Chapter 6
Fugacities in Liquid Mixtures: Excess Functions

Calculation of fugacities from volumetric properties was discussed in Chap. 3; many of the relations derived there [in particular, Eqs. (3-14) and (3-53)] are general and may be applied to condensed phases as well as to the gas phase. However, often it is not practical to do so because the necessary integrations require that volumetric data be available at constant temperature and constant composition over the entire density range from the ideal-gas state (zero density) to the density of the condensed phase, including the two-phase region. It is a tedious task to obtain such data for fluid mixtures and very few data of this type have been reported. A more useful alternate method, therefore, is needed for calculation of fugacities in liquid solutions. Such a method is obtained by defining an ideal liquid solution and by describing deviations from ideal behavior in terms of excess functions; these functions yield the familiar activity coefficients that give a quantitative measure of departure from ideal behavior.

The fugacity of component i in a liquid solution is most conveniently related to the mole fraction x_i by

(6-1)

where γ_i is the activity coefficient and is the fugacity of i at some arbitrary condition known as the standard state. At any composition, the activity coefficient depends on the choice of standard state and the numerical value of γ_i has no significance unless the numerical value of is also specified.

Because the choice of standard state is arbitrary, it is convenient to choose such that γ_i assumes values close to unity and when, for a range of conditions, γ_i is exactly equal to unity, we say that the solution is ideal.¹ However, because of the intimate relation between the activity coefficient and the standard-state fugacity, the definition of solution ideality (γ_i = 1) is not complete unless the choice of standard state is clearly indicated. Either of two choices is frequently used. One of these leads to an ideal solution in the sense of Raoult’s law and the other leads to an ideal solution in the sense of Henry’s law.

¹ The two expressions “ideal solution” and “ideal mixture” are both in common usage. They are equivalent.

6.1 The Ideal Solution

The history of modern science has shown repeatedly that a quantitative description of nature can often be achieved most successfully by first idealizing natural phenomena, i.e., by setting up a simplified model, either physical or mathematical, that crudely describes the essential behavior while neglecting details. (One of the outstanding characteristics of great contributors to modern science has been their ability to distinguish between what is essential and what is incidental.) The behavior of nature is then related to the idealized model by various correction terms that can be interpreted physically and that sometimes can be related quantitatively to those details in nature that were neglected in the process of idealization.

An ideal liquid solution is one where, at constant temperature and pressure, the fugacity of every component is proportional to some suitable measure of its concentration, usually the mole fraction. That is, at some constant temperature and pressure, for any component i in an ideal solution,

(6-1a)

where ℜ_i is a proportionality constant dependent on temperature and pressure but independent of x_i.

We notice at once from Eq. (6-1) that, if we let = ℜ_i, then γ_i = 1. If Eq. (6la) holds for the entire range of composition (from x_i = 0 to x_i = 1), the solution is ideal in the sense of Raoult’s law. For such a solution it follows from the boundary condition at x_i = 1 that the proportionality constant ℜ_i is equal to the fugacity of pure liquid i at the temperature of the solution.² For this case, if the fugacities are set equal to partial pressures, we then obtain the familiar relation known as Raoult’s law.

² The standard-state fugacity of pure liquid i at system temperature is usually taken either at the saturation pressure of pure i, or else at P, the total pressure of the mixture. The latter choice is more common, especially at Sow or moderate pressures. At high pressures, special care must be taken in specifying the pressure of the standard state.

In many cases, the simple proportionality between and x_i holds only over a small range of composition. If x_i is near zero, it is still possible to have an ideal solution according to Eq. (6-la), without, however, equating ℜ_i to the fugacity of pure liquid i. We call such a solution an ideal dilute solution leading to the familiar relation known as Henry’s law.

The strict definition of an ideal solution requires that Eq. (6-la) must hold not only at a special temperature and pressure of interest but also at temperatures and pressures in their immediate vicinity. This feature leads to an important conclusion concerning heat effects and volume changes of mixing for a solution ideal in the sense of Raoult’s law. For such a solution, we have, at any T, P and x,

(6-2)

where, for convenience, we have deleted superscript L. We now use two exact thermodynamic relations:³

³ Equation. (6-3) is a form of the Gibbs-Helmhcltz equation.

(6-3)

and

(6-4)

where is the partial molar enthalpy of component i in the liquid phase, h_i is the enthalpy of pure liquid i, is the enthalpy of pure i in the ideal-gas state, is the partial molar volume of i, and υ_i, is the molar volume of pure i, both in the liquid phase, all at system temperature T and pressure P. Upon substitution of Eq. (6-2) we find that

(6-5)

and

(6-6)

Because the partial molar enthalpy and partial molar volume of component i in an ideal solution are, respectively, the same as the molar enthalpy and molar volume of pure i at the same temperature and pressure, it follows that the formation of an ideal solution occurs without evolution or absorption of heat and without change of volume.

Mixtures of real fluids do not form ideal solutions, although mixtures of similar liquids often exhibit behavior close to ideality. However, all solutions of chemically stable nonelectrolytes behave as ideal dilute solutions in the limit of very large dilution. The correction terms that relate the properties of real solutions to those of ideal solutions are called excess functions.

6.2 Fundamental Relations of Excess Functions

Excess functions are thermodynamic properties of solutions that are in excess of those of an ideal (or ideal dilute) solution at the same conditions of temperature, pressure, and composition.⁴ For an ideal solution all excess functions are zero. For example, G^E, the excess Gibbs energy, is defined by

⁴ Most excess functions, but not all, are extensive. However, it follows from the definition that we cannot have an excess pressure, temperature, or composition. See R. Missen, 1969, Ind. Eng. Ghent. Fundam., 8: 81.

(6-7)

Similar definitions hold for excess volume V^E, excess entropy S^E, excess enthalpy H^E, excess internal energy U^E, and excess Helmholtz energy A^E. Relations between these excess functions are exactly the same as those between the total functions:

(6-8)

(6-9)

(6-10)

Also, partial derivatives of extensive excess functions are analogous to those of the total functions. For example,

(6-11)

(6-12)

(6-13)

Excess functions may be positive or negative; when the excess Gibbs energy of a solution is greater than zero the solution is said to exhibit positive deviations from ideality, whereas if it is less than zero the deviations from ideality are said to be negative.

Partial molar excess functions are defined in a manner analogous to that used for partial molar thermodynamic properties. If M is an extensive thermodynamic property, then , the partial molar M of component i, is defined by

(6-14)

where n_i is the number of moles of i and where subscript n_j designates that the number of moles of all components other than i are kept constant. Similarly,

(6-15)

Also, from Euler’s theorem, we have that

(6-16)

It then follows that

(6-17)

For our purposes, an extensive excess property is a homogeneous function of the first degree in the mole numbers.⁵

⁵ Many, but not all, extensive excess properties can be defined in this way. See Redlich, 1968, "Fundamental Thermodynamics since Caratheodory," Rev. Mod. Phys., 40: 556; and O. Redlich, 1976, Thermodynamics: Fundamentals, Applications, (Amsterdam: Elsevier).

For phase-equilibrium thermodynamics, the most useful partial excess property is the partial molar excess Gibbs energy that is directly related to the activity coefficient. The partial molar excess enthalpy and partial molar excess volume are related, respectively, to the temperature and pressure derivatives of the activity coefficient. These relations are summarized in the next section.

6.3 Activity and Activity Coefficients

The activity of component i at some temperature, pressure, and composition is defined as the ratio of the fugacity of i at these conditions to the fugacity of i in the standard state, that is a state at the same temperature as that of the mixture and at some specified condition of pressure and composition:

(6-18)

where p⁰ and x⁰ are, respectively, an arbitrary but specified pressure and composition.

The activity coefficient γ_i is the ratio of the activity of i to some convenient measure of the concentration of i, usually the mole fraction⁶

⁶ For electrolyte solutions, it is often more convenient to use molality instead of mole fraction (see Sec. 9.1). For polymer solutions, mole fractions are not useful; instead, weight fractions or volume fractions are more appropriate.

(6-19)

The relation between partial molar excess Gibbs energy and activity coefficient is obtained by first recalling the definition of fugacity. At constant temperature and pressure, for a component i in solution,

(6-20)

Next, we introduce the partial molar excess function by differentiation of Eq. (6-7) at constant T, P, and n_j:

(6-21)

Substitution then gives

(6-22)

and substituting Eq. (6-la), we obtain

(6-23)

It follows from Eq. (6-la) that an ideal solution is one where the activity is equal to the mole fraction; if we set the standard-state fugacity f_i⁰ equal to ℜ_i we then have

(6-24)

But for an ideal solution [Eq. 6-la)], f_i is equal to ℜ_ix_i and therefore, for an ideal solution, γ_i = 1 and a_i = x_i. Substitution of Eq. (6-24) into Eq. (6-23) gives the important and useful result⁷

⁷ A shorter but equivalent derivation of Eq. (6-25) follows from writing
                                                        
Because f_i = y_i x_i f_i⁰, and f_i(ideal) = x_i f_i⁰ we obtain =RT In γ_i.

(6-25)

Substitution into Eq. (6-17) gives the equally important relation⁸

⁸ Equations. (6-25) and (6-26) are related. Equation. (6-26) can be rewritten as where n_i is the number of moles of component i and n_T is the total number of moles. Differentiating with respect to n_i at constant T, P, and all other mole numbers n_j,
                                                  
where the summation is over all components, including i. The Gibbs-Duhem equation asserts that this summation is zero, yielding Eq. (6-25).

(6-26)

where g^E is the molar excess Gibbs energy. Equations. (6-25) and (6-26) are used repeatedly in the remainder of this chapter as well as in later chapters.

We now want to consider the temperature and pressure derivatives of the activity coefficient. Let us first discuss the case where the excess Gibbs energy is defined relative to an ideal solution that is ideal over the entire range of composition in the sense of Raoult’s law. In this case,

(6-27)

and

(6-28)

Using Eq. (6-3), differentiation with respect to temperature at constant P and x gives

(6-29)

where is the partial molar enthalpy of i minus the molar enthalpy of pure liquid i at the same temperature and pressure. Differentiation with respect to pressure at constant T and x gives⁹

⁹ In some cases ℜ_i is set equal to the fugacity of pure liquid i at temperature T and at its own saturation pressure P_i^s (rather than the total pressure P, that may vary with the composition in an isothermal mixture). In that case Eq. (6-29) is not affected significantly because the enthalpy of a pure liquid is usually a weak function of pressure. However, Eq. (6-30) becomes

(6-30a)

(6-30)

where is the partial molar volume of i minus the molar volume of pure liquid i at the same temperature and pressure.

Now let us consider the case where the excess Gibbs energy is defined relative to an ideal dilute solution. It is useful to define excess functions relative to an ideal dilute solution whenever the liquid mixture cannot exist over the entire composition range, as happens, for example, in a liquid mixture containing a gaseous solute. If the critical temperature of solute 2 is lower than the temperature of the mixture, then a liquid phase cannot exist as x₂ → 1, and relations based on an ideal mixture in the sense of Raoult’s law can be used only by introducing a hypothetical standard state for solute 2.

However, relations based on an ideal dilute solution avoid this difficulty. The proportionality constant ℜ₂ is not determined from the pure-component boundary condition x₂ = 1, but rather from the boundary condition of the infinitely dilute solution, i.e., x₂ → 0. For an ideal dilute solution, we have for the solute 2^l0

¹⁰ For any binary system, Henry’s ccmstant H_2,1 depends on both temperature and pressure. At temperatures remote from the solvent’s critical temperature, unless the pressure is large, the effect of pressure on H_2,1 is often negligible (see Sec. 10.3).

(6-31)

where H_2,1 is Henry’s constant for solute 2 in solvent 1.

However, for the solvent (component 1, present in excess) we obtain the same result as before:

(6-32)

For the solute, the activity coefficient is given by

(6-33)

Substituting into Eq. (6-25) and differentiating with respect to temperature we obtain, as before,

(6-34)

However, now has a different meaning; it is given by

(6-35)

where is the partial moiar enthalpy of solute 2 in an infinitely dilute solution.

The effect of pressure on the activity coefficient of the solute is given by

(6-36)

where the partial molar volume of solute 2.¹¹ The derivatives with respect to temperature and pressure of the activity coefficient for component 1 (the solvent) are the same as those given by Eqs. (6-29) and (6-30).

¹¹ The derivation of Eq. (6-36) requires that Henry’s constant in Eq. (6-31) be evaluated at the temperature of the solution and at some pressure that usually is the saturation pressure of pure 1. The relationship between Henry’s constant at system pressure P and Henry’s constant at pressure p_i^s is given by

If, in Eq. (6-31), H_2,1(P) is used rather than H_2,1 (p₁³), then Eq. (6-36) must be replaced by

(6-36a)

where is the partial molar volume of solute 2 at infinite dilution.

For all components in a mixture, the partial molar excess Gibbs energies (and the activity coefficients) are related to one another by a fundamental relation known as the Gibbs-Duhem equation, discussed in Sec. 6.6.

6.4 Normalization of Activity Coefficients

It is convenient to define activity in such a way that for an ideal solution, activity is equal to the mole fraction or, equivalently, that the activity coefficient is equal to unity. Since we have distinguished between two types of ideality (one leading to Raoult’s law and the other leading to Henry’s law), it follows that activity coefficients may be normalized (that is, become unity) in two different ways.

If activity coefficients are defined with reference to an ideal solution in the sense of Raoult’s law [Eq. (6-27)], then for each component i the normalization is

(6-37)

Because this normalization holds for both solute and solvent, Eq. (6-37) is called the symmetric convention for normalization.

However, if activity coefficients are defined with reference to an ideal dilute solution [Eq. (6-31)], then

(6-38)

Because solute and solvent are not normalized in the same way, Eq. (6-38) gives the unsymmetric convention for normalization. To distinguish between symmetrically and unsymmetrically normalized activity coefficients, it is useful to denote with an

asterisk (*) the activity coefficient of a component that approaches unity as its mole fraction goes to zero. With this notation, Eq. (6-38) becomes

(6-38a)

The two methods of normalization are illustrated in Fig. 6-1. In the dilute region (x₂ << 1), γ₂^*=1 and the solution is ideal;¹² however, γ₂ ≠ 1 and therefore, while the dilute solution is ideal in the sense of Henry’s law, it is not ideal in the sense of Raoult’s law.

¹² It can readily be shown from the Gibbs-Duhem equation that for the region where γ₂^* = 1, Y₁ = 1.

Figure 6-1 Normalization of activity coefficients.

Figure 6-2 shows experimental activity coefficient data for typical aqueous solutions that follow the symmetric convention (γ_methanol → 1 as x_methanol → 1) and the unsymrnetric convention (γ* _NaCl → 1 as x_NaCl → 0) for normalization of activity coefficients. Methanol activity coefficients become essentially constant for low solute concentrations. (The activity coefficient of methanol at infinite dilution in water at 25°C is γ^∞ = 1.74). In contrast, ideal behavior of aqueous NaCl solutions is approached as the solution becomes infinitely dilute.

Figure 6-2 Symmetric and unsymmetric activity coefficient conventions. Experimental data at 25°C for the activity coefficients of methanol in water and sodium chloride in water (Ragal et ai, 1994). Solid lines are smooth data and dashed lines are extrapolations.

Symmetric normalization of activity coefficients is easily extended to solutions containing more than two components; the activity coefficient of any component approaches unity as its mole fraction goes to unity. However, as discussed elsewhere,¹³ extension of unsymmetric normalization of activity coefficients to multicomponent solutions requires care.

¹³ H. C. Van Ness and M. M Abbott, 1979, AIChEJ., 25: 645.

In binary mixtures, activity coefficients that are normalized symmetrically are easily related to activity coefficients that are normalized unsymmetrically. The definitions of γ₂ and γ₂^* are

(6-39)

(6-40)

Therefore,

(6-41)

Because

(6-42)

we obtain

(6-43)

Substitution in Eq. (6-41) gives

(6-44)

By a similar argument, we can also show that

(6-45)

Both Eqs. (6-44) and (6-45) relate to each other the two activity coefficients of the solute, one normalized by the symmetric convention and the other by the unsymmetric convention. However, Eq. (6-44) is much more useful than Eq. (6-45) because the limit given on the right side of Eq. (6-44) corresponds to a real physical situation, whereas the limit on the right side of Eq. (6-45) corresponds to a situation that is hypothetical (physically unreal) whenever component 2 cannot exist as a pure liquid at the temperature of the solution.

6.5 Activity Coefficients from Excess Functions in Binary Mixtures

At a fixed temperature, the molar excess Gibbs energy g^E of a mixture depends on the composition of the mixture and, to a smaller extent, on pressure. At low or moderate pressures, well removed from critical conditions, the effect of pressure is negligible; it is therefore not considered in this section.¹⁴

¹⁴ The effect of pressure on g^E is important only if the pressure is large, as discussed in See. 12.4.

We now consider a binary mixture where the excess properties are taken with reference to an ideal solution wherein the standard state for each component is the pure liquid at the temperature and pressure of the mixture. In that case, any expression for the molar excess Gibbs energy must obey the two boundary conditions:

g^E = 0       when       x₁ = 0

g^E = 0       when       x₂ = 0

Two-Suffix Margules Equation. The simplest nontrivial expression that obeys these boundary conditions is

(6-46)

where A is an empirical constant with units of energy, characteristic of components 1 and 2, that depends on the temperature but not on composition.

Equation (6-46) immediately gives expressions for activity coefficients γ₁ and γ₂ by substitution in the relation between activity coefficient and excess Gibbs energy [Eq. (6-25)]:

(6-47)

where n_i is the number of moles of i and n_T is the total number of moles. Remembering that x₁ = n₁/n_T and x₂ = n₂/n_T we obtain

(6-48)

(6-49)

Equations (6-48) and (6-49), often called the two-suffix Margules equations, provide a good representation for many simple liquid mixtures, i.e., for mixtures of molecules that are similar in size, shape, and chemical nature. The two equations are symmetric: When In γ₁ and In γ₂ are plotted against x₂ (or x₁), the two curves are mirror images. At infinite dilution, the activity coefficients of both components are equal:

(6-50)

(6-51)

Coefficient A may be positive or negative, and while it is in general a function of temperature, it frequently happens that for simple systems over a smail temperature range, A is nearly constant. For example, vapor-liquid equilibrium data of Pool et al. (1962), for argon/oxygen are well represented by the two-suffix Margules equations as shown in Fig. 6-3. At 83.8 K, A is 148.1 J mol^-1 and at 89.6 K, A is 141.0J mol^-1 For this simple system, A is a weak function of temperature.

Figure 6-3 Applicability of two-suffix Margutes equation to simple binary mixtures.

Another binary system whose excess Gibbs energy is well represented by the two-suffix Margules equation is the benzene/cyclohexane system studied by Scatchard et al. (1939). Their results are also shown in Fig. 6-3. The variation of A with temperature is again not large, although not negligible; at 30, 40, and 50°C, the values of A are, respectively, 1268, 1185, and 1114 J mol^-1. In nonpolar solutions, A frequently falls with rising temperature.

Equation. (6-46) is a very simple relation; in the general case, a more complex equation is needed to represent adequately the excess Gibbs energy of a binary solution. Because the boundary conditions given just before Eq. (6-46) must be obeyed regardless of the complexity of the solution, one convenient extension of Eq.(6-46) is to write a series expansion:

(6-52)¹⁵

¹⁵ For binary mixtures containing one highly associating component, often better representation of the composition dependence of g^E (or any other excess function) is obtained by multiplying the Redlich-Kister expansion by a skewing factor:

where K is a constant to be determined from experimental data. If K = 0, Eq. (6-52) is recovered.

where B, C, D,… are additional, temperature-dependent parameters that must be determined from experimental data. Equation (6-55) is the Redlich-Kister expansion and, upon using Eq. (6-47), we obtain these expressions for the activity coefficients

(6-53)

(6-54)

where

a(1) = A + 3B + 5C + 7D	a(2) = A – 3B – 5C – 7D
b(1) = –4(B + 4C + 9D)	b(2) = 4(B – 4C + 9D)
c(1) = 12(C + 5D)	c(2) = 12(C – 5D)
d(1) = –32D	d(2) = 32D

The number of parameters (A, B, C,…) that should be used to represent the experimental data depends on the molecular complexity of the solution, on the quality of the data, and on the number of data points available. Typical vapor-liquid equilibrium data reported in the literature justify no more than two or at most three constants; very accurate and extensive data are needed to warrant the use of four or more empirical parameters.

The Redlich-Kister expansion provides a flexible algebraic expression for representing the excess Gibbs energy of a liquid mixture. The first term in the expansion is symmetric in x and gives a parabola when g^E is plotted against x. The odd-powered correction terms [first (B), third (D),…] are asymmetric in x and therefore tend to skew the parabola either to the left or right. The even-powered correction terms [second (C), fourth (E),…] are symmetric in x and tend to flatten or sharpen the parabola. To illustrate, we show in Fig. 6-4 the first three terms of the Redlich-Kister expansion for unit value of the coefficients A, B, and C.

Figure 6-4 Contributions to g^E in Redlich-Kister equation (for A = B = C = 1).

Redlich-Kister equations provide not only a convenient method for representing liquid-phase activity coefficients, but also for classifying different types of liquid solutions. From experimental data, γ₁ and γ₂ are calculated and these are then plotted as log(γ₁/γ₂) versus x₁ as shown in Figs. 6-5, 6-6, and 6-7 taken from Redlich et al. (1952). Strictly, Eqs. (6-52), (6-53), and (6-54) apply to isothermal data, but they are often applied to isobaric data; provided that the temperature does not change much with composition, this practice does not introduce large errors. From Eqs. (6-53) and (6-54) we obtain, after some rearrangement,

(6-55)

For simple solutions, B = C = D = … = 0, and a plot of log(γ₁/γ₂) versus x₁ gives a straight line, as shown in Fig. 6-5 for the system n-hexane/toluene. In this case a good representation of the data is obtained with A/RT = 0.352.

Figure 6-5 Activity-coefficient ratio for a simple mixture. Experimental data for n-hexane (1)/toluene (2) at 1.013 bar. The line is drawn to satisfy the area (consistency) test given by Eq. (6-92).

Data for a somewhat more complicated solution are shown in Fig. 6-6. In this case the plot has some curvature and two parameters are required to represent it adequately; they are A/RT = 0.433 and B/RT = 0.104. Although the systems in Figs. 6-5 and 6-6 are similar (both are mixtures of a paraffinic and an aromatic hydrocarbon), it may appear surprising that two parameters are needed for the second system, while only one is required for the first. It is likely that two parameters are required for the second system because the difference in the molecular sizes of the two components is much larger in the isooctane/benzene system than it is in the n-hexane/toluene system. At 25°C, the ratio of molar volumes (paraffin to aromatic) in the system containing benzene is 1.86, while it is only 1.23 in the system containing toluene. The effect of molecular size on the representation of activity coefficients is shown more clearly by Wohl’s expansion discussed in Sec. 6.10.

Figure 6-6 Activity-coefficient ratio for a mixture of intermediate complexity. Experimental data for benzene (1)/isooctane (2) at total pressures ranging from 0.981 to 1.013 bar.

Finally, Fig. 6-7 shows data for a highly complex solution. The plot is not only curved but has a point of inflection; four parameters are required to give an adequate representation. This mixture, containing an alcohol and a saturated hydrocarbon, is complex because the degree of hydrogen bonding of the alcohol is strongly dependent on the composition, especially in the region dilute with respect to alcohol.

Figure 6-7 Activity-coefficient ratio for a highly complex mixture. Experimental data for ethanol (1)/methylcyc!ohexane (2) in the region 30-35°C.

The number of parameters in Eq. (6-55) required to represent activity coefficients of a binary mixture gives an indication of the apparent complexity of the mixture, thereby providing a means for classification. If the number of required parameters is large (four or more parameters), the mixture is classified as a complex solution, and if it is small (one parameter), the mixture is classified as a simple solution. Most solutions of nonelectrolytes commonly encountered in chemical engineering are of intermediate complexity, requiring two or three parameters in the Redlich-Kister expansion.

Classification of solutions as determined by the number of required parameters is necessarily arbitrary. When a good fit of experimental data is obtained with one or two parameters, we cannot necessarily conclude that the mixture is truly simple in a molecular sense; in many cases a one- or two-parameter equation may fortuitously provide an adequate representation. For example, binary mixtures of two alcohols (such as methanol/ethanol) are certainly highly complex because of the various types of hydrogen bonds that can exist in such mixtures; nevertheless, it is frequently possible to represent the excess Gibbs energies of such solutions with a one- or two-parameter equation.

Similarly, the system acetic acid/water is surely complex. However, as shown in Fig. 6-8, the excess Gibbs energy is nearly a parabolic function of the composition. This apparent simplicity is due to cancellation; when we look at excess enthalpy and excess entropy, the mixture’s complexity becomes apparent.

Figure 6-8 Excess functions for the acetic acid (1)/water (2) system at 25°C (R. Haase etal., 1973, Z Naturforschung, 28a: 1740).

Numerous equations have been proposed for expressing analytically the composition dependence of the excess Gibbs energies of binary mixtures. Most of these equations are empirical and we mention several of the better known ones in Sec. 6.10 but some have at least a little theoretical basis and we discuss a few of these in Chap. 7. However, before continuing our presentation of equations for representing excess Gibbs energies of nonideal liquid mixtures, we first want to discuss how the Gibbs-Duhem equation can be used to reduce experimental effort and to test experimental data for thermodynamic consistency.

6.6 Activity Coefficients for One Component from Those of the Other Components

In a mixture, the partial molar properties of the components are related to one another by one of the most useful equations in thermodynamics, the Gibbs-Duhem equation. This equation says that at constant temperature and pressure,

(6-56)

where is any partial molar property. Equation (6-56) holds for ideal as well as real solutions and can be rewritten in terms of excess partial properties:

(6-57)

A derivation and more detailed discussion of this equation is given in App. D.

There are two important applications of Eq. (6-57). First, in the absence of complete experimental data on the properties of a mixture, Eq. (6-57) frequently may be used to calculate additional properties; for example, in a binary solution, if experimental measurements over a range of concentration yield activity coefficients of only one component, activity coefficients of the other component can be computed for the same concentration range. Second, if experimental data are available for a directly measured partial molar property for each component over a range of composition, it is then possible to check the data for thermodynamic consistency; if the data satisfy the Gibbs-Duhem equation, they are thermodynamically consistent and it is likely that they are reliable, but if they do not, it is certain that they are incorrect.

While the Gibbs-Duhem equation is applicable to all partial excess properties, it is most useful for the partial molar excess Gibbs energy that is directly related to the activity coefficient by Eq. (6-25). In terms of activity coefficients, Eq. (6-57) is

(6-58)

Equation (6-58) is a differential relation between the activity coefficients of all the components in the solution. Hence, in a solution containing m components, data for activity coefficients of m – l components may be used to compute the activity coefficient of the mth component. To illustrate, consider the simplest case, i.e., a binary solution for which isothermal data are available for one component and where the pressure is sufficiently low to permit neglect of the effect of pressure on the liquid-phase activity coefficient. For this case, the Gibbs-Duhem equation may be written

(6-59)

Assume now that data have been obtained for γ₁ for various x₁ By graphical integration of Eq. (6-59) one may then obtain values of γ₂. A simpler procedure, less exact in principle but much easier to use, is to curve-fit the data for γ₁ to an algebraic expression using x₁ (or x₂) as the independent variable. Once such an expression has been obtained, the integration of Eq. (6-59) may be performed analytically. For this purpose it is convenient to rewrite Eq. (6-59):

(6-60)

Assume now, as first suggested in 1895 by Margules, that the data for γ₁ can be represented by an empirical equation of the form¹⁶

¹⁶ We need β_k >1 to avoid singularities in In γ₂ when x₂ = 1.

(6-61)

where α_k and β_k are empirical constants to be determined from the data. Substituting (6-61) into (6-60) yields

(6-62)

Integrating Eq. (6-62) gives

(6-63)

where I is a constant of integration. To eliminate Inγ₁ in Eq. (6-63), we substitute Eq. (6-61):

(6-64)

To evaluate I it is necessary to impose a suitable boundary condition. If component 2 can exist as a pure liquid at the temperature of the solution, then it is common to use pure liquid component 2 at that temperature as the standard state for γ₂; in that case,

γ₂ = l when x₂ = 1.

The constant of integration then is

(6-65)

and the expression for Inγ₂ becomes

(6-66)

Equation (6-66) is a general relation and it is strictly for convenience that one customarily uses only positive integers for β_k. To illustrate the use of Eq. (6-66), suppose that the data for γ₁ can be adequately represented by Eq. (6-61) terminated after the fourth term (k = 2, 3, 4) with β_k = K. Equation (6-61) becomes

(6-67)

known as the four-suffix Margules equation.¹⁷ Coefficients α₂, α₃, and α₄ must be found from the experimental data that give γ₁ as a function of the mole fraction. When this four-suffix Margules equation for Inγ₁ is substituted into Eq. (6-66), the result for γ₂ is

¹⁷ The n-sufflx Margules equation gives Inγ₁ as a polynomial in x₂ of degree n.

(6-68)

The important feature of Eq. (6.68) is that γ₂ is given in terms of constants that are determined exclusively from data for γ₁.

In a binary system, calculating the activity coefficient of one component from data for the other is a common practice whenever the two components in the solution differ markedly in volatility. In that event, the measurements frequently give the activity coefficient of only the more volatile component and the activity of the less volatile component is found from the Gibbs-Duhem equation. For example, if one wished to have information on the thermodynamic properties of some high-boiling liquid (such as a polymer) dissolved in, say, benzene, near room temperature, then the easiest procedure would be to measure the activity (partial pressure) of the benzene in the solution and to compute the activity of the other component from the Gibbs-Duhem equation as outlined above; in this case it would not be practical to measure the extremely small partial pressure of the high-boiling component over the solution.

While the binary case is the simplest, the method just discussed can be extended to systems of any number of components. In these more complicated cases the computational work is greater but the theoretical principles are the same.

In carrying out numerical work, it is important that activity coefficients be calculated as rigorously as possible from the equilibrium data. We recall the definition of activity coefficient:

(6-69)

Quantities f_i and f_i⁰ must be computed with care; in vapor-liquid equilibrium, gas-phase corrections for both of these quantities are frequently important. For solutions of liquids, it is useful to use the pure component as the standard state and, to simplify calculations, it is common practice to evaluate γ_i from the data by the simplified expression

(6-70)

where P_i^s is the saturation (vapor) pressure of pure i. Equation (6-70), however, is only an approximate form of Eq. (6-69) because it neglects gas-phase corrections and the Poynting factor. In some cases, the approximation is justified but before Eq. (6-70) is used, it is important to inquire if the simplifying assumptions apply to the case under consideration. For mixtures of strongly polar or hydrogen-bonding components, or for mixtures at cryogenic temperatures, gas-phase corrections may be significant even at pressures near or below 1 bar.

The Gibbs-Duhem equation interrelates the activity coefficients of all the components in a mixture. In a binary mixture, Inγ₁ and Inγ₂ are not independent, as indicated by Eq. (6-59) and the subsequent relations, Eqs. (6-67) and (6-68). The interrelation of the activity coefficients can be used to reduce experimental effort, as indicated in the next two sections, and to test experimental data for thermodynamic consistency, as indicated in Sec. 6.9.

6.7 Partial Pressures from Isothermal Total-Pressure Data

Complete description of vapor-liquid equilibrium for a system gives equilibrium compositions of both phases as well as temperature and total pressure. In a typical experimental investigation, temperature or total pressure is held constant; in a system of m components, complete measurements require that for each equilibrium state, data must be obtained for either the temperature or pressure and for 2(m – 1) mole fractions. Even in a binary system this represents a significant experimental effort; it is advantageous to reduce this effort by utilizing the Gibbs-Duhem equation for calculation of at least some of the desired information.

We consider a procedure for calculating partial pressures from isothermal total-pressure data for binary systems. According to this procedure, total pressures are measured as a function of the composition of one of the phases (usually the liquid phase) and no measurements are made of the composition of the other phase. Instead, the composition of the other phase is calculated from the total-pressure data with the help of the Gibbs-Duhem equation. The necessary experimental work is thereby much reduced.

Numerous techniques have been proposed for making this kind of calculation and it is not necessary to review all of them here. However, to indicate the essentials of these techniques, one representative and useful procedure is given below.

Barker’s Numerical Method.¹⁸ The total pressure for a binary system is written

¹⁸ Barker (1953); Abbott and Van Ness (1977).

(6-71)

where P_i^s' is the “corrected” vapor pressure of component i:

(6-72)

(6-73)

where δ₁₂ is related to the second virial coefficients by

(6-74)

At constant temperature, activity coefficients γ₁ and γ₂ are functions only of composition.

Equation (6-71) is rigorous provided we assume that the vapor phase of the mixture, as well as the vapors in equilibrium with the pure components, are adequately described by the volume-explicit virial equation terminated after the second virial coefficient; that the pure-component liquid volumes are incompressible over the pressure range in question; and that the liquid partial molar volume of each component is invariant with composition. The standard states for the activity coefficients in Eq. (6-71) are the pure components at the same temperature and pressure as those of the mixture.¹⁹

¹⁹ From these assumptions it follows that

(6-75)

and

(6-76)

Barker’s method is used to reduce experimental data that give the variation of total pressure with liquid composition at constant temperature. One further relation is needed in addition to Eqs. (6-71) to (6-74), and that is an equation relating the activity coefficients to mole fractions. This relation may contain any desired number of undetermined numerical coefficients that are then found from the total pressure data as shown below. For example, suppose we assume that

(6-77)

where α and β are unknown constants. Then, from the Gibbs-Duhem equation [Eq. (6-68)] it follows that

(6-78)

Equations (6-71) to (6-78) contain only two unknowns, α and β. (It is assumed that values for the quantities υ^L, B, P^s, and δ₁₂ are available. It is true that y is also unknown but oace α and β are known, y can be determined.)

In principle, Eqs. (6-71) to (6-78) could yield α and β using only two points on the experimental P-x curve. In practice, however, more than two points are required; we prefer to utilize all reliable experimental points and then optimize the values of α and β to give the best agreement between the observed total pressure curve and that calculated with parameters α and β.

The calculations are iterative because y_l and y₂ can only be calculated after α and β have been determined; the method of successive approximations must be used. In the first approximation, γ₁ and γ₂ are set equal to zero in Eqs. (6-72) and (6-73). Then α and β are found, and immediately thereafter y₁ and y₂ are computed (from the first approximation of the parameters α and β) using Eqs. (6-75) to (6-78). The entire calculation is then repeated except that the new values of y_l and y₂ are now used in Eqs. (6-72) and (6-73). We proceed in this way until the assumed and calculated values of y₁ and y₂ are in agreement; usually three or four successive approximations are sufficient. The form of Eqs. (6-77) and (6-78) is arbitrary; we may use any desired set of equations with any desired number of constants, provided that the two equations satisfy the Gibbs-Duhem equation.

Although Barker’s numerical method is too complicated for manual calculation, it can easily be programmed for a computer capable of rapidly transforming isothermal P-x data to isothermal y-x data. An illustration of Barker’s method is provided by Hermsen (1963) who measured isothermal total vapor pressures for the benzene/cyclopentane system at 25, 35, and 45°C. Hermsen assumed that the excess Gibbs energy of this system is described by a two-parameter expansion of the Redlich-Kister type:²⁰

²⁰ Coefficients A′ and B′ are dimensionless.

(6-79)

where subscript 1 refers to benzene and subscript 2 to cyclopentane. Activity coefficients are obtained from Eq. (6-79) by differentiation according to Eq. (6-25). They are

(6-80)

and

(6-81)

As indicated by Eqs. (6-72) and (6-73), Barker’s method requires the molar liquid volumes of the pure liquids and the three second virial coefficients B₁₁, B₂₂, and B₁₂. For the benzene/cyclopentane system these are given in Table 6-1.

Table 6-1 Second virial coefficients and liquid moiar volumes* for benzene (1) and cyoiopentane (2) (cm³ mol^-1).

Hermsen’s calculated and experimental results are given in Table 6-2a. At a given temperature, the total pressure and the liquid composition were determined. Vapor-phase compositions were not measured. From the experimental measurements, optimum values of A′ and B′ were found such that the calculated total pressures [Eq. (6-71)] reproduce as closely as possible the experimental ones. The optimum values are given in Table 6-2b.

Table 6-2a Experimental and calculated results for the system benzene (1)/cyclopentane (2).*

Table 6-2b Constants in Eq. (6-79) for the system benzene (1)/cyclopentane (2).*

In the benzene/cyclopentane system, deviations from ideality are not large and only weakly asymmetric; therefore, a two-parameter expression for the excess Gibbs energy is sufficient. For other systems, where the excess Gibbs energy is large or strongly asymmetric, it is advantageous to use a more flexible expression for the excess Gibbs energy, or, for example, to include higher terms in the Redlich-Kister expansion. To illustrate, Orye (1965) has measured total pressures for five binary systems containing a hydrocarbon and a polar solvent and he has reduced these data with Barker’s method, using a three-parameter Redlich-Kister expansion for the excess Gibbs energy. Orye’s results at 45°C are given in Table 6-3, and a typical plot of total and partial pressures is shown for one of the systems in Fig. 6-9. To justify three parameters, the number of experimental determinations of total pressure must necessarily be larger than that required to justify only two parameters. Whereas measurements for six compositions at any one temperature were sufficient for Hermsen to fix two parameters for the moderately nonideal benzene/cyclopentane system, Orye’s measurements included about 15 compositions to specify three parameters for the more strongly nonideal hydrocarbon/polar-solvent systems.

Table 6-3 Excess Gibbs energies of five binary systems obtained from total-pressure measurements at 45°C.*

Figure 6-9 Total and partial pressures of toluene (1)/2,3-butanedione (2) at 45°C.

Total-pressure measurements at constant temperature are particularly convenient for obtaining excess Gibbs energies of those binary liquid mixtures whose components have similar volatilities. Such measurements can be made rapidly; with care and experience, they can be very accurate.

A detailed discussion of data reduction is given by Van Ness and coworkers (Byer et al, 1973; Abbott and Van Ness, 1975) who emphasize that isothermal P-x measurements often provide the best source of experimental data. For many typical mixtures, it is not necessary to measure y.

For data reduction, Van Ness and coworkers used a four-suffix Margules equation written in the form

(6-82)

From P-x measurements for 14 binary systems at 30°C, constants A’, B’, and D’ were obtained; results are shown in Table 6-4. The 14 systems represent a wide variety with some systems showing strong positive deviations from Raoult’s law (g^E >0) and others showing strong negative deviations (g^E <0). One system (No. 6) exhibits ideal behavior.

Table 6-4 Excess Gibbs energy from P-x measurements at 30°C.* [Constants in Eq. (6-82)].

Figure 6-10 indicates a particularly convenient way to illustrate the nonidealities of the 14 systems; the coordinates of the plot are suggested by the Redlich-Kister expansion [Eq. (6-52)]: for simple mixtures, the plot gives a straight horizontal line; for slightly complex mixtures, the line is straight but not horizontal; and for complex mixtures, the line is curved. These three cases correspond, respectively, to those shown in Figs. 6-5, 6-6, and 6-7.

Figure 6-10 Excess Gibbs energies for 14 binary systems listed in Table 6-4.

6.8 Partial Pressures from Isobaric Boiling-Point Data

As indicated in the preceding section, the Gibbs-Duhem equation can be used to convert isothermal P-x data for a binary system into y-x data; similarly, it can be used to convert isobaric T-x data into y-x data. However, the latter calculation is often less useful because the Gibbs-Duhem equation for an isobaric, nonisothermal system (see App. D) contains a correction term proportional to the enthalpy of mixing and this correction term is not always negligible. The isothermal, nonisobaric Gibbs-Duhem equation also contains a correction term (that is proportional to the volume change on mixing), but in mixtures of two liquids at low pressures this term may safely be neglected.

Rigorous reduction of isobaric T-x data, therefore, requires data for the enthalpy of mixing at the boiling point of the solution. Such data are almost never at hand and if the object of a particular study is to obtain accurate isobaric y-x data, then it is usually easier to measure the y-x data directly in an equilibrium still than to obtain enthalpy-of-mixing data in addition to the T-x data. However, for approximate results, sufficient for some practical applications, boiling-point determinations may be useful because of experimental simplicity; it is a simple matter to place a liquid mixture in a manostated flask and to measure the boiling temperature. We now discuss briefly how isobaric T-x data may be reduced to yield an isobaric y-x diagram.

We assume that in the Gibbs-Duhem equation the correction term for nonisothermal conditions may be neglected. Further, we assume for simplicity that the gas phase is ideal and that the two-suffix Margules equation is adequate for the relation between activity coefficient and mole fraction:

(6-83)

We assume that A is a constant independent of temperature, pressure, and composition. The Gibbs-Duhem equation then gives

(6-84)

The problem now is to find parameter A from the T-x data. Once A is known, it is a simple matter to calculate the y-x diagram. To be consistent with the approximate nature of this calculation, we here use the simplified definition of the activity coefficient as given by Eq. (6-70).

To find A, we write

(6-85)

From the T-x data and from the vapor-pressure curves of the pure components, everything in Eq. (6-85) is known except A. Unfortunately, Eq. (6-85) is not explicit in A but for any point on the T-x diagram, a value of A may be found by trial and error. Thus, in principle, the boiling point for one particular mixture of known composition is sufficient to determine A. However, to obtain a more representative value, it is preferable to measure boiling points for several compositions of the mixture, to calculate a value for A for each boiling point, and then either to use an optimum average value in the subsequent calculations or, if the data warrant doing so, to reject Eqs. (6-83) and (6-84) and, instead, to use a two- (or three-) parameter equation for relating activity coefficients to composition.

To illustrate, we consider boiling-point data for the diisopropyl ether/2-propanol system obtained at atmospheric pressure. Table 6-5 gives experimental boiling points. From Eq. (6-85) we find an average value of A = (3.18±0.13) kJ mol^-1. When this average value is used, a y-x diagram is obtained, as shown by the line in Fig. 6-11. The points represent the experimental y-x data of Miller and Bliss (1940). In this case, agreement between the observed and calculated y-x diagram is good, but one should not assume that this will always be the case.

Table 6-5 Boiling points of diisopropyi ether/2-propanol mixtures at 1.013 bar.*

Figure 6-11 Vapor-liquid equilibrium for the diisopropyS ether (1)/2-propanol (2) system at 1.013 bar.

The method outlined above is rough and we can make many refinements; for example, we may use additional coefficients in the expressions for the activity coefficients or we may correct for vapor-phase nonideality (Adler and Adler, 1973). But these refinements are frequently not worthwhile when the temperature correction in the Gibbs-Duhem equation is neglected. It appears to be unavoidable that, unless the enthalpy of mixing can be estimated with at least fair accuracy, the boiling-point method for obtaining the y-x diagram is necessarily an approximation.

Techniques for estimating the isothermal or isobaric y-x diagram with the help of the Gibbs-Duhem equation have received a large amount of attention in the literature and many variations on this theme have been proposed. One particularly popular topic is concerned with the use of azeotropic data; if a binary system has an azeotrope, and if we know its composition, temperature, and pressure, we can compute the two constants of any two-constant equation for the activity coefficients and thereby calculate the entire x-y diagram (Miller and Bliss, 1940). This method assumes validity of the isothermal, isobaric Gibbs-Duhem equation. Good results are often obtained provided that the azeotropic mole fraction is in the interval 0.3 to 0.7.

While Van Ness (Byer et al., 1973; Abbott and Van Ness, 1975) has stressed the relation between experimental P-x data and the Gibbs-Duhem equation, an interesting variation has been proposed by Christian et al. (1960), who describes apparatus and calculations for measuring P-y rather than P-x data to obtain the y-x diagram. A method for computing equilibrium phase compositions from dew point (T-y) data has been given by Bellemans (1959), and calculations using (P-x) data for ternary systems have been described by McDermott and Ellis (1965). All of these techniques have but one aim: To reduce the experimental effort needed to characterize liquid-mixture properties. Such techniques are useful but they have one serious limitation: Results obtained by data reduction with the Gibbs-Duhem equation cannot be checked for thermodynamic consistency because the method of calculation already forces the results to be thermodynamically consistent. Thus the Gibbs-Duhem equation may be used either to extend limited data or to test more complete data, but not both. In the next section we briefly discuss the basic principles for performing such tests.

6.9 Testing Equilibrium Data for Thermodynamic Consistency

The Gibbs-Duhem equation interrelates activity coefficients of all components in a mixture. Therefore, if data are available for all activity coefficients, these data should obey the Gibbs-Duhem equation; if they do not, the data cannot be correct. If they do obey the Gibbs-Duhem equation, the data are probably, although not necessarily, correct; it is conceivable that a given set of incorrect data may fortuitously satisfy the Gibbs-Duhem equation, but this is not likely. Unfortunately, there are many phase-equilibrium data in the literature that do not satisfy the Gibbs-Duhem equation and therefore must be incorrect.

To illustrate, we consider the simplest case: a binary solution of two liquids at low pressure for which isothermal activity-coefficient data have been obtained. For this case, the Gibbs-Duhem equation is Eq. (6-59).

A theoretically simple technique is to test the data directly with Eq. (6-59); that is, plots are prepared of Inγ₁ versus x₁ and Inγ₂ versus x₂ and slopes are measured. These slopes are then substituted into Eq. (6-59) at various compositions to see if the Gibbs-Duhem equation is satisfied. While this test appears to be both simple and exact, it is of little practical value; because experimental data inevitably exhibit some scatter, it is difficult to measure slopes with sufficient accuracy. Hence the “slope method” provides at best a rough measure of thermodynamic consistency that can only be applied in a semiquantitative manner. For example, if, at a given composition, d lnγ₁/dx₁ is positive, then d lnγ₂/dx₂ must also be positive, and if d lnγ₁/dx₁ is zero, d lnγ₂/dx₂ must also be zero. The slope method can therefore be used easily to detect serious errors in the equilibrium data.

For quantitative purposes it is much easier to use an integral rather than a differential (slope) test. While integral tests are popular and used often, unfortunately, they do not provide a stringent criterion for thermodynamic consistency. The most widely used integral test was proposed by Redlich and Kister (1948) and also by Herington (1947) and is derived below.

The molar excess Gibbs energy is related to activity coefficients by

(6-86)

Differentiating with respect to x₁ at constant temperature and pressure gives

(6-87)

Noting that dx₁ = -dx₂ and substituting the Gibbs-Duhem equation [Eq. (6-59)], we obtain

(6-88)

Integration with respect to x₁ gives

(6-89)

If the pure liquids at the temperature of the mixture are used as the standard states,

(6-90)

and

(6-91)

Equation (6-89) therefore becomes

(6-92)

Equation (6-92) provides what is called an area test of phase-equilibrium data. A plot of ln(γ1/γ2) versus x₁ is prepared; a typical plot of this type is shown in Fig. 6-7. Because the integral on the left-hand side of Eq. (6-92) is given by the area under the curve shown in the figure, the requirement of thermodynamic consistency is met if that area is zero, i.e., if the area above the x-axis is equal to that below the x-axis. These areas can be measured easily and accurately with a plaaimeter and thus the area test is a particularly simple one to carry out.

Unfortunately, however, the area test has little value for deciding whether or not a set of activity-coefficient data is, or is not, thermodynamically consistent. As indicated by Eq. (6-92), the area test uses the ratio of γ₁ to γ₂; when this ratio is calculated, the pressure cancels out:

(6-93)

Therefore, the area test does not utilize what is probably the most valuable (and usually the most accurate) measurement, total pressure P²¹. As pointed out by Van Ness (1995), the area test is severely limited because (except for minor corrections - see footnote 21) the only data needed to construct the plot are x and y data and the ratio of the two pure-component vapor pressures, . Therefore, for isothermal data, the area test does little more than determine whether or not the vapor pressure ratio is appropriate to the set of measured x-y values. A plot of ln(γ₁/γ₂) versus x₁ may be sensitive to scatter in the x-y data but it tells us nothing about the internal consistency of such data.

²¹ A minor qualification is necessary because P does enter into the calculation of fugacity coefficients φ₁ and, sometimes, standard-state fugacity f₁⁰. However, these are secondary effects that become negligible at low pressure. The primary effect of pressure is not included in Eq. (6-92).

The only meaningful way to check thermodynamic consistency of experimental data consists of three steps: First, measure all three quantities P, x, and y at constant T; next, select any two of these measured quantities and predict the third using the Gibbs-Duhem equation; finally, compare the predicted third quantity with the measured quantity.

Van Ness and coworkers (Byer et al., 1973; Abbott and Van Ness, 1975) have illustrated such a procedure by examining experimental P-x-y data (Fried et al., 1967) for the system pyridine (l)/tetrachloroethylene (2) at 60°C; a plot of these data is shown in Fig. 6-12. In this case, the total pressures are so low that all fugacity coefficients can be set equal to unity and standard-state fugacity f_i⁰ is equal to vapor pressure P_i^s.

Figure 6-12 Experimental vapor-liquid equilibrium data for the system pyridine (1)/tetrachioroethy!ene (2) at 60°C.

Van Ness and coworkers fit the data to Eq. (6-82) in three ways, using

1. P-x-y data.

2. x-y data only.

3. P-x data only.

Results of data reduction are shown in Tables 6-6 and 6-7. Table 6-6 indicates that the coefficients in the expression for g^E depend significantly on the choice of data²² used but, more important for testing data, deviations between calculated and measured quantities also differ markedly, as indicated in Table 6-7.

²² Nicolaides and Eckert (1978).

Table 6-6 Constants for Eq. (6-82) for the system pyridine/tetrachloroethylene at 60°C (Byer et al., 1973).

Table 6-7 Deviations between calculated and measured quantities for the system pyridine/tetrachioroethylene at 60°C (Byer et al., 1973).

The first horizontal row in Table 6-7 tells us nothing about thermodynamic consistency because all three measured quantities were used to obtain the coefficients in Eq. (6-82). The deviations merely tell us to what extent that equation can fit the total experimental data. In the second horizontal row, we must look at the deviations in pressure, because that quantity was not used in determining the coefficients of Eq. (6-82); similarly, in the third horizontal row, we must look at deviations in y.

Experimentally, it is much easier to measure pressure with high accuracy; accurate measurements of y are usually more difficult. When P-x data are used, the root-mean-square (RMS) error in y is probably within the experimental error of y; but when x-y data are used, the RMS error in P is probably larger than the experimental error. The thermodynamic consistency of these data, therefore, is reasonably good but not as good as it might be, probably because of inaccuracies in y.

While RMS values are useful for assessing thermodynamic consistency, a more useful procedure is to plot ΔP (or Δy) versus x to determine if the deviations are primarily of one sign or if they scatter uniformly about zero. If the deviations show a clear trend, they may well be suspect. However if the deviations scatter about zero, without bias, and if the deviations are small, the data are likely to be of good quality.

It is not easy to devise a truly significant test for thermodynamic consistency because it does not appear to be possible to define an unambiguous quantitative criterion of consistency; all experimental data have some uncertainty and the judgment of “good” or “bad” depends on the system, the experimental method, the standards set by the judge, and most important, by the ultimate use of the data; for some applications, rough approximations are sufficient, while for others very high accuracy is essential. The problem is further complicated by the question of how different experimental measurements are weighted; some measurements are worth more than others because experimental conditions vary; for example, in typical binary vapor-liquid experiments, data at the two ends of the composition scale are likely to have the lowest accuracy (unless special precautions are observed), but it is precisely these measurements that give the most valuable information for determining the excess Gibbs energy.

The literature is rich with articles on testing for thermodynamic consistency because it is much easier to test someone else’s data than to obtain one’s own in the laboratory. Much (but by no means ail) of this literature is obscured by excessive use of statistics. It has been said that “Patriotism is the last refuge of a scoundrel.” Similarly, we might say that “Statistics is the last refuge of a poor experimentalist” or, in a more positive way, that a gram of good data is worth more than a ton of consistency tests.

6.10 Wohl’s Expansion for the Excess Gibbs Energy

In Sec. 6.5 we discussed briefly some expressions for the excess Gibbs energy of binary solutions. We now continue this discussion with a general method for expressing excess Gibbs energies as proposed by Wohl (1946). One of the main advantages of this method is that some rough physical significance can be assigned to the parameters that appear in the equations; as a result, and as shown later in Sec. 6.14, Wohl’s expansion can be extended systematically to multicomponent solutions.

Wohl expresses the excess Gibbs energy of a binary solution as a power series in z₁ and z₂, the effective volume fractions of the two components:

(6-94)

where

Wohl’s equation contains two types of parameters, q’s and a’s. The q’s are effective volumes, or cross sections, of the molecules; q_i is a measure of the size of molecule i, or of its “sphere of influence” in the solution. A large molecule has a larger q than a small one and, in solutions of nonpolar molecules of similar shape, it is often a good simplifying assumption that the ratio of the q’s is the same as the ratio of the pure-component liquid molar volumes. The a’s are interaction parameters whose physical significance, while not precise, is in a rough way similar to that of virial coefficients. Parameter a₁₂ is a constant characteristic of the interaction between molecule 1 and molecule 2; parameter a₁₁₂ is a constant characteristic of the interaction between three molecules, two of component 1 and one of component 2, and so on. The probability that any nearest-neighbor pair of two molecules consists of one molecule of component 1 and one molecule of component 2 is assumed to be 2z₁z₂; similarly, the probability that a triplet of three nearest-neighbor molecules consists of molecules 1, 1, and 2 is assumed to be 3z₁²z₂, and so forth. Thus, there is a crude analogy between Wohl’s equation and the virial equation of state, but it is no more than an analogy because, while the virial equation has an exact theoretical basis, Wohl’s equation cannot be derived from any rigorous theory without drastic simplifying assumptions.

When, as in Eq. (6-94), the excess Gibbs energy is taken with reference to an ideal solution in the sense of Raoult’s law, only interactions involving at least two dissimilar molecules are contained in Eq. (6-94); that is, terms of the type z₁², z₁³,… and z₂², z₂³,… do not explicitly appear in the expansion. This is a necessary coasequence of the boundary condition that g^E must vanish as x₁ or x₂ becomes zero.

However, if g^E is taken relative to an ideal dilate solution that is dilute in, say, component 2, then Wohl’s expansion takes the form

(6-95)²³

²³ The minus signs before the coefficients are arbitrarily introduced for convenience. For most mixtures a₂₂ in Eq. (6-95) is a positive number. The asterisk on g^E* indicates that g^E* is taken relative to an ideal dilute solution.

In this case a₂₂ is the self-interaction coefficient characteristic of the interaction between two molecules of component 2, a₂₂₂ is the self-interaction coefficient characteristic of the interaction between three molecules of component 2, and so on. Because g^E* in Eq. (6-95) refers to a solution very dilute in component 2, it is not interaction between molecules 2 and molecules 1 but rather interaction between molecules of component 2 that cause deviation from ideal behavior and hence a nonvanishing g^E*.

Equation (6-94) is a well-known expression for mixtures whose components can exist as pure liquids at the solution temperature. Equation (6-95) is not known well but it is sometimes useful for solutions of gases or solids in liquids (see Chaps. 10 and 11).

van Laar Equation. To illustrate the generality of Eq. (6-94), we consider first the case of a binary solution of two components that are not strongly dissimilar chemically but that have different molecular sizes. An example is a solution of benzene (molar volume 89 cm³ mol^-1 at 25°C) and isooctane (molar volume 166 cm³ mol^-1 at 25°C). We make the simplifying assumption that interaction coefficients a₁₁₂, a_l22,… and higher may be neglected; i.e., Wohl’s expression is truncated after the first term. In that case, Eq. (6-94) becomes

(6-96)

that is the van Laar equation. From Eq. (6-25), expressions for the activity coefficients can be found. They are²⁴

²⁴ A′ and B′ are dimensionless.

(6-97)

and

(6-98)

where A′ = 2q₁a₁₂ and B′ = 2q₂a₁₂.

Equations (6-97) and (6-98) are the familiar van Laar equations commonly used to represent activity-coefficient data. These equations include two empirical constants, A′ and B′; the ratio of A′ to B′ is the same as the ratio of the effective volumes q₁ and q₂ and it is also equal to the ratio of In γ₁^∞ to γ₂^∞. Whereas Eqs. (6-97) and (6-98) contain only two parameters, Eq. (6-96) appears to be a three-parameter equation. However, from the empirically determined values of A′ and B′ it is not possible to find a value of the interaction coefficient a₁₂ unless some independent assumption is made concerning the value of q₁ or q₂. For practical purposes it is not necessary to know the values of q₁ and q₂ separately because it is only their ratio that is important.

Figure 6-13 gives activity coefficients for the benzene/isooctane system at 45°C. The data of Weissman and Wood (1960) are well represented by the van Laar equations with A′ = 0.419 and B′ = 0.745.

Figure 6-13 Application of van Laar’s equations to a mixture whose components differ appreciably in molecular size.

The derivation of the van Laar equations suggests that they should be used for solutions of relatively simple, preferably nonpolar liquids but empirically it has been found that these equations are frequently able to represent activity coefficients of more complex mixtures. In such mixtures, the physical significance of the van Laar constants is more obscure; the constants must be regarded as essentially little more than empirical parameters in a thermodynamically consistent equation. The van Laar equations are widely used; they have become popular for applied work because of their flexibility and because of their mathematical simplicity relative to the many other equations that have been proposed.²⁵ In the special case where van Laar constants A′ and B′ are equal, the van Laar equations are identical to the two-suffix Margules equations [Eqs. (6-48) and (6-49)].

²⁵ As pointed out by Abbott and Van Ness (1982), var, Laar equation implies that when the experimental data are plotted in the form x₁x₂/g^E versus x₁, a straight line should be obtained.

Whenever Wohl’s expansion is used as the basis for an expression giving the activity coefficient as a function of composition, the resulting equation holds for a fixed temperature and pressure; that is, the constants in the equation are not dependent on composition but instead are functions of temperature and pressure. Because the effect of pressure on liquid-phase properties is usually small (except at high pressures and at conditions near critical), the pressure dependence of the constants can usually be neglected; however, the temperature dependence is often not negligible. Although many industrial operations (e.g., distillation) are conducted at constant pressure rather than constant temperature, there is nevertheless a strong temptation to assume that the constants in equations such as those of van Laar are temperature independent. Physically, this assumption is most unreasonable; as indicated by Eq. (6-29), the activity coefficient of a component in solution is independent of temperature only in an athermal solution, i.e., one where the components mix isothermally and isobarically without evolution or absorption of heat. For practical applications, however, this assumption is often tolerable provided that the temperature range in question is not large. For example, Fig. 6-14 shows activity coefficients for the propanol/water system calculated from the data of Gadwa (1936) at a constant pressure of 1.013 bar. These activity coefficients are well represented by the van Laar equations with A′ = 2.60 and B′ = 1.13. In this particular case the assumption of temperature-invariant constants appears to be a good approximation because at a constant pressure of 1.013 bar, the boiling temperature varies only from 87.8 to 100°C.

Figure 6-14 Application of van Laar’s equations to an isobaric system. In this system the temperature varies only from 87.8 to 100°C.

When, as frequently happens, experimenta! data are insufficient to specify the temperature dependence of the activity coefficient, either one of two simplifying approximations is usually made. The first, mentioned in the preceding paragraph, is to assume that at constant composition the activity coefficient is invariant with temperature; the second is to assume that at constant composition In γ is proportional to the reciprocal of the absolute temperature. The first assumption is equivalent to assuming that the solution is athermal (h ^E = 0), and the second is equivalent to assuming that the solution is regular (s^E = 0). Real solutions are neither athermal nor regular but more often than not, the assumption of regularity provides a better approximation to the effect of temperature on the activity coefficient than does the assumption of athermal behavior.

A general empirical procedure is to assume that at constant x,

(6-99)

where c and d are empirical constants that depend on composition. When c = 0, we recover regular-solution behavior and when d = 0, we recover athermal-solution behavior. Unfortunately, in typical cases, the magnitude of c is similar to that of dT^-l.

In vapor-liquid equilibrium, the effect of temperature on γ is often not of major importance because the effect of temperature on the standard-state fugacity (essentially, the vapor pressure in most cases) is usually much stronger than that on the activity coefficient. Therefore, vapor-liquid equilibria are often insensitive to changes in γ with T. However, in liquid-liquid equilibria, the standard-state fugacity plays no role and for such equilibria, small effects of temperature on the activity coefficients can have a large influence on the liquid-liquid phase diagram.

Margules Equations. Next we consider a binary solution of two components whose molecular sizes are not much different. In that case we assume that q₁ = q₂ in Wohl’s expansion. Neglecting terms higher than the fourth power in the mole fraction, and again using Eq. (6-25) to obtain expressions for the activity coefficients, we now obtain

(6-100)

(6-101)

where

A′ = q(2a₁₂ + 6a₁₂₂ + 12a₁₁₁₂ - 6a₁₁₂₂)

B′ = q(6a₁₂₂ - 6a₁₁₂ - 8a₁₂₂₂ + 24a₁₁₂₂)

C′ = q(12a₁₁₁₂ + 12a₁₂₂₂ - 18a₁₁₂₂)

To simplify matters, and because experimental data are usually limited, it is common to truncate the expansions after the cubic terms, i.e., to set C′ = 0; in that case the equation is called the three-suffix Margules equation that has two parameters. Only in those cases where the data are sufficiently precise and plentiful is the expansion truncated after the quartic terms and we then have a four-suffix Margules equation with three parameters as given in Eqs. (6-100) and (6-101). On the other hand, if the mixture is a simple one, containing similar components, it is sometimes sufficient to retain only the quadratic term (two-suffix Margules equation).

Although the assumption q₁ = q₂ suggests that Margules equations should be used only for mixtures whose components have similar molar volumes, it is nevertheless used frequently for all sorts of liquid mixtures, regardless of the relative sizes of the different molecules. The primary value of the Margules and van Laar equations lies in their ability to serve as simple empirical equations for representing experimentally determined activity coefficients with only a few constants. When, as is often the case, experimental data are scattered or scarce, these equations can be used to smooth the data and, more important, they serve as an efficient too! for interpolation and extrapolation with respect to composition.²⁶

²⁶ As pointed cut by Abbott and Van Ness (1982), the three-suffix Margules equation implies that when the experimental data are plotted in the form g^E/x₁ x₂ versus x₁, a straight line should be obtained.

The three-suffix Margules equations have been used to reduce experimental vapor-liquid equilibrium data for many systems; to illustrate, Fig. 6-15 shows results for three binaries at 50°C: acetone/methanol, acetone/chloroform, and chloroform/methanol (Severns et al., 1955). Each of these binaries has an azeotrope at 50°C.

Figure 6-15 shows that the thermodynamic properties of these three systems differ markedly from one another; in the acetone/methanol system there are strong positive deviations from ideality, while in the acetone/chloroform system there are equally strong negative deviations; in the chloroform/methanol system there are very large positive deviations at the chloroform-rich side, and at the methanol-rich side the activity coefficient of chloroform exhibits unusual behavior because it goes through a maximum.²⁷ Despite these large differences, the three-suffix Margules equations give a good representation of the data for all three systems. The Margules constants,²⁸ as obtained from the experimental data at 50°C, are

²⁷ One of the advantages of the three-suffix Margules equations is that they are capable of representing maxima or minima; two-suffix Margules equations and van Laar equations cannot do so. However, maxima and minima are only rarely observed in isothermal plots of activity coefficient versus mole fraction.

²⁸ These constants pertain to Eqs. (6-100) and (6-101) with C′= 0.

Figure 6-15 Activity coefficients for three binary systems at 50°C. Lines calculated from three-suffix Margules equations.

Scatchard-Hamer Equation. For one additional example of the flexibility of Wohl’s expansion we consider the case where the series is truncated after the third-order terms but where, instead of assuming that q₁ = q₂, we assume that

where υ₁ and υ₂ are, respectively, the molar volumes of the pure liquids at the temperature of the solution. Truncating Wohl’s expansion after the cubic terms, we then obtain expressions for the activity coefficients first proposed by Scatchard and Hamer (1935). They are

(6-102)

(6-103)

where A′ = v₁ (2a₁₂ + 6a₁₁₂ - 3a₁₂₂) and B′ = v₁ (6a₁₂₂ - 6a₁₁₂).

The Scatchard-Hamer equations use only two adjustable parameters but they have not received much attention in the extensive literature on vapor-liquid equilibria. They are only slightly more complex than the popular van Laar equations and the three-suffix Margules equations. This is unfortunate because in the general case the assumptions of Scatchard and Hamer appear more reasonable than those of van Laar or those in the Margules equation truncated after the cubic terms.

The behavior of the Scatchard-Hamer equations may be considered as intermediate between that of the van Laar equations and that of the three-suffix Margules equations. All three equations contain two adjustable parameters, and if for each of these equations we arbitrarily determine these parameters from values of γ^∞₁ and γ^∞₂, we can then compare the three equations as shown in Fig. 6-16.

Figure 6-16 Activity coefficients according to three, two-parameter equations with γ^∞₁ = 10 and γ^∞₂ = 2.15. For the Scatchard-Hamer equation, υ₂/υ₁ = 2/3.

6.11 Wilson, NRTL, and UNIQUAC Equations

Many equations have been proposed for the relation between activity coefficients and mole fractions and new ones appear every year.²⁹ Some, but not all, of these can be derived from Word’s general method. In particular, three equations that are useful for many practical calculations cannot be obtained by Word’s formulation.

²⁹ See, for example, E. HaSa, J. Pick, V. Fried and O. Vilim, 1967, Vapor-Liquid Equilibrium, 2^nd Ed., trans. G. Standart, Part 1, (Oxford: Pergamon Press); C. Black, 1959, AIChEJ. 5: 249; M. Hiranuma and K. Honma, 1975, Ind. Eng. Chem. Process Des. Dev., 14: 221. A particularly simple but unusual equation has been proposed by H. Mauser, 1958, Z Elektrochem., 62: 895. There are many, many others, too many to list here.

Wilson Equation. Based on molecular considerations, Wilson (1964) presented the following expression for the excess Gibbs energy of a binary solution:

(6-104)

The activity coefficients derived from this equation are

(6-105)

(6-106)

In Eq. (6-104) the excess Gibbs energy is defined with reference to an ideal solution in the sense of Raoult’s law; Eq. (6-104) obeys the boundary condition that g^E vanishes as either x_l or x₂ becomes zero.

Wilson’s equation has two adjustable parameters, Λ₁₂ and Λ₂₁. In Wilson’s derivation, these are related to the pure-component molar volumes and to characteristic energy differences by

(6-107)

(6-108)

where υ_i, is the molar liquid volume of pure component i and the λ’s are energies of interaction between the molecules designated in the subscripts. To a fair approximation, the differences in the characteristic energies are independent of temperature, at least over modest temperature intervals. Therefore, Wilson’s equation gives not only an expression for the activity coefficients as a function of composition but also an estimate of the variation of the activity coefficients with temperature. This may provide a practical advantage in isobaric calculations where the temperature varies as the composition changes. For accurate work, (λ₁₂-λ₁₁) and (λ₂₁ - λ₂₂) should be considered temperature-dependent but in many cases this dependence can be neglected without serious error.

Wilson’s equation appears to provide a good representation of excess Gibbs energies for a variety of miscible mixtures. It is particularly useful for solutions of polar or associating components (e.g., alcohols) in nonpolar solvents. The three-suffix Margules equation and the van Laar equation are usually not adequate for such solutions. For a good data fit, an equation of the Margules type or a modification of van Laar’s equation (Black, 1959) may be used, but such equations require at least three parameters and, more important, these equations are not readily generalized to multicomponent solutions without further assumptions or ternary parameters.

A study of Wilson’s equation by Orye (1965a) shows that for approximately 100 miscible binary mixtures of various chemical types, activity coefficients were well represented by Wilson equation; in essentially all cases this representation was as good as, and in many cases better than, the representation given by the three-suffix (twoconstant) Margules equation and by the van Laar equation. Similar conclusions were obtained by Gmehling et al.,³⁰ who report Wilson parameters for many binary systems.

³⁰ J. Gmehling, U. Onken, and W. Arlt, DECHEMA Chemistry Data Series, starting in 1977.

To illustrate, Table 6-8 gives calculated and experimental vapor compositions for the nitromethane/carbon tetrachloride system. The calculations were made twice, once using the van Laar equation and once using Wilson’s equation; in both cases, required parameters were found from a least-squares computation using experimental P-x data at 45°C given by Brown and Smith (1957). In both calculations the average error in the predicted vapor compositions is not high, but for the calculation based on van Laar’s equation it is almost three times as large as that based on Wilson’s equation.

Table 6-8 Calculated vapor compositions from fit of P-x data at 45°C [nitromethane (1)/carbon tetrachloride (2)].

A similar calculation is shown in Fig. 6-17 for the ethanol/isooctane system. Wilson and van Laar parameters were calculated from the isothermal vapor- pressure data of Kretschmer (1948). In this case, the Wilson equation is much superior to the van Laar equation that erroneously predicts an immiscible region for this system at 50°C.

Figure 6-17 Vapor-liquid equilibrium for the ethanol (1)/isooctane (2) system at 50°C. Lines calculated from P-x data. The van Laar equations erroneously predict partial immiscibility.

For isothermal solutions that do not exhibit large or highly asymmetric deviations from ideality, Wilson’s equation does not offer any particular advantages over the more familiar three-suffix Margules or van Laar equations, although it appears to be as good as these. For example, isothermal vapor-liquid equilibrium data for the cryogenic systems argon/nitrogen and nitrogen/oxygen are represented equally well by all three equations.

Wilson’s equation has two disadvantages that are not serious for many applications. First, Eqs. (6-105) and (6-106) are not useful for systems where the logarithms of the activity coefficients, when plotted against x, exhibit maxima or minima. (Van Laar equations are also not useful for this case). Such systems, however, are not common. The second and more serious disadvantage of Wilson’s equation lies in its inability to predict limited miscibility. When Wilson’s equation is substituted into the equations of thermodynamic stability for a binary system (see next section), no parameters Λ₁₂ and Λ₂₁ can be found that indicate the existence of two stable liquid phases.³¹ Wilson’s equation, therefore, should be used only for liquid systems that are completely miscible or else for those limited regions of partially miscible systems where only one liquid phase is present.

³¹ For partially miscible systems, Wilson (1964) suggested that the right-hand side of Eq. (6-104) be multiplied by a constant greater than unity. This suggestion, not only introduces a third parameter but, more important, creates difficulties when the equation is applied to ternary (or higher) systems (see Sec. 6.15).

NRTL Equation. The basic idea in Wilson’s derivation of Eq. (6-104) follows from the concept of local composition that is discussed further in Sec. 7.7. This concept was also used by Renon (1968) in his derivation of the NRTL (nonrandom, twoliquid) equation; however, Renon’s equation, unlike Wilson’s, is applicable to partially miscible as well as completely miscible systems. The NRTL equation for the excess Gibbs energy is

(6-109)

where

(6-110)

(6-111)

The significance of g_ij is similar to that of λ_ij in Wilson’s equation; g_ij is an energy parameter characteristic of the i-j interaction. Parameter α₁₂ is related to the nonrandomness in the mixture; when α₁₂ is zero, the mixture is completely random and Eq. (6-109) reduces to the two-suffix Margules equation. The NRTL equation contains three parameters, but reduction of experimental data for a large number of binary systems indicates that α₁₂ varies from about 0.20 to 0.47; when experimental data are scarce, the value of α₁₂ can often be set arbitrarily; a typical choice is α₁₂ = 0.3. From Eq. (6-109), the activity coefficients are

(6-112)

(6-113)

For moderately nonideal systems, the NRTL equation offers no advantages over the simpler van Laar and three-suffix Margules equations. However, for strongly nonideal mixtures, and especially for partially immiscible systems,³² the NRTL equation often provides a good representation of experimental data if care is exercised in data reduction to obtain the adjustable parameters. For example, consider the nitro-ethane/isooctane system studied by Renon; below 30°C, this system has a miscibility gap. Reduction of liquid-liquid equilibrium data below 30°C and vapor-liquid equilibrium data at 25 and 45°C gave the results shown in Fig. 6-18. The parameters (g₁₂ - g₂₂) and (g₂₁ - g₁₁) appear to be linear functions of temperature, showing no discontinuities in the region of the critical solution temperature.

³² See App. E.

Figure 6-18 Parameters in NRTL equation for the nitroethane (1)/isooctane (2) system calculated from vapor-liquid and liquid-liquid equilibrium data.

Renon’s and Wilson’s equations are readily generalized to multicomponent mixtures as discussed in Sec. 6.15.

UNIQUAC Equation. A critical examination of the derivation of the NRTL equation shows that this equation, like those obtained from Word’s expansion, is more suitable for h^E than g^E (Renon and Prausnitz, 1969). Further, because experimental data for typical binary mixtures are usually not sufficiently plentiful or precise to yield three meaningful binary parameters, attempts were made (Abrams, 1975; Maurer, 1978; Anderson, 1978; Kemeny and Rasmussen, 1981) to derive a two-parameter equation for g^E that retains at least some of the advantages of the equation of Wilson without restriction to completely miscible mixtures. Abrams derived an equation that, in a sense, extends the quasichemical theory of Guggenheim (see Sec. 7.6) for nonrandom mixtures to solutions containing molecules of different size. This extension was therefore called the universal quasi-chemical theory or, in short, UNIQUAC. As discussed in Sec. 7.7, the UNIQUAC equation for g^E consists of two parts, a combinatorial part that attempts to describe the dominant entropic contribution, and a residual part that is due primarily to intermolecular forces that are responsible for the enthalpy of mixing. The combinatorial part is determined only by the composition and by the sizes and shapes of the molecules; it requires only pure-component data. The residual part, however, depends also on intermolecular forces; the two adjustable binary parameters, therefore, appear only in the residual part. The UNIQUAC equation is

(6-114)

For a binary mixture,

(6-115)

(6-116)

where the coordination number z is set equal to 10. Segment fraction, Φ*, and area fractions, θ and θ′, are given by

(6-117)

(6-118)

(6-119)

Parameters r, q, and q′ are pure-component molecular-structure constants depending on molecular size and external surface areas. In the original formulation, q = q′. To obtain better agreement for systems containing water or lower alcohols, q′ values for water and alcohols were adjusted empirically by Anderson (1978) to give an optimum fit to a variety of systems containing these components. For alcohols, the surface of interaction q′ is smaller than the geometric external surface q, suggesting that intermolecular attraction is dominated by the OH group (hydrogen bonding). Table 6-9 presents a few values of these structural parameters. For fluids other than water or lower alcohols, q = q′.

Table 6-9 Some structural parameters for UNIQUAC equation.*

For each binary mixture, there are two adjustable parameters, τ₁₂ and τ₂₁. These, in turn, are given in terms of characteristic energies Δu₁₂ and Δu₂₁, by

(6-120)

(6-121)

For many cases, Eqs. (6-120) and (6-121) give the primary effect of temperature on τ₁₂ and τ₂₁, Characteristic energies Δu₁₂ and Δu₂₁ are often only weakly dependent on temperature. Table 6-10 gives some binary parameters a₁₂ and a₂₁.

Table 6-10 Some binary parameters for UNIQUAC equation.*

Activity coefficients γ₁ and γ₂ are given by

(6-122)

(6-123)

where

(6-124)

(6-125)

The UNIQUAC equation is applicable to a wide variety of nonelectrolyte liquid mixtures containing nonpolar or polar fluids such as hydrocarbons, alcohols, nitriles, ketones, aldehydes, organic acids, etc. and water, including partially miscible mixtures. With only two adjustable binary parameters, it cannot always represent high-quality data with high accuracy, but for many typical mixtures encountered in chemical practice, UNIQUAC provides a satisfactory description.³³

³³ UNIQUAC parameters are given for many binary systems by J. Gmehling, U. Onken, and W. Arlt, DECHEMA Chemistry Data Series, since 1977. See also Prausnitz et al, 1980.

The main advantages of UNIQUAC follow first, from its (relative) simplicity, using only two adjustable parameters, and second, from its wide range of applicability. To illustrate, some results are shown in Figs. 6-19 to 6-24.

Figure 6-19 shows experimental and calculated phase equilibria for the acetonitrile/benzene system at 45°C. This system exhibits moderate, positive deviations from Raoult’s law. The high-quality data of Brown and Smith (1955) are very well represented by the UNIQUAC equation.

Figure 6-19 Moderate positive deviations from ideality. Vapor-liquid equilibria for the acetonitrile (1)/benzene (2) system at 45°C.

Figure 6-20 shows the isothermal data of Edwards (1962) for n-hexane and nitro-ethane. This system also exhibits positive deviations from Raoult’s law; however, these deviations are much larger than those shown in Fig. 6-19. At 45°C, the mixture shown in Fig. 6-20 is only 15°C above its critical solution temperature (see Sec. 6.13).

Figure 6-20 Strong positive deviations from ideality. Vapor-iiquid equilibria for the n- hexane (1)/nitroethane (2) system at 45°C.

The acetone/chloroform system, shown in Fig. 6-21, exhibits strong negative deviations from Raoult’s law because of hydrogen bonding between the single hydrogen atom of chloroform and the carbonyl oxygen of acetone.

Figure 6-21 Negative deviations from ideality. Vapor-liquid equilibria for the acetone (1)/chloroform (2) system at 50°C.

Figure 6-22 shows a fit of the UNIQUAC equation to the isobaric data of Nakanishi et al. (1967) for the methanol/diethylamine system; this system also exhibits strong negative deviations from Raoult’s law. The UNIQUAC equation correctly reproduces a weak minimum in the activity coefficient of methanol. Agreement with experiment is not as good as that in previous examples because the data are somewhat scattered, particularly near the azeotrope.

Figure 6-22 Vapor-liquid equilibria for a mixture containing two hydrogen-bonding components. Temperature-composition diagram for the system methanol (1)/diethylamine (2) at 0.973 bar.

At moderate pressures, vapor-phase nonideality is usually small in comparison to liquid-phase nonideality. However, when associating carboxylic acids are present, vapor-phase nonideality may dominate. These acids dimerize appreciably in the vapor phase even at low pressures; fugacity coefficients are well removed from unity. To illustrate, Fig. 6-23 shows observed and calculated vapor-liquid equilibria for two systems containing an associating component. In Fig. 6-23(a), both components strongly associate with themselves and with each other. In Fig. 6-23(b), only one of the components associates strongly. For both systems, representation of the data is very good. However, the interesting feature of these systems is that, whereas the fugacity coefficients are significantly remote from unity, the activity coefficients show only minor deviations from ideal-solution behavior. Figures 5-35 and 5-36 indicate that the fugacity coefficients show marked departure from ideality. In these systems, the major contribution to nonideality occurs in the vapor phase. Failure to take into account these strong vapor-phase nonidealities would result in erroneous activity-coefficient parameters, a₁₂ and a₂₁, leading to poor prediction of multicomponent equilibria.

Figure 6-23 Vapor-liquid equilibria for binary systems containing two carboxylic acids or one carboxylic acid and one ketone. Temperature-composition diagrams and activity coefficients at 1.013 bar for the systems: (a) formic acid (1)/acetic acid (2); (b) propionic acid (1)/methyl isobutyl ketone (2).

6.12 Excess Functions and Partial Miscibility

In preceding sections, we have been concerned with mixtures of liquids that are completely miscible. We now consider briefly the thermodynamics of binary liquid systems wherein the components are only partially miscible.³⁴

³⁴ See App. E.

At a fixed temperature and pressure, a stable state is that which has a minimum Gibbs energy. Thermodynamic stability analysis tells us that a liquid mixture splits into two separate liquid phases if upon doing so, it can lower its Gibbs energy. To fix ideas, consider a mixture of two liquids. 1 and 2, whose calculated Gibbs energy of mixing at constant temperature and pressure is given by the heavy line T₁ in Fig. 6-24. If the composition of the mixture is that corresponding to point a, then the molar Gibbs energy of that mixture is

(6-126)

However, if the mixture splits into two separate liquid phases, one having mole fraction and the other having mole fraction , then the Gibbs energy change upon mixing is given by point b and the molar Gibbs energy of the two-phase mixture is

(6-127)

Mole fractions x₁ and x₂ in Eq. (6-127) represent the overall composition and they are the same as those in Eq. (6-126).

It is evident from Fig. 6-24 that point b represents a lower Gibbs energy of the mixture than does point a. Therefore, at temperature T ₁ the liquid mixture having overall composition x₁ splits into two liquid phases having mole fractions and . Point b represents the lowest possible Gibbs energy that the mixture may attain subject to the restraints of fixed temperature, pressure, and overall composition x₁.

Figure 6-24 Molar Gibbs energy of mixing and T-x diagram of a mixture at constant pressure: T₁, partially miscible; T₂, totally miscible.

A decrease in the Gibbs energy of a binary liquid mixture due to the formation of another liquid phase can occur only if a plot of the Gibbs energy change of mixing against mole fraction is, in part, concave downward. Therefore, the condition for instability of a binary liquid mixture is

(6-128)³⁵

³⁵ In Eqs. (6-128) and (6-129) A: stands for either x₁ or x₂.

or

(6-129)

In the T-x diagram shown in the lower part of Fig. 6-24, T^c is the critical solution temperature (see Sec. 6.13). At temperatures T > T^c, the mixture is completely miscible because for all mole fractions _T.P > 0 At T < T^c the mixture is partially miscible because in part of the mole fraction range _T.P < 0 The binodal curve is the boundary between the one-phase region and the two-phase region. Within the two-phase region, the spinodal curve [ _T,P = 0] distinguishes the unstable region [ _T,P < 0] from the metastable region [ _T,P > 0]. If the overall mole fraction of the mixture falls within the unstable region, spontaneous demixing occurs when going from the one-phase to the two-phase region.

Let us now introduce an excess function into Eq. (6-128). As before, we define the excess Gibbs energy of a mixture relative to the Gibbs energy of an ideal mixture in the sense of Raoult’s law:

(6-130)

Substituting into Eq. (6-128), we obtain for instability

(6-131)

For an ideal solution, g^E = 0 for all x and in that event the inequality is never obeyed for any values of x₁ and x₂ in the interval zero to one. Therefore, we conclude that an ideal solution is always stable and cannot exhibit phase splitting.

Suppose now that the excess Gibbs energy is not zero but is given by the simple expression

(6-132)

where A is a constant dependent only on temperature. Then

(6-133)

and substitution in Eq. (6-131) gives

(6-134)

Multiplying both sides by -1 inverts the inequality sign, and the condition for instability becomes

(6-135)

The smallest value of A that satisfies inequality (6-135) is

(6-136)

and, therefore, instability occurs whenever

(6-137)

The borderline between stability and instability of a liquid mixture is called incipient instability. This condition corresponds to a critical state and it occurs when the two points of inflection shown in Fig. 6-24 merge into a single point. Incipient instability, therefore, is characterized by the two equations

(6-138)

and

(6-139)

An equivalent, but more useful characterization of incipient instability is provided by introducing into Eqs. (6-138) and (6-139) the activity function given by

(6-140)

We then obtain for incipient instability

(6-141)

and

(6-142)

We now want to illustrate graphically instability, incipient instability, and stability in a binary liquid mixture. Figure 6-25 gives a plot of activity versus mole fraction as calculated from the simple excess Gibbs energy expression given by Eq. (6-132); the activity is given by

Figure 6-25 Activity of component 1 in a binary liquid solution for different values of A/RT. Curve (3) shows incipient instability.

(6-143)

When A/RT > 2, the curve has a maximum and a minimum; for this case there are two stable liquid phases whose compositions are given by and as shown schematically in Fig. 6-25.³⁶ When A/RT = 2, the maximum and minimum points coincide and we have incipient instability. For A/RT < 2, only one liquid phase is stable.

³⁶ Compositions and are found from simultaneous solution of two equilibrium relations: (γ₁x₁)’=(γ₁x₁)” and (γ₂x₂)’=(γ₂x₂)” plus material balances x’₁ +x’₂ = 1 and x”₁ + x”₂ = 1. See App. E.

A plot similar to Fig. 6-25 is shown in Fig. 6-26 that presents experimental results reported by Butler (1937) for four binary aqueous systems containing methyl, ethyl, n-propyl, or n-butyl alcohol at 25°C. Methyl and ethyl alcohols are completely miscible with water and when their activities are plotted against their mole fractions, there is no point of inflection. Propyl alcohol is also completely miscible with water at this temperature but just barely so; the plot of activity versus mole fraction almost shows a point of inflection. Butyl alcohol, however, is miscible with water over only small ranges of concentration. If a continuous line were drawn through the experimental points shown in Fig. 6-26, it would necessarily have to go through a maximum and a minimum, analogous to an equation-of-state isotherm on P-V coordinates in the two-phase region.

Figure 6-26 Activities of four alcohols in binary aqueous solution at 25°C. Data from Butler (1937).

6.13 Upper and Lower Consolute Temperatures

As indicated in the preceding section, the condition for instability of a binary liquid mixture depends on the nonideality of the solution and on the temperature. In the simplest case, when the excess Gibbs energy is given by a one-parameter equation such as Eq. (6-132), the temperature T^c for incipient instability is given by

(6-144)

Temperature T^c is the consolute temperature;³⁷ when the excess Gibbs energy is given by a temperature-independent one-parameter Margules equation, T^c is always a maximum, but in general it may be a maximum (upper) or a minimum (lower) temperature on a T-x diagram, as shown in Fig. 6-27.

³⁷ Or, alternatively, the critical solution temperature.

Some binary systems have both upper and lower consolute temperatures. Upper critical solution temperatures are more common than lower critical solution temperatures, although the latter are sometimes observed in mixtures of components that form hydrogen bonds with one another (e.g., aqueous mixtures of amines). Such phase behavior is often observed in polymer solutions, as discussed in Sec. 8.2. In many simple liquid mixtures, parameter A is a weak function of temperature and therefore vapor-liquid-equilibrium measurements obtained at some temperature not far from T^c may be used to estimate the consolute temperature. However, it sometimes happens that A depends on temperature in a complicated way and, unless data are taken very near T^c, Eq. (6-144) provides only an approximation even in those cases where the simple one-parameter equation is adequate for the excess Gibbs energy. Figure 6-27 illustrates four possible cases of phase stability corresponding to the four types of temperature dependence for parameter A shown in Fig. 6-28. In Fig. 6-27, in case (a), T^c is a maximum and in case (b) it is a minimum; in case (c), as temperature rises, there is first a minimum and then a maximum, and in case (d), as temperature rises, there is first a maximum and then a minimum.

Figure 6-27 Phase stability in four binary liquid mixtures.

Figure 6-28 shows that when A/RT falls with increasing temperature, we may obtain an upper T^c; if A/RT rises with increasing temperature, we may obtain a lower T^c. If A/RT goes through a maximum, we may obtain first, a lower T^c followed by an upper T^c and we may obtain the reverse when A/RT goes through a minimum.

Figure 6-28 Phase stability in four binary liquid mixtures. For each, the excess Gibbs energy is given by a two-suffix Margules equation.

When the excess Gibbs energy is given by Eq. (6-132), we always find that the composition corresponding to the consolute temperature is x_l= x₂= 1/2. However, when the excess Gibbs energy is given by a function that is not symmetric with respect to x_l and x₂, the coordinates of the consolute point are not at the composition midpoint.

For example, if the excess Gibbs energy is given by van Laar’s equation in the form³⁸

³⁸ For a discussion of how different models for g^E affect critical mixing, see J. Wisniak, 1983, Chem. Eng. Sci., 38: 969. The same author also analyses the influence of a third component on the mutual solubilities of two liquids (1984, Chem. Eng. Sci, 39: 111).

(6-145)

we find, upon substitution into Eqs. (6-141) and (6-142) that the coordinates of the consolute point are

(6-146)

In Sec. 6.12 we indicated that when the excess Gibbs energy is assumed to follow a two-suffix Margules expression as given by Eq. (6-132), incipient instability occurs when A = 2RT. A two-suffix Margules equation, however, is only a rough approximation for many real mixtures and a more accurate description is given by including higher terms. If we write the excess Gibbs energy in the three-parameter Redlich-Kister series,

(6-147)

and substitute this series into the equations of incipient instability, we obtain the results given in Figs. 6-29 and 6-30, taken from Sham (1963). Figure 6-29 gives the effect of the coefficient B (when C = 0) on the maximum value of A/RT for complete miscibility. Coefficient B reflects the asymmetry of the excess Gibbs energy function (see Sec. 6.5), and we see that the maximum permissible value of A/RT decreases below 2 as the asymmetry rises. Figure 6-30 gives the effect of both coefficients B and C on the maximum value of A/RT for complete miscibility. As discussed in Sec. 6.5, the third term in the Redlich-Kister expansion is symmetric in the mole fraction; it affects the flatness or steepness of the excess Gibbs energy curve. Positive values of C make the excess Gibbs energy curve flatter, and from Fig. 6-30 we see that for small values of B, positive values of C increase the maximum permissible value of A/RT beyond 2 to 2.4.

Figure 6-29 Effect of Redlich-Kister coefficient B on maximum values of A/RT for complete miscibility.

Figure 6-30 Effect of Redlich-Kister coefficient Con maximum values of A/RT for complete miscibility.

Shain’s calculations show that whereas large positive values of the constant A favor limited miscibility, values of the constant B of either sign increase the tendency toward limited miscibility for a given value of A; however, small positive values of the constant C tend to decrease the tendency for phase separation. These calculations offer a useful guide but they must not be taken too seriously because they are based on a twofold differentiation of the excess Gibbs energy function. Extremely accurate data are required to assign precise quantitative significance to a function based on the second derivative of such data.

In concluding this section, we note that the thermodynamics of phase stability in binary liquid systems does not require that instability can occur only if the excess Gibbs energy is positive. In principle, a binary liquid mixture may be only partially miscible even though it has a negative excess Gibbs energy. However, such behavior is unlikely because the composition dependence of a negative excess Gibbs energy would have to be unusual to satisfy the condition of instability, Eq. (6-131). A good review of this subject is given, by Sørensen et al. (1979, 1979a, 1979b 1980). A compilation of liquid-liquid equilibrium data is given by Sørensen and Arlt (1979).

Appendix E gives some further discussion on liquid-liquid equilibrium.

6.14 Excess Functions for Multicomponent Mixtures

So far in this chapter, we have primarily considered binary mixtures. We now turn to a discussion of mixtures containing more than two components, with particular attention to their excess Gibbs energies.

One of the main uses of excess functions for describing the thermodynamic properties of liquid mixtures lies in establishing thermodynamically consistent relations for multicomponent mixtures containing any desired number of components; from these relations we can then calculate activity coefficients needed to find liquid-phase fugacities. Expressions for the excess functions require a number of constants and many of these can be evaluated from binary data alone; in some cases, it is possible to obtain all the required constants from binary data. Application of excess functions to mixtures of more than two components, therefore, provides a significant laborsaving device that minimizes the experimental effort required to describe a mixture of many components.

Wohl’s Equation. To illustrate the utility of excess functions, we consider first a ternary mixture as described by Wohl’s method (see Sec. 6.10). Extension to systems containing more than three components will then be evident.

As for a binary solution, the Gibbs energy is again a summation of two-body, three-body, etc., interactions. When the excess Gibbs energy is relative to an ideal solution in the sense of Raoult’s law, we have

(6-148)

Two-Suffix Margules Equation. First, consider a simple case: Suppose that components 1, 2, and 3 are chemically similar and of approximately the same size. We assume that q₁ = q₂ = q₃ that all three-body terms (and higher) may be neglected. Equation (6-148) simplifies to

(6-149)

The activity coefficients follow from differentiation as given by Eq. (6-25). They are

(6-150)

(6-151)

(6-152)

where A’₁₂ = 2qa₁₂, A’₁₃ = 2qa₁₃, and A’₂₃ = 2qa₂₃

Equations (6-150), (6-151), and (6-152) possess a great advantage: All the constants may be obtained from binary data without further assumptions. Constant 2qa₁₂ is given by data for the 1-2 binary; constants 2qa₁₃ and 2qa₂₃ are given, respectively, by data for the 1-3 and 2-3 binaries. Thus, by assumption, no ternary data are required to calculate activity coefficients for the ternary mixture.

Van Laar Equation. A somewhat more realistic, but still simplified, model of a ternary solution is provided by again assuming that all three-body terms (and higher) in Eq. (6-148) may be neglected, but this time we do not assume equality of all q terms. These assumptions lead to the van Laar equations for a ternary mixture. The molar excess Gibbs energy is given by

(6-153)

To simplify notation, let

Upon differentiation according to Eq. (6-25), the activity coefficient of component 1 is

(6-154)

Expressions for γ₂ and γ₃ are of exactly the same form as that for γ₁. To obtain γ₂, Eq. (6-154) should be used with this change of all subscripts on the right-hand side: Replace 1 with 2; replace 2 with 3; and replace 3 with 1. To obtain γ₃, replace 1 with 3; replace 2 with 1; and replace 3 with 2. Notice that if q₁ = q₂ = q₃ then Eq. (6-154) reduces to Eq. (6-150).

All parameters in Eq. (6-154) may be obtained from binary data, as indicated by Eqs. (6-97) and (6-98).³⁹

³⁹ Wohl’s expansion requires that parameter q_i depend only on pure-component properties of component i; therefore, the three sets of binary parameters are not independent because, according to the definitions given immediately after Eq. (6-153),

Three-Suffix Margules Equation. The three-suffix Margules equations can also be extended to a ternary mixture by Wohl’s method, but now all the constants cannot be found from binary data alone unless an additional assumption is made. We assume that the q’s are equal to one another but we retain three-body terms in the expansion for excess Gibbs energy; higher-body terms are neglected. Equation (6-148) becomes:

(6-155)

All constants appearing in Eq. (6-155) can be obtained from binary data except the last one, qa₁₂₃ This last constant is characteristic of the interaction between three different molecules: one of component 1, one of component 2, and one of component 3. It is a ternary constant and, in principle, can be obtained only from ternary data.

To simplify notation, let

and

All constants of type A’_ij can be determined from binary data alone.⁴⁰ However, constant Q’ requires information on the ternary mixture because it is a function of α₁₂₃.

⁴⁰ Notice that constants A’₁₂ and A’₂₁ are simply related to constants A’ and B’ as defined after Eqs. (6-100) and (6-101) for the case where four-body (and higher) interactions are neglected. The relations are: A’₁₂ = A’ + B’ and A’₁₂ = A’+B’/2. Similar relations can be written for the 1-3 and 2-3 mixtures.

Consider now the binary van Laar equations [Eqs. (6-97) and (6-98)] at infinite dilution. Algebraic substitution gives

This interdependence of parameters suggests that experimental data for the 1-3 binary and for the 1-2 binary can be used to predict (in part) behavior of the 2-3 binary. Unfortunately, such prediction is rarely reliable.

The activity coefficient for component 1 is given by

(6-156)

Expressions for γ₂ and γ₃ can be obtained from Eq. (6-156) by a change of all subscripts on the right-hand side. For γ₂ replace 1 with 2; replace 2 with 3; and 3 with 1. For γ₃ replace 1 with 3; 2 with 1; and 3 with 2.

If no ternary data are available, it is possible to estimate a₁₂₃ by a suitable assumption. A reasonable but essentially arbitrary assumption is to set Q’ = 0. An extensive study of Eq. (6-156) has been made by Adler et al. (1966).

In principle, only one experimental ternary point is required to determine qa₁₂₃. In practice, however, it is not advisable to base a parameter on one point only. For accurate work it is best to measure vapor-liquid equilibria for several ternary compositions that, in addition to the binary data, can then be used to evaluate a truly representative ternary constant.

The paragraphs above have shown how Wohl’s method may be used to derive expressions for activity coefficients in a ternary mixture; exactly the same principles apply for obtaining expressions for activity coefficients containing four, five, or more components. The generalization of Eqs. (6-149) and (6-153) to mixtures containing any number of components shows that all the constants may be calculated from binary data alone. However, the generalization of Eq. (6-155) to solutions containing any number of components shows that the constants appearing in the expressions for the activity coefficients must be found from data on all possible constituent ternaries as well as binaries. For the generalization of Eq. (6-155), data on quaternary, quinternary, etc. mixtures are not needed.

To illustrate the applicability of the three-suffix Margules equation to ternary systems, we consider three strongly nonideal ternary systems at 50°C studied by Severns et al. (1955). They are:

I. Acetone/methyl acetate/methanol.

II. Acetone/chloroform/methanol.

III. Acetone/carbon tetrachloride/methanol.

Margules constants for the three ternary systems are given in Table 6-11. For system I a good representation of the ternary data was obtained by using binary data only and setting Q’ = 0 in Eq. (6-156). In system II the ternary data required a small but significant ternary constant Q’ = -0.368 that, if it had been neglected, would introduce some, but not serious, error. However, in system III the ternary data required an appreciable ternary constant Q’ = 1.15 that cannot be neglected because it is of the same order of magnitude as the various A’_ij constants for this system.

Table 6-11 Three-suffix Margules constants for three ternary systems at 50°C.*

Kohler Equation. Kohler’s model permits prediction of thermodynamic properties of multicomponent liquid solutions from binary data only. As opposed to the Wilson, NRTL, and UNIQUAC equations, Kohler’s model does not impose restrictions on the functional form of the binary excess Gibbs energy expressions nor does it limit the number of adjustable binary parameters. A general polynomial expansion permits all types of binary data (such as vapor-liquid and liquid-liquid equilibria, excess enthalpy, etc.) to be optimized simultaneously to obtain one self-consistent expression for the binary excess Gibbs energy at all compositions and temperatures.

For a ternary system of components 1, 2, and 3, Kohler relates g^E of the ternary mixture to the g^E’s of the three constituent binaries by:

(6-157)

where x_i is the liquid-phase mole fraction of component i. Any equation (not necessarily the same for each binary) can be used to represent of the binary ij system.

Suppose that g^E of the binary liquid mixtures is given by the following Legendre polynomial expansion:

(6-158)

where a_i b_i, and c_i are adjustable parameters; upper limit N is chosen as required to fit the experimental data available for the binary mixture; and F_i(x₂ - x_l) is a Legendre polynomial of order i⁴¹. Using standard thermodynamics, we obtain from Eq. (6-157) expressions for several excess properties:

⁴¹ A polynomial expansion in (x₂ - x₁)ⁱ such as that of Redlich-Kister [Eq. (6-52)] can also be used. However, Legendre polynomials offer several advantages (e.g. their terms are independent and therefore may be truncated from the series to yield reasonable approximations) as pointed out by A. D. Pelton and C. W. Bale, 1986, Metall. Trans., 17A: 1057.

(6-159)

(6-160)

(6-161)

In Eq. (6-161), F’_i(x₂-x₁) is the first derivative of F_i(x₂ - x₁) with respect to (x₂-x₁). Therefore, using one simultaneous least-square regression analysis for all data available for each binary (VLE and LLE data, h^E, , γ^∞, etc.), one self-consistent set of parameters a_i, b_i and c_i is obtained. Because the number of parameters in Eq. (6-157) is unconstrained, the data can be fitted as precisely as justified by the extent and quality of the data.

An application of Kohler’s model to ternary and quaternary systems is provided by Mier et al. (1994, 1994a) and by Talley et al. (1993). Figure 6-31 compares predictions of Kohler’s method (with the coefficients for Eq. (6-157) in Table 6-12) with experimental liquid-liquid equilibria at 318.15 K for the ternary system benzene/acetonitrile/n-heptane. In the calculations shown in Fig. 6-31 no ternary data were used in the optimization procedure. The predicted curve is good for ternary compositions near the partially miscible binary but becomes increasingly poor as the plait point is approached. As with other models (Wilson, NRTL, UNIQUAC), the predicted plait point is in error because Kohler’s model also neglects the contribution of concentration fluctuations that are important near critical conditions. A reasonable (but not rigorous) attempt to take such fluctuations into account has been presented by de Pablo (1988, 1989, 1990).

Table 6-12 Coefficients of Eq. (6-157) for the constituent binaries of the ternary system benzene/acetonitrile/n-heptane.*

Figure 6-31 Liquid-liquid equilibria at 318.15 K for the ternary benzene/acetonitrile/n-heptane. • - - - • Experimental tie line.—— Calculated from Kohler’s model with the binary parameters in Table 6-12.

6.15 Wilson, NRTL, and UNIQUAC Equations for Multicomponent Mixtures

The equations discussed in Sec. 6.11 are readily extended to as many components as desired without any additional assumptions and without introducing any constants other than those obtained from binary data.

Wilson Equation. For a solution of m components, Wilson’s equation is

(6-162)

where

(6-163)

(6-164)

The activity coefficient for any component k is given by

(6-165)

Equation (6-165) requires only parameters that can be obtained from binary data; for each possible binary pair in the multicomponent solution, two parameters are needed.

Orye (1965a) has tested Eq. (6-165) for a variety of ternary systems, using only binary data, and finds that for most cases good results are obtained. For example, Fig. 6-32 compares calculated and observed vapor compositions for the acetone/methyl acetate/methanol system at 50°C. A similar comparison is also shown for calculations based on the van Laar equation. No ternary data were used in either calculation; the binary constants used are given in Table 6-13. For this ternary system, the Wilson equations give a much better prediction than the van Laar equations but, as indicated in Table 6-11, the three-suffix Margules equations (using binary data only) can also give a good prediction because no ternary constant is required.

Table 6-13 Parameters for Wilson and van Laar equations for the system acetone (1)/methyl acetate (2)/methanol (3) at 50°C (Orye, 1965a).

Figure 6-32 Experimental and calculated vapor compositions for the ternary system acetone/methyl acetate/methanol at 50°C. Calculations use only binary data.

Similar calculations for the ternary system acetone/methanol/chloroform are shown in Fig. 6-33, and again Wilson’s equations, based on binary data only, give a better prediction than van Laar’s equations. However, for this system the three-suffix Margules equations cannot give as good a prediction when only binary data are used because, as shown in Table 6-11, a significant ternary constant is required.

Figure 6-33 Experimental and calculated vapor compositions for the ternary system acetone/methanol/chloroform at 50°C. Calculations use only binary data.

A final example of the applicability of Wilson’s equation is provided by Orye’s calculations for the system ethanol/methylcyclopentane/benzene at 1.013 bar. Wilson parameters were found from experimental data for the three binary systems (Myers, 1956; Sinor and Weber, 1960; Wehe and Coates, 1955); vapor compositions in the ternary system were then calculated for six cases and compared with experimental results as shown in Table 6-14. Wilson’s equation again provides a good description for this ternary that has large deviations from ideal behavior.

Table 6-14 Calculated vapor compositions for the system ethanol (1)/methylcyclopentane (2)/benzene (3) at 1.013 bar using Wilson parameters obtained from binary data only.

NRTL Equation. For a solution of m components, the NRTL equation is

(6-166)

where

(6-167)

(6-168)

The activity coefficient for any component i is given by

(6-169)

Equations (6-166) and (6-169) contain only parameters obtained from binary data.

For nine ternary systems shown in Table 6-15, Renon (1968) predicted ternary vapor-liquid equilibria with Eq. (6-166) using binary data only. He also calculated ternary equilibria with Word’s equation [Eq. (6-156)] both with and without a ternary constant. Table 6-15 indicates that Eq. (6-166) gives a good prediction of multicomponent equilibrium from binary equilibrium data alone.

Table 6-15 Comparison of NRTL and Wohl’s equations for prediction of ternary vapor-liquid equilibria.

UNIQUAC Equation. For a multicomponent system, the UNIQUAC equation for the molar excess Gibbs energy is given by the sum of

(6-170)

and

(6-171)

where segment fraction Φ* and area fractions θ and θ’ are given by

and

The coordination number z is set equal to 10. For any component i, the activity coefficient is given by

(6-172)

where

(6-173)

Equation (6-172) requires only pure-component and binary parameters.

Using UNIQUAC, Table 6-16 summarizes vapor-liquid equilibria predictions for several representative ternary systems and one quaternary system. Calculated results agree well with experimental pressures (or temperatures) and vapor-phase compositions.

The largest errors in predicted compositions occur for the systems acetic acid/formic acid/water and acetone/acetonitrile/water, where experimental uncertainties are significantly greater than those for other systems.

Moderate errors in the total pressure calculations are evident for the systems chloroform/ethanol/n-heptane and chloroform/acetone/methanol. Here strong hydrogen bonding between chloroform and alcohol creates unusual deviations from ideality; for both alcohol/chloroform systems, the activity coefficients show well-defined extrema. Because extrema are often not well reproduced by the UNIQUAC equation, these binaries are not represented as well as the others. The overall ternary deviations are similar to those for the worst-fitting binaries, methanol/chloroform and ethanol/chloroform. In spite of the relatively large deviations in calculated pressure, predicted vapor compositions agree well with the experimental data of Severns et al. (1955). Fortunately, extrema in activity coefficients are rare in binary systems.

Predictions for the other isobaric systems show good agreement. Excellent agreement is obtained for the system carbon tetrachloride/methanol/benzene, where the binary data are of superior quality.

The results shown in Table 6-16 suggest that UNIQUAC can be used with confidence for typical multicomponent systems of nonelectrolytes, provided that good experimental binary data are available to determine reliable binary parameters.

Table 6-16 Prediction of multicomponent vapor-liquid equilibrium with UNIQUAC equation using binary data only.*

While Wilson’s equation is not applicable to liquid mixtures with miscibility gaps, the NRTL equation and the UNIQUAC equation may be used to describe such mixtures. However, when these equations are applied to ternary (or higher) systems, it is often not possible to predict multicomponent liquid-liquid equilibria using only experimental binary data.

Reduction of typical binary vapor-liquid or liquid-liquid experimental data does not yield a unique set of binary parameters; several sets may reproduce the data equally well within experimental error. Multicomponent vapor-liquid equilibrium calculations are not highly sensitive to the choice of binary parameters, but multicomponent liquid-liquid equilibrium calculations depend strongly on that choice. As discussed in App. E, only a few ternary data are needed to guide the selection of “best” binary parameters. When that selection is made with care, the NRTL equation or the UNIQUAC equation can often represent ternary (or higher) liquid-liquid equilibria with good accuracy.

To obtain improved correlation of ternary liquid-liquid equilibria, Nagata (1989) introduced adjustable ternary parameters in the residual contribution of the UNIQUAC equation:

with τ_jki ≠ 0 only for i ≠ j ≠ k, i.e. three adjustable ternary parameters for each ternary system.

Figure 6-34 gives an example of Nagata’s modification of the UNIQUAC equation applied to liquid-liquid equilibria of the three ternaries forming the quaternary system acetonitrile/aniline/n-heptane/benzene (Nagata and Tamura, 1998). Calculated results with the binary and ternary parameters in Table 6-17 agree well with experiment. However, if calculations were carried out using only the binary energy parameters a₁₂ and a_21’ large deviations from experiment would be observed, particularly in the critical region. Good agreement with experiment is obtained only if ternary parameters are used, determined from optimization of ternary LLE data; however, UNIQUAC here is not a predictive method.

Figure 6-34 Experimental (•) and calculated (UNIQUAC) liquid-liquid equilibria at 298 K for the ternary systems aniline/acetonitrile/n-heptane, aniline/n-heptane/benzene, and n-heptane/acetonitrile/benzene. •––––• Experimental tie line. —— Calculated from UNIQUAC equation with binary and ternary parameters obtained from binary VLE data and ternary LLE data in Table 6-17.

Table 6-17 Binary and ternary interaction parameters for modified UNIQUAC equation (Nagata, 1989) obtained, respectively, from fitting of binary vapor-liquid equilibrium data and ternary liquid-liquid equilibrium data (Nagata and Tamura, 1998).

6.16 Summary

This chapter is concerned with activity coefficients to calculate fugacities of components in a liquid mixture. In nonideal mixtures, these activity coefficients depend strongly on composition. At conditions remote from critical, they often depend only weakly on temperature and very weakly on pressure. Unless the pressure is high, we can usually neglect the effect of pressure on activity coefficients.

The numerical value of an activity coefficient is meaningful only when the standard-state fugacity to which it refers is clearly specified. The excess Gibbs energy g^E for a mixture is defined in terms of all activity coefficients in that mixture [Eq. (6-26)]; using the Gibbs-Duhem relation, individual activity coefficients are then related to g^E through Eq. (6-25).

It is advantageous to construct an analytical function for g^E because that function provides a useful method for interpolating and extrapolating activity-coefficient data. For a binary mixture, some physical or mathematical model is used to express g^E as a function of composition, using a few (typically two) adjustable binary parameters that may be temperature dependent. When these binary parameters are evaluated from limited binary activity-coefficient data, the physical or mathematical model predicts liquid-phase activity coefficients at new compositions (and to a lesser extent, at new temperatures) where experimental data are not available.

Numerous models for g^E have been proposed. Only a few of these are reviewed in this chapter.

For those binary mixtures where the pure components do not differ much in volatility, the best method for obtaining activity-coefficient data is to measure the total pressure as a function of liquid-phase composition at constant temperature. It is not necessary to measure the vapor composition; that composition can be calculated from the total-pressure data using the Gibbs-Duhem equation.

When a function for g^E has been established for a binary mixture and the parameters are known, that function may be used to determine if the binary system has a miscibility gap; if it does, the function can be used to calculate mutual solubilities. Inverting, experimental mutual-solubility data can be used to fix the binary parameters (see App. E).

For practical applications, the most important use of the g^E function follows from extension to multicomponent (ternary or higher) systems. Such extension requires primarily binary parameters; in some cases a few ternary parameters are desirable but in many cases binary parameters alone are sufficient for vapor-liquid equilibria. Therefore, the g^E function is a “scale-up” tool, in the sense that fugacities in ternary (and higher) mixtures can be calculated using only experimental information for binary systems. This “scale-up” provides a very large reduction in the experimental effort required to characterize a multicomponent liquid mixture.

The methods discussed in Chaps. 5 and 6 can be used directly to establish a computerized technique for calculating multicomponent vapor-liquid equilibria as required, for example, in the design of separation equipment such as distillation columns. Details, including computer programs, are given elsewhere (Prausnitz et al, 1980); the essential steps, however, are easily stated.

To fix ideas, suppose we have a liquid mixture containing m components at pressure P. Mole fractions x₁, x₂,…, x_m are known. We want to find vapor-phase mole fractions y₁, y₂,…, y_m and the equilibrium temperature T. To find these m + 1 unknowns, we need m + 1 independent equations:

One material balance:

m equations of equilibrium:

for every component i.

For the standard-state fugacity f_i⁰ we usually choose the fugacity of pure liquid i at system temperature and pressure (see Chap, 3). For the vapor-phase fugacity coefficient φ_i we choose some vapor-phase equation of state (as discussed in Chap. 5) and then use Eq. (3-53). For the liquid-phase activity coefficient we use some equation for the excess Gibbs energy (as discussed in this chapter) and then use Eq. (6-25).

Simultaneous solution of the m + 1 equations is, in principle, straightforward, using iterative techniques with a computer. Since we do not know mole fractions y_i or temperature T, we must iterate with respect to several quantities. However, iteration with respect to T is our primary concern; calculated results are strongly sensitive to T because, while φ_i and γ_i depend only weakly on temperature, f_i⁰ is usually a very strong function of temperature. On the other hand, the only quantity that depends on y_i is φ_i, and that dependence is usually weak.

Similar considerations are used to calculate liquid-liquid equilibria. Unlike vapor-liquid equilibria, liquid-liquid equilibria are highly sensitive to small changes in the constants that appear in the equation for molar excess Gibbs energy g^E. In many cases it is not possible to predict g^E for a ternary system (using only binary data) with sufficient accuracy for liquid-liquid equilibrium calculations; typically some ternary data are required.

Detailed numerical methods are given in Prausnitz et al. (1980) and are not of concern here. However, it is important to recognize that the methods discussed in Chaps. 5 and 6 are essentially sufficient to solve many practical problems in multi-component phase equilibrium.

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Problems

1. Experimental studies have been made on the isothermal vapor-liquid equilibrium of a ternary mixture. The measured quantities are liquid mole fractions x_1, x_2, x_3; vapor mole fractions y₁, y_2, y_3; total pressure P; and absolute temperature T. From these measurements indicate how to calculate the activity coefficient of component 1. All components are liquids at temperature T; the saturation pressure of component 1 is designated by P₁ ^s. The molar liquid volume of component 1 is .

For the vapor phase, assume that the volume-explicit, truncated virial equation of state holds:

All necessary virial coefficients B_ij (i = 1, 2, 3; j = 1, 2, 3) are available.

2. Consider a solution of two similar liquids which are miscible in all proportions over a wide range of temperature. The excess Gibbs energy of this solution is adequately represented by the equation

g^E = Ax₁ x₂

where A is a constant depending only on temperature.

Over a wide range of temperature the ratio of the vapor pressures of the pure components is constant and equal to 1.649. Over this same range of temperature the vapor phase may be considered ideal.

We want to find out whether or not this solution has an azeotrope. Find the range of values A may have for azeotropy to occur.

3. There is something unusual about the system hexafluorobenzene (1)/benzene (2). The following data were obtained at 343 K (x and y refer to the liquid- and vapor-phase mole fractions). Plot isothermal x-y and P-x-y diagrams. What is unusual about this binary system?

Data are as follows:

4. Total-pressure data are available for the entire concentration range of a binary solution at constant temperature. At the composition x₁ = a, the total pressure is a maximum. Show that at the composition x₁ = a, this solution has an azeotrope, i.e., that the relative volatility at this composition is unity. Assume that the vapor phase is ideal.

5. Consider a liquid mixture of components 1, 2, 3, and 4. The excess Gibbs energies of all the binaries formed by these components obey relations of the form

where A_ij is the constant characteristic of the i-j binary.

Derive an expression for the activity coefficient of component 1 in the quaternary solution.

6. Construct T-x₁ and y₁- x₁ diagrams for the cyclohexanone (1)/phenol (2) system at 30 kPa. Available data are:

Vapor-phase nonidealities may be neglected.

7. Limited vapor-liquid equilibrium data have been obtained for a solution of two slightly dissimilar liquids A and B over the temperature range 20 to 100°C. From these data it is found that the variation of the limiting activity coefficients (symmetric convention) with temperature can be represented by the empirical equation

where γ^∞ is the activity coefficient at infinite dilution. Estimate the enthalpy of mixing of an equimolar mixture of A and B at 60°C.

8. The partial molar enthalpy of water in concentrated sulfuric acid solutions containing less than 20 moles H₂O per mol H₂SO_4’ at 293 K and 1 bar, is given by

where subscripts w and A stand for water and acid, respectively, x_A is the mole fraction of acid, and the reference state for expressing is pure liquid water at the temperature and pressure of the mixture. For a certain application, 2 mol H₂O and 1 mol H₂SO₄ are mixed isothermally in an open vessel, which is equipped with cooling coils. The flow of the cooling medium is controlled so that the entire mixing process occurs isothermally at 293 K.

(a) Calculate the infinite dilution partial molar enthalpy of H₂O in H₂SO₄ at 293 K.

(b) Calculate how much heat must be removed during the mixing process.

9. (a) Vapor-phase spectroscopic data clearly show that sulfur dioxide and normal butene-2 form a complex. However, thermodynamic data at 0°C show that liquid mixtures of these components exhibit slight positive deviations from Raoult’s law. Is this possible? Or do you suspect that there may be experimental error?

(b) Qualitatively compare the excess Gibbs energies of mixtures containing sulfur dioxide and butene-2 with those containing sulfur dioxide and isobutene. Explain.

10. From the total-pressure data below compute the y-x diagram for ethyl alcohol (1)/chloroforrm (2) system at 45°C. Assume ideal gas behavior. Data are as follows:

Compare your computed results with the experimental data of Scatchard and Raymond (1938, J. Am. Chem. Soc., 60: 1275).

11. For the system 2-propanol (1)/water (2) at 45°C, experimental data indicate that at infinite dilution the activity coefficients are Using Wilson’s equation, and assuming ideal-gas behavior, construct the pressure-composition P-x-y diagram for this system at 45°C (see App. F). Compare with the experimental data of Sada and Morisue (1975, J. Chem. Eng. Jap., 8: 191).

Data are as follows:

12. You want to estimate the y-x diagram for the liquid mixture carbon tetrabromide (1)/nitroethane (2) at 0.5 bar total pressure. You have available boiling-point T-x data for this system at 0.5 bar as well as pure-component vapor-pressure data as a function of temperature. Explain how you would use the available information to construct the desired diagram. Assume that a computer is available. Set up the necessary equations and define all symbols used. State all assumptions made. Are there any advantages in using the Wilson equation for the solution of this problem? Explain.

13. For the binary mixture 2-butanone/cyclohexane:

(a) Identify UNIFAC groups for each component.

(b) Using UNIFAC calculate the activity of each component for an equimolar mixture of 2-butanone/cyclohexane at 75°C.

(c) Plot the y-x diagram for this binary at 75°C.

At 75°C and 1 bar pressure, the activity coefficients for 2-butanone (1) and cyclohexane (2) are:

Antoine’s constants for vapor pressure correlation are available in Reid et al., 1979, Properties of Gases and Liquids, New York: McGraw-Hill.

14. Because of fire hazards, large quantities of flammable liquids are often stored in outdoor tanks. Consider a mixture of n-hexane and nitroethane, stored outdoors, in a chemical plant located in a northern climate. It is undesirable to have phase separation in the storage tank because, when pumped back into the plant for participation in a chemical process, it would be necessary to remix the two liquid phases to obtain the original composition.

The lowest outside temperature is estimated to be -40°C. Is it likely that an equimolar mixture will separate into two liquid phases at this temperature? To answer this question, use the UNIQUAC equation. Consider possible advantages of a graphical solution. UNIQUAC parameters are as follows:

15. If one found data in the literature for the enthalpy of mixing Δ_mix^h and the entropy change of mixing Δ_mix^s (not excess entropy) at a particular T and P for a pair of liquids miscible in all proportions and upon plotting Δ_mixh/RT and Δ_mixs/R versus x₁ on the same graph found that the curves crossed, would one have reason to question the validity of the data? Explain.

16. At 300 K, some experimental data are available for dilute liquid mixtures of components 1 and 2. When 1 is dilute in an excess of 2, Henry’s constant H_1.2 = 2 bar. When 2 is dilute in an excess of 1, Henry’s constant H_2.1 = 1.60 bar. Estimate the vapor composition which is in equilibrium with an equimolar liquid mixture of 1 and 2 at 300 K. Assume that the vapor is an ideal gas. At 300 K, the pure-component vapor pressures are:

17. A ternary liquid mixture at 300 K contains components 1, 2, and 3; all liquid-phase mole fractions are equal to 1/3. The vapor pressures of the pure components (in kPa) at 300 K are P₁^s = 53.3, P₂^s = 40, and P₃^s = 53.3. Estimate the composition of the vapor in equilibrium with this mixture.

The following binary data are available:

1-2 binary: (infinite dilution) = 1.3 at 320 K.

1-3 binary: This binary forms an azeotrope at 300 K when x₁ = x₃ = 1/2 and P = 60 kPa.

2-3 binary: At 270 K, this binary has an upper consolute temperature when x₃ = x₂ =1/2.

Clearly state all assumptions made.

18. At 25°C, a binary liquid mixture contains nonpolar components 1 and 2. Data for the dilute regions of this mixture indicate that = 9.3 and = 4.7. At 25°C, are liquids 1 and 2 miscible in all proportions or is there a miscibility gap?

19. It is reported in the literature that the excess Gibbs function for a binary system of A and B as determined from vapor-liquid equilibria is given by the following relations:

The vapor pressures of pure A and pure B are given by

where P^s is in bar and T in K. Making reasonable assumptions, determine:

(a) Whether this system forms an azeotrope at any of the listed temperatures, and if so, the azeotropic compositions.

(b) Whether this system forms an azeotrope at 760 mmHg. Make your reasoning clear.

(c) Another literature source gives the enthalpy of mixing for this system at 50°C:

Is this equation completely consistent with the data given for g^E/RT? If not, give some indication of the degree of inconsistency.