Radioactive Half-Life Calculator

Radioactive half-life: N(t) = N₀·(1/2)^(t/T)

Radioactive decay obeys N(t) = N₀·(1/2)^(t/T), which is the same thing as N(t) = N₀·e^(−λt) once you set λ = ln(2)/T. Each half-life cuts the count in half, so after n of them you are left with 1/2ⁿ of the original nuclei. Take N₀ = 100 and t = 3·T: that leaves 12.5 nuclei. The half-lives span an enormous range. Carbon-14 sits at T = 5,730 years (Libby, Nobel 1960), which makes it datable out to roughly 50,000 years. Uranium-238, with T = 4.47 Ga, underpins the geochronology of zircons. Technetium-99m (T = 6 h) is the workhorse of nuclear medicine scintigraphy. And Caesium-137 (T = 30 years) was behind the 1987 Goiânia incident and still lingers in Chernobyl soils.

Applications

Radiocarbon dating in archaeology (Libby method, Nobel 1960), U-Pb and K-Ar dating in geology, nuclear medicine (Tc-99m diagnostics, I-131 thyroid therapy), industrial gauging, and the management of radioactive waste, which in Brazil falls under CNEN and Law 10.308/2001.

FAQ

Half-life vs mean lifetime? The mean lifetime is τ = 1/λ = T/ln(2) ≈ 1.443·T. Half-life marks the point where 50 % has decayed, whereas τ is the point where 1/e ≈ 36.8 % is still around.

Do temperature or pressure change half-life? No. Decay happens in the nucleus, so for all practical purposes it doesn't care about chemical or thermodynamic conditions (electron capture being a rare exception).

How many half-lives until "safe"? A common rule of thumb puts it at 10 half-lives, after which about 0.1 % of the activity remains. With long-lived isotopes like U-238, though, what counts as safe hinges on the activity level rather than on how much time has gone by.