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.

Throughout this text we have presented a molecular foundation for the principles of thermodynamics that has considered many highly simplified molecular models of a variety of systems, including idealized gases, solutions, polymers, and crystals. It might not be immediately obvious how we extend these ideas to more realistic models that, for example, might entail structured molecules with complex potential energy functions including bonded, electrostatic, and van der Waals energies all at once. Even the seemingly simple task of developing the thermodynamics of a monatomic fluid at high densities, such as liquid argon, can be challenging owing to the difficulty of treating the detailed pairwise interactions in the configurational partition function.

Two routes enable one to move beyond the general statistical-mechanical considerations of Chapters 16–19 to solve molecular models of nontrivial complexity. The first is the large collection of mathematical approximations and conceptual approaches that comprises the framework of statistical-mechanical theory. These techniques often give closed-form but approximate analytical expressions for the properties of a system that are valid in certain limits (e.g., the high-density one), or, alternatively, sets of equations that can be solved using standard numerical tools. The particular approaches are usually system-specific because they hinge on simplifications motivated by the physics of the interactions at hand. We will not discuss this body of work in any detail, but refer the reader to the excellent introductory texts by Hill and McQuarrie.

It is a familiar fact that pure substances tend to exist in one of three distinct states: solid, liquid, and gas. Take water, for example. As ice is heated at atmospheric pressure, it suddenly melts into liquid at a specific temperature. As the liquid continues to be heated, it eventually reaches a temperature at which it spontaneously vaporizes into a gas. These transitions are discontinuous; they occur at specific state conditions or particular combinations of T and P. At exactly those conditions, the system can exist in more than one form such that two (or more) phases are in equilibrium with each other.

Although we are typically familiar with phase behavior at atmospheric pressure, most substances experience a diverse set of phases over a broad range of pressures. Pure substances often have more than one crystalline phase, depending on the pressure. Figure 10.1 shows a schematic representation of a P–T phase diagram of water that illustrates the kind of complex behavior that can exist. In the case of mixtures, there are even more possibilities for phase equilibrium: for example, one can have equilibrium between two liquids of different compositions, or among multiple solid and liquid phases.

In this chapter we extend the fundamental properties of solutions introduced in Chapter 12 to a variety of cases in which a more advanced analysis is required. The uninterested reader may wish to skip these topics. Here we first examine in detail the phenomenology and mathematical description of liquid–vapor equilibrium in multicomponent systems, including nonideal ones. Subsequently, we consider two important classes of systems – polymers and strong electrolytes – for which the ideal solution never provides a reasonable model. We provide the basic models and conceptual underpinnings in these analyses; however, excellent elaborations on these topics can be found in the references noted at the end of the chapter.

Phenomenology of multicomponent vapor–liquid equilibrium

Because multicomponent vapor–liquid equilibrium is so important to the chemical process industries, we elaborate on the ideas initially developed in Chapter 12. Before we consider the mathematics of this problem, let us begin by describing some essential phenomenology. The behavior upon heating a pure liquid at constant pressure P is familiar: its temperature increases until the boiling temperature is reached, at which Pvap(Tb) = P. The liquid vaporizes entirely at Tb and then the temperature subsequently increases again as the vapor is heated.