For an ideal binary liquid solution with `p_(A)^(@) gt p_(B)^(@),` which is a relation between `X_(A)` (mole fraction of A in liquid phase) and `Y_(A)` (mole fraction of A in vapour phase) is correct, `X_(B) and Y_(B)` are mole fractions of B in liquid and vapour phase respectively?

Options

A. `X_A = Y_A`
B. `X_A gt Y_A`
C. `(X_A)/(X_B) lt (Y_A)/(Y_B)`
D. `X_A, Y_A, X_B and Y_B` cannot be correlated

Answer

To solve the problem, we need to establish the relationship between the mole fractions of components A and B in both the liquid and vapor phases of an ideal binary liquid solution, given that the vapor pressure of A (P₀A) is greater than that of B (P₀B).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have a binary liquid solution with components A and B.
- The vapor pressure of A (P₀A) is greater than the vapor pressure of B (P₀B), i.e., P₀A > P₀B.

2. **Applying Raoult's Law**:
- According to Raoult's Law, the partial pressure of each component in the vapor phase can be expressed as:
- \( P_A = P_0A \cdot X_A \) (for component A)
- \( P_B = P_0B \cdot X_B \) (for component B)
- Where \( X_A \) and \( X_B \) are the mole fractions of A and B in the liquid phase, respectively.

3. **Total Pressure in the System**:
- The total pressure (P_T) of the system is the sum of the partial pressures:
\[ P_T = P_A + P_B = P_0A \cdot X_A + P_0B \cdot X_B \]

4. **Relating Mole Fractions in Vapor Phase**:
- The mole fractions of A and B in the vapor phase can be expressed as:
- \( Y_A = \frac{P_A}{P_T} \)
- \( Y_B = \frac{P_B}{P_T} \)

5. **Substituting for Partial Pressures**:
- Substituting the expressions for \( P_A \) and \( P_B \) into the equations for \( Y_A \) and \( Y_B \):
\[ Y_A = \frac{P_0A \cdot X_A}{P_0A \cdot X_A + P_0B \cdot X_B} \]
\[ Y_B = \frac{P_0B \cdot X_B}{P_0A \cdot X_A + P_0B \cdot X_B} \]

6. **Setting Up the Ratio**:
- We can set up the ratio of the mole fractions:
\[ \frac{Y_A}{Y_B} = \frac{P_0A \cdot X_A}{P_0B \cdot X_B} \]
- Given that \( P_0A > P_0B \), this implies that:
\[ \frac{Y_A}{Y_B} > \frac{X_A}{X_B} \]

7. **Conclusion**:
- Rearranging gives us the relationship:
\[ \frac{X_A}{X_B} < \frac{Y_A}{Y_B} \]
- This indicates that the mole fraction of A in the liquid phase is less than the mole fraction of A in the vapor phase, which is consistent with the fact that A has a higher vapor pressure than B.

### Final Answer:
The correct relationship is:
\[ \frac{X_A}{X_B} < \frac{Y_A}{Y_B} \]