For an ideal binary liquid solution with `P_(A)^@gtP_(B)^@" "x_(A)`and `y_(A)` represent the mole fraction of A in liquid phase and vapour phase respectively whereas `x_(B)` and `y_(B)` represent the mole fraction of B in liquid phase and vapour phase respectively, therefore which of the following relation is correct?

Options

A. `x_(A)=y_(A)`
B. `x_(A)gty_(A)`
C. `(x_(A))/(x_(B))lt(y_(A))/y_(B)`
D. `(x_(A))/(x_(B))gt(y_(A))/y_(B)`

Explanation

To solve the problem regarding the relations in an ideal binary liquid solution, we will use Raoult's Law and the definitions of mole fractions in both the liquid and vapor phases.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- Let \( P_A^0 \) and \( P_B^0 \) be the vapor pressures of pure components A and B, respectively.
- Let \( x_A \) and \( x_B \) be the mole fractions of A and B in the liquid phase, respectively. Since it is a binary solution, \( x_A + x_B = 1 \).
- Let \( y_A \) and \( y_B \) be the mole fractions of A and B in the vapor phase, respectively. Similarly, \( y_A + y_B = 1 \).

2. **Applying Raoult's Law**:
- For component A:
\[ P_A = x_A P_A^0 \]
- For component B:
\[ P_B = x_B P_B^0 \]

3. **Total Pressure of the Solution**:
- The total pressure of the solution, \( P_{\text{solution}} \), is the sum of the partial pressures:
\[ P_{\text{solution}} = P_A + P_B = x_A P_A^0 + x_B P_B^0 \]

4. **Using the Definitions of Vapor Phase**:
- The mole fraction of A in the vapor phase can be expressed as:
\[ y_A = \frac{P_A}{P_{\text{solution}}} = \frac{x_A P_A^0}{x_A P_A^0 + x_B P_B^0} \]
- Similarly, for component B:
\[ y_B = \frac{P_B}{P_{\text{solution}}} = \frac{x_B P_B^0}{x_A P_A^0 + x_B P_B^0} \]

5. **Relating the Mole Fractions**:
- From the definitions of \( y_A \) and \( y_B \), we can derive:
\[ \frac{y_A}{y_B} = \frac{x_A P_A^0}{x_B P_B^0} \]
- Rearranging gives us:
\[ \frac{y_A}{y_B} = \frac{x_A}{x_B} \cdot \frac{P_A^0}{P_B^0} \]

6. **Conclusion**:
- Therefore, the correct relation that can be established for an ideal binary liquid solution is:
\[ \frac{y_A}{y_B} = \frac{x_A}{x_B} \cdot \frac{P_A^0}{P_B^0} \]