Draw the \(d\) -orbital splitting diagrams for the octahedral complex ions of each of the following. a. \(\mathrm{Zn}^{2+}\) b. \(\mathrm{Co}^{2+}\) (high and low spin) c. \(\mathrm{Ti}^{3+}\)

The first step is to identify the electron configurations for the ions \(\mathrm{Zn}^{2+}\), \(\mathrm{Co}^{2+}\), and \(\mathrm{Ti}^{3+}\). a. \(\mathrm{Zn}\) is element 30, so its electron configuration is \(\mathrm{[Ar]3d^{10}4s^2}\). Therefore, \(\mathrm{Zn}^{2+}\) has an electron configuration of \(\mathrm{[Ar]3d^{10}}\). b. \(\mathrm{Co}\) is element 27, so its electron configuration is \(\mathrm{[Ar]3d^7 4s^2}\). Therefore, \(\mathrm{Co}^{2+}\) has an electron configuration of \(\mathrm{[Ar]3d^7}\). c. \(\mathrm{Ti}\) is element 22, so its electron configuration is \(\mathrm{[Ar]3d^2 4s^2}\). Therefore, \(\mathrm{Ti}^{3+}\) has an electron configuration of \(\mathrm{[Ar]3d^1}\).

In an octahedral field, the crystal field splits the five degenerate d-orbitals into two sets. The \(\mathrm{d_{z^2}}\) and \(\mathrm{d_{x^2-y^2}}\) orbitals, which constitute the e_g set, experience a higher energy due to their direct interaction with the ligands. Meanwhile, the \(\mathrm{d_{xy}}\), \(\mathrm{d_{yz}}\), and \(\mathrm{d_{xz}}\) orbitals, which constitute the t_2g set, experience a lower energy because they lie between the ligands. The energy difference between the e_g and t_2g orbitals is called the crystal field splitting energy, denoted by Δ.

Now, we will populate the d-orbitals with the electrons using the Aufbau principle and Hund's rule. We will also consider the high-spin and low-spin cases for \(\mathrm{Co}^{2+}\). a. For \(\mathrm{Zn}^{2+}\) with an electron configuration of \(\mathrm{[Ar]3d^{10}}\), all the d-orbitals are fully occupied. b. For \(\mathrm{Co}^{2+}\) with an electron configuration of \(\mathrm{[Ar]3d^7}\): - High spin: The electrons occupy the t_2g and e_g orbitals singly as much as possible before pairing up. - Low spin: The electrons occupy the t_2g orbitals and pair up before moving to the e_g orbitals. c. For \(\mathrm{Ti}^{3+}\) with an electron configuration of \(\mathrm{[Ar]3d^1}\), only one d-orbital within the t_2g set is occupied.

We are now ready to draw the d-orbital splitting diagrams for the octahedral complex ions: a. \(\mathrm{Zn}^{2+}\): Since all the d-orbitals are fully occupied, the diagram will show the t_2g orbitals filled with a total of six electrons and the e_g orbitals filled with a total of four electrons. b. \(\mathrm{Co}^{2+}\): - High spin: The diagram will show the three t_2g orbitals singly occupied with electrons and two e_g orbitals singly occupied with the remaining electrons. - Low spin: The diagram will show the three t_2g orbitals each with a pair of electrons and one empty e_g orbital set. c. \(\mathrm{Ti}^{3+}\): The diagram will show one electron in a t_2g orbital and all other d-orbitals unoccupied.