### Calculating Partial Molar Properties: Graphical Methods

Start with the summation equation for partial molar
properties:

$v = \sum_j x_j \bar V_j$

For a binary mixture this gives:

$v = x_1 \bar V_1 + x_2 \bar V_2 = x_1 \bar V_1 +
(1-x_1)\bar V_2$

Taking the derivative with respect to $x_1$ of both sides:

$\frac{dv}{dx_1} = \bar V_1 - \bar V_2$

Multiplying both sides by $x_1$ and then adding and subtracting
$x_2\bar V_2$ gives:

$x_1\frac{dv}{dx_1} = x_1\bar V_1 - x_1\bar V_2 + (x_2\bar
V_2 - x_2 \bar V_2) = (x_1\bar V_1 + x_2\bar V_2) - (x_1\bar
V_2 - x_2 \bar V_2) = v - \bar V_2$

When we rearrange this, we note that $v$ versus $x_1$ gives a
straight line:

$v = \bar V_2 + x_1\frac{dv}{dx_1}$

##### Note:

The slope of this line will be the local slope (often called the
tangent line) of the $v$ curve, and the intercept of the line will
be $\bar V_2$.

##### Procedure:

At the desired concentration ($y_a$ value) go up to the $v$
curve (black). Draw a tangent line. Follow the tangent line to the
y-intercept to get $\bar V_b$. Note that you go to an "a"
concentration to get a "b" partial molar quantity. Also, note that
the $\bar V_a$ and $\bar V_b$ curves are shown for display
purposes only, they are not typically known.

##### Note:

Using this procedure at $y_a = 1$ will yield the "infinite
dilution" value for $\bar V_b$ ($\bar V_b^\infty$, that is the
value of $\bar V_b$ where $y_b$ = 0).

##### Outcome:

Calculate partial molar properties using graphical methods.