Half-Life and Decay Curves

A single radioactive nucleus has no memory and no clock. It might decay in the next second or sit stable for a thousand years, and nothing about its past tells you which. That unpredictability is real, not just a gap in our knowledge. But a sample of atoms is not one atom, it is trillions of them, and each one has the same fixed probability of decaying per unit time. With numbers that large, randomness at the individual level averages into near-perfect regularity at the group level. That regularity is the half-life: the time for half of whatever nuclei remain to decay, no matter how much or how little you started with.

Because the same fraction decays every half-life, the amount remaining follows N = N0 (1/2)^t/half-life, where N0 is the starting amount and t is elapsed time. Plot surviving nuclei against time and you get a curve that drops fast at first, then levels off, cutting itself in half again and again: after one half-life you have 50 percent left, after two 25 percent, after three 12.5 percent, and so on, never quite reaching zero. This is exponential decay, the same mathematical shape whether you are tracking atoms, radioactive dose, or drug concentration in the body.

This idea connects directly to carbon-14 dating, which reads age off this exact curve. Carbon-14 decays at a fixed, known half-life while stable carbon-12 does not change. Measuring the ratio of remaining carbon-14 to carbon-12 in a bone tells you what fraction of the original carbon-14 survives, and the decay equation converts that fraction straight into elapsed time since the organism died.