# 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:

• they must (collectively) satisfy the G-D relation
• they must approach a value of 1 when the solution approaches conditions that would behave in the same manner as the ideal solution reference
##### 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