Radioactive Dating — Question 7

Question 7

An isochron separates age from the initial daughter ratio

Co-genetic closed minerals share initial $y_0=D_0/S$. Let $x=P/S$ and $y=D/S$ today; all parent decays produce D and S is stable. Measurements give $(x,y)=(1,.250)$ and $(3,.650)$.

1. Derive the isochron equation and infer its slope and intercept.

2. Find age in half-life units and predict y for a third mineral with x=5.

Question 7 - Solution

An isochron separates age from the initial daughter ratio

(a) Method and calculation

$$D=D_0+P(e^{\lambda t}-1)\Rightarrow\boxed{y=y_0+x(e^{\lambda t}-1)}.$$$$\boxed{m=(.650-.250)/(3-1)=.200,\quad y_0=.250-.200=.0500}.$$

(b) Interpretation and further calculation

$$\boxed{t/T_{1/2}=\ln(1+m)/\ln2=\ln1.2/\ln2=.26303}.$$$$\boxed{y(5)=.0500+.200(5)=1.050}.$$

Physical check

A third consistent point supports, but does not by itself prove, shared initial composition and closure. Mixing unrelated materials can also create apparent lines.