Problem 62 - 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 \(3+\) ion: (a) Fe, (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?

Step-by-Step Solution

Answer

The electronic configurations of the 3+ ions are: Fe3+: \(3d^5\); Mo3+: \(4d^2\); Co3+: \(3d^6\). In an octahedral weak-field complex, the d-orbitals are filled as follows: Fe3+: \(t_{2g}^5 e_g^0\), Mo3+: \(t_{2g}^2 e_g^0\), and Co3+: \(t_{2g}^6 e_g^0\). There are no unpaired electrons in Fe3+ and Co3+, while Mo3+ has 2 unpaired electrons.

Step 1: (Step 1: Find the electronic configuration of each element)

First, we'll write the electronic configuration for each metal atom: (a) Fe: Z=26, electronic configuration: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^6\) (b) Mo: Z=42, electronic configuration: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^1 4d^5\) (c) Co: Z=27, electronic configuration: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^7\)

Step 2: (Step 2: Determine the electronic configuration of each element's 3+ ion)

Now, we'll find the electronic configuration of each 3+ ion: (a) Fe3+: \(1s^2 2s^2 2p^6 3s^2 3p^6 3d^5\) (b) Mo3+: \(1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 4d^2\) (c) Co3+: \(1s^2 2s^2 2p^6 3s^2 3p^6 3d^6\)

Step 3: (Step 3: Draw the crystal-field energy-level diagram for the d-orbitals of an octahedral complex)

For an octahedral complex, the d-orbitals split into two categories: \(e_g\) orbitals (dx²-y² and d~z²~) at higher energy levels, and \(t_{2g}\) orbitals (dxy, d~x~z, and d~y~z) at lower energy levels. In a weak-field complex, the energy gap between these orbitals isn't significant, so the electrons will occupy low-energy orbitals following Hund's Rule, prioritizing maximum unpaired electrons filling the orbitals in order.

Step 4: (Step 4: Show the placement of d electrons for each 3+ ion, assuming a weak-field complex)

We'll now fill the d-electrons based on the weak-field energy split in an octahedral complex: (a) Fe3+: Since we have 5 d-electrons, they will completely fill the \(t_{2g}\) orbitals, leaving 0 unpaired electrons. The electron configuration for Fe3+ will be: \(t_{2g}^5 e_g^0\) (b) Mo3+: Having 2 d-electrons, they will fill the two lowest-energy orbitals in the \(t_{2g}\) set. The electron configuration for Mo3+ will be: \(t_{2g}^2 e_g^0\) (c) Co3+: With 6 d-electrons, they will occupy all the orbitals in the \(t_{2g}\) set and one electron will be in the \(e_g\) set. The electron configuration for Co3+ will be: \(t_{2g}^6 e_g^0\)

Step 5: (Step 5: Find the number of unpaired electrons in each ion)

Lastly, we'll identify any unpaired electrons present in each 3+ ion: (a) Fe3+: 0 unpaired electrons (b) Mo3+: 2 unpaired electrons (c) Co3+: 0 unpaired electrons Hence, there are no unpaired electrons in Fe3+ and Co3+, while the Mo3+ ion has 2 unpaired electrons.

Key Concepts

Electronic Configuration

Electronic configuration is a fundamental concept that helps us understand how electrons are distributed across the different shells and subshells in an atom. Each electron in an atom occupies a specific region in space called an orbital, and the distribution is based on specific principles:

• Aufbau Principle: Electrons occupy the lowest energy orbital available.
• Pauli Exclusion Principle: No two electrons can have the same set of four quantum numbers, meaning an orbital can hold a maximum of two electrons with opposite spins.
• Hund’s Rule: Electrons will fill degenerate orbitals (orbitals with the same energy) singly first, and only pair up when all these orbitals are half-filled.

The electronic configuration of transition metals often involves filling the 3d orbitals after the 4s orbital. For example, iron (Fe) with atomic number 26 has the configuration: \[ 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^6 \]This configuration changes when the metal forms an ion. To form a 3+ ion, iron loses three electrons, primarily from the higher energy 4s and then the 3d orbitals, resulting in:\[ 1s^2 2s^2 2p^6 3s^2 3p^6 3d^5 \] for Fe\(^{3+}\). This step adjusts how the electrons fill the d orbitals and affects the electronic properties of the ion.

Transition Metals

Transition metals are elements found in the d-block of the periodic table, specifically in groups 3 to 12. These metals are unique due to their ability to form variable oxidation states by losing different numbers of electrons from their s and d orbitals.Some key characteristics of transition metals include:

• Variable Oxidation States: Transition metals can form several stable oxidation states. For example, iron can form both Fe\(^{2+}\) and Fe\(^{3+}\) ions.
• Colored Compounds: Many compounds of transition metals are colored due to d-d electron transitions. These transitions are influenced by the metal's oxidation state and the nature of the ligands surrounding it.
• Formation of Complexes: Transition metals can form complex compounds with ligands due to the presence of vacant d orbitals that can accept electron pairs.

Each transition metal's electronic configuration can change dramatically once it forms a complex, especially when transitioning to oxidation states like 3+, as seen in the elements such as Fe, Mo, and Co. The original electron configuration of these metals needs to be adjusted according to their ion form and the crystalline field effects.

Unpaired Electrons

Unpaired electrons in an atom or ion strongly influence its magnetic properties. These are electrons in orbitals that do not have a paired electron with the opposite spin. The presence of unpaired electrons causes materials to exhibit paramagnetism—a condition where materials are attracted to external magnetic fields.To determine the unpaired electrons in d-block transition metals, we can utilize the crystal field theory, especially when they form complexes:

• Crystal Field Splitting: In an octahedral complex, the five d-orbitals split into two energy levels: three lower-energy orbitals (\(t_{2g}\): d\(_{xy}\), d\(_{xz}\), d\(_{yz}\))and two higher-energy orbitals (\(e_{g}\): d\(_{x^2-y^2}\), d\(_{z^2}\)).
• Weak Field Complexes: In weak field complexes, the electron pairing energy is higher compared to the splitting energy; electrons fill all lower energy orbitals singly before pairing starts. For example, in Fe\(^{3+}\), all five d electrons fill the lower energy \(t_{2g}\) orbitals, leading to no unpaired electrons.

Understanding the placement of electrons in these split d-orbitals, especially for ions like Fe\(^{3+}\), Mo\(^{3+}\), and Co\(^{3+}\), is crucial to predicting and explaining the behavior of these ions in different chemical environments.