I've been searching for information on this, and just not coming up with what I'm expecting to find. So I have concluded I am either asking the wrong question (i.e., phrasing it wrong), am barking up the wrong tree altogether, or am asking about something that is just not easy to do. Here's an outline of the kind of problem I have in mind.
Let's say I have a given sample of a radioactive isotope. For our example let's say it is 1 g of Radon-222, which has a half-life of 3.8235 days. Let's say I'm interested in calculating the total decay energy from the sample over the period of one week.
OK, I know that from the half life that 0.28 g of the sample should remain after that time, which means that 0.72 g will have decayed. Radon-222 mostly decays to 5.590 MeV alphas, so I can say that this is 1.95e+21 decayed atoms x 5.590 MeV = ~1e22 MeV of alpha put out by the source over that period of time.
Except that Rn-222 will turn into Po-218, which has a half-life of 3.1 minutes, which is significant for the time period under discussion. So we've created 0.72 g of Po-218 and some point over the course of that week, and we'd need to then say how much of that has decayed as well and add those alphas (and a small number of betas) into that total output as well. And that will require knowing when the Po-218 was created, because it was being created continuously over the course of the week according to the half-life's exponential decay. And it seems in principle we could make a pretty good stab at guessing when this was created just based on the probabilities.
And once we've done the Po-218, we need to look at its daughter product, Pb-214, which has a half-life of 26.8 min, and so is also significant for the time period under question. And so on until we hit something stable enough to not matter for the time period we're asking about (a week in the case of this example — so by the time we get to something with a half-life measured in some multiple of that, we could probably ignore it).
I'm more a computer guy than a math guy, but this is starting to look like calculus, which is fine, I can make a script do this if I know the calculations I'm supposed to be doing, and I can feed it a list of isotopes, half-lives, and decay modes (and their probabilities and energies). But I have had a hard time finding the right equations and am not super interested in re-inventing the wheel, because I know this kind of thing must be very old-hat.
So anyone who can point me in the right direction (or tell me that I'm looking at this from the totally wrong angle) would be super helpful. And I am fine with something that is back-of-the-envelope in terms of its "resolution" in time — I am just trying to get things in the right order of magnitude, and ideally would like this approach to work with a lot of different isotopes (even if it requires having a master table of different decay products for the limited number of decay chains I am interested in).
TLDR;: I have a sample of Rn-222. How do I calculate the total energy of its various decay modes over the course of a given amount of time, including its daughter products?