Question

For each of the following metals, write the electronic configuration of the atom and its \(2+\) ion: (a) \(\mathrm{Mn},(\mathrm{b}) \mathrm{Ru}\), (c) 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?

Short answer

The electronic configurations for the given elements and their 2+ ions are: Mn: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^5\), Mn 2+: \(1s^2 2s^2 2p^6 3s^2 3p^6 3d^5\); Ru: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^6\), Ru 2+: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 4d^6\); Rh: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^1 4d^8\), Rh 2+: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 4d^6\). In strong-field octahedral complexes, their crystal field energy diagrams are Mn 2+: \(t_{2g}^5 e_g^0\), Ru 2+: \(t_{2g}^6 e_g^0\), and Rh 2+: \(t_{2g}^6 e_g^0\). Consequently, there are 5 unpaired electrons in Mn 2+, 2 unpaired electrons in Ru 2+, and 2 unpaired electrons in Rh 2+ octahedral complexes.

Step-by-step solution

Determine the atomic numbers

To find the electronic configuration, we need to know the atomic numbers of the given elements. The atomic numbers can be found using a periodic table. For Mn (manganese), the atomic number is 25. For Ru (ruthenium), the atomic number is 44. For Rh (rhodium), the atomic number is 45.

Write the electronic configuration

Using the atomic numbers, determine the electronic configurations of the atoms and their 2+ ions: Mn (atomic number 25): Electronic configuration: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^5\) Mn 2+ ion: Remove 2 electrons from 4s orbital: \(1s^2 2s^2 2p^6 3s^2 3p^6 3d^5\) Ru (atomic number 44): Electronic configuration: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^6\) Ru 2+ ion: Remove 2 electrons from 5s and 4d orbitals: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 4d^6\) Rh (atomic number 45): Electronic configuration: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^1 4d^8\) Rh 2+ ion: Remove 2 electrons from 5s and 4d orbitals: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 4d^6\)

Draw the crystal-field energy diagram

For an octahedral complex in a strong field, the d orbitals split into two energy levels: \(t_{2g}\) (lower energy - 3 orbitals) and \(e_g\) (higher energy - 2 orbitals). Now, let's fill the d electrons in these levels for each ion: Mn 2+: 5 d electrons: \(t_{2g}^5 e_g^0\) Ru 2+: 6 d electrons: \(t_{2g}^6 e_g^0\) Rh 2+: 6 d electrons: \(t_{2g}^6 e_g^0\)

Determine the number of unpaired electrons

The number of unpaired electrons can be found from the crystal-field energy diagram: Mn 2+: \(t_{2g}^5 e_g^0\) - 5 unpaired electrons Ru 2+: \(t_{2g}^6 e_g^0\) - 2 unpaired electrons Rh 2+: \(t_{2g}^6 e_g^0\) - 2 unpaired electrons So, in the octahedral complexes of Mn 2+, Ru 2+, and Rh 2+ ions, there are 5, 2, and 2 unpaired electrons, respectively, in strong-field complexes.

Key concepts

Atomic Numbers

Every element in the periodic table is assigned a unique number known as the atomic number. This number represents the number of protons found in the nucleus of an atom of that element. For instance, manganese (Mn) has an atomic number of 25, meaning it has 25 protons. Similarly, ruthenium (Ru) has an atomic number of 44, and rhodium (Rh) has an atomic number of 45. The atomic number is crucial because it determines the identity of the element and also dictates the electronic configuration of the atom, which in turn influences its chemical behavior. Understanding atomic numbers is the first step in writing electronic configurations, especially when transitioning from the atom to its ionic form.

Octahedral Complex

An octahedral complex refers to a coordination compound where six ligands symmetrically surround a central metal ion. This arrangement forms an octahedron shape, which is a common geometry in transition metal chemistry. In octahedral complexes, the central metal ion can interact with the surrounding ligands in such a way that it influences the distribution of the d electrons. It's important to note that the octahedral shape can impact the energy levels of the d orbitals, leading to what is called crystal-field splitting. Understanding octahedral complexes helps in predicting magnetic properties and electronic transitions of metal ions, especially in the context of strong-field or weak-field ligands.

Crystal-Field Energy Levels

When a metal ion is surrounded by ligands in an octahedral complex, the d orbitals split into two energy levels due to crystal-field splitting. This results from the electrostatic interaction between the metal ion and the ligands.

• The lower energy set, called the \( t_{2g} \) level, includes three orbitals.
• The higher energy set, known as the \( e_g \) level, consists of two orbitals.

For strong-field ligands, the energy gap between \( t_{2g} \) and \( e_g \) is large, causing the d electrons to fill the lower \( t_{2g} \) level first. This affects the arrangement of electrons and can alter properties, such as color and magnetism of the complex.

Unpaired Electrons

Unpaired electrons refer to electrons in an atom or ion that do not have a partner with opposite spin in the same orbital. These electrons play a significant role in determining the magnetic properties of a substance. In the context of crystal-field theory and octahedral complexes:

• Mn 2+ ion has 5 unpaired d electrons because its configuration in a strong field is \( t_{2g}^5 e_g^0 \).
• Ru 2+ and Rh 2+ ions each have 2 unpaired d electrons with the configuration \( t_{2g}^6 e_g^0 \).

Generally, the presence of unpaired electrons indicates paramagnetism, where the substance is attracted to a magnetic field. Conversely, a lack of unpaired electrons results in diamagnetism, where the substance is slightly repelled by a magnetic field. Understanding the number of unpaired electrons helps chemists predict and explain the magnetic behavior of different materials.