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# Half-life plot

## Video transcript

phosphorus-32 is radioactive and undergoes beta decay so we talked about beta decay in the last video here's our beta particle and the phosphorus is going to turn into sulfur so let's say we started with four milligrams of phosphorus 32 and we wait fourteen point three days and we see how much of our phosphorus is left you're going to find two milligrams of your phosphorus left the rest has turned into sulfur and this is the idea of half-life so let's look at the definition for half-life here it's the time it takes for one half of your radioactive nuclei to decay so if we start with four milligrams right and we lose half of that right then we're left with two milligrams and it took fourteen point three days for this to happen so fourteen point three days is the half-life of phosphorous 32 and this is the symbol for half-life so fourteen point three days is the half-life for phosphorous 32 so the half-life depends on what you're talking about so if you're talking about like your rhenium at 238 the half-life is different it's approximately four point four seven times 10 to the ninth in years so that's obviously much longer than phosphorus-32 we're going to stick with phosphorous 32 in this video and we're going to actually start with four milligrams every time in this video just to help us understand what half-life is next let's let's graph the rate of decay of phosphorous 32 and so let's let's look at our graph here and on the y-axis let's do the amount of phosphorous 32 and we're working in milligrams here so this will be in milligrams on the x-axis let's do time and since the half-life is in days it just makes it easier to do this in days all right we're going to start with four milligrams of our sample so let's go ahead and mark this off this be one milligram two milligram three and or so we're going to start with four milligrams so when time is time is equal to zero we have four milligrams so let's go ahead and mark this off so one two three and four so we wait fourteen point eight three days so this is fourteen point three days and half of our sample should be left so what's half of four it's of course two and so we can go ahead and graph our next data point so there should be two milligrams left after fourteen point three days so that's our point all right we wait another fourteen point three days so we wait another half-life so after two half-lives that should be twenty eight point six days all right so we know that after twenty eight point six days that's another half-life so what's half of two it's one of course so that's our next point so after twenty eight point six days we should have one milligram of our sample let's wait another half-life so twenty eight point six plus fourteen point three right should be forty two point nine so that's our next point and what's half of one it's point five course so in here it's about point five and so it gives us an idea about where our next data point is and we could keep going but this is enough to give you an idea of what the graph looks like right so if I think about this graph right this is this is exponential decay so that's what we're talking about when we're talking about radioactive decay here I'll talk a little bit more about exponential decay in the next video but this just helps you understand what's happening alright so as you increase the number of half-lives right you can see the amount of radioactive material is decreasing alright let's do let's do a very simple problem here so if you start with four milligrams of phosphorus 32 how much is left after fifty-seven point two days all right so if you're waiting fifty-seven point two days all right well the half-life of phosphorous 32 is fourteen point three days so how many half-lives is that fifty-seven point two days divided by fourteen point three days would give us how many half-lives and that's four so there are four half lives so four half lives here so we're starting with four milligrams so one very simple way of doing this is to think about what happens after each half-life so four milligrams if we wait one half-life goes to two milligrams wait another half-life goes to 1 milligram wait another half-life goes to 0.5 milligrams and if we wait one more half-life then that would go to 0.25 milligrams and so that would be our answer because that's four half-lives right so here's one half-life 2 3 & 4 which is how many we needed to account for all right so that's one way to do the math another way would be starting with 4 milligrams all right we need to multiply that by 1/2 all right and that would give us 2 and multiply by 1/2 again and 1/2 again and 1/2 again so that's four half-lives right so this represents our four half-lives and that's the same thing as as as going 4 times 1/2 to the fourth power which mathematically is 4 times 1 over 16 so that's four sixteenths so that's the same thing as 1/4 and so that's point two five milligrams and so it doesn't really matter how you do the math right there are lots of ways to do it you should get the same answer right you could also get this on the graph if you had a decent graph so after four half lives all right so after four half-lives you would be you would be over here somewhere and so you could just find where that is all right so let me use red all right so you could find where that is on your graph all right and then go over to here all right so that would be it's approximately right here and then read that off your graph and that looks like about 0.25 milligrams as well so we'll talk more about talk more about graphing in the next video