Qualitatively draw the crystal field splitting for a trigonal bi- pyramidal complex ion. (Let the \(z\) axis be perpendicular to the trigonal plane.)

In a trigonal bipyramidal complex, the d-orbitals of the central metal ion are relevant: \(d_{z^2}\), \(d_{xy}\), \(d_{xz}\), \(d_{yz}\), and \(d_{x^2-y^2}\).

Now we need to look at how the ligands interact with the d-orbitals. In a trigonal bipyramidal complex, the axial ligands are aligned along the z-axis, while the equatorial ligands are in the xy plane. Due to this arrangement, the following interactions occur: - The \(d_{z^2}\) orbital interacts strongly with the axial ligands. - The \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals interact with the equatorial ligands. - The \(d_{xz}\) and \(d_{yz}\) orbitals interact weakly with both axial and equatorial ligands.

Based on the interaction strengths determined in Step 2, we can order the d-orbitals by their energy levels: 1. The \(d_{z^2}\) orbital will be at the highest energy level due to its strong interaction with the axial ligands. 2. The \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals will be at the next highest energy level due to their interaction with the equatorial ligands. 3. The \(d_{xz}\) and \(d_{yz}\) orbitals will be at the lowest energy level due to their weak interaction with both axial and equatorial ligands.

Finally, we will draw the crystal field splitting diagram for the trigonal bipyramidal complex, with the energy levels on the vertical axis and the d-orbitals on the horizontal axis. 1. Place the \(d_{z^2}\) orbital at the highest energy level (top of the diagram). 2. Place the \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals at the next highest energy level, one on each side of the \(d_{z^2}\) orbital. 3. Place the \(d_{xz}\) and \(d_{yz}\) orbitals at the lowest energy level, one on each side of the pair of orbitals above them. The final diagram should show a pyramid-like arrangement, with the \(d_{z^2}\) orbital at the top, the \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals in the middle, and the \(d_{xz}\) and \(d_{yz}\) orbitals at the base, representing the crystal field splitting of the trigonal bipyramidal complex ion.