Which aqueous solution has minimum freezing point?

Options

A. `0.01 M NaCl`
B. `0.005 M C_(2)H_(5)OH`
C. `0.005 M Mgl_(2)`
D. `0.005 M MgSO_(4)`

Explanation

To determine which aqueous solution has the minimum freezing point, we can use the formula for depression in freezing point, which is given by:

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

Where:
- \(\Delta T_f\) = depression in freezing point
- \(i\) = van 't Hoff factor (number of particles the solute dissociates into)
- \(K_f\) = freezing point depression constant (which is the same for all aqueous solutions)
- \(m\) = molality of the solution

Since \(K_f\) is constant for all solutions, we can focus on maximizing the product \(i \cdot m\) to find the solution with the minimum freezing point.

### Step-by-Step Solution:

1. **Identify the Solutions and Their Molarities:**
- Option A: 0.01 molar NaCl
- Option B: 0.005 molar C2H5OH
- Option C: 0.005 molar MgI2
- Option D: 0.005 molar MgSO4

2. **Calculate the van 't Hoff factor (\(i\)) for each solute:**
- **Option A (NaCl):**
- NaCl dissociates into Na\(^+\) and Cl\(^-\).
- \(i = 2\) (since it produces 2 particles).
- **Option B (C2H5OH):**
- C2H5OH does not dissociate in solution.
- \(i = 1\).
- **Option C (MgI2):**
- MgI2 dissociates into Mg\(^{2+}\) and 2 I\(^-\).
- \(i = 3\) (since it produces 3 particles).
- **Option D (MgSO4):**
- MgSO4 dissociates into Mg\(^{2+}\) and SO4\(^{2-}\).
- \(i = 2\) (since it produces 2 particles).

3. **Calculate \(i \cdot m\) for each solution:**
- **Option A:**
\[ i \cdot m = 2 \cdot 0.01 = 0.02 \]
- **Option B:**
\[ i \cdot m = 1 \cdot 0.005 = 0.005 \]
- **Option C:**
\[ i \cdot m = 3 \cdot 0.005 = 0.015 \]
- **Option D:**
\[ i \cdot m = 2 \cdot 0.005 = 0.010 \]

4. **Compare the values of \(i \cdot m\):**
- Option A: 0.02
- Option B: 0.005
- Option C: 0.015
- Option D: 0.010

5. **Determine the solution with the maximum \(i \cdot m\):**
- The highest value is \(0.02\) from Option A (0.01 molar NaCl).

6. **Conclusion:**
- The aqueous solution with the minimum freezing point is **Option A: 0.01 molar NaCl**.