### Calculating Partial Molar Properties: EOS

Another method of calculating partial molar properties is to use
equations of state.

If we have a mixture that is well described by a volume-explicit
virial equation of state, so that:

$z = \frac{Pv}{RT} = 1 + B_{mix}P$

##### Note:

Here $v$ is the "total solution property", so it represents the
specific volume of the actual mixture of the fluids.

If we assume that we have a binary mixture of materials "1" and
"2", we have discussed in the past that $B_{mix}$ is related to
binary interactions between molecules of types 1 and 2:

$v = \frac{RT}{P} + y_1^2B_{11} + 2 y_1y_2B_{12} +
y_2^2B_{22}$

We know that, in order to calculate a partial molar property, we
need to start with an *extensive* property, so we multiply
by $n_{tot} = n_1+n_2$, so

$V = n_{tot}v = (n_1+n_2)\frac{RT}{P} + \frac{n_1^2B_{11} + 2
n_1n_2B_{12} + n_2^2B_{22}}{(n_1+n_2)}$

Taking the derivative with respect to $n_1$ will give us $\bar
V_1$ so

$\bar V_1 = \left (\frac{\partial V}{\partial n_1}\right
)_{P,T,n_2} = \frac{RT}{P} + \frac{2n_1B_{11} + 2
n_2B_{12}}{(n_1+n_2)^2}-\frac{n_1^2B_{11} + 2 n_1n_2B_{12} +
n_2^2B_{22}}{(n_1+n_2)^2}$

which we can simplify to

$\bar V_1 = \frac{RT}{P} + 2y_1B_{11} + 2 y_2B_{12}
-B_{mix}$

##### Note:

We can then calculate $\bar V_2$ either following the same
procedure or the **Gibbs-Duhem** equation.

##### Outcome:

Calculate partial molar properties using equations of state (and
mixing relations)