Which one of the following salts would have the same value of van't Hoff factor as that of \(Al(NO_3)_3\)?

\(K_3[Fe(CN)_6]\)

The correct answer is option 2. \(K_3[Fe(CN)_6]\).

The van't Hoff factor (\(i\)) indicates the number of particles into which a solute dissociates in solution. It is crucial in colligative properties such as boiling point elevation and freezing point depression. The factor is calculated based on the dissociation of a solute in a solvent.

\(Al(NO_3)_3\)

Chemical Formula: \(Al(NO_3)_3\)

Dissociation in Water: \(Al(NO_3)_3 \rightarrow Al^{3+} + 3NO_3^-\)

Ions Formed: 1 \(Al^{3+}\) ion, 3 \(NO_3^-\) ions

Total Number of Ions: 1 + 3 = 4

van't Hoff Factor (\(i\)): 4

Analysis of Other Salts:

1. \(Al_2(SO_4)_3\)

Chemical Formula: \(Al_2(SO_4)_3\)

Dissociation in Water: \(Al_2(SO_4)_3 \rightarrow 2Al^{3+} + 3SO_4^{2-}\)

Ions Formed: 2 \(Al^{3+}\) ions, 3 \(SO_4^{2-}\) ions

Total Number of Ions: 2 + 3 = 5

van't Hoff Factor (\(i\)): 5

2. \(K_3[Fe(CN)_6]\)

Chemical Formula: \(K_3[Fe(CN)_6]\)

Dissociation in Water: \(K_3[Fe(CN)_6] \rightarrow 3K^+ + [Fe(CN)_6]^{3-}\)

Ions Formed: 3 \(K^+\) ions, 1 \([Fe(CN)_6]^{3-}\) ion

Total Number of Ions: 3 + 1 = 4

van't Hoff Factor (\(i\)): 4

3. \(K_2SO_4\)

Chemical Formula: \(K_2SO_4\)

Dissociation in Water: \(K_2SO_4 \rightarrow 2K^+ + SO_4^{2-}\)

Ions Formed: 2 \(K^+\) ions, 1 \(SO_4^{2-}\) ion

Total Number of Ions: 2 + 1 = 3

van't Hoff Factor (\(i\)): 3

4. \(KCl\)

Chemical Formula: \(KCl\)

Dissociation in Water: \(KCl \rightarrow K^+ + Cl^-\)

Ions Formed: 1 \(K^+\) ion, 1 \(Cl^-\) ion

Total Number of Ions: 1 + 1 = 2

van't Hoff Factor (\(i\)): 2

Thus, the salt that has the same van’t Hoff factor as \(Al(NO_3)_3\) is:2. \(K_3[Fe(CN)_6]\)