Radioactive decay and carbon-14 dating

A radioactive nucleus decays with a probability per unit time that is the same for every nucleus: it follows that the fraction of nuclei surviving after a time $\tau$ (half-life) is reduced by half. After $n$ halvings there remain $N_0/2^n$ nuclei.

Example — Carbon-14 dating

$^{14}$C has half-life $\tau\approx 5730$ years. In a living being the concentration of $^{14}$C relative to $^{12}$C is about $r_0=1{,}3\cdot 10^{-12}$. In a bone relic a concentration $r=0{,}25\,r_0$ is measured. How old is it?

From the model: $r(t)/r_0 = (1/2)^{t/\tau}$. Hence $(1/2)^{t/\tau} = 0{,}25 = (1/2)^2 \implies t/\tau = 2 \implies t = 2\tau \approx 11\,460 \text{ years}.$

Example — Non-integer case: $0{,}3\,r_0$

$(1/2)^{t/\tau}=0{,}3 \implies t/\tau = \log_{1/2}(0{,}3) = -\dfrac{\ln 0{,}3}{\ln 2} = \dfrac{\ln(10/3)}{\ln 2} \approx 1{,}737$, hence $t\approx 1{,}737\cdot 5730 \approx 9\,950 \text{ years}$.

Links

Topics: Exponential function
Concepts: Exponential decay · Half-life
Functions: Exponential function
Skills: Modelling · Using formulae