Which of the following aqueous solution has minium freezing point?

Options

A. `0.005 m MgSO_(4)`
B. `0.005 m C_H_(5)OH`
C. `0.00 m MgI_(2)`
D. `0.01 m NaCl`

Explanation

To determine which aqueous solution has the minimum freezing point, we need to analyze the given solutions based on their molality and van't Hoff factor (i). The freezing point depression formula is given by:

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

Where:
- \(\Delta T_f\) = change in freezing point
- \(K_f\) = freezing point depression constant (depends on the solvent)
- \(m\) = molality of the solution
- \(i\) = van't Hoff factor (number of particles the solute dissociates into)

The freezing point of the solution can be calculated as:

\[ T_f = T_{f0} - \Delta T_f \]

Where \(T_{f0}\) is the freezing point of the pure solvent. A higher value of \(\Delta T_f\) indicates a lower freezing point for the solution.

### Step-by-Step Solution:

1. **Identify the Solutions**:
- \(0.005 \, \text{M} \, \text{MgSO}_4\)
- \(0.005 \, \text{M} \, \text{C}_2\text{H}_5\text{OH}\)
- \(0.005 \, \text{M} \, \text{MgI}_2\)
- \(0.005 \, \text{M} \, \text{NaCl}\)

2. **Calculate the van't Hoff factor (i)**:
- For **MgSO₄**: It dissociates into \( \text{Mg}^{2+} + \text{SO}_4^{2-} \) → \( i = 2 \)
- For **C₂H₅OH**: It does not dissociate → \( i = 1 \)
- For **MgI₂**: It dissociates into \( \text{Mg}^{2+} + 2\text{I}^{-} \) → \( i = 3 \)
- For **NaCl**: It dissociates into \( \text{Na}^{+} + \text{Cl}^{-} \) → \( i = 2 \)

3. **Calculate the effective molality (m × i)**:
- For **MgSO₄**:
\[ m \cdot i = 0.005 \cdot 2 = 0.01 \]
- For **C₂H₅OH**:
\[ m \cdot i = 0.005 \cdot 1 = 0.005 \]
- For **MgI₂**:
\[ m \cdot i = 0.005 \cdot 3 = 0.015 \]
- For **NaCl**:
\[ m \cdot i = 0.005 \cdot 2 = 0.01 \]

4. **Determine the solution with the highest effective molality**:
- **MgSO₄**: 0.01
- **C₂H₅OH**: 0.005
- **MgI₂**: 0.015 (highest)
- **NaCl**: 0.01

5. **Conclusion**:
The solution with the highest effective molality (and thus the lowest freezing point) is **MgI₂** with an effective molality of 0.015. Therefore, the aqueous solution with the minimum freezing point is:

\[ \text{MgI}_2 \]