# CfE: Calculating Partial Molar Properties: Gibbs-Duhem Relation

### Calculating Partial Molar Properties: Gibbs-Duhem Relation

Noting that the partial molar volumes, $\bar V_i$, may be summed to yield the total solution (extensive) volume:

$V = \sum_i \bar n_iV_i$

Taking the derivative of both sides:

$dV = d \left [ \sum_i \bar n_iV_i \right ] = \sum_i \left ( \bar V_i dn_i + n_i d \bar V_i \right )$

recall that, from the definition of partial molar properties:

$dV = \sum_i \bar V_i dn_i$

so this can be eliminated from our equation to give the Gibbs-Duhem relation:

$\sum_i n_i d\bar V_i = 0$

##### Definition:

The Gibbs-Duhem equation relates the partial molar property values of one of the components in a mixture to that of the other species. This makes sense since the reason that we do not use the pure component specific volumes ($\bar V_i \ne v_i$) is specifically because the interaction a-a is different from a-b (but it should not be different from b-a!).

Using the Gibbs-Duhem relation for a binary mixture goes as follows:

$d\bar V_a n_a + d\bar V_b n_b = 0$

if we divide both sides by $dx_a$ and $n_{tot} = (n_a+n_b)$,

$x_a \frac{d\bar V_a}{d x_a} + x_b \frac{d\bar V_b}{ x_a} = 0$

if we note that $x_b = 1- x_a$, we can rearrange and integrate to get

$\bar V_b = \int d\bar V_b = -\int \left ( \frac{x_a}{1-x_a} \right ) \left (\frac{d\bar V_a}{d x_a} \right ) dx_a$

##### Note:

If we know an expression for $\bar V_a$ (or at least how the slope changes with $x_a$), we can evaluate this integral to get $\bar V_b$.

##### Outcome:

Calculate partial molar properties using the Gibbs-Duhem equation

##### Test Yourself:

If the partial molar volume of component "a" in an a-b mixture is known to be given by:

$\bar V_a = 50x_a + 150x_b + 30x_ax_b$

determine $\bar V_b$ using the Gibbs-Duhem equation.