Question

The vapour pressure of pure benzene and methyl benzene at \( 27^\circ \text{C} \) is given as 80 Torr and 24 Torr, respectively. The mole fraction of methyl benzene in vapour phase, in equilibrium with an equimolar mixture of those two liquids (ideal solution) at the same temperature, is \( \dots  \times 10^{-2} \) (nearest integer).

Answer 1

Correct Answer: 23

To find the mole fraction of methyl benzene in the vapor phase, we need to use Raoult’s Law and Dalton’s Law of Partial Pressures, given that it's an ideal solution with pure benzene and methyl benzene. Assume the mole fractions of benzene (\(X_{\text{Benzene}}\)) and methylbenzene (\(X_{\text{Methylbenzene}}\)) in the liquid phase are both 0.5, as the mixture is equimolar.

Step 1: Apply Raoult’s Law.

Raoult’s Law states that the partial pressure of a component in an ideal solution is the mole fraction of the component in the liquid phase multiplied by its vapor pressure as a pure liquid:

\(P_{\text{Benzene}}=X_{\text{Benzene}} \cdot P_{\text{Benzene}}^{\text{pure}}\)

\(P_{\text{Methylbenzene}}=X_{\text{Methylbenzene}} \cdot P_{\text{Methylbenzene}}^{\text{pure}}\)

Substitute the given values:

\(P_{\text{Benzene}}=0.5 \times 80=40 \, \text{Torr}\)

\(P_{\text{Methylbenzene}}=0.5 \times 24=12 \, \text{Torr}\)

Step 2: Apply Dalton’s Law of Partial Pressures to find the total pressure.

The total vapor pressure \(P_{\text{Total}}\) is given by the sum of the partial pressures:

\(P_{\text{Total}}=P_{\text{Benzene}}+P_{\text{Methylbenzene}}=40+12=52 \, \text{Torr}\)

Step 3: Determine the mole fraction in the vapor phase.

The mole fraction of a component in the vapor phase (\(Y\)) is the ratio of its partial pressure to the total pressure:

\(Y_{\text{Methylbenzene}}=\frac{P_{\text{Methylbenzene}}}{P_{\text{Total}}}=\frac{12}{52}\)

Calculating this gives:

\(Y_{\text{Methylbenzene}} \approx 0.2308\)

To express this as \( \times 10^{-2} \), multiply by 100:

\(Y_{\text{Methylbenzene}} \times 100 = 23.08\)

Rounding to the nearest integer, the solution is 23 × 10^-2.

This value is within the specified range (23,23), confirming its correctness.

Answer 2

The mole fraction $x_m$ of methyl benzene in the vapor phase can be calculated using Raoult's Law for ideal solutions: 
\[ x_m = \frac{P_m}{P_1 + P_2} \] 
where:  $P_m$ is the partial vapor pressure of methyl benzene in the vapor phase, given as 24 Torr. 
$P_1$ and $P_2$ are the vapor pressures of pure benzene and methyl benzene, respectively.  
The mole fraction $x_m$ is:
\[ x_m = \frac{24}{80 + 24} = \frac{24}{104} \approx 0.2308 \]
Thus, the mole fraction is $23 \times 10^{-2}$. The correct answer is (23).