Normal Modes of Vibration in Benzene Molecule:

Benzene is a planar molecule with six carbon atoms and six hydrogen atoms. The carbon atoms are arranged in a hexagonal ring with alternating double bonds. The normal modes of vibration in the benzene molecule can be determined using group theory and the concept of symmetry.

1. Symmetry Elements of Benzene Molecule:

The benzene molecule has several symmetry elements, including:

- C6 rotation axis: A six-fold rotation axis passing through the center of the ring
- C2 rotation axis: A two-fold rotation axis perpendicular to the plane of the ring
- Six C2 axes of symmetry: Each bisects the angle between two adjacent C-C bonds
- Mirror planes: Perpendicular to the C6 axis and bisecting the C-C bonds

2. Character Tables and Irreducible Representations:

Using group theory, we can construct a character table for the benzene molecule based on its symmetry elements. The character table lists the symmetry operations and their corresponding irreducible representations.

The irreducible representations are labeled with Greek letters, such as Γ, Σ, and Π. Each mode of vibration in the molecule corresponds to an irreducible representation.

3. Normal Modes of Vibration:

The benzene molecule has 30 normal modes of vibration, which can be classified into the following irreducible representations:

- 6 A1 modes: In-phase stretching of all C-C bonds
- 6 A2 modes: Out-of-phase stretching of adjacent C-C bonds
- 12 E modes: Combination of in-phase and out-of-phase stretching of C-C bonds and bending of C-H bonds
- 6 B2 modes: In-phase bending of the ring

Each normal mode of vibration has a characteristic frequency, which can be measured experimentally using techniques such as infrared spectroscopy or Raman spectroscopy.

Conclusion:

In summary, the benzene molecule has 30 normal modes of vibration, which can be determined using group theory and the concept of symmetry. These modes correspond to different combinations of in-phase and out-of-phase stretching and bending of the C-C and C-H bonds in the molecule. The normal modes of vibration can be used to study the structural and electronic properties of the molecule.