# Nuclear Physics reaction equilibrium

1. Feb 25, 2010

### koab1mjr

1. The problem statement, all variables and given/known data

238U with half life of 4.5e9 years decays by alpha and beta(-) to stable 206Pb in a decay chain

a) how many beta decays are involved
b) What is the rate of alpha particle production in a rock containing 1.0g of 238U with all of the daughters in equilibirum?
c) Calculate the global inventory of 222Rn with half life 3.8 days if there are 4.9e9tons of 238U

2. Relevant equations
$$\alpha$$ = $$\lambda$$n(t)
n(t) = n0e^-$$\lambda$$t

3. The attempt at a solution

I am sort of stuck on the first one which is crtical for part B. I know this is probably a popular decay series but my instructor said you should be able to solve without looking it up. My problem is that there are a couple combos of alpha and beta decays that would work right, just that one is more popular than the other. Anyway I was trying to do something where I use 2 equations one for the change in neutrons and the other for the change in protons and solve a simple linear system but that has not work

my equations were
4 $$\alpha$$ - $$\beta$$ = 22 = neutrons lost
2 $$\alpha$$ + $$\beta$$ = 10 = protons lost

but i get a fractional answer for this so this is obviously not correct.

Far as part b goes the decay chain is going to be large, is the solution just going to be one huge differential equation? Equilibrium means that the production and decay for all the members of the chain is balanced. I am not sure how to solve such an equation. Each daughter will havea different $$\lambda$$ but somehow you need to solve to get the same activity.

Any help much appreciated