Answer: Find Wavelength of H-atom Induced Radiation

[I like Serena]

Is the wavelength 1025.73 nm?

I get 102.518 nm, but that is close enough, if at least you convert it to actual nanometers!
Is that more or less than the exposed radiation?

 

[Saitama]

I get 102.552 nm, but that is close enough, if at least you convert it to actual nanometers!
Is that more or less than the exposed radiation?

Your answer is less than the exposed radiation and mine is more than the exposed radiation.

 

[I like Serena]

Your answer is less than the exposed radiation and mine is more than the exposed radiation.

My bad, again I didn't check, and I didn't expect the problem to be so sharply defined.

I've looked up the proper value of the Rydberg constant for Hydrogen, which is:
R_H = 1.09678 x 10⁷ m^-1.

Filling this in, I get 102.573 nm.
So you were right!

 

[Saitama]

My bad, again I didn't check, and I didn't expect the problem to be so sharply defined.

I've looked up the proper value of the Rydberg constant for Hydrogen, which is:
R_H = 1.09678 x 10⁷ m^-1.

Filling this in, I get 102.573 nm.
So you were right!

Now what we have to do next?

 

[I like Serena]

Now what we have to do next?

Well, my question was: what is the highest energy level the electron can jump to.
Can you answer that?

And after that, what are the possibilities for the electron falling back to a lower energy level?

And after that, what are the corresponding induced wavelengths?

And after that, what is the lowest possible induced wavelength?

 

[Saitama]

Well, my question was: what is the highest energy level the electron can jump to.
Can you answer that?

And after that, what are the possibilities for the electron falling back to a lower energy level?

And after that, what are the corresponding induced wavelengths?

And after that, what is the lowest possible induced wavelength?

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?

 

[DiracRules]

I don't understand what do you mean by "If not, the electron could jump to another orbital if the energy gap between the initial and ending orbital is exactly the energy of the incoming electron."

Sorry I mistyped. I meant "If not, the electron could jump to another orbital if the energy gap between the initial and ending orbital is exactly the energy of the incoming photon."

To find the minimum wavelength of the induced radiation you just have to realize that emission is the absorption process seen backward: if the electron absorbs a photon with, say, 1J and gets excited, then it will come back to the ground state emitting a 1J photon. So, since you found that the maximum the energy, the lower the wavelength, you just have to calculate whether or not the incoming photon will bounce the electron in the ground state in an excited state.

If it gets excited, then it will emit the same amount of energy it receive. If not, it won't emit anything.

 

[Saitama]

Sorry I mistyped. I meant "If not, the electron could jump to another orbital if the energy gap between the initial and ending orbital is exactly the energy of the incoming photon."

To find the minimum wavelength of the induced radiation you just have to realize that emission is the absorption process seen backward: if the electron absorbs a photon with, say, 1J and gets excited, then it will come back to the ground state emitting a 1J photon. So, since you found that the maximum the energy, the lower the wavelength, you just have to calculate whether or not the incoming photon will bounce the electron in the ground state in an excited state.

If it gets excited, then it will emit the same amount of energy it receive. If not, it won't emit anything.

How do you find the energy of orbital?

I don't think i our case that the electron would bounce to the excited state since i calculated the energies as you said in a previous post.

 

[DiracRules]

How do you find the energy of orbital?

I don't think i our case that the electron would bounce to the excited state since i calculated the energies as you said in a previous post.

That's not correct.

You can use the Bohr equation for the energy:
$E_n=\frac{R_H}{n^2}$
where you can put $R_H=13.6\,\,eV$.

Now, you said correctly that the ground state is 13.6 eV, whilst the energy of incoming photon is 12.1 eV.

So if it gets excited, it will jump to a state whose energy $E_n$ is the remnant from incoming photon and the energy of the level. This is possible if the Bohr equation is verified. You have to check the one and only condition of that equation: that n is an integer.

 

[Saitama]

That's not correct.

You can use the Bohr equation for the energy:
$E_n=\frac{R_H}{n^2}$
where you can put $R_H=13.6\,\,eV$.

Now, you said correctly that the ground state is 13.6 eV, whilst the energy of incoming photon is 12.1 eV.

So if it gets excited, it will jump to a state whose energy $E_n$ is the remnant from incoming photon and the energy of the level. This is possible if the Bohr equation is verified. You have to check the one and only condition of that equation: that n is an integer.

How it will get excited when the energy of incoming photon is 12.1 eV?

 

[DiracRules]

How it will get excited when the energy of incoming photon is 12.1 eV?

Because the actual energy of the electron is -13.6eV, while the energy of the photon is +12.1eV.

Remember that all the bounded levels have negative energies. It will get excited if there exists a level at $E_0 + E_{photon}$...

 

[Saitama]

Because the actual energy of the electron is -13.6eV, while the energy of the photon is +12.1eV.

Remember that all the bounded levels have negative energies. It will get excited if there exists a level at $E_0 + E_{photon}$...

I thought of using the negative sign before but i got confused.
Anyways what i have to do next?

 

[DiracRules]

Have you checked whether the number fits with Bohr equation?

Here are the steps:
1) Find the possible energy of the excited state (I told you how to do in the previous post)
2) See if it fits in Bohr equation for energy, that is: calculate $n$ and see if there is something wrong.
3) Make your deductions...

 

[Saitama]

Have you checked whether the number fits with Bohr equation?

Which number?

(This question is making me mad. Btw, it's getting late night here, i am off to bed )

 

[DiracRules]

Which number?

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.

 

[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?