A solution of urea (mol. Mass `60 g mol^(-1)`) boils of `100.18^(@)C` at one one atmospheric pressure. If `k_(f)` and `K_(b)` for water are `1.86` and `0.512 K kg mol^(-1)` respectively, the above solution will freeze at:

Options

A. `-6.54^@C`
B. `6.54^@C`
C. `0.654^@C`
D. `-0.654^@C`

Answer

To solve the problem, we need to find the freezing point of the urea solution using the given data about its boiling point elevation and the freezing point depression constants for water.

### Step-by-step Solution:

1. **Identify the Given Data:**
- Molecular mass of urea = 60 g/mol
- Boiling point of the solution = 100.18 °C
- Boiling point of pure water = 100 °C
- \( K_f \) (freezing point depression constant for water) = 1.86 °C kg/mol
- \( K_b \) (boiling point elevation constant for water) = 0.512 °C kg/mol

2. **Calculate the Boiling Point Elevation (\( \Delta T_b \)):**
\[ \Delta T_b = \text{Boiling point of solution} - \text{Boiling point of pure solvent} = 100.18 °C - 100 °C = 0.18 °C \]

3. **Use the Relationship Between Freezing Point Depression and Boiling Point Elevation:**
The relationship between the freezing point depression (\( \Delta T_f \)) and boiling point elevation (\( \Delta T_b \)) can be expressed as:
\[ \frac{\Delta T_f}{\Delta T_b} = \frac{K_f}{K_b} \]

4. **Substitute the Known Values:**
\[ \frac{\Delta T_f}{0.18} = \frac{1.86}{0.512} \]

5. **Calculate \( \Delta T_f \):**
\[ \Delta T_f = 0.18 \times \frac{1.86}{0.512} \]
\[ \Delta T_f = 0.18 \times 3.6328125 \approx 0.654 °C \]

6. **Determine the Freezing Point of the Solution:**
The freezing point depression is given by:
\[ \Delta T_f = T_f (\text{pure solvent}) - T_f (\text{solution}) \]
Since the freezing point of pure water is 0 °C, we can set up the equation:
\[ 0.654 °C = 0 °C - T_f (\text{solution}) \]
Rearranging gives:
\[ T_f (\text{solution}) = 0 °C - 0.654 °C = -0.654 °C \]

7. **Final Result:**
The freezing point of the urea solution is approximately:
\[ T_f (\text{solution}) \approx -0.65 °C \]

### Conclusion:
The freezing point of the urea solution is approximately -0.65 °C.