Is there something special about the halfway point? Does the decay happen at a steady pace or exponentially?
In: 910
Draw a line.
Now Draw a midpoint.
Draw a midpoint of half of that line.
Draw a midpoint of half of that line.
Keep going and let me know when you run out of midpoints.
Practically, you’ll eventually run out of space depending how thick your lead is. But for decay, you’ll be waiting for the heat death of the universe to see that very last atom finally poop out a proton.
I use a dice analogy for this. Imagine your atom’s decay is measured as a tub of 100 dice. You roll all 100 dice at once, and if it lands on a 1 then it has decayed and anything else it hasn’t. It might take ten rolls to have 50 of these dice decay, but when there’s only 50 then it might take ten rolls for only 25 to land on 1. The fewer there is, the longer it takes for large amounts of decay to occur. This relates to half life, because the “ten rolls” are equivalent to the half-life being ten (whatever time measurements). It takes 10 for the number of dice to half, and then it takes another 10 for it to half again. In actual physics terms, this is because the greater the number of neutrons, the more unstable it is. Also, decay occurs spontaneously. These two factors mean that when the atoms have lots of neutrons, the decay will happen faster, but as there are less neutrons the decay happens at a slower rate, because it is less unstable and therefore the chance that the same amount of neutrons will “land on a 1” is lower. Just like with the dice, the first ten rolls 50 decay, but the next ten rolls only 25 decay. It happens at a slower rate. So to answer your question, there can’t really be a full life because it’s not a case of “the first ten rolls 50 decay and the next 10 rolls the other 50 decay” it’s exponential and spontaneous, so there’s no way to really know the full life. The half life is the most accurate way we have to measure the rate of decay.
I hate these questions that can’t be explained like someone is 5. the absolute simplest explanation is that it comes from the fact that “half” is an easy reference.
Many many things in all of existence follow the pattern of what’s known as “exponential growth and decay”. Anything with a “population” tends to follow the same type of mathematical formula, wherein the “rate of change” is proportional to the current population size.
The explanation can’t get any simpler than that before getting into the math of it, but that concept — that the rate of change of a population is proportional to its current size — applies all over the place: the volume of an evaporating puddle; the birth and death rate of a country’s population; the number of atoms in a volume of radioactive substance.
Before I get downvoted for not sticking to the “ELI5” spirit, I’ll make the disclaimer that I’m going to go into high school level math now.
let’s write the above principle as an equation:
ΔP/Δt ∝ P
which is that that rate of change in P (the population size) is the ratio of any given change in population for a given change in time, and that ratio is proportional to the current population.
let’s make that delta (change) in population for an associated change in time really really small, and set it equal to the population at time t, multiplied by some constant of proportionality:
dP(t)/dt = k × P(t)
it’s a really big assumption to say that k is constant, but it’s rarely not constant, and another term for it being constant is to say it’s “time-invariant”. it may be dependent on other things, but what’s important is that it *isn’t* dependent on time. that’s a whole subject by itself, and way more complicated than this topic
anyway, there are a few things we can say about the above equation, which is (again, another discussion) known as a “First Order Differential Equation”.
1. we know that at time t=0 (the time we first start looking at the population, not the beginning of all time), the population is non-zero;
2. if the rate of change is positive, meaning it’s growing, clearly k > 0, and if the rate is negative, meaning the population is shrinking, k < 0. why? because dP/dt has to be constant if k and P are constant, and we know that dP/dt is a line with that must point up, down, or stay flat because remember it’s a change in P relative to a change in t (remember from geometry that a line has a slope of (y2-y1)/(x2-x1), and here we’re just using P and t instead of y and x).
3. most important of all — and this comes from basic calculus, which is what we’re talking about here — we see that the derivative of the population function is equal to the population function itself (multiplied by a constant), and there is only one function that satisfies that relationship: e^x. or since we’re using time, e^t.
Using all the info above, we can solve that equation (moving k to the other side and cutting to the chase) as:
P(t) = P(0) × e^(kt)
The reason radioactive decay uses “half life” is because it’s easy enough to measure the time it takes for an amount of a substance to decrease by half, and with that information we can solve for k, and then apply that to any change in population to predict the time it will take to get there:
0.5 × P0 = P0 × e^(k × t_hl)
P0 cancels and solving for k you get:
k = ln(0.5) / t_hl
where t_hl is the “half life” time, or the time it takes for the population to decrease to half it’s previous size.
but obviously if you’re watching a volume of water evaporate you wouldn’t wait until it’s half the volume so you could just use some other ratio of current volume to original volume, like if you start with a volume of 1 liter and you measure the time it takes to go down to 900 ml. So in the above equation, you’d set the left side to 0.9 and use a time of t_90%.
you get the idea.
if I have a 1kg block of Plutonium-241, it will decrease to 500 grams of Pu-241 (with less than a gram of a bunch of other elements’ isotopes but one again that’s a whole other discussion) in 14.4 years. But you’re not going to be near it nearly that long. So knowing it’s half life, how long until it decreases by 1 gram?
k = ln(0.5 [kg]) / 14.4 [years] = -0.048
0.999 [kg] = 1[kg] × e^(-0.048 × t)
t = ln(0.999) / -0.048 = 0.02 years = 7 days, 14 hours
yes I know, not ELY5 but like I said, it can’t be ELY5, but hope this helps
edit: I can’t fucking believe you people downvoted me (ok yes i can because i said it would happen). last time I contribute here.
The half life is essentially the time it takes for each atom to have a 50% chance of decaying. You can almost imagine that each atom is flipping a coin every half life (really it would be continuously flipping a coin, but we’ll ignore that). So after two half-lives, each has a 75% chance of decaying, and after three half-lives it’s 87.5%, but it never reaches 100% (just like how no matter how many times you flip a coin, you’re never 100% certain that you’ll get even a single heads)
Half life is logarithmic, not linear.
Half of 1000 = 500.
Half of 500 = 250.
Half of 250 = 125.
You can see the issue here. You’ll never reach zero.
Try thinking of it this way:
When talking about a radioactive element, there is a X% chance that a specific atom will decay in the next minute. For all intents and purposes, this is a completely random and unpredictable event.
The half-life value is then spit out from this. For example, Let’s assume a radio-active element has a 1/6 chance of decaying in 1 minute. We can simulate this by rolling a die. If you have a large block of this element with 1,000 atoms in it, then we can roll a die 1,000 times to simulate which will decay and which wont. Every time we roll a 1, that atom decays, all other rolls, nothing happens.
After 1 minute we will have ~167 atoms that have decayed and 833 that haven’t. In the next minute we only roll the die for the 833 remaining that can still decay. After the second minute we have 694 atoms left. After the 3rd minute: 579, and after the 4th: 482. This means the half-life is somewhere around 4 minutes. Because we have fewer original atoms at the end of each minute fewer atoms will decay. If you graph the values it will look like a curve getting ever smaller, but not quite hitting zero.
Since we are only rolling the die again for the atoms remaining and each one has a chance of decaying or not, we will continue to get smaller and smaller numbers. In the above example with 1,000 atoms it would actually take about 43 minutes before we likely didn’t have any of the original atoms left. This is with a pretty quick half-life and an extremely small number of atoms. In the real world half-lives are typically longer and you will have a sample with something like 10^20+ atoms in it so there is likely to always be some of the original still there.
In theory you could calculate an elements quarter-life, tenth-life, or two-thirds-life. Half-life is just the easier to grasp and understand. Full-life isn’t really a thing since mathematically you never actually get to 0 atoms left, you just keep getting closer to it.
Saying that a radioactive material has a full life of X would imply that, after X, there will be none of it left. That’s not how it works. After two half lives, you still have 1/4 of the original radioactive atoms left. After another half life, you have 1/8. There is no amount of time where you can guarantee that there will be no atoms left. It’s not like lifespan for humans or animals, where you can confidently say that there is nobody alive today who was alive in 1870. There’s no amount of time you can wait and then confidently say that there are no atoms of a radioactive material left. If you want to talk about the lifetime of a radioactive material, you have to define some non-zero percentage of the original material where you will say there is none left below that threshold. Depending how you define that threshold, there might still be some atoms of the material left, even at the point where you’re saying its lifetime is over.
Half lives only work because of how many atoms there are. If you have one single atom, you have no idea how long it will take to decay, all you know is that after one half.life it has a 50% chance to decay. When you have billions of atoms and you know that any single atom has a 50% chance to decay after one half life, then after one half life half of the atoms should have decayed.
Half lives don’t work by communicating with each other and making sure exactly half of all the atoms decay, it’s just a random chance for each individual atom. It’s a result of statistics, not some special force ensuring this takes place.
Lets say you have 100 coins and you flip them. Every time you get a tails that coin is removed and then you repeat until you get rid of all the coins. On the first flip you would expect to remove about 50 coins. Let’s assume it was actually 50 so now there are 50 left. If you flip again you wouldn’t expect to get 50 tails because the coins don’t care what happened last, every time you flip a coin it’s 50/50 that you get heads or tails. Instead you would probably get around 25 tails and so on.
With decay you can think of it in the same way, if you observe for a second there is some probability that a given nucleus will decay and in the next second it has the same probability. Most of the time that probability is very small over a second time scale (or even a year time scale) so it’s easier to just talk about how long it takes half of the nuclei to decay. In reality a lot of the time I’m working on something to do with decay the first thing I do is turn it back into that probability because the math is just simpler.
It’s also worth noting the number of nuclei we deal with. For that original situation I proposed with only 100 coins you wouldn’t be that surprised if you get a lot more or less than 50 tails. It probably wouldn’t be 90 tails but something like 60-70 isn’t that unreasonable. When you’re talking about a radioactive decay though the number of nuclei gets to the scale of 10^23 and at that point getting to that half value is going to be really consistent. There are so many trials that your odds of getting a really unexpected result plummets to the point of being basically impossible.
It’s less of a half-life and more of a life-of-half.
All of substance doesn’t half-decay, rather, half of it full-decays.
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