(a) Sketch a diagram that shows the definition of the crystal-field splitting energy \((\Delta)\) for an octahedral crystal-field. (b) What is the relationship between the magnitude of \(\Delta\) and the energy of the \(d\)-\(d\) transition for a \(d^{1}\) complex? (c) Calculate \(\Delta\) in \(\mathrm{k} J / \mathrm{mol}\) if a \(d^{1}\) complex has an absorption maximum at 545 \(\mathrm{nm} .\)

To draw the diagram, we should first understand that in an octahedral complex, there are 6 ligands surrounding the central metal ion. The ligands approach along the \(x\), \(y\), and \(z\) axes, causing the \(d\) orbitals to split into two energy levels. The \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals have higher energy (collectively called \(e_g\) orbitals) and the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals have lower energy (collectively called \(t_{2g}\) orbitals). Sketch the diagram accordingly.

The crystal-field splitting energy, represented as \(\Delta\), is the difference in energy between the lower-energy \(t_{2g}\) orbitals and the higher-energy \(e_g\) orbitals. In a \(d^{1}\) complex, there is only one electron in the \(d\) orbitals, which occupies the lower-energy \(t_{2g}\) orbitals. The energy of the \(d\)-\(d\) transition corresponds to the energy required for the electron to transition from the \(t_{2g}\) orbitals to the \(e_g\) orbitals. Therefore, the energy of the \(d\)-\(d\) transition is equal to the crystal-field splitting energy, \(\Delta\).

To calculate \(\Delta\), we need to convert the absorption wavelength to energy using the equation: Energy = \(\frac{hc}{\lambda}\) where \(h\) is the Planck constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), \(c\) is the speed of light (\(3 \times 10^8 \mathrm{m/s}\)), and \(\lambda\) is the wavelength (545 \(\mathrm{nm}\)). First, convert the wavelength from nanometers to meters: \(\lambda = 545\,\mathrm{nm} \times \frac{1\,\mathrm{m}}{1 \times 10^9\,\mathrm{nm}} = 545 \times 10^{-9}\,\mathrm{m}\) Now, calculate the energy using the converted wavelength: Energy = \(\frac{(6.626 \times 10^{-34}\, \mathrm{Js})(3 \times 10^8\, \mathrm{m/s})}{545 \times 10^{-9}\, \mathrm{m}} = 3.64 \times 10^{-19}\,\mathrm{J}\) Now we have the energy in joules. We will convert it to kJ/mol: \(\Delta = \frac{3.64 \times 10^{-19} \mathrm{J}}{1 \times 10^{-3}\,\mathrm{kJ/J}} \times \frac{6.022 \times 10^{23}\,\mathrm{mol^{-1}}}{1\,\mathrm{mol}} = 218.7\,\mathrm{kJ/mol}\) So, the crystal-field splitting energy, \(\Delta\), is \(218.7\,\mathrm{kJ/mol}\).