Answer: Find Wavelength of H-atom Induced Radiation

[Saitama]

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

Homework Equations

The Attempt at a Solution

What does the question mean by "induced radiation"?

 

[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

Homework Equations

The Attempt at a Solution

What does the question mean by "induced radiation"?

The Hydrogen atom may be excited by the radiation to change to an excited state. The atom then drops bag to its original state, giving off a photon. That is the induced radiation.

 

[Saitama]

The Hydrogen atom may be excited by the radiation to change to an excited state. The atom then drops bag to its original state, giving off a photon. That is the induced radiation.

Thanks for explaining, but how should i start?

 

[PeterO]

Thanks for explaining, but how should i start?

Suppose the atom was excited to the second level by these Photons, What wavelengths of induced radiation might you get? And what might that tell you?

it is 12:40 pm here I am off to bed.

 

[pmsrw3]

The fundamental idea is that you never get more energy out than you put in.

 

[Saitama]

Suppose the atom was excited to the second level by these Photons, What wavelengths of induced radiation might you get? And what might that tell you?

it is 12:40 pm here I am off to bed.

I am taking hydrogen atom for example.
If i apply Rydberg formula, i get a wavelength of \frac{4}{3R}.

But i still don't understand what i have to do?

(Good night PeterO)

 

[I like Serena]

Hi Pranav-Arora!

Fill in R?
What is the corresponding wavelength?
Is it more or less than the exposed wavelength?
That is, does this induced radiation contain more or less energy than the exposed energy?

 

[Saitama]

Hi Pranav-Arora!

Fill in R?
What is the corresponding wavelength?
Is it more or less than the exposed wavelength?
That is, does this induced radiation contain more or less energy than the exposed energy?

I still don't get it.

 

[I like Serena]

I still don't get it.

R is the Rydberg constant.
In e.g. wikipedia you can find its value.

What you get is the wavelength of induced radiation.
The shorter the wavelength the higher the energy.
Induced radiation will always have a wavelength longer than the wavelength of the exposed radiation (can't create energy out of nothing).

What is the value of R?
What do you get if you fill it in?
Is it bigger or smaller than the exposed radiation?

Once you have that we can continue.

 

[Saitama]

R is the Rydberg constant.
In e.g. wikipedia you can find its value.

What you get is the wavelength of induced radiation.
The shorter the wavelength the higher the energy.
Induced radiation will always have a wavelength longer than the wavelength of the exposed radiation (can't create energy out of nothing).

What is the value of R?
What do you get if you fill it in?
Is it bigger or smaller than the exposed radiation?

Once you have that we can continue.

Hi!
I know what is "R".
\frac{4}{3R}, this value came because i assumed that electron jumped to the second level and came back. In the question, it's not specified that to which level electron jumps.

 

[I like Serena]

Hi!
I know what is "R".
\frac{4}{3R}, this value came because i assumed that electron jumped to the second level and came back. In the question, it's not specified that to which level electron jumps.

No, it is not specified.
The electron could jump to the second level and back, or it could jump to the third level and back, or it could jump to the third level, fall back to the second level and fall back the the first level, etcetera.

Your job is to find the highest level it could jump to, deduce which possible radiations could come out, and decide which one fits the question.

 

[Saitama]

No, it is not specified.
The electron could jump to the second level and back, or it could jump to the third level and back, or it could jump to the third level, fall back to the second level and fall back the the first level, etcetera.

Your job is to find the highest level it could jump to, deduce which possible radiations could come out, and decide which one fits the question.

What is the highest level it could jump to?

 

[I like Serena]

What is the highest level it could jump to?

\infty

 

[PeterO]

What is the highest level it could jump to?

MORE IMPORTANT: HOW DOES THE ENERGY OF A WAVE COMPARE TO WAVELENGTH.

Does higher energy mean longer wavelength or does higher energy mean shorter wavelength?

 

[PeterO]

\infty

Can it get there with this incoming radiation?

 

[Saitama]

This question is getting off my mind.
Let's begin from starting.

PeterO, you said that "The Hydrogen atom may be excited by the radiation to change to an excited state. The atom then drops back to its original state, giving off a photon. That is the induced radiation."

Doesn't the atom absorbs a photon when it goes to the excited state?

 

[I like Serena]

Yes, the atom absorbs a photon when it goes to the excited state.
This works out as an electron going to a higher energy level (there are only specific discrete energy levels).
After that the electron falls back to a lower energy level, giving off an induced photon.
The energy of the induced photon is equal to the difference in energy levels of the electron.

 

[I like Serena]

Can it get there with this incoming radiation?

No it can't.
To solve the problem the highest level must be calculated, based on the wavelength of the incoming radiation, and the wavelengths given by the Rydberg formula for the difference in energy levels that an electron can be in.

 

[Saitama]

Yes, the atom absorbs a photon when it goes to the excited state.
This works out as an electron going to a higher energy level (there are only specific discrete energy levels).
After that the electron falls back to a lower energy level, giving off an induced photon.
The energy of the induced photon is equal to the difference in energy levels of the electron.

So how would it help in solving this problem?

 

[I like Serena]

The rydberg formula gives you the wavelengths of the possible induced photons.
(Btw, the related energy of a photon of a wavelength lambda is E = h c / lambda.)
An induced photon can not have more energy than the photon that excited the atom.

Can you give me the wavelength belonging to an induced photon if the electron falls back from the second energy level to the first energy level (the ground state)?

 

[Saitama]

Can you give me the wavelength belonging to an induced photon if the electron falls back from the second energy level to the first energy level (the ground state)?

The wavelength is 4/(3R).

 

[PeterO]

This question is getting off my mind.
Let's begin from starting.

PeterO, you said that "The Hydrogen atom may be excited by the radiation to change to an excited state. The atom then drops back to its original state, giving off a photon. That is the induced radiation."

Doesn't the atom absorbs a photon when it goes to the excited state?

Yes it does - and it absorbs all the energy, not just part of it [as it can if struck by an electron for example]

 

[PeterO]

So how would it help in solving this problem?

For a start that is an incredible question, but further.

Suppose the energy levels for the Hydrogen atom were

Ground State - 0
1st Excited - 20 Joules
2nd Excited - 30 Joules
3rd excited - 35 Joules
4th excited - 38 Joules.

That tells us the atom can absorb Photon energy of 20, 30, 35 or 38 ...

Suppose it absorbed 35 Joules and was thus in its 3rd excited state.
It would then drop to a lower level or all the way to Ground state [it will get there finally].

It could give off a photon of 35, 15 or 5 Joules going to a lower level.
You will probably ask so 15 means it went from the 3rd level to the 1st level.

If it had gone to the 1st level, it will give off ANOTHER photon of 20 Joules on the way to Ground state.

If it had gone to the 2nd level [giving of the 5 Joules remember] It will then either give off 30 J on the way to Ground state, or 10 Joules on the way to second level [whence a 20 J photon will follow].

So having absorbed an incoming 35 Joule photon, we can expect to see at least one of the following energy photons given off:
35 J, 30 J, 20 J, 15 J, 10 J, 5 J

If we shone light on a whole bunch of atoms, we can expect lots of them to each do one of the possibilities so that all those enrgies would be seen at the same time.

Now with real atoms, the energies are not measured in whole Joules but tiny fractions, and are not so evenly spaced.

NOW MOST IMPORTANTLY, how would the wavelengths of those photons given off be related? Which one would have the longest wavelength? - which would have the shortest? That final bit is the key to answering this question!

 

[I like Serena]

The wavelength is 4/(3R).

Hmm, I seem to recall you already gave that answer.
Actually, I was hoping for a number...

 

[Saitama]

NOW MOST IMPORTANTLY, how would the wavelengths of those photons given off be related? Which one would have the longest wavelength? - which would have the shortest? That final bit is the key to answering this question!

Thanks for your explanation Peter!
Since Energy is inversely proportional to wavelength, therefore 1st excited state would have the longest wavelength and the 4th excited state would have the least wavelength.

Hmm, I seem to recall you already gave that answer.
Actually, I was hoping for a number...

Filling in R, i get 1216\dot{A}.

 

[I like Serena]

Filling in R, i get 1216\dot{A}.

Good!
I didn't actually check your number, but I'll assume it is right.
This is a longer wavelength than the incoming radiation, so the electron can and will be excited to the 2nd energy level.

Is there enough energy to excite it to the 3rd level (you would need to apply the Rydberg formula again)?
Since if not, the only induced radiation you can get is radiation of this same wavelength.
Otherwise their are several possibities.

Btw, can you convert angstroms to nanometers?

 

[Saitama]

Good!
I didn't actually check your number, but I'll assume it is right.
This is a longer wavelength than the incoming radiation, so the electron can and will be excited to the 2nd energy level.

Is there enough energy to excite it to the 3rd level (you would need to apply the Rydberg formula again)?
Since if not, the only induced radiation you can get is radiation of this same wavelength.
Otherwise their are several possibities.

Btw, can you convert angstroms to nanometers?

This wavelength is longer than the exposed radiation, so how it can jump to the second level? Is it possible that an electron is exposed to x nm and jumps to the second level and emits a radiaiton of more than 'x nm'?(nm-nanometre)

Yes i can convert angstrom to nanometres.

 

[I like Serena]

This wavelength is longer than the exposed radiation, so how it can jump to the second level? Is it possible that an electron is exposed to x nm and jumps to the second level and emits a radiaiton of more than 'x nm'?(nm-nanometre)

Yes, the wavelength emitted *has to* be at least the wavelength of the exposed radiation.
Note that a longer wavelength corresponds to a lower photon energy.

Yes i can convert angstrom to nanometres.

So what's the number in nanometers?

 

[Saitama]

So what's the number in nanometers?

121.6 nm.

 

[I like Serena]

121.6\dot{A}.

That is the right number!
(But the wrong unit! You're still specifying angstroms! )

Does it match with one of your answers?

And to repeat my question, is it possible the electron is excited to energy level 3?
Because if so, then other wavelengths might be induced.

 

[Saitama]

That is the right number!
(But the wrong unit! You're still specifying angstroms! )

Edited!

Does it match with one of your answers?

Yes it matches the (c) option but the answer is (a) option.

And to repeat my question, is it possible the electron is excited to energy level 3?
Because if so, then other wavelengths might be induced.

No, it's not possible.

 

[I like Serena]

No, it's not possible.

Right! :)

Yes it matches the (c) option but the answer is (a) option.

Interesting.

The (a) option is literally true, since the induced radiation cannot have a shorter wavelength than the exposed radiation.

But option (c) is also right afaik, since this is the only radiation that would be induced, and the answer is a little sharper.

 

[DiracRules]

Yes it matches the (c) option but the answer is (a) option.

The answer is (a). That's because you performed some miscalculation.

You can solve this exercise almost without any calculation.
I think it's better to think in the proper energy unit for this problem, eV (electron volt). You will handle with shorter numbers and the thinking will be easier. However, you can take what I say and do it in Joules, it will (and has to) be the same.

To get the energy in electron volt you just have to think that 1\,\,eV=1.6\cdot 10^{-19}\,\,J.

So, to get an idea, start by transforming the incoming photon energy into eV and then the ground state energy of hydrogen atom.

Then compare these two energies. Is the energy of the incoming photon higher or lower than the ground state? If it is higher, the electron will be bounced outside the atom (the atom will be ionized). 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.

Can you post here the comparison between the ground state energy of hydrogen and the energy of the incoming photon? (Either in joule or eV, as you wish...)
What does this comparison say to you? Remember that, after the fourth/fifth energy level, you can consider the spectrum nearly continuous, at least for a rough calculus.
Can you solve the exercise now?

 

[I like Serena]

And to repeat my question, is it possible the electron is excited to energy level 3?
Because if so, then other wavelengths might be induced.

No, it's not possible.

Ah, I see the problem now.
As DiracRules already said, you miscalculated.

Can you calculate the induced wavelength if the electron falls back from the 3rd energy level to the 1st?

 

[Saitama]

So, to get an idea, start by transforming the incoming photon energy into eV and then the ground state energy of hydrogen atom.

Then compare these two energies. Is the energy of the incoming photon higher or lower than the ground state? If it is higher, the electron will be bounced outside the atom (the atom will be ionized). 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.

Can you post here the comparison between the ground state energy of hydrogen and the energy of the incoming photon? (Either in joule or eV, as you wish...)
What does this comparison say to you? Remember that, after the fourth/fifth energy level, you can consider the spectrum nearly continuous, at least for a rough calculus.
Can you solve the exercise now?

Hi DiracRules!
The energy of the incoming photon is around 12.1 eV and the ground state energy of the H-atom is 13.6 eV. The energy is much less, so i don't think electron will be bounced outside the atom.

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."

Can you calculate the induced wavelength if the electron falls back from the 3rd energy level to the 1st?

Is the wavelength 102.573 nm?

 

[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.