(a) Sketch a diagram that shows the definition of the crystalfield splitting energy \((\Delta)\) for an octahedral crystal-field. \((\mathbf{b})\) 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{kJ} / \mathrm{mol}\( if a \)d^{1}\( complex has an absorption maximum at \)545 \mathrm{nm}$.

(a) In an octahedral crystal field, the five degenerate d-orbitals are split into two levels: three lower-energy orbitals (T₂g: dxy, dyz, and dxz) and two higher-energy orbitals (Eg: dx²-y² and d³z²-r²). The energy difference between these levels is the crystal field splitting energy (∆). (b) The energy of the d-d transition for a d¹ complex is equal to the crystal field splitting energy (∆). (c) Using the given absorption maximum (545 nm), we can calculate the energy of the d-d transition and the value of ∆ as follows: \( E = \dfrac{hc}{\lambda} \) Substitute the values \( h = 6.626 \times 10^{-34} \, \text{Js} \), \( c = 2.998 \times 10^8 \, \text{m/s} \), and \( \lambda = 545 \times 10^{-9} \, \text{m} \) to calculate E. Next, convert E to kJ/mol using Avogadro's number (NA): \( \Delta (\text{kJ/mol}) = \dfrac{E (\text{J}) \times N_\text{A}}{10^3} \) Substitute \( N_\text{A} = 6.022 \times 10^{23} \, \text{mol}^{-1} \) and the calculated E value to find the crystal field splitting energy (∆) in kJ/mol.