Crystal Field Theory and Absorption Wavelength

The energy required for an electron to transition to a higher energy level in a transition metal complex is directly related to the Crystal Field Splitting Energy ($\Delta_o$). This energy is inversely proportional to the wavelength of light absorbed ($\lambda$). The relationship is given by the equation:

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

where \(h\) is Planck's constant and \(c\) is the speed of light. Therefore, a complex with a larger $\Delta_o$ absorbs light of shorter wavelength, and a complex with a smaller $\Delta_o$ absorbs light of longer wavelength.

Spectrochemical Series and Ligand Strength

The strength of a ligand, which determines the magnitude of $\Delta_o$, is given by the spectrochemical series. For the ligands present in the given complexes, the order of increasing ligand strength (and thus increasing $\Delta_o$) is:

• $Cl^-$ (Chloride)
• $H_2O$ (Water)
• $NH_3$ (Ammonia)
• $CN^-$ (Cyanide)

This order corresponds to:

$ [CrCl_6]^{3-} < [Cr(OH_2)_6]^{3+} < [Cr(NH_3)_6]^{3+} < [Cr(CN)_6]^{3-} $

The ligands are listed in increasing order of their ability to split d-orbitals, meaning $\Delta_o$ increases in this sequence.

Determining Wavelength Order

Since wavelength ($\lambda$) is inversely proportional to $\Delta_o$, the increasing order of absorption wavelength will be the reverse of the order of increasing $\Delta_o$. This means the complex with the strongest ligand ($CN^-$) will have the shortest absorption wavelength, and the complex with the weakest ligand ($Cl^-$) will have the longest absorption wavelength.

The complexes are:

• i) $[Cr(NH_3)_6]^{3+}$
• ii) $[CrCl_6]^{3-}$
• iii) $[Cr(OH_2)_6]^{3+}$
• iv) $[Cr(CN)_6]^{3-}$

Based on the ligand strength order ($CN^- > NH_3 > H_2O > Cl^-$), the order of increasing $\Delta_o$ is:

$ \Delta_o([Cr(CN)_6]^{3-}) > \Delta_o([Cr(NH_3)_6]^{3+}) > \Delta_o([Cr(OH_2)_6]^{3+}) > \Delta_o([CrCl_6]^{3-}) $

And the corresponding order of increasing absorption wavelength ($\lambda$) is:

$ \lambda([Cr(CN)_6]^{3-}) < \lambda([Cr(NH_3)_6]^{3+}) < \lambda([Cr(OH_2)_6]^{3+}) < \lambda([CrCl_6]^{3-}) $

Mapping this to the roman numerals:

$ \text{iv} < \text{i} < \text{iii} < \text{ii} $

Final Answer Mapping

The increasing order of wavelength of absorption is: iv) $[Cr(CN)_6]^{3-}$ < i) $[Cr(NH_3)_6]^{3+}$ < iii) $[Cr(OH_2)_6]^{3+}$ < ii) $[CrCl_6]^{3-}$.

This corresponds to option C.