Order of Absorption Band Wavelength Maxima in Cobalt Complexes

To determine the correct order of the wavelength maxima ($\lambda_{\text{max}}$) of the absorption bands for the given cobalt complexes, we use the principles of Crystal Field Theory (CFT).

1. Relationship Between Energy and Wavelength

The absorption of light in the UV-Visible region by coordination complexes corresponds to $d$-$d$ transitions. The energy ($\Delta_o$) required for this transition is related to the wavelength ($\lambda$) of the absorbed light by Planck's equation:

$E = \Delta_o = \frac{hc}{\lambda}$

Thus, the absorption wavelength is inversely proportional to the crystal field splitting energy ($\Delta_o$):

$\lambda \propto \frac{1}{\Delta_o}$

2. Oxidation State and Spectrochemical Series

In all four given complexes, the central metal ion is cobalt in the $+3$ oxidation state ($\text{Co}^{3+}$):

• $[\text{Co}(\text{CN})_6]^{3-}$: $\text{Co}^{3+}$
• $[\text{Co}(\text{NH}_3)_6]^{3+}$: $\text{Co}^{3+}$
• $[\text{Co}(\text{NH}_3)_5(\text{H}_2\text{O})]^{3+}$: $\text{Co}^{3+}$
• $[\text{Co}(\text{NH}_3)_5(\text{Cl})]^{2+}$: $\text{Co}^{3+}$

Since the metal ion and its oxidation state are identical, the magnitude of $\Delta_o$ depends solely on the strength of the ligands in the coordination sphere.

According to the Spectrochemical Series, the relative field strengths of the ligands present in these complexes follow the order:

$\text{Cl}^- < \text{H}_2\text{O} < \text{NH}_3 < \text{CN}^-$

3. Comparing Crystal Field Splitting Energies ($\Delta_o$)

Using the rule of average environment for the mixed-ligand complexes:

1. $[\text{Co}(\text{CN})_6]^{3-}$ contains six strong $\text{CN}^-$ ligands, giving it the largest $\Delta_o$.
2. $[\text{Co}(\text{NH}_3)_6]^{3+}$ contains six $\text{NH}_3$ ligands.
3. $[\text{Co}(\text{NH}_3)_5(\text{H}_2\text{O})]^{3+}$ has one weaker $\text{H}_2\text{O}$ ligand replacing an $\text{NH}_3$ ligand, which reduces $\Delta_o$ compared to $[\text{Co}(\text{NH}_3)_6]^{3+}$.
4. $[\text{Co}(\text{NH}_3)_5(\text{Cl})]^{2+}$ has one even weaker $\text{Cl}^-$ ligand replacing an $\text{NH}_3$ ligand, reducing $\Delta_o$ further.

Therefore, the order of crystal field splitting energy ($\Delta_o$) is:

$\Delta_o\left([\text{Co}(\text{CN})_6]^{3-}\right) > \Delta_o\left([\text{Co}(\text{NH}_3)_6]^{3+}\right) > \Delta_o\left([\text{Co}(\text{NH}_3)_5(\text{H}_2\text{O})]^{3+}\right) > \Delta_o\left([\text{Co}(\text{NH}_3)_5(\text{Cl})]^{2+}\right)$

4. Order of Absorption Wavelength Maxima ($\lambda_{\text{max}}$)

Since $\lambda_{\text{max}}$ is inversely proportional to $\Delta_o$, reversing the inequality yields the correct order of absorption wavelength:

$[\text{Co}(\text{CN})_6]^{3-} < [\text{Co}(\text{NH}_3)_6]^{3+} < [\text{Co}(\text{NH}_3)_5(\text{H}_2\text{O})]^{3+} < [\text{Co}(\text{NH}_3)_5(\text{Cl})]^{2+}$

Hence, the correct option is (A).