The correct order of the wavelength maxima of the absorption band in the ultraviolet-visible region for the given complexes is

The correct option is:
[Co(CN)₆]³⁻ < [Co(NH₃)₆]³⁺ < [Co(NH₃)₅(H₂O)]³⁺ < [Co(NH₃)₅Cl]²⁺

Step-by-Step Explanation:

1. Crystal Field Splitting Energy (${\Delta }_o$) and Wavelength of Absorption ($\lambda$):
According to the crystal field theory, the absorption of light in transition metal complexes corresponds to the excitation of an electron from a lower-energy d-orbital to a higher-energy d-orbital (d-d transition). The energy change ($E$) associated with this transition is equal to the crystal field splitting energy (${\Delta }_o$) for octahedral complexes.
The relation between energy ($E$) and wavelength ($\lambda$) of the absorbed light is given by Planck's equation:
$E={\Delta }_o=\frac{hc}{\lambda }$
From this relation, it is clear that the wavelength of absorption ($\lambda$) is inversely proportional to the crystal field splitting energy (${\Delta }_o$):
$\lambda \propto \frac{1}{{\Delta }_o}$

2. Spectrochemical Series of Ligands:
The magnitude of crystal field splitting (${\Delta }_o$) depends on the strength of the ligand. Strong-field ligands cause a larger splitting (higher ${\Delta }_o$), while weak-field ligands cause a smaller splitting (lower ${\Delta }_o$).
The arrangement of the relevant ligands in the spectrochemical series in increasing order of field strength is:
${\text{Cl}}^{-}<{\text{H}}_2\text{O}<{\text{NH}}_3<{\text{CN}}^{-}$

3. Analyzing the Given Complexes:
All four complexes contain cobalt in the +3 oxidation state (${\text{Co}}^{3+}$), so the central metal ion is identical in all cases. Therefore, the splitting energy is determined solely by the strength of the coordinated ligands:
• In ${\left[\text{Co}{\left(\text{CN}\right)}_6\right]}^{3-}$, cyanide (${\text{CN}}^{-}$) is a very strong-field ligand, resulting in the largest crystal field splitting (${\Delta }_o$).
• In ${\left[\text{Co}{\left({\text{NH}}_3\right)}_6\right]}^{3+}$, ammonia (${\text{NH}}_3$) is a moderately strong-field ligand, but weaker than ${\text{CN}}^{-}$.
• In ${\left[\text{Co}{\left({\text{NH}}_3\right)}_5\left({\text{H}}_2\text{O}\right)\right]}^{3+}$, replacing one ${\text{NH}}_3$ with a weaker ligand ${\text{H}}_2\text{O}$ decreases the overall average ligand field strength compared to the hexaammine complex.
• In ${\left[\text{Co}{\left({\text{NH}}_3\right)}_5\text{Cl}\right]}^{2+}$, chloride (${\text{Cl}}^{-}$) is a weak-field ligand, which reduces the average ligand field strength even further.

4. Determining the Order:
The order of crystal field splitting energy (${\Delta }_o$) for the complexes is:
${\left[\text{Co}{\left({\text{NH}}_3\right)}_5\text{Cl}\right]}^{2+}<{\left[\text{Co}{\left({\text{NH}}_3\right)}_5\left({\text{H}}_2\text{O}\right)\right]}^{3+}<{\left[\text{Co}{\left({\text{NH}}_3\right)}_6\right]}^{3+}<{\left[\text{Co}{\left(\text{CN}\right)}_6\right]}^{3-}$
Taking the inverse relation for the absorption wavelength maximum (${\lambda }_{\text{max}}$), the order becomes reversed:
${\left[\text{Co}{\left(\text{CN}\right)}_6\right]}^{3-}<{\left[\text{Co}{\left({\text{NH}}_3\right)}_6\right]}^{3+}<{\left[\text{Co}{\left({\text{NH}}_3\right)}_5\left({\text{H}}_2\text{O}\right)\right]}^{3+}<{\left[\text{Co}{\left({\text{NH}}_3\right)}_5\text{Cl}\right]}^{2+}$