Bohr model radii

if you didn't watch the last video because there's 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 assume 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 alright so we calculated the radius of this circle and we said this was equal to R 1 so R 1 turned out to be 5 point 3 times 10 to the negative 11th 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 1 for example if you wanted to calculate R 1 again so so the first allowed radius using the Bohr model is equal to 1 squared times R 1 and so obviously 1 squared is 1 so R 1 is equal to 5 point 3 times 10 to the negative 11 meters and so when n is equal to 1 right 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 all right so this is a very important 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 alright so 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 alright times Planck's constant divided by 2 pi M times R and we figured this out using using Bohr's assumption for quantized angular momentum and 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 right because we're going to take we're going to take this rate and we're going to plug it in down here and then we know what these other numbers are right so we said n was equal to one so we're talking about and is equal to one so we're going to plug in a 1/2 here alright so this would be a 1 the velocity is equal to 1 times Planck's constant 6.6 2 6 times 10 to the negative 34 all right divided by divided by 2 pi times M and we're talking about the electron right so n was the mass of our electron which is 9 point 1 1 times 10 to the negative 31st kilograms and finally for n is equal to 1 this was our allowed radius so we can plug this in for our radius 5.3 times 10 to the negative 11 all right so if you do all that math all right 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 so approximately equal to approximate sign 2.2 times 10 to the sixth and your units should work out 2 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 all right so this is the the radius of the smallest orbit allowed using the Bohr model but you can have other radii and we can calculate the radii have larger orbits using this equation right so we're just going to use different values for n so we started off with n is equal to 1 let's see it's the same equation and let's do n is equal to 2 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 1 let's do n is equal to 2 here so n is equal to 2 so let's go ahead and plug in 2 so we'd have 2 squared times R 1 so R 2 all right the second allowed radii or the second allowed radius I should say is equal to 4 times are 1 so we're thinking about a picture right let's say this is the nucleus here and then this tiny little radius here is our one we want to sketch in the second allowed one to be four times that so I'm just going to approximate let's say that that radius is four times at least this is R 2 which is equal to 4 times R 1 and so we sketch in the radius of this this next radius here this is next allowed radius using the Bohr model we could figure out mathematically what that's equal to because we know R 1 is equal to five point three times 10 to the negative 11 meters and so if you do that calculation 4 times that number gives you approximately two point one two times 10 to the negative 10 meters all right so this is our our second our second radius right so when n is equal to 2 let's do one more so when n is equal to 3 so let's get a little bit more room here so when n is equal to 3 right this radius would be equal to 3 squared times R 1 so once again we're just taking 3 and plugging it into here and so 3 squared is of course 9 so this would be equal to 9 times R 1 so our next allowed radius would be 9 times R 1 and I'm sure I won't get this accurate but you just it's a lot bigger all right so the R 3 is equal to 9 times R 1 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 9 times 5 point 3 times 10 to the negative 11 meters will give you approximately 4 point 7 7 times 10 to the negative 10 meters and so these are the different allowed radii using the Bohr model so you could say that the orbit radii are quantized only certain radii are allowed so you couldn't get something in between right you couldn't you couldn't have something like in between and here according to the Bohr model so this is not possible and these radii are associated with different energies and that's going to be really important and that's really why we're doing these calculations so we're getting these these different radii here and each one of these 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