Anderson Video - Hydrogen Atom

Professor Anderson
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Let's think again about the hydrogen atom for a second, this is one of our favorites because it's number one on the periodic table and all others are 2 or lower. So what is the hydrogen atom look like? Looks like this, positive charge, negative charge, electron swirling around the proton, at a speed V. And let's pretend that this is moving around at a constant speed and so this is uniform circular motion which we know and love. Later on when you get to quantum, you're going to realize it's not really a point particle going around like a tennis ball on a string but this is an okay model to start with. There is of course a force, F, that is holding it in that circle, that is Coulomb's force, and there is a distance R between the electron and the proton. So in a hydrogen atom let's put in some numbers, R is about half an angstrom, 10 to the minus 10 meters. Let's calculate the acceleration of the electron. Okay, how do we do that? Well, this thing is moving in a circle, if you are moving in a circle you have centripetal acceleration, we know that from last term, right? Anytime you're moving in a circle you have a centripetal acceleration and it's not because your speed is changing, it's because the direction of your velocity is changing. Your velocity was this way and then later on it's this way and then so forth. If you're changing the direction of your velocity that is an acceleration but we know what acceleration is and we know how it relates to force, right? Force is equal to MA, and we know what the force is in this case, it is Coulomb's law, so it is K q1 q2 over R squared. And now we can solve this thing for a. A is equal to K q1 q2 divided by MR squared and now we have all those numbers, so let's plug them in. We've got a K, which is 8 point 9 9 times 10 to the 9. We've got a q1 which is an electron. An electron has charge one point six times ten to the minus 19 coulombs. There's another one that's a proton but that also has the charge of the electron,. And we need to put a minus sign out in front of the whole thing because one of those is negative, the electron is negative, and then we have the mass. It's the mass of the thing that's moving in the circle, so it's the mass of the electron, and you can look up the mass of the electron it's nine point one times ten to the minus 31 kilograms. Remember kilograms is SI units. And then we have zero point five times ten to the minus ten is our distance, and we have to square that. Okay, and now you guys should pull out your calculator and plug in those numbers and tell me what you get and we're going to approximate it here. So we have a nine times ten to the nine and then we have one point six squared, has got to be close to 3. 10 the minus nineteen squared is 10 to the minus 38 and then we're gonna divide by nine times ten to the minus thirty-one and then we have 0.5 squared which gets us a quarter and ten to the minus ten squared becomes 10 to the minus 20. So if we punch all these in what should we get? Well 9s cancel out, 3 divided by 1/4 is the same as 3 times 4, we're a little bit heavy so let's lower it to not 12, but how about 10, and we've got a 10 to the 9, a 10 to the minus 38, that's a 10 to the minus twenty nine and then down in the bottom we have a 10 to the--I better write this down, 10 to the minus twenty nine, that's still on the screen over there? Divided by, that is another 10 and so that becomes 10 to the minus 30, and then we have a 10 to the minus 50, five-o. All right, good. And let's see what we get here. So we have a 10 times 10 to the minus twenty nine which becomes 10 to the minus 28 divided by a 10 to the minus 50 and so we end up with 10 to the 22. Is that right? Did we do that right? What'd you guys get on your calculator? Anybody punch in the numbers? Yeah? Okay, oh that's right because we had a ten right there. I forgot that one. Okay so you said negative one point what? Zero times 10 to the 23rd, okay. That's right and we had the negative sign hanging out in the front, negative sign is just the direction. Good. 10 to the 23 meters per second squared, that is the acceleration specifically it's the centripetal acceleration of the electron in its orbit. Which is really high that's a very big value, right? Gravity, how fast does gravity accelerate things downwards? 9.8 meters per second squared. This is twenty two orders of magnitude bigger, that is a very big number. What about the speed? We said that the electron is orbiting at speed V, what is that orbital speed? Let's calculate it. So Newton's law in circular motion. What did we have? We had the sum of the forces in the radial direction have to add up to MV squared over R and those forces have to equal the mass times the acceleration and therefore the acceleration, specifically centripetal acceleration, is V squared over R. We're talking about uniform circular motion, we're not accelerating the electron tangentially or slowing it down and so we just have some triple acceleration, we now have a number for that. This we're only worried about the magnitude, not the direction so we put in a positive number here and we can solve this now for V. V is R AC square root it. And we know R, we said R was a half an angstrom, 0.5 times 10 to the minus 10, and put some parentheses around that. AC we just found, it's 10 to the 23 SI units, we'll take the square root and that should do it. And if you punch that into your calculator, tell me what you get. It looks like it should be about 1 over root 2 and then we've got a 10 to the 13, but that's too hard to do. So if we make it this, 1/2 times 10 to the 13, that's exactly the same as 5 times 10 to the 14, and square root of 10 to the 14 is 10 to the 7, square root of 5--I'm sorry this should be 12, right? Good, it should be 12. Square root of 10 to the 12 is 10 to the 6, square root of 5 is a little bit bigger than 2, right? So I'm gonna say this is about 2.3 times 10 to the 6. And did anybody punch it into your calculator? Okay, Okay, so this should be a 2.2, so we were close. Two point two times 10 to the 6 meters per second. Okay, which is cruising, right? Electron is really really moving very fast in that hydrogen atom. Why? Because it's very close to the proton there's a really strong force on it and so to keep in its orbit it has be moving really fast. We remember that from gravitation, when the space shuttle is in a low orbit, it's moving very fast, if it goes up to a higher orbit, it moves slower. So electrons that are near the nucleus have to move very fast, when they get further away they can move quite a bit slower. Okay, that's a pretty fast number right? Speed of light is 3 times 10 to the 8th, meters per second, so this is like one hundredth the speed of light.
Let's think again about the hydrogen atom for a second, this is one of our favorites because it's number one on the periodic table and all others are 2 or lower. So what is the hydrogen atom look like? Looks like this, positive charge, negative charge, electron swirling around the proton, at a speed V. And let's pretend that this is moving around at a constant speed and so this is uniform circular motion which we know and love. Later on when you get to quantum, you're going to realize it's not really a point particle going around like a tennis ball on a string but this is an okay model to start with. There is of course a force, F, that is holding it in that circle, that is Coulomb's force, and there is a distance R between the electron and the proton. So in a hydrogen atom let's put in some numbers, R is about half an angstrom, 10 to the minus 10 meters. Let's calculate the acceleration of the electron. Okay, how do we do that? Well, this thing is moving in a circle, if you are moving in a circle you have centripetal acceleration, we know that from last term, right? Anytime you're moving in a circle you have a centripetal acceleration and it's not because your speed is changing, it's because the direction of your velocity is changing. Your velocity was this way and then later on it's this way and then so forth. If you're changing the direction of your velocity that is an acceleration but we know what acceleration is and we know how it relates to force, right? Force is equal to MA, and we know what the force is in this case, it is Coulomb's law, so it is K q1 q2 over R squared. And now we can solve this thing for a. A is equal to K q1 q2 divided by MR squared and now we have all those numbers, so let's plug them in. We've got a K, which is 8 point 9 9 times 10 to the 9. We've got a q1 which is an electron. An electron has charge one point six times ten to the minus 19 coulombs. There's another one that's a proton but that also has the charge of the electron,. And we need to put a minus sign out in front of the whole thing because one of those is negative, the electron is negative, and then we have the mass. It's the mass of the thing that's moving in the circle, so it's the mass of the electron, and you can look up the mass of the electron it's nine point one times ten to the minus 31 kilograms. Remember kilograms is SI units. And then we have zero point five times ten to the minus ten is our distance, and we have to square that. Okay, and now you guys should pull out your calculator and plug in those numbers and tell me what you get and we're going to approximate it here. So we have a nine times ten to the nine and then we have one point six squared, has got to be close to 3. 10 the minus nineteen squared is 10 to the minus 38 and then we're gonna divide by nine times ten to the minus thirty-one and then we have 0.5 squared which gets us a quarter and ten to the minus ten squared becomes 10 to the minus 20. So if we punch all these in what should we get? Well 9s cancel out, 3 divided by 1/4 is the same as 3 times 4, we're a little bit heavy so let's lower it to not 12, but how about 10, and we've got a 10 to the 9, a 10 to the minus 38, that's a 10 to the minus twenty nine and then down in the bottom we have a 10 to the--I better write this down, 10 to the minus twenty nine, that's still on the screen over there? Divided by, that is another 10 and so that becomes 10 to the minus 30, and then we have a 10 to the minus 50, five-o. All right, good. And let's see what we get here. So we have a 10 times 10 to the minus twenty nine which becomes 10 to the minus 28 divided by a 10 to the minus 50 and so we end up with 10 to the 22. Is that right? Did we do that right? What'd you guys get on your calculator? Anybody punch in the numbers? Yeah? Okay, oh that's right because we had a ten right there. I forgot that one. Okay so you said negative one point what? Zero times 10 to the 23rd, okay. That's right and we had the negative sign hanging out in the front, negative sign is just the direction. Good. 10 to the 23 meters per second squared, that is the acceleration specifically it's the centripetal acceleration of the electron in its orbit. Which is really high that's a very big value, right? Gravity, how fast does gravity accelerate things downwards? 9.8 meters per second squared. This is twenty two orders of magnitude bigger, that is a very big number. What about the speed? We said that the electron is orbiting at speed V, what is that orbital speed? Let's calculate it. So Newton's law in circular motion. What did we have? We had the sum of the forces in the radial direction have to add up to MV squared over R and those forces have to equal the mass times the acceleration and therefore the acceleration, specifically centripetal acceleration, is V squared over R. We're talking about uniform circular motion, we're not accelerating the electron tangentially or slowing it down and so we just have some triple acceleration, we now have a number for that. This we're only worried about the magnitude, not the direction so we put in a positive number here and we can solve this now for V. V is R AC square root it. And we know R, we said R was a half an angstrom, 0.5 times 10 to the minus 10, and put some parentheses around that. AC we just found, it's 10 to the 23 SI units, we'll take the square root and that should do it. And if you punch that into your calculator, tell me what you get. It looks like it should be about 1 over root 2 and then we've got a 10 to the 13, but that's too hard to do. So if we make it this, 1/2 times 10 to the 13, that's exactly the same as 5 times 10 to the 14, and square root of 10 to the 14 is 10 to the 7, square root of 5--I'm sorry this should be 12, right? Good, it should be 12. Square root of 10 to the 12 is 10 to the 6, square root of 5 is a little bit bigger than 2, right? So I'm gonna say this is about 2.3 times 10 to the 6. And did anybody punch it into your calculator? Okay, Okay, so this should be a 2.2, so we were close. Two point two times 10 to the 6 meters per second. Okay, which is cruising, right? Electron is really really moving very fast in that hydrogen atom. Why? Because it's very close to the proton there's a really strong force on it and so to keep in its orbit it has be moving really fast. We remember that from gravitation, when the space shuttle is in a low orbit, it's moving very fast, if it goes up to a higher orbit, it moves slower. So electrons that are near the nucleus have to move very fast, when they get further away they can move quite a bit slower. Okay, that's a pretty fast number right? Speed of light is 3 times 10 to the 8th, meters per second, so this is like one hundredth the speed of light.