Statistical Thermodynamics for Beginners

Howard D Stidham

• Editore: WSPC
• Anno: 2017
• ISBN:
  9789813149953
• Argomento: Naturwissenschaften
• Categoria: Mathematische & Computerphysik

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This book is the distilled essence of the author teaching statistical mechanics to juniors, seniors and graduate students for over 50 years in various course settings. It uses a unique approach that leads naturally into the development of all possible ensembles. Much of the later chapters on polymers has previously been available only in the literature. Throughout the book, the assumption is made that the reader is still relatively raw, and mathematical detail is provided that other books leave to the abilities of the reader. While this produces a plethora of equations that mature scientists would regard as unnecessary, it is intended to help those just coming into the field and who want to get the idea without suffering hours of agony wondering, "where did that come from?".

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Chapter 1

Review of Classical Mechanics and Thermodynamics

1.1.Introduction

Spectroscopy and statistical mechanics developed nearly simultaneously. Max Planck developed the theory of the black body radiator, introducing for the first time the idea of quantized oscillators. A little later, Albert Einstein took the idea of quantized oscillation and developed an essentially correct theory of the heat capacity of substances. Independently of the European intellectual ferment, J. Willard Gibbs in the USA introduced the ensemble and investigated its properties. By 1906, all of these theoretical constructs were in the literature, waiting for applications. Classical mechanics was fully developed, but quantum mechanics took another 20 years to appear. Thermodynamics had been long since summarized, and produced many exact equivalences. For example, the constant pressure and constant volume heat capacities of substances are related by a well-known formula involving commonly measured quantities such as the bulk modulus of elasticity (the reciprocal of the isothermal compressibility) and the coefficient of thermal expansion. However, thermodynamics provided no insight into the reasons for the behavior of bulk matter, only relationships between various coefficients. Quantum mechanics provided a basis for understanding spectroscopy, but the understanding was, and is, model dependent. Thermodynamics is model independent. The object of statistical mechanics is to provide a model independent bridge between the behavior of bulk matter and that of the manifold quantum systems of which bulk matter is composed. In order to do this, it is necessary to assume that the ensemble average is equivalent to the long time average of a measurement, something called the ergodic hypothesis. This is a reasonable assumption, but it is one that is unprovable. Ultimately, the justification for the use of the ergodic hypothesis is that the statistical mechanical formulas to which it leads actually do produce quantitative agreement with experiment. It is interesting that the quantum behavior of isolated molecules is completely suppressed in bulk matter. The reason for this is partly that individual quantum systems fluctuate about a central value when in bulk, and that it takes an astronomical number of molecules to make up one mole of bulk matter. The fact that this number is so large is fortunate, as it leads to statistically valid averaging procedures that solve the problem of calculating the properties of bulk matter from those of the quantum systems on which these are based.

This book assumes a reasonable introduction to thermodynamics, classical and quantum mechanics on the part of the reader. The book starts with a brief review of classical mechanics, then of thermodynamics. The next several chapters are concerned with the postulates of statistical mechanics. Applications begin with gases, heteratomic diatomic molecules, homonuclear diatomic molecules and polyatomic molecules, in which the quantum mechanics and spectroscopy are regarded as completed studies. Three chapters of graded difficulty follow on polymers, then further investigations of the structure of statistical mechanics with the introduction of phase space and quantum statistics. Imperfect gases and liquids are briefly considered, and applications to chemical equilibrium, adsorption, and solid state finish the book.

1.2.Review of Classical Mechanics

The classical equations of motion are familiar in Cartesian coordinates, but usually less so in either LaGrange’s or Hamilton’s formulation. A brief study of the relationship between these different ways of essentially saying the same thing is often profitable.

In Cartesian coordinates, the kinetic energy and any potential energy that may be acting in the system are expressed in either coordinates x_i or velocities v_i. It is convenient to express the veloci...