16.3 Half-Life

Okay, today's lesson is on half-life, specifically nuclear decay. It's more mathematical in nature, something many of you may have encountered in Math 30 if you've taken it. Essentially, we're examining how much of an element decays over time. This is where the concept of half-life comes in. It refers to the amount of time it takes for half of the neutrons or nuclei of an element to transmute. For instance, in alpha and beta decay, we saw how a chemical can transform into something else. We're trying to determine how long it would take for half of that element to remain the same. The subsequent half-lives would be half of the previous one, resulting in a quarter, and so on. It's important to note that each element has a different time period for its half-life. Some decay almost instantly, within 10 to the negative 22 seconds, while others take thousands or millions of years, so 10 to the 28 seconds. The half-life is quite different for any chemical, and the thing to remember is that the number of atoms still stays the same. We're not changing protons and neutrons and so on. All we're doing is sort of transmuting them from one element to another. So, if we're trying to measure the half-life of carbon, for example, one half-life of carbon-14, the elements are staying the same. It's just that carbon-14 is translating or transmuting into something different. So, the half-life of carbon-14 is fifty-seven theories. That means that after 5730 years, half of those original nucleons would be gone. So, if you started with 100 atoms, in 5,700 years, you'd have 50. Then, in another 5,700 years, you'd have 25. And in another 5,700 years, you'd have 12 and a half. On it, I would just keep getting cut half and half as time goes on. So because every element has its own sort of time frame, we can use that half-life and work backwards to figure out how old certain things are. So if you find a dinosaur bone, we can measure the carbon-14 and figure out how much of it is left and what it should have been. Then we can backtrack to figure out how old that dinosaur was when it was still alive. As it undergoes decay, we can usually measure that activity in becquerels, which is the amount of decay per second. We measure this using a Geiger counter, which is the little beeper thing you often see in movies used to scan for nuclear decay. As time goes on, the number of nuclei that decay drop, so the activity also drops. Here's a graph to illustrate this: Showing the same thing we just talked about. So, if we have a hundred original nuclei in one half-life period, then it would be 50. And then I'd go to 25, into 12, and 6, and so on. It just keeps getting cut in half and half. So we can actually use a math formula to calculate this. The formula is n = (1/2)^n, where n represents the number that is left. This can be measured in atoms, mass grams, or any kind of unit. It really doesn't matter. And we just take the original amount times 1/2 to the exponent of the number of any half-lives they're on. So, for a question like we just looked at for carbon-14, we'd actually write it as 57,30/T, depending on the time. So, if you had ten thousand four hundred years, you... Divide that, and I would give you two half-lives, and so on. So let's do an example. In this case, he says we've got 765 grams of polonium, and the half-life is 138 days. If we use our formula, we'd write it as: 765 (our original amount) / 2 (half-life) = 382.5 Now, for our "n" value, we want to know 3.8 years later. So, we have to divide that by 138 days to figure out how many half-lives are on your calculator. If you go 3.8 years times up by 365 days in a year, that gives us 1387. Divide those, and you can see it's roughly ten half-lives. So, we'd have: 765 times 0.5 (to the power of 10) = 23.789 grams And that's the amount of polonium you'd have left after 3.8 years. 0.5 to the power of 10, or you can just type in 1387 divided by 138, which gives us 0.72 left over. So, this was originally within grams, meaning we only have 0.72 grams remaining. Now, what happened to the other 764.3 grams? The rest of it got transmuted. Therefore, what we have left is only the amount of polonium, which is 0.72 grams. However, the rest of the material could be any other kind of chemical, such as uranium or something else. The remaining substance would have been transmuted into a different element or elements. Here's one more example: One, we've got lead-212 and this has been 75 grams of it. So we have 75 as their starting amount. It says we have a half-life of 10.6 hours, so that means we have 10.6 on the bottom. This question is sort of asking how long it will take to get 9.3 grams left over. As you can see in this case, what we're doing is solving it for the time. So, to solve this equation mathematically, just like any other equation, what we want to do first is divide by the 75. Okay, so divided by the 75. If we go 9.3 divided by 75, that gives you 0.124, and that would equal half to the time over 10.6. This is where I run into a problem if. You haven't taken Math 30 yet, so we can't solve for that exponent unless we use what we call logs. So, what you want to do to solve this is just do the log of both sides. So, we'd have log(0.124) = log(1/2)^T. We can divide this 10^T over 10.6 comes down in front. That's one of the log rules, so we get T = T/10.6. So, we can cross multiply, and you'd have 10.6 * log(0.124) / log(0.5). You can just type that directly in your. calculator, and it should be good. So, log in 31.9 hours, then we'd be down to that amount. So, we know that the half-life is ten point six. If we can just use some logic, ten point six hours, we had half, half of 75, that's about 37. Then, in another ten point six, it would be like 21 hours, and we'd have half of that. So, we're down to about 18. Then, another ten point six, and now we're getting close to the 30, 31, 32 hours. They'd have half of that, which is about nine. So, it does make sense that it took three half-lives to go ten point six, ten point six, ten point. Six, which is roughly thirty-one hours, would get us down to three halves of the original amount. And that's all.