Sketch a \(d\) -orbital energy diagram for the following. a. a linear complex ion with ligands on the \(x\) axis b. a linear complex ion with ligands on the \(y\) axis

In this case, we need to consider the energy levels of the d-orbitals with respect to the x-axis. The five d-orbitals are: 1. \(d_{xy}\) 2. \(d_{yz}\) 3. \(d_{xz}\) 4. \(d_{x^2 - y^2}\) 5. \(d_{z^2}\) When the complex has linear geometry and ligands on the x-axis, the splitting of the d-orbitals occurs as follows: 1. \(d_{zy}\) and \(d_{zx}\) remain non-bonding orbitals, as they contain no interaction with the ligands on the x-axis. 2. \(d_{xy}\), \(d_{x^2 - y^2}\), and \(d_{z^2}\) interact with the ligands on the x-axis and experience an increase in energy. Thus, the energy diagram for this complex will have the following arrangement: - \(d_{zy}\) and \(d_{zx}\) are at the same lower energy level. - \(d_{xy}\), \(d_{x^2 - y^2}\), and \(d_{z^2}\) are at the same higher energy level.

In this case, we need to consider the energy levels of the d-orbitals with respect to the y-axis. The five d-orbitals are the same as stated before. When the complex has linear geometry and ligands on the y-axis, the splitting of the d-orbitals occurs as follows: 1. \(d_{yz}\) and \(d_{xy}\) remain non-bonding orbitals, as they contain no interaction with the ligands on the y-axis. 2. \(d_{xz}\), \(d_{x^2 - y^2}\), and \(d_{z^2}\) interact with the ligands on the y-axis and experience an increase in energy. Thus, the energy diagram for this complex will have the following arrangement: - \(d_{yz}\) and \(d_{xy}\) are at the same lower energy level. - \(d_{xz}\), \(d_{x^2 - y^2}\), and \(d_{z^2}\) are at the same higher energy level.