Answer: Find Wavelength of H-atom Induced Radiation

[Saitama]

What I mean is: if you put in the equation E_n=\frac{R_H}{n^2} the values of E_n and R_H, is there an integer that fits well in n? If so, then the electron can get to the excited state, else it can't.
That's all.

What should be the value of E_n?

 

[I like Serena]

Is the highest energy level infinity?
If so, then the lowest possible induced wavelength is 91.17 nm.

But how would i calculate the possibilities for the electron falling back to a lower energy level and the corresponding induced wavelengths?

You seem to be missing something here, but I don't understand what it is you are missing.

The highest energy level is indeed infinity, but that level cannot be reached since there is not enough energy in the incoming photon.

Btw, I've been checking up on this problem.
As DiracRules stated earlier, the photon can only be absorbed if its energy matches one of the jump-energies of the electron (almost) exactly.
After that the most energetic photon that can be emitted is a photon of this same wavelength.
A less energetic photon would have a longer wavelength.

 

[Saitama]

You seem to be missing something here, but I don't understand what it is you are missing.

The highest energy level is indeed infinity, but that level cannot be reached since there is not enough energy in the incoming photon.

Btw, I've been checking up on this problem.
As DiracRules stated earlier, the photon can only be absorbed if its energy matches one of the jump-energies of the electron (almost) exactly.
After that the most energetic photon that can be emitted is a photon of this same wavelength.
A less energetic photon would have a longer wavelength.

In this question, electron cannot jump to the second level too since the energy of the incoming photon is 12.1 eV.
Am i right?

 

[I like Serena]

In this question, electron cannot jump to the second level too since the energy of the incoming photon is 12.1 eV.
Am i right?

Yep.
The electron will jump to the 3rd level.

 

[Saitama]

Yep.
The electron will jump to the 3rd level.

Why it cannot jump to infinity.

 

[I like Serena]

Why it cannot jump to infinity.

Not enough energy in the incoming photon.
It would need 2 incoming photons, but only one can be absorbed at a time, after which a photon is emitted, before a new photon is absorbed.

 

[Saitama]

But i am not able to understand why the answer is (a) option?

 

[I like Serena]

But i am not able to understand why the answer is (a) option?

An incoming photon is absorbed, making the electron jump from the first to the third energy level.
This electron falls back to the ground state and emits a photon of the same wavelength as the absorbed photon.
Doesn't that match with answer (a)?

 

[PeterO]

But i am not able to understand why the answer is (a) option?

Where is your global view? If this were a test paper question, you should be solving it in about 1 minute.

If an atom is excited to an upper level, the energy of the emitted photon could be the same as the energy of incoming radiation - if the drop to ground state is done in one step - or smaller, if the drop back goes via lower levels.

Those lower energy emissions have a longer wavelength, so the SHORTEST wavelength [that is what we were asked about in the question] is the same as the incoming radiation.

Option (a) is the nanometre equivalent of the Angstrom description of the incoming radiation - so would be the answer.

Note: the only other possibility was that the incoming radiation was not an exact match for one of the excited state - in which case it would be elastically scattered, and we would see option (a) for that reason.

To paraphrase the question:

"Did you know that the energy of a photon given off by an excited atom is no higher than the energy of the incoming radiation exciting the atoms?" - in combination with "Did you know that minimum wavelength corresponds to maximum energy?"

Other than recognising that the Angstrom wavelength corresponded to one of the nonometre wavlengths, no calculations were necessary in the question [as befits the idea that you should have completed the question in one minute]

 

[Saitama]

Sorry for the late reply but i thought i would first go through our discussion.

An incoming photon is absorbed, making the electron jump from the first to the third energy level.
This electron falls back to the ground state and emits a photon of the same wavelength as the absorbed photon.
Doesn't that match with answer (a)?

Why doesn't the electron jump to the second level?

Where is your global view? If this were a test paper question, you should be solving it in about 1 minute.

If an atom is excited to an upper level, the energy of the emitted photon could be the same as the energy of incoming radiation - if the drop to ground state is done in one step - or smaller, if the drop back goes via lower levels.

Those lower energy emissions have a longer wavelength, so the SHORTEST wavelength [that is what we were asked about in the question] is the same as the incoming radiation.

Option (a) is the nanometre equivalent of the Angstrom description of the incoming radiation - so would be the answer.

Note: the only other possibility was that the incoming radiation was not an exact match for one of the excited state - in which case it would be elastically scattered, and we would see option (a) for that reason.

To paraphrase the question:

"Did you know that the energy of a photon given off by an excited atom is no higher than the energy of the incoming radiation exciting the atoms?" - in combination with "Did you know that minimum wavelength corresponds to maximum energy?"

Other than recognising that the Angstrom wavelength corresponded to one of the nonometre wavlengths, no calculations were necessary in the question [as befits the idea that you should have completed the question in one minute]

What do you mean by "elastically scattered"?

Yes i know that the energy of a photon given off by an excited atom is no higher than the incoming radiation, but is the energy given off always equal to that of incoming radiation?

And yes i know that energy is inversely proportional to wavelength.

What I mean is: if you put in the equation E_n=\frac{R_H}{n^2} the values of E_n and R_H, is there an integer that fits well in n? If so, then the electron can get to the excited state, else it can't.
That's all.

I still don't understand what should i put the value of E_n.

 

[PeterO]

What do you mean by "elastically scattered"?

Yes i know that the energy of a photon given off by an excited atom is no higher than the incoming radiation, but is the energy given off always equal to that of incoming radiation?

And yes i know that energy is inversely proportional to wavelength.

Just answering you questions about my response.

Elastically scattered means the incoming photon comes back out without losing any of its energy - naturally it has the same energy , so same wavelength as when it went in.

No the energy of an emitted photon is not always the same as the incoming, could be less than the incoming photon - but only if the atom was excited to the 2nd or higher level. [I gather, from some computations in this thread, that in this case it actually gets excited to the 3rd energy level - important for you to realize that I did NOT need to know that in order to answer the question!]

OK so you knew that energy is inversely proportional to wavelength for a photon. In that case you should have been able to answer the question - if you had recognised what the question was asking!

The question asked "what is the shortest wavelength photon emitted?".
The inverse expression between energy and wavelength means that question could be re-written as "what is the largest energy photon emitted?"

Since you knew that any emitted photon would be the same, or lower energy, you should have recognised that you were after the same photon that went in. - Option (a)

Arguably, the question is really testing whether you can convert Angstroms to nanometres!

NOW, had all the options been longer than the incoming radiation, you would have had to work out which energy level the atom could be excited to [apparently the 3rd level], then calculate the energy and wavelength of the radiations for drops to intermediate levels to make you selection.

As I said before: that would make it a 5-10 minute question rather than a 1 minutes question - so inappropriate for the multiple choice sections of most tests/exams

 

[I like Serena]

Why doesn't the electron jump to the second level?

You already noted before:

In this question, electron cannot jump to the second level too since the energy of the incoming photon is 12.1 eV.
Am i right?

And you were right.

 

[Saitama]

You already noted before:



And you were right.

So why it jumps to the third level?

 

[I like Serena]

So why it jumps to the third level?

The difference in energy between the first and the third level corresponds (almost) exactly to the energy of the incoming photon.
Btw, only when there is an (almost) exact match will the photon be absorbed.

 

[PeterO]

So why it jumps to the third level?

Don't forget that it is all but irrelevant that it jumps to the 3rd level!

 

[Saitama]

The difference in energy between the first and the third level corresponds (almost) exactly with the energy of the incoming photon.
Btw, only when there is an (almost) exact match will the photon be absorbed.

The energy difference between the first and the third level is 10.2eV but it doesn't match with the energy of the incoming photon.

 

[I like Serena]

The energy difference between the first and the third level is 10.2eV but it doesn't match with the energy of the incoming photon.

I didn't calculate or check the energy of the energy levels or the energy of the incoming photon.

However, you already calculated before that the wavelength corresponding to the first and third level is (almost) equal to the wavelength of the incoming radiation.

 

[Saitama]

I didn't calculate or check the energy of the energy levels or the energy of the incoming photon.

However, you already calculated before that the wavelength corresponding to the first and third level is (almost) equal to the wavelength of the incoming radiation.

Sorry it's my mistake, its not 10.2 eV.
But i still don't get why (a) is the minimum wavelength?

 

[PeterO]

sorry it's my mistake, its not 10.2 ev.
But i still don't get why (a) is the minimum wavelength?

PLEASE: Minimum wavelength = maximum energy !

Maximum emitted energy = incoming energy

i thought you had confirmed both those facts earlier ??

EDIT: Have you realized that Option (a) is the nanometre equivalent of the Angstrom Unit wavelength of the incoming radiation?

 

[I like Serena]

Sorry it's my mistake, its not 10.2 eV.
But i still don't get why (a) is the minimum wavelength?

Can you be a little bit more expansive please?
Provide a little more detail about what you do not get?
And perhaps on what you do get?

 

[Saitama]

PLEASE: Minimum wavelength = maximum energy !

Maximum emitted energy = incoming energy

i thought you had confirmed both those facts earlier ??

EDIT: Have you realized that Option (a) is the nanometre equivalent of the Angstrom Unit wavelength of the incoming radiation?

Yes i have confirmed those facts.
And yes i have realized that option (a) is the nm equivalent of the Angstrom Unit wavelength of the incoming radiation.

Can you be a little bit more expansive please?
Provide a little more detail about what you do not get?
And perhaps on what you do get?

I don't get why we checked the wavelength when transition takes place between 1st level and 2nd level and between 1st level and 3rd level. And when we found the matching wavelength, we stopped at third level and we didn't proceed further to the fourth level.

 

[PeterO]

Yes i have confirmed those facts.
And yes i have realized that option (a) is the nm equivalent of the Angstrom Unit wavelength of the incoming radiation.



I don't get why we checked the wavelength when transition takes place between 1st level and 2nd level and between 1st level and 3rd level. And when we found the matching wavelength, we stopped at third level and we didn't proceed further to the fourth level.

I can't understand why you are checking any levels at all. Decay from any excited level will be either the same as the incoming radiation [one big drop] or SMALLER - several smaller drops.
You are trying to find the biggest drop.

 

[Saitama]

I can't understand why you are checking any levels at all. Decay from any excited level will be either the same as the incoming radiation [one big drop] or SMALLER - several smaller drops.
You are trying to find the biggest drop.

I too don't understand why i was made to check for the levels?

 

[PeterO]

Homework Statement

H-atom is exposed to electromagnetic radiation of \lambda=1025.6 \dot{A} and excited atom gives out induced radiations. What is the minimum wavelength of these induced radiation:
(a)102.6 nm
(b)12.09 nm
(c)121.6 nm
(d)810.8 nm

Just in case you have forgotten the original question.

The two words I highlighted confirmed that the H-Atom was excited to a higher level, so the incoming radiation had an energy matching one of the excited levels.

The decay back to ground state will be either in one big jump - straight back to ground state - or a series of cascades via lower levels, if there are any.

The minimum wavelength of induced radiation corresponds to the largest energy given off.

The largest energy of induced radiation is from the single, big jump. That energy is the same as the incoming radiation.

So the answer is the same wavelength as the incoming - the nm version of the Angstoms.

Peter

 

[PeterO]

I too don't understand why i was made to check for the levels?

I didn't realize you felt someone made you check the levels? Certainly I never did. On the contrary I repeatedly said it was unnecessary.

EDIT: But you repeatedly claimed you didn't get why option (a) was the minimum wavelength !??

 

[Saitama]

Just in case you have forgotten the original question.

The two words I highlighted confirmed that the H-Atom was excited to a higher level, so the incoming radiation had an energy matching one of the excited levels.

The decay back to ground state will be either in one big jump - straight back to ground state - or a series of cascades via lower levels, if there are any.

The minimum wavelength of induced radiation corresponds to the largest energy given off.

The largest energy of induced radiation is from the single, big jump. That energy is the same as the incoming radiation.

So the answer is the same wavelength as the incoming - the nm version of the Angstoms.

Peter

Oh Wow! Thank you so much! I got it now!

(Sorry for irritating you and ILS for such an easy question. )

 

[I like Serena]

I don't get why we checked the wavelength when transition takes place between 1st level and 2nd level and between 1st level and 3rd level. And when we found the matching wavelength, we stopped at third level and we didn't proceed further to the fourth level.

The transition does not take place between 1st and 2nd level, since the energy signature does not match.
We stopped at 3rd level, since we found a match in the energy signature, so there was no point in continuing to the 4th level.

Ultimately the point of doing the question, and perhaps doing a little more work than necessary, is to get an understanding of how the atom, excitation, absorption, emission, and quantum states work.
Do you feel you have learned something? And that you can do more questions like this one?

 

[Saitama]

The transition does not take place between 1st and 2nd level, since the energy signature does not match.
We stopped at 3rd level, since we found a match in the energy signature, so there was no point in continuing to the 4th level.

Ultimately the point of doing the question, and perhaps doing a little more work than necessary, is to get an understanding of how the atom, excitation, absorption, emission, and quantum states work.
Do you feel you have learned something? And that you can do more questions like this one?

Yes, i think i have learned a lot from this discussion but still one question arises in my mind.
What would happen if there was no energy match? How then we would find out the minimum wavelength of the induced radiation?

 

[I like Serena]

What would happen if there was no energy match? How then we would find out the minimum wavelength of the induced radiation?

Then the photon would not be absorbed and there would be no induced radiation.

 

[Saitama]

Why the photon won't be absorbed?

 

[I like Serena]

Why the photon won't be absorbed?

Dunno. ;)
Experiments done by great scientists say so.
That's why it's called "quantum" physics.
Energy can only exist and be transferred in a specific "quantum".

 

[Saitama]

Dunno. ;)
Experiments done by great scientists say so.
That's why it's called "quantum" physics.
Energy can only exist and be transferred in a specific "quantum".

Thanks for your explanation!

 

[PeterO]

Why the photon won't be absorbed?

For photon to be absorbed, it has to be an exact energy match.

If the incoming "radiation" had been a stream of electrons - particles - then they could collide and pass on part of their energy only [not unlike a pair of billard balls colliding].

With photons, they either pass on ALL their energy or NONE of their energy.

[and the underlying principle is that the atom can only accept specific discrete amounts of energy]

EDIT: The energy of a photon is related to its frequency [or wavelength]. The frequency is determined by the source of the radiation. If the photon later interacts with an atom it is either absorbed totally, or scattered without energy loss. The collision cannot change the frequency of the radiation - so cannot take part of the photon energy.

 

[anshul arora]

when electron exite from a particular state to another stae by absorbing energy, then the electron tries to return to the orginal exited state and if possible to the ground state. these radiations that are emitted by the electron to return back to the ground state are called induced radiations.

in this questions energy supplied to the hydrogen atom is 12400/1028 ev = 12.06ev.

(note- energy gap between two shells is 12.06 ev hence electron will exite from ground state.
this is because energy is 2nd shell of hydorgen atom is -3.4ev and energy of elctron at ∞ is 0 then if electron would have have excited from 2nd energy level then max energy level would be 3.4 and would never reach 12.06 ev)

energy of electon at excited energy level = -13.6-(-12.06)ev = -1.54ev

calculating n, -1.54 = -13.6*z²/n² (where z=1)
gives n = 3

therefore, electrons can transist to all below possible energy level to produce all possible photons.
3→1
3→2
2→1
solving for λ, you will observe that the minimum wavelength is observed when electron will transist from 3 to 1 which is equal to 102.6 nm

 

[anshul arora]

answer is (a)