Question

Consider the pseudo-octahedral complex ion of \(\mathrm{Cr}^{3+}\) , where A and \(\mathrm{B}\) represent ligands. Ligand A produces a stronger crystal field than ligand B. Draw an appropriate crystal field diagram for this complex ion (assume the A ligands are on the \(z\) -axis).

Short answer

In the pseudo-octahedral complex of \(\mathrm{Cr}^{3+}\) with A and B ligands, A ligands produce a stronger crystal field than B ligands and are located on the z-axis. The crystal field diagram shows a larger splitting energy (\(\Delta\)) between the \(t_{2g}\) and \(e_g\) orbitals due to the presence of A ligands on the z-axis. The \(t_{2g}\) group comprises the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals, whereas the \(e_g\) group includes \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals. The diagram depicts two A ligands on the z-axis and three B ligands in the xy plane, causing the larger splitting due to the stronger crystal field produced by the A ligands.

Step-by-step solution

Identify the geometry of the complex ion

Pseudo-octahedral means the arrangement of six ligands around a central metal ion resembles the octahedral shape with some distortions. In this case, four ligands are found in the plane, and two other ligands are located above and below the central metal ion (on the z-axis).

Identify and label crystal field splitting

In an octahedral complex, the d-orbitals (\(d_{xy}\), \(d_{xz}\), \(d_{yz}\), \(d_{x^2-y^2}\), and \(d_{z^2}\)) are split into two groups: \(t_{2g}\) and \(e_g\). The \(t_{2g}\) group includes \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals, whereas the \(e_g\) group includes \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals.

Place the A and B ligands in the proposed geometry

Since A ligands produce a stronger crystal field than B ligands, and A ligands are on the z-axis, the corresponding crystal field diagram should have a larger splitting energy (\(\Delta\)) between \(t_{2g}\) and \(e_g\) orbitals. Place three B ligands in the xy plane and the two A ligands on the z-axis.

Draw crystal field diagram for the distorted octahedral complex

To draw the crystal field diagram: 1. Plot the energy levels for the \(t_{2g}\) and \(e_g\) orbitals. 2. Indicate the larger splitting energy for the A ligands on the z-axis (i.e., the \(\Delta\) between \(t_{2g}\) and \(e_g\) is larger for A ligands). 3. Label the orbitals that belong to the \(t_{2g}\) and \(e_g\) groups. 4. Place the B ligands in the plane and A ligands above and below the central metal ion. 5. Draw arrows to show the field splitting by the A and B ligands (considering the larger splitting produced by the A ligands). The crystal field diagram will show two A ligands on the z-axis (higher repulsion with \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals) and three B ligands in the xy plane. This causes a larger splitting between \(t_{2g}\) and \(e_g\) orbitals due to the stronger crystal field of the A ligands.

Key concepts

Pseudo-octahedral Complex

In coordination chemistry, a pseudo-octahedral complex describes a molecular arrangement that mimics the typical octahedral shape but with some distortions. Imagine a metal ion at the center surrounded by six ligands. These ligands are usually placed along the three-dimensional axes.

In a standard octahedral complex, these ligands are symmetrically placed, with four in a plane and two along the vertical axis. However, in a pseudo-octahedral complex, like the one in the problem involving \( \mathrm{Cr}^{3+} \), there might be different types of ligands or distortions affecting this symmetry. This can occur when certain ligands are stronger or weaker, altering their positions and influencing the overall geometry.

• This specific arrangement alters the energy and interaction for the central metal ion.
• The differences in ligands and their placement are crucial in determining properties like color and magnetism.

Ligand Field Splitting

Ligand field splitting involves the separation of the five degenerate d-orbitals when ligands come close to a metal ion. These ligands create a magnetic field that influences the metal's d-orbitals. In an octahedral complex, this splitting divides the d-orbitals into two groups: the \( t_{2g} \) and \( e_g \) orbitals.

The energy difference between these two groups is called the splitting energy, denoted by \( \Delta \). This value depends on the ligands involved. More strongly interacting ligands, like A in our exercise, produce a greater \( \Delta \).

• Stronger ligands lead to more significant splitting and stabilization.
• The placement of ligands (such as along the z-axis for ligand A in the exercise) can increase or decrease this energy gap further.

d-Orbital Splitting

When six ligands surround a metal ion, the d-orbitals experience different repulsions. This is because not all d-orbitals are oriented towards approaching ligands the same way. In octahedral coordination, the d-orbitals are split into two levels: \( t_{2g} \), consisting of \( d_{xy} \), \( d_{yz} \), \( d_{xz} \) orbitals; and \( e_g \), composed of \( d_{x^2-y^2} \) and \( d_{z^2} \) orbitals.

• \( e_g \) orbitals face the ligands directly, resulting in higher energy due to repulsion.
• \( t_{2g} \) orbitals are offset from the ligand axis, experiencing less repulsion and having lower energy.

Understanding these energy variations is crucial as they determine how electrons will populate the orbitals, affecting the complex’s properties.

Crystal Field Diagram

A crystal field diagram visually represents the energy levels of d-orbitals in a complex due to ligand interaction. It helps us understand how these electrons populate due to the impact of ligand fields, like in our chromium pseudo-octahedral complex.

When drawing such a diagram for a complex, you must indicate the splitting of d-orbitals caused by ligand field interactions. Specifically, in this exercise:

• The \( t_{2g} \) level should be closer to the metal's base energy, while the \( e_g \) should be higher due to stronger ligand interactions.
• We illustrate stronger \( A \) ligands at the z-axis, showing increased splitting, and weaker \( B \) ligands in the plane.
• The resulting diagram allows us to map out electron placements and predict properties like color absorption.

Creating this diagram necessitates careful consideration of ligand strength and geometry to accurately depict the orbital energy levels.