Question

$\mathbf{\beta }$-Pinene $\mathbf{\left(}{\mathbf{C}}_{10}{\mathbf{H}}_{16}\mathbf{\right)}$and $\mathbf{\alpha }$-terpineol $\mathbf{\left(}{\mathbf{C}}_{10}{\mathbf{H}}_{18}\mathbf{O}\mathbf{\right)}$are used in cosmetics to provide a “fresh pine” scent. At $\mathbf{367}\mathbf{\text{K}}$, the pure substances have vapor pressures of $\mathbf{100}\mathbf{.3}\mathbf{\text{torr}}$and $\mathbf{\text{9.8torr}}$respectively. What is the composition of the vapor (in terms of mole fractions) above a solution containing equal masses of these compounds at $\mathbf{367}\mathbf{\text{K}}$? (Assume ideal behaviour.)

Short answer

Composition of vapor (in terms of mole fractions) is $0.92$ and $0.077$.

Step-by-step solution

Mole fraction:

Number of moles is defined as the mass of solute per molar mass of solute.

$\mathbf{\text{numberofmoles}}\mathbf{=}\frac{\mathbf{\text{mass\hspace{0.17em}of\hspace{0.17em}solute}}}{\mathbf{\text{molarmass\hspace{0.17em}of\hspace{0.17em}solute}}}$

Moles of fraction is defined as the moles of solute per total moles.

$\mathbf{\text{moles\hspace{0.17em}fraction}}\mathbf{=}\frac{\mathbf{\text{moles\hspace{0.17em}of\hspace{0.17em}solute}}}{\mathbf{\text{total\hspace{0.17em}moles}}}$

^{Consider the equal mass of $\beta$-pinene and $\alpha$-terpineol is $\text{10g}$.}

^{Molecular mass of $\beta$-pinene is $\text{136.23}\raisebox{1ex}{$\text{g}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.$and molecular mass of $\alpha$-terpineol is $\text{154.25}\raisebox{1ex}{$\text{g}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.$.}

Calculate the moles of fraction:

Determine the number of moles of $\beta$-pinene as below.

$\begin{array}{rcl}\text{moles\hspace{0.17em}of\hspace{0.17em}}\beta -pinene& =& \frac{\text{mass\hspace{0.17em}of}\beta -pinene}{\text{molar\hspace{0.17em}mass\hspace{0.17em}of\hspace{0.17em}}\beta -pinene}\\ & =& \frac{\text{10g}}{\text{136.23}{\displaystyle \raisebox{1ex}{$\text{g}$}\!\left/ \!\raisebox{-1ex}{$\text{mol}$}\right.}}\\ & =& 0.073\text{mol}\end{array}$

Define the number of moles of $\alpha$-terpineol as follow.

$\begin{array}{rcl}\text{moles\hspace{0.17em}of\hspace{0.17em}}\alpha -terpineol& =& \frac{\text{mass\hspace{0.17em}of\hspace{0.17em}}\alpha -terpineol}{\text{molar\hspace{0.17em}mass\hspace{0.17em}of\hspace{0.17em}}\alpha -terpineol}\\ & =& \frac{\text{10g}}{\text{154.25g}/\text{mol}}\\ & =& \text{0.064mol}\end{array}$

Calculate the total moles as follow.

$\begin{array}{rcl}\text{Total\hspace{0.17em}moles}& =& 0.73\text{moles}+0.64\text{moles}\\ & =& 1.37\text{moles}\end{array}$

Determine the moles of fraction for $\beta$-pinene as below.

$\begin{array}{rcl}\text{moles\hspace{0.17em}fraction\hspace{0.17em}of\hspace{0.17em}}\beta -pinene& =& \frac{molesof\beta -pinene}{totalmoles}\\ & =& \frac{\text{0.73mol}}{\text{1.37mol}}\\ & =& 0.53\end{array}$

Define the moles of fraction for $\alpha$-terpineol as follow.

$\begin{array}{rcl}\text{moles\hspace{0.17em}fraction\hspace{0.17em}of\hspace{0.17em}}\alpha -terpineol& =& \frac{\text{moles\hspace{0.17em}of\hspace{0.17em}}\alpha -terpineol}{\text{total\hspace{0.17em}moles}}\\ & =& \frac{\text{0.64mol}}{\text{1.37mol}}\\ & =& 0.46\end{array}$

Partial vapor pressure of solution:

Partial vapor pressure of solution containing $\beta$-pinene can be calculated by multiplying mole fraction of $\beta$-pinene and the given partial vapor pressure of pure $\beta$-pinene $\left(100.3\text{torr}\right)$.

$\begin{array}{rcl}{P}_A& =& {\chi }_A\times {P_A}^{\circ }\\ & =& 0.53\times 100.3\text{torr}\\ & =& 53.159\text{torr}\end{array}$

Partial pressure of solution containing $\alpha$-terpineol can be calculated by multiplying the mole fraction of $\alpha$-terpineol and given partial vapor pressure of $\alpha$-terpineol $\left(9.8\text{torr}\right)$.

$\begin{array}{rcl}{P}_B& =& {\chi }_B\times {P_B}^{\circ }\\ & =& 0.46\times 9.8\text{torr}\\ & =& 4.508\text{torr}\end{array}$

Composition of vapor in terms of mole fraction:

Composition of vapor of a solution in terms of mole fraction can be calculated by dividing partial vapor pressure one component by sum of partial vapor pressure of both components.

Sum of partial vapor pressure of both solution

Composition of vapor for -pinene is,

$\begin{array}{rcl}P& =& P_A+P_B\\ & =& 53.569\text{torr}+4.508\text{torr}\\ & =& 58.077\text{torr}\end{array}$

Composition of vapor for $\beta$-pinene is,

role="math" localid="1663336947199" $\begin{array}{rcl}\frac{P_B}{P}& =& \frac{53.569\text{torr}}{58.077\text{torr}}\\ & =& 0.922\end{array}$

Composition of vapor for $\alpha$-terpineol is,

$\begin{array}{rcl}\frac{P_A}{P}& =& \frac{4.508\text{torr}}{58.077\text{torr}}\\ & =& 0.077\end{array}$

Hence, the composition of vapor (in terms of mole fractions) is$0.92$ and $0.077$.