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.

The topical organization of this book is as follows.

Chapter 1 is the basic introductory material which illustrates the richness of the subject, spanning applications to a wide range of problems in science and engineering. This chapter also provides the introduction to the basics of model building and shows the relationships among models of various levels of hierarchy. The basic vocabulary is introduced, and the physical properties needed in transport problems are discussed. The link between continuum and molecular models is indicated. The chapter concludes with a brief note on the historical development of the subject.

Chapter 2 illustrates the formulation of model equations for many common transport problems using a basic control-volume-balance type of approach. All three modes of transport are illustrated so that the student can grasp the similarities. Some “standard” problems are illustrated. This chapter is written assuming no significant earlier background knowledge in this field, and is therefore useful to bring such students up to speed.

The next few chapters, Chapters 3–6, provide the detailed framework for the analysis of momentum transport problems. The kinematics of flow are reviewed in Chapter 3, while the kinetics of flow are discussed in Chapter 4, leading to the derivation of the differential equations for the stress field and the velocity field in Chapter 5.

Transport of ions or charged species is encountered in a wide variety of processes. For example, sodium chloride is ionized in water and exists as sodium and chloride ionpairs. Thus the diffusion of sodium chloride in reality involves the diffusion of positively charged sodium ions and negatively charged chloride ions. Such systems are of importance in electrochemical reaction engineering. Examples can be found, for example, in electro-winning of metals, fuel cells, batteries, etc. Equally important is the transport of charged species in charged membranes or solids carrying a net surface charge. An example of such a system is the ion-exchange membrane, as is widely used in water purification, metal recovery, and selectivity modulation of chemical reactors. Transport of charged species in biological membranes is another example of transport across charged membranes. In addition, a number of techniques to separate large molecules such as DNA and proteins involve application of an electric field. Electrophoresis is such an example. The ionic-transport effects are also important in the study of the surface properties of colloids.

The key feature to be included in the model for transport in such systems is the charge migration due to the electric field. The constitutive equation for diffusion in such systems is the classical Nernst–Planck equation, which is discussed in Section 22.1.2.

Mathematical prerequisites

The ability to find general solutions to second-order differential equations is needed, as well as the procedure to find the integration constants and find solutions to specific problems.

The chapter aims essentially to illustrate the solution to some common and well-studied fluid-flow problems in laminar flow. The problems analyzed are mainly those where the Navier–Stokes (N–S) equations can be simplified and analytical solutions are possible. Steady-state flow problems are discussed. The types of problems examined here and a brief classification of flow problems are presented below.

Flows can be classified into external flows and internal flows. External flow is in a semi-infinite domain and the flow is always developing and of boundary-layer nature. Thus there is a region near the solid surface called a boundary layer where the major changes in velocity can be anticipated. The thickness of this boundary layer increases as one moves along the flow direction. Further the flow is 2D (or even 3D) and both components of the velocity are needed to describe the flow correctly. The solution of the N–S equations are therefore more complex than those for internal flows, and for boundary-layer problems they can be classified as parabolic partial differential equations. Details of external flow are postponed until a later chapter. However, some key results that are also useful for study of heat and mass transfer in boundary layers are summarized.