0.1 M aqueous solution of `K_4[Fe(CN)_6]` will have the same freezing point as 0.1 M aqueous solution of

Options

A. `Fe(SO_4)_3`
B. `Al_2(SO_4)_3`
C. `AlCl_3`
D. `K_3PO_4`

Answer

To solve the question, we need to determine which 0.1 M aqueous solution will have the same freezing point as a 0.1 M aqueous solution of \( K_4[Fe(CN)_6] \).

### Step-by-Step Solution:

1. **Understanding Freezing Point Depression**:
The freezing point depression (\( \Delta T_F \)) is given by the formula:
\[ \Delta T_F = i \cdot K_F \cdot m \]
where:
- \( i \) = van 't Hoff factor (number of particles the solute dissociates into)
- \( K_F \) = freezing point depression constant (depends on the solvent)
- \( m \) = molality of the solution (in this case, we are given molarity, which is the same for comparison)

2. **Determine the van 't Hoff Factor for \( K_4[Fe(CN)_6] \)**:
The compound \( K_4[Fe(CN)_6] \) dissociates in solution as follows:
\[ K_4[Fe(CN)_6] \rightarrow 4K^+ + [Fe(CN)_6]^{4-} \]
This results in a total of 5 particles (4 potassium ions and 1 hexacyanoferrate ion). Therefore, the van 't Hoff factor \( i \) for \( K_4[Fe(CN)_6] \) is:
\[ i = 5 \]

3. **Finding a Compound with the Same van 't Hoff Factor**:
We need to find another compound that also dissociates into 5 particles when dissolved in water.

- **Option 1: \( FeSO_4 \cdot 3H_2O \)**:
\[ FeSO_4 \cdot 3H_2O \rightarrow Fe^{2+} + SO_4^{2-} \]
This gives us 2 particles, so \( i = 2 \).

- **Option 2: \( Al_2(SO_4)_3 \)**:
\[ Al_2(SO_4)_3 \rightarrow 2Al^{3+} + 3SO_4^{2-} \]
This gives us a total of 5 particles (2 aluminum ions and 3 sulfate ions), so \( i = 5 \).

- **Option 3: \( AlCl_3 \)**:
\[ AlCl_3 \rightarrow Al^{3+} + 3Cl^{-} \]
This gives us a total of 4 particles, so \( i = 4 \).

- **Option 4: \( K_3PO_4 \)**:
\[ K_3PO_4 \rightarrow 3K^+ + PO_4^{3-} \]
This gives us a total of 4 particles, so \( i = 4 \).

4. **Conclusion**:
The only compound that has the same van 't Hoff factor \( i = 5 \) as \( K_4[Fe(CN)_6] \) is \( Al_2(SO_4)_3 \). Thus, a 0.1 M aqueous solution of \( Al_2(SO_4)_3 \) will have the same freezing point as a 0.1 M aqueous solution of \( K_4[Fe(CN)_6] \).

### Final Answer:
0.1 M aqueous solution of \( Al_2(SO_4)_3 \) will have the same freezing point as 0.1 M aqueous solution of \( K_4[Fe(CN)_6] \).

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