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Current time:0:00Total duration:12:31
Throughout the video, Sal writes "G" for "gram" when he means "g."

Half-life and carbon dating

Video transcript

in the last video we saw all sorts of different types of isotopes of atoms experiencing radioactive decay and turning it to other items or releasing different types of particles but the question is when does a atom or nucleus decide to decay let's say I have a bunch of let's say these are all atoms I have a bunch of atoms here now let's say we're talking about the type of decay where an atom turns into another atom so your proton number is going to change your atomic number is going to change so it could be it could either be beta decay which would release electrons from the neutrons in terms of protons or maybe positron emission turning protons into neutrons but that's not what's relevant here let's say we have a collection of atoms and normally when we have anything of any small amount of any element we really have huge amounts of atoms of that element and we've talked about moles that you know one gram of carbon-12 sorry twelve grams 12 grams of carbon-12 carbon-12 that has that's that has one mole of carbon-12 in it one mole of carbon-12 and what is one mole of carbon-12 that's 6.02 times 10 to the 23rd carbon-12 atoms this is this is a a ginormous number this is more than than weaken that my head can really grasp around how large of a number this is so and this is only when we have 12 grams 12 grams is not a large mass for example one kilogram is about two pounds so this is about what one one only say one fiftieth of a pound if I'm doing but this is this is not a lot of mass right here and pounds is obviously for us I should well you get the idea on earth this is not well any where mass is invariant this is not a tremendous amount so with that said let's go back to the question of how do we know if one of these guys are going to decay in some way and maybe not carbon-12 maybe we're talking about carbon 14 or something how do we know that they're going to decay and the answer is you don't they all have some probability of decaying at any given moment for a certain type of element or a certain type of isotope of an element there's some probability that one of them will dick okay that you know maybe this guy'll decay ii and then nothing happens for a long time a long time then all of a sudden two more guys decay two more two guys decay and so like everything in chemistry and a lot of what we're just trying to deal with in physics and quantum mechanics everything is probabilistic I mean maybe if we really got a detail on on the configurations of the nucleus maybe we could get it a little bit better in terms of our probabilities but we don't know what's going on inside of the nucleus so all we can do is describe some probabilities to something reacting now you could say okay what's the probability of any given molecule reacting in one second or you could define it that way but we're used to dealing things on the macro level on on dealing with you know huge huge amounts of atoms so what we do is we come up with terms that help us get our head around this and one of those terms is the term half life half life and let me erase this stuff down here so I have a description and what we're going to learn we're going to hopefully get an intuition of what half life means so I wrote a a radio a decay reaction right here where you have carbon-14 it decays into nitrogen-14 and Lily we could just a little bit of review you go from six protons to seven protons your mass changes the same so one of the neutrons must have turned into a proton and that is what happens and it does that by releasing an electron which is also called a beta particle we could have written this is minus one charge relatively zero mass it does have some mass but they write zero is just kind of notation so this is beta decay beta decay this is just review but the way we think about half-life is people have studied carbon and they said look if I if I start off with 10 grams if I have just a block of carbon that's 10 grams that's 10 grams if I wait carbons half carbon 14s half life this is a specific isotope of carbon remember isotopes are they all isotopes that there's carbon can come in twelve within an atomic mass number of 12 or with 14 or I mean there are different isotopes of different elements and they all the the atomic number defines the carbon because it has six protons carbon-12 has six protons carbon-14 has six protons but they have a different number of neutrons so when you have the same element varying number of neutrons that's an isotope so the carbon-14 version or this this isotope of carbon let's say we start with 10 grams if they say that it's half-life is five thousand seven hundred and forty years that means that if on year on day one we start off with 10 grams of pure carbon-14 after five thousand seven hundred and forty years half of this will have turned into nitrogen-14 by beta decay and you might say oh okay so maybe let's see let me make nitrogen magenta right there so you might say okay maybe that half turns into nitrogen and I've actually seen this drawn this way in some chemistry classes or physics classes and my major question is how does this half know that it must turn into nitrogen and how does this half know that it must stay as carbon and the answer is they don't know and it really shouldn't be drawn this way so let me redraw it so this is our original block of of our carbon-14 what happens over that five thousand seven hundred forty years is that probabilistically a bunch some of these guys just start turning into started turning into nitrogen randomly at random points at random points and over five thousand seven hundred forty years you determine that there's a 50% chance that any one of these carbon atoms will turn into a nitrogen atom so they're over after five thousand seven hundred forty years the half-life of carbon 50% chance that any of the guys at our carbon will turn to nitrogen so if you go back after a half-life half of the atoms will now will now be nitrogen so now you have after one half-life so let's ignore this so we started with this all ten grams were carbon ten grams of c14 this is after one half-life one half-life and now we have five grams of c14 and we have five grams of nitrogen 14 fair enough let's think about what happens after another half-life well we said that after during a half-life 5,740 years in the case of carbon-14 well different elements have different half-life if they're radioactive over five thousand seven hundred forty years there's a 50% and if I look just look at any one atom there's a 50% chance that it'll decay so if we go to another half-life if we go another half-life from there I had five grams of carbon-14 so let me let me actually copy and paste this one this is what I started with now after another half-life Gig nor all my little actually let me erase some of this up here let me clean it up a little bit after what one half-life what happens well still these this I now I'm left with 5 grams of carbon-14 those 5 grams of carbon-14 every one of those atoms still has over the next 5,700 whatever that number was five thousand seven hundred forty years after 5,740 years all of those once again have a 50% chance and by the law of large numbers half of them will have converted into nitrogen-14 so we'll have even more conversion into nitrogen-14 so now half of that five grams so now we're only left with 2.5 grams of c14 and how much nitrogen 14 well we have another two and a half what two nitrogen so now we have seven and a half grams of nitrogen 14 and we could keep going further and four further future and go into the future and we'll always be after every half-life 5,740 years we will have half of our of the carbon that we started with but we'll always have an infinitesimal amount of carbon but let me ask you a question let's say i'm just staring at one carbon atom let's say i just have this one carbon atom you know it's got this nucleus with its its c 14 so it's got six protons one two three four five six it's got its eighth it's got its eight neutrons it's got its it's got its six electrons one two three four five six whatever what is the probability well what's going to happen I won't listen what's going to happen after let's go what's going to happen after one second well I don't know it'll probably still be carbon but there's some probability that after one second it will have already turned into nitrogen-14 what's going to what's going to happen after 1 billion years 1 billion years well after 1 billion years I'll say well you know it'll probably have turned into carbon nitrogen 14 at that point but I'm not sure this might be the one ultra stable nucleus that just happened to kind of go against the odds and stay carbon 14 so after one half-life if you're just looking at one atom after 5,740 years that's a-- you don't know whether this turned into a nitrogen rod this exact atom you just know that it had a 50% chance of turning into a nitrogen now if you look at it over a huge number of atoms I mean if you start approaching you know Avogadro's number or anything a larger I erased that then all of a sudden you can use the law of large numbers to say okay on average if each of those atoms must have had a 50% chance and if I have gazillions of them half of them will have turned into nitrogen I don't know which half but half of them will turn into it so you might get a question like I start with oh I don't know let's say I start with 80 grams of of something with a of let's just call it X and it has a half-life of two years I'm just making up this compound two year half-life half-life and then let's say we go into a time machine we look back at our sample and let's say we only have 10 grams of our sample left and we want to know how much time has passed by so 10 grams left of X how much time you know X is decaying the whole time how much time has passed let's think about it we're starting at time 0 with grams after two years two years how much are we going to have left we're going to have 40 grams so T equals two then after two more years how many we're going to have we're gonna have 20 grams so this is T equals three I'm sorry that's T equals four years and then after two more years two more years two more I only have half of that left again so now I'm going to have 10 grams left and that's where I am and this is T equals six so if you if you know you have some compound you're starting off with 80 grams you know it has a two-year half-life you get in a time machine and then you you didn't build your time machine well you don't know how well it calibrates against time but you just look at your comp at your sampling so you only have 10 grams left you know that one one two three half-lives have gone by and you could also think about it this way 1/2 to the third power because every time you have half of your original sample that's the number of half-lives after three half-lives you'll have 1/8 of your original sample and that's what we have here we have 1/8 of a Teague Rams and then and this is just when you're doing it with it discrete you know when you're right at the half-life point in the next video we're going to explore what if I ask you a question how many of the particles or how many grams will you have at exactly 10 days or at two and a half years and we'll do that in the next video