Radioactive Dating — Question 4

Question 4

A branching decay clock uses the total decay constant

A parent has total half-life T and branch probability b=.100 for producing stable daughter D. Other branches produce different nuclides. Initially D=0, the system stays closed, and the present number ratio is D/P=.150.

1. Derive the age relation accounting for branching.

2. Find age in units of T and compare with the unbranched formula.

Question 4 - Solution

A branching decay clock uses the total decay constant

(a) Method and calculation

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

(b) Interpretation and further calculation

$$\boxed{t/T=\ln(1+.150/.100)/\ln2=1.32193}.$$$$\boxed{t_{\mathrm{naive}}/T=\ln(1+.150)/\ln2=.20163}\quad\text{too young}.$$

Physical check

The branching probability controls daughter production, while all branches deplete the parent. Replacing the total half-life with a partial one would miscount parent survival.