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. 450 mm Hg 0.5 , 0.5
B. 450 mm Hg 0.4 , 0.6
C. 500 mm Hg 0.4 , 0.6
D. 500 mm Hg 0.5 , 0.5

Explanation

To solve the problem, we will use Raoult's law and the concept of mole fractions. Let's go through the solution step by step.

### Step 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, \( X_B = 0.5 \)

### Step 2: Calculate Mole Fraction of Liquid A
Using the relationship for mole fractions in a binary mixture:
\[ X_A + X_B = 1 \]
Substituting the value of \( X_B \):
\[ X_A + 0.5 = 1 \implies X_A = 1 - 0.5 = 0.5 \]

### Step 3: Calculate Total Vapor Pressure Using Raoult's Law
According to Raoult's law, the total vapor pressure \( P_T \) of the solution is given by:
\[ P_T = P^0_A \cdot X_A + P^0_B \cdot X_B \]
Substituting the known values:
\[ P_T = (400 \, \text{mm Hg} \cdot 0.5) + (600 \, \text{mm Hg} \cdot 0.5) \]
Calculating each term:
\[ P_T = 200 \, \text{mm Hg} + 300 \, \text{mm Hg} = 500 \, \text{mm Hg} \]

### Step 4: Calculate Mole Fraction of Components in Vapor Phase
To find the mole fractions of A and B in the vapor phase, we first calculate the partial pressures of A and B.

**Partial Pressure of A:**
\[ P_A = P^0_A \cdot X_A = 400 \, \text{mm Hg} \cdot 0.5 = 200 \, \text{mm Hg} \]

**Partial Pressure of B:**
\[ P_B = P^0_B \cdot X_B = 600 \, \text{mm Hg} \cdot 0.5 = 300 \, \text{mm Hg} \]

### Step 5: Calculate Mole Fraction of A in Vapor Phase (Y_A)
Using the total vapor pressure:
\[ Y_A = \frac{P_A}{P_T} = \frac{200 \, \text{mm Hg}}{500 \, \text{mm Hg}} = 0.4 \]

### Step 6: Calculate Mole Fraction of B in Vapor Phase (Y_B)
Using the relationship for mole fractions in the vapor phase:
\[ Y_A + Y_B = 1 \implies Y_B = 1 - Y_A = 1 - 0.4 = 0.6 \]

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

### Summary
The vapor pressure of the final solution is \( 500 \, \text{mm Hg} \), and the mole fractions of components A and B in the vapor phase are \( 0.4 \) and \( 0.6 \), respectively.

---