# At 20°C the vapor pressure of benzene (C6H6) is 75 torr, and that of toluene (C7H8) is 22 torr. Assume that benzene and toluene form an ideal solution. What is the composition in mole fractions of a solution that has a vapor pressure of 40. torr at 20°C?

Nov 11, 2015

${\chi}_{\text{benzene}} = 0.34$
${\chi}_{\text{toluene}} = 0.66$

#### Explanation:

The idea here is that the vapor pressures of benzene and toluene will contribute to the total vapor pressure of the solution proportionally to their respective mole fraction - this is known as Raoult's Law.

Mathematically, you can express this by the following equation

${P}_{\text{sol" = chi_"benzene" * P_"benzene"^@ + chi_"toluene" * P_"toluene}}^{\circ}$, where

${P}_{\text{sol}}$ - the vapor pressure of the solution
${\chi}_{\text{benzene}}$ - the mole fraction of benzene
${P}_{\text{benzene}}^{\circ}$ - the vapor pressure of pure benzene

Now, mole fraction is defined as the ratio between the number of moles of a component of a solution and the total number of moles present in the solution.

Since you only have two components to this solution, benzene and toluene, you can say that

${\chi}_{\text{benzene" + chi_"toluene}} = 1$

Let's say that the mole fraction of benzene is $x$ and that of toluene is $y$. You can say that

$x = 1 - y$

and

${P}_{\text{total" = x * P_"benzne"^@ + (1-x) * P_"toluene}}^{\circ}$

$40. \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{torr"))) = x * 75color(red)(cancel(color(black)("torr"))) + (1-x) * 22color(red)(cancel(color(black)("torr}}}}$

$40. = 75 x + 22 - 22 x$

$40 - 22 = 53 x \implies x = \frac{18}{53} = 0.3396$

This means that you have

y = 1 - 0.3396 = 0.6604#

The mole fractions of benzene and toluene in the solution will thus be - rounded to two sig figs

${\chi}_{\text{benzene" = color(green)(0.34)" }}$ and $\text{ "chi_"toluene} = \textcolor{g r e e n}{0.66}$