General properties of solids

The term “solids” denotes materials that generally have the following properties. From a microscopic perspective, the molecules in a solid are in a condensed, closely packed state, and they vibrate around a fixed equilibrium position. That is, molecules can be considered tethered near a specific location in space, since their diffusion is very slow relative to the time scales of observation. From a macroscopic point of view, solids have an elastic modulus. This means that the application of a stress to the material produces a strain as well as an opposing force that tends to return the solid to its original, unstrained state once the stress is removed. This contrasts with viscous behavior in which an applied stress results in continuous, permanent deformation, such as the flow of a liquid.

Generally speaking, there are two primary classes of solids. Crystalline solids are equilibrium states of matter in which the microscopic structure has a well-defined geometric pattern with long-range order: a crystalline lattice. In contrast to crystals, amorphous solids have no long-range order, meaning that they lack a lattice structure and regular positioning of the molecules. Glasses and many polymeric materials are amorphous. Frequently these systems are not at equilibrium, but evolve very slowly in time and are metastable with respect to a crystalline phase. They might be considered liquids of extremely high viscosity that are slowly en route to crystallization. However, typically the time scale to reach equilibrium is so long (perhaps longer than the age of the universe) that for all practical purposes the amorphous state appears solid and stable. Thus, in an empirical sense, often we can treat such systems as in quasi-equilibrium.

Microstate probabilities

In Chapter 3 we discussed the way the world works at a microscopic level: the interactions and laws governing the time evolution of atoms and molecules. We found that an important, unifying perspective for both quantum and classical descriptions is the concept of energy. Now we take a macroscopic point of view. What happens when many (~1023) molecules come together, when we cannot hope to measure individual atomic properties but can probe only bulk material ones? As the title of this chapter suggests, the relevant concept at the macro-resolution is entropy.

Remember that from a macroscopic perspective, we care about macrostates, that is, states of a system characterized by a few macroscopic variables, like E, V, N, T, or P. Empirical measurements generally establish values for properties that are the net result of many atomic interactions averaged over time, and we are thus able to describe a system only in terms of these large-scale, reduced-information metrics that smooth over the molecular world. The statement that a system is at one specific macrostate actually implies that it is evolving through a particular ensemble of many microscopic configurations.

We will focus on classical isolated systems because these offer the simplest introductory perspective. Let us imagine that a closed, insulated container of molecules evolves in time.

New thermodynamic potentials from baths

We began our discussion of equilibrium by considering how the entropy emerges in isolated systems, finding that it is maximized under constant E, V, N conditions. In practical settings, however, systems are not often isolated and it is difficult to control their energy and volume in the presence of outside forces. Instead, it is easier to control so-called field parameters like temperature and pressure.

In this chapter, we will discuss the proper procedure for switching the independent variables of the fundamental equation for other thermodynamic quantities. In doing so, we will consider non-isolated systems that are held at constant temperature, pressure, and/or chemical potential through coupling to various kinds of baths. In such cases, we find that entropy maximization requires us to consider the entropy of both the system and its surroundings. Moreover, new thermodynamic quantities will naturally emerge in this analysis: additional so-called thermodynamic potentials.

To achieve conditions of constant temperature, pressure, or chemical potential, one couples a system to a bath. As discussed before, a bath is a large reservoir that can exchange energy, volume, or particles with the system of interest. While exchanging volume/energy/particles alters the system state, such changes are so minuscule for the bath that it is essentially always at the same equilibrium condition.

In Chapters 4 and 5 we introduced the molecular basis for the first two laws of thermodynamics. In particular, we developed a physical interpretation for the entropy in terms of microstates. However, we have yet to address two subtle questions regarding this relationship. Is there any such thing as an absolute value of the entropy? That is, does it make sense to identify an exact numerical value of S for a particular system and a particular state point, as opposed to the more modest calculation of a change in entropy between two state points? Furthermore, what is the behavior of the entropy and other thermodynamic functions as the temperature approaches absolute zero?

As we shall see in this chapter, both of these questions cannot be addressed using the first two laws alone. Instead, we must introduce a third law of thermodynamics that provides a context for understanding absolute entropies and absolute zero. The third law is not conceptually as straightforward as the others, first because it is not needed in many practical calculations far away from absolute zero, and second and more importantly, because it can be presented in several quite different ways. In particular, we will describe a number of distinct formulations of the third law and attempt to provide some molecular interpretation for their rationale. Unlike the first and second laws, however, there are many subtleties associated with these formulations that in some cases remain actively discussed. We will not attempt to synthesize and reconcile the details here, but present only the main ideas. The interested reader is referred to the references at the end of the chapter for further information.

Until now, most of what we have discussed has involved general relationships among thermodynamic quantities that can be applied to any system, such as the fundamental equation, reversibility, Legendre transforms, and Maxwell equations. In this and the coming chapters, we begin to investigate properties of specific types of substances. We will mostly consider very simple models in which only the essential physics is included; these give insight into the basic behaviors of solids, liquids, and gases, and actually are sufficient to learn quite a bit about them. Of course, there are also many detailed theoretical and empirical models for specific systems, but very often these theories simply improve upon the accuracy of the approaches rather than introduce major new concepts and qualitative behaviors.

Statistical mechanics provides a systematic route to state- and substance-specific models. If one can postulate a sufficiently simple description of the relevant atomic interaction energetics, the entropy or free energy can be determined in fundamental form. Ultimately our strategy for most of these simple models will be to determine the chemical potential μ(T, P) in single-component systems or μ(T, P, {x}) for multicomponent ones, where {x} gives the mole fractions. In both cases, knowledge of the chemical potentials does indeed give a fundamental perspective, allowing us to extract all of the intensive thermodynamic properties. Moreover, for problems involving phase equilibrium, the chemical potential is the natural starting point, as we will see in Chapter 10.

Like so many texts, this book grew out of lecture notes and problems that I developed through teaching, specifically, graduate thermodynamics over the past seven years. These notes were originally motivated by my difficulty in finding a satisfactory introductory text to both classical thermodynamics and statistical mechanics that could be used for a quarter-long course for first-year chemical engineering graduate students. However, as the years pressed forward, it became apparent that there was a greater opportunity to construct a new presentation of these classic subjects that addressed the needs of the modern student. Namely, few existing books seem to provide an integrated view of both classical and molecular perspectives on thermodynamics, at a sufficient level of rigor to address graduate-level problems.

It has become clear to me that first-year graduate students respond best to a molecular-level “explanation” of the classic laws, at least upon initial discussion. For them this imparts a more intuitive understanding of thermodynamic potentials and, in particular, entropy and the second law. Moreover, students’ most frequent hurdles are conceptual in nature, not mathematical, and I sense that many older presentations are inaccessible to them because concepts are buried deep under patinas of unnecessarily complex notation and equations.

Thermodynamics is a remarkable subject, both in its pervasiveness throughout the pure and engineering sciences, and in the striking simplicity and elegance of its principles. Indeed, it is hard to underestimate the significance of thermodynamics to virtually any physical problem of interest, even if its role appears only indirectly through derivative theories or models. As a testament to its importance, Einstein made the rather potent statement that thermodynamics is “the only physical theory of universal content concerning which I am convinced that within the framework of the applicability of its basic concepts, it will never be overthrown.”

At the same time, thermodynamics can be surprisingly difficult to grasp at a fundamental level, even for the experienced student. Unlike many other advanced scientific subjects, its main challenges are not mathematical in nature; a working knowledge of multivariate calculus is usually quite sufficient. Instead, the most difficult aspects of thermodynamics are its conceptual underpinnings. Students often struggle with the seemingly simple task of how to begin thinking about a problem, finding it difficult to answer questions such as the following. What constitutes the system? What is constant or constrained? What thermodynamic variables are equal across a boundary? What assumptions and models are reasonable? All of these questions precede the analytical analysis and are concerned with how to transform the physical problem into a mathematical one. When this is done, the solutions often present themselves in a rather straightforward manner, at least for an introductory treatment.

Quantum theory

To conceptualize the molecular origins of thermodynamic equilibrium, one must first understand the elemental ways by which molecules interact. How does the world really work? What are the most fundamental principles that form the basis of reality as we know it?

Currently our understanding of reality rests upon two principal concepts in physics: quantum theory and relativity. Both of these have been subjected to stringent experimental tests over the past century, and their combination has in part led to a deep understanding of elementary particles. There still remain some incompatibilities between the two, namely in understanding the nature of gravity, and there have been intense efforts to find new fundamental physical explanations. However, for the purposes of our discussion, we will focus solely on quantum theory since for nearly all of the models and systems that we will discuss one can safely avoid considerations of relativistic effects.

Quantum mechanics describes the complete time evolution of a system in a quantum sense, in a manner analogous to what Newtonian mechanics does for classical systems. It is most easily described in terms of a system of fundamental particles, such as electrons and protons.