The complexes \(\left[\mathrm{CrBr}_{6}\right]^{3-}\) and \(\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\) are both known. (a) Draw the \(d\) -orbital energy-level diagram for octahedral \(\mathrm{Cr}(\mathrm{III})\) complexes. (b) What gives rise to the colors of these complexes? (c) Which of the two complexes would you expect to absorb light of higher energy?

To draw the d-orbital energy-level diagram for octahedral Cr(III) complexes, we must first know the electron configuration of Cr(III). Chromium has an atomic number of 24, and its ground state electron configuration is [Ar] 3d^5 4s^1. In the Cr(III) state, it loses 3 electrons, resulting in the following electron configuration: [Ar] 3d^3. In an octahedral complex, the d-orbitals split into two sets due to the crystal field: the t₂g set (dxy, dyz, dxz) and the e_g set (dz^2, dx^2-y^2). The t₂g orbitals are lower in energy than the e_g orbitals. The Cr(III) ion has 3 electrons to distribute among the d-orbitals. These electrons will fill the t₂g orbitals first, following Hund's rule. The resulting d-orbital energy-level diagram is as follows: (1) t₂g: ↑↑↑ (three spin-up electrons) (2) e_g : (empty)

The colors of these complexes arise from the electronic transitions between the d-orbitals. When light is absorbed, an electron from a lower energy d-orbital (t₂g) is promoted to a higher energy d-orbital (e_g). The energy difference between these orbitals (∆E) corresponds to the energy of the absorbed photon and depends on the ligands surrounding the metal ion. Since visible light covers a range of energies (and hence colors), the absorbed light's energy (color) will determine the color of the complex as perceived by the human eye. The complementary color of the absorbed light will be observed as the color of the complex.

To determine which of the two complexes would absorb light of higher energy, we must compare the ligand-field splitting energies in both cases. Different ligands cause different degrees of splitting, which is summarized in the spectrochemical series, a list of ligands ordered by their ability to split d-orbitals: I^- < Br^- < Cl^- < F^- < OH^- < H_2O < NH_3 < en < NO_2^- < CN^- In the given complexes, the Cr ion is coordinated to Br^- in the first complex and NH_3 in the second complex. According to the spectrochemical series, NH_3 causes a larger splitting than Br^-. A larger splitting means that the energy gap between the t₂g and e_g orbitals will be greater in the [Cr(NH₃)₆]³⁺ complex compared to the [CrBr₆]³⁻ complex. Therefore, the [Cr(NH₃)₆]³⁺ complex would absorb light of higher energy, as the electronic transitions in this complex require a higher energy to promote an electron from the t₂g orbitals to the e_g orbitals.