# Rotational energies

1. May 11, 2005

### Pengwuino

Ok so i have 5 wavelengths of the rotational spectrum of a certain molecule. I need to find the moment of inertia.

I have the equation down to I=(Hbar * wavelength)/(hc)

Do i just use the shortest wavelength to figure out the moment of intertia? No radius was given.

2. May 13, 2005

### ehild

How did you get this formula? Check it. The unit of I should be mass times length squared and you have "second".

ehild

3. May 13, 2005

### Pengwuino

The book gave me the equation and i figured what 'I' would be. Its an HCl module molecule.

4. May 13, 2005

### Staff: Mentor

I don't know where that equation comes from--the units don't make sense.

Treating the HCL molecule as a rigid rotor, the allowable rotational energy levels are:
$$E_J = J (J +1) \frac {\hbar^2}{2I}$$

You should be able to relate the wavelengths to transistions between levels.

5. May 13, 2005

### ehild

To Pendwuino:

It has to be known that only such transitions are allowed where J changes by +1 or -1. If you have absorption spectrum the spectrum lines correspond to the transitions from J to J+1. In an emission spectrum, it is the opposite, the molecule emits a photon while it gets back from the J+1-th rotational level to the J-th one.

The energy of a the photon emitted is

$$hf=((J+2)(J+1)-J(J+1))\frac{\hbar^2}{2I}=(J+1)\frac{\hbar^2}{I}$$
The emission spectrum of a two-atomic molecule consists of equidistant spectral lines, which correspond to transitions on to the levels J=0, J=1...and so on. The difference between the frequencies of two closest lines is
$$\Delta f = \frac{h}{4\pi^2I}$$
You know the wavelength of the spectral lines. Calculate the frequencies from the wavelengths
$$f=c/\lambda$$. Sort the frequencies and calculate the difference between the subsequent ones. Take the average: and calculate I from it.

ehild