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. `0.763`
C. `0.237`
D. `1`

Explanation

To find the mole fraction of toluene in the vapor phase that is in equilibrium with a solution of benzene and toluene having a mole fraction of toluene of 0.500, we can use Raoult's Law.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Mole fraction of toluene in the solution, \( x_T = 0.500 \)
- Mole fraction of benzene in the solution, \( x_B = 1 - x_T = 1 - 0.500 = 0.500 \)
- Vapor pressure of pure benzene, \( P^0_B = 119 \, \text{torr} \)
- Vapor pressure of pure toluene, \( P^0_T = 37.0 \, \text{torr} \)

2. **Calculate the Partial Pressures:**
- According to Raoult's Law, the partial pressure of each component in the vapor phase is given by:
\[ P_B = x_B \cdot P^0_B \]
\[ P_T = x_T \cdot P^0_T \]
- Calculate \( P_B \):
\[ P_B = 0.500 \cdot 119 = 59.5 \, \text{torr} \]
- Calculate \( P_T \):
\[ P_T = 0.500 \cdot 37.0 = 18.5 \, \text{torr} \]

3. **Calculate Total Pressure:**
- The total pressure \( P_{total} \) is the sum of the partial pressures:
\[ P_{total} = P_B + P_T = 59.5 + 18.5 = 78.0 \, \text{torr} \]

4. **Calculate Mole Fraction of Toluene in Vapor Phase:**
- The mole fraction of toluene in the vapor phase \( y_T \) can be calculated using the formula:
\[ y_T = \frac{P_T}{P_{total}} \]
- Substitute the values:
\[ y_T = \frac{18.5}{78.0} \approx 0.236 \]

### Final Answer:
The mole fraction of toluene in the vapor phase is approximately \( 0.236 \).