Which one of the following salts will have the same value of van't hoff factor (i) 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 salt has the same van't Hoff factor (i) as \( K_4[Fe(CN)_6] \), 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 \( K_4[Fe(CN)_6] \)

The dissociation of \( K_4[Fe(CN)_6] \) can be represented as follows:

\[ K_4[Fe(CN)_6] \rightarrow 4K^+ + [Fe(CN)_6]^{4-} \]

From this dissociation, we can see that:

- 4 potassium ions (\( K^+ \))
- 1 hexacyanoferrate ion (\( [Fe(CN)_6]^{4-} \))

Thus, the total number of ions (n) produced is:

\[ n = 4 + 1 = 5 \]

Therefore, the van't Hoff factor \( i \) for \( K_4[Fe(CN)_6] \) is:

\[ i = n = 5 \]

### Step 2: Analyze the given options

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

1. **Option 1: \( Al_2(SO_4)_3 \)**

The dissociation can be represented as:

\[ Al_2(SO_4)_3 \rightarrow 2Al^{3+} + 3SO_4^{2-} \]

Here, the total number of ions (n) is:

\[ n = 2 + 3 = 5 \]

Thus, the van't Hoff factor \( i \) is:

\[ i = 5 \]

2. **Option 2: \( NaCl \)**

The dissociation is:

\[ NaCl \rightarrow Na^+ + Cl^- \]

Here, the total number of ions (n) is:

\[ n = 1 + 1 = 2 \]

Thus, the van't Hoff factor \( i \) is:

\[ i = 2 \]

3. **Option 3: \( Al(NO_3)_3 \)**

The dissociation is:

\[ Al(NO_3)_3 \rightarrow Al^{3+} + 3NO_3^{-} \]

Here, the total number of ions (n) is:

\[ n = 1 + 3 = 4 \]

Thus, the van't Hoff factor \( i \) is:

\[ i = 4 \]

4. **Option 4: \( Na_2SO_4 \)**

The dissociation is:

\[ Na_2SO_4 \rightarrow 2Na^+ + SO_4^{2-} \]

Here, the total number of ions (n) is:

\[ n = 2 + 1 = 3 \]

Thus, the van't Hoff factor \( i \) is:

\[ i = 3 \]

### Conclusion

Among the given options, only \( Al_2(SO_4)_3 \) has a van't Hoff factor \( i \) equal to 5, which is the same as that of \( K_4[Fe(CN)_6] \).

**Final Answer: Option 1: \( Al_2(SO_4)_3 \)**