Question

Which of the following salts would have the same value of the van't Hoff factor as that of \(\text{Al}(\text{NO}_3)_3\) in aqueous solution?
Assume complete dissociation of electrolytes.
(A) \(\text{K}_2\text{SO}_4\)
(B) \(\text{KCl}\)
(C) \(\text{Al}_2(\text{SO}_4)_3\)
(D) \(\text{K}_3[\text{Fe}(\text{CN})_6]\)
Choose the correct answer from the options given below:

• (A) and (B) only
• (B) and (C) only
• (D) only
• (C) and (D) only

Hint:

Always remember that ions inside the square brackets \([\dots]\) of a coordination sphere do not ionize in water.
Only external counter ions dissociate along with the complex ion entity itself.

Answer 1

The Correct Option is C

Concept:
The van 't Hoff factor (\(i\)) accounts for the extent of dissociation or association of a solute in solution.
For an electrolyte that undergoes complete ionic dissociation in aqueous medium, the van 't Hoff factor equals the total number of ions produced per formula unit of the substance:
\[ i = n \] where \(n\) is the number of constituent cations and anions released into solution.

Step 1: Dissociation of Aluminum Nitrate:
Aluminum nitrate, \(\text{Al}(\text{NO}_3)_3\), dissociates completely in water as follows:
\[ \text{Al}(\text{NO}_3)_3(\text{aq}) \rightarrow \text{Al}^{3+}(\text{aq}) + 3\text{NO}_3^-(\text{aq}) \] The total number of ions produced per formula unit is:
\[ n = 1 \, (\text{for } \text{Al}^{3+}) + 3 \, (\text{for } \text{NO}_3^-) = 4 \] Therefore, the van 't Hoff factor for \(\text{Al}(\text{NO}_3)_3\) is \(i = 4\).

Step 2: Dissociation of the Given Salts:
Let us evaluate the dissociation behavior and calculate the value of \(i\) for each salt:
- Compound (A): \(\text{K}_2\text{SO}_4\) dissociates as:
\[ \text{K}_2\text{SO}_4 \rightarrow 2\text{K}^+ + \text{SO}_4^{2-} \implies n = 2 + 1 = 3 \implies i = 3 \] - Compound (B): \(\text{KCl}\) dissociates as:
\[ \text{KCl} \rightarrow \text{K}^+ + \text{Cl}^- \implies n = 1 + 1 = 2 \implies i = 2 \] - Compound (C): \(\text{Al}_2(\text{SO}_4)_3\) dissociates as:
\[ \text{Al}_2(\text{SO}_4)_3 \rightarrow 2\text{Al}^{3+} + 3\text{SO}_4^{2-} \implies n = 2 + 3 = 5 \implies i = 5 \] - Compound (D): \(\text{K}_3[\text{Fe}(\text{CN})_6]\) is a coordination compound. The complex coordination sphere remains intact in solution, releasing only the ionizable counter-ions:
\[ \text{K}_3[\text{Fe}(\text{CN})_6] \rightarrow 3\text{K}^+ + [\text{Fe}(\text{CN})_6]^{3-} \implies n = 3 + 1 = 4 \implies i = 4 \]

Step 3: Comparison and Matching:
Comparing the van 't Hoff factors shows that only salt (D), \(\text{K}_3[\text{Fe}(\text{CN})_6]\), has \(i = 4\), which is identical to that of \(\text{Al}(\text{NO}_3)_3\).
Final Answer:
Thus, the salt with the same van 't Hoff factor is (D) only, corresponding to option (C).