Solving Bohr Atom Problem (Qs a-e)

[StillAnotherDave]

Homework Statement

Help with parts (c) and (e)

Relevant Equations

Given in the screenshot

Hello folks,

I've managed questions (a) and (b) but don't get what to do with part (c). Normally you would equate the velocity equations v²=e²/4πεmr=(n²h²)/(m²r² ). This let's you isolate the radius and use it to calculate E_n. But I can't see how you could do this for v given in (a) and (b).

Also, part (e). If E_n = E₁n then the energy levels simply scale linearly ... that can't be right?

 

[StillAnotherDave]

Okay. I managed to solve part (c) by isolating r².

But part (e) is still unclear ...

 

[ehild]

Okay. I managed to solve part (c) by isolating r².

But part (e) is still unclear ...

Remember, this is not a "real"atom where electric force acts between the proton and electron. You derived that En = E_in .Are those energy levels equidistant?

 

[StillAnotherDave]

They would be linear (equidistant) and positive - i.e. n=1 is 1eV; n=2 is 2eV; n=3 is 3eV ... is that correct?

But wouldn't n = ∞ need to be 0eV?

It makes the remaining parts of the question (f) and (g) very confusing:

 

[StillAnotherDave]

Clearly this is ludicrous ... help!

 

[DrClaude]

They would be linear (equidistant) and positive - i.e. n=1 is 1eV; n=2 is 2eV; n=3 is 3eV ... is that correct?

Yes.

But wouldn't n = ∞ need to be 0eV?

The choice of zero of energy is arbitrary. For the hydrogen atom, the ionization limit (separated particles) is often chosen as the 0, but this is arbitrary. One could also take the ground state to be 0.

 

[StillAnotherDave]

Okay ... so for question (f), could photons not be emitted for any energy multiple of E₁? And how would the ionisation energy be determined?

 

[DrClaude]

Okay ... so for question (f), could photons not be emitted for any energy multiple of E₁?

Yes.

And how would the ionisation energy be determined?

How is it determined for the hydrogen atom?

 

[StillAnotherDave]

The lowest energy level, n=1 at E₁ is given as 1.0 eV ... so that would be the ionisation energy?

 

[StillAnotherDave]

Okay ... so for question (f), could photons not be emitted for any energy multiple of E1?

Yes.

But this doesn't seem to make sense for a 3 mark question ...

 

[DrClaude]

The lowest energy level, n=1 at E₁ is given as 1.0 eV ... so that would be the ionisation energy?

No. Again, how is the ionization energy calculated in hydrogen?

But this doesn't seem to make sense for a 3 mark question ...

I'm not responsible the grading scheme . I guess some explanation is expected.

 

[StillAnotherDave]

Ionisation works by removing an electron from it's most outer bound shell ... but if the energy is increasing as E_n=E₁n then essentially you're saying you can't remove the electron ... ?

 

[StillAnotherDave]

As far as I understand ionization energy, it depends on which state the electron is in.

 

[DrClaude]

You're almost there, so I'll give you the answer. The ionization energy is the difference in energy between the last bound state and the ground state. Here, you have
$$
\lim_{n\rightarrow \infty} E_n = \infty
$$
so it is impossible to remove the electron.

By the way, you are basically creating a model for the harmonic oscillator, which indeed has no limit in energy for bound states.
https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator