Question on Emission Spectra/Spectral Series | Atomic Physics

[warhammer]

Homework Statement

Describe Frank-Hertz experiment. What is the significance of the minimum voltage? If this experiment is performed with hydrogen gas and electrons of energy 4 eV are used, which observed spectral series is emitted?

Relevant Equations

E= hc/λ

(I need help with the 2nd part as I can answer the theory part properly).

For E=4 eV we can find the wavelength of emitted photon.

E= 4 eV = 6.4087e-19 J

Using E= hc/λ we get λ=310 nm (approx)

My doubt is that this should fall in the Balmer Series but we know that the lowest wavelength value for this series is around 364nm (this info is not supplied within the question as can be seen but it was given in the book and verified from the Internet). But my value is about 310 nm and I cannot bracket this into any Spectral Series. Have I done something wrong?

I request someone to please have a look at my solution and please guide me accordingly so that I can rectify my mistakes!

 

[kuruman]

Your theory assumes that the electron loses all of its energy when it collides with a hydrogen atom. Is that necessarily true?

 

[warhammer]

Your theory assumes that the electron loses all of its energy when it collides with a hydrogen atom. Is that necessarily true?

No no. That is not true. There should be some atomic recoil as well due to the collision. But according to Serway Moses & Moyer in Modern Physics little energy is needed to conserve momentum so one can generalise equating photon's energy to atomic energy level separation..

 

[kuruman]

No no. That is not true. There should be some atomic recoil as well due to the collision. But according to Serway Moses & Moyer in Modern Physics little energy is needed to conserve momentum so one can generalise equating photon's energy to atomic energy level separation..

That's not what I am talking about. Describe in your own words the Franck & Hertz experiment. I suspect that you do not fully understand it. Specifically, explain the role of the 4 eV electrons in the emission of spectral lines.

 

[warhammer]

That's not what I am talking about. Describe in your own words the Franck & Hertz experiment. I suspect that you do not fully understand it. Specifically, explain the role of the 4 eV electrons in the emission of spectral lines.

Franck & Hertz experiment apart from existence of atomic spectra was another proof of existence of quantised energy levels existing within the atomic structure. In the set up, electrons are maintained at a given minimum potential so that their energy is not lower than this particular value. An accelerating potential is applied to the set up which causes increasing number of electrons to arrive at the anode and consequently a current flow is established. This goes on until at specific values (called critical potential) the current suddenly drops. However when the voltage is again increased the situation resumes to normal until a point is reached (which is in multiples of the values encountered during first drastic fall) where the current again drops to zero.

This interpreted as the fact that atoms require impinging electrons to have specific energy levels that can excite them sufficiently. For instance in the original experiment, electrons when accelerated to a potential of 4.9ev attain KE of 4.9eV on reaching the grid (part of setup). Here it undergoes inelastic collision with mercury atoms because this energy level is sufficient to excite the atom. When we increase the voltage again, the electrons still have some energy left even after atomic excitation that they reach the anode and we register current flow. However again at a value of 9.8eV this energy is sufficient to excite the atoms from their 1st excited state to 2nd and we register our second fall in current value owing to inelastic collisions that electrons suffer with atoms.

 

[kuruman]

OK then, let's continue talking about the original experiment. You said that a 4.9 eV electron would lose all its energy. Suppose that an electron energy impinging on the atom has 5.2 eV of energy. Can it not lose 4.9 eV to excite the atom and keep going with an initial energy of 0.3 eV?

 

[warhammer]

OK then, let's continue talking about the original experiment. You said that a 4.9 eV electron would lose all its energy. Suppose that an electron energy impinging on the atom has 5.2 eV of energy. Can it not lose 4.9 eV to excite the atom and keep going with an initial energy of 0.3 eV?

Yes yes, it can certainly do that!

I think I get your drift (hopefully ). I should find the energy that corresponds near to 4eV using the concept of shortest possible wavelength from Lyman/Balmer series. And then compare those values with the 4eV in order to ascertain which series I will be getting.. Is this correct?

 

[kuruman]

You got my drift. The 4 eV energy determines which series cannot be excited because there isn't enough energy.

 

[TSny]

If you use hydrogen gas in the Frank-Hertz experiment, the gas would consist of hydrogen molecules rather than individual hydrogen atoms. So you would be observing the energy quantization of molecules rather than atoms. Maybe you are supposed to assume that the gas is individual atoms for the sake of the exercise. But then, the bombarding electrons do not have enough energy to excite a hydrogen atom from the ground state to an excited state. So, the problem is a little confusing to me. Do you think that they expect you to be familiar with the energy levels of hydrogen molecules?

 

[kuruman]

But then, the bombarding electrons do not have enough energy to excite a hydrogen atom from the ground state to an excited state.

That is true. However the electrons will have enough energy to kick an electron to the continuum from some excited state ($n=?$ is the question) which is then repopulated by electrons of higher $n$. This would result in the emission of photons characteristic to a spectral series.

 

[TSny]

That is true. However the electrons will have enough energy to kick an electron to the continuum from some excited state ($n=?$ is the question) which is then repopulated by electrons of higher $n$. This would result in the emission of photons characteristic to a spectral series.

As I understand it, the temperature of the gas in the Frank-Hertz tube is roughly room temperature (kT about 1/40 eV). So, for hydrogen atoms, essentially all of the atoms would initially be in the ground state. The bombarding electrons (4 eV) would not be able to cause excitations from the ground state. For hydrogen molecules, the bombarding electrons would also not have enough energy to cause electronic excitations, but they would have enough energy to cause vibrational and rotational excitations of the molecules.

 

[kuruman]

Again, all that is true and I am not disagreeing with you. This is a hypothetical experiment using hydrogen instead of mercury which will not work in a real experiment for the reasons you mentioned plus more if you look. A qualitative answer is expected and I tried to guide OP in the direction I saw as the best. Please correct me if you see a better path for OP to follow.

 

[PeroK]

So, the assumption is that the hydrogen gas consists of atoms in a distribution of energy states?

 

[kuruman]

That would be my assumption if I interpret the spirit of the question correctly: testing the understanding of the Bohr atom as a model for the F&H experiment.

 

[warhammer]

You got my drift. The 4 eV energy determines which series cannot be excited because there isn't enough energy.

Okay. So I have worked it out in the following manner:

Lyman Series-
Shortest Wavelength is 911.7 A with Energy=2.18E-18 J or 13.6 eV (higher than 4eV)

Longest Wavelength is 1215.7 A with Energy=1.63E−18J or 10.22 eV (higher than 4eV)

Balmer Series-
Shortest Wavelength is 3647 A with Energy=5.45E−19J or 3.4 eV (lower than 4eV)

Longest Wavelength is 6565 A with Energy=3.03E−19J or 1.9 eV (lower than 4eV)

Paschen Series-
Shortest Wavelength is 8189 A with Energy=2.427E−19 J or 1.5 eV (lower than 4eV)

Longest Wavelength is 18750 A with Energy=1.06E-19 J or 0.6 eV (lower than 4eV)

It is obvious that remaining series will be having lesser energies than 4eV.

So the answer to the question "If this experiment is performed with hydrogen gas and electrons of energy 4 eV are used, which observed spectral series is emitted?" should be - Balmer Series with the Energy 3.4eV right since we pick the one which has the highest possible energy lower than/equivalent to given threshold which is 4eV here? (Obviously others are possible too but considering for such a simplistic case)

 

[TSny]

Suppose we assume that we are dealing with hydrogen atoms that are already in various excited states before being hit by the 4 eV electrons. As a specific example, suppose an atom is initially in a state with n = 2. You should be able to show that a 4 eV electron has enough energy to excite this atom into any energy level with n > 2. Suppose the atom gets excited up to the n = 5 level. Now there are many possibilities for how this excited atom can emit photons to reduce its energy. One possibility is for it to first make a transition from the n = 5 level to the n = 3 level (a Paschen line). Then the atom might transition from n = 3 to n = 2 (a Balmer line). Finally, the atom could jump from the n = 2 level to the n = 1 level (a Lyman line).

Are there any spectral lines in any of the spectral series of atomic hydrogen that could not be produced in principle by similar processes starting from the n = 2 level?

Now suppose that we are dealing with hydrogen atoms that are all in the ground state initially (n = 1). What spectral lines could be produced in principle?

 

[warhammer]

First of all, sorry for an extremely delayed response.

You should be able to show that a 4 eV electron has enough energy to excite this atom into any energy level with n > 2.

So what I tried here was to see what is the energy difference between any 2 given states. For instance, I found the energy difference between first and second state (-3.4-(-13.6))= 10.2 eV. Similarly energy differences between 2nd and 3rd, 2nd and 4th, 2nd and 5th etc. was 1.89 eV, 2.55 eV, 2.856 eV and so on. Now all these values are lesser than stipulated 4eV so I should be able to excite from n=2 to any level until the Energy Difference exceeds 4eV. Is this correct?

I am sorry but I am having a hard time to figure this out. We are using Arthur Beiser's book but it is also bit unclear in this regard. I have tried and tried but its getting extremely confusing.. I think this is wrong as well..

Okay. So I have worked it out in the following manner:

Lyman Series-
Shortest Wavelength is 911.7 A with Energy=2.18E-18 J or 13.6 eV (higher than 4eV)

Longest Wavelength is 1215.7 A with Energy=1.63E−18J or 10.22 eV (higher than 4eV)

Balmer Series-
Shortest Wavelength is 3647 A with Energy=5.45E−19J or 3.4 eV (lower than 4eV)

Longest Wavelength is 6565 A with Energy=3.03E−19J or 1.9 eV (lower than 4eV)

Paschen Series-
Shortest Wavelength is 8189 A with Energy=2.427E−19 J or 1.5 eV (lower than 4eV)

Longest Wavelength is 18750 A with Energy=1.06E-19 J or 0.6 eV (lower than 4eV)

It is obvious that remaining series will be having lesser energies than 4eV.

So the answer to the question "If this experiment is performed with hydrogen gas and electrons of energy 4 eV are used, which observed spectral series is emitted?" should be - Balmer Series with the Energy 3.4eV right since we pick the one which has the highest possible energy lower than/equivalent to given threshold which is 4eV here? (Obviously others are possible too but considering for such a simplistic case)

I request you to please guide me here, a bit more deeply and directly.. I will be extremely obliged and grateful

 

[kuruman]

In my opinion you have answered the question sufficiently well. If the incoming electron doesn't have enough energy to promote an atomic electron to next higher state, the atomic electron will stay put and there will be no transition. There is nothing deeper than energy conservation.

 

[warhammer]

In my opinion you have answered the question sufficiently well. If the incoming electron doesn't have enough energy to promote an atomic electron to next higher state, the atomic electron will stay put and there will be no transition. There is nothing deeper than energy conservation.

Thank you so much for your response sir. So just to confirm sir, is my post #15 fine or #17.. Or are they essentially equivalent approaches..

 

[kuruman]

You have presented arguments why the Lyman series must be excluded and why the Balmer series can be included. What can you say about the Paschen, Pfund and Humphreys series? On the basis of your criteria for exclusion or inclusion, what's to be done with them and why?

 

[warhammer]

You have presented arguments why the Lyman series must be excluded and why the Balmer series can be included. What can you say about the Paschen, Pfund and Humphreys series? On the basis of your criteria for exclusion or inclusion, what's to be done with them and why?

I think due to requisite energy being available, all the other series too must be present. However I thought that this being a simplistic case and considering the fact that Balmer Series operating in visible region against Paschen, Pfund et al operating in Infrared Regions (former being more energetic than latter) may be more "dominant", so to speak. But I'm unsure whether this assertion is even appropriate or not

 

[TSny]

Is this a problem from the Beiser textbook or a problem given to you by your instructor? If it's from your instructor, then you might consider asking for clarification regarding the initial state of the hydrogen atoms before they are hit by the 4 eV electrons.

If the initial state of the atoms is assumed to be the ground state, then you can show that no excitations can be produced by the 4 eV electron beam. So, no spectral lines are observed.

However, if you can assume that initially there are a significant number of atoms in the $n = 2$ level, then the electron beam can excite these atoms to any level $n > 2$. Then you can argue that all spectral series could be observed in principle.

So, one way to answer the question would be to discuss both of these scenarios and show the reasoning and calculations involved in each case.

I don't have access to any recent editions of Beiser. I was able to access a copy of the second edition of his Concepts of Modern Physics (published in 1973! ). At the end of chapter 4, I found the following problem

If you would like to test your understanding, try working this problem. We can check your answer.

 

[kuruman]

I don't have access to any recent editions of Beiser. I was able to access a copy of the second edition of his Concepts of Modern Physics (published in 1973! ).

That's the edition I used to have. Alas, someone borrowed it and didn't return it. Maybe "minimum" needs to be inserted before "potential difference $\dots~$"?

 

[TSny]

That's the edition I used to have. Alas, someone borrowed it and didn't return it.

Although I never used any of Beiser's texts, I did occasionally browse through some of them. I thought they were pretty good.

Maybe "minimum" needs to be inserted before "potential difference $\dots~$"?

Yes.

 

[warhammer]

I am again sorry for a late response. I had actually sent in a reply the next day itself but for some reason it looks like the Post Reply button didn't work! Consequently got very curious as to why I didn't hear from the PF Mentors!

Is this a problem from the Beiser textbook or a problem given to you by your instructor? If it's from your instructor, then you might consider asking for clarification regarding the initial state of the hydrogen atoms before they are hit by the 4 eV electrons.

If the initial state of the atoms is assumed to be the ground state, then you can show that no excitations can be produced by the 4 eV electron beam. So, no spectral lines are observed.

However, if you can assume that initially there are a significant number of atoms in the $n = 2$ level, then the electron beam can excite these atoms to any level $n > 2$. Then you can argue that all spectral series could be observed in principle.

So, one way to answer the question would be to discuss both of these scenarios and show the reasoning and calculations involved in each case.

I don't have access to any recent editions of Beiser. I was able to access a copy of the second edition of his Concepts of Modern Physics (published in 1973! ). At the end of chapter 4, I found the following problem

View attachment 314369

If you would like to test your understanding, try working this problem. We can check your answer.

No this is not a problem from a Beiser textbook neither assigned by my instructor. It was in an old problem set on some printed sheets discovered while sifting books in the library (caught my attention due to the weird and a bit different language of the question!).

For the problem I think the Hydrogen atoms here are unexcited (in ground state config given by n=1) while Balmer Series is specified for a transition from n=3 to n=2 state. So essentially given the conditions, we need to accelerate the electrons to atleast n=3 from n=1 for Balmer Emission to occur, which gives us ΔE=-1.5-(-13.6)eV=12.1eV ~ 12.1 V must be the PD.PS- Is Beiser a biblical tier introductory text for Modern Physics? Because it is solidly amazing to see that the book is still being published after 1973 and being grinded by several students! And you all too share some sort of acquaintance with it

 

[TSny]

For the problem I think the Hydrogen atoms here are unexcited (in ground state config given by n=1) while Balmer Series is specified for a transition from n=3 to n=2 state. So essentially given the conditions, we need to accelerate the electrons to atleast n=3 from n=1 for Balmer Emission to occur, which gives us ΔE=-1.5-(-13.6)eV=12.1eV ~ 12.1 V must be the PD.

Looks good. I agree with your solution.

PS- Is Beiser a biblical tier introductory text for Modern Physics? Because it is solidly amazing to see that the book is still being published after 1973 and being grinded by several students! And you all too share some sort of acquaintance with it

A search for Beiser books at Amazon shows that he is the author of a number of texts. The text The Mainstream of Physics was published in 1962. The latest edition of the modern physics text appears to be the 6th edition published in 2002. He's a coauthor of the text Physical Universe which is now in its 17th edition (2022). Impressive.

 

[warhammer]

Looks good. I agree with your solution.

Thank you for being so patient with me sir. Really feel elated to finally get the hang of this concept, even if it was trivial

A search for Beiser books at Amazon shows that he is the author of a number of texts. The text The Mainstream of Physics was published in 1962. The latest edition of the modern physics text appears to be the 6th edition published in 2002. He's a coauthor of the text Physical Universe which is now in its 17th edition (2022). Impressive.

Indeed. Physical Universe seems to be on a bull run of some sort. Although in addition to his age as made available online, he appears to be a recluse. Information on him is extremely sparse without any portraits (he doesn't seem to be the individual whose photo tends to appear when searched online). But a fine author nonetheless.