An ideal solution has two components A is more volatile than B, i.e.`P_A^(@)gtP_B^(@)` and also` P_A^(@) gt P_("total")`. If `X_(A) andY_(A)` are mole fraction of components A in liquid and vapour phases, than :

Options

A. (a) `X_(A) = Y_(A)`
B. (b) `X_(A) gt Y_(A)`
C. (c) `X_(A) lt Y_(A)`
D. (d) Data insuffcient

Answer

To solve the problem, we need to establish a relationship between the mole fractions \(X_A\) (of component A in the liquid phase) and \(Y_A\) (of component A in the vapor phase) using Raoult's law and Dalton's law. Let's break it down step by step.

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

### Step 2: Understand Dalton's Law of Partial Pressures
Dalton's law states that the partial pressure of a component in a mixture is also given by:
\[ P_A = Y_A \cdot P_T \]
where:
- \(Y_A\) is the mole fraction of component A in the vapor phase,
- \(P_T\) is the total pressure of the solution.

### Step 3: Equate the Two Expressions for Partial Pressure
From both laws, we can equate the expressions for \(P_A\):
\[ X_A \cdot P^0_A = Y_A \cdot P_T \]

### Step 4: Rearranging the Equation
Rearranging the equation gives us:
\[ \frac{X_A}{Y_A} = \frac{P_T}{P^0_A} \]

### Step 5: Analyze the Given Conditions
We are given that:
- \(P^0_A > P^0_B\) (component A is more volatile than B),
- \(P^0_A > P_T\) (the vapor pressure of A is greater than the total pressure of the solution).

Since \(P^0_A > P_T\), it follows that:
\[ \frac{P_T}{P^0_A} < 1 \]

### Step 6: Substitute the Inequality
From the rearranged equation:
\[ \frac{X_A}{Y_A} = \frac{P_T}{P^0_A} < 1 \]
This implies:
\[ X_A < Y_A \]

### Conclusion
Thus, we can conclude that the mole fraction of component A in the vapor phase \(Y_A\) is greater than the mole fraction of component A in the liquid phase \(X_A\):
\[ Y_A > X_A \]

### Final Answer
The correct relationship is:
\[ Y_A > X_A \]

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