Radioactive decay, half-life and carbon dating
How exponential decay works, what a half-life really measures, and how carbon-14 turns leftover atoms into an age estimate.
Why decay is exponential
A radioactive nucleus has no memory and no schedule. In any short slice of time each surviving atom has the same fixed chance of decaying, independent of how long it has already lasted. When you add that up across a huge population, a constant fraction disappears in every equal interval, which is exactly what produces an exponential curve. That is why the decay law is written as a power of one half rather than a straight line: the amount lost each year shrinks because there are fewer atoms left to lose.
Half-life versus the decay constant
The half-life T is the time for a sample to fall to half of its starting amount, an intuitive stopwatch figure. The decay constant lambda is the same fact stated as a rate, lambda = ln(2) divided by T, giving the fraction that decays per unit time. A short half-life means a large decay constant and a fast, hot source, while a long half-life means a tiny constant and a nearly stable one. This calculator reports both so you can move between the everyday number and the physics one.
How carbon-14 dating uses half-life
Living things absorb carbon-14 from the atmosphere and hold it at a roughly steady level while alive. Once an organism dies it stops taking in fresh carbon, so its carbon-14 decays with a half-life near 5730 years while ordinary carbon-12 stays put. Measuring the ratio that survives tells you how many half-lives have passed, which the elapsed-time mode of this tool turns into an age. One half-life leaves 50 percent, two leaves 25 percent, and by about ten half-lives, roughly 57,000 years, too little remains to date reliably.
Where the simple model stops
Pure exponential decay assumes a single isotope, no fresh input and a perfectly known half-life. Real measurements add complications: atmospheric carbon-14 has varied over history, so raw radiocarbon ages are adjusted with calibration curves, and contamination can push a date younger or older. Decay chains, where one isotope decays into another that also decays, need more than one half-life to describe. Use this calculator for the clean underlying math and lean on published calibration data when an exact date matters.