The mole fraction of toluene in vapour phase which is in equlibrium with a solution of benzene and toluene having a mole fraction of toluene `0.500` is (vapour pressure of pure benzene and pure toluene are 119 torr and 37.0 torr respectively at the same temperature).

Options

A. 0.5
B. 0.75
C. 0.625
D. 0.237

Answer

To find the mole fraction of toluene in the vapor phase that is in equilibrium with a solution of benzene and toluene, we can follow these steps:

### Step 1: Identify the given values
- Mole fraction of toluene in the solution, \( X_A = 0.500 \)
- Mole fraction of benzene, \( X_B = 1 - X_A = 1 - 0.500 = 0.500 \)
- Vapor pressure of pure toluene, \( P^0_A = 37.0 \, \text{torr} \)
- Vapor pressure of pure benzene, \( P^0_B = 119.0 \, \text{torr} \)

### Step 2: Calculate the total vapor pressure of the solution
According to Raoult's Law, the total vapor pressure \( P \) of the solution is given by:
\[ P = P^0_A \cdot X_A + P^0_B \cdot X_B \]

Substituting the known values:
\[ P = (37.0 \, \text{torr} \cdot 0.500) + (119.0 \, \text{torr} \cdot 0.500) \]
\[ P = 18.5 \, \text{torr} + 59.5 \, \text{torr} = 78.0 \, \text{torr} \]

### Step 3: Calculate the partial pressures of toluene and benzene
The partial pressure of toluene \( P_A \) and benzene \( P_B \) can be calculated as follows:
\[ P_A = P^0_A \cdot X_A = 37.0 \, \text{torr} \cdot 0.500 = 18.5 \, \text{torr} \]
\[ P_B = P^0_B \cdot X_B = 119.0 \, \text{torr} \cdot 0.500 = 59.5 \, \text{torr} \]

### Step 4: Calculate the mole fraction of toluene in the vapor phase
The mole fraction of toluene in the vapor phase \( Y_A \) can be calculated using the formula:
\[ Y_A = \frac{P_A}{P} \]
Substituting the values:
\[ Y_A = \frac{18.5 \, \text{torr}}{78.0 \, \text{torr}} \approx 0.237 \]

### Final Answer
The mole fraction of toluene in the vapor phase is approximately \( 0.237 \).

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