When Does Alpha Emission Peak in the Bi-210 to Po-210 to Pb-208 Decay Chain?

[ghetom]

Hi I'm trying to do a question on nuclear decay chains. The question is:

Bi-210 decays to Po-210 by beta decay (half life = 7.2 days), and this decays by alpha decay to Pb-208 (half life = 200 decays). If A substance is initially pure Bi-210, when does the alpha emmision peak?

So far I've got
Bi(0)=1, Po(0)=Pb(0)=o

Bi(t) = Bi(0)exp(-Lt) (L being the decay constant, t_1/2 = ln(2)/L)

and dPo(t)/dt = L(Bi(0)exp(-Lt)) -L'Po(t)

Obviously I just need to find when Po(t) is at a maximum, to find the corresponding maximum in alpha emmision, but I can't solve for Po(t)can anyone help?

 

[Astronuc]

Well, think on mean Bi-210 => to Po-210 => Pb-206. Half-lives are 7.2 and 200 days respectively, although reliable sources indicate half-life of Po-210 is 140 d (Hyperphysics) or 138.376 d (BNL NNDC).
http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/radser.html#c3

This is at the bottom end of the U-238 decay series.


Also - apparently Bi-210 can decay to Tl-206, which beta decays to Pb-206, but it's almost 100% alpha.

To solve the problem, make sure one has the correct half-lives or decay constants.

One has to develop two differential equations. One deals with the change (decay) in amount of Bi-210, which is also the production rate of Po-210. The other equation deals with the production and decay of Po-210 (production of Pb-206). The peak alpha emission from Po-210 occurs when the amount of Po-210, or when the rate of change is zero (dP/dt = 0), or when production rate = decay rate.

This is called transient equilibrium.