Question

For a 1.0 molal solution containing the non-volatile solute urea, the elevation in boiling point is 2.0 K while the depression in freezing point in a 3.0 molal solution having the same solvent is 4.0 K. If the ratio \( \frac{K_b}{K_f} = \frac{1}{X} \), what is the value of \(X\)?

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{2}{3} \)
• D. \( \frac{3}{2} \)

Hint:

For colligative properties: \( \Delta T_b = K_b m \) and \( \Delta T_f = K_f m \). Always substitute values carefully before taking ratios.

Answer 1

The Correct Option is C

Step 1: Use elevation in boiling point formula.
\[ \Delta T_b = K_b \cdot m \]
Given:
\[ \Delta T_b = 2.0,\quad m = 1.0 \]
\[ 2.0 = K_b \cdot 1 \Rightarrow K_b = 2. \]

Step 2: Use depression in freezing point formula.
\[ \Delta T_f = K_f \cdot m \]
Given:
\[ \Delta T_f = 4.0,\quad m = 3.0 \]
\[ 4.0 = K_f \cdot 3 \Rightarrow K_f = \frac{4}{3}. \]

Step 3: Find ratio \( \frac{K_b}{K_f} \).
\[ \frac{K_b}{K_f} = \frac{2}{\frac{4}{3}}. \]
\[ = 2 \times \frac{3}{4} = \frac{6}{4} = \frac{3}{2}. \]

Step 4: Compare with given expression.
\[ \frac{K_b}{K_f} = \frac{1}{X}. \]
So,
\[ \frac{1}{X} = \frac{3}{2}. \]

Step 5: Solve for \(X\).
\[ X = \frac{2}{3}. \]

Step 6: Final conclusion.
\[ \boxed{\frac{2}{3}} \]