Question

Draw a molecular orbital energy level diagram for each of these species: \(\mathrm{He}_{2}, \mathrm{HHe}, \mathrm{He}_{2}^{+} .\) Compare their relative stabilities in terms of bond orders. (Treat HHe as a diatomic molecule with three electrons.)

Short answer

The relative stabilities in terms of bond orders are lowest for \(\mathrm{He}_{2}\) (bond order: 0), and highest for \(\mathrm{HHe}\) and \(\mathrm{He}_{2}^{+}\) (both with bond order: 0.5). \(\mathrm{HHe}\) is likely unstable due to large difference in electronegativities. However, under normal conditions, \(\mathrm{He}_{2}^{+}\) would be the most stable.

Step-by-step solution

Draw the molecular orbital diagram for each species

Molecular orbital diagrams for \(\mathrm{He}_{2}, \mathrm{HHe}, \mathrm{He}_{2}^{+}\) can be drawn by taking into consideration that all these are essentially diatomic molecules. The molecular orbitals are formed by overlap of atomic orbitals. In the case of \(\mathrm{He}_{2}\), two 1s orbitals from two helium atoms overlap to form molecular orbitals- one bonding orbital (lower energy) and one antibonding orbital (higher energy). It has two electrons in both the bonding and antibonding orbitals. For \(\mathrm{HHe}\), one 1s orbital from hydrogen combines with one 1s orbital from helium to create bonding and antibonding molecular orbitals. This molecule has two electrons in the bonding orbital and one in the antibonding orbital. Lastly, for \(\mathrm{He}_{2}^{+}\), the process is similar to \(\mathrm{He}_{2}\), but here there is only one electron in the antibonding orbital, because this positive ion has one less electron.

Calculate bond orders

The bond order can be calculated by taking the difference of the number of electrons in bonding and antibonding orbitals, divided by 2. \(\mathrm{He}_{2}\) has a bond order of 0 ([2-2]/2), which infers no bond. \(\mathrm{HHe}\) has a bond order of 0.5 ([2-1]/2) and \(\mathrm{He}_{2}^{+}\) also has a bond order of 0.5 ([2-1]/2). The higher the bond order, the more stable the molecule.

Determine their relative stabilities

The bond order gives an idea about the stability of molecules. Greater the bond order, greater is the bond strength and hence the stability. \(\mathrm{He}_{2}\) has a bond order of 0 which infers that this molecule is not stable. \(\mathrm{HHe}\) and \(\mathrm{He}_{2}^{+}\) both have bond orders of 0.5 which infers that they have similar stabilities. However, \(\mathrm{HHe}\) would not exist in nature under normal conditions due to the large difference in electronegativities of hydrogen and helium.

Key concepts

Diatomic Molecules

Diatomic molecules consist of two atoms, which may be either of the same or different chemical elements. These two atoms bind together to form a discrete, stable unit. They are held by chemical bonds which involve the sharing or transfer of electrons from one atom to another. The simplest and most well-known diatomic molecules are the diatomic elements, like hydrogen (H₂), oxygen (O₂), and nitrogen (N₂). Each molecule's electron arrangement is central to understanding its chemical properties, and the molecular orbital energy level diagram is a key tool for visualizing this.

In the MO energy level diagram, the atomic orbitals of each atom are shown to merge into a set of molecular orbitals, where the electrons are no longer assigned to individual atoms but are considered to be shared between them. The resulting molecular orbitals are categorized into bonding orbitals, which stabilize the molecule due to the attraction between the nuclei and shared electrons, and antibonding orbitals, which have the opposite effect because they contain electrons that are found mostly in areas outside the space between the nuclei.

For instance, in the exercise solution, we see different diatomic molecular configurations such as He₂, HHe, and He₂⁺. Each exhibits a unique set up in their molecular orbital diagrams according to the number and types of constituent atoms and the number of electrons each has to contribute to the bonding and antibonding orbitals.

Bond Order

The bond order is a concept in molecular orbital theory that reflects the number of chemical bonds between a pair of atoms. In a simple sense, a higher bond order suggests a stronger, more stable bond. It is calculated by subtracting the number of electrons in antibonding molecular orbitals (ABMO) from the number of electrons in bonding molecular orbitals (BMO), and then dividing the result by two.

Using the formula \[\begin{equation}Bond\text{ }Order = \frac{(Number\text{ }of\text{ }electrons\text{ }in\text{ }BMO) - (Number\text{ }of\text{ }electrons\text{ }in\text{ }ABMO)}{2},\text{we can determine the bonding characteristics of diatomic molecules.}\end{equation}\]For example, a bond order of 0, as calculated for He₂, suggests no bond is present, which implies that a He₂ molecule has no stability and would not exist under normal conditions. In contrast, both HHe and He₂⁺ have a bond order of 0.5, indicating a single weak bond, which makes these species more stable than He₂, albeit still relatively unstable in general chemical contexts.

Molecular Stability

Molecular stability refers to the tendency of a molecule to remain in its current state without reacting or breaking down. A molecule's stability is influenced by several factors, including electron configuration, bond strength, and the presence of full or half-full orbitals. The bond order is directly proportional to the bond's strength; thus, it serves as a good indicator of molecular stability. Generally, molecules with higher bond orders are more stable.

In molecular orbital theory, stability can also be evaluated by the distribution of electrons in bonding versus antibonding molecular orbitals. Molecules with a greater number of electrons in bonding orbitals compared to antibonding orbitals will have a positive bond order and are typically more stable. In the exercise's context, He₂ was found to have a bond order of 0, which implies instability as the electrons completely negate each other's stabilizing effects being equal in number in bonding and antibonding orbitals. On the other hand, both HHe and He₂⁺ had a bond order of 0.5 each, signifying marginal stability. Nonetheless, it's critical to note that factors like atomic size, electronegativity differences, and the overall energy of the system also impact the thermodynamic feasibility and actual existence of these species.