Question

the lowest frequency of a rotaional absorption 14N 16O is the transition from J=0 to J=1 , it is a microwave transition and occurs at v=102.216 GHz(this is the technnical frequency v not the angular frequency w)

a) rotational spectra are often used to find the exact geometries of molecules. Lets do that for simple case by finding the bond distance in teh molecule in A

b) Predict the transition energies for J=1 to J=2 TO j=3 IN CM^-1

Answer 1

In pure rotational spectroscopy for a simple diatomic molecule, the energy levels are given by

E_J = BJ(J+1)

where J is the rotational quantum number,

B is the rotational constant for the particular molecule given by B = h² / 8π²I in Joules, where I is the moment of inertia, given by I = μr² - where r is the bond length of this particular diatomic molecule and μ is the reduced mass, given by μ = m₁m₂ / m₁ + m₂.

The rotation constant, B of a molecule is characteristic to the structure of molecule, and independent of J.

From the rotational frequency one can get force constant B and using the line spacing the bond length can be estimated.

The greater the polarity in molecule the higher its moment of inertia and B becomes smaller. If the velocity of light is given in cm/s units the unit of B is cm^-1.

Most energy level transitions in spectroscopy are governed by selection rules. In pure rotational spectroscopy, the selection rule is ΔJ = ±1. Based on the selection rule, the energy gap between each level increases by 2B as the energy levels we consider increase by J = 1. This leads to the line spacing of 2B in the spectrum.