Question

Draw the molecular orbital electron configuration energy diagram of \(\mathrm{O}_{2}\) and determine the bond order.

Short answer

The bond order of \(\mathrm{O}_{2}\) is 2.

Step-by-step solution

Determine the Number of Electrons

Oxygen (\(\mathrm{O}_{2}\)) has 8 electrons per atom, totaling 16 electrons for the molecule. We will be using these electrons to fill the molecular orbitals.

Sketch the Energy Levels for Molecular Orbitals

Begin by sketching an energy diagram with sigma \(\sigma\) orbitals at both the lowest (bonding) and highest (antibonding) positions, and pi \(\pi\) orbitals in between. The order from low to high is: \(\sigma_{1s}\), \(\sigma^{*}_{1s}\), \(\sigma_{2s}\), \(\sigma^{*}_{2s}\), \(\sigma_{2p_z}\), \(\pi_{2p_x}\) = \(\pi_{2p_y}\), \(\pi^{*}_{2p_x}\) = \(\pi^{*}_{2p_y}\), \(\sigma^{*}_{2p_z}\).

Fill the Molecular Orbitals with Electrons

Place the 16 total electrons into the molecular orbitals according to the order defined above. Fill each orbital according to Hund's rule, starting from the lowest energy level. The configuration is: \(\sigma_{1s}^{2}\sigma^{*}_{1s}^{2}\sigma_{2s}^{2}\sigma^{*}_{2s}^{2}\sigma_{2p_z}^{2}\pi_{2p_x}^{2}\pi_{2p_y}^{2}\pi^{*}_{2p_x}^{1}\pi^{*}_{2p_y}^{1}\).

Calculate the Bond Order

The bond order is calculated as \((\text{number of electrons in bonding orbitals} - \text{number of electrons in antibonding orbitals})/2\). In this case, there are 10 bonding electrons (\sigma_{1s}^{2}\sigma_{2s}^{2}\sigma_{2p_z}^{2}\pi_{2p_x}^{2}\pi_{2p_y}^{2}) and 6 antibonding electrons (\sigma^{*}_{1s}^{2}\sigma^{*}_{2s}^{2}\pi^{*}_{2p_x}^{1}\pi^{*}_{2p_y}^{1}), so the bond order is \((10 - 6)/2 = 2\).

Key concepts

Oxygen Molecule

Oxygen, commonly found in its diatomic form as \(\mathrm{O}_{2}\), is a fundamental component of the Earth's atmosphere, crucial for the survival of living organisms. In chemistry, understanding the structure of the oxygen molecule involves analyzing its molecular orbitals.
\(\mathrm{O}_{2}\) consists of two oxygen atoms bonded together, each contributing electrons to form molecular orbitals. This process involves combining atomic orbitals to create a more stable electronic structure. The study of these molecular orbitals provides insight into the molecule's properties, such as its bond stability and reactivity.

Bond Order Calculation

Bond order is a measure indicating the stability of a bond. It defines how many bonds exist between two atoms within a molecule. In a molecular orbital context, bond order can tell us about the strength and length of the bond.
To calculate the bond order, we use the formula: \[\text{Bond Order} = \frac{\text{Number of bonding electrons} - \text{Number of antibonding electrons}}{2}\] For \(\mathrm{O}_{2}\), it's necessary to identify which electrons are in bonding orbitals and which are in antibonding orbitals:

• Bonding electrons are found in lower energy bonding orbitals, such as \(\sigma_{1s}^{2}\,\sigma_{2s}^{2}\,\sigma_{2p_z}^{2}\,\pi_{2p_x}^{2}\,\pi_{2p_y}^{2}\).
• Antibonding electrons are in higher energy, destabilizing orbitals like \(\sigma^{*}_{1s}^{2}\,\sigma^{*}_{2s}^{2}\,\pi^{*}_{2p_x}^{1}\,\pi^{*}_{2p_y}^{1}\).

Plugging these into the formula gives us a bond order of 2, indicating a double bond between the two oxygen atoms.

Electron Configuration

Electron configuration refers to the distribution of electrons among the molecular orbitals. For \(\mathrm{O}_{2}\), we must fill the molecular orbitals in order of increasing energy, obeying the rules of quantum mechanics.
According to Hund's rule, electrons fill degenerate orbitals singly before pairing up to minimize repulsion. The electron configuration for \(\mathrm{O}_{2}\) is:

• \(\sigma_{1s}^{2}\) - Indicates two electrons in the \(\sigma_{1s}\) orbital.
• \(\sigma_{1s}^{*2}\) - Two electrons in the antibonding \(\sigma^{*}_{1s}\) orbital.
• \(\sigma_{2s}^{2}\) - Two electrons in the \(\sigma_{2s}\) orbital.
• \(\sigma_{2s}^{*2}\) - Two in the antibonding \(\sigma^{*}_{2s}\) orbital.
• \(\sigma_{2p_z}^{2}\) - Two electrons in the bonding \(\sigma_{2p_z}\) orbital.
• \(\pi_{2p_x}^{2} = \pi_{2p_y}^{2}\) - Two electrons in each degenerate \(\pi_{2p_x}\) and \(\pi_{2p_y}\) orbital.
• \(\pi_{2p_x}^{*1} = \pi_{2p_y}^{*1}\) - One electron in each degenerate antibonding \(\pi^{*}_{2p_x}\) and \(\pi^{*}_{2p_y}\) orbital.

This electron configuration is key for explaining the paramagnetic nature of \(\mathrm{O}_{2}\), resulting from unpaired electrons in the \(\pi^{*}\) orbitals.

Energy Diagram

A molecular orbital energy diagram is a visual representation that shows the energy levels of the orbitals in a molecule. In an energy diagram, we list molecular orbitals from lowest to highest energy. This arrangement helps us understand how electrons fill these orbitals.
For \(\mathrm{O}_{2}\), the energy levels can be drawn starting with the \(\sigma_{1s}\) bonding orbital at the lowest energy, followed by the \(\sigma^{*}_{1s}\) antibonding orbital slightly higher, \(\sigma_{2s}\), and \(\sigma^{*}_{2s}\).Next are the \(\sigma_{2p_z}\) bonding orbital and the degenerate \(\pi_{2p_x}\) and \(\pi_{2p_y}\) orbitals. Finally, at the highest energy, we have the \(\pi^{*}_{2p_x}\), \(\pi^{*}_{2p_y}\), and \(\sigma^{*}_{2p_z}\) antibonding orbitals.
These energy levels reflect the stability and strength of the bonds in the molecule. The more electrons a molecule has in lower-energy bonding orbitals, the more stable and less reactive it typically is.