In which of the following solution the depression in freezing point is lowest ?

Options

A. 0.2 M urea and 0.2 M glucose
B. `0.1M Al_2(SO_4)_3 and 0.1M Na_2SO_4`
C. `0.1MKNO_3 and 0.2 M Ba (NO_3)_2`
D. `0.1 M Ca(NO_3)_2 and 0.1M Ba (NO_3)_2`

Answer

To determine which solution has the lowest depression in freezing point, we can follow these steps:

### Step 1: Understand the Concept of Depression in Freezing Point
The depression in freezing point (\( \Delta T_f \)) is a colligative property that depends on the number of solute particles in a solution. It can be calculated using the formula:
\[ \Delta T_f = i \cdot K_f \cdot m \]
where:
- \( i \) = van 't Hoff factor (number of particles the solute dissociates into)
- \( K_f \) = freezing point depression constant (specific to the solvent)
- \( m \) = molality of the solution

### Step 2: Identify the van 't Hoff Factor for Each Solute
1. **Urea (NH₂CO NH₂)**: Urea is a non-electrolyte and does not dissociate in solution, so \( i = 1 \).
2. **Glucose (C₆H₁₂O₆)**: Glucose is also a non-electrolyte and does not dissociate, so \( i = 1 \).
3. **Aluminum Sulfate (Al₂(SO₄)₃)**: This dissociates into 2 Al³⁺ and 3 SO₄²⁻ ions, so \( i = 2 + 3 = 5 \).
4. **Sodium Sulfate (Na₂SO₄)**: This dissociates into 2 Na⁺ and 1 SO₄²⁻ ion, so \( i = 2 + 1 = 3 \).
5. **Potassium Nitrate (KNO₃)**: This dissociates into K⁺ and NO₃⁻ ions, so \( i = 1 + 1 = 2 \).
6. **Barium Nitrate (Ba(NO₃)₂)**: This dissociates into Ba²⁺ and 2 NO₃⁻ ions, so \( i = 1 + 2 = 3 \).
7. **Calcium Nitrate (Ca(NO₃)₂)**: This dissociates into Ca²⁺ and 2 NO₃⁻ ions, so \( i = 1 + 2 = 3 \).

### Step 3: Compare the van 't Hoff Factors
- Urea: \( i = 1 \)
- Glucose: \( i = 1 \)
- Aluminum Sulfate: \( i = 5 \)
- Sodium Sulfate: \( i = 3 \)
- Potassium Nitrate: \( i = 2 \)
- Barium Nitrate: \( i = 3 \)
- Calcium Nitrate: \( i = 3 \)

### Step 4: Determine the Solution with the Lowest Depression in Freezing Point
The depression in freezing point is lowest when \( i \) is the smallest. Here, both urea and glucose have the smallest \( i \) value of 1. Therefore, they will have the lowest depression in freezing point.

### Conclusion
The solutions with the lowest depression in freezing point are urea and glucose.