Question

Sketch the \(d\) -orbital energy level diagram for a typical octahedral complex.

Short answer

In an octahedral complex, the d-orbitals split into two energy levels: the lower-energy t2g set (( d_{xy}, d_{xz}, d_{yz} )) and the higher-energy eg set (( d_{z^2}, d_{x^2 - y^2} )). The energy gap between them is labeled as ( \(\Delta_o \)).

Step-by-step solution

Understand the Crystal Field Splitting in an Octahedral Complex

In an octahedral crystal field, the five degenerate d-orbitals (orbitals with the same energy) split into two groups because of the interactions with the ligands. The two different energy levels formed are the higher-energy 'eg' set (( d_{z^2}, d_{x^2 - y^2} )) and the lower-energy 't2g' set (( d_{xy}, d_{xz}, d_{yz} )).

Draw the Energy Level Diagram

Start by drawing a vertical line to represent the energy axis. At the central part of this axis, indicate the original energy level of the five degenerate orbitals before the crystal field is applied. Then, draw two sets of lines: three lower lines to represent the t2g set of orbitals and two upper lines to represent the eg set of orbitals. Show the energy gap (( \(\Delta_o \)) which is the crystal field splitting energy between these two sets.

Label the Orbitals

Label the three lower lines as t2g (( d_{xy}, d_{xz}, d_{yz} )) and the two higher lines as eg (( d_{z^2}, d_{x^2 - y^2} )).

Add Electrons if Necessary

If the exercise specifies the number of electrons in the d-orbitals, then add them accordingly to the t2g and eg levels. Fill the t2g set of orbitals first because they have the lower energy, and then fill the eg set according to Hund's rule and the Pauli exclusion principle.

Key concepts

Crystal Field Splitting

Crystal field splitting occurs when metal ions form complexes with ligands, molecules or ions that can donate a pair of electrons. In an octahedral complex, the d-orbitals split into two energy levels due to the electrostatic interactions with ligands positioned along the axes.

This splitting results in three d-orbitals (the t2g set: d_xy, d_xz, and d_yz) having lower energy and two (the e_g set: d_z², d_x²-y²) with higher energy. The energy difference between these sets is denoted as Δ_o (octahedral splitting energy).

This concept is key to understanding the behavior of transition metal complexes, including their magnetic properties and the color they exhibit in solutions. It's caused by the specific geometry of the octahedral arrangement, where ligands are symmetrically arranged around the central metal ion.

Octahedral Complex

An octahedral complex forms when six ligands symmetrically surround a central metal ion at equal distances. The shape resembles an octahedron, a polyhedron with eight faces.

The understanding of an octahedral complex is pivotal as it frequently occurs in coordination chemistry. It influences the crystal field splitting and dictates the arrangement of an electron in d-orbitals. Due to the octahedral ligand field, electron repulsion is minimized when lower energy t2g orbitals are filled before higher energy e_g orbitals.

Hund's Rule

Electron Distribution in Orbtials

Hund's rule addresses the distribution of electrons across orbitals of the same energy. It states that electrons will fill degenerate orbitals singly as far as possible before pairing up.

Within an octahedral complex, once the t2g orbitals are singly filled, additional electrons then pair up in these orbitals to minimize electron-electron repulsions. Following Hund's rule results in the most stable arrangement with the highest possible spin states for these electrons.

Pauli Exclusion Principle

The Pauli exclusion principle is a fundamental concept in quantum mechanics that restricts the placement of electrons in orbitals.

It states that no two electrons can have the same set of quantum numbers within an atom, meaning each orbital can hold a maximum of two electrons with opposite spins. When filling d-orbitals in a crystal field such as an octahedral complex, this principle ensures that electrons are distributed properly across the orbitals, maintaining the uniqueness of their quantum states.