Radioactive Dating — Question 9

Question 9

A reservoir offset changes the fraction present at closure

An organism begins with carbon-14 fraction .800 relative to the reference atmosphere because of its reservoir. At measurement the fraction is .200. Assume closed-system decay with half-life 5730 y and no later contamination.

1. Calculate the elapsed time since closure using the correct initial fraction.

2. Find the apparent age assuming an initial fraction of 1 and derive the reservoir age offset.

Question 9 - Solution

A reservoir offset changes the fraction present at closure

(a) Method and calculation

$$f=f_0e^{-\lambda t}\Rightarrow\boxed{t=\lambda^{-1}\ln(f_0/f)}.$$$$\boxed{t=5730\log_2(.800/.200)=11460\,\mathrm y}.$$

(b) Interpretation and further calculation

$$\boxed{t_{\mathrm{app}}=5730\log_2(1/.200)=13305\,\mathrm y}.$$$$\boxed{\Delta t=t_{\mathrm{app}}-t=5730\log_2(1/.800)=1845\,\mathrm y}.$$

Physical check

The offset reflects the initial isotope ratio, not an additional period after the organism stopped exchanging carbon. A real reservoir correction may vary in space and time.