Band theory of solids | Class 12 (India) | Physics | Khan Academy

Key Concepts

• Pauli's Exclusion Principle
• Atomic Orbitals
• Molecular Orbitals
• Energy Levels
• Bonding Orbital
• Antibonding Orbital
• Energy Bands
• Energy Continuum
• Band Theory of Solids

From Atomic Orbitals to Molecular Orbitals

• Pauli's Exclusion Principle: No two electrons in an atom can have the same set of quantum numbers (identical energy levels).
• Single Atom Configuration: In a single atom (e.g., Sodium with 11 electrons), electrons fill discrete energy levels according to Pauli's Exclusion Principle.
• Two Atoms Close Together: When two atoms approach each other, their atomic orbitals overlap.
• Molecular Orbital Formation: Overlapping atomic orbitals transform into molecular orbitals, which are unique energy levels for the entire molecule.
• Example: Two 1s atomic orbitals (one from each Sodium atom) combine to form a 1s molecular orbital.
• Number of Energy Levels: The number of energy levels in a molecular orbital equals the number of overlapping atomic orbitals. Two atomic orbitals result in two energy levels within the molecular orbital.
• Bonding and Antibonding Orbitals: The lower energy level is called the bonding orbital, and the higher energy level is called the antibonding orbital (denoted with a star, e.g., 1s*).
• Pauli's Exclusion Principle Compliance: Electrons from both atoms can now occupy the two energy levels within the molecular orbital without violating Pauli's Exclusion Principle.

Expanding to Multiple Atoms and Solids

• Three Atoms: If three atoms are close together, three atomic orbitals overlap, resulting in a molecular orbital with three energy levels.
• Solid Formation: As more atoms are added (e.g., forming a solid), the molecular orbital has more energy levels.
• Solid Example: A solid made of Sodium with 10^23 atoms will have a molecular orbital with 10^23 energy levels.
• Energy Continuum: With a vast number of energy levels (10^23), the gaps between them become extremely tiny, effectively creating a continuous range of available energies.
• Energy Bands: This continuous range of energies is called an energy band. The word "band" signifies that the small gaps between the individual energy levels are ignored, and the entire range is treated as one continuous energy.
• Examples: 1s band, 2s band, 3s band, 3p band.

Energy Bands and Electron Capacity

• N Atoms and N Levels: If there are N atoms, there will be N energy levels within the corresponding energy band.
• Electron Capacity: Each energy level can hold two electrons. Therefore, an energy band with N levels can hold 2N electrons.
• P Level Capacity: A single p level can hold six electrons. Therefore, the 2p band can hold 6N electrons.

Gases vs. Solids: Discrete Levels vs. Energy Bands

• Gases: In gases, atoms are far apart and behave like single atoms. Electrons have discrete energy levels, and transitions between levels require a "jump."
• Solids: In solids, atoms are close together, forming energy bands. Within these bands, electrons can possess any continuous energy.

Band Theory of Solids

• Definition: The concept of energy bands and continuous energy levels in solids is known as the Band Theory of Solids.
• Application: This theory explains how free electrons are generated and why some materials are conductors (with readily available free electrons) while others are not.

Synthesis/Conclusion

The video explains how the interaction of atomic orbitals leads to the formation of energy bands in solids. Starting with the Pauli Exclusion Principle and the behavior of electrons in single atoms, it progresses to show how overlapping atomic orbitals create molecular orbitals with multiple energy levels. As the number of atoms increases (approaching solid-state conditions), these energy levels become so numerous that they form continuous energy bands. This Band Theory of Solids is crucial for understanding the electrical properties of materials, particularly the availability of free electrons for conduction.