Question

In the vapor over a pentane–hexane solution at ${\mathbf{25}}^o\mathbf{C}$, the mole fraction of pentane is equal to 0.15. What is the mole fraction of pentane in the solution?

Short answer

The mole fraction of pentane in the solution is ${\chi }_P=0.050$.

Step-by-step solution

Step 1:

Equalize the equation

$\begin{array}{l}\frac{{\chi }_P{P^{\circ }}_p}{{\chi }_P}=\frac{{\chi }_H{P^{\circ }}_H}{{\chi }_H}\\ \frac{{\chi }_P{P^{\circ }}_p}{{\chi }_P}=\frac{(1-{\chi }_P){P^{\circ }}_H}{{\chi }_H}\end{array}$

Step 2:

Multiply the whole equation by 0.15 and 0.85.

$\begin{array}{c}\frac{{\chi }_P\times 511torr}{0.15}=\frac{(1-{\chi }_P)\times 150torr}{0.85}\\ 0.85\times 511torr\times {\chi }_P=0.15\times 150torr\times (1-{\chi }_P)\\ 434.35torr\times {\chi }_P=22.5torr-22.5torr\times {\chi }_P\\ {\chi }_P=\frac{22.5torr}{456.85}\\ {\chi }_P=0.050\end{array}$