FEC: Gibbs-Duhem and Modeling for Activity Coefficients

Gibbs-Duhem and the Activity Coefficient

Recall that the Gibbs-Duhem equation relates partial molar properties. Here we are specifically interested in applying this to the partial molar Gibbs free energy (or chemical potential), so:

$0 = \sum_i n_i d[\bar G_i] = \sum_i n_i d\mu_i$

Recalling the definition of fugacity ($\mu_i -\mu_i^o = RT\ln[\frac{\hat f_i}{\hat f_i^o}]$) and the fact that our reference values are constants, we can plug them into this equation to get:

$0 = \sum_i n_i d\mu_i= \sum_i n_i d[\ln(\hat f_i)]$

Using the expression for a liquid phase fugacity ($\hat f_i = x_i\gamma_if_i^o$) we get:

$0 = \sum_i n_i d[\ln(\hat f_i)]= \sum_i n_i \left [d[\ln(x_i)] + d[\ln(\gamma_i)] + d[\ln(f_i^o)] \right ]$

Again, we recall that our reference fugacity ($f_i^o$) is a constant (so that derivative goes to zero) and that the sum of the $x_i$'s is also a constant (1!) leads us to the fact that the sum of those derivatives must be zero, so we can reduce this to:

$0 = \sum_i n_i \left [d[\ln(\gamma_i)]$

which is essentially the Gibbs-Duhem relation applied to activity coefficients.

Note

We did not mention whether we were considering Henry or Lewis-Randall states as our reference point because it does not matter! In fact, this equation holds true even if we have a mixture of reference states (that is, we choose Henry for some components and Lewis-Randall for others).

Now that we know that activity coefficients must be related to each other, we can list the known qualities of activity coefficients:

Important

This second quality sounds complex, but it simply means that the activity coefficient for the concentrated component must approach 1 as the composition of that component approaches 1 (provided that we chose a Lewis-Randall or "everyone is like me" reference state for that component). Similarly, the activity coefficient should approach 1 for a Henry reference state component as that component's composition goes to zero (because a Henry reference state is like saying "no one is like me").

Margules and other activity coefficient correlations

Provided that empirical correlations satisfy these two conditions, they are valid models for activity coefficient dependence on composition.

Definition

One of the most common correlations for activity coefficients is the Margules model. The "two suffix" version has only 1 fitting parameter and looks (for a binary mixture) like

$\ln (\gamma_1) = A x_2^2$ and $\ln (\gamma_2) = A x_1^2$

The "three suffix" version has 2 fitting parameters so it can now accomodate differing "infinite dilution" behavior for the two components and looks (for a binary mixture) like

$\ln (\gamma_1) = x_2^2[A_{12} +2(A_{21}-A_{12})x_1]$ and $\ln (\gamma_2) = x_1^2[A_{21} +2(A_{12}-A_{21})x_2]$

Outcome

Use the Gibbs-Duhem equation to relate activity coefficients in a mixture