Half-Life and Activity

Activity (A) — the number of decays per second, measured in becquerels (Bq), where 1 Bq is one decay per second.

The decay constant (λ) — the probability that a given nucleus decays per unit time, measured in s⁻¹. It links activity to the number of undecayed nuclei:

A = λ N

Exponential decay — because the number decaying per second is proportional to the number remaining:

N = N₀ e^(−λt) and A = A₀ e^(−λt)

This is the same mathematics as capacitor discharge, with λ playing the role of 1/RC. Everything learned there transfers: the ln-plot is a straight line of gradient −λ, and the quantity falls by the same fraction in every equal interval.

Half-life (t₁⁄₂) — the average time for half the nuclei in a sample to decay, or for the activity to halve:

t₁⁄₂ = ln 2 / λ = 0.693 / λ

A large λ means a short half-life — an unstable isotope decays fast and is intensely active but soon gone. A long half-life means a feeble but persistent source. The two cannot be had together: a given number of nuclei releases the same total, and the half-life only sets how quickly.

Example: this trade-off drives the choice of medical isotopes. Technetium-99m has a half-life of six hours — long enough to image a patient, short enough that it is essentially gone within a day or two. Carbon-14's 5730 years makes it useless medically but ideal for dating.

Radioactive dating — a sample's remaining fraction of a radioactive isotope gives its age. Carbon-14 is absorbed by living things and stops being replenished at death, so its decline dates organic material back around 50 000 years. For rocks, the ratio of uranium-238 to its stable decay product lead-206 works over billions of years.

Assumptions dating relies on — that the initial proportion is known, that the half-life is constant, and that nothing has entered or left the sample since. Each is a genuine limitation rather than a technicality; carbon dating, for instance, needs calibration because atmospheric carbon-14 has not been constant.