Radioactive Dating — Question 2

Question 2

A stable daughter counts the parents that have disappeared

A mineral initially contains radioactive parent P and no daughter D. Each parent decay produces one stable daughter, and the mineral remains closed. The measured number ratio is D/P=3.00; the parent half-life is $5.00\times10^8$ y.

1. Derive age as a function of D/P without knowing the initial parent number.

2. Evaluate the age and the surviving fraction of the original parent population.

Question 2 - Solution

A stable daughter counts the parents that have disappeared

(a) Method and calculation

$$P=P_0e^{-\lambda t},\quad D=P_0-P\Rightarrow D/P=e^{\lambda t}-1.$$$$\boxed{t=\lambda^{-1}\ln(1+D/P)}.$$

(b) Interpretation and further calculation

$$\boxed{t=\frac{5.00\times10^8}{\ln2}\ln4=1.00\times10^9\,\mathrm y}.$$$$\boxed{P/P_0=P/(P+D)=1/(1+3)=.250}.$$

Physical check

Number ratios, not raw mass ratios, enter this expression. Initial daughter or subsequent loss would violate the assumptions.