Equimolal solutions `A` and `B` show depression in freezing point in the ratio `2:1`. A remains in the normal state in solution. `B` will be

Options

A. Normal in solution
B. Dissociated in solution
C. Associated in solution
D. Hydrolysed in solution

Explanation

To solve the problem, we need to analyze the information given about the two equimolal solutions A and B and their freezing point depression.

### Step-by-Step Solution

1. **Understanding Freezing Point Depression**:
The depression in freezing point (\( \Delta T_f \)) is given by the formula:
\[ \Delta T_f = i \cdot K_f \cdot m \]
where:
- \( i \) is the van 't Hoff factor (number of particles the solute breaks into),
- \( K_f \) is the cryoscopic constant of the solvent,
- \( m \) is the molality of the solution.

2. **Given Information**:
- The depression in freezing point for solutions A and B is in the ratio \( 2:1 \):
\[ \frac{\Delta T_{fA}}{\Delta T_{fB}} = \frac{2}{1} \]
- Both solutions are equimolar, meaning their molalities (\( m \)) are the same.

3. **Setting Up the Equations**:
For solution A:
\[ \Delta T_{fA} = i_A \cdot K_f \cdot m \]
For solution B:
\[ \Delta T_{fB} = i_B \cdot K_f \cdot m \]

4. **Substituting into the Ratio**:
Using the ratio of the freezing point depressions:
\[ \frac{i_A \cdot K_f \cdot m}{i_B \cdot K_f \cdot m} = \frac{2}{1} \]
The \( K_f \) and \( m \) cancel out:
\[ \frac{i_A}{i_B} = 2 \]

5. **Analyzing Solution A**:
The problem states that solution A remains in the normal state in solution. This implies that:
\[ i_A = 1 \]
(since it does not dissociate or associate).

6. **Finding \( i_B \)**:
Substituting \( i_A \) into the ratio:
\[ \frac{1}{i_B} = 2 \implies i_B = \frac{1}{2} \]

7. **Interpreting the Van 't Hoff Factor**:
The van 't Hoff factor \( i_B = \frac{1}{2} \) indicates that solute B is associated in solution, meaning it forms pairs or aggregates, resulting in fewer particles than expected.

8. **Conclusion**:
Since \( i_B < 1 \), it indicates that B is associated in solution.

### Final Answer:
B will be **associated in solution** (Option C).