The vapour pressures of pure liquids A and B are 400 and 600 mm Hg respectively at 298 K. On mixing the two liquids, the sum of their initial volumes is equal to the volume of the final mixture. The mole fraction of liquid B is 0.5 in the mixture. The vapour pressure of the final solution, the mole fractions of components A and B in vapour phase, respectively are :

Options

A. 500 mmHg, 0.5, 0.5
B. 450 mmHg, 0.4, 0.6
C. 450 mmHg, 0.5, 0.5
D. 500 mmHg, 0.4, 0.6

Answer

To solve the problem, we need to find the vapor pressure of the final solution and the mole fractions of components A and B in the vapor phase.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Vapor pressure of pure liquid A, \( P^0_A = 400 \, \text{mm Hg} \)
- Vapor pressure of pure liquid B, \( P^0_B = 600 \, \text{mm Hg} \)
- Mole fraction of liquid B in the mixture, \( X_B = 0.5 \)

2. **Calculate Mole Fraction of Liquid A:**
Since the sum of the mole fractions must equal 1:
\[ X_A = 1 - X_B = 1 - 0.5 = 0.5 \]

3. **Calculate Total Vapor Pressure of the Mixture:**
The total vapor pressure \( P_{\text{total}} \) of the solution can be calculated using Raoult's Law:
\[ P_{\text{total}} = X_A \cdot P^0_A + X_B \cdot P^0_B \]
Substituting the values:
\[ P_{\text{total}} = (0.5 \cdot 400) + (0.5 \cdot 600) \]
\[ P_{\text{total}} = 200 + 300 = 500 \, \text{mm Hg} \]

4. **Calculate the Mole Fraction of A in the Vapor Phase:**
The mole fraction of A in the vapor phase, \( Y_A \), can be calculated using the formula:
\[ Y_A = \frac{P_A}{P_{\text{total}}} \]
Where \( P_A \) is the partial vapor pressure of A:
\[ P_A = X_A \cdot P^0_A = 0.5 \cdot 400 = 200 \, \text{mm Hg} \]
Now substituting into the equation:
\[ Y_A = \frac{200}{500} = 0.4 \]

5. **Calculate the Mole Fraction of B in the Vapor Phase:**
The mole fraction of B in the vapor phase, \( Y_B \), can be found using:
\[ Y_B = 1 - Y_A = 1 - 0.4 = 0.6 \]

### Final Results:
- Total vapor pressure of the final solution: \( P_{\text{total}} = 500 \, \text{mm Hg} \)
- Mole fraction of A in vapor phase: \( Y_A = 0.4 \)
- Mole fraction of B in vapor phase: \( Y_B = 0.6 \)

### Summary of Results:
- **Total Pressure:** 500 mm Hg
- **Mole Fraction of A in Vapor Phase:** 0.4
- **Mole Fraction of B in Vapor Phase:** 0.6