Question

The lobes of which \(d\) orbitals point directly between the ligands in (a) octahedral geometry, (b) tetrahedral geometry?

Short answer

The d orbitals that point directly between the ligands in (a) octahedral geometry are \(d_{xy}, d_{xz}\), and \(d_{yz}\), while in (b) tetrahedral geometry, they include the small lobe of \(d_{z^2}\) and \(d_{x^2 - y^2}\).

Step-by-step solution

Identify the d orbitals

There are 5 d orbitals that we need to consider: \(d_{xy}, d_{xz}, d_{yz}, d_{x^2-y^2}\), and \(d_{z^2}\). Each orbital has a unique orientation in space, which we need to visualize in order to understand their positions relative to the ligands in the given geometries.

Octahedral Geometry

In octahedral geometry, there are six ligands surrounding the central atom, forming an octahedron. The ligands are located at points along the x, y, and z axes. Let's evaluate the orientation of each d orbital: - \(d_{xy}\): The lobes of this orbital lie between the x and y axes, pointing directly between the ligands. - \(d_{xz}\): The lobes of this orbital lie between the x and z axes, pointing directly between the ligands. - \(d_{yz}\): The lobes of this orbital lie between the y and z axes, pointing directly between the ligands. - \(d_{x^2-y^2}\): The lobes of this orbital lie along the x and y axes, pointing at the ligands. - \(d_{z^2}\): The lobes of this orbital lie along the z axis, pointing at the ligands. Considering the above orientations, the orbitals that point directly between the ligands in octahedral geometry are \(d_{xy}, d_{xz}\), and \(d_{yz}\).

Tetrahedral Geometry

In tetrahedral geometry, there are four ligands surrounding the central atom, forming a tetrahedron. The ligands are located roughly at points between the x, y, and z axes. Let's evaluate the orientation of each d orbital: - \(d_{xy}\): The lobes of this orbital lie between the x and y axes, pointing at the ligands. - \(d_{xz}\): The lobes of this orbital lie between the x and z axes, pointing at the ligands. - \(d_{yz}\): The lobes of this orbital lie between the y and z axes, pointing at the ligands. - \(d_{x^2-y^2}\): The lobes of this orbital lie along the x and y axes, pointing directly between the ligands. - \(d_{z^2}\): The lobes of this orbital lie along the z axis, with a small lobe pointing directly between the ligands. Considering the above orientations, the orbitals that point directly between the ligands in tetrahedral geometry are the small lobe of \(d_{z^2}\) and \(d_{x^2 - y^2}\). In conclusion, the d orbitals that point directly between the ligands are: - Octahedral geometry: \(d_{xy}, d_{xz}\), and \(d_{yz}\) - Tetrahedral geometry: The small lobe of \(d_{z^2}\) and \(d_{x^2 - y^2}\)