Step 1: Understanding the Concept:
Colligative properties are properties of solutions that depend only on the number of solute particles, not their identity.
Elevation in boiling point (\(\Delta T_b\)) and depression in freezing point (\(\Delta T_f\)) are two such properties.
Both are directly proportional to the molality of the solution.
Since urea is a non-electrolyte, it does not dissociate in water, so the van't Hoff factor (\(i\)) is \(1\).
For the same solution, the molality is the same for both properties, allowing us to establish a ratio relationship between boiling point elevation and freezing point depression based on their respective constants.
Step 2: Key Formula or Approach:
1. \(\Delta T_b = K_b \cdot m\).
2. \(\Delta T_f = K_f \cdot m\).
3. Ratio: \(\frac{\Delta T_f}{\Delta T_b} = \frac{K_f}{K_b}\).
Step 3: Detailed Explanation:
First, calculate the elevation in boiling point from the given data:
Pure water boils at \(100^{\circ}\)C.
\[ \Delta T_b = T_{\text{solution}} - T_{\text{pure}} = 100.18 - 100 = 0.18^{\circ}\text{C} \]
Now, use the ratio formula to find \(\Delta T_f\):
\[ \Delta T_f = \Delta T_b \cdot \frac{K_f}{K_b} = 0.18 \cdot \frac{1.86}{0.52} \]
Calculate the numerical ratio: \(1.86 / 0.52 \approx 3.577\).
\[ \Delta T_f = 0.18 \cdot 3.577 \approx 0.6438^{\circ}\text{C} \]
The freezing point of pure water is \(0^{\circ}\)C. Freezing point depression means the solution will freeze at a lower temperature.
\[ T_{\text{freezing}} = 0 - 0.6438 = -0.6438^{\circ}\text{C} \]
Rounding to the given options, we get \(-0.64^{\circ}\)C.
Step 4: Final Answer:
The freezing point of the urea solution is \(-0.64^{\circ}\)C.