Chapter 2
Classical Thermodynamics of Phase Equilibria

Thermodynamics as we know it today originated during the middle of the nineteenth century, and while the original thermodynamic formulas were applied to only a limited class of phenomena (such as heat engines), they have, as a result of suitable extensions, become applicable to a large number of problems in both physical and biological sciences. From its Greek root (therme, heat; dynamis, force), one might well wonder what “thermodynamics” has to do with the distribution of various components between various phases. Indeed, the early workers in thermodynamics were concerned only with systems of one component, and it was not until the monumental work of J. Willard Gibbs that thermodynamic methods were shown to be useful in the study of multicomponent systems. It was Gibbs who first saw the generality of thermodynamics. He was able to show that a thermodynamic procedure is possible for a wide variety of applications, including the behavior of chemical systems.

This chapter briefly reviews the essential concepts of the classical thermodynamic description of phase equilibria.¹ It begins with a combined statement of the first and second laws as applied to a closed, homogeneous system, and proceeds toward the laws of equilibrium for an open, heterogeneous system. For our purposes here, we exclude surface and tensile effects, acceleration, or change of position in an external field, such as a gravitational or electromagnetic field (other than along a surface of constant potential); for simplicity, we also rule out chemical and nuclear reactions.² We are then left with the classical problem of phase equilibrium that considers internal equilibrium with respect to three processes:

¹ More complete discussions are given in references listed at the end of this chapter.

² However, see the final two paragraphs of App. A.

1. Heat transfer between any two phases within the heterogeneous system.

2. Displacement of a phase boundary.

3. Mass transfer of any component in the system across a phase boundary.

The governing potentials in the first two processes are temperature and pressure, respectively, and we assume prior knowledge of their existence; the governing potential for the third process, however, is considered not to be known a priori, and it is one of the prime responsibilities of classical thermodynamics of phase equilibria to “discover” and exploit the appropriate “chemical potential.”³ A heterogeneous system that is in a state of internal equilibrium is a system at equilibrium with respect to each of these three processes.

³ This was first done by Gibbs in 1875.

The chapter continues with some discussion of the nature of the chemical potential and the need for standard states, and then introduces the auxiliary functions fugacity and activity. The chapter concludes with a very simple example of how the thermodynarnic equations of phase equilibrium may be applied to obtain a physically useful result.

2.1 Homogeneous Closed Systems

A homogeneous system is one with uniform properties throughout; i.e., a property such as density has the same value from point to point, in a macroscopic sense. A phase is a homogeneous system. A closed system is one that does not exchange matter with its surroundings, although it may exchange energy. Thus, in a closed system not undergoing chemical reaction, the number of moles of each component is constant. This constraint can be expressed as

(2-1)

where n₁ is the number of moles of the ith component and m is the number of components present.

For a homogeneous, closed system, with the qualifications given previously, taking into account interactions of the system with its surroundings in the form of heat transfer and work of volumetric displacement, a combined statement of the first and second laws of thermodynamics is (Denbigh, 1981)

(2-2)

where dU, dS, and dV are, respectively, small changes in energy, entropy, and volume of the system resulting from the interactions; each of these properties is a state function whose value in a prescribed state is independent of the previous history of the system. For our purposes here, the surroundings are considered to be two distinct bodies: a constant-volume heat bath, also at constant, uniform temperature T_B, in thermal contact only with the system; and another external body, at constant, uniform pressure P_E, in "volumetric" contact only with the system through a movable, thermally insulated piston.

Because Eq. (2.2) is our starting point, it is important to have a better understanding of its significance and range of validity, even though we do not attempt to develop or justify it here. However, before proceeding, we need to discuss briefly three important concepts: equilibrium state, reversible process, and state of internal equilibrium.

By an equilibrium state we mean one with no tendency to depart spontaneously, having in mind certain permissible changes or processes, i.e., heat transfer, work of volume displacement and, for open systems (next section), mass transfer across a phase boundary. In an equilibrium state, the properties are independent of time and of previous history of the system; further, they are stable, that is, not subject to "catastrophic" changes on slight variations of external conditions. We distinguish an equilibrium state from a steady state, insisting that in an equilibrium state there are no net fluxes of the kind under consideration (heat transfer, etc.) across a plane surface placed anywhere in the system.

In thermodynamics, we are normally concerned with a finite change in the equilibrium state of a system or a variation in an equilibrium state subject to specified constraints. A change in the equilibrium state of a system is called a process. A reversible process is one where the system is maintained in a state of virtual equilibrium throughout ihu process; a reversible process is sometimes referred to as one connecting a series of equilibrium states. This requires that the potential difference (between system and surroundings) causing the process to occur be only infinitesimal; then the direction of the process can be reversed by an infinitesimal increase or decrease, as the case may be, in the potential for the system or the surroundings. Any natural or actual process occurs irreversibly; we can think of a reversible process as a limit approached but never attained. The inequality in Eq. (2.2) refers to a natural (irreversible) process and the equality to a reversible process.

By a "single-phase system in a state of internal equilibrium" we mean one that is homogeneous (uniform properties) even though it may be undergoing an irreversible process as a result of an interaction with its surroundings. In practice, such a state may be impossible to achieve, but the concept is useful for a discussion of the significance of Eq. (2.2), to which we now return.

If the interaction of the system with its surroundings occurs reversibly (reversible heat transfer and reversible boundary displacement), the equality sign of Eq. (2.2) applies; in that event, T_B = T, the temperature of the system, and P_E= P, the pressure of the system. Hence, we may write

(2-3)

The first term on the right is the heat absorbed by the system (TdS = δ>Q_rev), and the second term is the work done by the system (δWrev> PdV). The form of this equation implies that the system is characterized by two independent variables or degrees of freedom, here represented by S and V.

If the interaction between system and surroundings occurs irreversibly, the inequality of Eq. (2.2) applies:

(2-4)

In this case δW = P_E dV, but δQ ≠ T_BdS. However, if the system is somehow maintained in a state of internal equilibrium during the irreversible interaction, that is, if it has uniform properties, then it is a system characterized by two independent variables and Eq. (2.3) applies. Hence, this equation may be applicable whether the process is externally reversible or irreversible. However, in the latter situation the terms TdS and PdV can no longer be identified with heat transfer and work, respectively.

To obtain the finite change in a thermodynamic property occurring in an actual process (from equilibrium state 1 to equilibrium state 2), the integration of an equation such as Eq. (2.3) must be done over a reversible path using the properties of the system. This results in an equation of the form

(2-5)

Because U is a state function, this result is independent of the path of integration, and it is independent of whether the system is maintained in a state of internal equilibrium or not during the actual process; it requires only that the initial and final states be equilibrium states. Hence the essence of classical (reversible) thermodynamics lies in the possibility of making such a calculation by constructing a convenient, reversible path to replace the actual or irreversible path of the process that is usually not amenable to an exact description.

Equation (2.3) represents the fundamental thermodynarnic relation (Gibbs, 1961). If U is considered to be a function of S and V, and if this function U is known, then all other thermodynarnic properties can be obtained by purely mathematical operations on this function. For example, T = (∂U/dS)_v and P = -(∂U/∂V)_s. While another pair of independent variables could be used to determine U, no other pair has this simple physical significance for the function U. We therefore call the group of variables U, S, V afundamental grouping.

An important aspect of Eq. (2.2) is that it presents U as a potential function. If the variation dU is constrained to occur at constant S and V, then

(2-6)

Equation (2.6) says that at constant S and V, U tends toward a minimum in an actual or irreversible process in a closed system and remains constant in a reversible process. Because an actual process is one tending toward an equilibrium state, an approach to equilibrium at constant entropy and volume is accompanied by a decrease in internal energy. Equation (2.6), then, provides a criterion for equilibrium in a closed system; we shall make use of this criterion later.

Other extensive thermodynarnic potentials for closed systems and other fundamental groupings can be obtained by using different pairs of the four variables P, V, T, and S as independent variables on the right-hand side of Eq. (2.3). Partial Legendre transformations (Callen, 1985) enable us to use three other pairs while retaining the important property of a fundamental equation. For example, suppose we wish to interchange the roles of P and V in Eq. (2.3) to have P as an independent variable. We then define a new function, viz. the original function, U, minus the product of the two quantities to be interchanged with due regard for the sign of the term in the original equation. That is, we define

(2-7)

where H, the enthalpy of the system, is a state function because it is defined in terms of state functions. Differentiation of Eq. (2.7) and substitution for dU in Eq. (2.3) gives

(2-8)

and the independent variables are now S and P. The role of H as a potential for a dosed system at constant S and P means that

(2-9)

Similarly, to interchange T and S (but not P and V) in Eq. (2.3), we define the Helmholtz energy

(2-10)

giving

(2-11)

and

(2-12)

In this case, the independent variables or constraints are T and V. Finally, to interchange both T and S and P and V in Eq. (2.3) so as to use T and P as the independent variables, we define the Gibbs energy

(2-13)

giving

(2-14)

and

(2-15)

Table 2-1 gives a summary of the four fundamental equations and the roles of U, H, A, and G as thermodynamic potentials. Also included in the table are a set of identities resulting from the fundamental equations and the set of equations known as Maxwell relations. These relations are obtained from the fundamental equations by the application of Euler’s reciprocity theorem that takes advantage of the fact that the order of differentiation in forming second partial derivatives is immaterial for continuous functions and their derivatives.

Table 2-1 Some important thermodynamic relations for a homogeneous closed system.

2.2 Homogeneous Open Systems

An open system can exchange matter as well as energy with its surroundings. We now consider how the laws of thermodynamics for a closed system can be extended to apply to an open system.

For a closed homogeneous system, we considered U to be a function only of S and V; that is,

(2-16)

In an open system, however, there are additional independent variables. For these, we can use the mole numbers of the various components present. Hence, we must now consider U as the function

(2-17)

where m is the number of components. The total differential is then

(2-18)

where subscript n_i refers to all mole numbers and subscript n_j to all rnole numbers other than the ith. Because the first two derivatives in Eq. (2-18) refer to a closed system, we may use the identities of Table 2-1. Further, we define the function μ_i as

(2-19)

We may then rewrite Eq. (2-18) in the form

(2-20)

Equation (2-20) is the fundamental equation for an open system corresponding to Eq. (2-3) for a closed system. The function μ_i is an intensive quantity and we expect it to depend on temperature, pressure, and composition of the system. However, our primary task is to show that (μ_i is a mass or chemical potential, as we might suspect from its position in Eq. (2-20) as a coefficient of dn_i, just as T (the coefficient of dS) is a thermal potential and P (the coefficient of dV) is a mechanical potential. Before doing this, however, we consider other definitions of μ_i and the corresponding fundamental equations for an open system in terms of H, A, and G. Using the defining equations for H, A, and G [Eqs. (2-7), (2-10), and (2-13)], we may substitute for dU in Eq. (2-20) in each case and arrive at the following further three fundamental equations for an open system:

(2-21)

(2-22)

(2-23)

From the definition of μ_i given in Eq. (2-19) and from Eqs. (2-20) to (2-23), it follows that

(2-24)

There are thus four expressions for μ_i where each is a derivative of an extensive property with respect to the amount of the component under consideration, and each uses a fundamental grouping of variables: U, S, V; H, S, P; A, T, V; and G, T, P. The quantity μ_i is the partial molar Gibbs energy, but it is, not the partial molar internal energy, enthalpy, or Helmholtz energy. This is because the independent variables T and P, chosen for defining partial molar quantities, are also the fundamental independent variables for the Gibbs energy G.

2.3 Equilibrium in a Heterogeneous Closed System

A heterogeneous, closed system is made up of two or more phases with each phase considered as an open system within the overall closed system. We now consider the conditions where the heterogeneous system is in a state of internal equilibrium with respect to the three processes of heat transfer, boundary displacement, and mass transfer.⁴

⁴ We negiect here “special” effects such as surface forces; semipeimeable membranes; and electric, magnetic or gravitational forces.

We already have four criteria with different sets of constraints for equilibrium in a closed system as given by the third set of equations in Table 2-1 with the equal sign in each case. However, these are in terms of the four extensive thermodynamic potentials U, H, A, and G. We can obtain more useful criteria in terms of the intensive quantities T, P, and μ_i. We expect that, to have thermal and mechanical equilibrium in the system, temperature and pressure must be uniform throughout the entire heterogeneous mass. If μ_i is the intensive potential governing mass transfer, we expect that μ_i, must also have a uniform value throughout the whole heterogeneous system at equilibrium with respect to this process. Gibbs first gave the proof of this in 1875. He used the function U as a starting point rather than H, A, or G, probably because of the symmetry in the expression for dU in Eq. (2-20); each differential on the right is the differential of an extensive quantity and each coefficient is an intensive quantity. This means that the uniformity of all intensive potentials at equilibrium can be proved by consideration of only one function U. Details of this proof are given in App. A.

The general result for a closed, heterogeneous system consisting of π phases and m components is that at equilibrium with respect to the processes described earlier,

(2-25)

(2-26)

(2-27)

where the superscript in parentheses denotes the phase and the subscript denotes the component. This set of equations provides the basic criteria for phase equilibrium for our purposes. In the next two sections, we consider the number of independent variables (degrees of freedom) in systems of interest to us.

2.4 The Gibbs-Duhem Equation

We may characterize the intensive state of each phase present in a heterogeneous system at internal equilibrium by its temperature and pressure, and the chemical potential of each component present - a total of m + 2 variables. However, these are not all independently variable, and we now derive an important relation, known as the Gibbs-Duhem equation, that shows how the variables are related.

Consider a particular phase within the heterogeneous system as an open, homogeneous system. The fundamental equation in terms of U [Eq. (2-20)] is

(2-28)

We may integrate this equation from a state of zero mass (U = S = V = n₁ =… = n_m = 0) to a state of finite mass (U, S, V, n₁,…, n_m) at constant temperature, pressure, and composition; along this path of integration, all coefficients, including all μ_i in Eq. (2-28), are constant; integration gives

(2-29)

This equation may be regarded as expressing U as a function of T, P, composition, and the size of the system. The path of integration amounts to adding together little bits of the phase, each with the same temperature, pressure, and composition, to obtain a finite amount of phase. Because U is a state function, the result expressed by Eq. (2-29) is independent of the path of integration. Differentiation of this equation to obtain a general expression for dU comparable to that in Eq. (2-28) gives

(2-30)

Comparing Eqs. (2-28) and (2-30), we have

(2-31)

Equation (2-31) is the Gibbs-Duhem equation, a fundamental equation in the thermo-dynamics of solutions used extensively in Chap. 6. For now we note that it places a restriction on the simultaneous variation of T, P, and the μ_i, for a single phase. Hence, of the m + 2 intensive variables that may be used to characterize a phase, only m + 1 are independently variable; a phase has m + 1 degrees of freedom.

2.5 The Phase Rule

When we consider the number of degrees of freedom in a heterogeneous system, we need to take into account the results of the preceding two sections. If the heterogeneous system is not in a state of internal equilibrium, but each phase is, the number of independent variables is Π(m + 1), because for each phase there are m + 1 degrees of freedom; a Gibbs-Duhem equation applies to each phase. However, if we stipulate that the entire system is in a state of internal equilibrium, then among the Π(m +1) variables there are (Π - l)(m + 2) equilibrium relations given by Eqs. (2-25) to (2-27). Thus the number of degrees of freedom, F, is the number of intensive variables used to characterize the system minus the number of relations or restrictions connecting them:

(2-32)

In the type of system we have been considering, the number of components m is equal to the number of independently variable species in a chemical sense, because we have ruled out chemical reaction and all special restrictions.⁵

⁵ See the final paragraph of App. A.

2.6 The Chemical Potential

The task of phase-equilibrium thermodynamics is to describe quantitatively the distribution at equilibrium of every component among all the phases present. For example, in distillation of a mixture of toluene and hexane we want to know how, at a certain temperature and pressure, the toluene (or hexane) is distributed between the liquid and the gaseous phases; or in extraction of acetic acid from an aqueous solution using benzene, we want to know how the acetic acid distributes itself between the two liquid pfiases. Gibbs obtained the thermodynamic solution to the phase-equilibrium problem many years ago when he introduced the abstract concept chemical potential. The goal of present work in phase-equilibrium thermodynamics is to relate the abstract chemical potential of a substance to physically measurable quantities such as temperature, pressure, and composition.

To establish the desired relation, we must immediately face one apparent difficulty: We cannot compute an absolute value for the chemical potential but must content ourselves with computing changes in the chemical potential that accompany any arbitrary change in the independent variables temperature, pressure, and composition. This difficulty is apparent rather than fundamental; it is really no more than an inconvenience. It arises because the relations between chemical potential and physically measurable quantities are in the form of differential equations that, upon integration, give only differences. These relations are discussed in more detail in Chap. 3, but one example is useful here.

For a pure substance i, the chemical potential is related to the temperature and pressure by the differential equation

(2-33)

where S_i is the molar entropy and v_i the molar volume. Integrating and solving for (μ_i, at some temperature T and pressure P, we have

(2-34)

where superscript r refers to some arbitrary reference state.

In Eq. (2-34) the two integrals on the right side can be evaluated from thermal and volumetric data over the temperature range T^r to T and the pressure range P^r to P. However, the chemical potential μ_i(T^r, P^r) is unknown. Hence, the chemical potential at T and P can only be expressed relative to its value at the arbitrary reference state designated by T^r and P^r.

Our inability to compute an absolute value for the chemical potential complicates the use of thermodynamics in practical applications. This complication follows from a need for arbitrary reference states that are commonly called standard states. Successful application of thermodynamics to real systems frequently is based on a judicious choice of standard states, as shown by examples discussed in later chapters. For the present it is only necessary to recognize why standard states arise and to remember that they introduce a constant into our equation. This constant need not give us concern because it must always cancel out when we compute for some substance the change of chemical potential that results from a change of any, or all, of the independent variables.⁶

⁶ Standard states are reference points; we use these frequently in daily life. For example, when traveling in Europe, the senior author is often asked “Where is Berkeley?” A possible reply could be “Berkeley is 2000 miles west of Podunk, Iowa.” In that statement, Podonk is the standard state. But this standard state is not useful because no one in Europe knows where Podunk, Iowa might be. A more useful reply is “Berkeley is ten miles east of San Francisco.” Now San Francisco is the standard state. San Francisco is a better standard state not only because it is close to Berkeley but perhaps more important, essentially everyone in Europe knows where San Francisco is.

2.7 Fugacity and Activity

The chemical potential does not have an immediate equivalent in the physical world and it is therefore desirable to express the chemical potential in terms of some auxiliary function that might be more easily identified with physical reality. A useful auxiliary function is obtained by the concept fugacity.

In attempting to simplify the abstract equation of chemical equilibrium, G. N. Lewis first considered the chemical potential for a pure, ideal gas and then generalized to all systems the result he obtained for the ideal case. From Eq. (2-33),

(2-35)

Substituting the ideal-gas equation,

(2-36)

and integrating at constant temperature,

(2-37)

Equation (2-37) says that for an ideal gas, the change in chemical potential, in isothermally going from pressure P⁰ to pressure P, is equal to the product of RT and the logarithm of the pressure ratio P/P⁰. Hence, at constant temperature, the change in the abstract thermodynamic quantity p, is a simple logarithmic function of the physically real quantity, pressure. The essential value of Eq. (2-37) is that it simply relates a mathematical abstraction to a common, intensive property of the real world. However, Eq. (2-37) is valid only for pure, ideal gases; to generalize it, Lewis defined a function f, called fugacity,⁷ by writing for an isothermal change for any component in any system, solid, liquid, or gas, pure or mixed, ideal or not,

⁷ From the Latin fuga, meaning flight or escape.

(2-38)

While either μ⁰_i or f_i⁰ is arbitrary, both may not be chosen independently; when one is chosen, the other is fixed.

For a pure, ideal gas, the fugacity is equal to the pressure, and for a component i in a mixture of ideal gases, it is equal to its partial pressure y_iP. Because all systems, pure or mixed, approach ideal-gas behavior at very low pressures, the definition of fugacity is completed by the limit

(2-39)

where y_i is the mole fraction of i.

Lewis called the ratio f/f ⁰ the activity, designated by symbol a. The activity of a substance gives an indication of how “active” a substance is relative to its standard state because it provides a measure of the difference between the substance’s chemical potential at the state of interest and that at its standard state. Because Eq. (2-38) was obtained for an isothermal change, the temperature of the standard state must be the same as that of the state of interest. The compositions and pressures of the two states, however, need not be (and indeed usually are not) the same.

The relation between fugacity and chemical potential provides conceptual aid in performing the translation from thermodynamic to physical variables. It is difficult to visualize the chemical potential, but the concept of fugacity is less so. Fugacity is a “corrected pressure”; for a component in a mixture of ideal gases it is equal to the partial pressure of that component. The ideal gas is not only a limiting case for thermodynamic convenience but corresponds to a well-developed physical model based on the kinetic theory of matter. The concept of fugacity, therefore, helps to make the transition from pure thermodynamics to the theory of intermolecular forces; if the fugacity is a “corrected pressure”, these corrections are due to nonidealities that can be interpreted by molecular considerations.

The fugacity provides a convenient transformation of the fundamental equation of phase equilibrium, Eq. (2-27). For phases α and β, respectively, Eq. (2-38) is

(2-40)

and

(2-41)

Substituting Eqs. (2-40) and (2-41) into the equilibrium relation, Eq. (2-27), yields

(2-42)

We now consider two cases. First, suppose that the standard states for the two phases are the same; i.e., suppose

(2-43)

In that case, it follows that

(2-44)

Equations (2-42), (2-43), and (2-44) give a new form of the fundamental equation of phase equilibrium:

(2-45)

Second, suppose that the standard states for the two phases are at the same temperature but not at the same pressure and composition. In that case, we use the exact relation between the two standard states:

(2-46)

Substituting Eq. (2-46) into Eq. (2-42), we again have

Equation (2-45) gives an useful result. It tells us that the equilibrium condition in terms of chemical potentials can be replaced without loss of generality by an equation that says, for any species i, the fugacities must be the same in all phases. (The condition that the activities must be equal holds only for the special case where the standard states in all phases are the same.) Equation (2-45) is equivalent to Eq. (2-27); from a strictly thermodynamic point of view, one is not preferable to the other. However, from the viewpoint of one who wishes to apply thermodynamics to physical problems, an equation that equates fugacities is often more convenient than one that equates chemical potentials. In much of our subsequent discussion, therefore, we regard Eqs. (2-25), (2-26), and (2-45) as the three fundamental equations of phase equilibrium.

Most of the chapters to follow present in detail relations between fugacity and independent variables temperature, pressure, and composition. However, before discussing the details of these relations, it is desirable to give a preview of where we are going, to present an illustration of how the various concepts in this chapter can, in at least one very simple case, lead to a relation possessing immediate physical utility.

2.8 A Simple Application: Raoult’s Law

Consider the equilibrium distribution of a component in a binary system between a liquid phase and a vapor phase. We seek a simple relation describing the distribution of the components between the phases, i.e., an equation relating x, the mole fraction in the liquid phase, to y, the mole fraction in the vapor phase. We limit ourselves to a very simple system, whose behavior can be closely approximated by the assumption of several types of ideal behavior.

For component 1, the equilibrium equation says

(2-47)

where superscript V refers to the vapor and superscript L to the liquid. We now have the problem of relating the fugacities to the mole fractions. To solve this problem, we make two simplifying assumptions, one for each phase:

Assumption 1. The fugacity f₁^V, at constant temperature and pressure, is proportional to the mole fraction y₁. That is, we assume

(2-48)

where f_{pure 1}^V is the fugacity of pure component 1 as a vapor at the temperature and pressure of the mixture.

Assumption 2. The fugacity f₁^V, at constant temperature and pressure, is proportional to the mole fraction x₁. That is, we assume

(2-49)

where f_{pure 1}^V is the fugacity of pure component 1 as a liquid at the temperature and pressure of the mixture.

Assumptions 1 and 2 are equivalent to saying that both vapor-phase and liquid-phase solutions are ideal solutions; Eqs. (2-47) and (2-48) are statements of the Lewis fugacity rule. These assumptions are valid only for very limited conditions as discussed in later chapters. For mixtures of similar components, however, they provide reasonable approximations based on the naive but attractive supposition that the fugacity of a component in a given phase increases in proportion to its mole fraction in that phase.

Upon substituting Eqs. (2-48) and (2-49) into (2-47), the equilibrium relation now becomes

(2-50)

Equation (2-50) gives an ideal-solution relation using only mole fractions and pure-component fugacities. It is the basis of the original K charts (K = y/x = f ^L/f ^V) used in the petroleum industry. Equation (2-50) can be simplified further by introducing two additional assumptions.

Assumption 3. Pure component 1 vapor at temperature T and pressure P is an ideal gas. It follows that

(2-51)

Assumption 4. The effect of pressure on the fugacity of a condensed phase is negligible at moderate pressures. Further, we assume that the vapor in equilibrium with pure liquid 1 at temperature T is an ideal gas. It follows that

(2-52)

where P₁^s is the saturation (vapor) pressure of pure liquid 1 at temperature T.

Substituting Eqs. (2-51) and (2-52) into (2-50) we obtain

(2-53)

Equation (2-53) is the desired, simple relation known as Raoult’s law.

Equation (2-53) is of limited utility because it is based on severe simplifying assumptions. The derivation of Raoult’s law has been given here only to illustrate the general procedure whereby relations in thermodynamic variables can, with the help of physical arguments, be translated into useful, physically significant, equations. In general, this procedure is considerably more complex but the essential problem is always the same: How is the fugacity of a component related to the measurable quantities temperature, pressure and composition? It is the task of molecular thermodynamics to provide useful answers to this question. All of the chapters to follow are concerned with techniques for establishing useful relations between the fugacity or chemical potential of a compoeent in a phase and physicochemical properties of that phase. To establish such relations, we rely heavily on classical thermodynamics but we also utilize, when possible, concepts from statistical mechanics, molecular physics, and physical chemistry.

References

Bett, K. E., J. S. Rowlinson, and G. Saville, 1975, Thermodynamics for Chemical Engineers. Cambridge: The MIT Press.

Callen, H. B., 1985, Thermodynamics and an Introduction to Thermostatistics, 2^nd Ed. New York: John Wiley & Sons.

Denbigh, K. G., 1981, The Principles of Chemical Equilibrium, 4^th Ed., Chaps. 1 and 2. Cambridge: Cambridge University Press.

Gibbs, J., 1961, The Scientific Papers of J. Willard Gibbs, Vol. I, pp. 55-100. New York: Dover Publications.

Guggenheim, E. A., 1967, Thermodynamics, 5^th Ed., Chap. 1. Amsterdam: North-Holland.

Klotz, I. M. and R. M. Rosenberg, 1994, Chemical Thermodynamics: Basic Theory and Methods, 5^th Ed. New York: John Wiley & Sons.

Kyle, B., 1991, Chemical and Process Thermodynamics, 2^nd Ed. Englewood Cliffs: Prentice-Hall.

Pitzer, K. S., 1995, Thermodynamics, 3^rd Ed. New York: McGraw-Hill.

Redlich, O., 1976, Thermodynamics: Fundamentals, Applications. Amsterdam: Elsevier.

Rowlinson, J. S. and F. L. Swinton, 1982, Liquids and Liquid Mixtures, 3^rd Ed. London: Butterworths.

Prigogine, I. and R. Defay, 1954, Chemical Thermodynamics (Trans./Rev. by D. H. Everett). London: Longmans & Green.

Sandler, S. I., 1989, Chemical and Engineering Thermodynamics, 2^nd Ed. New York: John Wiley & Sons.

Smith, J. M., H. C. Van Ness, and M. M. Abbott, 1996, Introduction to Chemical Engineering Thermodynamics, 5^th Ed. New York: McGraw-Hill.

Tester, J. W. and M. Modell, 1996, Thermodynamics and Its Applications, 3^rd Ed. Englewood Cliffs: Prentice-Hall.

Van Ness, H. C. and M. M. Abbott, 1982, Classical Thermodynamics of Nonelectrolyte Solutions. New York: McGraw-Hill.

Winnick, J., 1997, Chemical Engineering Thermodynamics. New York: John Wiley & Sons.

Problems⁸

⁸ Appendix J gives same fundamental constants and conversion factors to SI units.

1. The volume coefficient of expansion of mercury at 0°C is 18×10^–5(^°C)^–1. The coefficient of compressibility κ is 5.32 ×, 10^-6 (bar)^-1. If mercury were heated from 0°C to 1°C in a constant-volume system, what pressure would be developed?

2. Find expressions for (δS/δV)_T, (δS/δP)_T, (δU/δV)_T, (δU/δP)_T and (δH/δP)_T for a gas whose behavior can be described by the equation

Also find expressions for ΔS, ΔU, ΔH, ΔG, and ΔA for an isothermal change.

3. If the standard entropy of liquid water at 298.15 K is 69.96 J K^-1 mol^-1, calculate the entropy of water vapor in its standard state (i.e., an ideal gas at 298.15 K and 1 bar). The vapor pressure of water is 3168 Pa at 298.15 K and its enthalpy of vaporization is 2.436 kJ g^-1.

4. The residual volume a is the difference between the ideal-gas volume and the actual gas volume. It is defined by the equation

For a certain gas, α has been measured at 100°C and at different molar volumes; the results are expressed by the empirical equation α = 2 – (3/ν²), where ν is in L mol^-1. The velocity of sound w is given by the formula

where g_c is a dimensional constant equal to 1 kg m N^-1 s^-2. Calculate the velocity of sound for this gas at 100°C when its molar volume is 2.3 liter, using k = 1.4. The molar mass is 100 g mol^-1.

5. A gas at 350°C and molar volume 600 cm³ mol^-1 is expanded in an isentropic turbine. The exhaust pressure is atmospheric. What is the exhaust temperature? The ideal-gas heat capacity at constant pressure is c^o_p. The P-V-T properties of the gas are given by the van der Waals equation, with a = 56 × 10⁵ bar (cm³ mol^-1)² and b = 45 cm³ mol^-1.

6. Show that when the van der Waals equation of state is written in the virial form,

the second virial coefficient is given by

7. The second virial coefficient B of a certain gas is given by

where a and b are constants.

Compute the change in internal energy for this gas in going, at temperature τ, from very low pressure to a pressure π. Use the equation

8. Consider the equation of state

where n and m are constants for any gas. Assume that carbon dioxide follows this equation. Calculate the compressibility factor of carbon dioxide at 100°C and at a volume of 6.948 dm³ kg^-1.

9. The volumetric behavior of a gas is satisfactorily described by the equation of state

where a and b are constants. To a very good approximation, the ideal-gas heat capacity of the gas, c^o_p, is temperature-independent. Derive an analytical expression for the molar internal energy of this gas in terms of temperature and molar volume. As a reference state, use one of temperature T₀ = 273 K and molar volume tending to infinity. Under reference state conditions the gas behaves practically as an ideal gas. Constants that may appear in the desired expression for internal energy are a, b, R, c^o_p and T₀.

10. Consider an aqueous mixture of sugar at 25°C and 1 bar pressure. The activity coefficient of water is found to obey a relation of the form

where is normalized such that y_w → 1 as x_w → 1 and A is an empirical constant dependent only on temperature. Find an expression for γ_s, the activity coefficient of sugar normalixed such that γ_s → 1 as x_w → (or as x_s → 0). The mole fractions x_w and x_s refer to water, and sugar, respectively.

11. Consider a binary liquid solution of components 1 and 2. At constant temperature (and low pressure) component 1 follows Henry’s law for the mole fraction range 0 ≤ x₁ ≤ a. Show that component 2 follows Raoult’s law for the mole fraction range (1 - a) ≤ x₂ ≤ 1.

12. Using only data given in the steam tables, compute the fugacity of steam at 320°C and 70 bar.

13. The inversion temperature is the temperature where the Joule-Thomson coefficient changes sign and the Boyle temperature is the temperature where the second virial coefficient changes sign. Show that for a van der Waals gas the low-pressure inversion temperature is twice the Boyle temperature.

A gas, designated by subscript 1, is to be dissolved in a nonvolatile liquid. At a certain pressure and temperature the solubility of the gas in the liquid is x₁ (where x is the mole fraction). Assume that Henry’s law holds. Show that the change in solubility with temperature is given by

where

at the same pressure and temperature. Based on physical reasoning alone, would you expect to be positive or negative?