Problem 95 - Chemistry An Atoms First Approach

Question

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

Step-by-Step Solution

Answer

In a trigonal bipyramidal complex, the crystal field splitting diagram consists of three energy levels for the d-orbitals. The highest-energy level is occupied by the \(d_{z^2}\) orbital, followed by the \(d_{x^2-y^2}\) orbital, and the lowest-energy level contains the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals, all having the same energy. The energy difference between the orbital levels represents the crystal field splitting.

Step 1: Identify the d-orbitals in a trigonal bipyramidal complex

First, we need to know the five d-orbitals in a trigonal bipyramidal complex: \(d_{z^2}\), \(d_{x^2-y^2}\), \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\).

Step 2: Analyze the axial and equatorial ligand's positions

In a trigonal bipyramidal complex, three ligands (A, B, and C) are in the equatorial plane, while the other two ligands (D and E) are in the axial position, perpendicular to the z-axis.

Step 3: Determine the interaction between the ligands and d-orbitals

Now we will analyze how the different d-orbitals interact with the ligands. The \(d_{z^2}\) orbital has a high electron density along the axial ligands (D and E) and will experience a strong repulsion, making it the highest-energy orbital. The \(d_{x^2-y^2}\) orbital has its electron density in between the equatorial ligands (A, B, and C), causing it to experience less repulsion and have lower energy. The \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals all have electron density in between both axial and equatorial ligands, experiencing minimum repulsion and forming the lowest-energy group of orbitals.

Step 4: Sketch the crystal field splitting diagram

To draw the crystal field splitting diagram for a trigonal bipyramidal complex, plot the energy (y-axis) against the d-orbitals (x-axis). The highest-energy level will be occupied by the \(d_{z^2}\) orbital, followed by the \(d_{x^2-y^2}\) orbital, and finally, the lowest-energy level will consist of the three orbitals \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) (all having the same energy). The energy difference between the orbital levels represents the crystal field splitting.