Problem 61 - Chemistry: The Central Science in SI Units, Global Edition

Question

For each of the following metals, write the electronic configuration of the atom and its \(2+\) ion: \((\mathbf{a}) \mathrm{Mn},(\mathbf{b}) \mathrm{Ru},(\mathbf{c}) \mathrm{Rh} .\) 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 \(2+\) ion, assuming a strong-field complex. How many unpaired electrons are there in each case?

Step-by-Step Solution

Answer

The electronic configurations of the given metal atoms Mn, Ru, Rh and their 2+ ions are: a. Mn : \(3d^{5}4s^{2}\); Mn2+ : \(3d^{5}\) b. Ru : \(4d^{7}5s^{2}\); Ru2+ : \(4d^{6}\) c. Rh : \(4d^{8}5s^{2}\); Rh2+ : \(4d^{7}\) In an octahedral strong-field complex, d-orbitals split into higher energy \(e_g\) orbitals (\(d_{z^2}\) and \(d_{x^2-y^2}\)) and lower energy \(t_{2g}\) orbitals (\(d_{xy}\), \(d_{xz}\), and \(d_{yz}\)). The electrons pair up in the lower energy orbitals before occupying the higher energy ones. In the strong-field octahedral complexes for each 2+ ion, the number of unpaired electrons are: a. Mn2+: 5 b. Ru2+: 4 c. Rh2+: 5

Step 1: Writing the electron configuration of the atoms and ions

First, let's write the electronic configuration of each metal atom: a. Mn : \(1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^{2}3d^{5}\) b. Ru : \(1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{10}4s^{2}4p^{6}5s^{2}4d^{7}\) c. Rh : \(1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{10}4s^{2}4p^{6}5s^{2}4d^{8}\) Now, we'll write the electronic configuration of the respective 2+ ions: a. Mn2+ : \(1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^{0}3d^{5}\) b. Ru2+ : \(1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{10}4s^{0}4p^{6}5s^{0}4d^{6}\) c. Rh2+ : \(1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{10}4s^{0}4p^{6}5s^{0}4d^{7}\)

Step 2: Drawing the crystal-field energy-level diagram for d orbitals in an octahedral complex

For an octahedral complex, the d orbitals split into two energy levels: \(e_g\) (the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals) and \(t_{2g}\) (the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals). The splitting between these levels is called the crystal-field splitting energy (\(\Delta\)).

Step 3: Placing d electrons and determining unpaired electrons for each 2+ ion in strong-field complex

In a strong-field complex, the crystal-field splitting energy is higher than the pairing energy, leading to the pairing of electrons in the lower energy orbitals before occupying the higher energy orbitals. a. Mn2+ : For Mn2+ ion, there are five d-electrons, they will fill the lower energy orbitals (\(t_{2g}\)) pairwise with all containing one unpaired electron. Unpaired electrons: 5 b. Ru2+ : For Ru2+ ion, there are six d-electrons. The first five will fill the \(t_{2g}\) orbitals, and the sixth electron will pair up in one of the \(t_{2g}\) orbitals. Unpaired electrons: 4 c. Rh2+ : For Rh2+ ion, there are seven d-electrons. The first five will fill the \(t_{2g}\) orbitals, and the two remaining electrons will fill up the \(e_g\) orbitals with one unpaired electron in each. Unpaired electrons: 5 To summarize, the number of unpaired electrons for Mn2+, Ru2+, and Rh2+ are 5, 4, and 5, respectively.

Key Concepts

Crystal-field theory

Crystal-field theory is a model that provides insight into the electronic structures and properties of coordination compounds, particularly those involving transition metals. It explains how the energies of a metal's d orbitals are affected by the presence of ligands that create an electric field around the metal ion. This theory helps us understand the color, magnetism, and stability of metal complexes. In an octahedral complex, where six ligands surround a central metal ion, the d orbitals split into two different energy levels known as \( e_g \) and \( t_{2g} \). This splitting occurs because ligands approach the metal along axes, affecting orbital energies differently.

• The energy difference between these two sets of orbitals is called the crystal-field splitting energy, denoted as \( \Delta \).
• The way d electrons fill these split orbitals depends on the strength of the field generated by the ligands.

Understanding crystal-field theory is crucial for predicting the electronic behavior of transition metal ions in complexes.

Octahedral complexes

Octahedral complexes are a common type of coordination compound where a central metal ion is surrounded symmetrically by six ligands. These complexes exhibit unique characteristics due to their geometry and the crystal field effects they experience. First, it's important to understand the geometry:

• The metal ion lies at the center with ligands at the vertices of an octahedron.
• This arrangement leads to specific electronic configurations and properties.

In the context of crystal-field theory, the octahedral arrangement causes the d orbitals of the central metal ion to split into two energy groups:

• The higher-energy \( e_g \) set includes \( d_{x^2-y^2} \) and \( d_{z^2} \) orbitals.
• The lower-energy \( t_{2g} \) set comprises \( d_{xy} \), \( d_{xz} \), and \( d_{yz} \) orbitals.

The manner in which d electrons occupy these orbitals depends on the nature of the ligands and can lead to various electronic and magnetic properties, such as the number of unpaired electrons.

Transition metals

Transition metals are elements found in the d-block of the periodic table, known for their ability to form various oxidation states and complex ions. Characteristics of transition metals include their partially filled d orbitals, which contribute to unique chemical and physical properties. These properties include:

• Variable oxidation states due to similar energies of 4s and 3d orbitals.
• The ability to form complex ions with different electronic configurations based on their environment.
• Exhibiting magnetic properties caused by unpaired d electrons.

Transition metals play a key role in catalysis, electronics, and biological systems because of their versatile bonding and electronic properties. In coordination chemistry, these metals form the core of complexes, influencing the color and magnetism of the compounds greatly due to their d orbital configurations.

Unpaired electrons

Unpaired electrons play a significant role in determining the magnetic properties and reactivity of transition metal complexes. An electron in a d orbital is unpaired if it does not have a partner with an opposite spin. Understanding unpaired electrons involves a few key points:

• Unpaired electrons are primarily responsible for the paramagnetic nature of a compound, which means it is attracted to an external magnetic field.
• The number of unpaired electrons can vary depending on the electron configuration and the crystal-field splitting within the complex.
• Complexes with more unpaired electrons tend to have higher magnetic moments and greater magnetic susceptibility.

In strong-field ligands, the pairing energy is lower, often leading to fewer unpaired electrons as electrons prefer to pair up rather than occupy higher energy orbitals. Understanding how electrons are distributed in orbitals helps predict the behavior and properties of transition metal complexes in various environments.