Problem 45 - Chemistry The Science in Context

Question

How many unpaired electrons are there in the following transition metal ions in an octahedral field? High-spin \(\mathrm{Fe}^{2+}, \mathrm{Cu}^{2+}, \mathrm{Co}^{2+},\) and \(\mathrm{Mn}^{3+}\)

Step-by-Step Solution

Answer

Answer: Fe²⁺ has 4 unpaired electrons, Cu²⁺ has 1 unpaired electron, Co²⁺ has 3 unpaired electrons, and Mn³⁺ has 4 unpaired electrons.

Step 1: Determine electron configurations for the metal ions

First, let's find the electron configurations of the neutral atoms: Fe: \(\mathrm{[Ar] 3d^6 4s^2}\) Cu: \(\mathrm{[Ar] 3d^{10} 4s^1}\) Co: \(\mathrm{[Ar] 3d^7 4s^2}\) Mn: \(\mathrm{[Ar] 3d^5 4s^2}\) Now, let's remove the appropriate number of electrons to create the ions: Fe\(^{2+}\): \(\mathrm{[Ar] 3d^6}\) Cu\(^{2+}\): \(\mathrm{[Ar] 3d^9}\) Co\(^{2+}\): \(\mathrm{[Ar] 3d^7}\) Mn\(^{3+}\): \(\mathrm{[Ar] 3d^4}\)

Step 2: Apply octahedral crystal field theory

Now, we'll apply the octahedral crystal field theory by distributing the electrons among the d orbitals (labelled as \(t_{2g}\) and \(e_g\)) following Hund's rule, which states that electrons will fill singly before pairing up. Fe\(^{2+}\) (3d\(^6\)): The distribution is \(t_{2g}^4 e_g^2\). Thus, there are 4 unpaired electrons. Cu\(^{2+}\) (3d\(^9\)): The distribution is \(t_{2g}^6 e_g^3\). There is 1 unpaired electron. Co\(^{2+}\) (3d\(^7\)): The distribution is \(t_{2g}^5 e_g^2\). Thus, there are 3 unpaired electrons. Mn\(^{3+}\) (3d\(^4\)): The distribution is \(t_{2g}^4 e_g^0\). Thus, there are 4 unpaired electrons.

Step 3: List the number of unpaired electrons in each ion

We found the number of unpaired electrons for each ion as follows: Fe\(^{2+}\): 4 unpaired electrons Cu\(^{2+}\): 1 unpaired electron Co\(^{2+}\): 3 unpaired electrons Mn\(^{3+}\): 4 unpaired electrons

Key Concepts

Transition Metal Ions

Transition metal ions are ions of elements that belong to the d-block of the periodic table. These elements are known for their ability to form colorful compounds, exhibit variable oxidation states, and have magnetic properties, thanks to the special arrangement of their electron d-orbitals. When these metals form ions, they do so by losing electrons from their outermost shell, which usually includes their 4s electrons before they begin to lose any from the 3d orbital.

Understanding the electron configuration of these ions is crucial for predicting their behavior in various chemical environments, including their magnetic nature and the types of bonds they can form with other elements. Transition metal ions play a significant role in many biological processes and are often central in catalysis in both nature and industrial processes. Therefore, learning their properties is fundamental to understanding a wide range of chemical reactions.

Octahedral Field

An octahedral field refers to an environment around a central metal ion where six ligands (molecules or ions that donate at least one pair of electrons to the metal ion) are symmetrically arranged at the corners of an octahedron. In crystal field theory, this arrangement causes a splitting of the metal ion's d-orbital energies into two different levels: a lower-energy set called the t_2g orbitals and a higher-energy set known as the e_g orbitals.

The energy gap between these sets is critical for understanding the color, magnetism, and stability of the resulting complexes. In an octahedral field, ligands approach along the axes, causing greater repulsion and, therefore, higher energy for the e_g orbitals which directly lie along these axes. The specific splitting pattern greatly influences the distribution of electrons and hence the properties of the transition metal ion in question.

Unpaired Electrons

Unpaired electrons refer to the electrons in an atom or ion that do not have a corresponding electron with the opposite spin in the same orbital. Since electrons are fermions, they obey the Pauli exclusion principle, which states that no two electrons can have the same set of quantum numbers within an atom. As a result, electrons will pair up in orbitals only when they must, preferring to remain unpaired if possible.

Having unpaired electrons is what gives substances their magnetic properties. In magnetic-field-related applications, materials with several unpaired electrons - like certain transition metal ions - are of particular interest. The number of unpaired electrons can be determined using electron configurations and Hund's rule, and it's essential for predicting the magnetic nature of a compound, whether diamagnetic (no unpaired electrons) or paramagnetic (one or more unpaired electrons).

Hund's Rule

Hund's rule is an important principle when considering the electronic configuration of atoms and ions. It states that electrons occupy orbitals of the same energy in a way that maximizes the number of electrons with the same spin. Essentially, within a given sublevel, one electron goes into each orbital before any pairing occurs.

This rule is based on the idea of minimizing electron repulsions within a given subshell to lower the overall energy. Following Hund's rule leads to the most stable and lowest energy configuration for an atom or ion. When applied in the context of crystal field theory, Hund's rule helps explain the distribution of electrons among the split d-orbitals in a transition metal complex and is essential for determining the number of unpaired electrons.

Electron Configuration

Electron configuration is a description of the distribution of electrons in an atom or ion and is based on the specific sequence in which orbitals are filled according to their increasing energy levels. The typical order is defined by the Aufbau principle, which suggests that electrons are added to the lowest energy orbitals first, before moving to higher energy levels. Each orbital can hold a maximum of two electrons, with opposite spins.

When describing the electron configuration of transition metal ions, the d-orbitals are a primary focus because they are in the process of being filled across the series. Transition metals have unique properties and complex behaviors due to the d-orbitals' ability to accommodate up to 10 electrons, leading to a wide variety of possible electron configurations. The correct electron configuration not only helps in understanding the color and magnetic properties of the compounds but also in predicting reactivity and the type of chemical bonding.