Question

Arrange the following species in order of increasing stability: \(\mathrm{Li}_{2}, \mathrm{Li}_{2}^{+}, \mathrm{Li}_{2}^{-}\). Justify your choice with a molecular orbital energy level diagram.

Short answer

The order of increasing stability is \(\mathrm{Li}_{2}^{-}\), \(\mathrm{Li}_{2}^{+}\), \(\mathrm{Li}_{2}\).

Step-by-step solution

Understand Molecular Orbital Theory

To determine the stability of diatomic molecules and ions, we need to use Molecular Orbital (MO) Theory. This involves examining the electrons in bonding and antibonding orbitals and calculating the bond order.

Determine Electron Configuration for Each Species

For each lithium species, calculate the total number of electrons:- \(\mathrm{Li}_{2}\) has 2 electrons.- \(\mathrm{Li}_{2}^{+}\) has 1 electron (a positive charge indicates 1 less electron).- \(\mathrm{Li}_{2}^{-}\) has 3 electrons (a negative charge indicates 1 extra electron).

Fill Molecular Orbitals

In lithium molecules, electrons fill the \(\sigma_{1s}\) and \(\sigma_{1s}^{*}\) orbitals. For each species:- \(\mathrm{Li}_{2}\): Two electrons fill the \(\sigma_{1s}\) orbital.- \(\mathrm{Li}_{2}^{+}\): One electron in the \(\sigma_{1s}\) orbital.- \(\mathrm{Li}_{2}^{-}\): Two electrons in \(\sigma_{1s}\) and one in \(\sigma_{1s}^{*}\).

Calculate Bond Order

The bond order can be calculated as \(\text{Bond Order} = \frac{1}{2}(\text{Number of electrons in bonding MO} - \text{Number of electrons in antibonding MO})\).- For \(\mathrm{Li}_{2}\): \(\text{Bond Order} = \frac{1}{2}(2-0) = 1\).- For \(\mathrm{Li}_{2}^{+}\): \(\text{Bond Order} = \frac{1}{2}(1-0) = 0.5\).- For \(\mathrm{Li}_{2}^{-}\): \(\text{Bond Order} = \frac{1}{2}(2-1) = 0.5\).

Arrange by Stability

The stability of a molecule is proportional to its bond order. Higher bond order means more stability. Thus, the order of increasing stability is \(\mathrm{Li}_{2}^{-}\), \(\mathrm{Li}_{2}^{+}\), \(\mathrm{Li}_{2}\).

Key concepts

Electron Configuration

Electron Configuration is a fundamental concept in chemistry that describes how electrons are distributed in molecular orbitals of atoms or ions. To understand this, you often start by calculating the total number of electrons present. This total is influenced by the charge of the species:

• A neutral molecule, like \(\mathrm{Li}_2\), will have all its electrons contributed by the two lithium atoms.
• A positively charged ion (like \(\mathrm{Li}_{2}^{+}\)) has one fewer electron, as the charge indicates loss of an electron.
• Conversely, a negatively charged ion (such as \(\mathrm{Li}_{2}^{-}\)) has an additional electron added due to its negative charge.

Once the total number of electrons is determined, they must be filled into the molecular orbitals according to their energy levels. This is governed by the Aufbau principle, starting from the lowest energy level. For simple molecules like lithium, the involved orbitals are primarily the \(\sigma_{1s}\) and \(\sigma_{1s}^{*}\) orbitals.
In the case of \(\mathrm{Li}_{2}\), both electrons fill the \(\sigma_{1s}\) orbital, making it stable. For \(\mathrm{Li}_{2}^{+}\), just one electron is present, whereas \(\mathrm{Li}_{2}^{-}\) has electrons both in the bonding and antibonding orbitals.

Bond Order

In molecular orbital theory, Bond Order plays a crucial role in understanding the strength and stability of a bond between atoms. It can be calculated using the formula:\[\text{Bond Order} = \frac{1}{2}(\text{Number of electrons in bonding MO} - \text{Number of electrons in antibonding MO})\]The Bond Order is a measure of the number of electron pairs holding two atoms together. Generally, a higher bond order means a stronger, more stable bond. Let's consider the lithium species:

• For \(\mathrm{Li}_{2}\): Two electrons are in the bonding \(\sigma_{1s}\) orbital, and none are in the antibonding \(\sigma_{1s}^{*}\). This results in a bond order of 1.
• For \(\mathrm{Li}_{2}^{+}\): A single electron resides in the \(\sigma_{1s}\) orbital, giving a bond order of 0.5.
• In \(\mathrm{Li}_{2}^{-}\), two electrons occupy the bonding orbital and one the antibonding, also yielding a bond order of 0.5.

The Bond Order directly influences molecular stability, as seen here.

Molecular Stability

The stability of a molecule is intricately linked to its Bond Order. Higher bond orders generally indicate more stable molecules due to the greater number of electron pairs keeping the atoms bound tightly together. Let’s explore why this matters:

• A molecule with zero bond order does not exist because there is no net bonding force.
• Higher bond orders suggest more overlap in atomic orbitals.
• Greater overlap means more energy is needed to break the bond, increasing the molecule's stability.

Considering the lithium species discussed, \(\mathrm{Li}_{2}\) with a bond order of 1, is the most stable form. This means the diatomic molecule is energetically more favorable compared to its charged counterparts. Both \(\mathrm{Li}_{2}^{+}\) and \(\mathrm{Li}_{2}^{-}\) have lesser stability with a bond order of 0.5, implying they are less likely to exist or persist under standard conditions. Therefore, the stability in increasing order is \(\mathrm{Li}_{2}^{-}\), \(\mathrm{Li}_{2}^{+}\), \(\mathrm{Li}_{2}\). This order highlights the critical role bond order plays in molecular stability.