Radiocarbon Dating/Example/7000 BCE

Example of Radiocarbon Dating
It was reported in the newspaper that a prehistoric village had been dated using radiocarbon dating.

The carbon-$14$ test had been used to measure the amount of radioactivity still present in the organic material found in the ruins.

It was determined that a settlement existed in that place as long ago as 7000 BCE.

Given that:
 * the estimated half-life of carbon-$14$ used was $5600$ years
 * the newspaper report dated from about $1960$

the carbon-$14$ test must have shown that approximately $32 \%$ or $33 \%$ of carbon-$14$ was still present in the organic material at the time of discovery.

Proof
From First-Order Reaction, we have:


 * $x = x_0 e^{-k t}$

where:
 * $x$ is the quantity of carbon-$14$ at time $t$
 * $x_0$ is the quantity of carbon-$14$ at time $t = 0$
 * $k$ is a positive number.

By definition of half-life, when $x = \dfrac {x_0} 2$, we have $t = 5600$.

So:
 * $e^{-5600 k} = \dfrac 1 2$

So:
 * $k = \dfrac {\ln 0.5} {-5600} = \dfrac {\ln 2} {5600}$

After $7000 + 1960$ years:

So there is between $32 \%$ and $33 \%$ remaining.