Question

How many electrons does it take to fill (a) a \(\sigma\) bonding MO; (b) a \(\pi\) antibonding \(\mathrm{MO} ;\) (c) the MOs formed from combination of the \(1 s\) orbitals of two atoms?

Short answer

Each \(\backslash sigma\) bonding MO or \(\backslash pi\) antibonding MO needs 2 electrons to be filled completely. The MOs from two \(1s\) orbitals hold 4 electrons total.

Step-by-step solution

- Understanding Molecular Orbitals (MOs)

Molecular orbitals (MOs) are formed by the combination of atomic orbitals. These can be bonding, antibonding, or nonbonding.

- Definition of \(\backslash sigma\) Bonding MO

A \(\backslash sigma\) bonding MO results from the head-on overlap of atomic orbitals along the axis connecting the nuclei. It can hold up to two electrons with opposite spins.

- Calculate Electrons for \(\backslash sigma\) Bonding MO

To fill a \(\backslash sigma\) bonding molecular orbital completely, it requires 2 electrons.

- Definition of \(\backslash pi\) Antibonding MO

A \(\backslash pi\) antibonding MO is formed by the side-by-side overlap of p orbitals. It also can host up to two electrons with opposite spins.

- Calculate Electrons for \(\backslash pi\) Antibonding MO

To fill a \(\backslash pi\) antibonding orbital completely, it requires 2 electrons.

- Combination of Two \(1s\) Orbitals

When two \(1s\) orbitals combine, they form one bonding \(\backslash sigma_{1s}\) and one antibonding \(\backslash sigma^{*}_{1s}\) molecular orbital.

- Calculate Electrons for Combination of Two \(1s\) Orbitals

Each \(\backslash sigma_{1s}\) or \(\backslash sigma^{*}_{1s}\) orbital can hold 2 electrons. Therefore, the MOs formed from the combination of \(1s\) orbitals of two atoms can collectively hold a total of 4 electrons (2 in the bonding \(\backslash sigma_{1s}\) and 2 in the antibonding \(\backslash sigma^{*}_{1s}\)).

Key concepts

sigma bonding MO

A \(\sigma\) bonding molecular orbital (MO) arises from the straightforward head-on overlap of atomic orbitals along the internuclear axis—the imaginary line that connects the nuclei of two bonding atoms. This type of molecular orbital is characterized by a cloud of electron density directly between the two nuclei, which acts to hold them together strongly.

The \(\sigma\) bonding MO can accommodate up to two electrons, which must have opposite spins—this is in line with the Pauli Exclusion Principle. These electrons are shared between the two atoms and contribute to the bond's overall strength, lowering the potential energy of the system and stabilizing the molecule.

• The \(\sigma\) bonding orbital is paramount in single bonds, such as the bond found in diatomic hydrogen (H₂).
• To completely fill a \(\sigma\) bonding MO, it requires exactly 2 electrons.

pi antibonding MO

A \(\pi\) antibonding MO results from the side-by-side overlap of p orbitals, leading to a molecular orbital with electron density situated above and below the internuclear axis. Unlike \(\sigma\) orbitals, \(\pi\) orbitals involve lateral or sideways interactions.

In the \(\pi\) antibonding MO, there is a node—a region of zero electron density—directly between the two nuclei. This node results in an area of repulsion between the two atoms, which means that placing electrons in this orbital tends to destabilize the molecule.

• Similar to the \(\sigma\) bonding MO, the \(\pi\) antibonding MO can hold a maximum of 2 electrons, which must have opposite spins.
• Therefore, to fully occupy a \(\pi\) antibonding MO, 2 electrons are needed.

1s orbitals

The combination of two \(1s\) orbitals from two atoms results in two distinct molecular orbitals: one bonding and one antibonding. Specifically, these are the \(\sigma_{1s}\) (bonding \(\sigma\) orbital) and \(\sigma^{*}_{1s}\) (antibonding \(\sigma\) orbital).

Each of these orbitals can hold up to 2 electrons with opposite spins.

• The bonding \(\sigma_{1s}\) orbital results from the constructive interference of the two \(1s\) orbitals, leading to increased electron density between the nuclei and a stable bond.
• The antibonding \(\sigma^{*}_{1s}\) orbital, on the other hand, originates from the destructive interference, creating a node between the nuclei and causing instability if occupied.

Consequently, the molecular orbitals (MOs) formed from the combination of \(1s\) orbitals of two atoms can collectively hold up to 4 electrons: 2 in the bonding \(\sigma_{1s}\) orbital and 2 in the antibonding \(\sigma^{*}_{1s}\) orbital.