Formula for Radiocarbon Dating

Theorem
Let $Q$ be a quantity of a sample of dead organic material (usually wood) whose time of death is to be determined.

Let $t$ years be the age of $Q$ which is to be determined.

Let $r$ denote the ratio of the quantity of carbon-14 remaining in $Q$ after time $t$ to the quantity of carbon-14 in $Q$ at the time of its death.

Then the number of years that have elapsed since the death of $Q$ is given by:
 * $t = -8060 \ln r$

Proof
Let $x_0$ denote the ratio of carbon-14 to carbon-12 in $Q$ at the time of its death.

Let $x$ denote the ratio of carbon-14 to carbon-12 in $Q$ after time $t$.

Thus:
 * $r = \dfrac x {x_0}$

It is assumed that the rate of decay of carbon-14 is a first-order reaction.

Hence we use:

Chemical analysis tells us that after $10$ years, $99.876 \%$ of the carbon-14 that was present in the organic matter still remains.

Thus at time $t = 10$, we have:

Hence the result.