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 solve the problem of determining which pairs of solutions have the same van't Hoff factor (i), we need to analyze each pair and calculate the number of ions produced upon dissociation.

### Step-by-Step Solution:

1. **Understanding the van't Hoff Factor (i)**:
The van't Hoff factor (i) is defined as the number of particles (ions or molecules) into which a solute dissociates in solution. For complete dissociation, it is given by the total number of ions formed.

2. **Analyzing the First Pair: K₄[Fe(CN)₆] and FeSO₄**:
- **K₄[Fe(CN)₆]** dissociates into:
- 4 K⁺ ions + 1 [Fe(CN)₆]⁴⁻ complex ion
- Total ions = 4 + 1 = 5
- **FeSO₄** dissociates into:
- 1 Fe²⁺ ion + 1 SO₄²⁻ ion
- Total ions = 1 + 1 = 2
- **Conclusion**: i for K₄[Fe(CN)₆] = 5, i for FeSO₄ = 2 (not the same).

3. **Analyzing the Second Pair: K₄[Fe(CN)₆] and (NH₄)₂SO₄·6H₂O**:
- **K₄[Fe(CN)₆]** dissociates as above: i = 5.
- **(NH₄)₂SO₄·6H₂O** dissociates into:
- 2 NH₄⁺ ions + 1 SO₄²⁻ ion
- Total ions = 2 + 1 = 3
- **Conclusion**: i for K₄[Fe(CN)₆] = 5, i for (NH₄)₂SO₄ = 3 (not the same).

4. **Analyzing the Third Pair: NaCl and BaCl₂**:
- **NaCl** dissociates into:
- 1 Na⁺ ion + 1 Cl⁻ ion
- Total ions = 1 + 1 = 2
- **BaCl₂** dissociates into:
- 1 Ba²⁺ ion + 2 Cl⁻ ions
- Total ions = 1 + 2 = 3
- **Conclusion**: i for NaCl = 2, i for BaCl₂ = 3 (not the same).

5. **Analyzing the Fourth Pair: FeSO₄ and KCl·MgCl₂·6H₂O**:
- **FeSO₄** dissociates as above: i = 2.
- **KCl·MgCl₂·6H₂O** dissociates into:
- 1 K⁺ ion + 1 Mg²⁺ ion + 2 Cl⁻ ions from MgCl₂
- Total ions = 1 + 1 + 2 = 4
- **Conclusion**: i for FeSO₄ = 2, i for KCl·MgCl₂ = 4 (not the same).

6. **Final Analysis**:
- The only pair where the van't Hoff factor is the same is:
- **(NH₄)₂SO₄·6H₂O and K₄[Fe(CN)₆]** both yield i = 5.
- Therefore, the correct options are **B** and **D**.

### Final Answer:
The pairs of solutions with the same van't Hoff factor are **B** and **D**.