Bohr's model of the hydrogen atom
Bohr model energy levels (derivation using physics)
- If we continue with our Bohr model, the next thing we have to talk about are the different energy levels. And so we're gonna be talking about energy in this video, and once again, there's a lot of derivation using physics, so you can jump ahead to the next video to see what we come up with in this video, to see how it's applied. Alright, so we need to talk about energy, and first, we're going to try to find the kinetic energy of the electron, and we know that kinetic energy is equal to: 1/2 mv squared, where "m" is the mass of the electron, and "v" is the velocity. So, if our electron is going this way around, if it's orbiting our nucleus, so this is our electron, the negative charge, the velocity vector, it'd be tangent at this point. And we know that this electron is attracted to the nucleus. We have one proton in the nucleus for a hydrogen atom, using the Bohr model, and we know, we know, that if we're doing the Bohr model, there's a certain radius associated with where that electron is. So we know the electron is also attracted to the nucleus. There's an electric force, alright, so this electron is pulled to the nucleus, this is an attractive force. This is the electric force, this is a centripetal force, the force that's holding that electron in a circular orbit around the nucleus here. And, once again, we talked about the magnitude of this electric force in an earlier video, and we need it for this video, too. We're gonna use it to come up with the kinetic energy for that electron. So the electric force is given by Coulomb's Law, the magnitude of the electric force is equal to K, which is a constant, "q1", which is, let's say it's the charge on the proton, times "q2", charge on the electron, divided by "r squared", where "r" is the distance between our two charges. We know that Newton's Second Law: force is equal to the mass times the acceleration. We're talking about the electron here, so the mass of the electron times the acceleration of the electron. The electric force is a centripetal force, keeping it in circular motion, so we can say this is the "centripetal acceleration". Alright, let's go ahead and write down what we know. "K" is a constant, we'll write that in here, "q1", "q1" is the charge on a proton, which we know is elemental charge, so it would be positive "e"... "q2" is the charge on the electron. The charge on the electron is the same magnitude as the charge on the proton, but it's a negative value. So we have negative "e", is the charge on the electron, divided by "r squared", is equal to the mass of the electron times the centripetal acceleration. So, centripetal acceleration is equal to "v squared" over "r". So, we did this in a previous video. We're gonna do the exact same thing we did before. We only care about the magnitude of the electric force because we already know the direction is always going to be towards the center, and therefore, we only care... we don't care about this negative sign here. We can also cancel one of the "r"s. So if we don't care about... if we only care about the magnitude, on the left side, we get: Ke squared over r is equal to mv squared, on the right side. And you can see, we're almost to what we want. Our goal was to try to find the expression for the kinetic energy, that's 1/2 mv squared. Here, we have mv squared, so if we multiply both sides by 1/2, right, multiply both sides by 1/2, now we have an expression for the kinetic energy of the electron. So: 1/2 mv squared is equal to the kinetic energy. So we know the kinetic energy is equal to: 1/2 Ke squared over r Alright, so we will come back to the kinetic energy. Next, we're gonna find the potential energy. So the potential energy of that electron. And that potential energy is given by this equation in physics. So the electrical potential energy is equal to: "K", our same "K", times "q1", so the charge of one... so we'll say, once again, that's the charge of the proton, times the charge of the electron, divided by the distance between them. So again, it's just physics. So let's plug in what we know. This would be equal to K. "q1", again, "q1" is the charge on the proton, so that's positive "e", and "q2" is the charge on the electron, so that's negative "e", negative "e", divided by "r". This time, we're going to leave the negative sign in, and that's a consequence of how we define electrical potential energy. So we get: negative Ke squared over r So we define the electrical potential energy equal to zero at infinity. And so we need to keep this negative sign in, because it's actually important. Alright, so now we have the electrical potential energy, and we have the kinetic energy. And to find the total energy associated with that electron, the total energy associated with that electron, the total energy would be equal to: so, E-total is equal to the kinetic energy, plus the potential energy. So this would be the electrical potential energy. So let's plug in those values. We found the kinetic energy over here, 1/2 Ke squared over r, so we plug that into here, and then we also found the electrical potential energy is: negative Ke squared over r, so we plug that in, and now we can calculate the total energy. So we get some more room... The total energy is equal to: 1/2 Ke squared over r, our expression for the kinetic energy, and then, this was plus, and then we have a negative value, so we just write: minus Ke squared over r So, if you think about the math, this is just like 1/2 minus one, and so that's going to give you negative 1/2. 1/2 - 1 = -1/2 So "negative 1/2 Ke squared over r" is our expression for the total energy. So this is the total energy associated with our electron. Alright, let's find the total energy when the radius is equal to r1. What we talked about in the last video. The radius of the electron in the ground state. And r1, when we did that math, we got: 5.3 times 10 to the negative 11 meters. And so, we're going to be plugging that value in for this r. So we can calculate the total energy associated with that energy level. And remember, we got this r1 value, we got this r1 value, by doing some math and saying, n = 1, and plugging that into our equation. The radius for any integer, n, is equal to n squared times r1. So when n = 1, we plugged it into here and we got our radius. So let's go ahead and plug that in. Let's do the math, actually. So, we're going to get the total energy for the first energy level, so when n = 1, it's equal to negative 1/2 times K, which is nine times 10 to the 9th, times the elemental charge. Alright, so we just took care of K, E is the magnitude of charge on a proton or an electron, which is equal to 1.6 times 10 to the negative 19 Coulombs, we're going to square that, and then put that over the radius, which was 5.3 times 10 to the negative 11 meters. And to save time, I won't do that math here, but if you do that calculation, if you do that calculation, the energy associated with the ground state electron of a hydrogen atom, is equal to: negative 2.17 times 10 to the negative 18 and the units would be joules. So if you took the time to do all those units, you would get joules here. So that's the lowest energy state, the ground state. The energy is negative, and I'll talk more about what the negative sign means in the next video. Alright, so we could generalize this energy. We could say, here we did it for n = 1, but we could say that: E at any integer "n", is equal to, then put an "r sub n" here. Let me just re-write that equation. So we could generalize this and say: the energy at any energy level is equal to negative 1/2 Ke squared, r n. Okay, so we could now take this equation, right here, the one we talked about and actually derived in the earlier video, and plug all of this in for our "n". So we're gonna plug all of that into here. So let's get some more room, and continue... So the energy at an energy level "n", is equal to negative 1/2 Ke squared, over, right? So we're gonna plug in "n squared r1" here. So this would be: n squared r1 We can re-write that. This is the same thing as: negative 1/2 Ke squared over r1 times one over n squared. So I just re-wrote this in a certain way because I know what all of this is equal to. I know what negative 1/2 Ke squared over r1 is equal to. We just did the math for that. Alright, so this is negative 1/2 Ke squared over r1. And so we got this number: this is the energy associated with the first energy level. And so we can go ahead and plug that in. We can plug in this number. We can take this number and plug it in for all of this. So that's what all of that is equal to. So, here's another way to write our energy. So, energy is equal to: negative 2.17 times 10 to the negative 18 and then this would be: times one over n squared. So we can just put it over n squared like that. And then we could write it in a slightly different way. Since that's equal to E1, we could just make it look even shorter here. We could just say that: The energy at energy level n... So: the energy at energy level n is equal to the energy associated with the first energy level divided by n squared. Either one of these is fine. So we could write it like this, or we could write it like this, it doesn't really matter which one you use, but we're gonna be using these equations, or this equation, it's really the same equation, in the next video, and we're gonna come up with the different energies, the different energies at different energy levels. So we're gonna change what "n" is and come up with a different energy. So energy is quantized. Now, this is really important to think about this idea of energy being quantized. And this is one reason why the Bohr model is nice to look at, because it gives us these quantized energy levels, which actually explains some things, as we'll see in later videos. So the next video, we'll continue with energy, and we'll take these equations we just derived, and we'll talk some more about the Bohr model of the hydrogen atom.
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