Problem 52 - 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

(a) In octahedral geometry, the \(d_{xy}, d_{yz}\), and \(d_{xz}\) orbitals point directly between the ligands. (b) In tetrahedral geometry, the \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals point directly between the ligands.

Step 1: Understand the geometries of the d orbitals

There are five d orbitals: \(d_{xy}, d_{yz}, d_{xz}, d_{x^2-y^2},\) and \(d_{z^2}\). The \(d_{xy}, d_{yz},\) and \(d_{xz}\) orbitals have four lobes in the xy, yz, and xz planes, respectively. The \(d_{x^2-y^2}\) orbital has four lobes in the x and y directions, and the \(d_{z^2}\) orbital has a doughnut-shaped lobe in the xy plane and two additional lobes along the z-axis. Step 2: Analyze the octahedral geometry

Step 2: Analyze the octahedral geometry

In octahedral geometry, there are six ligands surrounding the central atom, with each ligand located at the corners of an octahedron. The x, y, and z axes bisect the angles formed by the ligands, which means the ligands are placed along the diagonals of the axes. Step 3: Identify the d orbitals for octahedral geometry

Step 3: Identify the d orbitals for octahedral geometry

Since the ligands are located along the diagonals of the axes in octahedral geometry, the d orbitals that point directly between the ligands are the ones that have lobes between these diagonals. In this case, the \(d_{xy}, d_{yz}\), and \(d_{xz}\) orbitals have lobes pointing directly between the ligands in octahedral geometry. Step 4: Analyze the tetrahedral geometry

Step 4: Analyze the tetrahedral geometry

In tetrahedral geometry, there are four ligands surrounding the central atom, with each ligand located at the corners of a tetrahedron. The ligands are not located along the x, y, and z axes but rather between these axes. Step 5: Identify the d orbitals for tetrahedral geometry

Step 5: Identify the d orbitals for tetrahedral geometry

Since the ligands are located between the axes in tetrahedral geometry, the d orbitals that point directly between the ligands are the ones that have lobes along the axes. In this case, the \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals have lobes pointing directly between the ligands in tetrahedral geometry. #Answer#: (a) For octahedral geometry, the \(d_{xy}, d_{yz}\), and \(d_{xz}\) orbitals have lobes pointing directly between the ligands. (b) For tetrahedral geometry, the \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals have lobes pointing directly between the ligands.