Draw an MO energy-level diagram for beryllium metal, and show the population of the MOs for the following two cases. (a) The 2s and 2p bands are well separated in energy. (b) The 2s and 2p bands overlap in energy. Which diagram agrees with the fact that beryllium has a high electrical conductivity? Explain.

Verified step by step guidance

1. Identify the electron configuration of a single beryllium atom, which is 1s^2 2s^2. Focus on the valence electrons (2s^2) for constructing the MO diagram.
2. For case (a), where the 2s and 2p bands are well separated: Draw two separate energy levels for the 2s and 2p orbitals. The 2s orbital will be lower in energy compared to the 2p orbital. Fill the 2s orbital with the two electrons from each beryllium atom, indicating that the 2s band is completely filled and the 2p band is empty.
3. For case (b), where the 2s and 2p bands overlap: Draw the 2s and 2p orbitals closer in energy, indicating overlap. Electrons can now partially occupy both the 2s and 2p orbitals, forming a mixed band that may not be completely filled, allowing for some electron mobility.
4. Compare the electrical conductivity implications of both cases. In case (a), with a completely filled 2s band and an empty 2p band, there are no available states for electrons to move to under an electric field, suggesting lower conductivity. In case (b), the overlap allows some electrons to be in the 2p band, providing available states for movement under an electric field, suggesting higher conductivity.
5. Conclude which case agrees with the high electrical conductivity of beryllium. Since beryllium is known for its good electrical conductivity, case (b) with overlapping 2s and 2p bands, allowing electron mobility, is more consistent with this property.

Key Concepts

Molecular Orbital Theory

Molecular Orbital (MO) Theory describes how atomic orbitals combine to form molecular orbitals that can be occupied by electrons. In this theory, electrons are delocalized over the entire molecule rather than being confined to individual atoms. Understanding MO diagrams is crucial for predicting the electronic structure and properties of materials, including conductivity.

Energy Bands in Solids

In solid-state physics, energy bands are ranges of energy levels that electrons can occupy. The 2s and 2p bands refer to the energy levels associated with the s and p orbitals of the atoms in a solid. The separation or overlap of these bands influences the material's electrical properties, such as conductivity, with overlapping bands typically allowing for easier electron movement.

Electrical Conductivity

Electrical conductivity is the ability of a material to conduct electric current, which depends on the availability of free charge carriers (electrons or holes). In metals like beryllium, high conductivity is often associated with overlapping energy bands, allowing electrons to move freely under an applied electric field. This concept is essential for understanding why certain MO diagrams correlate with high conductivity.