For an ideal binary liquid solutions with `P_(A)^(@)gtP_(B)^(@)`, which relation between `chi_(A)` (mole fraction of `A` in liquid phase) and `Y_(A)` (mole fraction of `A` in vapour phase) is correct :

Options

A. `chi_(A)=Y_(A)`
B. `chi_(A)gtY_(A)`
C. `(chi_(A))/(chi_(B))lt(Y_(A))/(Y_(B))`
D. `chi_(A),Y_(A),chi_(B)` and `Y_(B)` cannot be correlated

Explanation

To solve the problem regarding the relationship between the mole fraction of component A in the liquid phase (\(x_A\)) and the mole fraction of component A in the vapor phase (\(y_A\)) for an ideal binary liquid solution where \(P^0_A > P^0_B\), we can follow these steps:

### Step 1: Understand Raoult's Law
According to Raoult's Law, the partial vapor pressure of a component in a solution is given by:
\[ P_A = P^0_A \cdot x_A \]
where \(P_A\) is the partial pressure of component A, \(P^0_A\) is the vapor pressure of pure component A, and \(x_A\) is the mole fraction of component A in the liquid phase.

### Step 2: Express the Total Vapor Pressure
The total vapor pressure of the solution (\(P_{solution}\)) can be expressed as:
\[ P_{solution} = P_A + P_B \]
where \(P_B\) is the partial pressure of component B.

### Step 3: Apply Raoult's Law for Component B
Similarly, for component B, we have:
\[ P_B = P^0_B \cdot x_B \]
where \(x_B\) is the mole fraction of component B in the liquid phase.

### Step 4: Relate Vapor Fractions
The mole fraction of component A in the vapor phase (\(y_A\)) can be expressed as:
\[ y_A = \frac{P_A}{P_{solution}} \]
Substituting \(P_A\) from Raoult's Law:
\[ y_A = \frac{P^0_A \cdot x_A}{P_{solution}} \]

### Step 5: Substitute Total Vapor Pressure
Now, substituting \(P_{solution}\) from the expressions for \(P_A\) and \(P_B\):
\[ P_{solution} = P^0_A \cdot x_A + P^0_B \cdot x_B \]
Thus, we can write:
\[ y_A = \frac{P^0_A \cdot x_A}{P^0_A \cdot x_A + P^0_B \cdot x_B} \]

### Step 6: Establish the Relationship
Now we can analyze the relationship between \(y_A\) and \(x_A\). Since \(P^0_A > P^0_B\), it implies that:
\[ \frac{y_A}{y_B} = \frac{P^0_A}{P^0_B} \cdot \frac{x_A}{x_B} \]
Given that \(P^0_A > P^0_B\), we can conclude that:
\[ y_A > y_B \cdot \frac{x_A}{x_B} \]

### Final Step: Conclusion
This implies that if \(P^0_A > P^0_B\), then:
\[ y_A > x_A \]
Thus, the correct relationship is established.