Chapter 1
The Phase-Equilibrium Problem

We live in a world of mixtures – the air we breathe, the food we eat, the gasoline in our automobiles. Wherever we turn, we find that our lives are linked with materials that consist of a variety of chemical substances. Many of the things we do are concerned with the transfer of substances from one mixture to another; for example, in our lungs, we take oxygen from the air and dissolve it in our blood, while carbon dioxide leaves the blood and enters the air; in our coffee maker, water-soluble ingredients are leached from coffee grains into water; and when someone stains his tie with gravy, he relies on cleaning fluid to dissolve and thereby remove the greasy spot. In each of these common daily experiences, as well as in many others in physiology, home life, industry, and so on, there is a transfer of a substance from one phase to another. This occurs because when two phases are brought into contact, they tend to exchange their constituents until the composition of each phase attains a constant value; when that state is reached, we say that the phases are in equilibrium. The equilibrium compositions of two phases are often very different from one another, and it is precisely this difference that enables us to separate mixtures by distillation, extraction, and other phase-contacting operations.

The final, or equilibrium, phase compositions depend on several variables, such as temperature and pressure, and on the chemical nature and concentrations of the substances in the mixture. Phase-equilibrium thermodynamics seeks to establish the relations among the various properties (in particular, temperature, pressure, and composition) that ultimately prevail when two or more phases reach a state of equilibrium wherein all tendency for further change has ceased.

Because so much of life is concerned with the interaction between different phases, it is evident that phase-equilibrium thermodynamics is a subject of fundamental importance in many sciences, physical as well as biological. It is of special interest in chemistry and chemical engineering because so many operations in the manufacture of chemical products consist of phase contacting: Extraction, adsorption, distillation, leaching, and absorption are essential unit operations in chemical industry; an understanding of any one of them is based, at least in part, on the science of phase equilibrium.

Equilibrium properties are required for the design of separation operations; these, in turn, are essential parts of a typical chemical plant, as shown in Fig. 1-1. In this plant, the central part (stage II) is the chemical reactor and it has become frequent practice to call the reactor the heart of the plant. But, in addition, a plant needs a mouth (stage I) and a digestive system (stage III). Prior to reaction, the reactants must be prepared for reaction; because the raw materials provided by nature are usually mixtures, separation is often required to separate the desired reactants from other unwanted components that may interfere with the reaction. Downstream from the reactor, separation is necessary to separate desired from undesired products and, because reaction is rarely complete, it is also necessary to separate the unreacted reactants for recycle.

Figure 1-1 Schematic of a chemical plant.

Figure 1-1 illustrates why separations are so important in chemical engineering. In a typical large-scale chemical plant, the investment for separation operations is in the neighborhood of 50% and often it is appreciably more.

1.1 Essence of the Problem

We want to relate quantitatively the variables that describe the state of equilibrium of two or more homogeneous phases that are free to interchange energy and matter. By a homogeneous phase at equilibrium we mean any region in space where the intensive properties are everywhere the same.¹ Intensive properties are those that are independent of the mass, size, or shape of the phase; we are concerned primarily with the intensive properties temperature, density, pressure, and composition (often expressed in terms of mole fractions). We want to describe the state of two or more phases that are free to interact and that have reached a state of equilibrium. Then, given some of the equilibrium properties of the two phases, our task is to predict those that remain.

¹ We are here neglecting all special forces such as those due to gravitational, electric, or magnetic fields, surface forces. etc.

Figure 1-2 illustrates schematically the type of problem that phase-equilibrium thermodynamics seeks to solve. We suppose that two multicomponent phases, α and β, have reached an equilibrium state; we are given temperature T of the two phases and mile fractions of phase α. Our task, then, is to find mole fractions of phase β and pressure P of the system. Alternatively, we might know and p and be asked to find and T, or our problem might be characterized by other combinations of known and unknown variables. The number of intensive properties that must be specified to fix unambiguously the state of equilibrium is given by the Gibbs phase rule. In the absence of chemical reactions, the phase rule is:

Figure 1-2 Statement of problem.

Number of independent intensive properties = Number of components – Number of phases + 2

For example, for a two-component, two-phase system, the number of independent intensive properties is two. In such a system the intensive properties of interest usually are T, and P.² Two of these, any two, must be specified before the remaining two can be found.

² Because for each phase, and are not additional variables in this ease.

How shall we go about solving the problem illustrated in Fig. 1-2? What theoretical framework is available to give us a basis for finding a solution? When this question is raised, we turn to thermodynamics.

1.2 Application of Thermodynamics to Phase-Equilibrium Problems

One of the characteristics of modern science is abstraction. By describing a difficult, real problem in abstract, mathematical terms, it is sometimes possible to obtain a simple solution to the problem not in terms of immediate physical reality, but in terms of mathematical quantities that are suggested by an abstract description of the real problem. Thermodynamics provides the mathematical language that enables us to obtain an abstract solution of the phase-equilibrium problem.

Application of thermodynamics to phase equilibria in multicomponent systems is shown schematically in Fig. 1-3. The real world and the real problem are represented by the lower horizontal line, while the upper horizontal line represents the world of abstraction. The three-step application of thermodynamics to a real problem consists of an indirect mental process; instead of attempting to solve the real problem within the world of physically realistic variables, the indirect process first projects the problem into the abstract world, then seeks a solution within that world, and finally projects this solution back to physical reality. The solution of a phase-equilibrium problem using thermodynamics requires three steps:

Figure 1-3 Three-step application of thermodynamics to phase-equilibrium problems.

I. The real problem is translated into an abstract, mathematical problem.

II. A solution is found to the mathematical problem.

III. The mathematical solution is translated back into physically meaningful terms.

The essential feature of step I is to define appropriate and useful mathematical functions to facilitate step II. The profound insight of Gibbs, who in 1875 defined such a function – the chemical potential - made it possible to achieve the goal of step II; the mathematical solution to the phase-equilibrium problem is given by the remarkably simple result that at equilibrium, the chemical potential of each component must be the same in every phase.

The difficult step is the last one, step III. Thanks to Gibbs, steps I and II present no further problems and essentially all work in this field, after Gibbs, has been concerned with step III. From the viewpoint of a formal theoretical physicist, the phase-equilibrium problem has been solved completely by Gibbs’ relation for the chemical potentials. A pure theoretician may require nothing further, but someone who is concerned with obtaining useful numerical answers to real problems must face the task of translating the abstract results of step II into the language of physical reality.

Our concern in this book is concentrated almost exclusively on step III. In Chap. 2 we briefly review some of the important concepts that lead to Gibbs’ equation, i.e., that for any component, the chemical potential must be the same in all equilibrated phases. In a sense, we may call Chap. 2 historical because it reproduces, in perhaps more modern terminology, work that was completed many years ago. However, in all the remaining chapters, we address the contemporary problem of how quantitatively to relate the chemical potential to the primary variables temperature, pressure, and composition. We should point out at once that this problem, designated by step III, is mostly outside the realm of classical thermodynamics and. therefore, much of the material in later chapters cannot be called thermodynamics in the strict sense. Classical thermodynamics by itself gives us important but also severely limited information on the relation between the abstract chemical potential and the real, experimentally accessible quantities temperature, pressure, and composition. For quantitative, numerical results, classical thermodynamics is not sufficient. Step III must also utilize concepts from statistical thermodynamics, molecular physics, and physical chemistry.

To solve problems of the type illustrated in Fig. 1-2, we must make the transition from what we have, i.e., the abstract thermodynamic equation of equilibrium, toward what we want, i.e., quantitative information about temperature, pressure, and phase compositions. Thanks to Gibbs, the thermodynamic equation of equilibrium is now well known and we need not concern ourselves with it except as a place to start. In any problem concerning the equilibrium distribution of some component i between two phases α and β, we must always begin with the relation

(1-1)

where μ is the chemical potential. It is then that our problem begins; we must now ask how is related to T, P, and and similarly, how is related to T, P, and . To establish these relations, it is convenient to introduce some auxiliary functions such as fugacity and activity. These functions do not solve the problem for us, but they facilitate our efforts to find a solution because, in many cases, they make the problem somewhat easier to visualize; fugacity and activity are quantities much closer to our physical senses than the abstract concept chemical potential. Suppose, for example, that phase α is a vapor and phase β is a liquid. Then, as discussed in subsequent chapters, Eq. (1-1) can be rewritten

(1-2)

where, in the vapor phase, y_i is the mole fraction and φ_i is the fugacity coefficient, and in the liquid phase, x_i is the mole fraction, γ_i is the activity coefficient, and is the fugacity of component i at some fixed condition known as the standard state.

The details of Eq. (1-2) are not important just now; they are covered later. What is important to note is the procedure whereby the highly abstract Eq. (1-1) has been transformed into the not-quite-so-abstract Eq.(1-2). Equation (1-2), unlike Eq. (1-1), at least contains explicitly three of the variables of interest, x_i, y_i, and P. Equation (1-2) is no more and no less fundamental than Eq. (1-1); one is merely a mathematical transformation of the other, and any claim Eq. (1-2) may have to being more useful is only a consequence of a fortunate choice of useful auxiliary functions in the transformation.

Much of this utility comes from the concept of ideality. If we define mixtures with certain properties as ideal mixtures, we then find, as a result of our choice of auxiliary functions, that the equation of equilibrium can be simplified further; for example, for a mixture of ideal gases φ_i = 1, and for ideal liquid mixtures at low pressures, γ_i = 1 when is given by the saturation pressure of pure liquid i at the temperature of interest. We thus find that some of the auxiliary functions (such as φ_i and γ_i) are useful because they are numerical factors, frequently of the order of unity, that establish the connection between real mixtures and those that, by judicious choice, have been defined as ideal mixtures.

From the viewpoint of formal thermodynamics, Eq. (1-2) is no better than Eq. (1-1); but from the viewpoint of experimental chemistry and chemical engineering, Eq. (1-2) is preferable because it provides a convenient frame of reference.

In the general case we cannot assume ideal behavior and we must then establish two relations, one for φ_i and γ_i:

(1-3)

(1-4)

In Chaps. 3, 5, and 12, we discuss in detail what we can say about function F_φ in Eq. (1-3). In Chap. 4, we digress with a brief discussion of the nature of intermolecular forces, because the functional relationships of both Eqs. (1-3) and (1-4) are determined by forces that operate between molecules. In Chaps. 6, 7 and 12, we are concerned with function F_γ in Eq. (1-4), and in Chaps. 10 and 11, primary attention is given toward determination of a useful , such that activity coefficient γ is often close to unity. Chapter 8 discusses activity coefficients in systems containing polymers and Chap. 9 is devoted to liquid solutions of solutes that dissociate into ions.

Before discussing in detail various procedures for calculating fugacities and other useful auxiliary functions, we first give in Chap. 2 a brief survey of steps I and II indicated in Fig. 1-3.