Vapour pressure of pure benzene is 119 torr and that of toluene is `37.0` torr at the same temperature. Mole fraction of toluene in vapour phase which is in equilibrium with a solution of benzene and toulene having a mole fraction of toluene `0.50`, will be

Options

A. `0.137`
B. `0.205`
C. `0.237`
D. `0.435`

Answer

To solve the problem of finding 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: Understand the Given Data
- Vapor pressure of pure benzene (P_B^0) = 119 torr
- Vapor pressure of pure toluene (P_A^0) = 37 torr
- Mole fraction of toluene in the liquid phase (X_A) = 0.50

### Step 2: Calculate the Mole Fraction of Benzene in the Liquid Phase
Since the total mole fraction must equal 1, we can find the mole fraction of benzene (X_B):
\[ X_B = 1 - X_A = 1 - 0.50 = 0.50 \]

### Step 3: Use Raoult's Law
According to Raoult's Law, the partial vapor pressure of each component can be calculated as:
- For toluene (A):
\[ P_A = P_A^0 \cdot X_A \]
- For benzene (B):
\[ P_B = P_B^0 \cdot X_B \]

### Step 4: Substitute Values into the Equations
Substituting the values we have:
- For toluene:
\[ P_A = 37 \, \text{torr} \cdot 0.50 = 18.5 \, \text{torr} \]
- For benzene:
\[ P_B = 119 \, \text{torr} \cdot 0.50 = 59.5 \, \text{torr} \]

### Step 5: Calculate Total Vapor Pressure (P_T)
The total vapor pressure (P_T) is the sum of the partial pressures:
\[ P_T = P_A + P_B = 18.5 \, \text{torr} + 59.5 \, \text{torr} = 78.0 \, \text{torr} \]

### Step 6: Calculate the Mole Fraction of Toluene in the Vapor Phase (Y_A)
Using the relation for mole fraction in the vapor phase:
\[ Y_A = \frac{P_A}{P_T} \]
Substituting the values:
\[ Y_A = \frac{18.5 \, \text{torr}}{78.0 \, \text{torr}} \approx 0.237 \]

### Conclusion
The mole fraction of toluene in the vapor phase (Y_A) is approximately **0.237**.

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