Question

For the following pairs of molecular orbitals, indicate the one you expect to have the lower energy, and state the reason for your choice. (a) \(\sigma_{1 s}\) or \(\sigma_{1 s}^{*} ;\) (b) \(\sigma_{2 s}\) or \(\sigma_{2 p}\) (c) \(\sigma_{1 s}^{*}\) or \(\sigma_{2 s} ;\) (d) \(\sigma_{2 p}\) or \(\sigma_{2 p}^{*}\)

Short answer

The orbitals with lower energy are \(\sigma_{1s}\), \(\sigma_{2s}\), \(\sigma_{2s}\), and \(\sigma_{2p}\). In each pair, the one without the '*' superscript has lower energy because anti-bonding orbitals (indicated by '*') have more energy. Among 's' and 'p' types, 's' usually has lower energy.

Step-by-step solution

Understand Molecular Orbits

Molecular orbitals are obtained by combining atomic orbitals, according to quantum physics. The type of atomic orbitals that create the molecular orbitals is indicated by the subscript (e.g., 's' or 'p'). The superscript '*' indicates that the orbital is an anti-bonding orbital with higher energy level. Therefore, for each pair of molecular orbitals, it can be judged which has lower energy based on these rules.

Compare The First Pair

Compare \(\sigma_{1s}\) and \(\sigma_{1s}^{*}\). The second one is an anti-bonding orbital, and the first one is not. Anti-bonding orbitals generally have higher energy than bonding orbitals. Therefore, \(\sigma_{1s}\) is assumed to have lower energy.

Compare The Second Pair

Compare \(\sigma_{2s}\) and \(\sigma_{2p}\). Both are molecular orbitals, but 's' orbitals usually have lower energy than 'p' orbitals. Therefore, \(\sigma_{2s}\) is assumed to have lower energy.

Compare The Third Pair

Compare \(\sigma_{1s}^{*}\) and \(\sigma_{2s}\). The first one is an anti-bonding orbital and the second one is not. Although the first one is of '1s' type and the second one is of '2s' type, the anti-bonding orbital will typically still have higher energy. Therefore, \(\sigma_{2s}\) is assumed to have lower energy.

Compare The Fourth Pair

Compare \(\sigma_{2p}\) and \(\sigma_{2p}^{*}\). The second one is an anti-bonding orbital, and the first one is not. Anti-bonding orbitals generally have higher energy than bonding orbitals. Therefore, \(\sigma_{2p}\) is assumed to have lower energy.

Key concepts

Anti-bonding Orbitals

In the realm of molecular orbitals, anti-bonding orbitals play a crucial role. Recognized by the asterisk (*) notation, these orbitals result from the destructive interference of atomic orbitals. When atomic orbitals combine in this unfriendly manner, they create an area of reduced electron density between the nuclei of two atoms, indicating higher energy. As a rule of thumb, anti-bonding orbitals always have more energy compared to their bonding counterparts.
When electrons occupy anti-bonding orbitals, they increase the overall energy of a molecule, thereby weakening or destabilizing the bond. For example, in the comparison between \(\sigma_{1s}\) and \(\sigma_{1s}^{*}\), the \(\sigma_{1s}^{*}\) orbital is an anti-bonding one, thus it has a higher energy.
Anti-bonding orbitals can significantly influence the stability and reactivity of molecules. Because of their higher energy, molecules with electrons in anti-bonding orbitals often exhibit unique properties and behavior. Understanding how these orbitals function is essential for grappling with molecular structure and reactivity in chemistry.

Bonding Orbitals

Bonding orbitals are fundamental when it comes to understanding molecular connections. Created through the constructive interference of atomic orbitals, bonding orbitals enable electron density to increase between atomic nuclei, leading to a stable configuration. These orbitals are usually at a lower energy level compared to anti-bonding orbitals.
In practical terms, when electrons inhabit bonding orbitals, they contribute to a stable bond, lowering the molecule's overall energy. This principle is evident in the case of \(\sigma_{1s}\) and \(\sigma_{1s}^{*}\) orbitals—the \(\sigma_{1s}\) orbital has lower energy as it is a bonding orbital.
Bonding orbitals not only depict a stronger connection between atoms but also provide insight into the geometric and electronic structure of molecules. They simplify the understanding of molecular symmetry, allowing scientists to predict physical properties and reactivity based on molecular orbital configurations.

Energy Levels

Understanding energy levels in molecular orbitals is key to predicting and explaining chemical behavior. Energy levels indicate the relative stability and behavior of orbitals. Typically, lower energy orbitals are deemed more stable and are thus filled first.
With molecular orbitals, the energy levels are determined by the type and nature of the orbital. For instance, 's' orbitals usually have lower energy than 'p' orbitals. This hierarchy becomes apparent when comparing orbitals such as \(\sigma_{2s}\) with \(\sigma_{2p}\), where the \(\sigma_{2s}\) is of a lower energy.
Moreover, energy levels are profoundly affected by whether orbitals are bonding or anti-bonding. Bonding orbitals, with their increased electron density between atoms, are lower in energy. Conversely, anti-bonding orbitals, marked by an asterisk, possess higher energy due to reduced electron density.
Knowing the arrangement of energy levels helps in deducing the electronic structure and predicting how a molecule might behave during chemical reactions.