Among the following 0.10 m aqueous solutions, which one will exhibit the largest freezing point depression?

Options

A. KCl
B. `C_(6)H_(12)O_(6)`
C. `Al_(2)(SO_(4))_(3)`
D. `K_(2)SO_(4)`

Explanation

To determine which of the given 0.10 m aqueous solutions will exhibit the largest freezing point depression, we can use the formula for freezing point depression:

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

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

Since all solutions are 0.10 m, the only variable that will affect the freezing point depression is the van 't Hoff factor (\(i\)).

Let's analyze the provided solutes:

1. **C₆H₁₂O₆ (Glucose)**:
- Glucose does not dissociate in solution, so \(i = 1\).

2. **Al₂(SO₄)₃ (Aluminum sulfate)**:
- Aluminum sulfate dissociates into 2 Al³⁺ ions and 3 SO₄²⁻ ions. Therefore, \(i = 2 + 3 = 5\).

3. **K₂SO₄ (Potassium sulfate)**:
- Potassium sulfate dissociates into 2 K⁺ ions and 1 SO₄²⁻ ion. Therefore, \(i = 2 + 1 = 3\).

4. **NaCl (Sodium chloride)** (assuming this is one of the options):
- Sodium chloride dissociates into 1 Na⁺ ion and 1 Cl⁻ ion. Therefore, \(i = 1 + 1 = 2\).

Now we can summarize the van 't Hoff factors for each solute:
- C₆H₁₂O₆: \(i = 1\)
- Al₂(SO₄)₃: \(i = 5\)
- K₂SO₄: \(i = 3\)
- NaCl: \(i = 2\) (if included)

Now we can calculate the freezing point depression for each solute using the formula:

1. **C₆H₁₂O₆**:
\[ \Delta T_f = 1 \cdot K_f \cdot 0.10 \]

2. **Al₂(SO₄)₃**:
\[ \Delta T_f = 5 \cdot K_f \cdot 0.10 \]

3. **K₂SO₄**:
\[ \Delta T_f = 3 \cdot K_f \cdot 0.10 \]

4. **NaCl**:
\[ \Delta T_f = 2 \cdot K_f \cdot 0.10 \]

From these calculations, it is clear that the solution with the largest van 't Hoff factor will have the largest freezing point depression.

**Conclusion**: The solution with the largest freezing point depression is **Al₂(SO₄)₃** with \(i = 5\).