Problem 16 - Chemistry and Chemical Reactivity

Question

Give the electron configurations for the ions \(\mathrm{Li}_{2}^{+}\) and \(\mathrm{Li}_{2}^{-}\) in molecular orbital terms. Compare the \(\mathrm{Li}-\mathrm{Li}\) bond order in these ions with the bond order in \(\mathrm{Li}_{2}\).

Step-by-Step Solution

Answer

\(\text{Li}_2\) has a bond order of 1, whereas \(\text{Li}_2^+\) and \(\text{Li}_2^-\) each have a bond order of 0.5.

Step 1: Understand Molecular Orbital Theory

The molecular orbital (MO) theory combines atomic orbitals to form molecular orbitals, which can be bonding, antibonding, or non-bonding. In diatomic molecules like \(\text{Li}_2\), we populate molecular orbitals according to the Aufbau principle following the order of increasing energy.

Step 2: Electron Configuration of \(\text{Li}_2\)

The lithium atom has electron configuration \([\text{He}] 2s^1\). In \(\text{Li}_2\), two lithium atoms combine, contributing a total of 2 electrons to the bonding interaction. These populate the \(\sigma(2s)\) bonding orbital, leading to the configuration: \((\sigma(2s))^{2}\).

Step 3: Calculate Bond Order of \(\text{Li}_2\)

The bond order is calculated using the formula: \(\text{Bond Order} = \frac{1}{2}(\text{Number of Bonding Electrons} - \text{Number of Antibonding Electrons})\). For \(\text{Li}_2\), there are 2 bonding electrons and 0 antibonding electrons, thus the bond order is \(\frac{1}{2}(2-0) = 1\).

Step 4: Electron Configuration of \(\text{Li}_2^+\)

\(\text{Li}_2^+\) involves the removal of one electron from \(\text{Li}_2\), resulting in one electron in the \(\sigma(2s)\) orbital: \((\sigma(2s))^{1}\).

Step 5: Calculate Bond Order of \(\text{Li}_2^+\)

With 1 bonding electron and 0 antibonding electrons, the bond order for \(\text{Li}_2^+\) is \(\frac{1}{2}(1-0) = 0.5\).

Step 6: Electron Configuration of \(\text{Li}_2^-\)

\(\text{Li}_2^-\) involves the addition of one electron to \(\text{Li}_2\), resulting in three electrons in the molecular orbitals. The configuration is \((\sigma(2s))^{2}(\sigma^*(2s))^{1}\).

Step 7: Calculate Bond Order of \(\text{Li}_2^-\)

With 2 bonding electrons and 1 antibonding electron, the bond order for \(\text{Li}_2^-\) is \(\frac{1}{2}(2-1) = 0.5\).

Step 8: Compare Bond Orders

\(\text{Li}_2 \) has a bond order of 1, while both \(\text{Li}_2^+\) and \(\text{Li}_2^-\) have a bond order of 0.5. This indicates that the Li-Li bond in \(\text{Li}_2\) is stronger than in \(\text{Li}_2^+\) or \(\text{Li}_2^-\).

Key Concepts

Electron Configuration

Electron configuration in the context of molecular orbital (MO) theory helps us understand how electrons are distributed in molecules. It involves arranging the electrons in specific orbitals, such as bonding or antibonding orbitals, according to principles like the Aufbau principle.
In

• \( \text{Li}_2 \): Both lithium atoms contribute one electron each, forming the configuration \((\sigma(2s))^{2}\), where both electrons occupy the bonding orbital.
• \( \text{Li}_2^+ \): One electron is removed, resulting in \((\sigma(2s))^{1}\). Here, one electron remains in the bonding orbital.
• \( \text{Li}_2^- \): An additional electron is added, leading to the configuration \((\sigma(2s))^{2}(\sigma^*(2s))^{1}\), where two electrons are in the bonding orbital, and one in the antibonding orbital.

These configurations illustrate how the addition or removal of electrons affects the distribution across molecular orbitals.

Bond Order

Bond order is an important concept in chemistry that tells us about the stability and strength of a bond. It is calculated using the formula: \[ \text{Bond Order} = \frac{1}{2}(\text{Number of Bonding Electrons} - \text{Number of Antibonding Electrons}) \]
For the lithium molecules:

• For \( \text{Li}_2 \), 2 bonding electrons and 0 antibonding electrons give a bond order of 1.
• \( \text{Li}_2^+ \) involves 1 bonding electron and 0 antibonding electrons, resulting in a bond order of 0.5.
• For \( \text{Li}_2^- \), with 2 bonding electrons and 1 antibonding electron, the bond order is 0.5.

A higher bond order indicates a stronger bond. Thus, \( \text{Li}_2 \) has the strongest bond among the species, compared to \( \text{Li}_2^+ \) and \( \text{Li}_2^- \).

Diatomic Molecules

Diatomic molecules consist of two atoms, which may be identical or different, forming a stable arrangement. In this context, we focus on lithium diatomic molecules like \( \text{Li}_2 \), \( \text{Li}_2^+ \), and \( \text{Li}_2^- \).

• These molecules are small and simple, allowing a clear illustration of concepts like molecular orbital theory.
• Diatomic molecules like \( \text{Li}_2 \) bond through overlapping atomic orbitals, filling molecular orbitals to achieve stability.
• By analyzing their bond orders and electron configurations, we gain insights into their stability and chemical behavior.

Diatomic molecules provide a foundation for understanding larger, more complex molecular interactions. They serve as a model for applying fundamental concepts in chemical bonding.