$[CrF_6]^{3-}$ Electronic Spectrum Analysis

The complex $[CrF_6]^{3-}$ contains chromium in the $+3$ oxidation state ($Cr^{3+}$), which has a $d^3$ electron configuration. In an octahedral crystal field, the ground state term symbol for a $d^3$ ion is $^4A_{2g}$.

d-d Transitions in Octahedral Complexes

The absorption bands observed in the electronic spectrum of transition metal complexes arise from d-d electronic transitions. For a $d^3$ octahedral complex, the allowed spin transitions from the ground state $^4A_{2g}$ typically occur to the following excited states, listed in increasing order of energy:

• $^4T_{2g}$
• $^4T_{1g}(F)$
• $^4T_{1g}(P)$

Wavelength and Energy Relationship

The energy ($E$) of absorbed photons is inversely proportional to the wavelength ($\lambda$) according to the equation $E = \frac{hc}{\lambda}$, where $h$ is Planck's constant and $c$ is the speed of light. Therefore:

• Longer wavelengths correspond to lower energy transitions.
• Shorter wavelengths correspond to higher energy transitions.

Matching Transitions to Absorption Bands

The absorption bands are given at $670$ nm, $440$ nm, and $290$ nm. We match these wavelengths to the energy order of the transitions:

1. Lowest Energy (Longest Wavelength): $670$ nm corresponds to the transition with the smallest energy gap, which is $^4A_{2g}\rightarrow^4T_{2g}$.
2. Intermediate Energy: $440$ nm corresponds to the transition with the next higher energy gap, which is $^4A_{2g}\rightarrow^4T_{1g}(F)$.
3. Highest Energy (Shortest Wavelength): $290$ nm corresponds to the transition with the largest energy gap, which is $^4A_{2g}\rightarrow^4T_{1g}(P)$.

Conclusion

Therefore, the absorption bands observed at $670$, $440$, and $290$ nm are, respectively, due to the transitions $^4A_{2g}\rightarrow^4T_{2g}$, $^4A_{2g}\rightarrow^4T_{1g}(F)$, and $^4A_{2g}\rightarrow^4T_{1g}(P)$.