For a ideal liquid solution with `P_A^(@)gtP_B^(@)`, which relation between` X_(A)` ((mole fraction of A in liquid phase) and `Y_(A)`(mole fraction of A in vapour phase) is correct ?

Options

A. `Y_(a) lt Y_(b)`
B. `X_(A) gtX_(B)`
C. `(Y_(A))/(Y_(B)) gt( X_(A))/(X_(B))`
D. `(Y_(A))/(Y_(B)) lt( X_(A))/(X_(B))`

Answer

To solve the problem, we need to establish the relationship between the mole fractions of component A in the liquid phase (X_A) and in the vapor phase (Y_A) for an ideal liquid solution where the vapor pressure of pure A (P_A^(@)) is greater than the vapor pressure of pure B (P_B^(@)).

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- Let \( X_A \) be the mole fraction of component A in the liquid phase.
- Let \( Y_A \) be the mole fraction of component A in the vapor phase.
- Given that \( P_A^(@) > P_B^(@) \).

2. **Using Raoult's Law**:
- According to Raoult's Law, the partial pressure of component A in the vapor phase (P_A) can be expressed as:
\[ P_A = X_A \cdot P_A^(@) \]
- Similarly, the partial pressure of component B can be expressed as:
\[ P_B = X_B \cdot P_B^(@) \]
- Where \( X_B \) is the mole fraction of component B in the liquid phase, and \( X_B = 1 - X_A \).

3. **Total Pressure**:
- The total pressure (P_total) of the solution is given by:
\[ P_{total} = P_A + P_B = X_A \cdot P_A^(@) + X_B \cdot P_B^(@) \]

4. **Mole Fraction in Vapor Phase**:
- The mole fraction of A in the vapor phase (Y_A) is given by:
\[ Y_A = \frac{P_A}{P_{total}} = \frac{X_A \cdot P_A^(@)}{X_A \cdot P_A^(@) + (1 - X_A) \cdot P_B^(@)} \]

5. **Simplifying the Expression**:
- Substitute \( P_B^(@) \) and simplify:
\[ Y_A = \frac{X_A \cdot P_A^(@)}{X_A \cdot P_A^(@) + (1 - X_A) \cdot P_B^(@)} \]

6. **Analyzing the Relationship**:
- Since \( P_A^(@) > P_B^(@) \), it implies that the contribution of \( P_A^(@) \) in the numerator is greater than that of \( P_B^(@) \) in the denominator when \( X_A \) is significant.
- This leads to the conclusion that \( Y_A > X_A \).

7. **Final Relation**:
- Therefore, we conclude that:
\[ Y_A > X_A \]
- This indicates that the mole fraction of A in the vapor phase is greater than that in the liquid phase.

### Conclusion:
The correct relationship between the mole fractions is:
\[ Y_A > X_A \]