Carbon-14 Dating of an Ancient Sample
An ancient bone fragment contains only 1/8 of the carbon-14 it held when the animal died. Carbon-14 has a half-life of 5730 years. Assuming no carbon exchange with the environment since death, estimate the age of the bone.
Given
Fraction of original carbon-14 remaining: 1/8
Half-life of carbon-14: t1/2 = 5730 yr
The sample has been a closed system since death (no exchange with the atmosphere).
Find
The age of the sample in years.
Principle
Each half-life halves the number of radioactive nuclei present. If a fraction (1/2)^n of the original amount remains, then n half-lives have elapsed. The age is the number of half-lives multiplied by the length of one half-life. This is the basis of radiocarbon dating for once-living materials.
Formula
Solution
Write the decay relation: fraction remaining = (1/2)^n, where n is the number of half-lives that have passed.
Set (1/2)^n = 1/8. Since (1/2)^1 = 1/2, (1/2)^2 = 1/4, and (1/2)^3 = 1/8, we get n = 3 half-lives.
Convert half-lives to years: age = n x t1/2 = 3 x 5730 yr = 17,190 yr.
Unit check
The number of half-lives n is dimensionless, and t1/2 is in years, so age = n x t1/2 comes out in years, as required.
What this means physically
About 17,190 years ago the organism died and stopped taking in fresh carbon, and its carbon-14 has since decayed through three half-lives. This is how radiocarbon dating converts a measured isotope ratio into an age: the fraction remaining tells you the number of half-lives, and the half-life converts that count into years.
A common error is assuming decay is linear — for example, thinking that losing half takes 5730 years and losing the other half takes another 5730 in a straight line down to zero. Radioactive decay is geometric: each half-life removes half of whatever is currently present, so the amounts follow 1, 1/2, 1/4, 1/8, ...