The mole fraction of toluene in vapour phase which is in equilibrium 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 110 torr and 37.0 torr respectively at the same temperature).

Options

A. 0.5
B. `0.763`
C. `0.237`
D. 1

Answer

To solve the problem, we will use Raoult's law to find the mole fraction of toluene in the vapor phase that is in equilibrium with a solution of benzene and toluene. Here are the steps to arrive at the solution:

### Step 1: Identify the given values
- Mole fraction of toluene in the liquid phase (XT) = 0.500
- Mole fraction of benzene in the liquid phase (XB) = 1 - XT = 1 - 0.500 = 0.500
- Vapor pressure of pure benzene (P0B) = 110 torr
- Vapor pressure of pure toluene (P0T) = 37.0 torr

### Step 2: Calculate the total vapor pressure using Raoult's law
According to Raoult's law, the total vapor pressure (P_total) of the solution is the sum of the partial pressures of each component:
\[ P_{\text{total}} = P_{0B} \cdot X_B + P_{0T} \cdot X_T \]

Substituting the known values:
\[ P_{\text{total}} = (110 \, \text{torr} \cdot 0.500) + (37.0 \, \text{torr} \cdot 0.500) \]

Calculating each term:
- For benzene: \( 110 \, \text{torr} \cdot 0.500 = 55.0 \, \text{torr} \)
- For toluene: \( 37.0 \, \text{torr} \cdot 0.500 = 18.5 \, \text{torr} \)

Now, adding these values together:
\[ P_{\text{total}} = 55.0 \, \text{torr} + 18.5 \, \text{torr} = 73.5 \, \text{torr} \]

### Step 3: Calculate the mole fraction of toluene in the vapor phase
The mole fraction of toluene in the vapor phase (Y_T) can be calculated using the formula:
\[ Y_T = \frac{P_{T} \cdot X_T}{P_{\text{total}}} \]

Where:
- \( P_T = P_{0T} \cdot X_T \)

Calculating \( P_T \):
\[ P_T = 37.0 \, \text{torr} \cdot 0.500 = 18.5 \, \text{torr} \]

Now substituting into the mole fraction formula:
\[ Y_T = \frac{18.5 \, \text{torr}}{73.5 \, \text{torr}} \]

Calculating \( Y_T \):
\[ Y_T = 0.251 \]

### Step 4: Conclusion
The mole fraction of toluene in the vapor phase is approximately 0.251.

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