In the last post, we concluded that only the vibrational and electronic energy of molecules, need to be considered while constructing a molecular energy diagram. In this post, we will study in detail, the molecular vibrations.

How does a molecule vibrate? How do transitions in vibrational energies occur? What rules apply to these transitions? Let us answer all these questions in the upcoming posts.

Vibrations in molecules – Harmonic or Anharmonic?

In post 170, we discussed harmonic oscillations. A harmonic oscillator is a system vibrating with an equilibrium position and experiencing a restorative force(F) proportional to the displacement(x), just like the motion of the spring.

F = -k x

When the potential energy of a molecule is plotted against the internuclear distance (or displacement), a harmonic oscillator produces a parabolic curve as shown below –

Vibrational levels in a harmonic oscillator.

Consider an imaginary molecule AB, which vibrates like a perfect harmonic oscillator. At the zero-point energy (indicated by O in the figure), the molecule is at equilibrium. At this point, the distance between the two molecules is the internuclear distance, r.

As the molecule absorbs energy, it starts vibrating like a spring. The internuclear distance(displacement,x) increases as the energy goes on increasing. With the increase in energy, the atoms start going farther apart and they are then pulled back by a restorative force (F). This force is directly proportional to the displacement (F ∝ x). Thus, we conclude that this system produces harmonic oscillations. The different vibrational energy levels are shown as v”(pronounced as v prime) series. As seen in the figure above, these vibrational energy levels are quantized i.e they take only certain discrete values of energy. For more info on quantization read post 23.

The lowest energy vibrational level is v”=0. This is the first vibrational energy level. From this point onwards, the energy increases, the displacement keeps on increasing too. This continues till a certain threshold level (v”=6), after which, if more energy is given to the molecule, it dissociates.

The harmonic vibrations give a perfect parabolic curve, with a potential well.

Note that the vibrational energy levels are equally spaced in the case of a harmonic oscillator.

The energy required to break a bond is called the bond dissociation energy(De). The subscript ‘e’ stands for equilibrium position. At this point, the potential energy is more than the bond dissociation energy. Thus, the bond between A and B ruptures. The bond dissociation energy is calculated from the zero-point energy till the point where the bond ruptures.

The harmonic vibrations give a perfect parabolic curve, with a symmetric potential well.

However, it is observed that molecules don’t behave as perfect harmonic oscillators. Thus, they are ANHARMONIC OSCILLATORS.

In the next post, we will study the anharmonic vibrations of molecules in greater detail. Till then,

Be a perpetual student of life and keep learning…

Good Day!

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