Is there something special about the halfway point? Does the decay happen at a steady pace or exponentially?
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So you have to understand how radiation works. There is no clock. Instead at any moment, a radioactive atom, could, just randomly, decay. It’s kind of like playing roulette every second, and if it stops at 00, it decays. Except that it’s a lot harder than getting an 00 in roullette (it’s just that this game is played many times a second). The rules of the game and when you decay or not are defined by quantum mechanics, and because of this it’s impossible to every predict when exactly an atom will decay. Could be now, could be in billions of years, could be in trillions of years, it is possible (but really improbable) it could be never.
But we can measure the odds of when eventually you’d probably win. Think of rolling a die. How many times would you have to roll it before you got a 6? Well who knows, we can’t say. But we can say the probability. If you want to learn of the math, you use the “Geometric Distribution” to see what is the probability that you win a game of chance after N plays, given you know the odds of winning a single game. The cummulative of this one will tell us the chance that you win given that you play *at most* N games, but you could win earlier. So to win a single time with 6 die, if you roll it only once you have 16.66%, but if you roll it up to 4 times, the chance that you win is 57.77%. If you roll it up to 50 times, the chance that you won at least once, is 99.99%, but it’s never 100% (though it can get veeerrryyy close, so much that most calculators would show it as 100%). And that’s the first problem with full life: you can’t every reach 100%, full life could be infinite, who knows.
We can do the same for the atom, we know that it’s playing this game every so much, and it has the chance of decaying. So we know how many games per second it plays, and we can use this to say what are the odds of it decaying in 1 second, or in 20 seconds, or in 10 years, or a million.
Now what happens when we have add sextillion atoms, all playing at the same time. It might sound like an insanely large number, but in a single drop of water there’s more than 600 sextillion molecules of water, each one with 3 atoms! So each atom plays their own game, and they all try. Some will get very lucky and decay very quickly, at this large numbers some are bound to. Others will play a bit more. But by the time we reach 50%, we’d expect that about 50% of the atoms have lost, yeah sometimes it’ll be higher, sometimes lower, but with a sextillion atoms, that is 1 followed by 21 zeros, the times that more get lucky or few do is going to be very very very low, for every lucky that wins early, there’s someone who wins later, and cancel each other out. This means that we can calculate after how much time the “win chance” is 50%, and this means that after this time, about half of the atoms should have decayed.
Phew. But here’s the crazy thing. If you don’t know how long the atoms have been playing, it doesn’t matter. Because it’s still playing you can know the odds that it’ll decay after X amount of time, and it’s the same at any one moment. Remember the dice game? When I said that your chance of winning in no more than 4 rolls is about 57.77%, if you lose the first three rolls, then the probability that you win in the remaining last roll is 16.67%! That’s because now that I know that you lose the first three rolls, the odds are different, I know I can’t bet on you winning on any of the first three.
So you get something weird here. Lets say the half life of some material is 1000 years. The time it takes for half of 1000g of that material to decay, is 1000 years, which means that in 1000 years 500g of material decayed. But the time it takes for half of the remaining 500g to decay is 1000 years: for 250 grams to decay! What gives here? Well remember that each atom is playing their own game, and once we know they didn’t decay after some time, we have to calculate the odds of them decaying in the same way. So you can see how then in 1000 more years it will become about 125g, and then in 1000 more (so now at 4000 years) 62.5g. And so it will keep going. After a very long time it will have decayed to the point that there’s only 1 atom left, and then that one will go away too, eventually. Basically you can’t have 1 atom. But remember it could also last forever, it’s very hard, very improbable, but it could happen.
And that’s why half-life is so useful as a number. Because it doesn’t matter how much material you have. You know that once the half-life passes, about half of the material should have decayed and the other half should still stick around.
And how is this useful? Well many materials divide into other radioactive materials with their own half-life. Say we have a rock with a material that has a half-life of 1000 years, and it decays into a material that is 100 and that decays into one that is stable. So we find a rock that was made of that first material, and we’re sure it was close to 100% of that at first. We can measure how much percentage it is of each one of the three materials, and make a formula of how much material would have decayed from material A to material B given T time, and then how much of B would decay into C as well. We put how much of the rock is A, B and C materials, and use those to calculate roughly how much time has passed. Of course there’s a chance that maybe some bit of A decayed faster or slower than expected, but over enough time, it should be pretty predictable. This is why Carbon-dating, which uses this, is so good at long times, but bad a short amounts.
Another relates to understanding how nuclear power-plants and their fuel work. We know that after the half-life the fuel will be 50% in the next phase, after a while we can start seeing that it will lose effectiveness, and we can calculate this and use it to know when it’s just not going to work well enough. But of course the fuel still has some radioactive material left, so it’s still problematic. Some people try to recycle it, remove the material that isn’t radioactive enough, keep what is still good, stick it together and get a new piece of fuel, but that one decays differently, because you don’t get 100% pure material, but some derivatives that are still good enough, but that changes the decay pattern. So you need to change how the power plant works, but many people think that you could have plants that use this fuel (because it’s cheaper) and then when that one runs out, recycle it again and make new plants that can use that one, and so on, until finally you have a material that isn’t that radioactive at all.
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