Question

Calculate the vapor pressure of a solution of 0.39 mol of cholesterol in 5.4 mol of toluene at ${\mathbf{32}}^{\mathbf{°}}\mathit{C}$. Pure toluene has a vapor pressure of 41 torr at${\mathbf{32}}^{\mathbf{°}}\mathit{C}$. (Assume ideal behavior.)

Short answer

The vapor pressure is 38 torr.

Step-by-step solution

Concept

To calculate the ${\mathbf{P}}_{\mathbf{solution}}\mathbf{,}$use the equation${\mathit{P}}_{\mathbf{s}\mathbf{o}\mathbf{l}\mathbf{u}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}}\mathbf{=}\mathbf{}{\mathit{X}}_{\mathbf{s}\mathbf{o}\mathbf{l}\mathbf{v}\mathbf{e}\mathbf{n}\mathbf{t}}\mathbf{}\mathbf{\times }\mathbf{}{\mathit{P}}_{\mathbf{s}\mathbf{o}\mathbf{l}\mathbf{v}\mathbf{e}\mathbf{n}\mathbf{t}}^{\mathbf{\circ }}$

This is known as the Raoult’s law.

Calculate mole fraction of toluene (Xtoluene)

$$\begin{gathered} \mathbf{Molesoftoluene}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{4}\mathbf{mol} \\ \mathbf{Molesofcholesterol}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{39}\mathbf{mol} \end{gathered}$$

$$\begin{gathered} {\mathbf{X}}_{\mathbf{toluene}}\mathbf{=}\frac{\mathbf{molesoftoluene}}{\mathbf{molesoftoluene}\mathbf{+}\mathbf{molesofcholesterol}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{5}\mathbf{.}\mathbf{4}}{\mathbf{5}\mathbf{.}\mathbf{4}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{39}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{932}\mathbf{.} \end{gathered}$$

Vapor pressure of solution

Using $P_{solution}=X_{solvent}\times P_{solvent}^{\circ }$ calculate the vapor pressure as shown below.

${\mathbf{X}}_{\mathbf{solv}e\mathbf{nt}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{932}$

role="math" localid="1663335854755" $$\begin{gathered} {\mathit{P}}_{\mathbf{s}\mathbf{o}\mathbf{l}\mathbf{v}\mathbf{e}\mathbf{n}\mathbf{t}}^{\mathbf{\circ }}\mathbf{=}\mathbf{}\mathbf{41}\mathbf{}\mathit{t}\mathit{o}\mathit{r}\mathit{r} \\ {\mathbf{P}}_{\mathbf{solution}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{932}\times \mathbf{41}\mathbf{torr} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{38}\mathbf{.}\mathbf{212}\mathbf{torr}\mathbf{.} \end{gathered}$$

Therefore, the vapor pressure of the solution is 38 torr.