In which of the following pairs of solutions will the values of the vant Hoff factor be the same?

Options

A. `0.05M K_(4)[Fe(CN)_(5)]` and `0.10MFeSO_(4)`
B. `0.10MK_(4)[Fe(CN)_(6)]` and `0.05M FeSO_(4)(NH_(4))_(2)SO_(4).6H_(2)O`
C. `0.20M NaCl` and `0.10M BaCl_(2)`
D. `0.05M FESO_(4) (NH_(4))_(2)SO_(4). 6H_(2)O` and `0.02M KCl. MgCl_(2).6H_(2)O`

Answer

To determine in which pairs of solutions the values of the van 't Hoff factor (i) will be the same, we first need to understand what the van 't Hoff factor is. The van 't Hoff factor is defined as the number of particles into which a solute dissociates in solution.

### Step-by-Step Solution:

1. **Identify the Solutions:**
We have the following pairs of solutions to analyze:
- Pair 1: 0.05 M K₄[Fe(CN)₆] and 0.10 M FeSO₄
- Pair 2: 0.10 M K₄[Fe(CN)₆] and 0.05 M FeSO₄
- Pair 3: NaCl and BaCl₂
- Pair 4: K₄[Fe(CN)₆] and another solution.

2. **Calculate the van 't Hoff factor for each solution:**
- **For K₄[Fe(CN)₆]:**
- K₄[Fe(CN)₆] dissociates into 4 K⁺ ions and 1 [Fe(CN)₆]⁴⁻ complex ion.
- Therefore, the van 't Hoff factor (i) = 4 + 1 = 5.

- **For FeSO₄:**
- FeSO₄ dissociates into 1 Fe²⁺ ion and 1 SO₄²⁻ ion.
- Therefore, the van 't Hoff factor (i) = 1 + 1 = 2.

- **For NaCl:**
- NaCl dissociates into 1 Na⁺ ion and 1 Cl⁻ ion.
- Therefore, the van 't Hoff factor (i) = 1 + 1 = 2.

- **For BaCl₂:**
- BaCl₂ dissociates into 1 Ba²⁺ ion and 2 Cl⁻ ions.
- Therefore, the van 't Hoff factor (i) = 1 + 2 = 3.

3. **Compare the van 't Hoff factors:**
- In Pair 1: K₄[Fe(CN)₆] has i = 5 and FeSO₄ has i = 2. (Not the same)
- In Pair 2: K₄[Fe(CN)₆] has i = 5 and FeSO₄ has i = 2. (Not the same)
- In Pair 3: NaCl has i = 2 and BaCl₂ has i = 3. (Not the same)
- In Pair 4: K₄[Fe(CN)₆] has i = 5. (Need another solution to compare)

4. **Conclusion:**
The pairs of solutions that have the same van 't Hoff factor are:
- K₄[Fe(CN)₆] (i = 5) and another solution that also has i = 5.
- NaCl (i = 2) and FeSO₄ (i = 2) do not match with BaCl₂.

### Final Answer:
The pairs of solutions with the same van 't Hoff factor are:
- K₄[Fe(CN)₆] and another solution with i = 5.