Number of IR Active Vibrational Normal Modes of CO₂

CO₂ is a linear molecule consisting of three atoms, one carbon atom and two oxygen atoms. The vibrational normal modes of CO₂ can be determined using group theory analysis.

Group Theory Analysis

The point group of CO₂ is D_∞h, which has the following symmetry elements:

- Identity (E)
- C_∞ axis
- σ_v plane
- σ_h plane

Using group theory analysis, we can determine the irreducible representations of the vibrational normal modes and their corresponding symmetries.

Irreducible Representations

The vibrational normal modes of CO₂ are classified into three irreducible representations:

- Γ_1g (symmetric stretch)
- Γ_2u (bending)
- Γ_3g (antisymmetric stretch)

IR Active Modes

To determine the IR active modes, we need to apply the selection rule: Δμ ≠ 0, where Δμ is the change in dipole moment.

The dipole moment of CO₂ is zero in its equilibrium geometry. However, in the presence of an electric field, the molecule can undergo a temporary dipole moment, which can be detected by IR spectroscopy.

Using group theory analysis, we can determine the symmetry of the dipole moment for each irreducible representation. The dipole moments for the Γ_1g and Γ_3g modes are not zero, while the dipole moment for the Γ_2u mode is zero.

Therefore, the IR active modes of CO₂ are the Γ_1g and Γ_3g modes, which correspond to the symmetric and antisymmetric stretches, respectively.

Conclusion

In summary, CO₂ has two IR active vibrational normal modes: the symmetric stretch (Γ_1g) and the antisymmetric stretch (Γ_3g). The bending mode (Γ_2u) is IR inactive.