# Quantum Chemistry/Example 31

Write an example problem related to the conversion between wavelength, frequency, energy and wavenumber of electromagnetic radiation. Show each step in detail and explain the conversion.

Example: Carbon monoxide shows a sharp peak in its IR spectrum at 2143 cm^{-1}.

a) Calculate the frequency and wavelength of the molecule's vibration.

The question gives us the wavenumber which can be converted to frequency through the equation:

where v is the frequency, ṽ is the wavenumber and c is the speed of light.

(2143cm^{-1}) * (2.99792458*10^{10}cm s^{-1})

= 6.42455237494*10^{13} s^{-1}

= 64.24 THz

To find the wavelength of the vibration we can use the equation:

=

where is the wavelength.

=

This equation can be rearranged to solve for wavelength:

=

=

= 4.666356*10^{-4} cm

This should be converted to nm since wavelength is usually written in nm.

λ = 4.666356*10^{-4} cm *

= 4.6*10^{3}nm

b) Calculate the energy of carbon monoxide at its ground state.

To find the energy of a molecule at its ground state we can use the equation:

where E is the energy, h is Planck's constant, is the frequency of the ground state, and is the quantum number of the energy level.

We can use this equation since carbon monoxide's vibrations act as a harmonic oscillator.

c) Calculate the wavelength of a photon that excites an electron from the ground state up 2 levels.

In this case, the quantum number v is equal to 2. We can use the same equation used in b) to first solve the change in energy caused by the photon.

Now that we have found the change in energy we can calculate the wavelength of the photon using the equation:

Rearrange the equation to solve for λ:

Convert the wavelength to nm.

Therefore, a photon that is able to cause a transition from ground state to v = 2 must have a wavelength of 1.86 * 10^{-6} nm.