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 problem of determining which 0.1 M aqueous solution will have the same freezing point as a 0.1 M aqueous solution of `K4[Fe(CN)6]`, we need to consider the concept of freezing point depression and the van 't Hoff factor (i).

### Step-by-Step Solution:

1. **Understand Freezing Point Depression**:
The freezing point depression (ΔTf) is given by the formula:
\[ \Delta T_f = i \cdot m \cdot K_f \]
where:
- \( i \) = van 't Hoff factor (number of particles the solute dissociates into)
- \( m \) = molality of the solution
- \( K_f \) = freezing point depression constant of the solvent

2. **Calculate the van 't Hoff Factor for `K4[Fe(CN)6]`**:
The compound `K4[Fe(CN)6]` dissociates in solution as follows:
\[ K4[Fe(CN)6] \rightarrow 4K^+ + [Fe(CN)6]^{4-} \]
This results in a total of 5 particles:
- 4 potassium ions (K⁺)
- 1 hexacyanoferrate ion ([Fe(CN)6]⁴⁻)

Therefore, the van 't Hoff factor \( i \) for `K4[Fe(CN)6]` is 5.

3. **Set Up the Equation for Other Solutions**:
We need to find another 0.1 M solution that also has an \( i \) value of 5 to have the same freezing point depression. The equation can be set up as:
\[ i_1 \cdot 0.1 = i_2 \cdot 0.1 \]
Since the molality (m) is the same for both solutions, we can compare their \( i \) values directly.

4. **Evaluate Possible Solutions**:
We need to evaluate the dissociation of other compounds to find which one also has an \( i \) value of 5.

- **Fe2(SO4)3**:
\[ Fe2(SO4)3 \rightarrow 2Fe^{3+} + 3SO4^{2-} \]
Total particles = 2 + 3 = 5, so \( i = 5 \).

- **Al2O3**:
\[ Al2O3 \rightarrow 2Al^{3+} + 3O^{2-} \]
Total particles = 2 + 3 = 5, so \( i = 5 \).

- **AlCl3**:
\[ AlCl3 \rightarrow Al^{3+} + 3Cl^{-} \]
Total particles = 1 + 3 = 4, so \( i = 4 \).

- **K3PO4**:
\[ K3PO4 \rightarrow 3K^{+} + PO4^{3-} \]
Total particles = 3 + 1 = 4, so \( i = 4 \).

5. **Conclusion**:
The solutions that will have the same freezing point as 0.1 M `K4[Fe(CN)6]` are:
- 0.1 M `Fe2(SO4)3`
- 0.1 M `Al2O3`

### Final Answer:
0.1 M aqueous solution of `Fe2(SO4)3` and `Al2O3` will have the same freezing point as 0.1 M aqueous solution of `K4[Fe(CN)6]`.