Bohr's model of the hydrogen atom
Emission spectrum of hydrogen
- [Voiceover] I'm sure that most of you know the famous story of Isaac Newton where he took a narrow beam of light and he put that narrow beam of light through a prism and the prism separated the white light into all the different colors of the rainbow. And so if you did this experiment, you might see something like this rectangle up here so all of these different colors of the rainbow and I'm gonna call this a continuous spectrum. It's continuous because you see all these colors right next to each other. So they kind of blend together. So that's a continuous spectrum If you did this similar thing with hydrogen, you don't see a continuous spectrum. So, if you passed a current through a tube containing hydrogen gas, the electrons in the hydrogen atoms are going to absorb energy and jump up to a higher energy level. When those electrons fall down to a lower energy level they emit light and so we talked about this in the last video. This is the concept of emission. If you use something like a prism or diffraction grating to separate out the light, for hydrogen, you don't get a continuous spectrum. You'd see these four lines of color. So, since you see lines, we call this a line spectrum. So this is the line spectrum for hydrogen. So you see one red line and it turns out that that red line has a wave length. That red light has a wave length of 656 nanometers. You'll also see a blue green line and so this has a wave length of 486 nanometers. A blue line, 434 nanometers, and a violet line at 410 nanometers. And so this emission spectrum is unique to hydrogen and so this is one way to identify elements. And so this is a pretty important thing. And since line spectrum are unique, this is pretty important to explain where those wavelengths come from. And we can do that by using the equation we derived in the previous video. So I call this equation the Balmer Rydberg equation. And you can see that one over lamda, lamda is the wavelength of light that's emitted, is equal to R, which is the Rydberg constant, times one over I squared, where I is talking about the lower energy level, minus one over J squared, where J is referring to the higher energy level. For example, let's say we were considering an excited electron that's falling from a higher energy level n is equal to three. So let me write this here. So we have an electron that's falling from n is equal to three down to a lower energy level, n is equal to two. All right, so it's going to emit light when it undergoes that transition. So let's look at a visual representation of this. Now let's see if we can calculate the wavelength of light that's emitted. All right, so if an electron is falling from n is equal to three to n is equal to two, I'm gonna go ahead and draw an electron here. So an electron is falling from n is equal to three energy level down to n is equal to two, and the difference in those two energy levels are that difference in energy is equal to the energy of the photon. And so that's how we calculated the Balmer Rydberg equation in the previous video. All right, let's go ahead and calculate the wavelength of light that's emitted when the electron falls from the third energy level to the second. So, we have one over lamda is equal to the Rydberg constant, as we saw in the previous video, is one point zero nine seven times ten to the seventh. The units would be one over meter, all right? One over I squared. So, I refers to the lower energy level, all right? So the lower energy level is when n is equal to two. So we plug in one over two squared. And then, from that, we're going to subtract one over the higher energy level. That's n is equal to three, right? So this would be one over three squared. So one over two squared minus one over three squared. Let's go ahead and get out the calculator and let's do that math. So one over two squared, that's one fourth, so that's point two five, minus one over three squared, so that's one over nine. So, one fourth minus one ninth gives us point one three eight repeating. And if we multiply that number by the Rydberg constant, right, that's one point zero nine seven times ten to the seventh, we get one five two three six one one. So let me go ahead and write that down. So now we have one over lamda is equal to one five two three six one one. So to solve for lamda, all we need to do is take one over that number. So one over that number gives us six point five six times ten to the negative seven and that would now be in meters. So we have lamda is equal to six point five six times ten to the negative seventh meters. So let's convert that into, let's go like this, let's go 656, that's the same thing as 656 times ten to the negative ninth meters. And so that's 656 nanometers. 656 nanometers, and that should sound familiar to you. All right, so let's go back up here and see where we've seen 656 nanometers before. 656 nanometers is the wavelength of this red line right here. So, that red line represents the light that's emitted when an electron falls from the third energy level down to the second energy level. So let's go back down to here and let's go ahead and show that. So we can say that a photon, right, a photon of red light is given off as the electron falls from the third energy level to the second energy level. So that explains the red line in the line spectrum of hydrogen. So how can we explain these other lines that we see, right? So we have these other lines over here, right? We have this blue green one, this blue one, and this violet one. So if you do the math, you can use the Balmer Rydberg equation or you can do this and you can plug in some more numbers and you can calculate those values. So those are electrons falling from higher energy levels down to the second energy level. So let's go ahead and draw them on our diagram, here. So, let's say an electron fell from the fourth energy level down to the second. All right, so that energy difference, if you do the calculation, that turns out to be the blue green line in your line spectrum. So, I'll represent the light emitted like that. And if an electron fell from the fifth energy level down to the second energy level, that corresponds to the blue line that you see on the line spectrum. And then, finally, the violet line must be the transition from the sixth energy level down to the second, so let's go ahead and draw that in. And so now we have a way of explaining this line spectrum of hydrogen that we can observe. And since we calculated this Balmer Rydberg equation using the Bohr equation, using the Bohr model, I should say, the Bohr model is what allowed us to do this. So the Bohr model explains these different energy levels that we see. So when you look at the line spectrum of hydrogen, it's kind of like you're seeing energy levels. At least that's how I like to think about it 'cause you're, it's the only real way you can see the difference of energy. All right, so energy is quantized. We call this the Balmer series. So this is called the Balmer series for hydrogen. But there are different transitions that you could do. For example, let's think about an electron going from the second energy level to the first. All right, so let's get some more room here If I drew a line here, again, not drawn to scale. Think about an electron going from the second energy level down to the first. So from n is equal to two to n is equal to one. Let's use our equation and let's calculate that wavelength next. So this would be one over lamda is equal to the Rydberg constant, one point zero nine seven times ten to the seventh, that's one over meters, and then we're going from the second energy level to the first, so this would be one over the lower energy level squared so n is equal to one squared minus one over two squared. All right, so let's get some more room, get out the calculator here. So, one over one squared is just one, minus one fourth, so that's point seven five and so if we take point seven five of the Rydberg constant, let's go ahead and do that. So one point zero nine seven times ten to the seventh is our Rydberg constant. Then multiply that by point seven five, right? So three fourths, then we should get that number there. So that's eight two two seven five zero zero. So let's write that down. One over the wavelength is equal to eight two two seven five zero. So to solve for that wavelength, just take one divided by that number and that gives you one point two one times ten to the negative seven and that'd be in meters. So the wavelength here is equal to one point, let me see what that was again. One point two one five. One point two one five times ten to the negative seventh meters. And so if you move this over two, right, that's 122 nanometers. So this is 122 nanometers, but this is not a wavelength that we can see. So 122 nanometers, right, that falls into the UV region, the ultraviolet region, so we can't see that. We can see the ones in the visible spectrum only. And so this will represent a line in a different series and you can use the Balmer Rydberg equation to calculate all the other possible transitions for hydrogen and that's beyond the scope of this video. So, here, I just wanted to show you that the emission spectrum of hydrogen can be explained using the Balmer Rydberg equation which we derived using the Bohr model of the hydrogen atom. So even thought the Bohr model of the hydrogen atom is not reality, it does allow us to figure some things out and to realize that energy is quantized.
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