Question

Use molecular orbital theory to explain why the \(\mathrm{Be}_{2}\) molecule does not exist

Short answer

According to Molecular Orbital Theory, after filling up the Molecular Orbital diagram for Be2, the bond order comes out to be 0 which indicates that no bond forms between the atoms. Hence, Be2 molecule does not exist.

Step-by-step solution

Discuss Be Atomic Orbitals

Beryllium (Be) has an atomic number of 4. Therefore, its electronic configuration is 1s² 2s². For Be2, the outermost orbitals of Be atoms (2s orbitals) overlap to form Molecular Orbitals.

Construct the MO diagram

In the MO diagram for this molecule, 2s orbitals from each Be atom combine to form two molecular orbitals, one bonding (marked as \(\sigma_{2s}\)) and one antibonding (marked as \(\sigma_{2s}^{*}\)). In a simple molecular orbital diagram like this, the bonding orbitals are placed below the corresponding antibonding orbitals.

Filling of the MOs

Two electrons each from both Be atoms, totaling 4 electrons, are to be filled in the molecular orbitals now. The first two electrons will go to the \(\sigma_{2s}\) orbital. The next two electrons will fill the \(\sigma_{2s}^{*}\) orbital.

Calculate Bond Order

The bond order is given by the formula: \[Bond Order = 0.5×(Number of electrons in bonding orbitals – Number of electrons in antibonding orbitals)\], Substituting in the values we obtained, \[Bond order = 0.5×(2-2) = 0\]. A bond order of 0 means that no bond forms between the atoms, indicating that the Be2 molecule does not exist.

Key concepts

Beryllium molecule

The Beryllium molecule, denoted as \( \mathrm{Be}_2 \), is a theoretical model that helps us understand molecular interaction using Molecular Orbital (MO) Theory. A single beryllium atom has an atomic number of 4, resulting in an electronic configuration of \( 1s^2 2s^2 \). When considering the potential molecule \( \mathrm{Be}_2 \), each beryllium atom contributes its 2s orbitals to form potential molecular orbitals through the overlap of these atomic orbitals.
This overlap concept is crucial because it provides a framework for how electrons can be shared or paired across the atoms in a molecule. However, with \( \mathrm{Be}_2 \), this molecular formation goes further to show the possible interaction—or lack thereof—between paired electrons in the outermost orbit of the beryllium atoms. This theoretical model allows us to visualize and predict why some molecules might not form, as in the case for \( \mathrm{Be}_2 \) where atomic interactions result in an unstable structure.
The non-existence of the beryllium molecule serves as an example of how MO Theory provides insights into why certain diatomic molecules are unstable.

Bond Order

In Molecular Orbital Theory, bond order is a key aspect that helps predict the stability of a molecule. Bond order provides a quantitative measure of the bonding between a pair of atoms. It is calculated using the formula:

• \[ \text{Bond Order} = 0.5\times(\text{Number of electrons in bonding orbitals} - \text{Number of electrons in antibonding orbitals}) \]

For the potential \( \mathrm{Be}_2 \) molecule, filling the molecular orbitals according to the rules results in both the \( \sigma_{2s} \) bonding and the \( \sigma_{2s}^{*} \) antibonding orbitals being filled with two electrons each.
This is significant because it means that the number of electrons in bonding molecular orbitals equals the number in antibonding orbitals, resulting in a bond order of zero. A bond order of zero suggests no net bonding interactions between the atoms, indicating that the \( \mathrm{Be}_2 \) molecule does not bind together effectively, making it not viable as a stable molecule.

Molecular Orbitals Diagram

The Molecular Orbitals (MO) Diagram is a simplified visual representation that explains how atomic orbitals combine to form molecular orbitals in a molecule. For \( \mathrm{Be}_2 \), the 2s orbitals from each Be atom merge to create two distinct molecular orbitals: a bonding orbital (\( \sigma_{2s} \)) and an antibonding orbital (\( \sigma_{2s}^{*} \)).
In these diagrams, bonding orbitals are typically shown lower in energy compared to antibonding orbitals. This lower energy indicates a more stable state for electrons filling these orbitals. Meanwhile, antibonding orbitals are at a higher energy level, reflecting an unstable electron state when occupied.
However, in \( \mathrm{Be}_2 \), both the \( \sigma_{2s} \) and \( \sigma_{2s}^{*} \) orbitals contain two electrons. This kind of electron filling pattern results in a bond order calculation that equals zero, as all the electron stabilization is negated by the destabilization of electrons in the antibonding orbital. As such, while the MO Diagram for \( \mathrm{Be}_2 \) elucidates the theoretical molecular interaction, it also highlights why this molecule doesn't occur naturally.