Problem 36 - Chemistry Central Science

Question

Explain why the \(d_{x y}, d_{x z}\), and \(d_{y z}\) orbitals lie lower in energy than the \(d_{z^{2}}\) and \(d_{x^{2}-y^{2}}\) orbitals in the presence of an octahedral arrangement of ligands about the central metalion.

Step-by-Step Solution

Answer

In the presence of an octahedral arrangement of ligands, the \(d_{xy}, d_{xz}\), and \(d_{yz}\) orbitals lie lower in energy than the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals because they experience less electron repulsion with the ligands. The \(d_{xy}, d_{xz}\), and \(d_{yz}\) orbitals have lobes oriented between the axes, while the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals have lobes along the axes, where ligands approach. Consequently, the orbitals with lobes along the axes face greater electron repulsion leading to higher energy levels and decreased stability compared to the orbitals with lobes between the axes.

Step 1: Understand the orientation of d-orbitals in octahedral field

First, we need to visualize the orientation of the five d‐orbitals in an octahedral field: \(d_{xy}, d_{xz}, d_{yz}, d_{z^2}\), and \(d_{x^2-y^2}\). The \(d_{xy}, d_{xz}\), and \(d_{yz}\) orbitals have lobes oriented between the axes, while the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals have lobes along the axes.

Step 2: Visualize octahedral arrangement of ligands

In an octahedral arrangement, ligands are approaching the metal ion from the positive and negative directions of the x, y, and z axes. So, there are ligands along all the axes.

Step 3: Examine electron repulsion

The energy of the orbitals is affected by the electron repulsion between the metal ion d-orbitals and the ligands' electron pairs. When ligands approach the metal ion, the d-orbitals with lobes along the axes (i.e., \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals) will experience greater electron repulsion than those with lobes between the axes (i.e., \(d_{xy}, d_{xz}\), and \(d_{yz}\) orbitals).

Step 4: Analyze stability and energy levels

Due to greater electron repulsion, the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals become less stable, resulting in higher energy levels compared to the \(d_{xy}, d_{xz}\), and \(d_{yz}\) orbitals. In an octahedral field, the orbitals with lobes along the axes lie at a higher energy level than the orbitals with lobes between the axes.

Step 5: Conclusion

In conclusion, the \(d_{xy}, d_{xz}\), and \(d_{yz}\) orbitals lie lower in energy than the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals in the presence of an octahedral arrangement of ligands about the central metal ion because they experience less electron repulsion with the ligands compared to the orbitals with lobes along the axes.

Key Concepts

d-Orbitals

To fully understand the distribution of energy levels amongst the various orbitals in transition metal complexes, we first delve into what d-orbitals are. d-Orbitals are a set of five electron orbitals — denoted as d_xy, d_xz, d_yz, d_z², and d_x²-y² — that are pertinent to the atoms of transition metals.

The orientation of these d-orbitals is spatially characteristic; orbitals like d_xy, d_xz, and d_yz exhibit lobes situated between the axes, whereas d_z² and d_x²-y² point along the x, y, and z axes. This orientation significantly influences how these orbitals interact with their surroundings, primarily the ligands that approach the metal ion, thus determining their energy levels in a coordination compound.

Ligand Field Theory

Ligand Field Theory (LFT) is a powerful tool used to explain the bonding, structure, and electronic spectra of coordination compounds. It's an adaptation of the molecular orbital theory which considers the interaction between transition metal d-orbitals and ligand groups.

Under LFT, when ligands – molecules or ions that donate at least one pair of electrons to a central atom – approach a transition metal ion, they generate an electrostatic field that affects the energy of the d-orbitals. The closeness of the ligands and the direction of their approach result in different spatial interactions with these d-orbitals, leading to what is known as crystal field splitting. It is the dissimilar energies of the d-orbitals after the crystal field has been established that is foundational in understanding various properties of coordination complexes.

Crystal Field Stabilization Energy

When discussing Crystal Field Theory (CFT), the concept of Crystal Field Stabilization Energy (CFSE) emerges as a crucial component in the understanding of coordination complexes. CFSE is the energy difference between the energy of the d-orbitals in a spherical field (where all are degenerate) and the energy in the actual field generated by the ligands.

CFSE offers insights into the overall stability of the complex: the greater the CFSE, the more stable the complex. This is a result of the distribution of electrons in the split d-orbitals, where electrons seek to fill the lower-energy d-orbitals (d_xy, d_xz, and d_yz) first, in line with the Aufbau principle, causing these orbitals to have lower energies in an octahedral field compared to those that experience greater repulsion (d_z² and d_x²-y²). This difference in energy is the physical manifestation of CFSE.

Electron Repulsion

At the heart of why different d-orbitals have different energies in the context of LFT and CFT lies the fundamental principle of electron repulsion. Each ligand contains electron pairs that, when they approach the metal ion, interact electrostatically with the electrons in the d-orbitals of the metal.

Orbitals with lobes directed along the axis where ligands approach (d_z² and d_x²-y²) will naturally experience a greater repulsive interaction with these approaching ligand electrons, compared to orbitals with lobes oriented between the axes (d_xy, d_xz, and d_yz). This repulsion raises the energy of the orbitals, making the ones along the axes less energetically favorable, thus higher in energy, in accordance with Pauli's Exclusion Principle and Coulomb's Law. Understanding this electron repulsion is key in explaining the crystal field splitting that occurs within octahedral coordination complexes.