Problem 50 - Chemistry: The Central Science

Question

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

Step-by-Step Solution

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 1: 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.

Step 2: 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 \(d_{yz}\).

Step 3: 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 \(d_{x^2 - y^2}\)