# Electron Energy Levels

by Jared Rovny
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00:01 To discuss this phenomenon and the way that these energy levels are related to each other, we're going to focus our discussion just on the hydrogen atom for now.

00:09 So in the past, in last few slides we've been talking about the hydrogen atom may be as an example of some of these properties.

00:17 But the properties we talked about were very general.

00:19 They are true for any atom that you are looking at.

00:21 The energy levels, changing energy levels, being in an excited state emitting photons, those are very general properties for any atom.

00:28 So now what I'm about to discuss, so be careful it's only true for the hydrogen atom specifically.

00:34 And what's true for the hydrogen atom is that the energy levels, the particular energies that the electrons have follow this equation here which says, "That the energy at a particular energy level." So we put a particular 'n' below the e is equal to -13.6 eV over 'n' square.

00:52 Where again 'n' is the energy level.

00:54 So for example if we were in the second energy level, we would put in two for 'n' and we would have -13.6 divided by 2 square or divided by 4.

01:04 The units of EV might be a little confusing.

01:08 We haven't seen these yet.

01:09 Remember that 'V' is a volt so these are electron volts.

01:13 So the V as a volt is a unit which we've already discussed has a unit of joules per coulomb or energy per charge.

01:20 How much energy is there per amount of charge.

01:23 This E letter is the word electron and we say eV often it's said as electron volt.

01:30 So this E is referring to the charge of an electron.

01:33 And we would know that the units of charge of a given electron are just coulombs because we're talking about charge.

01:39 And we've discussed the units of coulombs already.

01:41 And so the units of energy when we are discussing E V for our units, is just the joules per coulomb times coulombs.

01:49 It's just the coulomb will cancel in numerator and denominator.

01:52 We just have units of joules.

01:54 And so this is exactly an energy unit as we would expect.

01:57 It's just a new way of talking about an energy.

01:59 The reason we talk about the energy can be in these units because we're talking about very small energy.

02:06 So don't forget that the electron charge is a very, very tiny number.

02:10 So when you're talking about an eV, it's not the case that an eV is equal to 1 joule.

02:16 The 1 eV has units of joules.

02:19 So an EV is units of joules but it is a very, very small number of joules.

02:25 Namely exactly the charge of the electron.

02:28 So be careful with that.

02:29 It's sort of a tricky definition specially when you are introduced to it for the first time.

02:32 So go over it and look at the idea of an EV.

02:35 Go back and look at the charge of the electron that we introduced which was again a small number.

02:39 And note that an eV is this very small number the charge of the electron in units of joules.

02:45 The difference in the energy between one state and the other is something we've already introduced to find the energy of a photon that would get emitted.

02:55 So this delta e or the change in energy is just the difference in the two energy states.

03:01 So to find the change in energy for this atom, since we just introduce the equation for the hydrogen atom, we can just write down the difference.

03:09 Which would be -13.6 divided by N squared for one level minus the -13.6 divided by 'n' squared for the other level.

03:18 And this again if we're using this 13.6 number would be in units of eV.

03:23 And so for hydrogen atom you should be able to always calculate the equation the amount of energy that is emitted for a given photon, just by knowing what the two energy levels are.

03:33 And again these energy levels n will just be some integer n will be 1 or 2 or 3.

03:38 So you plugin those two numbers 1 or 2 or 3.

03:41 And the equation that we have written here and find the change in the energy in the atom which will again be equal to the energy of an emitted photon if your electron changed energy levels.

03:52 There's one last thing that's confusing potentially about this energy equation and that's the minus sign.

03:58 And why do we have a minus sign in front of our energy equation.

04:01 And the reason is this if we want to sort of look at this in a little more detail so that we can see it visually, we could plot the energy to see what it looks like.

04:09 It would look something like this.

04:11 We are on the x-axis, we have the energy levels.

04:13 So that would be the n the 1 or the 2 or the 3 as we go.

04:16 On the vertical axis we have plotted the energy itself.

04:19 The energy is decreasing in the sense that it's going towards 0 so it's decreasing in magnitude.

04:27 But it's going upwards.

04:28 It's going in the upward direction.

04:30 So you can see it's going in the positive direction but approaching the number 0.

04:34 The reason that we're using minus sign and have this down below the axis, is we can have the energy 0 point be wherever we would like to define it.

04:44 So we talked about when were introducing potential energy.

04:47 The fact that we can pick our 0 point wherever we would like it to be.

04:50 So in this case the electrons orbiting the nucleus, you would pick the 0 energy level to be in the electron is so far away from the nucleus, the energy level is so removed, we would say that there is no interaction.

05:04 There's no effect.

05:05 The electron, the negative electrons and the positive protons are not interacting because the distance is too great.

05:11 And so this is just a convention.

05:13 And we just say that when you're far, far, far away from the nucleus, we define that to be our 0 energy level.

05:19 At sort of an infinite distance, completely removed from the nucleus.

05:23 And then we define the rest of the energy relative to that point.

05:27 And the energy as you can see in this graph decreases as you go closer and closer or lower and lower towards the nucleus.

05:33 It's also true that as an electron gets closer and closer to the nucleus, this negative energy is referring to bound states.

05:41 So this is sort of a definition for this negative energy is that if an electron is far, far, far away from a nucleus, it doesn't have any effect.

05:49 But if it's getting closer to the nucleus, enters one of these energy levels, it becomes bound to the proton.

05:56 So as long as the electron have negative energy, we say that it is a bound electron to the nucleus.

06:03 There's another way to describe this which is a very, very common way to discuss the energy levels of electrons.

06:10 So you should certainly be familiar with this kind of diagram.

06:13 It's very important for an exam setting.

06:15 This is called an "Energy Level Diagram." What we do with an energy level diagram is simply plot these different end levels, these different electron energy levels going from the bottom to the top.

06:28 Starting with n equals to 1, the lowest energy level.

06:30 And then moving up to 2 and 3 and so on.

06:33 In principle we could go in until we reached an infinite sort of an energy level.

06:38 But again since the distance between the energy levels is decreasing as we go up and up and up, it's not as though the infinite energy level will actually has infinite energy.

06:48 In fact it's just the opposite that by definition remember we saw on the graph there that when something is far, far, far away from the proton, the n equals infinity level that the highest energy level you can get to is defined to be the 0 energy level.

07:02 Anything below that will have a negative energy.

07:04 And again we already saw the equation for what this energy is, it's always -13.6 eV divided by whatever energy level you're in 1 or 2 or 3 square.

The lecture Electron Energy Levels by Jared Rovny is from the course Electronic Structure.

### Included Quiz Questions

1. It increases, decreasing in magnitude by a factor of 4.
2. It increases, decreasing in magnitude by a factor of 2.
3. It decreases, increasing in magnitude by a factor of 4.
4. It decreases, increasing in magnitude by a factor of 2.
5. It decreases, decreasing in magnitude by a factor of 4.
1. Energy
2. Momentum
3. Electrons
4. Volts
5. Elementary Volts
1. 12.75 eV
2. 13.6 eV
3. 27.2 eV
4. -13.6 eV
5. 25.5 eV
1. It is unbound (freed).
2. It is entirely bound (orbiting).
3. It is as close to the nucleus as possible.
4. It would combine with the nucleus, causing radiation.
5. It is completely stationary in its orbit.

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