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Current time:0:00Total duration:11:30

uranium consists of two major isotopes so we have uranium-235 and uranium-238 and let's say you wanted to separate the two isotopes from each other one way to do it is to use a mass spectrometer and a variation of this was done in World War two to separate uranium isotopes you can separate them because they have different masses uranium 238 has more neutrons in uranium-235 and so therefore it has more mass all right so let's see how a mass spectrometer works so the first step is ionization so we're going to assume that we knock off one electron from each one of our isotopes here so we knock off an electron from each of these isotopes we're going to form a positive charge we're going to form a positive ions I'm a drawing imma draw in a positive charge right here the next step is to accelerate the ions over a potential difference and so we're going to accelerate the ions over potential difference Delta V and I should warn you there's going to be a whole lot of physics in this video so hopefully you are comfortable with with that all right so we accelerate the ions over a potential difference and that means we're going to get a final velocity of those ions so final velocity V so let's see how to calculate that final velocity well the amount of work that's done in order to accelerate the ions is equal to Q which is the charge on the ions times the potential difference so Q on the ion times Delta V which is the potential difference that's equal to the kinetic energy of the ion so that's one-half MV squared where m is the mass and V is the velocity all right let's solve for the velocity of the ion so let's solve for V just do some algebra so V squared would be equal to this would be 2 Q Delta V alright divided by M and so we could just take the square root of both sides so the velocity is equal to the square root of two times the charge and this is really Delta V but I'm just going to write I'm just going to write capital big V here now potential is different from velocity all right so so so this V is different from this V alright so this is divided by the mass that's the final velocity of the ion so when the ion enters this portion of the mass spectrometer it's moving in this direction with a velocity V this region the mass spectrometer has a magnetic field in it so let's say we have a uniform magnetic field that's pointing directly out of the page so it's coming straight at you and so this is meant to represent like if you're looking at the tip of an arrow like if you point an arrow directly at your eye right you would see the tip of the arrow pointing at you so this is the magnetic field and in physics we represent magnetic field with B all right the moving ion is going to experience a magnetic force due to the presence of that magnetic field and that magnetic force is equal to Q which is the charge of the ion once again V cross B so let's run through these things q is the charge on the ion V is the velocity of the ion and B is the magnetic field all right so the first thing that we're going to do is figure out the direction of the magnetic force on that ion and to do that we use one version of the right-hand rule so V is our first vector and that's the velocity vector and the velocity vector is in the plane of the page directed up and so you can see I have my fingers pointing in the direction of that velocity vector so it's flat in the plane of the page but it's pointing towards the top all right next we think about our second vector right that's our magnetic field vector which is coming directly out at us right so it's like pointing at us that's straight out of the page and so I curl my fingers in the direction of that second vector so if you think about if you think about my finger here alright think about like that being the tip of my finger right so we're curling it in the direction of this vector which is pointing out of the page all right finally finally once we finish curling our fingers here the thumb your thumb of your right hand is forced to point in the direction of the magnetic force on a positive charge all right so my thumb has to point to the right here so that's the direction of the magnetic force on this ion so when the ion enters the magnetic field it's going to experience a magnetic force pointed to the right the magnetic force is in the plane of the page so notice there's a 90-degree angle between the velocity vector and the magnetic field vector and so if the mat if the magnetic field weren't there the the ion would just continue moving in this direction but since straight up but since there is a magnetic force right it's going to cause the ion to deflect it's going to cause the ion to move in a cervical so the ion is going to move in a circle all right so it's going to move in this direction all right so I'm going to attempt to sketch in a semicircle here so you get the idea so there's my semicircle and a little bit of intuition about why the ion is going to move in this circular path but think about the ion at this point all right the velocity would be in this direction right and if you use your right hand rule you would see that the magnetic force would be pointing in this direction so once again 90 degrees to your velocity so the magnetic force always points towards the center of this circle and so that's a centripetal force all right so we have a circle here or a semicircle is what I've drawn of radius R all right so this is a this is a certain radius R let me go ahead and make that I mean let me just do a better radius than that so let's let's sketch that in so this is radius I'll make it lowercase R we can calculate we can calculate what that radius of the ion should be by using our equation for magnetic force so let's get some more space down here and let's rewrite our equation for magnetic force magnetic force is equal to QV cross B which is the same thing as Q V B sine theta where theta is the angle between your velocity vector and your magnetic field vector let's go back up here the velocity vector is in the plane of the page pointing up the magnetic field vector is coming straight out at us so that's 90 degrees between those two vectors and so let's let's get some more space down here we know that sine of theta that'd be sine of 90 degrees sine of 90 Riis is equal to one so this would just be Q V times B times one well force is equal to mass times acceleration right so Newton's second law and we know the ion is moving in a circular path so this would be the centripetal acceleration so we have QV B is equal to mass times the centripetal acceleration QV B is equal to the mass the centripetal acceleration is equal to V squared over R so we can cancel one of our V's so we get Q B is equal to M V over R and so we can solve for R R would be equal to M V this is velocity divided by Q B so now we have the radius of the circle and and we can go a little bit further we can take the velocity that we solved for earlier and we can plug it in here all right so let's go ahead and do that so this would be equal to M over Q B times the velocity which is square root of 2 Q member this was the potential difference over m all right to get rid of that square root we would have to square both sides so we square R so we get R squared and then we square all of this so this would be equal to M Squared and over Q squared B squared all right this would be 2 Q V over m so now we can cancel a few things we can cancel one of the MS and we can cancel one of the Q's so we get R squared is equal to M two times the potential difference divided by Q times B squared all right so take the square root to find the radius so we take the square root of both sides so we get R is equal to the square root of 2m times the potential difference V divided by Q B squared so finally we have we have the radius of our circle here and let's think about let's think about these things the magnetic field is constant there's no change in the magnetic field there's no change in the potential difference and if we assume that the charge on both ions is the same right the only thing that's different between those two ions is the mass right so we can say that we can say that if we increase this number I've increased the mass right we're going to increase the radius so if we increase the mass just looking at our equation we're going to increase the radius and I should point out that we have we have hiding in here right at M over Q ratio so M over Q ratio so M over Q is the mass to charge mass over charge ratio here and you'll see this written as M over Z and a lot of mass spectrometry examples so M over Z is the mass to charge ratio alright so for our purposes right we're just trying to think about how the mass affects the radius of the circle that the ion will move in so we've seen if you increase the mass you increase the radius so let's go back up here and let's look at our isotopes again alright and let's look at this circular path that we drew alright so if I if I wanted to draw the path for let me go ahead and just label this one right here so let's say that this hits where this ion hits this represents the u-235 ion right then with the smaller mass if we represent the one with the larger mass right so the u-238 has more mass that means that the radius of the circle is going to be greater so let me let me use blue here and we have a lot of things going on so we have blue so I'm going to draw a path with a bigger radius all right so I draw a path with a bigger radius again not the best drawing but we can see that with the bigger radius this represents the u-238 ion all right this is where those ions would hit in your mass spectrometer and so this allowed us to separate our ions based on mass and this final stage here the detection stage all right so there's a there's a lot that goes in - detecting these things and in modern mass spectrometers you're not going to separate isotopes there are better ways to do that these days but this is a nice way to introduce how a mass spectrometer works and in modern mass spectrometers you're going to use it to get very accurate masses all right and so will in some other videos we can talk about that