# CfE: Calculating Partial Molar Properties: Equations of State

### 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)