Atom Energy Levels & Absorption of Photons

[roeb]

Homework Statement

Say that an atom has the following energy levels:
E₁ = -9.20 eV
E₂ = -6.20 eV
E₃ = -3.70 eV
E₄ = -2.40 eV
E₅ = -1.60 eV

Suppose atoms in the E₂ level are exposed to incident radiation of wavelength 413.3 nm. Would this radiation be absorbed by atoms in the E₂ level?

I am looking for some clarification on this problem if possible...

Homework Equations

The Attempt at a Solution

E_2->3 = -2.5 eV
E_2->4 = -3.8 eV

E = hf = 1240/413.3 = 3 eV (radiation energy)

The atom starts out in the n = 2 energy level. According to my professor, the atom would NOT absorb the radiation because it's not exactly 2.5 eV (to go up to n = 3).

1) Why is it that the photon has enough energy to go up to n = 3 but the atom won't absorb the photon? Is it just because of quantization of energy? It seems strange to me that it wouldn't absorb.


My second question is if the wavelength is 496 nm then the photon will be absorbed by the n = 2 atom. Then the atom could emit wavelengths from 3->1, 2->1, and 3->2.
3->1 = 5.5 eV
2->1 = 3 eV
3->2 = 2.5 eV

2) Why could the atom emit 3->1 since it takes 5.5 eV and the atom only absorbed 2.5 eV? As well as 2->1 takes 3 eV?

 

[Doc Al]

1) Why is it that the photon has enough energy to go up to n = 3 but the atom won't absorb the photon?

But you just showed that it doesn't have enough energy to get to n = 3!

2) Why could the atom emit 3->1 since it takes 5.5 eV and the atom only absorbed 2.5 eV? As well as 2->1 takes 3 eV?

The atom was already in an excited state above the ground state, so it has more than enough energy.

 

[borgwal]

There *is* enough energy to get to n=3, as roeb correctly noted. However, this would be a far off-resonant transition, meaning that the probability of the atom absorbing the photon is very very small: energy would not be conserved. Some small part of the excess energy could go to the kinetic energy of the atom, so it's not the full 3eV that the electron of the atom has to absorb, but the difference of 0.5eV is much too large.

 

[Doc Al]

There *is* enough energy to get to n=3, as roeb correctly noted.

Oops! I misread what roeb had written. You are correct.