Question

Draw the \(d\) -orbital splitting diagrams for the octahedral complex ions of each of the following. a. \(\mathrm{Fe}^{2+}\) (high and low spin) b. \(\mathrm{Fe}^{3+}(\text { high spin })\) c. \(\mathrm{Ni}^{2+}\)

Short answer

The d-orbital splitting diagrams for the given octahedral complex ions are as follows: a. \(\mathrm{Fe}^{2+}(3d^6)\): High spin: \(t_{2g}\): ↑(1, ↓1), ↑(2, ↓2), ↑(3, ↓3); \(e_g\): ↑(4), ↑(5), ↑(6) Low spin: \(t_{2g}\): ↑(1, ↓1), ↑(2, ↓2), ↑(3, ↓3); \(e_g\): ↑(1) b. \(\mathrm{Fe}^{3+}(3d^5)\): (High spin) \(t_{2g}\): ↑(1, ↓1), ↑(2, ↓2), ↑(3, ↓3); \(e_g\): ↑(1), ↑(2) c. \(\mathrm{Ni}^{2+}(3d^8)\): (High spin) \(t_{2g}\): ↑(1, ↓1), ↑(2, ↓2), ↑(3, ↓3); \(e_g\): ↑(1, ↓1), ↑(2, ↓2), ↑(3)

Step-by-step solution

Identify the Electron Configuration of the Metal Ions

The electron configurations for the given metal ions are: a. \(\mathrm{Fe}^{2+}\): [Ar] \(3d^6\) b. \(\mathrm{Fe}^{3+}\): [Ar] \(3d^5\) c. \(\mathrm{Ni}^{2+}\): [Ar] \(3d^8\)

Understand the Energy Levels of d-Orbitals in Octahedral Complexes

In an octahedral complex, the five \(d\) orbitals split into two different energy levels: three degenerate orbitals with lower energy, usually written as \(t_{2g}\), and two degenerate orbitals with higher energy, written as \(e_g\).

Draw the Diagrams for \(\mathrm{Fe}^{2+}\) in High and Low Spin Cases

For high spin complexes, we fill up the \(d\) orbitals according to the Hund's rule. For low spin complexes, we try to pair electrons in the \(t_{2g}\) orbitals before we fill the \(e_g\) orbitals. a. \(\mathrm{Fe}^{2+}(3d^6)\): High spin: 1. Fill the \(t_{2g}\) orbitals with 3 electrons: ↑(1, ↓1), ↑(2, ↓2), ↑(3, ↓3) 2. Fill the \(e_g\) orbitals with 3 remaining electrons: ↑(4), ↑(5), ↑(6) Low spin: 1. Fill the \(t_{2g}\) orbitals with 5 electrons: ↑(1, ↓1), ↑(2, ↓2), ↑(3, ↓3) 2. Fill the \(e_g\) orbitals with 1 remaining electron: ↑(1)

Draw the Diagram for \(\mathrm{Fe}^{3+}\) in a High Spin Complex

b. \(\mathrm{Fe}^{3+}(3d^5)\): High spin: 1. Fill the \(t_{2g}\) orbitals with 3 electrons: ↑(1, ↓1), ↑(2, ↓2), ↑(3, ↓3) 2. Fill the \(e_g\) orbitals with 2 remaining electrons: ↑(1), ↑(2)

Draw the Diagram for \(\mathrm{Ni}^{2+}\)

c. \(\mathrm{Ni}^{2+}(3d^8)\): Since it's not specified if it's high or low spin, we will consider the \(\mathrm{Ni}^{2+}\) case in high spin. 1. Fill the \(t_{2g}\) orbitals with 3 electrons: ↑(1, ↓1), ↑(2, ↓2), ↑(3, ↓3) 2. Fill the \(e_g\) orbitals with 5 remaining electrons: ↑(1, ↓1), ↑(2, ↓2), ↑(3) Here, you have filled up the \(d\)-orbitals following basic principles of crystal field theory. These diagrams illustrate the splitting of the \(d\)-orbitals in octahedral complex ions for the given transition metal ions in high and low spin cases.

Key concepts

Octahedral Complex

An octahedral complex is a type of coordination compound where a central metal ion is surrounded by six ligands. These ligands are arranged symmetrically around the metal ion, forming an octahedral shape. Imagine connecting the corners of an octahedron to see this structure visually.
In this arrangement, the ligands approach the metal ion along the x, y, and z axes. This causes the five d orbitals of the metal ion to experience different levels of repulsion.

• Three of these orbitals face directly towards the ligands, experiencing higher repulsion and thus, higher energy. These are known as the \( e_g \) orbitals.
• The other two orbitals are placed between the axes, experiencing less repulsion and having lower energy. These form the \( t_{2g} \) orbitals.

High Spin and Low Spin

The terms high spin and low spin refer to the arrangement of electrons within the d orbitals of a transition metal ion in a complex. These arrangements are determined by the strength of the field created by the surrounding ligands, also known as the ligand field strength.

High Spin Complexes

In high spin complexes:

• The ligand field is weak.
• Electrons prefer to remain unpaired and occupy all the d orbitals to minimize repulsions as predicted by Hund's Rule.

This leads to a smaller energy difference between the \( t_{2g} \) and \( e_g \) orbitals, and more unpaired electrons.

Low Spin Complexes

In low spin complexes:

• The ligand field is strong.
• Electrons will pair up in the lower energy \( t_{2g} \) orbitals before occupying the higher energy \( e_g \) orbitals, which reduces unpaired electrons.

Increased pairing means a larger energy difference between the \( t_{2g} \) and \( e_g \) orbitals.

Crystal Field Theory

Crystal field theory (CFT) is a model that describes the breaking of degeneracies of electron orbitals in transition metal complexes due to the presence of a ligand field. Developed to explain the electronic structure of complexes, it relies on the electrostatic interactions between the central ion and surrounding ligands.
CFT explains how:

• The spatial arrangement of ligands creates an electrostatic field that impacts the d orbitals' energy levels.
• This results in the splitting of the otherwise degenerate d orbital into two groups, \( t_{2g} \) and \( e_g \), as seen in octahedral complexes.

This model helps predict various properties, such as:

• The color of the complex, due to the absorption of specific light wavelengths corresponding to electronic transitions.
• Magnetic properties stemming from the existence of unpaired electrons in either high spin or low spin configurations.

Transition Metal Ions

Transition metal ions are elements in the d-block of the periodic table, which have partially filled d orbitals. When they form complexes, these d orbitals play a pivotal role in determining the complex's properties.

Characteristics of Transition Metal Ions

• They can exist in multiple oxidation states. For example, iron can be \( ext{Fe}^{2+} \) or \( ext{Fe}^{3+} \), which influences their electronic configuration and reactivity.
• Their d orbitals allow for complex geometries and coordination with ligands. This results in splitting of energy levels, as explained by crystal field theory.
• They are often colorful due to electron transitions between split d orbitals.
• They exhibit magnetic properties, which vary depending on high spin vs low spin and the specific metal ion involved.

The behavior of transition metal ions in complexes impacts many areas of chemistry, from biological systems to industrial applications.