### 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:

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.