UNIFAC13 (short for UNIversal Functional Activity Coefficient model) is an extension of UNIQUAC with no user-adjustable parameters to fit to experimental data. Instead, all of the adjustable parameters have been characterized by the developers of the model based on group contributions that correlate the data in a very large database. The assumptions regarding coordination numbers, and so forth, are similar to the assumptions in UNIQUAC. The same strategy is applied:
The combinatorial term is therefore identical and given by Eqn. 13.54. The major difference between UNIFAC and UNIQUAC is that, for the residual term, UNIFAC considers interaction energies between functional groups (rather than the whole molecule). Interactions of functional groups are added to predict relative interaction energies of molecules. Examples are shown in Table 13.2. Each of the subgroups has a characteristic size and surface area; however, the energetic interactions are considered to be the same for all subgroups with a particular main group. Thus, representative interaction energies (aij) are tabulated for only the main functional groups, and it is implied that all subgroups will use the same energetic parameters. An illustrative sample of values for these interactions is given in Table 13.3. Full implementations of the UNIFAC method with large numbers of functional groups are typically available in chemical engineering process design software. A subset of the parameters is provided on the UNIFAC spreadsheet in the Actcoeff.xlsx spreadsheet included with the text. Knowing the values of these interaction energies permits estimation of the properties for a really impressive number of chemical solutions. The limitation is that we are not always entirely sure of the accuracy of these predictions.
Table 13.3. Selected VLE Interaction Energies aij for the UNIFAC Equation in Units of Kelvin
Although UNIFAC is closely related to UNIQUAC, keep in mind that there is no direct extension to a correlative equation like UNIQUAC. If you want to fit experimental data that might be on hand, you cannot do it within the defined framework of UNIFAC; UNIQUAC or NTRL is the preferred choice when adjustable parameters are desired. Although it is tedious to estimate the aij parameters of UNIQUAC or NRTL from UNIFAC, some implementations of chemical engineering process design software have included facilities for estimating UNIQUAC or NRTL parameters from UNIFAC. This approach can be useful for estimating interactions for a few binary pairs in a multicomponent mixture when most of the binary pairs are known from experimental data specific to those binary interactions.
The basic approach to understanding UNIFAC is the generalization of the methods for calculating the residual activity coefficient. Imagine the interactions of a CH3 group in a mixture of isopropanol (1) and component (2). The isopropanol consists of 2(CH3) + 1(CH) + 1(OH). Therefore, in the mixture, a CH3 will encounter CH3, CH, OH groups, and the groups of component (2), and the interaction energies depend on the number of each type of group available in the solution. Therefore, the interaction energy of CH3 groups can be calculated relative to a hypothetical solution of 100% CH3 groups. The mixture can be approximated as a solution of groups (SOG)14 (rather than a solution of molecules), and the interaction energies can be integrated with respect to temperature to arrive at chemical potential in a manner similar to the development of Eqn. 13.40.
Therefore, it is possible to calculate 
where 
is the chemical potential in a hypothetical solution of 100% CH3 groups and ΓCH3 is the activity coefficient of CH3 in the solution of groups. The chemical potential of CH3 groups in pure isopropanol (1), given by 
, will differ from 
because even in pure isopropanol CH3 will encounter a mixture of CH3, CH, and OH groups in the ratio that they appear in pure isopropanol, and therefore the activity coefficient of CH3 groups in pure isopropanol, 
, is not unity, where the superscript (1) indicates pure component (1). The difference that is desired is the effect of mixing the CH3 groups in isopropanol with component (2), relative to pure isopropanol,
Fig. 13.2 provides an illustration of the differences that we seek to calculate, with water as a component (2).
Figure 13.2. Illustration relating the chemical potential of CH3 groups in pure 2-propanol, a real solution of groups where water is component (2), and a hypothetical solution of CH3 groups. The number of groups sketched in each circle is arbitrary and chosen to illustrate the types of groups present. The chemical potential change that we seek is 
. We calculate this difference by taking the difference between the other two paths.
If the chemical potential of a molecule consists of the sum of interactions of the groups,
Therefore, we arrive at the important result that is utilized in UNIFAC,
where the sum is over all function groups in molecule (1) and 
is the number of occurrences of group m in the molecule. The activity coefficient formula for any other molecular component can be found by substituting for (1) in Eqn. 13.56. Note that Γm is calculated in a solution of groups for all molecules in the mixture, whereas 
is calculated in the solution of groups for just component (1). Note that we use uppercase letters to represent the group property analog of the molecular properties, with the following exceptions: Uppercase τ looks too much like T, so we substitute Ψ, and the aij for UNIFAC is understood to be a group property even though the same symbol is represented by a molecular property in UNIQUAC. The relations are shown in Table 13.4.
Table 13.4. Comparison of Group Variables and Molecular Variables for UNIFAC
lnΓm is calculated by generalizing the UNIQUAC expression for 
. Generalizing Eqn. 13.55 and supporting equations,
UNIFAC. See Actcoeff.xlsx, worksheets UNIFAC and UNIFACLLE, and MATLAB UnifacCaller.m.
where 
is the number of groups of type k in molecule i. Fortunately, the spreadsheet and programs provided with the textbook save us from doing the tedious calculations for UNIFAC, although an understanding of the principles is important.
Example 13.5. Calculation of group mole fractions
Calculate the group mole fraction for CH3 in a mixture of 60 mole% 2-propanol, 40 mole% water.
Solution
The two molecules are illustrated in Example 13.1 on page 500 and the group assignments are tabulated there. On a basis of 10 moles of solution, there are six moles of 2-propanol, and four moles of H20. The table below summarizes the totals of the functional groups.
The mole fraction of CH3 groups is then XCH3 = 12/28 = 0.429. The mole fractions of the other groups are found analogously and are also summarized in the table. The results are consistent with Eqn. 13.60 which is more easily programmed,
Actcoeff.xlsx, UNIFAC, and UnifacVLE/unifacCaller.m.
Example 13.6. Detailed calculations of activity coefficients via UNIFAC
Let’s return to the example for the IPA + water system mentioned in Example 13.1. Compute the surface fractions, volume fractions, group interactions, and summations that go into the activity coefficients for this system at its azeotropic conditions. The isopropyl alcohol (IPA) + water (W) system is known to form an azeotrope at atmospheric pressure and 80.37°C (xW = 0.3146)a.
Applying Eqn. 13.57,
Solution
(this calculation can be followed interactively in the UNIFAC spreadsheet):
The molecular size and surface area parameters are found by applying Eqn. 13.44. Isopropanol has 2 CH3, 1 OH, and 1 CH group. The group parameters are taken from Table 13.2.
For IPA: r = 2·0.9011 + 0.4469 + 1.0 = 3.2491; q = 2·0.8480 + 0.2280 + 1.2 = 3.124 For water: r = 0.920; q = 1.40
At xW = 0.3146, ΦW = 0.1150, and θW = 0.1706 using the same combinatorial contribution as UNIQUAC, Eqn. 13.54,
Note that these combinatorial contributions are computed on the basis of the total molecule. This is because the space-filling properties are the same whether we consider the functional groups or the whole molecules.
For the residual term, we break the solution into a solution of groups. Then we compute the contribution to the activity coefficients arising from each of those groups. We have four functional groups altogether (2CH3, CH, OH, H2O). We will illustrate the concepts by calculating 
and simply tabulate the results for the remainder of the calculations since they are analogous.
First, let us tabulate the energetic parameters we will need. We can summarize the calculations in tabular form as follows:
For pure isopropanol, we tabulate the mole fractions of functional groups, and calculate the surface fractions:
The same type of calculations can be repeated for the other functional groups. The calculation of 
is not necessary, since the whole water molecule is considered a functional group. Performing the calculations in the mixture, the mole fractions, Xj, need to be recalculated to reflect the compositions of groups in the overall mixture. Table 13.5 summarizes the calculations.
Table 13.5. Summary of Calculations for Mixture of Isopropanol and Water at 80.37°C and xw = 0.3146
The pure component values of InΓj(i) can be easily verified on the spreadsheet after unhiding the columns with the intermediate calculations or in MATLAB by removing the appropriate “;”. Entering values of 0 and 1 for the respective molecular species mole fractions causes the values of InΓj(i) to be calculated. (Note that values will appear on the spreadsheet computed for infinite dilution activity coefficients of the groups which do not exist in the pure component limits, but these are not applicable to our calculation so we can ignore them.) Subtracting the appropriate pure component limits gives the final row in Table 13.5. All that remains is to combine the group contributions to form the molecules, and to add the residual part to the combinatorial part.
a. Perry, R.H., Chilton, C.H. 1973. Chemical Engineers’ Handbook, 5 ed. New York, NY: McGraw-Hill, Chapter 13.
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