The correct order of the wavelength of light absorbed by the following complexes is,

A. [Co(NH₃)₆]³⁺
B. [Co(CN)₆]^3–
C. [Cu(H₂O)₄]²⁺
D. [Ti(H₂O)₆]³⁺
Choose the correct answer from the options given below:

The correct option is B < A < D < C.

To determine the correct order of the wavelength of light absorbed by the given coordination complexes, we can use Crystal Field Theory (CFT).

According to CFT, the energy of light absorbed ($E$) by a transition metal complex is equal to its crystal field splitting energy ($\Delta$). The relationship between energy and the wavelength ($\lambda$) of absorbed light is given by Planck's equation:
$E=\Delta =\frac{hc}{\lambda }$
This shows that the wavelength of light absorbed is inversely proportional to the crystal field splitting energy:
$\lambda \propto \frac{1}{\Delta }$

The magnitude of crystal field splitting energy ($\Delta$) depends on two primary factors:
1. Oxidation state of the metal ion: A higher oxidation state (e.g., +3 vs +2) increases the electrostatic attraction between the metal and the ligands, bringing them closer and increasing the splitting energy ($\Delta$).
2. Nature of the ligand: According to the spectrochemical series, strong-field ligands cause greater splitting ($\Delta$) than weak-field ligands. The relative field strength of the ligands involved is:
${\mathrm{CN}}^{-}>{\mathrm{NH}}_3>H_{2}O$

Let us analyze each complex:
• B. ${\left[\mathrm{Co}{\left(\mathrm{CN}\right)}_6\right]}^{3-}$: Cobalt is in the +3 oxidation state and is coordinated to ${\mathrm{CN}}^{-}$, which is a very strong field ligand. Thus, it has the highest splitting energy (${\Delta }_B$).
• A. ${\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_6\right]}^{3+}$: Cobalt is in the +3 oxidation state and is coordinated to ${\mathrm{NH}}_3$, which is a strong field ligand but weaker than ${\mathrm{CN}}^{-}$. Thus, its splitting energy (${\Delta }_A$) is high, but less than B.
• D. ${\left[\mathrm{Ti}{\left(H_{2}O\right)}_6\right]}^{3+}$: Titanium is in the +3 oxidation state coordinated to $H_{2}O$, which is a weak-field ligand. Its splitting energy (${\Delta }_D$) is lower than both A and B.
• C. ${\left[\mathrm{Cu}{\left(H_{2}O\right)}_4\right]}^{2+}$: Copper is in the +2 oxidation state coordinated to $H_{2}O$. Because of the lower oxidation state of the metal ion, this complex has the lowest splitting energy (${\Delta }_C$).

Comparing the crystal field splitting energy values:
${\Delta }_B>{\Delta }_A>{\Delta }_D>{\Delta }_C$

Taking the reciprocal relationship for the wavelength of light absorbed:
${\lambda }_B<{\lambda }_A<{\lambda }_D<{\lambda }_C$

Therefore, the correct order is B < A < D < C.