Isotope Half Life Time

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

How much of a radioactive isotope is left after a time t follows N(t) = N₀·(1/2)^(t/T), which is the same thing as N(t) = N₀·e^(−λt) once you set the decay constant λ = ln(2)/T. Take N₀ = 100 g and t = 2T, and you're down to 25 g. The reference half-lives cover an enormous spread, around 24 orders of magnitude: Carbon-14 T = 5,730 years (radiocarbon dating up to ~50,000 yr, Libby 1949), Uranium-238 T = 4.47 Ga (Earth-age geochronology), Potassium-40 T = 1.25 Ga (K-Ar dating), Technetium-99m T = 6 h (the nuclear-medicine workhorse), Iodine-131 T = 8.02 days (thyroid therapy), Caesium-137 T = 30.17 yr (Goiânia 1987, Chernobyl 1986), Plutonium-239 T = 24,110 yr (reactors and weapons). The mean lifetime works out to τ = T/ln(2) ≈ 1.443·T.

Applications

Radiocarbon dating in archaeology and paleoclimatology, U-Pb and K-Ar geochronology of rocks and meteorites, nuclear medicine (Tc-99m scintigraphy, I-131 therapy, F-18 in PET), radioactive waste classification (in Brazil under CNEN/Law 10.308/2001), industrial radiography, and even the smoke detector sitting on your ceiling (Am-241, T ≈ 432 yr).

FAQ

Can half-life be changed? Not really. Decay happens in the nucleus, and it doesn't care about temperature, pressure, or what chemistry surrounds the atom. The rare exceptions show up with electron-capture isotopes under extreme conditions.

How many half-lives until "safe"? A rule of thumb people use is 10 half-lives, which leaves roughly 0.1 % of the activity. With long-lived isotopes that rule loses meaning, since what counts as "safe" hangs on the absolute activity level rather than how much time has gone by.

Why use Tc-99m if it lasts only 6 hours? That short half-life is exactly the point: it gives high specific activity for imaging and then clears the body fast, which keeps the patient's dose low. Hospitals make it on-site from Mo-99 generators.