The vapour pressure of a solvent decreased by `10 mm` of `Hg` when a non-volatile solute was added to the solvent. The mole fraction of solute is `0.2`, what would be the mole fraction of solvent if the decrease in vapour pressure is `20 mm` of `Hg`.

Options

A. 0.8
B. 0.6
C. 0.4
D. 0.2

Explanation

To solve the problem, we will use the concept of relative lowering of vapor pressure, which states that the decrease in vapor pressure of a solvent when a non-volatile solute is added is directly proportional to the mole fraction of the solute in the solution.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We know that the vapor pressure of a solvent decreases by `10 mm Hg` when a non-volatile solute is added, and the mole fraction of the solute (Xb1) at that time is `0.2`.
- We need to find the mole fraction of the solvent (Xa2) when the vapor pressure decreases by `20 mm Hg`.

2. **Using the Formula for Relative Lowering of Vapor Pressure**:
- The relative lowering of vapor pressure can be expressed as:
\[ \frac{\Delta P}{P_0} = X_b \]
- Where:
- \(\Delta P\) is the decrease in vapor pressure,
- \(P_0\) is the original vapor pressure of the pure solvent,
- \(X_b\) is the mole fraction of the solute.

3. **Setting Up the Ratios**:
- For the first scenario (when \(\Delta P = 10 \, \text{mm Hg}\)):
\[ \frac{10}{P_0} = X_{b1} = 0.2 \]
- For the second scenario (when \(\Delta P = 20 \, \text{mm Hg}\)):
\[ \frac{20}{P_0} = X_{b2} \]

4. **Finding the Relationship Between the Two Scenarios**:
- Dividing the two equations gives:
\[ \frac{\Delta P_1}{\Delta P_2} = \frac{X_{b1}}{X_{b2}} \]
- Plugging in the known values:
\[ \frac{10}{20} = \frac{0.2}{X_{b2}} \]
- Simplifying:
\[ \frac{1}{2} = \frac{0.2}{X_{b2}} \]
- Cross-multiplying gives:
\[ X_{b2} = 0.2 \times 2 = 0.4 \]

5. **Finding the Mole Fraction of the Solvent**:
- The mole fraction of the solvent (Xa2) is given by:
\[ X_{a2} = 1 - X_{b2} \]
- Substituting the value of \(X_{b2}\):
\[ X_{a2} = 1 - 0.4 = 0.6 \]

### Final Answer:
The mole fraction of the solvent when the vapor pressure decreases by `20 mm Hg` is **0.6**.

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