynamics is here reexamined from the perspective of atomic-molecular theory. The thermodynamic concepts of entropy and equilibrium are thus invested with new meaning and implication, and one comes to see how thermodynamic magnitudes (e.g., gaseous heat capacities and equilibrium constants) can be calculated from spectroscopic data. Chapter 1 introduces and develops a statistical analysis of the concept of distribution—culminating in a very simple derivation of the Boltzmann distribution law and a demonstration of the relation of statistical and thermodynamic entropies. Chapters 2, 3, and 4 then treat in turn the formulation, the evaluation, and the application of partition functions. Compared with the first edition, the present second edition offers in its opening chapter an analysis that is both much simpler and more decisive. This chapter also provides a brief but convincing demonstration of one crucial point merely announced in the first edition, namely: the enormous odds by which some distributions are favored over others. The remaining chapters reflect complete reorganization and extensive rewriting of the corresponding material in the first edition, and further incorporate several major additions. These include some illuminating interpretations of partition functions and statistical entropies, a brief but substantial development of the statistics of indistinguishable units, an exploration of the “dilute-gas condition” under which these statistics reduce to a limiting form equivalent to “corrected” Boltzmann statistics, and a derivation of the Maxwell-Boltzmann molecular-speed distribution law in three dimensions. Enlarged by 70%, the set of problems should both challenge and reward the reader. This text has been designed to convey its message at either of two levels. First: if systematically supported by a series of exegetical lectures, the book may be used in strong introductory college chemistry courses. Years of experience in teaching such a course convince me that, wherever classical thermodynamics can be taught successfully, this much statistical thermodynamics can be taught at least as successfully—and with striking reflexive improvement in student understanding of classical thermodynamics. Second: the book lends itself to essentially independent study by more advanced students who seek a view of the subject that is less formal, and far less compressed, than that afforded by undergraduate physical-chemistry texts. Irrespective of background, the reader will find need for just three small bodies of prior knowledge. 1. A rudimentary knowledge of the calculus, such as is acquired by my own freshmen in the first two-thirds of an introductory course in college mathematics. 2. An elementary understanding of macroscopic thermodynamics, on which we draw for a few simple relations like dE = T dS – P dV. Many current texts for the introductory college chemistry course offer a quite sufficient foundation in this area. 3. A slight acquaintance with the energy-quantization conditions that permit calculation of molecular parameters from spectroscopic measurements. The inclusion of some abbreviated didactic material lends the present book a minimal self-sufficiency in this department, but one may well prefer the more ample background afforded, for example, by pp. 1-80 of G. M. Barrow’s Structure of Molecules (Benjamin, 1962). The argument constructed on these modest foundations comprises essentially all the statistical mechanics that appears in even the most sophisticated of undergraduate physical – chemistry textbooks. I have neglected internal rotation, and I have not pursued the argument into the realm of kinetics—where it readily yields some powerful new insights. More significant than these easily remediable omissions is one notable limitation: applying to assemblies of effectively independent units, the results here obtained from analysis of microcanonical ensembles cannot at once be extended to assemblies of strongly interacting units. Given a sufficiently enlarged background, one may easily approach these important assemblies by way of Gibbsian analyses (of canonical and ground canonical ensembles) that fully display both the great power and the great beauty of statistical mechanics. Among the many equations appearing herein, some are marked by letters to facilitate back-references in the immediately succeeding text, or in a problem. On the other hand, apart from facilitating such back-references, the numbers attached to thirty-odd equations signal a call for particular attention. As they proceed, readers would be well advised to compile their own lists of these numbered equations, each of which expresses an important idea and/or represents a useful computational tool. Cambridge, Massachusetts L.K.N. August 1973. Table of Contents Title Page Copyright Page Preface Acknowledgments 1 - The Statistical Viewpoint 2 - The Partition Function 3 - Evaluation of Partition Functions 4 - Applications Problems Index Acknowledgments I am obliged to the publishers cited below for permission to reproduce a number of figures taken from copyrighted works. As identified by the numbers assigned them in this book, the figures in question are: Figure 7, taken from p. 55 of J. D. Fast’s Entropy (Eindhoven, Holland: Philips Technical Library, 1962); Figure 16, from p. 68 of G. W. Castellan’s Physical Chemistry, 2nd ed. (Reading, Mass.: Addison-Wesley Publishing Co., 1971); Figure 19, from an article by R. K. Fitzgerel and F. H. Verhoek, Journal of Chemical Education 37, 547 (1960); Figure 22, from p. 142 of Malcolm Dole’s Introduction to Statistical Thermodynamics (Englewood Cliffs, N.J.: Prentice-Hall, 1954); Figures 23 and 24, from pp. 203 and 257 respectively of Statistical Thermodynamics by J. F. Lee, F. W. Sears, and D. L. Turcotte (Reading, Mass.: Addison-Wesley Publishing Co., 1963); and Figure 25, from p. 102 of Statistical Thermodynamics by R. H. Fowler and E. A. Guggenheim (Cambridge: University Press, 1939). I am happy to acknowledge that my first 18 pages have been developed on a pattern suggested by reading of the late Ronald W. Gurney’s ingenious introductory text. The arguments on pp. 49 – 53 and 61 – 62 owe some of their shape to constructive criticisms of the first edition forwarded to me by William C. Child, Jr., who has kindly supplied several more comments on a draft of this second edition. For a multitude of additional suggestions I am much indebted to Francis T. Bonner, Peter C. Jordan, and Lawrence C. Krisher. To Walter Kauzmann, the author of a splendid alternative to the present text, I am deeply grateful for his generous willingness to indicate