The basic concepts of classical thermodynamics can be summarized by invoking the following four postulates (Callen, 1985):

1. There exist particular states (called equilibrium states) of simple compressible systems that, macroscopically, are characterized completely by the internal energy, U, the volume, V, and the mole or particle numbers, of the chemical components.

2. There exists a function called the entropy, S, of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those which maximize the entropy for the composite isolated system.

3. The entropy of a composite system is additive over the constituent subsystems. Moreover, the entropy is a continuous, differentiable, and monotonically increasing function of the internal energy.

4. The entropy of any system vanishes in the state for which (i.e., at the zero of temperature).

Recall that a simple compressible system is defined as one that is macroscopically homogeneous, uncharged, and chemically inert, that is sufficiently large that surface effects can be neglected, and that is not acted on by electric, magnetic, or gravitational fields. Although these four basic postulates are restricted to simple compressible systems, they can readily be extended to more complex systems (Lewis and Randall, 1961).

My intention in this textbook is to provide a self-contained exposition of the fundamentals and applications of statistical thermodynamics for beginning graduate students in the engineering sciences. Especially within engineering, most students enter a course in statistical thermodynamics with limited exposure to statistics, quantum mechanics, and spectroscopy. Hence, I have found it necessary over the years to “start from the beginning,” not leaving out intermediary steps and presuming little knowledge in the discrete, as compared to the continuum, domain of physics. Once these things are done carefully, I find that good graduate students can follow the ideas, and that they leave the course excited and satisfied with their newfound understanding of both statistical and classical thermodynamics.

Nevertheless, a first course in statistical thermodynamics remains challenging and sometimes threatening to many graduate students. Typically, all their previous experience is with the equations of continuum mechanics, whether applied to thermodynamics, fluid mechanics, or heat transfer. For most students, therefore, the mathematics of probability theory, the novelty of quantum mechanics, the confrontation with entropy, and indeed the whole new way of thinking that surrounds statistical thermodynamics are all built-in hills that must be climbed to develop competence and confidence in the subject. For this reason, although I introduce the ensemble method at the beginning of the book, I have found it preferable to build on the related Maxwell–Boltzmann method so that novices are not confronted immediately with the conceptual difficulties of ensemble theory.

To this point, we have dealt exclusively with systems composed of independent particles and thus we have utilized the Maxwell–Boltzmann method of statistical thermodynamics. We know, however, that at a sufficiently high pressure or low temperature any gas will begin demonstrating nonideal behavior. For such real gases, and also for liquids, centralized forces arise among constituent particles owing to shorter intermolecular distances. Consequently, we are eventually confronted with systems composed of dependent rather than independent particles. Such systems mandate that we forsake the Maxwell–Boltzmann method and turn instead to a more robust computational procedure known as the Gibbs or ensemble method of statistical thermodynamics.

The Ensemble Method

We recall from Section 3.2 that an ensemble is a mental collection of a huge number of identical systems, each of which replicates macroscopically the thermodynamic system under investigation. Because such replication occurs at the macroscopic and not at the microscopic level, every member of the ensemble may be associated with a possibly different system quantum state. In essence, the independent particles required for the Maxwell–Boltzmann method are replaced with independent systems for the ensemble method. As a result, when using the latter, we inherently retain independent events proffered for statistical analyses, while accounting for the intermolecular forces needed to model real gases and liquids. In so doing, we shift our focus from a consideration of particle quantum states to a consideration of system quantum states.

As discussed previously, the utility of the Gibbs method rests on two fundamental postulates of statistical thermodynamics.

We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the εj's and gj's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics.

We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.

As indicated in Chapter 18, ensemble theory is especially germane when calculating thermodynamic properties for systems composed of dependent rather than independent particles. Potential applications include real gases, liquids, and polymers. In this chapter, we focus on the thermodynamic properties of nonideal gases. Our overall approach is to develop an equation of state using the grand canonical ensemble. From classical thermodynamics, equilibrium properties can always be determined by suitably operating on such equations of state. As shown later in this chapter, typical evaluations require an accurate model for the intermolecular forces underlying any macroscopic assembly. This requirement is endemic to all applications of ensemble theory, including those for liquids and polymers. As a matter of fact, by mastering the upcoming procedures necessary for the statistics of real gases, you should be prepared for many pertinent applications to other tightly-coupled thermodynamic systems.

The Behavior of Real Gases

As the density of a gas rises, its constituent particles interact more vigorously so that their characteristic intermolecular potential exercises a greater influence on macroscopic behavior. Accordingly, the gas becomes less ideal and more real; i.e., its particles eventually display greater contingency owing to enhanced intermolecular forces. This deviation from ideal behavior is reflected through a more complicated equation of state for real gases.

An equation of state, you recall, describes a functional relation among the pressure, specific volume, and temperature of a given substance.

Introduction

In the last chapter, we used the two-phase model of suspensions to consider several examples of sound propagation in suspensions. The examples were chosen to illustrate the main aspects of the propagation of sound waves, but they refer to some physically unrealistic conditions, such as the isothermal aerosol. Of course, the results obtained for those conditions may apply in limited portions of the frequency range; but, generally speaking, they leave out important effects. For example, as we saw in Chapter 6, the neglect of thermal effects on the pulsations of bubbles is not warranted at frequencies that are below resonance.

From a fundamental point of view, it is desirable to study sound propagation in suspensions without imposing such restricting conditions. To be sure, the two-phase model allows such a study, but the inclusion of all of the effects that play a role in the propagation would require the two complete sets of equations. While the system can be easily solved for linear motions, it is evident that its solution would generally require numerical computations and would produce results difficult to interpret. Furthermore, results valid for a wide range of frequencies would be obtained only if the particulate force and heat transfer rate are specified in their most general forms, thus increasing the complexity of the system.

In this chapter, we consider the propagation of sound waves in dilute suspensions using various techniques, models, and approximations.