Evaluation of Activity Coefficients

From Eqs. 9.2.46 and 9.2.50, the solute chemical potential is given by /t-b = C/b + pV — TS. In the dilute solution, we assume Ub and Ir are linear functions of xr as explained above. We also assume the dependence of 5b on xr is approximately the same as in an ideal mixture this is a prediction from statistical mechanics for a mixture in which aU molecules have similar sizes and shapes. Thus we expect the deviation of the chemical potential from ideal-dilute behavior, /tr = + RT InxB, can be 

If we equate this expression for /xr with the one that defines the activity coefficient, Mb = ln(yjc,B Xr) (Eq. 9.5.16), and solve for the activity coefficient, 

Thus we predict that at constant T and p, Yx,b is a linear function of xb at low xr. An ideal-dilute solution, then, is one in which xr is much smaller than RT/kx so that Yx,B is approximately 1. An ideal mixture requires the interaction constant kx to be 

By similar reasoning, we reach analogous conclusions for solute activity coefficients on a concentration or molahty basis. For instance, at low otr the chemical potential of B should be approximately Mm3 RT ln(wB/m°) + fcmWB, where k is a constant at a given T and p then the activity coefficient at low otr is given by 

The prediction from the theoretical argument above, that a solute activity coefficient in a dilute solution is a linear function of the composition variable, is borne out experimentally as illustrated in Fig. 9.10 on page 264. This prediction applies only to a nonelectrolyte solute for an electrolyte, the slope of activity coefficient versus molality approaches —oo at low molality (page 290). 

---

For succeeding evaluations of activity coefficients, the values of the mol fractions are normalized as... 

The UNIFAC method for evaluation of activity coefficients depends on the concept that a liquid mixture may be considered a solution of the structural units from which the molecules are formed rather than a solution of the molecules themselves. These structural units are called subgroups, and a few of them are listed in the second column of Table D.l. An identifying number, represented by k, is associated with each subgroup. The relative volume Rk and relative surface area Qk are properties of the subgroups, and values are listed in columns 4 and 5 of Table D.l. Also shown (column 6) are examples of the subgroup compositions of molecular species. When it is possible to construct a molecule... 

We could of course calculate f values by Eq. (11.67) for conditions of low-pressure VLE and combine them with experimental values of P, Tf Xj, and y, for the evaluation of activity coefficients by Eq. (11.66). However, at. low pressures (up to at least 1 bar), vapor phases usually approximate ideal gases, for which < , = 7 = 1, and the Poynting factor (represented by the exponential) differs from unity by only a few parts per thousand. Moreover, values of 4>, and tf>f differ significantly less from each other than from unity, and their influence in Eq. (11.67) tends to cancel. Thus the assumption that = 1 introduces little error for low-pressure VLE, and it reduces Eq. (11.66) to... 

Application of the Wilson equation for evaluation of activity coefficients requires knowledge of the liquid-phase composition. We therefore calculate x, by Eq. (12.27) ... 

Kontogeorgis, G.M. et ah. Improved models for the prediction of activity coefficients in nearly athermal mixtures. Part I. Empirical modifications of free-volume models. Fluid Phase Equilibria, 92, 35, 1994. Coutinho, J.A.P., Andersen, S.I., and Stenby, E.H., Evaluation of activity coefficient models in prediction of alkane SEE, Fluid Phase Equilibria, 103, 23, 1995. 

Figure 12.9 shows the experimental data points (Figure 12.8), curves based solely on the azeotrope evaluation of activity coefficients, and the curves determined from the UNIFAC model. The agreement is remarkably good. 

Kg. Except when we are particularly interested in the evaluation of activity coefficients, we shall treat as if it were independent of composition doing so greatly simplifies the discussion of equilibria. 

APPENDIX 2 Activity Coefficieius 994 a2A Properties of Activity Coefficients 994 a2B Experimental Evaluation of Activity Coefficients 995... 

Coutinho, J.A.P., Andersen, S.I., and Stenby, E.H., 1995. Evaluation of activity coefficient models in prediction of alkane SLE. Fluid Phase Equilib., 103 23. 

Determination, thus, of the fugacity coefficients through the virial equation and of the Poynting effect through Eq.9.11.14 allows the evaluation of activity coefficients from the experimental data. An appropriate computer program (GAMMA) is presented in Appendix E. 

---