Is there something special about the halfway point? Does the decay happen at a steady pace or exponentially?
In: 910
It’s because it’s exponential decay. In the equation it decays with a factor of 2. So if it takes 5 years it goes from 100%, to 50% then it will take 5 years to get halved again ang go from 50% to 25% of the original. Then again 5 years for the 25% remaining to be halved to 12..5%. Then 5 years 12.5% -> 6.25%, 5 more years for 6.25% -> 3.125% etc. If it was full life it would never decay. If it was 3rd life you couldn’t produce dates accurately because the decay doesn’t follow exponentially the same way it does with 2. 1 out of 2. Half life.
It’s not linear, but exponential decay.
After a few half life’s, the substance will still be around but in much smaller and often safe quantities, but it would take huge amounts of time to completely decay.
Zeno’s paradox is easy enough for a 5year old.
Go halfway to the wall. Go half of what remains. Go half of what remains. Go half of what remains…. Do it forever. Never reach the wall.
Radioactive materials are radioactive forever. In one half life’s time, activity diminishes by half. It never gets to the wall.
So there’s your answer. With this mathematical model for radiation, the full life for any substance is *forever*.
Now, all you 7 year olds might be smart enough to realize that if you keep halfing you eventually just get down to a single atom. One mole of a substance can be halved 80 times before this point is reached. So, a mole of radium225 ( about a spoonful) with halflife 14.9 days, will be utterly free of ra225 after 1192 days.
The fun don’t stop, though, because many radioactive substances decay into things that are themselves radioactive. On the way to turning into Pb (the stable last link of most heavy radioactive chains) you get a spectrum of decay products, many of which last much longer than ra225.
Every half life half of it goes away, so if you have 10lb, one half life gives you 5, a second half life gives you 2.5, next 1.25…. Goes on and on till it is theoretically gone.
So radioactive decay is probabilistic, technically an isotope could exist forever without decaying, it is just increasingly unlikely that this will happen.
Think of it like this, you have to roll 2000 6 sided dice, everytime these dice roll a 1 they are pulled out of the pool. Now we can ask the question what is the half-life of this pool of dice? Well each roll we can expect about 1/6 of the dice to disappear on the first roll we lose 1/6 of the dice but on the second roll we don’t have 2000 dice we have on average 1666, so after this roll we will have on average 1388 dice left, after the third roll we will have 1156 dice left, and finally after the 4th roll we will have 963 dice left.
So in this case we could say the half life of a set of 6 dice is about 4 rolls, but this is on average technically it is possible for more than half of the dice to still be there after 4 rolls, or even less than half, technically you did this experiment until you had no dice you could be rolling dice for all of eternity. Saying something could last for “maybe all of eternity” isn’t really a useful metric for these sorts of things and it is more useful for us to talk about the time it takes for them to reduce by half, because while dice aren’t really harmful but radioactivity and radiation sources are dangerous, and they will stay dangerous for quite a while and are proportionally dangerous to the amount of radioactive material is still there so a way to estimate the amount of radioactive material that still exists is very useful.
But we can take this further if we know the amount of time it takes for a radioactive substance to decay to another stable substance we can calculate the amount of half lives have occurred if we also measure the amount of each of those substances, like if we take the previous example and I said we started off 1 million dice and we now have around 500 dice left you could figure out the amount rolls have occurred:
First half-life we would have around 500k dice left and would take about 4 rolls
Second half life we would have 250k dice left and would take a total of 8 rolls
Third half life we would have 127.5k dice left and would take a total of 12 rolls
Fourth half life we would have 63750 dice left and would take a total of 16 rolls
Fifth half life we would have 31875 dice left and would take a total of 20 rolls.
Sixth half life we would have 15937 dice left and would take a total of 24 rolls.
Seventh half life we would have 7968 dice left and would take a total of 28 rolls.
Eighth half life we would have 3984 dice left and would take a total of 32 rolls.
Ninth half life we would have 1992 dice left and would take a total 36 rolls.
Tenth half life we would have 996 dice left and would take a total of 40 rolls.
And finally on the eleventh half life we would have 498 dice left and it would take on average about 44 rolls.
So the time it would take for 1 million dice to “decay” to 500 dice would take on average about 44 rolls.
And this is the idea behind carbon dating.
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.
Years ago I watched a weird cartoon.
There was a big friendly creature sitting at the top of a hill with a circular plate in front of it and a whole chocolate cake. It’s just about to tuck in when a little friend turns up with a plate in its hand.
The big friendly creature cuts the cake in half and gives half to its little friend. Fair’s fair, right? It’s just about to tuck in when another little friend shows up with an empty plate. The big friendly creature says “half for me, half for you”, cuts the cake in front of it in half and gives it to its friend. Except it’s now left with a quarter cake.
This keeps happening until the big friendly creature is paring down ever thinner wedges of cake. We pan out to see a queue of little friends leading all the way down the hill with no end in sight.
Weird right? I thought it was a bit short sighted of the big guy, and I thought it wasn’t fair that it never gets to eat anything while the first little guy is presumably chowing down on an entire half but anyway I digress…
“When will the cake run out?” is not a useful question in this case because the answer is “technically, never”. “When is there so little cake left there’s no point bothering with it?” is what we’re actually getting at, but that’s too subjective.
So if I just tell you how long each “round” takes (how long it takes the big friendly creature to cut and serve half of whatever’s left) you can take that information and do what you want with it. For example if you decide a 1/32 piece isn’t worth bothering with you can work out how long it will take to get there.
Latest Answers