Complete the exercises below. a. Sketch a diagram that shows the definition of the crystal-field splitting energy (∆) for an octahedral crystal-field. b. What is the relationship between the magnitude of ∆ and the energy of the d-d transition for a d¹ complex? c. Calculate ∆ in kJ/mol if a d¹ complex has an absorption maximum at 545 nm.

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1. Step 1: For part (a), understand that in an octahedral crystal field, the d-orbitals split into two sets: the lower energy t_{2g} set (d_{xy}, d_{xz}, d_{yz}) and the higher energy e_g set (d_{x^2-y^2}, d_{z^2}). Sketch a diagram showing these orbitals with the energy levels, and label the energy difference between them as the crystal-field splitting energy (∆).
2. Step 2: For part (b), recognize that the energy of the d-d transition in a d¹ complex is directly related to the crystal-field splitting energy (∆). The energy of the transition corresponds to the energy required to promote an electron from the lower energy t_{2g} orbitals to the higher energy e_g orbitals, which is equal to ∆.
3. Step 3: For part (c), use the relationship between wavelength (λ) and energy (E) of light, given by the equation E = h*c/λ, where h is Planck's constant (6.626 x 10^{-34} J·s) and c is the speed of light (3.00 x 10^8 m/s). Convert the wavelength from nanometers to meters for calculation.
4. Step 4: Calculate the energy of the absorbed light in joules using the formula from Step 3. Substitute the given wavelength (545 nm) into the equation to find the energy in joules per photon.
5. Step 5: Convert the energy from joules per photon to kilojoules per mole by using Avogadro's number (6.022 x 10^{23} mol^{-1}). This will give you the value of ∆ in kJ/mol.