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Bohr model energy levels (derivation using physics)

Video transcript

if we continue with our Bohr model the next thing we've talked about are the different energy levels and so we're going to talk 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 the to see what we come up with in this video and to see how its applied all right 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 one-half MV squared where m is the mass of the electron and V is the velocity so if our electron is going this way around right if it's orbiting our nucleus right so this is our electron a negative charge the velocity vector and be tangent at this point all right and we know this electron is attracted to the nucleus so we have one proton in the nucleus right for a hydrogen atom using the Bohr model and we know we know that we're doing the Bohr model there's a certain radius associated with where that electron is all right so we know the electron is also attracted to the nucleus there's an electric force right so this this electron is pulled to the nucleus there's an attractive force this is the electric force 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 - we're going to 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 as a distance between our two charges all right we know that Newton's second law force is equal to the mass times the acceleration we're talking about the electron here is that 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 so I'll write that in here q1 q1 is the charge on a proton right which we know is elemental charge so it would be positive II all right q2 is the charge on the electron the charge on the electron it's the same magnitude as a 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 going to 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 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 the only care about the magnitude on the left side we get K e squared over R is equal to M V squared on the right side and you can see we're almost to what we want our goal was to try to find expression for the kinetic energy that's one-half 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 one-half MV squared is equal to the kinetic energy so we know that 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 going to find the potential energy so the potential energy of that electron and and that potential energy is given by this equation in physics so the electrical potential energy is equal to K and our same K times Q 1 to the charge of 1 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 it's just physics all right so let's let's plug in what we know so this would be equal to K q1 again q1 is the charge on the proton so that's positive II all right and q2 is the charge on the electron so that's negative II right so we have negative II and 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 K e squared over R so we define the the electrical potential energy equal to zero at infinity and so we need to keep this negative sign in because it's actually important all right so now we have the electrical potential energy and we have the kinetic energy and to find the total energy associated with that electron all right the total energy associate with that electron the total energy would be equal to so e total is equal to the kinetic energy right plus the potential energy so this would be this would be the electrical potential energy so let's plug in those values so 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 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 this was plus and then we have a negative value so we could just write minus ke squared over R so if you think about that math oh this is just like 1/2 minus 1 and so that's going to give you that's going to give you negative 1/2 1/2 minus 1 is negative 1/2 so negative 1/2 ke squared over R is our expression for the total energy right so this is the this is the total energy associated with our electron alright let's find the total enter gee when the radius is equal to r1 so what we talked about in the last video so the radius the radius of the electron in the ground state right and r1 we did that when we did that math we got five point three 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 right with that energy level and remember we got this r1 value right we got this r1 value by doing some math and I'm saying n is equal to one and plugging that into our equation right the the radius for any integer n is equal to N squared times r1 so when n is equal to 1 we plugged it into here and we got and we got our radius all right 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 is equal to 1 is equal to negative 1/2 times K which is 9 times 10 to the 9th times the elemental charge right 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 right if you do that calculation the energy associated with the ground state electron in a hydrogen atom is equal to negative two point one seven 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 all right so that's the lowest energy stage the ground states the energy is negative and I'll talk more about what the negative sign means in the next video all right so we could we could generalize this energy right we could say here we did it for equal to one but we could say that e e at any integer n is equal to and put an R sub n here let me just rewrite that equation so we could generalize this and say the energy love the energy at any energy level right is equal to negative 1/2 ke squared right our n ok 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 going to 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 all right so we're going to plug in N squared R 1 here so this would be N squared R 1 we can we can rewrite that this is the same thing as negative 1/2 ke squared over R 1 times 1 over N squared so I just rewrote this in in a certain way because I know what all this is equal to I know what negative 1/2 ke squared over R 1 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 right this is the energy associated with the first energy level and so we can 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 all right so that's what all that is equal to so there's another way to write our energy so energy is equal to negative two point one seven 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 all right and then we could we could write it in a slightly different way since that's equal to e1 we could just make it look even 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 so either one of these either one of these is fine okay 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 going to be using these equations or this equation it's really the same equation in the next video and we're going to come up with the different energies right the different energies at different energy levels so we're going to change what n is right and come up with a different energy so energy is quantized this is really important to think about this idea about energy being quantized and this is this is one reason why the Bohr model is nice to to look at because it gives us these quantized energy levels which actually explain some things as we'll see in later videos so in 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