CfE: Calculating Partial Molar Properties: Graphical Methods

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 $\bar V_2$.

graphical partial molar v
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.