In the linear crystal-field shown here, the negative charges are on the z-axis. Using Figure 23.28 as a guide, predict which of the following choices most accurately describes the splitting of the d orbitals in a linear crystal-field? [Find more in Section 23.6.]                                                                                                                                                

Verified step by step guidance

1. Understand the geometry of the crystal field: In a linear crystal-field, the ligands are aligned along the z-axis, directly influencing the d orbitals along this axis.
2. Identify the d orbitals affected: The d orbitals that will be most affected are those that have lobes pointing along the z-axis, namely the $d_{z^2}$ orbital.
3. Consider the effect on other d orbitals: The $d_{x^2-y^2}$, $d_{xy}$, $d_{xz}$, and $d_{yz}$ orbitals will be less affected as their lobes do not point directly along the z-axis.
4. Predict the splitting pattern: The $d_{z^2}$ orbital will experience a different energy shift compared to the other four d orbitals due to its direct alignment along the z-axis.
5. Conclude the splitting: In a linear crystal-field, typically the $d_{z^2}$ orbital will have a different energy compared to the other four ($d_{x^2-y^2}$, $d_{xy}$, $d_{xz}$, and $d_{yz}$) which might remain degenerate or split further depending on the specific environment and ligand characteristics.

Key Concepts

Crystal Field Theory

Crystal Field Theory (CFT) explains how the arrangement of ligands around a central metal ion affects the energy levels of the d orbitals. In a crystal field, the presence of negative charges from ligands can cause the d orbitals to split into different energy levels, depending on the geometry of the arrangement. This theory is crucial for understanding the electronic structure and color properties of transition metal complexes.

Orbital Splitting in Linear Fields

In a linear crystal field, the d orbitals split into two distinct energy levels due to the orientation of the ligands along the z-axis. Specifically, the d_{z^2} orbital experiences a higher energy due to direct interaction with the ligands, while the d_{x^2-y^2} orbital is also affected but to a lesser extent. Understanding this splitting is essential for predicting the electronic transitions and magnetic properties of the complex.

Ligand Field Strength

Ligand field strength refers to the ability of ligands to influence the energy levels of d orbitals in a metal complex. Strong field ligands cause greater splitting of the d orbitals, leading to different electronic configurations and properties compared to weak field ligands. This concept is vital for predicting the stability, color, and reactivity of transition metal complexes in various environments.