Question

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

Short answer

In a trigonal bipyramidal complex ion, the d-orbitals of the metal ion split into two energy levels due to different electrostatic interactions with the ligands. The axial orbitals (\(d_{z^2}\) and \(d_{x^2 - y^2}\)) have higher energy (\(E_g\)) while the equatorial orbitals (\(d_{xy}\), \(d_{yz}\), and \(d_{xz}\)) have lower energy (\(T_{2g}\)). To draw the crystal field splitting diagram, start with a vertical line representing energy levels. Draw two horizontal lines for the two energy levels (\(E_g\) and \(T_{2g}\)). Label and draw the axial orbitals (\(d_{z^2}\) and \(d_{x^2 - y^2}\)) on the \(E_g\) level and the equatorial orbitals (\(d_{xy}\), \(d_{yz}\), and \(d_{xz}\)) on the \(T_{2g\) level. Make sure to indicate the energy splitting between the levels.

Step-by-step solution

- Understand the Trigonal Bipyramidal Geometry

In trigonal bipyramidal geometry, there are five ligands surrounding the central metal ion, forming a triangle in the equatorial XY plane (trigonal plane), and two axial bonds connecting the metal ion to ligands above and below the equatorial plane along the Z-axis. This arrangement causes the d-orbitals of the metal ion to split into two sets because of different electrostatic interactions with the ligands.

- Identify the d-orbitals

In a transition-metal ion, there are five d-orbitals - \(d_{xy}\), \(d_{yz}\), \(d_{xz}\), \(d_{x^2 - y^2}\), and \(d_{z^2}\). Since the Z-axis is perpendicular to the trigonal plane in a trigonal bipyramidal complex ion, we can classify these d-orbitals as: - Axial orbitals: \(d_{z^2}\) and \(d_{x^2 - y^2}\) - Equatorial orbitals: \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\)

- Determine the Crystal Field Splitting

In the trigonal bipyramidal complex ion, the axial orbitals and equatorial orbitals experience different electrostatic interactions with surrounding ligands. The axial orbitals face repulsion from two axial ligands along the z-axis, while the equatorial orbitals experience repulsion from the three equatorial ligands in the XY plane. This difference causes the d-orbitals to split into two energy levels: Higher energy level (\(E_g\)): Axial orbitals - \(d_{z^2}\) and \(d_{x^2 - y^2}\), Lower energy level (\(T_{2g}\)): Equatorial orbitals - \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\).

- Draw the Crystal Field Splitting Diagram

Based on this information, we can now draw the crystal field splitting diagram for a trigonal bipyramidal complex ion: 1. Start by drawing a vertical line to represent energy levels. 2. Draw two horizontal lines with space between to represent the two different energy levels: higher (\(E_g\)) and lower (\(T_{2g}\)). 3. Label the higher energy level line as "(\(E_g\))" and the lower energy line as "(\(T_{2g}\))". 4. On the higher energy level line, draw two boxes and label them \(d_{z^2}\) and \(d_{x^2 - y^2}\), representing the axial orbitals. 5. On the lower energy level line, draw three boxes and label them \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\), representing the equatorial orbitals. Make sure to indicate the energy splitting between these two levels clearly, and the overall diagram should represent the qualitative crystal field splitting for a trigonal bipyramidal complex ion.