UNIQUAC5 (short for UNIversal QUAsi Chemical model) builds on the work of Wilson by making three primary refinements. First, the temperature dependence of Ωij is modified to depend on surface areas rather than volumes, based on the hypothesis that the interaction energies that determine local compositions are dependent on the relative surface areas of the molecules. If the parameter qi is proportional to the surface area of molecule i,
where z = 10. The intermediate parameter τij = exp(– aij/T) is used for compact notation where τii = τjj = 1, aii = ajj = 0.6 In addition, when the energy equation 13.16 is written with Nc,j = zqj = 10qj for all j at all ρ, the different sizes and shapes of the molecules are implicitly taken into account. Qualitatively, the number of molecules that can contact a central molecule increases as the size of the molecule increases. Using surface fractions attempts to recognize the branching and overlap that can occur between segments in a polyatomic molecule. The inner core of these segments is not accessible, only the surface is accessible for energetic interactions. Therefore, the model of the energy is proposed to be proportional to surface area. Unfortunately, it is not straightforward to construct a more rigorous argument in favor of surface fractions from the energy equation itself. Inserting Eqns. 13.31 and 13.16 into Eqn. 13.17, the excess Helmholtz for a binary solution becomes
where θi is the surface area fraction, and θi = xiqi/(x1q1 + x2q2) for a binary. Analogous to Wilson’s equation, GE is calculated as AE, a good approximation. The first two terms represent (GE/RT)RES,
that can be compared with Eqn. 13.20 and the final term 
represents (GE/RT)COMB. The (GE/RT)COMB term is attributed to the entropy of mixing hard chains, and an approximate expression for this contribution is applied by Maurer and Prausnitz.7 This representation of the entropy of mixing traces its roots back to the work of Staverman8 and Guggenheim9 and was discussed more recently by Lichtenthaler et al.10 It is very similar to the Flory term, but it corrects for the fact that large molecules are not always large balls, but sometimes long “strings.” By noting that the ratio of surface area to volume for a sphere is different from that for a string, Guggenheim’s form (the form actually applied in UNIFAC and UNIQUAC) provides a simple but general correction, giving an indication of the degree of branching and nonsphericity. Nevertheless, the Staverman-Guggenheim term represents a relatively small correction to Flory’s term. As shown in Fig. 13.1, the extra correction of including the “surface to volume” parameter serves to decrease the excess entropy to some value between zero and the Flory-Huggins estimate. The combinatorial part of UNIQUAC 
for a binary system takes a form that can be compared with 13.21
Figure 13.1. Excess entropy according to the Flory-Huggins equation versus Guggenheim’s equation at V2/V1 = 1695 for a polymer solvent mixture.
The Guggenheim form of the excess entropy is based on the molecular volume fractions, Φj, and the surface fractions, θj. Instead of using macroscopic property data to calculate volume fractions and surface fractions they are based on relative molecule volumes, r, and relative molecule surface areas, q, for each type of molecule.
r and q are the relative volume and relative surface area of a molecule.
The relative molecular parameters r and q may be calculated from group size and surface area parameters using the concept of group contributions. The size/area parameters are ratios to the equivalent size/area for the -CH2– group in a long chain alkane.11 The group parameters are added in the same manner as the UNIFAC method discussed in the next section and as given in Table 13.2 on page 512, except the UNIQUAC r and q values for alcohols are typically not calculated by group contributions (see the footnote to Table. 13.2). In the table, the uppercase Rk parameter is for the group volume, and the uppercase Qk parameter is for group surface area. From these values, the molecular size (rj) and shape (qj) parameters may be calculated by multiplying the group parameter by the number of times each group appears in the molecule, and summing over all the groups in the molecule,
Table 13.2. Selected Group Parameters for the UNIFAC and UNIQUAC Equationsa
a. AC in the table means aromatic carbon. The main groups serve as categories for similar subgroups as explained in the UNIFAC section.
b. Alcohols are usually treated in UNIQUAC without using the group contribution method. Accepted UNIQUAC values for the set of alcohols [MeOH, EtOH, 1-PrOH, 2-PrOH, 1-BuOH] are r = [1.4311, 2.1055, 2.7799, 2.7791, 3.4543], q = [1.4320, 1.9720, 2.5120, 2.5080, 3.0520]. See Gmehling, J., Oken, U. 1977-Vapor-Liquid Equilibrium Data Collection. Frankfort, Germany: DECHEMA.
Rk and Qk are the relative volume and relative surface area of a functional group.
Care is necessary when subdividing a molecule into functional groups.
where 
is the number of groups of the kth type in molecule j. The subdivision of the molecule into groups is sometimes not obvious because there may appear to be more than one way to subdivide, but the conventions have been set forth in examples in the table and these conventions should be followed. The large number of possible functional groups is divided into main groups and further subdivided into structurally similar subgroups. Usually the functional groups include a nearest neighbor atom as part of the group. The group parameters are calculated from the van der Waals volume and van der Waals surface area. Note that the van der Waals volume and van der Waals area are not calculated from the van der Waals EOS. They are inferred from x-ray and other property data.12.
Example 13.4. Combinatorial contribution to the activity coefficient
In polymer solutions, it is not uncommon for the solubility parameter of the polymer to nearly equal the solubility parameter of the solvent, but the mixture is still nonideal. To illustrate, consider the case when 1 g of benzene is added to 1 g of pentastyrene to form a solution. Estimate the activity coefficient of the benzene (B) in the pentastyrene (PS) if δps = δB = 9.2 and Vps and VB are estimated using group contributions. Use the Flory activity model and group contributions of UNIQUAC/UNIFAC to estimate volume fractions.
Solution
Since δps = δB = 9.2, we can ignore the residual contribution. Therefore,
Benzene is composed of 6(ACH) groups @ 0.5313 R-units per group ⇒ VB ∝ 3.1878. Pentastyrene is composed of 25(ACH) + 1(ACCH2) + 4(ACCH) + 4(CH2) + 1(CH3) ⇒ Vps ∝ 21.17:
Note: The volume fraction is close to the weight fraction because they are so structurally similar.
Flory’s model (no energetic contribution) predicts that the partial pressure of benzene in the vapor, yBP = xBγBPBsat, would be about 12% less than the ideal solution model.
The parameters to characterize the volume and surface area fractions have already been tabulated, so no more adjustable parameters are really introduced by writing it this way. The only real problem is that including all these group contributions into the formulas makes hand calculations extremely tedious. Fortunately, computers and spreadsheets make this task much simpler. As such, we can apply the UNIQUAC method almost as easily as the van Laar method.
For a binary mixture, the activity equations become
See Actcoeff.xlsx, worksheets uniquac, uniquac5 MATLAB: uniquac.m.
Literature UNIQUAC parameter values, aij, typically are in K.
Like the Wilson equation, the UNIQUAC equation requires that two adjustable parameters be characterized from experimental data for each binary system. The inclusion of the excess entropy in UNIQUAC by Abrams et al. (1975) is more correct theoretically, but Wilson’s equation can be as accurate as the UNIQUAC method for many binary vapor-liquid systems, and much simpler to apply by hand. UNIQUAC supersedes the Wilson equation for describing liquid-liquid systems, however, because the Wilson equation is incapable of representing liquid-liquid equilibria as long as the λij parameters are held positive (as implied by their definition as exponentials, and noting that exponentials cannot take on negative values).
Extending Eqn. 13.40 to a multicomponent solution, the UNIQUAC equation becomes
Note that the leading sum is simply Flory’s equation. The first two terms are the combinatorial contribution and the last is the residual contribution. The parts can be individually differentiated to find the contributions to the activity coefficients,
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