If two substances A and B have `p_(A)^(@): p_(B)^(@) = 1:2` and have mole fraction in solution as 1:2 then mole fraction of A in vapour phase is

Options

A. 0.33
B. 0.25
C. 0.52
D. 0.2

Answer

To find the mole fraction of substance A in the vapor phase, we can follow these steps:

### Step 1: Understand the Given Ratios
We are given that the partial pressures of A and B in their pure states are in the ratio:
\[ p_A^0 : p_B^0 = 1 : 2 \]
This means we can express:
\[ p_A^0 = 1k \quad \text{and} \quad p_B^0 = 2k \]
for some constant \( k \).

We are also given the mole fractions in the solution:
\[ X_A : X_B = 1 : 2 \]
This means:
\[ X_A = \frac{1}{3} \quad \text{and} \quad X_B = \frac{2}{3} \]

### Step 2: Apply Raoult's Law
According to Raoult's Law, the partial pressure of each component in the solution can be expressed as:
\[ P_A = X_A \cdot p_A^0 \]
\[ P_B = X_B \cdot p_B^0 \]

Substituting the values we have:
\[ P_A = \left(\frac{1}{3}\right) \cdot (1k) = \frac{k}{3} \]
\[ P_B = \left(\frac{2}{3}\right) \cdot (2k) = \frac{4k}{3} \]

### Step 3: Calculate Total Pressure
The total pressure \( P \) in the solution is the sum of the partial pressures:
\[ P = P_A + P_B = \frac{k}{3} + \frac{4k}{3} = \frac{5k}{3} \]

### Step 4: Use Dalton's Law
According to Dalton's Law, the mole fraction of A in the vapor phase \( Y_A \) can be expressed as:
\[ Y_A = \frac{P_A}{P} \]

Substituting the values we calculated:
\[ Y_A = \frac{\frac{k}{3}}{\frac{5k}{3}} = \frac{1}{5} \]

### Step 5: Conclusion
Thus, the mole fraction of A in the vapor phase is:
\[ Y_A = 0.2 \]

### Final Answer
The mole fraction of A in the vapor phase is \( 0.2 \).

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