Problem 43 - Chemistry: The Central Science

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?

Step-by-Step Solution

Answer

The electronic configurations for the given atoms and their corresponding 2+ ions are as follows: - Mn: \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^5\) (atom), \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 3d^5\) (2+ ion) - Ru: \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 5s^2 \, 4d^7\) (atom), \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 4d^7\) (2+ ion) - Rh: \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 5s^2 \, 4d^8\) (atom), \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 4d^8\) (2+ ion) In an octahedral strong-field complex, the number of unpaired electrons are: - Mn 2+: 1 unpaired electron - Ru 2+: 1 unpaired electron - Rh 2+: 0 unpaired electrons

Step 1: Electronic Configuration of Mn Atom

The atomic number of Mn is 25. Thus, the electronic configuration of Mn is: \[1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^5\]

Step 2: Electronic Configuration of Mn 2+ Ion

To form the Mn 2+ ion, two electrons need to be removed. They are removed from the 4s orbital. Thus, the electronic configuration of Mn 2+ becomes: \[1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 3d^5\]

Step 3: Crystal-Field Energy-Level Diagram and Placement of d Electrons

In a strong-field complex with an octahedral geometry, \(d_{xy}, d_{yz}\), and \(d_{xz}\) orbitals are higher in energy, while \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals are lower in energy. Distribute 5 d electrons in the orbitals starting with lower energy ones: \[\begin{aligned} d_{z^2} &: \uparrow\downarrow \\ d_{x^2-y^2} &: \uparrow\downarrow \\ d_{xy} &: \uparrow \\ d_{xz} &: \\ d_{yz} &: \end{aligned}\]

Step 4: Number of Unpaired Electrons

From the energy-level diagram, we can see that there is 1 unpaired electron. (b) Ru

Step 5: Electronic Configuration of Ru Atom

The atomic number of Ru is 44. Thus, the electronic configuration of Ru is: \[1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 5s^2 \, 4d^7\]

Step 6: Electronic Configuration of Ru 2+ Ion

To form the Ru 2+ ion, two electrons need to be removed. They are removed from the 5s orbital. Thus, the electronic configuration of Ru 2+ becomes: \[1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 4d^7\]

Step 7: Crystal-Field Energy-Level Diagram and Placement of d Electrons

Distribute 7 d electrons in the orbitals, following the same order as in case of Mn: \[\begin{aligned} d_{z^2} &: \uparrow\downarrow \\ d_{x^2-y^2} &: \uparrow\downarrow \\ d_{xy} &: \uparrow\downarrow \\ d_{xz} &: \uparrow\downarrow \\ d_{yz} &: \uparrow \end{aligned}\]

Step 8: Number of Unpaired Electrons

From the energy-level diagram, we can see that there is 1 unpaired electron. (c) Rh

Step 9: Electronic Configuration of Rh Atom

The atomic number of Rh is 45. Thus, the electronic configuration of Rh is: \[1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 5s^2 \, 4d^8\]

Step 10: Electronic Configuration of Rh 2+ Ion

To form the Rh 2+ ion, two electrons need to be removed. They are removed from the 5s orbital. Thus, the electronic configuration of Rh 2+ becomes: \[1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 4d^8\]

Step 11: Crystal-Field Energy-Level Diagram and Placement of d Electrons

Distribute 8 d electrons in the orbitals, following the same order as in case of Mn: \[\begin{aligned} d_{z^2} &: \uparrow\downarrow \\ d_{x^2-y^2} &: \uparrow\downarrow \\ d_{xy} &: \uparrow\downarrow \\ d_{xz} &: \uparrow\downarrow \\ d_{yz} &: \uparrow\downarrow \end{aligned}\]

Step 12: Number of Unpaired Electrons

From the energy-level diagram, we can see that there are 0 unpaired electrons. In conclusion, the number of unpaired electrons for each metal's 2+ ion is as follows: - Mn 2+: 1 unpaired electron - Ru 2+: 1 unpaired electron - Rh 2+: 0 unpaired electrons

Key Concepts

Octahedral Complex

An octahedral complex refers to a coordination compound where six ligands are symmetrically arranged around a central transition metal ion. This structure forms an octahedron—imagine a shape with eight triangular faces. In such complexes, the transition metal cation is situated at the center while ligands occupy the corners of the octahedron.

In the context of transition metals, this geometric setup is particularly relevant because it influences the energy distribution of the metal's d orbitals. Different placements of ligands distend the d orbitals into distinct energy levels.

• Two of the d orbitals, namely, the \(d_{z^2}\) and \(d_{x^2-y^2}\), have lobes pointing directly along the ligand axes.
• These orbitals experience more repulsion and are pushed to higher energy levels.
• The remaining three orbitals—\(d_{xy}, d_{yz},\) and \(d_{xz}\)—align between the ligands. Thus, they experience less repulsion.

This separation and distinction in energies of the d orbitals are crucial in understanding electronic configurations and properties in coordination complexes.

Crystal-Field Theory

Crystal-field theory (CFT) is a model that describes the breaking of degeneracies of electronic orbitals, particularly notable in transition metal complexes. When transition metals form complexes, the surrounding ligands influence the energies of the metal's d orbitals, leading to a pattern of split energy levels. In an octahedral complex:

• The ligands create an electric field that affects the d orbital energies.
• This electric field causes the d orbitals to split into two sets; \(d_{z^2}\) and \(d_{x^2-y^2}\) are elevated in energy, while \(d_{xy}, d_{xz},\) and \(d_{yz}\) are lowered.

The degree of this splitting depends on the nature of the ligands and the specifics of the metal involved.

Understanding crystal-field splitting is vital as it determines factors like magnetic properties, colors of the complexes, and their electronic configurations. Whether an electron occupies the high-energy set of orbitals or fills the lower energy ones generates different levels of magnetic and spectral behavior.

Transition Metals

Transition metals are elements found in the d-block of the periodic table, known for their ability to form coordination complexes. Their defining feature is the incomplete filling of d subshells in one or more of their oxidation states.

A hallmark of transition metals is their electronic configuration, which allows them to display a wide range of oxidation states. This capability emerges due to shallow energy differences between their s and d orbitals.

• Their flexible electronic configurations support multiple bonding arrangements and complex formations with ligands.
• They form colorful compounds due to d-d transitions allowed by partly filled d orbitals.

Transition metals are significant in both biological systems and industrial applications. Their complexes often exhibit properties like variable oxidation states, paramagnetism, and catalytic functionalities, making them crucial for processes like oxygen transport in hemoglobin and catalysts in the Haber process for ammonia production.