Half-Life Calculator

Leave blank the value you want to find. With the defaults, 100 units decaying to 10 units in 50 time units, the half-life is 50 × ln 2 ÷ ln 10 ≈ 15.05 time units. Any unit of time works, as long as the time and the half-life use the same one. The quantity can be a mass, a number of atoms, a radioactivity reading or a drug concentration.

Definition and Formula

The half-life of a substance is the time it takes for half of it to decay or disappear. After one half-life, 50% remains; after two, 25%; after three, 12.5%; and after ten, a little under 0.1%. The idea is most closely tied to radioactive decay, where each unstable nucleus has a fixed chance of decaying in any instant, regardless of its age or of what its neighbors do. Because that chance is constant, the number of nuclei falls by the same fraction in every equal time interval, which is exponential decay.

The same mathematics describes many other processes: the elimination of a drug from the bloodstream, the discharge of a capacitor, the cooling of a hot object toward room temperature, and the decline of a chemical reactant in a first-order reaction. Doctors use a drug's half-life to choose dosing intervals, and archaeologists use the 5,730-year half-life of carbon-14 to date organic remains up to about 50,000 years old.

Three equivalent constants describe the rate. The half-life t½ is the most intuitive. The mean lifetime τ is the average time a single particle survives before decaying, and it is longer than the half-life by a factor of 1 ÷ ln 2 ≈ 1.4427. The decay constant λ is the probability of decay per unit time, the reciprocal of τ. They are related by t½ = τ ln 2 = ln 2 ÷ λ.

Half-lives span an enormous range, from fractions of a second for some artificial isotopes to billions of years for uranium-238, which is why some radioactive waste stays hazardous for millennia while medical tracers are chosen to fade within hours or days.

N(t) = N₀ · (1/2)^(t ÷ t½) = N₀ · e^(−t ÷ τ) = N₀ · e^(−λt)

N₀

the initial quantity

N(t)

the quantity remaining after time t

t½

the half-life

τ

the mean lifetime

λ

the decay constant

Derivation of the Relationship Between Half-Life Constants

Start from the exponential decay law N(t) = N₀e^(−λt). At t = t½, exactly half remains, so N₀ ÷ 2 = N₀e^(−λt½). Dividing by N₀ and taking natural logarithms gives −ln 2 = −λt½, so t½ = ln 2 ÷ λ. The mean lifetime is the average of t weighted by the decay rate, τ = ∫ t·λe^(−λt) dt from 0 to ∞, which works out to 1 ÷ λ. Combining the two gives t½ = τ ln 2.

Frequently asked questions