Which of the following salt has the same value of Van’t Hoff factor ‘i’ as that of `K_3 [Fe (CN)_6]` ?

Options

A. `Al_2(SO_4)_3`
B. `NaCl`
C. `Na_2SO_4`
D. `Al(NO_3)_3`

Answer

To determine which salt has the same value of Van’t Hoff factor ‘i’ as that of \( K_3[Fe(CN)_6] \), we need to first calculate the Van’t Hoff factor for \( K_3[Fe(CN)_6] \) and then compare it to the other options provided.

### Step-by-Step Solution:

1. **Understanding Van’t Hoff Factor (i)**:
The Van’t Hoff factor \( i \) is defined as the number of particles into which a solute dissociates in solution. It is calculated using the formula:
\[ i = \frac{\text{Number of particles after dissociation}}{\text{Number of particles before dissociation}} \]

2. **Calculate \( i \) for \( K_3[Fe(CN)_6] \)**:
- When \( K_3[Fe(CN)_6] \) dissolves in water, it dissociates into:
\[ 3K^+ + [Fe(CN)_6]^{3-} \]
- Initially, there is 1 particle (\( K_3[Fe(CN)_6] \)).
- After dissociation, there are 4 particles (3 \( K^+ \) ions and 1 \( [Fe(CN)_6]^{3-} \) ion).
- Therefore, the Van’t Hoff factor \( i \) for \( K_3[Fe(CN)_6] \) is:
\[ i = \frac{4}{1} = 4 \]

3. **Evaluate Other Salts**:
Now, we will check the Van’t Hoff factor \( i \) for the other salts provided.

- **Option A: \( Al_2(SO_4)_3 \)**:
- Dissociates into:
\[ 2Al^{3+} + 3SO_4^{2-} \]
- Total particles = 2 + 3 = 5.
- \( i = \frac{5}{1} = 5 \) (not equal to 4).

- **Option B: \( NaCl \)**:
- Dissociates into:
\[ Na^+ + Cl^- \]
- Total particles = 1 + 1 = 2.
- \( i = \frac{2}{1} = 2 \) (not equal to 4).

- **Option C: \( Na_2SO_4 \)**:
- Dissociates into:
\[ 2Na^+ + SO_4^{2-} \]
- Total particles = 2 + 1 = 3.
- \( i = \frac{3}{1} = 3 \) (not equal to 4).

- **Option D: \( Al(NO_3)_3 \)**:
- Dissociates into:
\[ Al^{3+} + 3NO_3^- \]
- Total particles = 1 + 3 = 4.
- \( i = \frac{4}{1} = 4 \) (equal to 4).

4. **Conclusion**:
The salt that has the same value of Van’t Hoff factor \( i \) as that of \( K_3[Fe(CN)_6] \) is \( Al(NO_3)_3 \).

### Final Answer:
The correct answer is \( Al(NO_3)_3 \).