Equimolal solutions `KCl` and compound `X` in water show depression in freezing point in the ratio of `4:1`, Assuming `KCl` to be completely ionized, the compound `X` in solution must

Options

A. Dissociate to the extent of `50%`
B. Hydrolyze to the extent of `80%`
C. Dimerize to the extent of `50%`
D. Trimerize to the extent of `75%`

Explanation

To solve the problem, we need to determine the degree of association or dissociation of compound X based on the depression in freezing point of equimolar solutions of KCl and compound X.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have equimolar solutions of KCl and compound X.
- The depression in freezing point (ΔTf) for KCl and compound X is in the ratio of 4:1.
- KCl is assumed to completely ionize in solution.

2. **Determine the Van 't Hoff Factor (i) for KCl**:
- KCl dissociates into K⁺ and Cl⁻ ions.
- Therefore, for 1 mole of KCl, it produces 2 moles of ions.
- The van 't Hoff factor (i) for KCl is:
\[ i_{KCl} = 2 \]

3. **Applying the Freezing Point Depression Formula**:
- The depression in freezing point is given by:
\[ \Delta Tf = i \cdot K_f \cdot m \]
- For KCl:
\[ \Delta Tf_{KCl} = i_{KCl} \cdot K_f \cdot m = 2 \cdot K_f \cdot m \]
- For compound X:
\[ \Delta Tf_{X} = i_{X} \cdot K_f \cdot m \]

4. **Setting Up the Ratio**:
- Given that:
\[ \frac{\Delta Tf_{KCl}}{\Delta Tf_{X}} = \frac{4}{1} \]
- Substituting the expressions for ΔTf:
\[ \frac{2 \cdot K_f \cdot m}{i_{X} \cdot K_f \cdot m} = 4 \]
- The \( K_f \) and \( m \) cancel out:
\[ \frac{2}{i_{X}} = 4 \]

5. **Solving for i_X**:
- Rearranging gives:
\[ i_{X} = \frac{2}{4} = \frac{1}{2} \]

6. **Interpreting the Result**:
- A van 't Hoff factor (i) of 1/2 indicates that the compound X is associating in solution.
- This means that for every 1 mole of compound X, it behaves as if it produces only 0.5 moles of particles.

7. **Conclusion**:
- Since the van 't Hoff factor (i) is 1/2, this suggests that compound X dimerizes to the extent of 50%.
- Therefore, the correct answer is **option C: dimerize to the extent of 50%**.