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.

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

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").

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

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

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