A Natural Nuclear Reactor — Question 5

Question 5

Extinct radioactive products leave stable descendants as evidence

A hypothetical fission product P decays entirely to stable D with half-life 30.0 yr. It was produced in a brief episode $2.00\times10^9$ yr ago. Initially there was no D; assume full retention and no other production.

1. Express the surviving P fraction using a base-ten logarithm.

2. Determine present D relative to initial P and explain why absence of P does not disprove ancient fission.

Question 5 - Solution

Extinct radioactive products leave stable descendants as evidence

(a) Method and calculation

$$N_P/N_0=2^{-t/T}\Rightarrow\log_{10}(N_P/N_0)=-(t/T)\log_{10}2.$$$$\boxed{\log_{10}(N_P/N_0)=-\frac{2.00\times10^9}{30.0}(.30103)=-2.00687\times10^7}.$$

(b) Interpretation and further calculation

$$\boxed{N_D=N_0-N_P\simeq N_0}.$$$$\boxed{N_P+N_D=N_0}\quad\text{the stable daughter preserves the atom count}.$$

Physical check

The surviving-parent fraction is effectively zero. Stable isotope anomalies can preserve evidence long after short-lived parents disappear.