Problem 67 - Chemistry The Molecular Science

Question

Use MO theory to predict the MO diagram, the number of bonds, and the number of unpaired electrons in (a) peroxide ion, \(\mathrm{O}_{2}^{2-}\) (b) \(\mathrm{B}_{2}^{+}\) (c) \(\mathrm{Li}_{2}^{+}\) (d) \(\mathrm{O}_{2}^{+}\)

Step-by-Step Solution

Answer

(a) Bond Order 1, 0 unpaired; (b) Bond Order 1.5, 1 unpaired; (c) Bond Order 0.5, 0 unpaired; (d) Bond Order 1.5, 1 unpaired.

Step 1: Understand Molecular Orbital Theory

Molecular Orbital (MO) Theory is used to describe the electronic structure of molecules by considering individual atomic orbitals combining to form molecular orbitals, which can hold two electrons of opposite spins. These molecular orbitals are ranked by energy, and electrons fill them according to the Aufbau principle and Hund’s Rule, similar to filling atomic orbitals.

Step 2: Construct MO Diagram for Peroxide Ion, \(\mathrm{O}_2^{2-}\)

For \(\mathrm{O}_2^{2-}\), start with the normal \(\mathrm{O}_2\) MO diagram. \(\mathrm{O}_2\) has 12 valence electrons, and the 2 additional electrons from the \(2-\) charge occupy the next available MO. This fills the antibonding \(\pi^*_{2p}\) orbitals (4 electrons).

Step 3: Analyze Bond Order for \(\mathrm{O}_2^{2-}\)

Bond order is calculated as: \(\text{Bond Order} = \frac{1}{2}(\text{number of bonding electrons} - \text{number of antibonding electrons})\). For \(\mathrm{O}_2^{2-}\), bonding electrons = 8, antibonding electrons = 6, so Bond Order = 1.

Step 4: Identify Unpaired Electrons for \(\mathrm{O}_2^{2-}\)

\(\mathrm{O}_2^{2-}\) has all paired electrons in its MO diagram, resulting in 0 unpaired electrons.

Step 5: Construct MO Diagram for \(\mathrm{B}_2^{+}\)

\(\mathrm{B}_2\) has an electron configuration up to \(\pi_{2p}\) orbitals with 6 valence electrons. Removing 1 electron from the neutral \(\mathrm{B}_2\) results in 5 electrons filling the molecular orbitals.

Step 6: Analyze Bond Order for \(\mathrm{B}_2^{+}\)

In \(\mathrm{B}_2^{+}\), bonding electrons = 4 and antibonding electrons = 1. Bond Order = \(\frac{1}{2}(4-1) = 1.5\).

Step 7: Identify Unpaired Electrons for \(\mathrm{B}_2^{+}\)

There is 1 unpaired electron in the \(\sigma_{2p}\) orbital in \(\mathrm{B}_2^{+}\).

Step 8: Construct MO Diagram for \(\mathrm{Li}_{2}^{+}\)

\(\mathrm{Li}_2\) has an electron configuration \(\sigma_{1s}^2 \sigma_{1s}^*2, \sigma_{2s}^2\) with 6 valence electrons. \(\mathrm{Li}_2^{+}\) loses 1 electron, resulting in 5 electrons.

Step 9: Analyze Bond Order for \(\mathrm{Li}_{2}^{+}\)

Bonding electrons = 3, antibonding electrons = 2, so Bond Order = \(\frac{1}{2}(3-2) = 0.5\).

Step 10: Identify Unpaired Electrons for \(\mathrm{Li}_2^{+}\)

All electrons in \(\mathrm{Li}_2^{+}\) are paired, leading to 0 unpaired electrons.

Step 11: Construct MO Diagram for \(\mathrm{O}_2^{+}\)

For \(\mathrm{O}_2\), subtract 1 electron for the \(+\) charge, leaving 11 electrons. Fill the MO diagram by removing 1 electron from the antibonding \(\pi^*_{2p}\) orbitals.

Step 12: Analyze Bond Order for \(\mathrm{O}_2^{+}\)

Components: 8 bonding and 5 antibonding electrons. Bond Order = \(\frac{1}{2}(8-5) = 1.5\).

Step 13: Identify Unpaired Electrons for \(\mathrm{O}_2^{+}\)

With 11 electrons in the \(\mathrm{O}_2^{+}\) MO diagram, there is 1 unpaired electron in the \(\pi^*\) orbitals.

Key Concepts

MO Diagram

Molecular Orbital Diagrams (MO Diagrams) provide a visual representation of the molecular orbital theory. They illustrate how atomic orbitals from individual atoms combine to form molecular orbitals for the entire molecule. Here's how to understand them:

• Atomic orbitals on the left and right of the diagram represent each atom in the molecule.
• Molecular orbitals are depicted in the center, showing bonding and antibonding interactions.
• Bonding orbitals are lower in energy compared to antibonding orbitals.
• Electrons are filled into these orbitals following the Aufbau principle, which prioritizes lower energy orbitals first, and Hund's rule, which maximizes unpaired electrons in equal energy orbitals.

These diagrams are essential for predicting the number of bonds and unpaired electrons in molecular species. For instance, in step 2 of the original solution, the MO diagram for the peroxide ion (\( \mathrm{O}_2^{2-} \)) starts with the configuration of normal oxygen, which contains 12 valence electrons. Adding the charge results in the excess electrons occupying the higher energy antibonding orbitals.

Bond Order

Bond order is a concept in molecular orbital theory that provides an idea of the stability and strength of a bond between atoms in a molecule. It can be determined using the formula:
\[\text{Bond Order} = \frac{1}{2}(\text{number of bonding electrons} - \text{number of antibonding electrons})\]A higher bond order typically indicates a stronger, more stable bond. For example:

• Peroxide ion (\( \mathrm{O}_{2}^{2-} \)), has 8 bonding and 6 antibonding electrons, giving a bond order of 1.
• In \( \mathrm{B}_{2}^{+} \), the bond order is calculated from 4 bonding electrons and 1 antibonding electron, resulting in a bond order of 1.5, indicating more stability and bond strength compared to peroxide ion.

Understanding bond orders help in assessing molecular stability, which is an important aspect of chemical reactivity and structure.

Unpaired Electrons

Unpaired electrons are electrons in a molecule’s molecular orbitals that do not have a pair with opposite spin. These are important because they can affect the chemical and magnetic properties of a molecule. Unpaired electrons result in a molecule being paramagnetic, which means it is attracted to a magnetic field.
In the case studies from the original exercise:

• The peroxide ion (\( \mathrm{O}_{2}^{2-} \)) has all paired electrons, hence it is diamagnetic (not attracted to a magnetic field).
• \( \mathrm{B}_{2}^{+} \) shows one unpaired electron, indicating it is paramagnetic due to its unpaired electron in the \( \sigma_{2p} \) orbital.
• Similarly, \( \mathrm{O}_{2}^{+} \) has one unpaired electron in its \( \pi^* \) orbitals, resulting in paramagnetism.

Recognizing unpaired electrons is crucial for predicting chemical interactions and understanding the magnetic nature of molecules.