Radiocarbon dating

Problem

A find shows a $^{14}\text{C}/^{12}\text{C}$ ratio equal to $1/8$ of that of a living sample. Knowing that $T_{1/2} = 5730$ years, estimate the age of the find. Use the decay graph to interpret the result.

Exponential decay curve: the dashed line marks the $1/8$ level at $t = 3\,T_{1/2}$.

Solution

Reading the ratio. The value $1/8$ is an exact power of $1/2$:

$\frac{1}{8} = \left(\frac{1}{2}\right)^{3}$

Each factor of $1/2$ corresponds to a half-life: exactly $3$ half-lives are needed for the $^{14}\text{C}$ to drop to an eighth of the initial value. On the graph this is the point where the curve meets the dashed line, at $t/T_{1/2} = 3$.

Age of the find.

$t = 3\cdot T_{1/2} = 3\cdot 5730\ \text{years} \approx \ev{t \approx 17\,190\ \text{years}}$

Links

Topics: Nuclear physics Concepts: Radioactive decay Skills: Reading graphs