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# Bohr model radii (derivation using physics)

## Video transcript

and the Bohr model of the hydrogen atom we have one proton in the nucleus so I draw on a positive charge here and a negatively charged electron orbiting the nucleus so kind of like the planets orbiting the Sun and even though the Bohr model is not reality it is useful for for a concept of the atom so it's useful to calculate say for example we can calculate the radius of this circle and we're actually going to do that in this video so it's worth going into some of the details but I should warn you that this is a lot of physics in this video as well so if you don't like physics you can jump to the next video where I show you the result of what we're going to calculate in this video so going back to the electron here let's say the electron is going around counterclockwise and so the velocity of that electron at this point is tangent to the circle so that's the direction of the velocity vector so the electron has mass M let's say and the electron is going to feel a force it's going to be attracted to the nucleus alright so opposite charges attract and so this negatively charged electron is going to feel a force towards the center of the circle so that's a centripetal force and in this case we're talking about the electric force all right so this is the electric force that's causing the the electron to move in a circle we can find the electric force by using Coulomb's law over here on the left this is Coulomb's law so the electric force is equal to K which is a constant times q1 which is one of the charges let's just say that q1 is the charge on the proton times the other charge q2 which we'll say is the charge on the electron divided by the distance between those two charges squared so this is Coulomb's law let's go ahead and plug in what we know so far so K is some constant which we'll get to later q1 I said was the charge on the proton and the charge in the proton will say is e for now so elemental charge q2 I said was the charge on the electron and that the electron has the same magnitude of charge as the proton but it's negative so we put in an egg II hear all right divided by the distance between the two charges squared so force is equal to mass times acceleration using Newton's second law so M is the mass of the electron alright and this is uh this would be the centripetal acceleration since we're talking about a centripetal force and we know that the centripetal acceleration is equal to V squared over R so we go ahead and plug in M times V squared over R immediately we can cancel out one of the R's and since we only care about the magnitude of the force right we know the direction of the electric force we don't really care about this negative sign so we can just say we only care about the magnitude of the electric force here so we can go ahead and simplify a little bit this is B K e squared over R on the left and on the right this is B M V squared so continuing with some more classical physics right next we're gonna talk about angular momentum which is a tricky concept so angular momentum is capital L and the one equation for it is R cross P where R is a vector and P is the linear momentum right so linear momentum is equal to the mass times the velocity so we're talking about the linear momentum of the electron so the mass of the electron times the velocity of the electron and let's go ahead and plug this in for angular momentum so we're going to take the angular momentum about the center of our circle here so the angular momentum at the center so R is a vector let me go ahead and draw our and so R is a vector all right it's the distance from the center to where our electron is so we have R right there this is the R vector so I put an R and it cross this is a cross product so this would be times the linear momentum so times P which is MV times the sine of the angle between the two vectors all right so let's let's think about the other vector here so the other vector is the momentum vector so we took care of the R effect or the momentum vector is in the same direction as the velocity all right so this is the direction of the velocity that's also the direction of linear momentum vector it's the angle between those two vectors all right the angle between those two vectors is obviously 90 degrees so sine of 90 is 1 and so we can just say the angular momentum is equal to R M V times 1 so niels bohr thought that this angular momentum should be quantized and so what he did was he set this angular momentum equal to some integer so like 1 2 or 3 or you can keep going but let's just say an integer n times H which is Planck's constant divided by 2 pi so this is what Bohr came up with and he took he took this and he solved for the velocity so let's go ahead and do that so on the right we're just going to solve for V all right so the velocity is equal to this would just be n times H divided by 2 pi M R so we just solve for V and then we're going to take that alright so we just solved for V and we're going to plug that into our other equation over here on the left and so let's go ahead and do that we would have K e squared over R and the right would have M times all of that right n times H over 2 pi M R and then we'd square all of that so let's go ahead and get some more room and let's continue with our algebra here so we'd have K e squared over R is equal to the mass times so square everything in parentheses so N squared H squared 4 PI squared M Squared R squared all right we can cancel a few things we can cancel out one of these M's here and we can cancel out one of these ours so now we would have on the left side ke squared is equal to N squared H squared over 4pi squared and we would have one M left and one are left all right so the goal of all this is to solve for the radius of that circle and so to solve for R we could start by multiplying both sides by 4pi squared M R so we would get K e squared 4 PI squared M R on the left side the right side we would get N squared H squared so we're going to solve for R all right so let's go ahead and do that so R would be equal to N squared H squared over all right this would be over K e squared 4 PI squared M all right so now next we're going to take all of this stuff and we are going to plug in what those numbers are so for example H is Planck's constant all right so we know what that is that's six point six two six times ten to the negative 34 and we're going to be squaring that number and that's going to be overall this okay all right if you take in physics you know that K is equal to nine times 10 to the ninth it's a constant e is elemental charge right the magnitude of charge on a proton or an electron is one point six times 10 to the negative 19 coulombs all right so we put that in there and that number needs to be squared as well so we have a 4pi squared in there remember M was the mass of the electron so you can look up the mass of electron it's nine point one one times ten to the negative 31st kilograms all right and so that's that's a lot of math and rather than take out the calculator and show you you can do that on yourself and you'll see that that number comes out to be this comes out to be this is equal to I'll put it down here 5.3 times 10 to the negative 11 and if you had time to do all the units you would get meters for this so go ahead and do that calculation yourself and you would see that you get that number that's a very important number let's plug that in all right to what we have so far on the left so the radius is equal to U and squared times that number now so 5.3 times 10 to the negative 11 and let's go ahead and plug in n is equal to 1 so an integer so this represents a ground state electron in hydrogen so if n is equal to 1 all right this would be this would be R 1 is equal to 1 squared times this number and so obviously that's very simple math we know we know that the radius right when n is equal to 1 the radius is equal to this number 5 point 3 times 10 to the negative 11 meters so let's go back up here let's go back up here to the picture so I can show you what we're talking about why that's an important value so this is what we just calculated we calculated we calculated this radius right for a ground state electron in hydrogen so we calculate this distance and we called it r1 all right so the idea of niels bohr right by quantizing angular momentum right that's going to limit your radii the different radii that you could have so let's go ahead and and generalize this equation so we could say R right for any for any integer n be equal to N squared times this number times R 1 so n squared times R 1 which would just calculate to be 5 point 3 times 10 to the negative 11 meters and so this is a very important right so R for any integer n is equal to n squared times R 1 and this means only certain radii are allowed because Niels Bohr quantized the angular momentum right so you have to have you have to have specific radii and we'll talk about the other radii in the next video so this video after all that physics we got this equation and we're going to use that to go into more detail about the Bohr model radii