Bateman equations

Let's say you put some Thorium 228 in a barrel then throw it in a swimming pool for storage. Thorium 228 being the radioactive faggot it is, it will want to decay. The thorium will decay into something new, which will in turn decay into something else, which will then decay into something else etc until it decays into something stable. This is called the decay chain, there are only 4 different chains.

How the fuck do you know what is in the barrel after 1 year? Or 10 years, or 5000 years?

And how do you know how radioactive it will be at any given time?

This is where the Bateman equations come in. With these equations you can say exactly what's in that barrel at any given time, even if the decay chain is 10 or 15 stages long. All you need to know is the decay constant of every compound in the decay chain (common knowledge) and how much you put in the first place (hurr).

Here they are:

a_m(t) = N_i^°∑ c_i^me^-λit

where

c_i^m = Π_j=1^m λ_j / Π_j=1;j≠i^m (λ_j - λ_i)

m being the compound,  λ being the decay constant, N_i^° being the initial number of nuclei in the barrel.

Sorry they look like shit, was too lazy to make a fancy PNG file with it laid out nicely. It was about time I started the "science" part of this blog so here it is. Hope you're having a nice week.