Question

For each of the following metals, write the electronic configuration of the atom and its \(3+\) ion: (a) Ru, (b) Mo, (c) 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?

Short answer

The electronic configurations of the given transition metals and their 3+ ions are: - Ru: \([Kr] 4d^7 5s^1\), Ru³⁺: \([Kr] 4d^4\) - Mo: \([Kr] 4d^5 5s^1\), Mo³⁺: \([Kr] 4d^3\) - Co: \([Ar] 3d^7 4s^2\), Co³⁺: \([Ar] 3d^6\) For an octahedral weak-field complex, the d electrons have the following placements: - Ru³⁺: \(\uparrow \downarrow\) in first and second \(t_{2g}\) orbitals, empty in others. - Mo³⁺: \(\uparrow \downarrow\) in first \(t_{2g}\) orbital, \(\uparrow\) in second \(t_{2g}\) orbital, empty in others. - Co³⁺: \(\uparrow \downarrow\) in all \(t_{2g}\) orbitals and one \(e_g\) orbital. The number of unpaired electrons for each 3+ ion is: - Ru³⁺: 0 - Mo³⁺: 1 - Co³⁺: 2

Step-by-step solution

Electronic configuration of the given transition metals

We will use the periodic table to find the atomic numbers of each transition metal and write their electronic configurations. Ruthenium (Ru): - Atomic number: 44 - Electronic configuration: \([Kr] 4d^7 5s^1\) Molybdenum (Mo): - Atomic number: 42 - Electronic configuration: \([Kr] 4d^5 5s^1\) Cobalt (Co): - Atomic number: 27 - Electronic configuration: \([Ar] 3d^7 4s^2\)

Electronic configuration of the 3+ ions

Now let's find the electronic configurations for the 3+ ions. Three electrons are removed from the highest n and l value orbitals. Ruthenium 3+ (Ru³⁺): - Electronic configuration: \([Kr] 4d^4\) Molybdenum 3+ (Mo³⁺): - Electronic configuration: \([Kr] 4d^3\) Cobalt 3+ (Co³⁺): - Electronic configuration: \([Ar] 3d^6\)

Draw the crystal-field energy-level diagram for an octahedral complex

In an octahedral complex, the d orbitals are split into two sets: \(t_{2g}\) (lower energy) and \(e_g\) (higher energy). They are separated by an energy gap called Δ. Since we are considering weak-field complexes, the Δ is small, and the electrons will fill the orbitals according to the Aufbau's principle. The energy-level diagram for an octahedral complex is: ``` e_g ----- Δ ----- t_{2g} ```

Show the placement of the d electrons for each 3+ ion

We will fill the electrons in the energy-level diagram according to the Aufbau's principle and Hund's rule. Here, \(t_{2g}\) has three d orbitals while \(e_g\) has two d orbitals. Ruthenium 3+ (Ru³⁺): - 4 d electrons: \(\uparrow \downarrow\) in first \(t_{2g}\) orbital, \(\uparrow \downarrow\) in second \(t_{2g}\) orbital, and empty in others. Molybdenum 3+ (Mo³⁺): - 3 d electrons: \(\uparrow \downarrow\) in first \(t_{2g}\) orbital, \(\uparrow\) in second \(t_{2g}\) orbital, and empty in others. Cobalt 3+ (Co³⁺): - 6 d electrons: \(\uparrow \downarrow\) in all \(t_{2g}\) orbitals and \(\uparrow \downarrow\) in one \(e_g\) orbital.

Determine the number of unpaired electrons in each case

Now we will count the number of unpaired electrons from the energy-level diagrams that we obtained in step 4. Ruthenium 3+ (Ru³⁺): - Number of unpaired electrons: 0 Molybdenum 3+ (Mo³⁺): - Number of unpaired electrons: 1 Cobalt 3+ (Co³⁺): - Number of unpaired electrons: 2

Key concepts

Understanding Crystal-Field Theory

Crystal-field theory (CFT) is a model that describes how the arrangement of ligands around a central metal ion within a complex influences the energies of its d-orbitals. It is particularly useful in understanding the electronic structures and properties of transition metal complexes.

In an octahedral complex, which consists of six ligands symmetrically arranged around the central metal ion, the d-orbitals split into two distinct energy levels due to the electrostatic interactions between the d-orbitals of the metal ion and the negative charges on the ligands. The lower energy set is denoted as \(t_{2g}\) and consists of three orbitals (\(d_{xy}\), \(d_{xz}\), and \(d_{yz}\)), while the higher energy set is called \(e_g\) and contains two orbitals (\(d_{z^2}\) and \(d_{x^2-y^2}\)).

The extent of this splitting, designated as \(\Delta\), varies based on the field strength of the ligands. Ligands that cause a large splitting are known as strong-field ligands, and those that cause a smaller splitting are referred to as weak-field ligands. In weak-field complexes, the gap \(\Delta\) is relatively small, allowing electrons to occupy the higher energy \(e_g\) orbitals as per the Aufbau principle, which states that electrons fill orbitals starting with the lowest energy first.

Octahedral Complex Ions Formation

Octahedral complex ions are a common type of coordination complexes where a central metal ion is surrounded by six ligands at the vertices of an octahedron. The electronic configuration of transition metal ions plays a pivotal role in the formation and characteristics of these complexes.

When transitioning from the electronic configuration of a metal atom to that of a \(3+\) ion, electrons are typically removed from the s-orbitals before the d-orbitals due to their higher energy. After the removal of electrons to form a \(3+\) ion, the configuration is solely based on the d-orbitals. In the provided examples: Ruthenium (Ru) forms Ru\(3+\) with a \([Kr] 4d^4\) configuration, Molybdenum (Mo) forms Mo\(3+\) with a \([Kr] 4d^3\) configuration, and Cobalt (Co) forms Co\(3+\) with a \([Ar] 3d^6\) configuration.

These configurations are the starting point for understanding how the d-electrons will fill the \(t_{2g}\) and \(e_g\) orbitals in an octahedral crystal-field. The weak-field assumption means that the electrons will first fill the \(t_{2g}\) orbitals before pairing up or moving to the \(e_g\) level.

Determining Unpaired Electrons

Unpaired electrons are electrons in an atom or ion that are not paired with another electron in the same orbital. These unpaired electrons are significant in understanding the magnetic properties of the compound and are often determined using Hund's Rule and the principles of crystal-field theory.

According to Hund's Rule, orbitals are singly occupied as far as possible before electrons start to pair up. For transition metals and their ions, particularly those forming weak-field octahedral complexes, electrons will enter into the lower energy \(t_{2g}\) orbitals first. Once these are fully or half-filled, electrons may then occupy the higher energy \(e_g\) orbitals.

Taking the 3+ ions in the given exercise: Ruthenium (Ru\(3+\)) with a \([Kr] 4d^4\) configuration will have all electrons paired and no unpaired electrons. Molybdenum (Mo\(3+\)) with a \([Kr] 4d^3\) configuration will have one unpaired electron. Similarly, Cobalt (Co\(3+\)) with a \([Ar] 3d^6\) configuration will have two unpaired electrons. Understanding the placement of these unpaired electrons within the energy-level diagram is crucial for predicting the magnetic and spectral properties of these complexes.