Which of the following 0.2m aqueous solutions will show minimum freezing point :-

Options

A. `CaCl_(2)`
B. `K_(2)SO_(4)`
C. `Al(NO_(3))_(3)`
D. KBr

Answer

To determine which of the following 0.2m aqueous solutions will show the minimum freezing point, we need to consider the concept of freezing point depression. The freezing point depression (\( \Delta T_f \)) can be calculated using the formula:

\[ \Delta T_f = i \cdot K_f \cdot m \]

Where:
- \( i \) = van 't Hoff factor (the number of particles the solute dissociates into)
- \( K_f \) = freezing point depression constant (a property of the solvent, which is water in this case)
- \( m \) = molality of the solution

Since the molality (\( m \)) is the same for all solutions (0.2m), and \( K_f \) is constant for water, the freezing point depression will depend solely on the van 't Hoff factor (\( i \)). Thus, to find the solution with the minimum freezing point, we need to maximize \( i \).

### Step-by-Step Solution:

1. **Identify the van 't Hoff factor (\( i \)) for each solute:**
- For \( \text{CaCl}_2 \):
- Dissociates into \( \text{Ca}^{2+} + 2\text{Cl}^- \)
- \( i = 1 + 2 = 3 \)

- For \( \text{K}_2\text{SO}_4 \):
- Dissociates into \( 2\text{K}^+ + \text{SO}_4^{2-} \)
- \( i = 2 + 1 = 3 \)

- For \( \text{Al(NO}_3\text{)}_3 \):
- Dissociates into \( \text{Al}^{3+} + 3\text{NO}_3^- \)
- \( i = 1 + 3 = 4 \)

- For \( \text{KBr} \):
- Dissociates into \( \text{K}^+ + \text{Br}^- \)
- \( i = 1 + 1 = 2 \)

2. **Compare the \( i \) values:**
- \( \text{CaCl}_2 \): \( i = 3 \)
- \( \text{K}_2\text{SO}_4 \): \( i = 3 \)
- \( \text{Al(NO}_3\text{)}_3 \): \( i = 4 \)
- \( \text{KBr} \): \( i = 2 \)

3. **Determine the maximum \( i \):**
- The highest \( i \) value is \( 4 \) for \( \text{Al(NO}_3\text{)}_3 \).

4. **Conclusion:**
- Since \( \Delta T_f \) is directly proportional to \( i \), the solution with the highest \( i \) will have the lowest freezing point. Therefore, the 0.2m aqueous solution of \( \text{Al(NO}_3\text{)}_3 \) will show the minimum freezing point.

### Final Answer:
The 0.2m aqueous solution that will show the minimum freezing point is **\( \text{Al(NO}_3\text{)}_3 \)**.

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