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 (ΔTf), we need to analyze the van 't Hoff factor (i) for each solution. The depression in freezing point is directly proportional to the van 't Hoff factor, meaning that the greater the number of particles in solution, the greater the depression in freezing point.

### Step-by-Step Solution:

1. **Identify the Solutions and Their Concentrations**:
- 0.2 M Urea
- 0.2 M Glucose
- 0.1 M Al2(SO4)3
- 0.1 M Na2SO4
- 0.1 M KNO3
- 0.2 M Ba(NO3)2
- 0.1 M Ca(NO3)2

2. **Calculate the van 't Hoff Factor (i) for Each Solution**:
- **Urea (0.2 M)**: Urea is a non-ionic compound, so it does not dissociate.
\[ i = 1 \]
- **Glucose (0.2 M)**: Glucose is also a non-ionic compound.
\[ i = 1 \]
- **Al2(SO4)3 (0.1 M)**: This dissociates into 2 Al³⁺ and 3 SO₄²⁻ ions.
\[ i = 2 + 3 = 5 \]
- **Na2SO4 (0.1 M)**: This dissociates into 2 Na⁺ and 1 SO₄²⁻ ion.
\[ i = 2 + 1 = 3 \]
- **KNO3 (0.1 M)**: This dissociates into 1 K⁺ and 1 NO₃⁻ ion.
\[ i = 1 + 1 = 2 \]
- **Ba(NO3)2 (0.2 M)**: This dissociates into 1 Ba²⁺ and 2 NO₃⁻ ions.
\[ i = 1 + 2 = 3 \]
- **Ca(NO3)2 (0.1 M)**: This dissociates into 1 Ca²⁺ and 2 NO₃⁻ ions.
\[ i = 1 + 2 = 3 \]

3. **Summarize the van 't Hoff Factors**:
- Urea: \( i = 1 \)
- Glucose: \( i = 1 \)
- Al2(SO4)3: \( i = 5 \)
- Na2SO4: \( i = 3 \)
- KNO3: \( i = 2 \)
- Ba(NO3)2: \( i = 3 \)
- Ca(NO3)2: \( i = 3 \)

4. **Determine the Solution with the Lowest i**:
- The lowest van 't Hoff factor (i) is 1, which corresponds to both Urea and Glucose.

5. **Conclusion**:
- Since the depression in freezing point is lowest for the solution with the lowest i, the solutions of Urea and Glucose both have the lowest depression in freezing point. Therefore, the correct answer is either Urea or Glucose.

### Final Answer:
The solution with the lowest depression in freezing point is **0.2 M Urea** (or **0.2 M Glucose**).