Use of Activity Coefficient Models to Correlate Data
The points in Figs. 9.5-4 and 9.5-5 represent smoothed values of the activity coefficients for both species in a benzene–2,2,4-trimethyl pentane mixture at 55°C taken from the vapor-liquid equilibrium measurements of Weissman and Wood (see Illustration 10.2-4). Test the accuracy of the one-constant Margules equation and the van Laar equations in correlating these data.
a. The one-constant Margules equation. From the data presented in Fig. 9.5-4 it is clear that the activity coefficient for benzene is not the mirror image of that for trimethyl pentane. Therefore, the one-constant Margules equation cannot be made to fit both sets of activity coefficients simultaneously. (It is interesting to note that the Margules form, R T \ln \gamma_{ i }=A_{ i } x_{ j }^{2}, will fit these data well if A_{1} \text { and } A_{2} are separately chosen. However, this suggestion does not satisfy the Gibbs-Duhem equation! Can you prove this?)
b. The van Laar equation. One can use Eqs. 9.5-10 and a single activity coefficient–composition data point (or a least-squares analysis of all the data points) to find values for the van Laar parameters. Using the data at x_{1}=0.6, \text { we find that } \alpha=0.415 \text { and } \beta=0.706. The activity coefficient predictions based on these values of the van Laar parameters are shown in Fig. 9.5-5. The agreement between the correlation and the experimental data is excellent. Note that these parameters were found using the MATHCAD worksheet ACTCOEFF on the website for this book, and discussed in Appendix B.III.
\begin{aligned}\alpha &=\left(1+\frac{x_{2} \ln \gamma_{2}}{x_{1} \ln \gamma_{1}}\right)^{2} \ln \gamma_{1} \\\beta &=\left(1+\frac{x_{1} \ln \gamma_{1}}{x_{2} \ln \gamma_{2}}\right)^{2} \ln \gamma_{2}\end{aligned} (9.5-10)