Which of the following salts will have the same value of van't Hoff factor (as that of `K_(4) [Fe(CN_(6))]`

Options

A. `Al_(2)(SO_(4))_(3)`
B. `NaCl`
C. `Al(NO_(3))_(3)`
D. `Na_(2)SO_(4)`

Explanation

To determine which of the given salts has the same van't Hoff factor (i) as that of \( K_4[Fe(CN)_6] \), we first need to calculate the van't Hoff factor for \( K_4[Fe(CN)_6] \).

### Step 1: Calculate the van't Hoff factor for \( K_4[Fe(CN)_6] \)

1. **Dissociation of \( K_4[Fe(CN)_6] \)**:
- When \( K_4[Fe(CN)_6] \) dissociates, it breaks down into:
- 4 \( K^+ \) ions
- 1 \( [Fe(CN)_6]^{4-} \) ion
- Therefore, the total number of ions produced is:
\[ 4 + 1 = 5 \]

2. **Van't Hoff factor (i)**:
- The van't Hoff factor \( i \) for \( K_4[Fe(CN)_6] \) is 5.

### Step 2: Analyze the given salts

Now, we will analyze the dissociation of each given salt to find which one also has a van't Hoff factor of 5.

1. **\( Na_2SO_4 \)**:
- Dissociation:
\[ Na_2SO_4 \rightarrow 2Na^+ + SO_4^{2-} \]
- Total ions = 2 + 1 = 3 (i = 3)

2. **\( NaCl \)**:
- Dissociation:
\[ NaCl \rightarrow Na^+ + Cl^- \]
- Total ions = 1 + 1 = 2 (i = 2)

3. **\( Al_2(SO_4)_3 \)**:
- Dissociation:
\[ Al_2(SO_4)_3 \rightarrow 2Al^{3+} + 3SO_4^{2-} \]
- Total ions = 2 + 3 = 5 (i = 5)

4. **\( KCl \)**:
- Dissociation:
\[ KCl \rightarrow K^+ + Cl^- \]
- Total ions = 1 + 1 = 2 (i = 2)

### Step 3: Conclusion

The only salt that has the same van't Hoff factor as \( K_4[Fe(CN)_6] \) is \( Al_2(SO_4)_3 \) since both have a van't Hoff factor of 5.

### Final Answer:
The salt that has the same van't Hoff factor as \( K_4[Fe(CN)_6] \) is \( Al_2(SO_4)_3 \).

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