Last updated: 2026-06-15

Understanding half-life

Half-life (t½) is the time required for exactly half of a given quantity of a substance to decay or transform. The concept applies to radioactive nuclides, first-order chemical reactions, pharmaceutical drug clearance, and any process that decays at a rate proportional to the current amount. The half-life is constant for a given nuclide or reaction — it does not depend on the starting quantity.

Carbon-14 has a half-life of about 5,730 years. Iodine-131, used in thyroid treatment, decays in roughly 8 days. Some synthetic nuclides last only microseconds; uranium-238 has a half-life of 4.5 billion years.

The decay formula

The amount remaining after any elapsed time follows:

where $N_0$ is the initial amount, $t$ is the elapsed time, $t_{1/2}$ is the half-life, and $N$ is the amount that remains. The exponent $n = t / t_{1/2}$ counts how many half-lives have passed.

Worked example

A sample contains 80 g of a radioactive isotope with a half-life of 10 days. How much remains after 30 days?

After three half-lives, 10 g remains — 12.5% of the original 80 g.

Half-life and the decay constant

The continuous form of the decay law uses the decay constant λ (lambda):

The two forms are equivalent when:

The mean lifetime of a single nucleus — the average time before it decays — is:

A larger λ means faster decay and a shorter half-life.

Half-life and radiocarbon dating

While an organism is alive, it continuously exchanges carbon with the atmosphere, keeping its carbon-14 to carbon-12 ratio equal to the atmospheric ratio. When the organism dies, no new carbon-14 enters, and the existing carbon-14 decays with a half-life of 5,730 years.

By measuring the surviving fraction of carbon-14 and applying the decay formula, scientists can estimate the age of organic material up to about 50,000 years. Beyond that, the remaining carbon-14 falls below reliable detection limits.

Half-life in medicine and pharmacology

Drug clearance often follows first-order kinetics: the concentration in the body decreases by half every half-life. This determines how frequently a drug must be taken to maintain a therapeutic level. After five half-lives, less than 3.2% of the initial dose remains, which is why five half-lives is the conventional threshold for calling a drug "cleared."