The wavelength of absorbed electromagnetic radiation for \(\mathrm{CoBr}_{4}^{2-}\) is \(3.4 \times 10^{-6} \mathrm{m} .\) Will the complex ion \(\mathrm{CoBr}_{6}^{4-} \mathrm{ab}\) sorb electromagnetic radiation having a wavelength longer or shorter than \(3.4 \times 10^{-6} \mathrm{m} ?\) Explain.

We know that the energy of absorbed electromagnetic radiation is inversely proportional to its wavelength. The equation that relates energy and wavelength is: \(E = \dfrac{hc}{\lambda}\) where \(E\) is the energy, \(h\) is the Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength. The energy absorbed by a complex ion corresponds to the crystal field splitting energy, which is the energy required to promote an electron from a lower energy level to a higher energy level.

The crystal field splitting energy depends on the charge of the metal ion and the nature of the ligands. In the case of the complex ions \(\mathrm{CoBr}_{4}^{2-}\) and \(\mathrm{CoBr}_{6}^{4-}\), the metal ion is cobalt (Co) in both cases. However, the charges of the complex ions are different, and the number of ligands (bromide ions, Br⁻) is also different. When the charge of the complex increases, the crystal field splitting energy also increases. This is because a higher charge on the metal ion results in stronger attraction between the metal ion and the ligands. Additionally, a higher number of ligands would also increase the crystal field splitting energy, as ligands contribute to splitting the energy levels.

We have established that a greater charge and a higher number of ligands would increase the crystal field splitting energy. Now, let's compare the two complex ions: - \(\mathrm{CoBr}_{4}^{2-}\): charge = 2-, ligands = 4 - \(\mathrm{CoBr}_{6}^{4-}\): charge = 4-, ligands = 6 The complex ion \(\mathrm{CoBr}_{6}^{4-}\) has both a higher charge and a higher number of ligands than \(\mathrm{CoBr}_{4}^{2-}\). Therefore, we can infer that the crystal field splitting energy would be higher for \(\mathrm{CoBr}_{6}^{4-}\) as compared to \(\mathrm{CoBr}_{4}^{2-}\).

Recall that the energy absorbed is inversely proportional to the wavelength, which means that a higher crystal field splitting energy would correspond to a shorter wavelength. In our comparison: - \(\mathrm{CoBr}_{4}^{2-}\): wavelength = \(3.4 \times 10^{-6}\,\mathrm{m}\) - \(\mathrm{CoBr}_{6}^{4-}\): ? Since the crystal field splitting energy for \(\mathrm{CoBr}_{6}^{4-}\) is higher than that of \(\mathrm{CoBr}_{4}^{2-}\), its absorbed wavelength would be shorter.

The complex ion \(\mathrm{CoBr}_{6}^{4-}\) will absorb electromagnetic radiation having a wavelength shorter than \(3.4 \times 10^{-6}\,\mathrm{m}\).