Question

Qualitatively draw the crystal field splitting of the $d$ orbitals in a trigonal planar complex ion. (Let the $z$ axis be perpendicular to the plane of the complex.)

Short answer

In a trigonal planar complex ion, the $d$ orbitals can be classified into three groups based on their interactions with ligands: $d_{xy}$ and $d_{x^2-y^2}$ have strong interactions, increasing their energy levels; $d_{xz}$ and $d_{yz}$ have weak interactions, keeping their energy levels relatively lower; and $d_{z^2}$ has no interaction, retaining the lowest energy level. The Crystal Field Splitting diagram for a trigonal planar complex ion should illustrate these energy levels, with $d_{xy}$ and $d_{x^2-y^2}$ at a higher energy, $d_{xz}$ and $d_{yz}$ at a lower energy, and $d_{z^2}$ at the lowest energy level.

Step-by-step solution

Understand the trigonal planar complex structure

A trigonal planar complex ion consists of a central metal atom/ion surrounded by three ligands in a equilateral triangular arrangement. The central axis is the z-axis, which is perpendicular to the plane of the complex.

Identify the $d$ orbitals in the orthogonal coordinate system

In the orthogonal coordinate system, there are five $d$ orbitals, which are $d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{x^2-y^2}$, and $d_{z^2}$.

Analyze the interaction of ligands with the $d$ orbitals

Since the ligands are in the xy-plane and the z-axis is perpendicular to the plane of the complex, we can classify the $d$ orbitals into two groups: 1. Orbitals lying in the xy-plane ($d_{xy}$ and $d_{x^2-y^2}$): These orbitals strongly interact with the ligands. 2. Orbitals lying along the xz and yz planes ($d_{xz}$ and $d_{yz}$): These orbitals weakly interact with the ligands. 3. Orbitals lying along the z-axis ($d_{z^2}$): This orbital does not interact with the ligands.

Determine the energy levels of the $d$ orbitals after splitting

Given the interaction of ligands with the $d$ orbitals, we can now determine the energy levels of the $d$ orbitals after the Crystal Field Splitting: 1. Orbitals lying in the xy-plane ($d_{xy}$ and $d_{x^2-y^2}$): Their energy will increase due to strong interaction with ligands. 2. Orbitals lying along the xz and yz planes ($d_{xz}$ and $d_{yz}$): Their energy will remain relatively lower due to weak interaction with ligands. 3. Orbitals lying along the z-axis ($d_{z^2}$): Its energy remains unchanged and remains lower.

Draw the Crystal Field Splitting diagram

Now that we know the energy levels of the $d$ orbitals after interacting with the ligands, we can draw the Crystal Field Splitting diagram: 1. Write the energy levels on the vertical axis. 2. Draw energy levels for the $d_{xy}$ and $d_{x^2-y^2}$ orbitals at a higher energy. 3. Draw energy levels for the $d_{xz}$ and $d_{yz}$ orbitals at a lower energy. 4. Draw the energy level for the $d_{z^2}$ orbital at the lowest energy. The resulting Crystal Field Splitting diagram will show the energy levels of the $d$ orbitals in a trigonal planar complex ion.