Question

A solution at $20^{\circ}$C is composed of $1.5\, mol$ of benzene and $3.5\, mol$ of toluene. If the vapour pressure of pure benzene and pure toluene at this temperature are $74.7$ torr and $22.3$ torr, respectively, then the total vapour pressure of the solution and the benzene mole fraction in equilibrium with it will be, respectively :

• A. 35.0 torr and 0.480
• B. 38.0 torr and 0.589
• C. 30.5 torr and 0.389
• D. 35.8 torr and 0.280

Answer 1

The Correct Option is B

To solve this problem, we will apply Raoult's law, which states that the vapor pressure of a solution is dependent on the vapor pressures of each chemical component and their mole fractions in the solution.

The formula for the total vapor pressure of a solution is:

P_{\text{total}} = X_{\text{benzene}} \times P^{\circ}_{\text{benzene}} + X_{\text{toluene}} \times P^{\circ}_{\text{toluene}}

where:

• X_{\text{benzene}} and X_{\text{toluene}} are the mole fractions of benzene and toluene respectively.
• P^{\circ}_{\text{benzene}} and P^{\circ}_{\text{toluene}} are the vapor pressures of pure benzene and pure toluene respectively.

Given:

• P^{\circ}_{\text{benzene}} = 74.7 torr
• P^{\circ}_{\text{toluene}} = 22.3 torr
• Moles of benzene: 1.5 mol
• Moles of toluene: 3.5 mol

Step 1: Calculate the total number of moles in the solution:

\text{Total moles} = 1.5 + 3.5 = 5.0 mol

Step 2: Calculate the mole fraction of benzene (X_{\text{benzene}}) and toluene (X_{\text{toluene}}):

X_{\text{benzene}} = \frac{1.5}{5.0} = 0.3

X_{\text{toluene}} = \frac{3.5}{5.0} = 0.7

Step 3: Calculate the total vapor pressure of the solution:

P_{\text{total}} = (0.3 \times 74.7) + (0.7 \times 22.3)

P_{\text{total}} = 22.41 + 15.61 = 38.02 torr

Step 4: Calculate the mole fraction of benzene in the vapor phase. Using Raoult's law:

The partial pressure of benzene is: P_{\text{benzene}} = X_{\text{benzene}} \times P^{\circ}_{\text{benzene}} = 0.3 \times 74.7 = 22.41 torr

The mole fraction of benzene in the vapor phase (Y_{\text{benzene}}) is:

Y_{\text{benzene}} = \frac{P_{\text{benzene}}}{P_{\text{total}}} = \frac{22.41}{38.02} \approx 0.589

Thus, the total vapor pressure of the solution is 38.0 torr and the mole fraction of benzene in the vapor phase is 0.589. Therefore, the correct answer is:

• 38.0 torr and 0.589