Elements of Statistical Thermodynamics

Second Edition

Leonard K. Nash

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486449784
• eBook ISBN: 9780486137469
• Topic: Physical Sciences
• Subtopic: Chemistry
• Index: Chemistry

This concise, elementary treatment illustrates the ways in which an atomic-molecular perspective yields new insights and powers operative in the realms of macroscopic thermodynamics. Starting with an analysis of some very simple microcanonical ensembles, it proceeds to the Boltzmann distribution law and a systematic exploration of the proper formulation, evaluation, and application of partition functions. The concepts of equilibrium and entropy thus acquire new significance, and readers discover how thermodynamic parameters may be calculated from spectroscopic data.
Encompassing virtually all of the forms of statistical mechanics customary to undergraduate physical chemistry books, this brief text requires prior acquaintance with only the rudiments of the calculus and a few of the simplest propositions of classical thermodynamics. Appropriate for introductory college chemistry courses, it further lends itself to use as a supplementary text for independent study by more advanced students.

1

The Statistical Viewpoint

In every change, however drastic it may appear, we surmise a “something” that remains constant. From the very beginning of the modern era, certain men (e.g., Descartes) have conceived that “something” in terms suggestive of what we would call energy. And energy—or, better, mass-energy-is surely conceived by us as a “something constant” enduring through all change. The energy concept thus gives quantitative expression to our firm conviction that “plus ça change, plus c’est la même chose.” But we have too another conviction scarcely less intense: the conviction that the future will not repeat the past, that time unrolls unidirectionally, that the world is getting on. This second conviction finds quantitative expression in the concept of entropy (from Gr. en, in + trope, turning). By always increasing in the direction of spontaneous change, entropy indicates the “turn,” or direction, taken by all such change.

From a union of the entropy and energy concepts, little more than a century ago, there was born a notably abstract science with innumerable concrete applications; a science of thermodynamics that combines magnificent generality with unfailing reliability to a degree unrivaled by any other science known to man. Yet, for all its immense power, thermodynamics is a science that fails to reward man’s quest for understanding. Yielding impressively accurate predictions of what can happen, thermodynamics affords us little or no insight into the why of those happenings. Thus it permits us to calculate what is the position of equilibrium in the system N₂—H₂—NH₃, for example, but it fails entirely to tell us why that is the equilibrium condition for this specific system.

To be sure, given that certain thermodynamic parameters (the “free energies”) are what they are, we readily see that a particular equilibrium condition is entailed. But we can find in thermodynamics no explanation of why the free energies are what they are. And in general, though thermodynamics teaches us to see important relations among the various macroscopic properties of a substance, so that many can be calculated from experimental measurements of a few, thermodynamics is powerless to produce from its own calculations numerical values for the few.

What is it about NH₃ that determines the magnitude of the free-energy characteristic of that compound? In principle this question should, we feel, be answerable. But we find scant prospect of any such answer in a classical thermodynamics which, focusing solely on the properties of matter in bulk, eschews all concern with the microcosmic constitution of matter. For consider that we can hope to explain the free energy of some substance only by showing how that particular free energy is entailed by the distinctive values of the atomic and/or molecular parameters of the substance. That is, given a (spectroscopic) determination of such parameters as the length, angle, and flexibility of the bonds in NH₃, we must be able to see that the free energy of NH₃ could not be other than it is. This will be possible only if we can bridge the gap between the microcosmic realm of atoms and molecules and the macroscopic realm of classical thermodynamics.

Statistical mechanics provides such a bridge, by teaching us how to conceive a thermodynamic system as an assembly of units. More specifically, it demonstrates that the thermodynamic parameters of the system are interpretable in terms of—and are indeed calculable from—the parameters descriptive of such constituent units as atoms and molecules. In a bounded system, the crucial characteristic of these microcosmic units is that their energies are “quantized.” That is, where the energies accessible to a macroscopic system form a virtual continuum of possibilities, the energies open to any of its submicroscopic components are limited to a discontinuous set of alternatives associated with integral values of some “quantum number.”

Perhaps the most familiar example of what is meant by quantization is presented by the Bohr interpretation of the hydrogen emission spectrum. This spectrum consists of a series of sharp “lines,” characterized by particular wavelengths. Each of these lines is supposed to arise in the emission by the hydrogen atom of an energy packet of some particular size. Such an energy packet is emitted when the atom passes from a state of higher energy to one of lower energy. From a study of the sizes of the emitted energy packets, one infers that the atom can exist only in a certain well-defined set of quantum states. The energy (ε_H) associated with any of these permissible states is given by the equation:

Here h symbolizes Planck’s universal constant, m and e respectively represent the mass and charge of the “orbital” electron in the hydrogen atom, and n is a quantum number that can assume any integral value within the range 1 to oo. The possible states of the hydrogen atom, each characterized by some integral value of the quantum number n, are thus linked with the discontinuous set of permissible energies given by the last equation—which expresses the energy-quantization condition for the hydrogen atom. Rather more complicated relations, involving additional quantum numbers, express analogous energy-quantization conditions applicable to other species of gas...

Table of contents

• Title Page
• Copyright Page
• Preface
• Table of Contents
• Acknowledgments
• 1 - The Statistical Viewpoint
• 2 - The Partition Function
• 3 - Evaluation of Partition Functions
• 4 - Applications
• Problems
• Index