An aqueous solution freezes at 272.4 K, while pure water freezes at 273 K, given Kf = 1.86 K kg `mol^(-1)`, and `K_(b) = 0.512` - k kg `mol^(-1)`, the molality of solution and boiling point of solution respectively will be–

Options

A. 0.322 and 373.16 K
B. 0.222 and 273.15 K
C. 0.413 and 400 K
D. 0.5 and 300.73 K

Explanation

To solve the problem, we need to find the molality of the solution and the boiling point of the solution based on the given freezing point and the constants \( K_f \) and \( K_b \).

### Step-by-Step Solution:

1. **Determine the Depression in Freezing Point (\( \Delta T_f \))**:
\[ \Delta T_f = T_f^{\text{pure}} - T_f^{\text{solution}} = 273 \, \text{K} - 272.4 \, \text{K} = 0.6 \, \text{K} \]

**Hint**: Remember that the depression in freezing point is calculated as the difference between the freezing point of pure solvent and the freezing point of the solution.

2. **Calculate the Molality (\( m \))**:
Using the formula for freezing point depression:
\[ \Delta T_f = K_f \times m \]
Rearranging for molality:
\[ m = \frac{\Delta T_f}{K_f} = \frac{0.6 \, \text{K}}{1.86 \, \text{K kg mol}^{-1}} \approx 0.322 \, \text{mol kg}^{-1} \]

**Hint**: Make sure to use the correct value of \( K_f \) and ensure your units are consistent when calculating molality.

3. **Determine the Elevation in Boiling Point (\( \Delta T_b \))**:
Using the formula for boiling point elevation:
\[ \Delta T_b = K_b \times m \]
Substituting the values:
\[ \Delta T_b = 0.512 \, \text{K kg mol}^{-1} \times 0.322 \, \text{mol kg}^{-1} \approx 0.165 \, \text{K} \]

**Hint**: The boiling point elevation is calculated similarly to the freezing point depression, using the boiling point constant \( K_b \).

4. **Calculate the Boiling Point of the Solution**:
The boiling point of pure water is 373 K. Therefore, the boiling point of the solution is:
\[ T_b^{\text{solution}} = T_b^{\text{pure}} + \Delta T_b = 373 \, \text{K} + 0.165 \, \text{K} \approx 373.165 \, \text{K} \]

**Hint**: Always remember to add the elevation in boiling point to the boiling point of the pure solvent to find the boiling point of the solution.

### Final Answers:
- **Molality of the solution**: \( 0.322 \, \text{mol kg}^{-1} \)
- **Boiling point of the solution**: \( 373.165 \, \text{K} \)