Question

the ratio of carbon-14 to carbon-12 in a prehistoric wooden artifact is measured to be one-eight of the ratio measured in a fresh sample of wood from the same region. the half life of carbon-14 is 5,715 years. Determine its age.

Answer 1

$1.715 \times 10^{4} \; \text{years}$

Carbon-14 is radioactive and decays spontaneously. Interaction between stellar radiation and atmospheric nitrogen ensures a constant carbon-14 to carbon-12 ratio in the earth atmosphere. Living plants undergo respiration and take in carbon constantly and maintain a constant carbon-14-to-carbon-12 ratio in their tissues. Carbon atoms in wood artifacts are no longer refreshed such that the relative abundance of carbon-14 decays exponentially.

It takes one half-life for the relative abundance of carbon-14 in a sample to drop by one half. It would thus take three consecutive half-lives for the mass of carbon-14 in the wooden artifact to decrease by a factor of eight.

$t_{1/2} = 5, 715 \; \text{years}$ as seen in the question. Three half-lives would thus correspond to

$3 \times 5,715 = 17145 = 1.715 \times 10^{4} \; \text{years}$

The age of this sample is therefore $1.715 \times 10^{4} \; \text{years}$ at most.