Nuclear Half Life: Calculations - So here's the equation for radium doing alpha decay to make radon. And the half-life for this process is 11 days. Our question is, if you start with a 120 gram sample of radium, how much will be left after 44 days? The first thing let's do is figure out how many half-lives 44 days is going to be. OK, so one half-life is 11 days. So 44 days is going to be four half-lives. OK, now that we know it's going to be four half-lives, let's just make kind of a little chart and figure out how much we're going to have at each step. So we're starting with 120 grams. After one half-life, we'll have half of that. So we'll have 60 grams. OK, that's the first half-life. Now, a second half-life, we'll go from 60 down to 30, two half-lives. Three, we'll go to 15 grams. And now our fourth half-life will give us 7.5 grams. So 7.5 grams is how much radium will have left after 44 days or four half-lives. Now, a lot of times, when people ask questions about half-lives, they want to know about percent and fractions too. So let's see how we'd answer this if we were asked what percentage will be left after 44 days, instead of what amount. This is really simple. It's actually easier than this. OK, so, for the percent left, we'll do the same thing, but we'll just assume that we start at 100%, OK? So 100% is where we start. After, one half-life, how much will be left? Well, half of it, which is 50%, so there's our first half-life. Now, our second half-life, we'll go from 50% down to 25%. Our third half-life, we'll go to 12.5%, OK? And then, for our fourth half-life, we'll go to 6.25%. And this is what percentage of the starting amount I'd have left after 44 days or four half-lives. Now, finally, if you were asked to find the fraction that was left after 44 days, here's how you'd do it. Keep in mind, again, this is going to be four half-lives. So, after one half-life, we'd have 1/2 left, OK? Now we multiply that. After two half-lives, we'd lose another half. So now we'd have 1/2 times 1/2, 1/4 left after two half-lives. We're going to lose another half. So now it's 1/2 times 1/2 times 1/2. We have 1/8 left. And, finally, a fourth half-life, we're going to have 1/16 of the original amount left. And, if you do 1 divided by 16 and turn that into a percent, it's 6.25. So that's how you can solve a problem like this for the actual amount, the percent, and for the fractional amount. Here's our next question. Hydrogen-3, which is also known as tritium, undergoes beta decay to make helium-3. And this process has a half-life of 12.3 years. OK, so an 80 gram sample of tritium decays, leaving 2.5 grams of tritium. How long would this take? OK, let's figure it out by just making a chart like we did before. So we're starting with 80 grams of tritium. One half-life is going to give us how much? 40 grams, OK, one half-life. Now we're going to do another half-life. Now it's down to 20 grams, two half-lives, down one more, 10 grams, another one, 5 grams, and, finally, down to 2.5 grams. So that is one, two, three, four five half-lives, OK, so five half-lives. And how much does each half-life take? Each half-life takes 12.5 years. So we're going to do 12-- I'm sorry, 12.3 years. So we're going to do 12.3 years times 5 is going to give us 61.5 years. That's how long this whole process would take. OK, now what if the question involved percentages, asking how long it would take if we were left with 3.1%? I'm just going to do this really fast so that you can do it on your own, all right? But we'd start with 100%, take that down 50%, and then another half-life would give us 25%, 12.5%, 6.25%, and, finally, 3.125%. That's pretty close to the 3.1 they're talking about, and it's the same answer. It's one, two, three, four, five half-lives, 61.5 years. Now, finally, what would happen if you were given this amount as a fraction, asking how long it would take to get down to 1/32 of the original amount? We'd just multiply 1/2's together and see how many we'll need. So we'd do 1/2 times 1/2. That's 1/4, 1/8, 1/16, 1/32. Each one of these 1/2's represent one half-life. So that's how we could get five half-lives with fractions, five t 1/2's, OK? Let's do one more. Here's the equation for thallium undergoing beta decay to make lead, but we don't know the half-life here. We're going to have to figure out what it is. So the question asks us, we start with 200 grams of thallium 207 here. After 20 minutes, there's only 12.5 grams of thallium left. What is the half-life of the decay process. As usual, let's make a chart that shows how much this is decaying. So we start with 200 grams. One half-life is going to knock us down to 100 grams. Now another half-life will take us down to 50 grams. Then we'll get down to 25 grams. And, finally, a fourth half-life will take it down to 12.5 grams. So we have one, two, three, four half-lives, four half-lives. Now it's said that this whole process to go from 200 grams down to 12.5 grams takes 20 minutes. And, in that 20 minutes, there have been four half-lives. So, to figure out the length of one half-life, we're just going to do 20, the total time, divided by 4 half-lives, which is going to give us 5 minutes for the length of one half-life. And, as I've shown you in the previous examples, you could also do this with percentages by starting at 100% and working your way down. Or, if the number had to do with fractions, just multiply 1/2 together for each half-life that you have. Now, the calculations that we've done for all these problems so far, you could probably do them in your head pretty well. You just take numbers and cut them in half a bunch of times and do some relatively simple math, but there are a bunch of half-life problems that require trickier math that use exponents and logarithms. So we'll now talk about those in the next lesson.