The vapour pressure of two pure isomeric liquids X and Y are 200 torr and 100 torr respectively at a given temperature. Assuming a solution of these components to obey Raoults law, the mole fraction of component X in vapour phase in equilibrium with the solution containing equal amounts of X and Y, at the same temperature is :

Options

A. 0.33
B. 0.5
C. 0.66
D. 0.8

Answer

To solve the problem, we need to find the mole fraction of component X in the vapor phase that is in equilibrium with a solution containing equal amounts of X and Y. We will use Raoult's law to do this.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Vapor pressure of pure liquid X, \( P^0_X = 200 \, \text{torr} \)
- Vapor pressure of pure liquid Y, \( P^0_Y = 100 \, \text{torr} \)
- The solution contains equal amounts of X and Y, which means the mole fraction of X and Y in the liquid phase is \( x_X = x_Y = 0.5 \).

2. **Calculate the Total Vapor Pressure of the Solution:**
According to Raoult's law, the total vapor pressure \( P \) of the solution can be calculated using the formula:
\[ P = P^0_X \cdot x_X + P^0_Y \cdot x_Y \]
Substituting the values:
\[ P = (200 \, \text{torr} \cdot 0.5) + (100 \, \text{torr} \cdot 0.5) \]
\[ P = 100 \, \text{torr} + 50 \, \text{torr} = 150 \, \text{torr} \]

3. **Calculate the Partial Pressure of Component X:**
The partial pressure of component X in the vapor phase can be calculated using:
\[ P_X = P^0_X \cdot x_X \]
Substituting the values:
\[ P_X = 200 \, \text{torr} \cdot 0.5 = 100 \, \text{torr} \]

4. **Calculate the Mole Fraction of Component X in the Vapor Phase:**
The mole fraction of component X in the vapor phase \( y_X \) can be calculated using the formula:
\[ y_X = \frac{P_X}{P} \]
Substituting the values:
\[ y_X = \frac{100 \, \text{torr}}{150 \, \text{torr}} = \frac{100}{150} = \frac{2}{3} \approx 0.6667 \]

5. **Final Result:**
The mole fraction of component X in the vapor phase is approximately \( 0.6667 \) or \( \frac{2}{3} \).

### Conclusion:
The mole fraction of component X in the vapor phase in equilibrium with the solution is \( \frac{2}{3} \) or approximately \( 0.67 \).