The depression of freezing points of `0.05` molal aqueous solution of the following compounds are measured:
(P) NaCl ` " " (Q) K_(2)SO_(4)`
(R) `C_(6)H_(12)O_(6) " " (S) Al_(2)(SO_(4))_(3)`
Which one of the above compounds will exhibit the maximum depression of freezing point ?

Options

A. (R)
B. (Q)
C. (S)
D. (P)

Answer

To determine which compound exhibits the maximum depression of freezing point in a 0.05 molal aqueous solution, we will analyze the van 't Hoff factor (i) for each compound. The depression of freezing point (ΔTf) is given by the formula:

\[ \Delta T_f = i \cdot K_f \cdot m \]

Where:
- \( \Delta T_f \) = depression of freezing point
- \( i \) = van 't Hoff factor (number of particles the solute dissociates into)
- \( K_f \) = freezing point depression constant (which is the same for all solutions in this case)
- \( m \) = molality of the solution (which is also the same for all solutions)

Since \( K_f \) and \( m \) are constant for all solutions, the depression of freezing point will depend solely on the van 't Hoff factor \( i \). The higher the value of \( i \), the greater the depression of the freezing point.

### Step-by-Step Solution:

1. **Identify the compounds and their dissociation:**
- (P) NaCl dissociates into Na⁺ and Cl⁻.
- (Q) K₂SO₄ dissociates into 2 K⁺ and SO₄²⁻.
- (R) C₆H₁₂O₆ (glucose) does not dissociate in solution.
- (S) Al₂(SO₄)₃ dissociates into 2 Al³⁺ and 3 SO₄²⁻.

2. **Calculate the van 't Hoff factor (i) for each compound:**
- For NaCl:
\[ i = 2 \quad (\text{Na}^+ + \text{Cl}^-) \]
- For K₂SO₄:
\[ i = 3 \quad (2 \text{K}^+ + \text{SO}_4^{2-}) \]
- For C₆H₁₂O₆:
\[ i = 1 \quad (\text{non-electrolyte, does not dissociate}) \]
- For Al₂(SO₄)₃:
\[ i = 5 \quad (2 \text{Al}^{3+} + 3 \text{SO}_4^{2-}) \]

3. **Compare the van 't Hoff factors:**
- NaCl: \( i = 2 \)
- K₂SO₄: \( i = 3 \)
- C₆H₁₂O₆: \( i = 1 \)
- Al₂(SO₄)₃: \( i = 5 \)

4. **Determine which compound has the maximum \( i \):**
- The maximum \( i \) is for Al₂(SO₄)₃, which has \( i = 5 \).

5. **Conclusion:**
- Since Al₂(SO₄)₃ has the highest van 't Hoff factor, it will exhibit the maximum depression of freezing point among the given compounds.

### Final Answer:
The compound that will exhibit the maximum depression of freezing point is (S) Al₂(SO₄)₃.

---