Radioactive Half-Life & Decay Calculator
Solve any unknown in the radioactive decay law — remaining amount, elapsed time, half-life or starting amount — and visualize the decay curve.
Solve for
time
time
Using N = N₀·(½)^(t/T½)
- 1
Exponent t ÷ T½
20 ÷ 10 = 2 - 2
(½)^(t/T½)
0.5^2 = 0.25 - 3
Remaining amount N
100 × 0.25 = 25
How does this calculator work?
Radioactive decay follows N = N₀·(1/2)^(t/T½), equivalently N₀·e^(−λt) with decay constant λ = ln2/T½. Choose any one unknown — remaining amount, elapsed time, half-life or starting amount — and this calculator solves it, reports the remaining fraction, decay constant and mean lifetime, and plots the decay curve over five half-lives.
Formula
How this is calculated
Radioactive decay is a first-order process: the rate at which nuclei decay is proportional to the number present. This gives the exponential law N = N₀·(1/2)^(t/T½), where N₀ is the initial amount, t is the elapsed time, T½ is the half-life (the time for half the sample to decay) and N is the amount remaining. The decay constant λ = ln2/T½ links the half-life to the continuous form N = N₀·e^(−λt), and the mean lifetime is τ = 1/λ = T½/ln2.
Use the selector to choose which quantity is unknown. Solving for N evaluates the formula directly. Solving for elapsed time rearranges to t = T½·log₂(N₀/N). Solving for half-life gives T½ = t·ln2 / ln(N₀/N). Solving for the initial amount gives N₀ = N·2^(t/T½). The remaining fraction N/N₀ and decay constant are always reported.
All time inputs (t and T½) must share the same units (seconds, years, etc.) and the decay constant comes out in inverse of those units. Amounts may be in any consistent unit — atoms, moles, grams or activity (Bq) — because only the ratio N/N₀ matters. Inputs must be positive; for time and half-life solutions the remaining amount must be strictly less than the initial amount, otherwise the logarithm is undefined and the calculator returns no result.
Frequently asked questions
The half-life T½ is the time for half a sample to decay. The mean lifetime τ = 1/λ = T½/ln2 ≈ 1.443·T½ is the average time a single nucleus survives, so it is always longer than the half-life.
The decay law depends only on the ratio N/N₀, so any unit (atoms, grams, becquerels) works as long as N and N₀ use the same one. Likewise t and T½ must use the same time unit so the exponent t/T½ is dimensionless.
Yes. Set N₀ and N to the desired ratio (for example 100 and 12.5 for one-eighth) and solve for elapsed time; the result will be the number of half-lives times T½.
Also known as
TG we-Calculate Editorial Team. (2026). Radioactive Half-Life & Decay Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/half-life-calculator
TG we-Calculate Editorial Team. "Radioactive Half-Life & Decay Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/half-life-calculator.
TG we-Calculate Editorial Team, "Radioactive Half-Life & Decay Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/half-life-calculator
@misc{wecalculate_half_life_calculator, title = {Radioactive Half-Life & Decay Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/half-life-calculator}}, year = {2026}, note = {TG we-Calculate} }
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