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# Half-life plot

## Video transcript

- [Voiceover] 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. Let's say we started with four milligrams of phosphorus-32. And we wait 14.3 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. Let's look at the definition for half-life here. It's the time it takes for 1/2 of your radioactive nuclei to decay. So, if we start with four milligrams, and we lose 1/2 of that, right, then we're left with two milligrams. And it took 14.3 days for this to happen. So 14.3 days is the half-life of phosphorus-32. And this is the symbol for half-life. So, 14.3 days is the half-life for phosphorus-32. The half-life depends on what you're talking about. So if you're talking about something like uranium-238, the half-life is different, it's approximately 4.47 times 10 to the ninth, in years. That's obviously much longer than phosphorus-32. We're going to stick with phosphorus-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 graph the rate of decay of phosphorus-32. Let's look at our graph here. On the Y-axis, let's do the amount of phosphorus-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. Alright, we're going to start with four milligrams of our sample. Let's go ahead and mark this off so this would be one milligram, two milligrams, three, and four. So we're going to start with four milligrams. So when time is equal to zero, we have four milligrams. Let's go ahead and mark this off. So one, two, three, and four. We wait 14.3 days, so this is 14.3 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. There should be two milligrams left after 14.3 days so that's our point. Alright, we wait another 14.3 days, so we wait another half-life, so after two half-lives, that should be 28.6 days. So we know that after 28.6 days, it's another half-life, so what's 1/2 of two, it's one, of course. So that's our next point. So after 28.6 days, we should have one milligram of our sample. Let's wait another half-life. 28.6 plus 14.3, should be 42.9. So that's our next point. And what's half of one? It's 0.5, of course, so, in here, that's about 0.5, and so that 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, this is exponential decay. That's what we're talking about when we're talking about radioactive decay here. We'll talk a little bit more about exponential decay in the next video. But this just helps you understand what's happening. So as you increase the number of half-lives, you can see the amount of radioactive material is decreasing. Alright, let's do a very simple problem here. If you start with four milligrams of phosphorus-32, how much is left after 57.2 days? So if you're waiting 57.2 days, well, the half-life of phosphorus-32 is 14.3 days. So, how many half-lives is that? 57.2 days divided by 14.3 days would give us how many half-lives, and that's four. So there are four half-lives, so four half-lives here. 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 one 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. So that would be our answer, because that's four half-lives. Here's one half-life, two, three, and four, which is how many we needed to account for. That's one way to do the math. Another way, would be starting with four milligrams, we need to multiply that by 1/2, and that would give us two, and then 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 going four, times 1/2 to the fourth power, which mathematically, is four times one over 16, so that's 4/16, so that's the same thing as 1/4, and so that's 0.25 milligrams. So it doesn't really matter how you do the math, there are lots of ways to do it. You should get the same answer. You could also get this on the graph if you had a decent graph. After four half-lives, you would be you would be over here somewhere. And so you could just find where that is. So let me use red, so you could find where that is on your graph, and then go over to here, so that would be approximately right here, and then read that off your graph. And that looks like about 0.25 milligrams as well. We'll talk more about graphing in the next video.