Problem 62 - Chemistry The Central Science In Si Units

Question

For each of the following metals, write the electronic con- figuration of the atom and its \(3+\) ion: \((\mathbf{a}) \mathrm{Fe},(\mathbf{b}) \mathrm{Mo},(\mathbf{c}) \mathrm{Co} .\) Draw the crystal-field energy-level diagram for the \(d\) orbitals of an octahedral complex, and show the placement of the \(d\) electrons for each \(3+\) ion, assuming a weak-field complex. How many unpaired electrons are there in each case?

Step-by-Step Solution

Answer

Fe: 5 unpaired; Mo: 3 unpaired; Co: 0 unpaired.

Step 1: Writing Electron Configuration of Atoms

The electron configurations of the neutral atoms are as follows: For iron (Fe): The atomic number is 26, so the configuration is \([\text{Ar}] \, 4s^2 \, 3d^6\). For molybdenum (Mo): The atomic number is 42, leading to \([\text{Kr}] \, 5s^1 \, 4d^5\) due to a half-filled d subshell. For cobalt (Co): The atomic number is 27, making the configuration \([\text{Ar}] \, 4s^2 \, 3d^7\).

Step 2: Writing Electron Configuration of the 3+ Ions

To find the electron configuration of the \(3+\) ions, remove three electrons from the atom, prioritizing removal from the highest principal energy level first. For \(\text{Fe}^{3+}\): Remove two electrons from \(4s\) and one from \(3d\), resulting in \([\text{Ar}] \, 3d^5\). For \(\text{Mo}^{3+}\): Remove the electron from \(5s\) and two from \(4d\), leading to \([\text{Kr}] \, 4d^3\). For \(\text{Co}^{3+}\): Remove two electrons from \(4s\) and one from \(3d\), yielding \([\text{Ar}] \, 3d^6\).

Step 3: Drawing the Crystal-Field Energy-Level Diagram for \(3+\) Ions in a Weak-Field Octahedral Complex

In an octahedral complex, the d orbitals split into two sets: \(t_{2g}\) (\(d_{xy}\), \(d_{xz}\), \(d_{yz}\)) and \(e_g\) (\(d_{x^2-y^2}\), \(d_{z^2}\)). For a weak-field situation, electrons will fill each orbital singly before pairing. For \(\text{Fe}^{3+}\): There are 5 d electrons, filling half of the \(t_{2g}\) orbitals. For \(\text{Mo}^{3+}\): There are 3 d electrons, filling each of the \(t_{2g}\) orbitals singly. For \(\text{Co}^{3+}\): There are 6 d electrons, with 3 filling \(t_{2g}\) singly, the fourth adding another electron into \(t_{2g}\), and the remaining 2 pairing in \(e_g\).

Step 4: Counting Unpaired Electrons

Determine the number of unpaired electrons in the \(3+\) ions. \(\text{Fe}^{3+}\): There are 5 electrons, all in \(t_{2g}\), resulting in 5 unpaired electrons. \(\text{Mo}^{3+}\): There are 3 electrons, all in \(t_{2g}\), resulting in 3 unpaired electrons. \(\text{Co}^{3+}\): There are 6 electrons, with 4 in \(t_{2g}\) (3 unpaired, 1 paired) and 2 in \(e_g\) (both paired), resulting in no unpaired electrons.

Key Concepts

Octahedral Complex

An octahedral complex is a common type of coordination compound where six ligands are symmetrically arranged around a central metal ion, forming the vertices of an octahedron. This arrangement maximizes the ligand-metal ion interactions, as the ligands can approach the metal from maximum symmetry positions.

In the case of a weak-field octahedral complex:

• The d orbitals of the central metal ion split into two distinct sets of energy levels due to the electrostatic interactions with the ligands.
• The three lower energy d orbitals form the t_2g set, which includes the d_xy, d_xz, and d_yz orbitals.
• The two higher energy d orbitals make up the e_g set, composed of d_x²-y² and d_z² orbitals.

These energy differences are key in predicting the behavior of electrons within the d orbitals.

Crystal-Field Theory

Crystal-field theory is a model that describes the electronic structure of transition metal complexes. It focuses on the interaction between the central metal ion and the surrounding ligands, treating the ligands as point charges in the simplified model.

The primary concepts behind this theory include:

• The splitting of d orbitals: In an octahedral complex, the presence of ligands disrupts the degeneracy of the d orbitals, splitting them into different energy levels. The t_2g orbitals have lower energy compared to the e_g orbitals.
• Weak-field versus strong-field ligands: In a weak-field complex, splitting is small, and electrons remain unpaired to a greater extent. In strong-field complexes, the energy gap is large enough to potentially force electron pairing within the lower energy orbitals.

This theory helps predict magnetic properties and spectral characteristics by examining how the d electron configuration changes in response to the field strength of the ligands.

Unpaired Electrons

Unpaired electrons in a d orbital refer to electrons that remain unpaired due to electron configuration and ligand strength. The presence of unpaired electrons is vital when determining the magnetic properties of a complex.

In a weak-field octahedral complex:

• For Fe³⁺ with 5 d electrons: All occupy the lower energy t_2g orbitals, resulting in 5 unpaired electrons.
• For Mo³⁺ with 3 d electrons: Each fills one t_2g orbital, keeping all 3 electrons unpaired.
• For Co³⁺ with 6 d electrons: An electron pairs up in the t_2g, and the e_g orbitals are fully paired, resulting in no unpaired electrons.

Unpaired electrons contribute to a complex's paramagnetic behavior, causing it to be attracted by a magnetic field. The exact physical manifestation of these electrons can be quantitatively determined using spectroscopic and magnetic studies.