Question

Consider an octahedral complex having a univalent negative ion at each vertex of the octagon. Explain why the \(\mathrm{d}_{(\mathrm{x}) 2-(\mathrm{y}) 2}\) orbital of the central atom is less stable relative to the \(\mathrm{d}_{\mathrm{xy}}\) orbital.

Short answer

In an octahedral complex, the d_(x^2-y^2) orbital is less stable relative to the d_xy orbital due to their different orientations with respect to the ligand positions. The d_(x^2-y^2) orbital has its lobes directed along the x and y axes and experiences more electron repulsion from negatively charged ligands, resulting in a higher energy level. In contrast, the d_xy orbital has its lobes in between the axes, experiencing lesser repulsion, and thus is more stable. This stability difference can be explained using crystal field theory, which shows that d_(x^2-y^2) is an \(e_g\) orbital with a higher energy level compared to the \(t_{2g}\) orbitals, which include the d_xy orbital.

Step-by-step solution

Understand the d orbitals' arrangements in an octahedral complex

In an octahedral complex, there are six ligands surrounding the central metal atom/ion. These ligands are located along the x, y, and z axes and can interact with the d orbitals of the central atom/ion. There are five d orbitals: d_xy, d_yz, d_xz, d_(x^2-y^2), and d_(z^2). The arrangement of these orbitals can influence their stability due to the repulsion between ligand electrons and the central atom's d electrons.

Analyze electron repulsion in different d orbitals

In an octahedral complex, the ligands are situated along the x, y, and z axes. The d_(x^2-y^2) orbital has lobes directed along the x and y axes, while the d_xy orbital has lobes situated between the x and y axes. Due to the different orientations of these orbitals, the electron repulsion they experience from the negatively charged ligands will be different, affecting their relative stability.

Apply crystal field theory to explain stability difference

Crystal field theory is used to describe the effect of negatively charged ligands' electron repulsion on the energy levels of the central atom/ion's d-orbitals. In an octahedral complex, d orbitals split into two groups based on their energy levels due to crystal field splitting. The d_(x^2-y^2) and d_(z^2) orbitals, collectively known as \(e_g\) orbitals, are at a higher energy level compared to the d_xy, d_yz, and d_xz orbitals, known as \(t_{2g}\) orbitals. This is because the d_(x^2-y^2) and d_(z^2) orbitals have their lobes directed along the x, y, and z axes, where the ligands are located, thus experiencing stronger electron repulsion with ligands. On the other hand, the d_xy, d_yz, and d_xz orbitals have their lobes in between the axes, experiencing lesser repulsion.

Conclude the explanation of the stability difference

In conclusion, the difference in stability between the d_(x^2-y^2) and d_xy orbitals in an octahedral complex can be attributed to their different orientations with respect to the ligand positions. The d_(x^2-y^2) orbital experiences more electron repulsion from the ligands due to its lobes' orientation along the x and y axes, resulting in a higher energy level and, consequently, a relatively less stable state than the d_xy orbital.

Key concepts

Crystal Field Theory

Crystal Field Theory (CFT) provides a model to explain the behavior of transition metal ions when they form complexes with ligands. It focuses on the interaction between the central metal ion's d orbitals and the approaching ligands. According to CFT, when ligands approach a central metal ion to form a complex, they create an electric field that affects the energies of the metal ion's electron orbitals.

For an octahedral complex, this electric field causes a splitting of the five d orbitals into two groups with different energy levels. The higher energy level is the eg set, comprised of the d_(x^2-y^2) and d_(z^2) orbitals, and the lower energy level is the t2g set, which includes the d_xy, d_yz, and d_xz orbitals. The splitting occurs because the eg orbitals point directly at the incoming ligands, leading to greater repulsion and higher energy, while the t2g orbitals have a more oblique orientation with respect to the ligands, resulting in less repulsion and lower energy. This concept is integral to understanding why certain orbital arrangements lead to greater stability in octahedral complexes.

d Orbitals

Understanding d orbitals is crucial when discussing the electronic structure of transition metals and their complexes. The d orbitals consist of five orbitals—d_xy, d_yz, d_xz, d_(x^2-y^2), and d_(z^2)—each having a distinct shape and orientation in space. These orbitals are significant because they participate in bonding with ligands in transition metal complexes.

In an octahedral arrangement, the shape and orientation of these orbitals influence their energy levels when interacting with ligands. The d_(x^2-y^2) and d_(z^2) orbitals are oriented along the Cartesian axes, and therefore, have higher energy levels due to stronger interactions with the ligands placed along those axes. In contrast, the other three orbitals (d_xy, d_yz, and d_xz) have orientations that are between the axes, leading to lower energy levels because the electron density in these orbitals does not directly face the ligands.

Electron Repulsion

Electron repulsion is a key factor in the stability of octahedral complexes. According to the principles of electrostatics, like charges repel each other. In the context of an octahedral complex, there is repulsion between the negatively charged electrons in the ligands and the electrons in the d orbitals of the central atom.

The extent of this repulsion varies based on the spatial arrangement of the orbitals. Orbitals such as d_(x^2-y^2), which have their electron density concentrated along the axes where the ligands are located, experience greater repulsion. This increased repulsion results in higher energy, making the orbital less stable compared to orbitals like d_xy, where the electron density is between the axes and experiences less direct repulsion from the ligands. Therefore, electron repulsion is a vital aspect to consider when examining the energy levels and stability of d orbitals in a transition metal complex.

Ligand Arrangement

Ligand arrangement is the three-dimensional pattern in which ligands are positioned around the central metal ion in a complex. The geometrical structure of the ligand set plays a decisive role in determining the stability and electronic properties of the complex.

In an octahedral complex, six ligands are symmetrically positioned along the x, y, and z axes, interacting with the d orbitals of the central metal ion. The arrangement is such that the d_(x^2-y^2) orbital's lobes point toward the ligands on the x and y axes, causing more electron repulsion and raising the energy level of the orbital, leading to decreased stability. This is why in crystal field splitting, eg orbitals, which include the d_(x^2-y^2) orbital, have higher energy than t2g orbitals. When designing or analyzing coordination compounds, chemists must carefully consider this arrangement to predict the complex's stability and reactivity.