Problem 60 - Chemistry The Central Science

Question

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

Step-by-Step Solution

Answer

The electronic configurations for the 3+ ions of Ru, Mo, and Co are \([Kr]4d^5\), \([Kr]4d^3\), and \([Ar]3d^6\), respectively. Assuming a weak-field complex in an octahedral crystal-field energy-level diagram, the electron placements are: 3 in t₂g and 2 in e_g for Ru³⁺, all 3 in t₂g for Mo³⁺, and 3 in t₂g and 3 in e_g for Co³⁺. The number of unpaired electrons in each ion is: 1 for Ru³⁺, 1 for Mo³⁺, and 2 for Co³⁺.

Step 1: Determine Electronic Configurations of the Metal Atoms

Find the electronic configurations for Ru, Mo, and Co using the periodic table. Use the Aufbau principle, Pauli Exclusion principle, and Hund's rule to distribute electrons. 1. Ru (Z = 44): \([Kr]5s^24d^6\) 2. Mo (Z = 42): \([Kr]5s^14d^5\) 3. Co (Z = 27): \([Ar]4s^23d^7\)

Step 2: Determine Electronic Configurations for 3+ Ions

Determine the electronic configurations for Ru³⁺, Mo³⁺, and Co³⁺ by removing 3 electrons from their corresponding atom. 1. Ru³⁺: Remove 2 electrons from 5s orbital and 1 electron from 4d orbital: \([Kr]4d^5\) 2. Mo³⁺: Remove 1 electron from 5s orbital and 2 electrons from 4d orbital: \([Kr]4d^3\) 3. Co³⁺: Remove 2 electrons from 4s orbital and 1 electron from 3d orbital: \([Ar]3d^6\)

Step 3: Draw Crystal-Field Energy-Level Diagram for the d Orbitals of Octahedral Complex

Draw the crystal-field energy-level diagram for the d orbitals of an octahedral complex and split the d orbitals into two sets: t₂g orbitals (lower energy) and e_g orbitals (higher energy). Under weak-field condition, electrons preferentially occupy the lower-energy orbitals (t₂g) before filling up the higher-energy orbitals (e_g).

Step 4: Place d Electrons for 3+ Ions in the Diagram

Place the electrons in the energy-level diagram as per the configurations in Step 2, using the weak-field assumption. 1. Ru³⁺ (\([Kr]4d^5\)): 5 d electrons ⇒ 3 in t₂g and 2 in e_g 2. Mo³⁺ (\([Kr]4d^3\)): 3 d electrons ⇒ all 3 in t₂g (no electron in e_g) 3. Co³⁺ (\([Ar]3d^6\)): 6 d electrons ⇒ all 3 in t₂g and 3 in e_g

Step 5: Count Unpaired Electrons

Count the number of unpaired electrons in each metal-ion's d orbitals. 1. Ru³⁺: 1 unpaired electron in the e_g set 2. Mo³⁺: 1 unpaired electron in the t₂g set 3. Co³⁺: 2 unpaired electrons in the e_g set So, there are 1, 1, and 2 unpaired electrons in the octahedral complexes of Ru³⁺, Mo³⁺, and Co³⁺, respectively.

Key Concepts

Crystal Field Theory

Crystal Field Theory (CFT) helps us understand how the \(d\) orbitals of transition metals are affected when surrounded by ligands in a complex. This theory explains the splitting of degenerate \(d\) orbitals into different energy levels, depending on the geometric arrangement of the surrounding ligands.

In an octahedral complex, six ligands approach the central metal ion. This interaction increases the energy of the \(d\) orbitals not uniformly. The \(d_{xy}, d_{yz},\) and \(d_{zx}\) orbitals experience less repulsion and form the lower-energy \(t_{2g}\) set. On the other hand, the \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals face greater repulsion, forming the higher-energy \(e_g\) set.

Due to this energy differential, electron placement within these orbitals can lead to different numbers of unpaired electrons, impacting the magnetic properties of the complex. Moreover, the extent of splitting (the difference in energy between \(t_{2g}\) and \(e_g\)) depends on whether the complex is a weak-field or strong-field ligand, influencing electron arrangement.

Transition Metals

Transition metals are elements found in the center block of the periodic table. They include groups 3 to 12. Known for their ability to form various oxidation states, these metals have incompletely filled \(d\) subshells in their atoms or ions.

The unique electron configuration of transition metals results in interesting chemical and physical properties, such as:

• Variable oxidation states: They can lose different numbers of electrons, forming various ionic states.
• Colored compounds: Often provide vibrant colors due to electronic transitions between \(d\) orbitals.
• Formation of complexes: Tend to bind with ligands forming coordinated complexes.
• Good conductors of heat and electricity: Due to their electron configurations.

These properties make transition metals valuable in numerous industrial applications, ranging from construction materials to catalysts in chemical reactions. The study of these metals and their behavior is crucial for advancing technological development.

Unpaired Electrons

Unpaired electrons are electrons positioned singly in an orbital, not paired with another electron of opposite spin. In the context of Crystal Field Theory, the number of unpaired electrons can affect several properties of the metal complex, including its magnetic behavior.

When electrons occupy degenerate orbitals, they tend to fill each orbital singly before pairing up (a rule known as Hund's rule). The presence of unpaired electrons in a metal complex often leads to paramagnetism, where the compound is attracted to an external magnetic field. Conversely, if all electrons are paired, the complex will be diamagnetic, meaning it will be repelled by a magnetic field.

The specific number of unpaired electrons not only determines the magnetic nature but also often influences the overall reactivity and stability of the compound. By examining the electronic configuration, especially the distribution of \(d\) electrons, chemists can predict these properties in metal ions.