The liquids X and Y from ideal solution having vapour pressures 200 and 100 mm Hg respectively. Calculate the mole fraction of component X in vapour phase in equilibrium with an equimolar solution of the two .

Explanation

To solve the problem, we need to calculate the mole fraction of component X in the vapor phase that is in equilibrium with an equimolar solution of two liquids, X and Y.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Vapor pressure of pure component X, \( P^0_X = 200 \, \text{mmHg} \)
- Vapor pressure of pure component Y, \( P^0_Y = 100 \, \text{mmHg} \)
- Since it is an equimolar solution, the mole fractions of X and Y in the liquid phase are:
\[ \chi_X = \chi_Y = 0.5 \]

2. **Calculate Partial Pressure of Component X:**
- The partial pressure of component X in the vapor phase can be calculated using Raoult's Law:
\[ P_X = P^0_X \times \chi_X \]
- Substituting the values:
\[ P_X = 200 \, \text{mmHg} \times 0.5 = 100 \, \text{mmHg} \]

3. **Calculate Partial Pressure of Component Y:**
- Similarly, the partial pressure of component Y can be calculated:
\[ P_Y = P^0_Y \times \chi_Y \]
- Substituting the values:
\[ P_Y = 100 \, \text{mmHg} \times 0.5 = 50 \, \text{mmHg} \]

4. **Calculate Total Pressure:**
- The total pressure in the system is the sum of the partial pressures:
\[ P_{Total} = P_X + P_Y \]
- Substituting the values:
\[ P_{Total} = 100 \, \text{mmHg} + 50 \, \text{mmHg} = 150 \, \text{mmHg} \]

5. **Calculate Mole Fraction of Component X in Vapor Phase:**
- The mole fraction of component X in the vapor phase can be calculated using the following formula:
\[ \chi_{X, vapor} = \frac{P_X}{P_{Total}} \]
- Substituting the values:
\[ \chi_{X, vapor} = \frac{100 \, \text{mmHg}}{150 \, \text{mmHg}} = \frac{2}{3} \approx 0.67 \]

### Final Answer:
The mole fraction of component X in the vapor phase is approximately \( 0.67 \).