Is there something special about the halfway point? Does the decay happen at a steady pace or exponentially?
In: 910
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.
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