Radioactive Dating — Question 6

Question 6

Two isotope clocks can test a shared closure history

Two independent radionuclides in one object are assumed to have known initial abundances and a common closure time. Isotope 1 has half-life 1000 y and surviving fraction .250. Isotope 2 has half-life 3000 y and surviving fraction .500.

1. Calculate the two model ages and test the shared closed-system assumption.

2. If isotope 1 gives the correct age, what surviving fraction should isotope 2 have, and what factor relates its actual initial abundance to the assumed one if initial abundance alone caused the discrepancy?

Question 6 - Solution

Two isotope clocks can test a shared closure history

(a) Method and calculation

$$\boxed{t_1=1000\log_2(4)=2000\,\mathrm y,\quad t_2=3000\log_2(2)=3000\,\mathrm y}.$$$$\boxed{t_1\ne t_2}\Rightarrow\text{at least one stated common-history assumption fails}.$$

(b) Interpretation and further calculation

$$\boxed{f_{2,\mathrm{expected}}=2^{-2000/3000}=.62996}.$$$$f_{2,\mathrm{measured}}=k f_{2,\mathrm{expected}}\Rightarrow\boxed{k=.500/.62996=.79370}.$$

Physical check

Discordance identifies a model inconsistency but does not uniquely diagnose its cause. The initial-abundance interpretation is imposed only for the second part.