Appendix F
Estimation of Activity Coefficients

Chapter 6 presents several equations for representing activity coefficients as a function of composition. These equations contain binary parameters that are obtained from fitting to binary data. While many binary systems have been studied experimentally (compilations have been published, for example, by Wichterle et al., 1973; many volumes by Gmehling and Onken, starting in 1977; Hirata et al,. 1975; many volumes by Maczynski et al,. starting in 1976; Ohe, 1989, and other titles in Physical Sciences Data Series; continuing compilations by DIPPR-AIChE, New York), there are also many other binaries where only fragmentary data, or no data at all, are available.

When interest is directed at a binary where data are absent or incomplete, one should not immediately dismiss the possibility of obtaining some data oneself. While making reliable experimental measurements is always a challenge, modern instrumentation often simplifies the experimental effort. For a typical binary mixture of nonelectrolytes, only a few experimental data are often sufficient to yield reasonable binary parameters. In any event, there is a vast difference between a few reliable data and none at all.

Estimation from Activity Coefficients at Infinite Dilution

For partially miscible binary mixtures, binary parameters can be obtained from mutual-solubility data, as discussed in App. E, and for binary mixtures that form azeotropes, binary parameters can be calculated from azeotropic data (Gmehling et al,. 1994). For all binary mixtures, regardless of the extent of miscibility and regardless of whether or not they form an azeotrope, binary parameters can be calculated from activity coefficients at infinite dilution. In many cases, when only limited experimental data are available, activity coefficients at infinite dilution provide the most valuable experimental information (Sandler, 1996). Fortunately, there are convenient experimental methods for measuring infinite-dilution activity coefficients (Dallinga et al,. 1993; Eckert and Sherman, 1996; Kojima et al,. 1997).

For liquid mixtures where the two components do not differ greatly in volatility, activity coefficients at infinite dilution can be obtained for both components by differential ebulliometry (Nicolaidis and Eckert, 1978; Kojima et al,. 1997). For liquid mixtures where one component is much more volatile than the other, the infinite-dilution activity coefficient of the more volatile component may be obtained by gas-liquid chromatography (Laub and Pecsok, 1978; Kojima et al,. 1997). In this technique, the chromatograph is used as the equilibrium cell; the heavy component is the substrate and an inert carrier gas flows over that substrate. A small amount of light component is injected and the important measured quantity is the retention time, i.e., the time that the light component is “retained” as a result of its contact with the heavy component. For example, Comanita et al. (1976) measured retention times yielding K factors¹ for propanol (2) at infinite dilution in n-hexadecane using helium as a carrier gas; their results are shown in Fig. F-1 From these data, the activity coefficient γ₂ is found from

¹ K = y/x, whew y is the vapor-phase mole fraction and x is the liquid-phase mole fraction

(F.1)

Here P is the total pressure, φ₂ is the fugacity coefficient of propanol in the carrier gas, and is the fugacity of pure liquid propanol, both at system temperature and pressure.

When component 2 is much more volatile than component 1, the activity coefficient at infinite dilution, , is found from a plot of γ₂ versus P, according to

(F.2)

From the data shown in Fig. F-1, the activity coefficients for propanol at infinite dilution are 4.32 (100°C), 3.24 (120°C), 2.90 (130°C), and 2.58 (140°C).

Figure F-1 K factors for propanol in n-hexadecane from gas-liquid chromatography. (Comanita et al, 1976).

In this example, measurements were made to obtain the infinite-dilution activity coefficient for the more volatile component, propanol; the chromatographic method cannot be used to measure also the infinite-dilution activity coefficient of the heavy component, n-hexadecane. Therefore, in this case, the experimental data can be used to obtain only one parameter, e.g., the χ parameter in the Flory-Huggins equation [Eq. (8-12)] or a one-parameter form of the UNIQUAC equation.²

² In the one-parameter form we assume that u₁₁ = -Δ_vapU₁lq₁ and u₂₂ = -Δ_vapU₂lq₂, where Δ_vapU is the energy of complete vaporization and q is a pure-component molecular-structure constant depending on molecular size and external surface area. At conditions remote from critical, Δ_vapU ≈ Δ_vapH – RT, where Δ_vapH is the enthalpy of vaporization. Further, we assume u₁₂ = u₂₁ = -(u₁₁u₂₂)^1/2(1 – c₁₂), where c₁₂ is the single adjustable binary parameter (Bruin, 1971).

Another convenient experimental method for obtaining activity coefficients at infinite dilution is by head-space analysis (Hackenberg and Schmidt, 1976; Kieckbusch and King, 1979; Kolb and Ettre, 1997). In this procedure, a chromatograph is used as an analytical tool to determine the vapor-phase concentration of a trace component in equilibrium with a liquid phase of known composition. Figure F-2 shows limiting activity coefficients for some organic solutes in aqueous sucrose solutions obtained by headspace analysis (Chandrasekaran and King, 1971). This experimental method can be used to obtain infinite-dilution activity coefficients for both components in a binary mixture if both pure components have a measurable vapor pressure.

Figure F-2 Activity coefficients at infinite dilution for organic solutes in aqueous sucrose solutions at 25°C obtained by gas-liquid chromatography (Chandrasekaran and King, 1971).

All the methods discussed above for determining infinite-dilution activity coefficients require a solute with appreciable volatility. For slightly volatile components, Trampe and Eckert (1993) have developed a method for measuring infinite dilution activity coefficients based on the accurate determination of the dew point of a mixture, because dew points are very sensitive to small amounts of low-volatile components.

When activity coefficients at infinite dilution are available for both components in a binary mixture, activity coefficients at intermediate compositions can be estimated easily. For a binary mixture, we choose some expression F for the molar excess Gibbs energy g^E containing two binary parameters A and B,

(F-3)³

³ See Sec. 6.10.

g^E = F(x, A, B)

where x is the mole fraction and parameters A and B depend on temperature. From Eq. (6-25) we then obtain

(F.4)

RT1nγ₁ = F₁(x,A,B)

(F.5)

RT1nγ₂ = F₂(x,A,B)

where functions F₁ and F₂ follow from the choice of function F in Eq. (F-3). At infinite dilution,

Therefore, experimental values for and are sufficient to yield binary parameters A and B by simultaneous solution of Eqs. (F-4) and (F-5). In some fortunate cases, simultaneous solution is not necessary; e.g., if the van Laar equation⁴ is used for Eq. (F-3), we obtain

⁴ See Eqs. (6-97) and (6-98).

(F.6)

(F.7)

In the general case, however, In and depend on both parameters A and B, requiring simultaneous solution. For example, if we choose Wilson’s equation⁵

⁵ See Eqs. (6-105) and (6-106).

(F.8)

(F.9)

where binary parameters Λ₁₂ and Λ₂₁ have replaced A and B. In this case, In and In depend on both binary parameters.

Once the binary parameters are known, activity coefficients can be predicted over the entire range of liquid composition. When experimental values for and are reliable, such predictions tend to be very good. To illustrate, Figure F-3 indicates that when only and are used, good agreement is obtained with experiment for the system acetone/benzene at 45°C (Lobien, 1982).

Figure F-3 Vapor-liquid equilibria for the system acetone (1)/benzene (2) at 45°C. Calculations based on Wilson’s equation using = 1.65 and = 1.57 (Lobien, 1982).

Estimation from Group-Contribution Methods

An attractive (but not always accurate) method for estimating activity coefficients is provided by equations based on the group-contribution concept presented in Sec. 7.8. In a group-contribution method, a molecule is divided (somewhat arbitrarily) into functional groups; molecule-molecule interactions are given by properly weighted sums of group-group interactions. A review of group-contribution methods is given by Fredenslund and Sørensen (1994) and by Gmehling (1998).

From the many different models proposed, ASOG and UNIFAC have the widest practical interest. For both methods, group-group interaction parameter tables are available, e.g. those by Tochigi et al. (1990, 1998) for ASOG and those by Hansen et al. (1991), Gmehling et al. (1993) and Fredenslund and Sørensen (1994) for UNIFAC. UNIFAC is successful for semiquantitative predictions of vapor-liquid equilibria for a wide variety of mixtures, including those containing polymers. It also has been used to predict activity coefficients at infinite dilution (Voutsas and Tassios, 1996; Möllmann and Gmehling, 1997; Zhang et al,. 1998). Some extensions of UNIFAC consider the solubilities of gases and light hydrocarbons in water (Li et al,. 1997) and aqueous solutions of biochemicals, such as sugars, imino acids and organic and inorganic salts (Kuramochi et al., 1997, 1998).

The most common application of UNIFAC is for prediction of VLE of ordinary solutions. To illustrate, Fig. F-4 compares UNIFAC predictions with experimental vapor-liquid equilibrium for the system methanol/water at 50°C. In the entire composition range agreement is very good. However, such good agreement is not typical.

Figure F-4 Vapor-liquid equilibria at 50°C for the system methanol/water. Experimental data from McGlashan and Williamson (1976). In this case agreement is very good but for many other cases it is not.

A second example is shown in Fig. F-5 for the system n-hexane/methyl ethyl ketone. Here agreement is not good, indicating that for numerous systems, UNIFAC gives only a semi-quantitative prediction.

Figure F-5 Vapor-liquid equilibria at 65°C for the system n-hexane/methyl ethyl ketone. Experimental data from Maripuri and Ratcliff (1972). Agreement is only semi-quantitative.

UNIFAC with parameters based on vapor-liquid equilibrium data does not yield reliable predictions for liquid-liquid equilibria. Estimates of liquid-liquid equilibria are only possible with UNIFAC parameters determined especially for describing such equilibria (Magnussen et al,. 1981; Gupte and Danner, 1987). A correlation restricted to water-hydrocarbon liquid-liquid equilibria was presented by Hooper et al. (1988).

UNIFAC has also been used to estimate binary constants in an equation of state. For example, in the quadratic mixing rule for constant a in equations of the van der Waals form, UNIFAC can be used to estimate ay (where i ≠ j).

Numerous authors have discussed group-contribution equations of state. Illustrative of their work is that of Gmehling and coworkers (Holderbaum and Gmehling, 1991; Fischer and Gmehling, 1996; Gmehling et al,. 1997; Li et al,. 1998). These authors developed a group-contribution equation of state based on the Soave-Redlich-Kwong equation for the prediction of gas solubilities and vapor-liquid equilibria at low and high pressures.

Oishi (1978) extended UNIFAC to polymer solutions by adding a free-volume contribution, as suggested by Prigogine-Flory-Patterson theory (Sec. 8.3), to account for the free-volume difference between polymer and solvent molecules. While this difference is usually insignificant for liquid mixtures of small molecules, it is important for polymer/solvent systems.

Another extension of UNIFAC to polymer solutions is given by Holten-Andersen et al. (1987) who obtain the free-volume contribution from an equation of state similar to that based on perturbed-hard-chain theory (Sec. 7.16). This equation of state has a simplified attraction term that contains a UNIQUAC-like expression for mixtures. Parameters are obtained from a group-contribution correlation. To illustrate, Table F-1 shows experimental and calculated infinite-dilution solvent activity coefficients for a variety of polymer/solvent systems. While there are some exceptions, agreement between experimental and calculated results is within 10%.

Table F-1 Experimental and predicted (γ₁ = a₁/w₁) (Holten-Andersen et al., 1987).

Goydan et al. (1989) have compared predictions of the UNIFAC free-volume model of Oishi and the Holten-Andersen equation of state for solvent activities in a variety of polymer solutions. They found that the Holten-Andersen correlation is somewhat more accurate, while Oishi’s model, also fairly accurate, is more widely applicable. In a similar investigation, High and Danner (1990) also found that both models predict solvent activities reasonably well.

Elbro et al. (1990) propose to include free-volume contributions to solvent activities in polymer solutions in a remarkably simple way. Starting with a generalized van der Waals partition function (Sec. 7.15), the combinatorial and free-volume terms are combined to give for an athermal solution:

(F.10)

where superscript fv denotes free volume and is the free-volume fraction of component i, given by

(F.11)

where v_i and are, respectively, the pure-component molar volumes and the hard-core volumes; x_i is the mole fraction of component i. In Eq. (F-11), if the volume differences (v_i – ) are replaced by the hard-core volumes , the free-volume fraction is identical to the segment fraction given in Eq. (8-4). In that case, Eq. (F-10) gives the combinatorial Flory-Huggins excess entropy, i.e. for a binary mixture Eq. (F-10) is then identical to Eq. (8-6).

For the activity coefficient of an athermal solution, Elbro obtained:

(F.12)

In Eq.(F-12), , where is the activity of component i due to combinatorial and free-volume effects.

Equation (F-12) reflects only the combinatorial and the free-volume contributions and, therefore, for real (non-athermal) polymer solutions a residual term has to be added, as discussed in Sec. 8.2. The total activity coefficient is given by

(F.13)

Elbro et al. (1990) adopted the UNIQUAC residual term that gives the residual activity coefficient⁶ for a binary ij,

⁶ See Eq. (6-122)

(F.14)

with

(F.15)

and

(F.16)

where x_i is the mole fraction, parameters q’ are pure-component molecular-structure constants depending on molecular size and external surface area, a_ij and a_ji are characteristic energy parameters dependent on temperature but often that dependence is small.

Parameters a_ij and a_ji may be estimated from UNIFAC. For polymer/solvent systems they also can be obtained from vapor-liquid-equilibrium data for binary mixtures of the solvent and small-molecule homologues of the polymer. To illustrate, Fig. F-6 compares calculated and experimental solvent activities as a function of solvent weight fraction for the system polystyrene/tetrachloromethane. The curves were calculated with energy parameters obtained from VLE at 293 K for the binary ethylbenzene/tetrachloromethane. Elbro’s model gives excellent results; when the simple Flory-Huggins combinatorial term (without free-volume correction) is combined with the UNIQUAC residual contribution, calculated results are poor. Figure F-6 shows the importance of free-volume effects when “scaling-up” data from a low-molecular-weight mixture to predict properties of a polymer solution. In general, Elbro’s model compares favorably with the Holten-Andersen equation of state and with Oishi’s method.

Figure F-6 Calculated and observed activities of tetrachloromethane in polystyrene at 293 K. in both calculations, the residual contribution was calculated with energy parameters obtained from VLE at 293 K for the binary ethylbenzene/tetrachloromethane (Elbro et al,. 1990).

Other group-contribution methods for polymer/solvent systems include those by Chen et al. (1990), High and Danner (1990a) and Lee and Banner (1996, 1996a, 1996b, 1997). At present all of the published group-contribution methods provide good estimates of solvent activities for those solvent/polymer systems where the polymer is readily soluble and where both components are nonpolar or weakly polar. Hydrogen bonded systems often require modifications with special parameters. At present, there is no reliable method for estimating liquid-liquid equilibria in polymer/solvent systems.

Group-contribution methods are attractive because they are based on the assumption of group independence; this assumption says, for example, that the properties of say, a carbonyl (C=O) group in a monoketone are the same as those in a diketone, and that the properties of this carbonyl group in a linear ketone (e.g. hexanone) are the same as those in a cyclic ketone (e.g. cyclohexanone). Also, a group-contribution method does not distinguish between hexanone-1 and hexanone-3; the position of the carbonyl group in the molecule is not considered. These assumptions are, of course, not correct. It is possible to relax these assumptions by introducing more groups, e.g. by distinguishing between a carbonyl group in a linear molecule and that in a cyclic molecule. Agreement with experiment can then be improved but at a high cost: more parameters are required; to obtain them we need more experimental data. In the limit, as we define more and more groups, the advantage of the group-contribution method is lost. A useful group-contribution method is a compromise between on the one hand, using many different groups as suggested by our knowledge of molecular structure, and on the other, using only a limited number of groups, as required by a limited bank of experimental data.

An exciting method for improving accuracy and for increasing the range of applicability of group-contribution methods is provided by quantum-chemical calculations to give group-group interaction parameters from first principles. Initial efforts toward that end have been reported by Sandler and coworkers (Wu and Sandler, 1991; Wolbach and Sandler, 1997).

The objective of group-contribution methods is to use existing phase equilibrium data to predict phase equilibria for systems where no data are available. While such predictions may be used for preliminary design purposes, group-contribution methods yield only approximate vapor-liquid equilibria. Whenever reliable experimental data are available, these should be used instead of group-contribution predictions. While Raoult’s law provides a zeroth approximation for phase equilibria, group-contribution methods often provide no more than a first approximation.

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