Radioactive Half-life
Compute remaining amount A = A₀·(1/2)^(t/T) for radioactive decay.
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.
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Compute the radioactive decay constant, λ = ln(2)/T½, from the half-life (T½). The constant λ is the probability of a nucleus decaying per unit time — the larger it is, the more unstable the isotope and the shorter its half-life. It links the half-life (time for half the nuclei to decay) to the activity and to the exponential decay law. Enter the isotope's half-life.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.