Question

Qualitatively draw the crystal field splitting for a trigonal bi- pyramidal complex ion. (Let the \(z\) axis be perpendicular to the trigonal plane.)

Short answer

In a trigonal bipyramidal complex ion, the \(d_{z^2}\) orbital has the highest energy level due to strong interaction with axial ligands, followed by the \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals interacting with equatorial ligands, and the \(d_{xz}\) and \(d_{yz}\) orbitals having the lowest energy level due to weak interaction with both axial and equatorial ligands. The crystal field splitting diagram shows a pyramid-like arrangement with the \(d_{z^2}\) orbital at the top, the \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals in the middle, and the \(d_{xz}\) and \(d_{yz}\) orbitals at the base.

Step-by-step solution

Identify the relevant d-orbitals in trigonal bipyramidal complex

In a trigonal bipyramidal complex, the d-orbitals of the central metal ion are relevant: \(d_{z^2}\), \(d_{xy}\), \(d_{xz}\), \(d_{yz}\), and \(d_{x^2-y^2}\).

Determine the relationships of d-orbitals to the ligands

Now we need to look at how the ligands interact with the d-orbitals. In a trigonal bipyramidal complex, the axial ligands are aligned along the z-axis, while the equatorial ligands are in the xy plane. Due to this arrangement, the following interactions occur: - The \(d_{z^2}\) orbital interacts strongly with the axial ligands. - The \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals interact with the equatorial ligands. - The \(d_{xz}\) and \(d_{yz}\) orbitals interact weakly with both axial and equatorial ligands.

Determine the crystal field splitting order for the d-orbitals

Based on the interaction strengths determined in Step 2, we can order the d-orbitals by their energy levels: 1. The \(d_{z^2}\) orbital will be at the highest energy level due to its strong interaction with the axial ligands. 2. The \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals will be at the next highest energy level due to their interaction with the equatorial ligands. 3. The \(d_{xz}\) and \(d_{yz}\) orbitals will be at the lowest energy level due to their weak interaction with both axial and equatorial ligands.

Draw the crystal field splitting diagram

Finally, we will draw the crystal field splitting diagram for the trigonal bipyramidal complex, with the energy levels on the vertical axis and the d-orbitals on the horizontal axis. 1. Place the \(d_{z^2}\) orbital at the highest energy level (top of the diagram). 2. Place the \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals at the next highest energy level, one on each side of the \(d_{z^2}\) orbital. 3. Place the \(d_{xz}\) and \(d_{yz}\) orbitals at the lowest energy level, one on each side of the pair of orbitals above them. The final diagram should show a pyramid-like arrangement, with the \(d_{z^2}\) orbital at the top, the \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals in the middle, and the \(d_{xz}\) and \(d_{yz}\) orbitals at the base, representing the crystal field splitting of the trigonal bipyramidal complex ion.

Key concepts

Trigonal Bipyramidal Complex

In coordination chemistry, a trigonal bipyramidal complex is a geometric arrangement of five ligands around a central metal atom. This complex features two axial positions that align along the z-axis and three equatorial positions that lie along the xy-plane. When forming a trigonal bipyramidal complex, the spatial orientation of the ligands creates a unique pattern of interactions with the central metal’s d-orbitals.

Understanding these spatial arrangements is crucial in predicting the physical properties and reactivity of the complex. Although this structure can occur naturally, it can also be synthesized in a lab for further study.

d-Orbital Splitting

In a trigonal bipyramidal complex, the d-orbitals of the central metal ion split into different energy levels due to the electrostatic interactions with the surrounding ligands. This phenomenon is known as d-orbital splitting. The presence of ligands affects the energies of specific d-orbitals in unique ways, causing differences in the energy levels among them.

For a trigonal bipyramidal complex:

• The \(d_{z^2}\) orbital is at the highest energy level because it experiences strong repulsion from the axial ligands.
• The \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals—interacting primarily with equatorial ligands—have intermediate energy levels.
• Lastly, the \(d_{xz}\) and \(d_{yz}\) orbitals have the lowest energy levels, as they experience weaker interactions with both axial and equatorial ligands.

The splitting pattern influences the electronic configuration and properties, such as color and magnetism, of the complex.

Ligand Interactions

Ligand interactions are essential in coordination chemistry as they determine the positioning and energy of d-orbitals. Ligands are molecules or ions that donate electron pairs to a central metal atom, forming coordinate bonds.

In a trigonal bipyramidal complex, interaction types include:

• **Axial Interactions:** Ligands aligned along the z-axis interact strongly with the \(d_{z^2}\) orbital due to direct approach and overlap.
• **Equatorial Interactions:** Those in the xy-plane influence the \(d_{xy}\) and \(d_{x^2-y^2}\) orbitals through lateral overlaps.
• **Mixed Interactions:** Both axial and equatorial ligands contribute weak repulsions to the \(d_{xz}\) and \(d_{yz}\) orbitals, leading to minimum energy changes.

These interactions decide the arrangement and energies of the d-orbitals and thus dictate the reactivity and properties of the chemical complex.

Coordination Chemistry

Coordination chemistry explores how metal atoms interact with surrounding molecules (ligands) in metal complexes. A central theme is understanding how ligands affect the geometry and electronic structure of complexes.

Key points in coordination chemistry include:

• It's not just about the number of ligands, but also their spatial arrangement, which influences the physical and chemical properties.
• Understanding crystal field theory enables chemists to predict the d-orbital splitting pattern, influencing properties like magnetism and color.
• The study of coordination complexes extends into diverse fields such as bioinorganic, medicinal, and industrial chemistry, showcasing their versatility.

Involving careful analysis of the ligand types, their donor properties, and the resulting structural geometry, coordination chemistry is integral to modern chemistry, deeply affecting both theoretical concepts and practical applications.