The statistical and classical thermodynamics framework developed in Chapters 1–8 of this text is based on analysis of systems at equilibrium. In Chapter 9 we explore the extension of this framework to systems that are not in equilibrium. This chapter focuses on systems that exhibit steady spatial variations of properties. Systems of this type are modeled as having local thermodynamic equilibrium and obeying a linear relation between fluxes and affinities. Analysis of microscale features of such linear systems is shown to link correlation moments and kinetic coefficients. The Onsager reciprocity relations are subsequently derived. Thermoelectric effects are examined as an example application of the nonequilibrium linear theory developed in this chapter.

Properties in Nonequilibrium Systems

The thermodynamic theoretical framework developed in previous chapters of this text is limited to analysis of equilibrium states. Often, however, it is the process that takes the system from one state to another that is of primary interest. Overall changes accomplished during the process can be determined by analyzing the initial and final states using equilibrium thermodynamics. If the process is very slow, it may be well approximated by a sequence of equilibrium states, and a quasistatic model may adequately predict the outcome of the process.

In many real processes, the departure from equilibrium is so severe that the quasistatic model is too inaccurate to be useful. The objective of this chapter is to develop thermodynamic tools that can be applied to irreversible processes in nonequilibrium systems.

Using the basic features of microscale energy storage discussed in Chapter 1, in Chapter 2 we develop the foundations of statistical thermodynamics. In doing so, we introduce the concepts of microstates and macrostates and properly account for the fact that particles in fluid systems are generally indistinguishable. The development of the theoretical framework in this and subsequent chapters considers a binary mixture of two particle types. The statistical machinery is applied first to a microcanonical ensemble of systems, each having a specified volume, number of particles, and total internal energy. Definitions of entropy and temperature emerge from this development. Application of the results to a monatomic gas is discussed.

Microstates and Macrostates

In this chapter we will construct a general statistical mechanics foundation on which we will develop a full equilibrium thermodynamic theory for systems composed of a large number of particles. In doing so we will make use of the information about energy storage derived from quantum theory in the previous chapter.

In analyzing systems of particles, we can deal with the state of a system at two levels: the microstate of the system and the macrostate of the system. The system microstate is the detailed configuration of the system at a microscopic level. To specify the microstate we would have to specify the quantum state (including the position) of each particle in the system. If we observe a system at a macroscopic level, we can, at best, distinguish some of the gross characteristics of the system.

The structure of this book is designed to facilitate coherent development of classical and statistical thermodynamic principles. The book begins with coverage of microscale energy storage mechanisms from a modern quantum mechanics perspective. This information is then incorporated into a statistical thermodynamics analysis of many-particle systems with fixed internal energy, volume, and number of particles. From this analysis emerges the definitions of entropy and temperature, the extremum principle form of the second law, and the fundamental relation for the system properties. The third chapter takes the concepts derived from the statistical treatment and uses mathematical techniques to expand the macroscopic thermodynamics framework. By the end of the third chapter, the full framework of classical thermodynamics is established, including definitions of all commonly used thermodynamic properties, relations among properties, different forms of the second law, and the Maxwell relations.

In the fourth chapter, statistical ensemble theory is covered, building on the initial statistical treatment in Chapter 2 and the expanded macroscopic framework developed in Chapter 3. The canonical ensemble and grand canonical ensemble formalisms are developed, and the relations developed from these formalisms are used to explore the significance of fluctuations in thermodynamics systems. By the end of the fourth chapter all the fundamental elements of classical and statistical thermodynamics have been established. Chapters 5–7 deal with applications of equilibrium statistical thermodynamics to solid, liquid, and gas phase systems.

The final three chapters of the text cover thermal phenomena that involve nonequilibrium and/or noncontinuum effects.

The basic elements of statistical thermodynamics were developed in Chapter 2. In this chapter, we digress briefly from development of the statistical theory to expand the theoretical framework using mathematical tools and macroscopic analysis. By doing so we more strongly link the statistical theory to classical thermodynamics and set the stage for alternative statistical viewpoints considered in Chapter 4.

Necessary Conditions for Thermodynamic Equilibrium

In the previous chapter, we have derived several important pieces of information about thermodynamic systems. The goal of this chapter is to expand the framework of macroscopic thermodynamic theory so that it can be applied effectively to a variety of system types. We will begin by summarizing the important ideas developed in the last chapter.

So far, we have taken the volume V, internal energy U, and particle numbers Na and Nb, to be intrinsic properties for any system we may consider. We subsequently defined the properties entropy S, temperature T, pressure P, and chemical potentials µa and µb. Our analysis of the statistical characteristics of thermodynamic systems has led to the conclusion that for a system with fixed U, V, Na, and Nb, equilibrium corresponds to a maximum value of the system entropy. This is referred to as the entropy maximum principle. The entropy of a composite system with an arbitrary number of subsystems is additive over the constituent subsystems. This is the additivity property of entropy.

Chapter 7 demonstrates the application of statistical thermodynamics theory to crystalline solids. Because of its relevance to electron transport in metallic crystalline solids, the electron gas theory for metals is also described in this chapter. This chapter provides only an introduction to the microscale thermophysics of solids. Readers interested in more comprehensive treatments of solid state thermophysics should consult the references cited at the end of this chapter.

Monatomic Crystals

Our objective here is to use statistical thermodynamics tools to evaluate thermodynamic properties of solid crystals. Our first goal is to derive a relation for the partition function Q. In doing so, we will specifically consider the structure of a monatomic crystal. One approach is to model the crystal as a system of regularly spaced masses and springs as indicated schematically in Figure 7.1. The springs represent the interatomic forces that each atom experiences. The mean locations of the masses are at regularly spaced lattice points.

Actually, each atom sits in a potential well whose minimum is at a lattice point. The potential well for each atom is usually very steep. Each atom vibrates about its equilibrium position with a small amplitude, which suggests that we can work with a Taylor series representation of the potential valid near the equilibrium point.

Rather than working with the potential for a single atom, we will consider the potential for the crystal as a whole, which we will designate as Φ.

Chapter 8 approaches the topics of phase equilibrium and phase transitions from a microscale perspective. Specifically, the roles of fluctuations and system stability in the onset of phase transitions are examined in detail. Using aspects of statistical thermodynamics theory developed in earlier chapters, the van der Waals model is used to demonstrate how fluctuations and system instability give rise to phase transitions in fluid systems. Binary fluid systems are considered, with pure fluid results being recovered when the mass of one species is set to zero. It is shown that critical exponents and the law of corresponding states can be deduced from the van der Waals model for pure fluids. Microscale aspects of solid–liquid transitions are also considered.

Fluctuations and Phase Stability

In the development of thermodynamics presented in the preceding chapters, we have identified different categories of substances according to the density of the substance and the nature of molecular interactions in the substance. Gases have low density and the molecules spend most of the time traveling through space with momentum and energy being exchanged between molecules only through brief collisions. In liquids, the molecules are free to roam about within the system but the density is much higher than in gases, with the mean distance between adjacent molecules being only one to two molecular diameters. Because the molecules are close to their neighbors, they continuously are subject to force interactions with nearby molecules, resulting in continuous exchange of momentum and energy.

This text differs from most statistical thermodynamics textbooks in that it does not deal exclusively with statistical aspects of thermodynamics. Instead, it attempts to weave together statistical and classical elements to develop the full theoretical framework of thermodynamics. Chapter 1 begins this development by establishing the basic features of energy storage at the atomic and molecular levels. It contains a very short introduction to basic aspects of quantum mechanics. The quantum models discussed in this chapter are models of energy storage modes found in common molecules. Conclusions regarding energy levels and their degeneracy for these modes of energy storage are cornerstones of the statistical thermodynamic theory developed in later chapters.

Microscale Energy Storage

Since this text is designed for graduate-level engineering instruction, it is likely that the reader has already encountered some elements of thermodynamics in previous courses and very likely that he or she has some idea of the usefulness of thermodynamic analysis for systems of scientific and technological interest. Mechanical or chemical engineers who are thoroughly versed in classical equilibrium thermodynamics may wonder what a statistical development of thermodynamics has to offer beyond the tools provided by classical thermodynamics.

In response to such an inquiry, we can identify two main benefits of developing a statistical thermodynamic theory. First, by design, statistical thermodynamics theory provides a link between macroscopic “classical” thermodynamic analysis of system behavior and the microscopic characteristics of the atoms, molecules, or subatomic particles that make up the system.

The ideal gases considered in Chapter 5 are arguably the simplest fluids encountered in real systems. The behavior of dense gases, liquids, and quantum fluids deviates strongly from that of an ideal gas. In this chapter, we examine how the thermodynamic framework developed in earlier chapters can be applied to such fluids. The van der Waals model for dense gases and liquids is explored in detail for pure and binary mixture systems. In doing so, we demonstrate that the statistical thermodynamic framework provides a link between microscopic characteristics of the molecules or particles and the macroscopic behavior of these fluids.

Behavior of Gases in the Classical Limit

In Chapter 5, we observed that if the temperature is high enough, we can replace the summation in the definition of the partition function with an integral to obtain the limiting form the partition function at high temperature. This reflects one of the fundamental characteristics of quantum theory, which is that classical behavior is attained in the limit of large quantum number. At high temperature, the average energy per molecule increases and this implies that the average quantum number for each energy storage mode is higher. Thus at higher temperatures, the mean behavior of the system is classical in nature.

In the previous sections of this text we have attacked the problem of determining the partition function by considering the problem from a quantum perspective.