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 answer is B < A < D < C.

Step-by-Step Explanation:
1. Relationship between Energy and Wavelength of Light Absorbed:
The energy of light absorbed ($\mathrm{\Delta E}$) by a transition metal complex is related to the wavelength of light absorbed ($\lambda$) by the Planck-Einstein relation:

$\mathrm{\Delta E}=\frac{hc}{\lambda }$

The energy transition corresponds to the crystal field splitting energy ($\Delta$). Therefore, the wavelength of light absorbed is inversely proportional to the crystal field splitting energy:

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

A larger crystal field splitting energy ($\Delta$) results in a shorter wavelength ($\lambda$) of absorbed light.

2. Factors Affecting Crystal Field Splitting Energy ($\Delta$):
The magnitude of $\Delta$ is determined by:
- Oxidation State of the Metal Ion: Higher oxidation states (e.g., +3 vs +2) result in greater electrostatic attraction, drawing ligands closer and increasing $\Delta$.
- Nature of the Ligands (Spectrochemical Series): Stronger field ligands produce larger splitting. The general order of ligand field strength is: CN^- > NH₃ > H₂O.

3. Analyzing the Given Complexes:
- B. [Co(CN)₆]^3-: Contains Co³⁺ and CN^- (a strong-field ligand). This complex exhibits the highest crystal field splitting energy due to the +3 metal charge and very strong ligands.
- A. [Co(NH₃)₆]³⁺: Contains Co³⁺ and NH₃ (a moderately strong-field ligand). The splitting energy is large, but less than that of the CN^- complex.
- D. [Ti(H₂O)₆]³⁺: Contains Ti³⁺ and H₂O (a weak-field ligand). The splitting energy is lower than the Co³⁺ complexes containing stronger ligands.
- C. [Cu(H₂O)₄]²⁺: Contains Cu²⁺ and H₂O. Since the metal has a lower oxidation state (+2), the electrostatic interactions are weaker, leading to the smallest crystal field splitting energy among the given complexes.

4. Determining the Order:
Comparing the splitting energies gives:

$\Delta \left(B\right)>\Delta \left(A\right)>\Delta \left(D\right)>\Delta \left(C\right)$

Taking the inverse to find the order of the wavelengths ($\lambda$):

$\lambda \left(B\right)<\lambda \left(A\right)<\lambda \left(D\right)<\lambda \left(C\right)$

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