Radioactive Half-life

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

Radioactive decay is a random, statistical process, and the count of surviving nuclei follows N(t) = N₀·(1/2)^(t/T). You can also write it as N(t) = N₀·e^(−λt), where the decay constant is λ = ln(2)/T. Once n half-lives have gone by, a fraction 1/2ⁿ of the original nuclei is still around. So if N₀ = 100 and t = 2·T, you are down to 25 nuclei. Take Carbon-14, with T = 5,730 years: that timescale is what lets archaeologists date finds back to roughly 50,000 years (Libby won the Nobel Prize for it in 1960). For dating zircons, geologists turn to Uranium-238 (T = 4.47 Ga) and U-235 (704 Ma). In medicine, Technetium-99m (T = 6 h) does most of the work in scintigraphy, while Iodine-131 (T = 8 days) is used against thyroid disorders. And Caesium-137 (T = 30 years) was behind the 1987 Goiânia accident and still shows up in soils contaminated by Chernobyl in 1986.

Applications

Radiocarbon dating in archaeology. U-Pb and K-Ar dating in geology. Nuclear medicine, where Tc-99m handles diagnostics and I-131 the therapy side. Industrial gauging. Radioactive waste management, which in Brazil falls to CNEN under Law 10.308/2001. And forensic work on nuclear materials.

FAQ

What's the difference between half-life and mean lifetime? The mean lifetime works out to τ = 1/λ = T/ln(2), which makes τ ≈ 1.443·T. Half-life marks the point where 50 % has decayed; τ marks where 1/e ≈ 36.8 % is still there.

Can temperature or pressure change half-life? No. Decay happens in the nucleus, and that's basically deaf to chemical or thermodynamic conditions. The rare exceptions involve electron capture.

How many half-lives until a material is "safe"? The usual rule of thumb is 10 half-lives, which brings you down to about 0.1 % of the original activity. For isotopes that live a very long time, like U-238, what counts as "safe" comes down to the activity level rather than just how much time has passed.