# Nuclear Physics - Uranium decay chain and Bateman equation

1. Feb 1, 2015

### Phruizler

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

Calculate the activity of $^{222}Rn$ in an ore sample containing 5g of natural uranium.

2. Relevant equations

$^{238}U$ decay chain (to Radon): $^{238}U\rightarrow^{234}Th\rightarrow^{234}Pa\rightarrow^{234}U\rightarrow^{230}Th\rightarrow^{226}Ra\rightarrow^{222}Ra$

Bateman equation (for activity of a daughter isotope after a series of decays):

$A_n = N_o\sum\limits_{i=1}^{n} c_ie^{-\lambda_it}$ where $c_i=\frac{\lambda_1\lambda_2 ... \lambda_n}{(\lambda_1 - \lambda_i)(\lambda_2 - \lambda_i) ... (\lambda_n - \lambda_i)} \qquad \left(i \neq n\right)$

3. The attempt at a solution

Well, I used the known decay constants for each isotope in the decay chain and plugged them into the Bateman equation. Unfortunately, $c_i$ turns out to be negative, giving me a negative number of atoms, which is obviously incorrect. The portion where I simply convert 5g of uranium into total number of atoms is fine. Anyone know where I'm going wrong? I'm assuming $t=0$ since time is seemingly irrelevant to the problem as stated.

EDIT: I guess I should add that I am getting my decay constants from Wolfram Alpha, but I don't see why it would be giving me anything but the correct values, as they match with other available literature.

Last edited: Feb 1, 2015
2. Feb 2, 2015

### Staff: Mentor

If you start with pure uranium at t=0 then your Radon decays at t=0 will be zero.
U-238 should be by far the most long-living isotope in your sample. If the sample is old enough (no time is given, so I guess you have to assume that), all other isotopes should be very close to their equilibrium concentrations.

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