We have to show that in a diatomic molecule consisting of two atoms of equal mass m separated by a distance d and rotating about an axis, the energy levels can be written as

(b) We have to calculate the energies of the lowest three levels of the molecule, for which the two atoms are 121 pm apart. The mass of the oxygen atom is 16.0 u.

Solution:             Click For PDF Version

(a)

Classically the energy of this system in terms of its angular momentum L can be expressed as

According to the quantization of angular momentum

Therefore, the energy values of the quantum rotator are

(b)

We will calculate next using this model the ground state energy in eV of the molecule with and .

We will use the following data and values:

The ground state of the molecule will correspond to . Substituting the above data, we get

The energies of the lowest three levels of the molecule will therefore be