Number of photons for multi-photon laser ionization

[Lion.P]

Homework Statement

The ionization potential of a nitrogen molecule is 15.5 eV.

How many photons from a beam of a 1054 nm high power laser must be absorbed by a nitrogen molecule simultaneously to cause multi-photon ionization?

Homework Equations

None were given.

The Attempt at a Solution

I wasn't too sure where to start

First I used E = hc/λ to find the energy of the beam:

E = (6.62608E-34 Js)(2.998E10 m/s)/(1054 nm)(1m/1E9nm) = 1.885 E-17 J

Next I converted the ionization potential to J/mol:

15.5 eV/atom * (1 kJ/mol)/(0.010364 eV/atom) * (1000 J/kJ) = 1.4956E6 J/mol

I tried dividing the two energies out and dividing by Avogadro number but that gives me a very small amount of photons & the units do not include photons. If anyone has an idea of where to look next or a hint I'd appreciate it.

Thanks

 

[anathema]

for your question! It looks like you're on the right track with your calculations. However, there are a few things you could consider to get a more accurate answer.

First, it's important to note that the ionization potential you have given is for a single nitrogen molecule, not a mole of nitrogen molecules. So, your calculation for the ionization energy per mole should actually be:

15.5 eV/atom * (1 kJ/mol)/(0.010364 eV/atom) * (1000 J/kJ) = 1.4956E6 J/mol

Also, keep in mind that the energy of a single photon is given by Planck's constant (h) times the frequency (ν), not the wavelength (λ). So, you should use the frequency of the laser beam (ν) to calculate the energy of a single photon:

E = h*ν = (6.62608E-34 Js)(2.998E10 m/s)/(1054 nm)(1m/1E9nm) = 5.945E-19 J

Now, to find the number of photons needed for multi-photon ionization, you can divide the ionization energy per mole by the energy of a single photon:

(1.4956E6 J/mol) / (5.945E-19 J/photon) = 2.517E24 photons/mol

Keep in mind that this is the number of photons needed for multi-photon ionization of a mole of nitrogen molecules. To find the number of photons needed for a single nitrogen molecule, you would need to divide by Avogadro's number (6.022E23 molecules/mol):

(2.517E24 photons/mol)/(6.022E23 molecules/mol) = 4.182 photons/molecule

So, to answer the original question, you would need approximately 4 photons from the high power laser beam to cause multi-photon ionization of a single nitrogen molecule.