Vibrational Frequency of a Diatomic Molecule

Vibrational levels of a diatomic molecule in terms of force constant and masses of atoms

Let us consider a diatomic molecule in which atoms having masses m₁ and m₂ oscillate against each other like a harmonic oscillator.

Let r_e be the equilibrium distance between two atoms and becomes r when stretched. Atoms shift by −x₁ to x₂ in two opposite directions so that the displacement-
x = r − r_e = −x₁ + x₂ = x₂ − x₁
The restoring force on each atom is proportional to displacement, x.
Restoring force constant = −k.x
where k is force constant.
If x₁ and x₂ be the distances of the center of the molecule from the centers of A and B atoms respectively. Then-

Therefore, the vibrational frequency (𝜈) of a diatomic molecule is related to the force constant (k) and reduced mass (µ).

The fundamental vibrational frequency of HCl is 2890 cm⁻¹. Calculate its force constant.

We know that-