Bohr's model of the hydrogen atom
Bohr model radii
- [Voiceover] If you didn't watch the last video because there was too much physics, I'll just quickly summarize what we talked about. We went over the Bohr model of the hydrogen atom, which has one proton in its nucleus, so here's the positive charge in the nucleus, and a negatively charged electron orbiting the nucleus. And even though this is not reality, the Bohr model is not exactly what's happening, it is a useful model to think about. And so we just assumed the electron was going in this direction So counter-clockwise around which gives our electron a velocity tangent to our circle, which we said was v in the last video. And in the last video, we calculated this radius. So we calculated the radius of this circle, and we said this was equal to r one. So r one turned out to be five point three times 10 to the negative eleventh meters, which is an important number. And we also derived this equation, right. So r for any integer n is equal to n squared times r one, for example, if you wanted to calculate r one again. So the first allowed radius using the Bohr model is equal to one squared times r one. And so obviously one squared is one so r one is equal to five point three times 10 to the negative eleven meters. And so when n is equal to one, we said this was an electron in the ground state, in the lowest energy state for hydrogen. We'll talk about energy states in the next two videos. So this is a very important number here. So this is, this number right here, is the radius of the smallest orbit in the Bohr model. In the previous video, we also calculated the velocity or we came up with an equation that you could use to calculate the velocity of that electron. If you go back to the previous video, you'll see the equation that we derived was the velocity is equal to the integer n times Planck's constant divided by two pi m times r, and we figured this out using Bohr's assumption for quantised angular momentum and the classical idea of angular momentum. So if we plug in some numbers here, we can actually solve for the velocity of this electron cause we're gonna take this radius and we're gonna plug it in down here and then we know what these other numbers are. So we said n was equal to one, so we're talking about n is equal to one so we're going to plug a one to here. So this will be a one. The velocity is equal to one times Planck's constant, six point six two six times 10 to the negative 34 divided by two pi times m. And we're talking about the electron so m was the mass of our electron, which is nine point one one times 10 to the negative 31st kilograms. And finally, for n is equal to one, this was our allowed radius so we can plug this in for our radius, five point three times 10 to the negative 11. So if you do all that math, I won't take the time to do it here, but you'll get a velocity approximately equal to, approximate sign, two point two times 10 to the sixth and your units should work out to meters per second so that's the velocity. So going by the Bohr model, you can calculate the radius of this circle here so you can calculate this radius, and you can also calculate the velocity. And,again, this isn't reality but we'll use these numbers in later videos so it's important to figure out where they came from. So this is the radius of the smallest orbit allowed using the Bohr model but you can have other radii, and we can calculate the radii of larger orbits using this equation. So we're just gonna use different values for n. So we started off with n is equal to one. Let's use the same equation and let's do n is equal to two. So let me go ahead and rewrite that equation down here. Let's get some room. So r for any integer n is equal to n squared times r one. Let's do n is equal to two here. So n is equal to two so let's go ahead and plug in two. So we'd have two squared times r one. So r two, the second allowed radii or the second allowed radius I should say, is equal to four times r one. So if we're thinking about a picture, let's say this is the nucleus here and then this tiny, little radius here is r one. If we wanted to sketch in the second allowed one, it would be four times that so I'm just going to approximate. Let's say that radius is four times that so this is r two, which is equal to four times r one. And so we sketch in the radius of this next radius here, this next allowed radius, using the Bohr model. We could figure out mathematically what that's equal to because we know r one is equal to five point three times 10 to the negative 11 meters. And so if you do that calculation, four times that number gives you approximately two point one two times 10 to the negative 10th meters. So this is our second radius when n is equal to two. Let's do one more when n is equal to three so let's get a little bit more room here. So when n is equal to three, this radius will be equal to three squared times r one. So once again, we're just taking three and plugging it into here and so three squared is, of course, nine. So this would be equal to nine times r one so our next allowed radius will be nine times r one. And I'm sure I won't get this accurate, but it's a lot bigger. So this will be r three is equal to nine times r one. So I won't even attempt to draw in that circle, but you get the idea. And we could do that math as well, so nine times five point three times 10 to the negative 11 meters would give you approximately four point seven seven times 10 to the negative 10th meters. And so these are the different allowed radii using the Bohr model so you can say that the orbit radii are quantised, only certain radii are allowed so you couldn't get something in between. You couldn't have something in between in here according to the Bohr model so this is not possible. And these radii are associated with different energies and that's gonna be really important and that's really why we're doing these calculations. So we're getting these different radii here and each one of these radii is associated with a different energy. So, again, more to come in the next few videos about energy, which is probably more important.
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