The aqueous solution of the following salts will be coloured in the case of:

\(Co(NO_3)_2\)

The correct answer is option 3. \(Co(NO_3)_2\).

Let us go through each salt and understand why its aqueous solution would be colored or colorless.

1. \( \text{Zn(NO}_3\text{)}_2 \) (Zinc nitrate)

Oxidation State: Zinc is in the +2 oxidation state.

Electron Configuration of \( \text{Zn}^{2+} \): \( [\text{Ar}] 3d^{10} \)

d-electron Count: Zinc in the +2 state has a fully filled d-subshell with 10 electrons.

Color: Transition metal ions are colored due to d-d transitions, which require unpaired d electrons. Since \( \text{Zn}^{2+} \) has no unpaired d electrons (it's \(d^{10}\)), it does not exhibit these transitions and thus forms a colorless solution.

2. \( \text{LiNO}_3 \) (Lithium nitrate)

Oxidation State: Lithium is in the +1 oxidation state.

Electron Configuration of \( \text{Li}^+ \): \( 1s^2 \)

d-electron Count: Lithium does not have any d electrons.

Color: Without d electrons, there are no d-d transitions. Hence, \( \text{Li}^+ \) forms a colorless solution.

3. \( \text{Co(NO}_3\text{)}_2 \) (Cobalt nitrate)

Oxidation State: Cobalt is in the +2 oxidation state.

Electron Configuration of \( \text{Co}^{2+} \): \( [\text{Ar}] 3d^7 \)

d-electron Count: Cobalt in the +2 state has 7 d electrons.

Color: The \( \text{Co}^{2+} \) ion has unpaired d electrons (it is \(d^7\)), which can undergo d-d transitions. These transitions absorb specific wavelengths of light, resulting in a colored solution. Typically, \( \text{Co}^{2+} \) solutions are pink or red.

4. Potash alum (\( \text{KAl(SO}_4\text{)}_2 \cdot 12\text{H}_2\text{O} \))

Oxidation State: Aluminum is in the +3 oxidation state.

Electron Configuration of \( \text{Al}^{3+} \): \( [\text{Ne}] \)

d-electron Count: Aluminum does not have any d electrons in the +3 state.

Color: Without d electrons, there are no d-d transitions. Thus, \( \text{Al}^{3+} \) forms a colorless solution.

Conclusion

Only \( \text{Co(NO}_3\text{)}_2 \) forms a colored aqueous solution because it contains \( \text{Co}^{2+} \) ions, which have unpaired d electrons capable of d-d transitions. The other salts (\( \text{Zn(NO}_3\text{)}_2 \), \( \text{LiNO}_3 \), and potash alum) either have fully filled d subshells or no d electrons, resulting in colorless solutions.

Therefore, the correct answer is: \(Co(NO_3)_2\)