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Chapter 21: Problem 94

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

Expert verified
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

01

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.
02

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 \(d_{z^2}\).
03

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.
04

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.
05

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.

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