Problem 94 - Chemistry An Atoms First Approach

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.)

Step-by-Step Solution

Answer

The crystal field splitting diagram for a trigonal planar complex ion can be qualitatively drawn as follows: Along the energy axis, place the \(d_{z^2}\) orbital at the zero level, indicating minimal interaction with the ligands. The strongly interacting orbitals, \(d_{xz}\) and \(d_{yz}\), are positioned at a higher energy level (+E), while the weaker interacting orbitals, \(d_{xy}\) and \(d_{x^2 - y^2}\), are at a lower energy level (-E).

Step 1: Understanding the geometry of the complex

The trigonal planar complex ion has a planar geometry, where the ligands are 120 degrees apart from one another. The z-axis is perpendicular to the plane of this complex.

Step 2: Identifying the d orbitals

There are five d orbitals: \(d_{xy}\), \(d_{xz}\), \(d_{yz}\), \(d_{x^2 - y^2}\) and \(d_{z^2}\).

Step 3: Analyzing the interaction between the d orbitals and the ligands

According to the geometry of the complex, the interaction between the ligands and d orbitals will depend on their orientation in relation to the z-axis and the plane of the complex. 1. For the \(d_{xz}\) and \(d_{yz}\) orbitals, the interaction with the ligands will be stronger as they will have a larger overlap with the ligand electron cloud. 2. The \(d_{xy}\) and \(d_{x^2 - y^2}\) orbitals will also have some interaction with the ligands due to their location in the plane of the complex, but the effect will be relatively weak. 3. The \(d_{z^2}\) orbital will have the least interaction with the ligands as it is primarily along the z-axis and perpendicular to the plane of the complex.

Step 4: Drawing the crystal field splitting diagram

Based on the interactions between the d orbitals and the ligands, we can now qualitatively draw the crystal field splitting diagram. 1. Draw the y-axis for energy on the left side of the diagram and label it as "energy". 2. For the energy levels, first draw a horizontal line in the middle of the diagram and label it as "0" on the energy axis. 3. Above the zero level, draw another horizontal line at a higher energy level and label it with a positive energy value (e.g., "+E") on the energy axis. This high level corresponds to the strongly interacting orbitals (\(d_{xz}\) and \(d_{yz}\)). 4. Below the zero level, draw two horizontal lines for the weaker interacting orbitals (\(d_{xy}\) and \(d_{x^2 - y^2}\)). Label the lower level with a negative energy value on the energy axis (e.g., "-E"). 5. Position the least interacting orbital (\(d_{z^2}\)) at the zero level. This indicates that this orbital is not significantly affected by the crystal field. 6. To complete the diagram, add labels for the orbitals near their respective energy levels: left-aligned labels for \(d_{xy}\), \(d_{x^2 - y^2}\), and \(d_{z^2}\) and right-aligned labels for \(d_{xz}\) and \(d_{yz}\).