Calculation of alpha particles from 226-Ra decay

[geojon]

I am trying to solve the following problem for my graduate course in isotope geochemistry. If 1 g of ²²⁶Ra is separated from its decay products and then placed in a vessel, how much helium (cm³) will accumulate in the vessel in 60 days?

Here is what I know. The decay series from ²²⁶Ra to Lead 206 (the final stable daughter) includes five alpha decays, and these will be the source of He2+ ions. I am assuming secular equilibrium because the half-life of ²²⁶Ra is >> 60 days. Ultimately, I know I will have to determine how many alpha particles are created in order to answer the question. However, I am not sure how to get started on the problem and was hoping someone could offer some help.

 

[DEvens]

It depends how accurate you want to get.

The full-blown solve-the-differential-equations version goes like this.

Ra-226 alpha decays into Rn-224.
Rn-224 alpha decays into Po-218.
Po-218 alpha into Pb-214.
Pb-214 beta decays into Bi-214.
Bi-214 beta decays into Po-214.
Po-214 alpha decays into Pb-210.
Pb-210 alpha decays into Pb-206. And there you can stop.

In each case, the equation you want to solve is like so. You have decays of the particular kind of isotope. You have production from the previous member of the chain.

$\frac{dN_i (t)}{dt} = - \lambda_i N_i(t) + \lambda_p N_p(t)$

Here $N_i$ is the specific isotope, and $N_p$ is the precursor. You get an equation like that for each isotope except Ra-226 which will not have a precursor, and Pb-206 which you will ignore. And the $\lambda$ comes from the decay equation and the half life for that isotope.

So you get this set of coupled differential equations. And at t=0 you have 1 g of Ra-226 and none of the rest. And you solve them. And you total up all of the cases where something alpha decays, and that adds one He to the total. That would mean you would need to solve this set of differential equations, probably numerically though it is possible to do in closed form. That closed form is fairly gnarly. If you want it, look up Bateman's equations. But consider if you want to do it numerically.

For a graduate homework, it seems reasonable. It will also give you, as a function of time, the concentrations of all of those other things.

Lesser accuracy, and an over estimate, is to simply assume all the daughter products decay right away.

Least accurate of all, and an under estimate, is to ignore the decay products and only do the Ra-226 decay.

 

[Vanadium 50]

I would not set this up as a set of coupled differential equations.

If you have a fast decay, with t << 60 days, you can assume it's immediate. If you have a slow decay, with t >> 60 days, you can assume it doesn't decay at all. That will enormously simplify the problem and will change the final answer by only a fraction of a percent.

 

[gleem]

Pb-210 alpha decays into Pb-206. And there you can stop.

OOP's Pb-210 beta decays to Bi-210 But since the decay of Pb-210 is so slow (H.L. = 22 yrs) compared to 60 days you can stop here for practical purposes and only consider the previous alpha decays( as DEvens suggested) of which there are six for each Ra-226 decay. The Bi 210 population will grow at almost a constant rate as Ra-226 decays. But that which is formed will decays about 0.5% over 60 days. So how may Ra-226 decays in 60 days?