Half-Life & Radioactive Decay Calculator

After every half-life, exactly half the radioactive atoms decay. So the amount remaining after time t is N = N × (1/2)^(t/T/), where N is the starting amount and T/ is the half-life. After 3 half-lives, you'd have (1/2)³ = 1/8 of the original. So 100 grams of a substance with a 5-year half-life leaves 12.5 grams after 15 years. The exponential nature means decay never quite reaches zero — it just gets vanishingly small. This same formula works for atoms, drug doses, and even capacitor discharge.

Enter starting amount N, half-life T/, and elapsed time t, and the calculator returns the remaining amount N = N × (1/2)^(t/T/). Some tools let you solve for any unknown — give two of the three quantities and it works out the third. Half-life and elapsed time must be in the same time unit (both years, both seconds, both whatever). For Carbon-14 with a 5730-year half-life, 1 g remaining after 17190 years means you started with 8 g. Reverse-solving like that is how radiocarbon dating works.

Half-life and decay constant are inversely related through T/ = ln(2)/λ, where ln(2) ≈ 0.693 and λ is the decay constant in inverse time units. So if λ = 0.1 per year, then T/ = 0.693/0.1 ≈ 6.93 years. The decay constant tells you the fraction of the sample that decays per unit time, while the half-life tells you how long until half is gone. They carry the same physical information in different forms. This conversion comes up whenever a problem mixes exponential decay equations with half-life data.

The conversion runs both ways: from half-life T/ to decay constant λ, use λ = ln(2)/T/; from λ to T/, use T/ = ln(2)/λ. Same time units throughout. So Iodine-131, with T/ = 8.02 days, has λ = 0.693/8.02 ≈ 0.0864 per day. Calculators do this in either direction at the click of a button. The decay constant is what plugs cleanly into N = N exp(−λt), the calculus form of the decay law. Both forms describe the same phenomenon, just suited to different problems.

Set (1/2)^n = 0.1 and solve for n. Take logs: n × log(0.5) = log(0.1), so n = log(0.1)/log(0.5) ≈ 3.32. So after roughly 3.32 half-lives, only 10% of the original remains. After 4 half-lives, you'd be down to 6.25%; after 5, down to 3.125%. This logarithmic relationship is handy for quick mental estimates: about 3 and a third half-lives gets you to 10%, and about 6.6 half-lives gets you to 1%. Useful for understanding nuclear waste decay timescales or dosimetry rules.

Rearrange the decay law to solve for t: t = T/ × log(N/N)/log(0.5), or equivalently t = T/ × log(N/N). If you started with 80 g of Cobalt-60 (T/ = 5.27 years) and now have 5 g, then t = 5.27 × log(80/5)/log(2) ≈ 21.1 years — exactly four half-lives. This kind of calculation is at the heart of radiometric dating, from carbon dating ancient artefacts to uranium-lead dating of rocks. Plug in carefully and don't forget to use the same log base in both numerator and denominator.

Half-life T/ is the time for half the sample to decay. Mean lifetime τ is the average time an individual atom survives, related to the decay constant by τ = 1/λ. The two are connected by T/ = τ × ln(2), so τ is always longer than T/ by a factor of about 1.443. For a substance with a 10-year half-life, the mean lifetime is roughly 14.43 years. Half-life is more intuitive for population statistics; mean lifetime appears more naturally in particle physics and the exponential N = N exp(−t/τ).