For an binary mixture of A and B with `p_(A)^(0)ltp_(B)^(0)`:
`x_(A)=` mole fraction of A in solution
`x_(B)=` mole fraction of B in solution
`y_(A)=` mole fraction of A in vapour phase
`y_(B)=` mole fraction of B in vapour phase solution

Options

A. `x_(A)gty_(A)`
B. `y_(B)gty_(A)`
C. `y_(B)gtx_(B)`
D. `y_(A)gty_(B)`

Explanation

To solve the problem regarding the binary mixture of components A and B, we need to analyze the relationships between the mole fractions in the liquid phase and the vapor phase. Here’s a step-by-step solution:

### Step 1: Define the Given Information
We have:
- \( P^0_A < P^0_B \) (the vapor pressure of A is less than that of B)
- \( x_A \) = mole fraction of A in the liquid solution
- \( x_B \) = mole fraction of B in the liquid solution
- \( y_A \) = mole fraction of A in the vapor phase
- \( y_B \) = mole fraction of B in the vapor phase

### Step 2: Write the Equations for Partial Pressures
The partial pressure of component A in the vapor phase can be expressed as:
\[ P'_A = P^0_A \cdot x_A \]
Similarly, for component B:
\[ P'_B = P^0_B \cdot x_B \]

### Step 3: Total Pressure in the System
The total pressure \( P_n \) in the system is the sum of the partial pressures:
\[ P_n = P'_A + P'_B = P^0_A \cdot x_A + P^0_B \cdot x_B \]

### Step 4: Expressing Mole Fractions in the Vapor Phase
Using the definitions of mole fractions in the vapor phase:
\[ y_A = \frac{P'_A}{P_n} = \frac{P^0_A \cdot x_A}{P^0_A \cdot x_A + P^0_B \cdot x_B} \]
\[ y_B = \frac{P'_B}{P_n} = \frac{P^0_B \cdot x_B}{P^0_A \cdot x_A + P^0_B \cdot x_B} \]

### Step 5: Setting Up the Ratio
Now, we can set up the ratio of the mole fractions:
\[ \frac{y_A}{y_B} = \frac{P^0_A \cdot x_A}{P^0_B \cdot x_B} \]

### Step 6: Analyzing the Ratios
Since \( P^0_A < P^0_B \), we can conclude that:
\[ \frac{P^0_A}{P^0_B} < 1 \]
Thus, we have:
\[ \frac{y_A}{y_B} < \frac{x_A}{x_B} \]

### Step 7: Conclusion
From the above relationship, we can derive that:
\[ x_A > y_A \quad \text{and} \quad x_B < y_B \]
This indicates that the mole fraction of A in the liquid phase is greater than that in the vapor phase, while the mole fraction of B in the liquid phase is less than that in the vapor phase.

### Final Answer
The correct conclusion is that:
- \( x_A > y_A \)
- \( x_B < y_B \)