Which of the following electrolytes has the same value of van't Hoff factor as that of `Al_(2)(SO_(4))_(3)`(if all are `100%`ionised?

Options

A. (a) `K_(2)SO_(4)`
B. (b) `K_(3)[Fe(CN)_(6)]`
C. (c) `Al(NO_(3))_(3)`
D. (d) `K_(4)[Fe(CN)_(6)]`

Explanation

To determine which electrolyte has the same van't Hoff factor (i) as that of \( Al_2(SO_4)_3 \) when fully ionized, we first need to calculate the van't Hoff factor for \( Al_2(SO_4)_3 \).

### Step 1: Calculate the van't Hoff factor for \( Al_2(SO_4)_3 \)

1. **Dissociation of \( Al_2(SO_4)_3 \)**:
- When \( Al_2(SO_4)_3 \) dissociates, it produces:
- 2 \( Al^{3+} \) ions
- 3 \( SO_4^{2-} \) ions
- Therefore, the total number of ions produced is:
\[ 2 + 3 = 5 \]

2. **Van't Hoff factor (i)**:
- The van't Hoff factor \( i \) is equal to the total number of ions produced upon dissociation.
- Thus, for \( Al_2(SO_4)_3 \):
\[ i = 5 \]

### Step 2: Evaluate the van't Hoff factor for each option

Now, we will evaluate the van't Hoff factor for each of the given options.

1. **Option A: \( K_2SO_4 \)**:
- Dissociation:
\[ K_2SO_4 \rightarrow 2 K^+ + SO_4^{2-} \]
- Total ions = 2 + 1 = 3
- \( i = 3 \)

2. **Option B: \( K_3Fe(CN)_6 \)**:
- Dissociation:
\[ K_3Fe(CN)_6 \rightarrow 3 K^+ + Fe(CN)_6^{3-} \]
- Total ions = 3 + 1 = 4
- \( i = 4 \)

3. **Option C: \( Al(NO_3)_3 \)**:
- Dissociation:
\[ Al(NO_3)_3 \rightarrow Al^{3+} + 3 NO_3^{-} \]
- Total ions = 1 + 3 = 4
- \( i = 4 \)

4. **Option D: \( K_4Fe(CN)_6 \)**:
- Dissociation:
\[ K_4Fe(CN)_6 \rightarrow 4 K^+ + Fe(CN)_6^{4-} \]
- Total ions = 4 + 1 = 5
- \( i = 5 \)

### Step 3: Compare van't Hoff factors

Now we compare the van't Hoff factors calculated:

- \( Al_2(SO_4)_3 \) has \( i = 5 \)
- \( K_2SO_4 \) has \( i = 3 \)
- \( K_3Fe(CN)_6 \) has \( i = 4 \)
- \( Al(NO_3)_3 \) has \( i = 4 \)
- \( K_4Fe(CN)_6 \) has \( i = 5 \)

### Conclusion

The electrolytes that have the same van't Hoff factor as \( Al_2(SO_4)_3 \) (which is 5) are:
- \( K_4Fe(CN)_6 \)

Thus, the correct answer is **Option D: \( K_4Fe(CN)_6 \)**.

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