radiocarbon dating
/ carbon dating / KAR-bon /
How can we tell that a wooden beam, a scrap of linen, or a charcoal hearth is two thousand or twenty thousand years old? The trick is to use a radioactive clock that nature winds up in every living thing. Radiocarbon dating reads that clock, and it is one of the most famous real-world uses of exponential decay.
While an organism is alive it constantly exchanges carbon with the air, so its tissues keep the same small, steady fraction of radioactive carbon-14 as the atmosphere. The moment it dies, the exchange stops and no new carbon-14 comes in; the carbon-14 already inside simply decays away by the law dN/dt = -lambda N, with a half-life of about 5730 years. So measure how much carbon-14 is left relative to the living level, set N/N0 = e^(-lambda t), and solve for the age: t = -(1/lambda) ln(N/N0) = (5730/ln 2) ln(N0/N). Less carbon-14 remaining means more half-lives have passed, means older.
The method assumes the atmospheric carbon-14 level was the same in the past as today, which is not exactly true — so raw dates are calibrated against tree rings and other records. And because only about ten half-lives' worth of signal survives, radiocarbon reaches back roughly fifty thousand years; older samples need isotopes with longer half-lives, such as uranium-lead. It is a clean illustration that a model is only as trustworthy as its assumptions.
A charcoal fragment contains 25% of the living carbon-14 level. Then e^(-lambda t) = 0.25 = (1/2)^2, so t equals two half-lives, about 2 × 5730 ≈ 11460 years old.
Read the remaining carbon-14 fraction, invert the decay law for the age.
Dates are only as good as the assumption that past atmospheric carbon-14 matched today's; that is why measured ages must be calibrated, and why the technique runs out beyond roughly 50000 years.