This radioactive decay calculator works the full decay family — remaining amount after a time, age from a surviving fraction, decay constant and mean lifetime — for any isotope whose half-life you know.
N(t) = N₀ · e^(−λt) with λ = ln 2 / t½. Keep your time units consistent — the preset times are labeled in years, days or hours, and your elapsed time must use the same unit as the half-life you enter.
Exponential Decay, One Law for Everything
Every unstable nucleus has the same per-unit-time probability of decaying, so populations follow N = N₀e^(−λt): the count halves every half-life (t½ = ln 2 / λ) and the average atom lives τ = 1/λ ≈ 1.44 half-lives. Our Half-Life Calculator solves the same law arranged around the half-life itself; this page adds the age-solving direction (radiocarbon dating's core arithmetic) plus λ and τ. The identical exponential shapes pharmacokinetics (our Caffeine Calculator) and compound interest with the sign flipped — one mathematics, many costumes. Part of our Chemistry Calculators collection.
Worked Examples (Verified)
- Carbon-14 dating: a sample at 25% of the modern ¹⁴C level is exactly 2 half-lives old: 2 × 5,730 = 11,460 years — the defaults above, run in "Elapsed time" mode
- The same sample after just 1,000 years: 88.6% remains — ¹⁴C dating is a long-baseline instrument, precise over millennia, which is why it transformed archaeology
- Iodine-131 (t½ 8.02 days, medical thyroid work): after 32 days (4 half-lives) 1/16 = 6.25% of the activity remains — waste-storage and dosing schedules live on this count
Reading the Three Time Constants
- Half-life (t½): the headline number — when half is gone.
- Decay constant (λ): the per-unit-time decay probability; activity A = λN is computed from it.
- Mean lifetime (τ = 1.443 × t½): the average atom's lifespan — longer than the half-life because the survivors' tail is long.
- Dating range honesty: ¹⁴C reaches ~50,000 years (fraction ~0.2%); beyond that, other clocks (K-Ar, U-Pb) take over with their billion-year half-lives.
Related Tools
- Half-Life Calculator — the half-life-arranged solver
- Exponential Growth Calculator — the same law, positive sign
- Chemistry Calculators — the full collection
Frequently Asked Questions
How do you calculate radioactive decay?
N = N₀ × (½)^(t/t½), equivalently N₀e^(−λt) with λ = ln2/t½. Two half-lives leave 25%; five leave 3.125%; ten leave about 0.1%.
How does radiocarbon dating work?
Living things hold atmospheric ¹⁴C levels; at death the intake stops and the clock starts. Measure the surviving fraction, invert the decay law (this calculator's age mode): 25% remaining = 11,460 years for ¹⁴C. Calibration curves correct for historical variation in atmospheric ¹⁴C.
What is the difference between half-life and mean lifetime?
Half-life is the median survival (half the atoms gone); mean lifetime τ is the average (longer, τ ≈ 1.44 t½) because a fraction of atoms survive many half-lives. Particle physics quotes lifetimes; nuclear dating quotes half-lives.
Can half-life be changed?
Essentially no — decay constants are famously indifferent to temperature, pressure and chemistry (electron-capture nuclides shift by fractions of a percent under extreme ionization). That indifference is what makes radioactive clocks trustworthy across geology.
How accurate is the age from one measurement?
Limited by the surviving fraction's measurement precision and contamination — a 1% contamination of modern carbon in a 40,000-year sample pulls its apparent age thousands of years younger. Labs publish uncertainties with every date for this reason.
Is this radioactive decay calculator free?
Yes — free, instant, no sign-up.