Binary LLE by Graphing the Gibbs Energy of Mixing

 shows the contributions to the Gibbs energy of a mixture for A₁₂ = 3 of Fig 14.1. The pure component Gibbs energies do not contribute to the curvature in the Gibbs energy of a mixture, and therefore are not needed for LLE calculations—we need just ΔG_mix. In principle, all that is required to make predictions of LLE partitioning is some method of calculating activity coefficients. In this section we use specific models (MAB and UNIFAC) to demonstrate calculation of LLE using ΔG_mix. The plotting/tangent line method can be extended to any activity coefficient model. This method is often the easiest method to use for binary solutions, though we show that it is subject to uncertainties from drawing/reading the tangent line.

The MAB and UNIFAC models are convenient for demonstrating the calculations, but there is a certain danger in applying too much confidence in such predictions. LLE is more sensitive to the accuracy of the activity coefficients than VLE. Furthermore, the empirical nature of UNIFAC means that the same parameter set, {a_mn}, is not generally accurate for both VLE and LLE, so a different predictive parameter set is used. As for the sensitivity problem, the best advice is not to take any predictions too seriously. They can be used as a guide to assess miscibility in a way that is slightly better than looking at solubility parameter tables, but should never be considered as a substitute for experimental data. With these cautions in mind, it is useful to show how LLE can be predicted using UNIFAC and MAB. We have provided the LLE parameters on the spreadsheet UNIFAC(LLE) within Actcoeff.xlsx, and within Matlab/gammaModels/unifacLLE.

 UNIFAC parameters for LLE differ from those for VLE.

 Actcoeff.xlsx–UNIFAC(LLE); MATLAB Ex14_03.m.

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Example 14.3. LLE predictions by graphing

Arce et al.^a give the compositions for the tie lines in the system water(1) + propanoic acid(2) + methylethylketone (MEK)(3) at 298 K and 1 bar. As limiting conditions, the mutual solubilities of water + MEK (1CH₃ + 1CH₃CO + 1CH₂) binary are also listed as x₁^α = 0.342, x₁^β = 0.922.

a. Use MAB to roughly estimate the water + MEK binary mutual solubilities to ± 5 mole%.

b. Use UNIFAC to roughly estimate the water + MEK binary mutual solubilities to ± 5 mole%.

Solution

a. A₁₂ = (50.13 – 0)(15.06 – 9.70)(90.1 + 18.0)/(4·8.314·298) = 2.931, virtually the same as the parameter used above.

Adding G^E/RT = A₁₂x₁x₂ and  gives ΔG_mix/(RT). Using the drawing tool shows  and 

b. Selecting the appropriate groups from the UNIFAC menu, then copying the values of the activity coefficient, we can develop Figs. 14.3 and 14.4 using increments of x_w = 0.05. In MATLAB we can set up a vector x₁ = 0:0.05:1, and then insert a loop into unifacCallerLLE.m. Noting  and programming the formula for ΔG_mix/(RT).

Figure 14.3. Gibbs energy of mixing in the water + MEK system as predicted by (a) MAB and (b) UNIFAC.

Figure 14.4. Activities of water and MEK as a function of mole fraction water as predicted by UNIFAC. The activity versus mole fraction plots will have a maximum when LLE exists. The dashed lines show the compositions where the activities of components are equal in both phases simultaneously.

Using the line drawing tool we obtain tangents at  and .

These are sufficiently precise for the problem statement as given above. Note how the MAB model results in symmetric estimates of the compositions, a serious deficiency for LLE, and UNIFAC happens to be fairly close.