Radioactive Decay Calculator
Enter the initial amount N₀, half-life T½ and elapsed time t; using N = N₀·(½)^(t/T½) the tool instantly computes the remaining amount and activity and the decayed fraction.
Input Data
Results
At a glance:Radioactive decay is the spontaneous change of unstable nuclei into more stable ones, emitting radiation (α, β, γ). A core feature is that the decay rate is proportional to the current number of undecayed nuclei, i.e. a 'first-order' (exponential) process, so the amount left after time t (since start) follows N = N₀·(½)^(t/T½), where N₀ is the initial amount (number of atoms, mass, or activity), T½ is the half-life (the time for the amount to halve) and t is the elapsed time. The 'half-life' is the most intuitive parameter: after 1 half-life, half remains; after 2, a quarter; after n half-lives, (½)ⁿ remains. The decayed amount is N₀ − N, and the decayed fraction is (N₀ − N)/N₀ = 1 − (½)^(t/T½). Because 'activity' (decays per unit time) is proportional to the number of undecayed nuclei, activity obeys the same law: A = A₀·(½)^(t/T½) — activity also halves every half-life. Using this tool's default: N₀ = 100, T½ = 5, t = 10, N = 100×(½)^(10/5) = 100×(½)² = 100×0.25 = 25, decayed 75%. Radioactive decay is everywhere: (1) radioactive dating — by measuring the remaining ratio of a parent-daughter pair and their known half-life, work back to the sample age, e.g. carbon-14 (T½ ≈ 5730 yr) for archaeological and paleo samples, uranium-lead for rocks (billions of years), potassium-argon for strata. (2) Nuclear medicine — half-life sets a drug's effective time; iodine-131 (T½ ≈ 8 days) for thyroid therapy, technetium-99m (T½ ≈ 6 h) for imaging, so they decay fast enough to limit dose. (3) Radiation safety — after accidents or waste disposal, decay computes how long until radiation falls below a safe threshold. (4) Carbon dating and environmental monitoring. Notes: first, time t and half-life T½ must use the same unit. Second, N₀, N can be mass, atom count or activity — as long as consistent, the formula works. Third, this is ideal behaviour; very long storage ignores cosmic-ray production and minor branches. Fourth, T½ and t non-negative. In short, this calculator lets you quickly obtain remaining amount and decayed fraction from N₀, T½ and t — a practical tool for nuclear chemistry, dating and medical isotope work.
Formula
Remaining amount: N = N₀·(½)^(t/T½).
Decayed amount: N₀ − N; decayed fraction = 1 − (½)^(t/T½).
Activity: A = A₀·(½)^(t/T½) (activity ∝ amount).
After n half-lives: (½)ⁿ remains.
$$N = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}$$How to Use
- Enter the initial amount N₀ (mass, atoms or activity).
- Enter the half-life T½ and the elapsed time t (same unit).
- The right panel instantly shows the remaining amount N and the decayed fraction (%).
Half-lives of common radioactive isotopes
| Isotope | Half-Life | Application |
|---|---|---|
| Carbon-14 (¹⁴C) | ≈ 5730 yr | Archaeological dating |
| Iodine-131 (¹³¹I) | ≈ 8.0 days | Thyroid therapy |
| Technetium-99m (⁹⁹ᵐTc) | ≈ 6.0 h | Medical imaging |
| Uranium-238 (²³⁸U) | ≈ 4.5×10⁹ yr | Geological dating |
Remaining fraction = (½)^(t/T½); activity decays the same way as amount.
Case Studies
Remaining amount after two half-lives
Initial amount 100 (units), half-life 5, elapsed 10 = two half-lives.
N = 100×(½)² = 25; decayed fraction = (100 − 25)/100 = 75%.
After each half-life the amount halves: 100 → 50 → 25.
Activity of a medical isotope
Technetium-99m activity A₀ at injection, half-life ≈ 6 h.
After 6 h (1 half-life) A = A₀/2; after 12 h A = A₀/4.
Fast decay limits patient radiation dose, so imaging is done within hours.
FAQ
Why does radioactive decay obey a half-life?
Because decay is a first-order (random, independent) process: the decay rate is proportional to the number of undecayed nuclei, giving the exponential law N = N₀·(½)^(t/T½). Each half-life halves the remaining amount, independent of how much is already there.
Is the half-life always constant?
For a given isotope under normal conditions, yes — half-life is an intrinsic nuclear property independent of temperature, pressure, chemical state, etc. (Only extreme astrophysics affects it.) This stability is the basis of radioactive dating.
How is activity related to amount?
Activity A (decays per second, Bq) is proportional to the number of undecayed nuclei N: A = λN, where λ is the decay constant. Since N decays exponentially, A does too: A = A₀·(½)^(t/T½). So activity halves every half-life, same as the amount.
How to use it for dating?
Measure the remaining parent isotope N and the daughter product in the sample; knowing the half-life T½, solve t = T½·log₂(N₀/N) for the elapsed time. Carbon-14 (5730 yr) is used for archaeological samples up to tens of thousands of years.
Why do medical isotopes need a suitable half-life?
Too short and it decays before reaching the target; too long and it leaves prolonged radiation in the body. Imaging agents (e.g. Tc-99m, 6 h) need short half-life to limit dose; therapy agents balance effect time and safety.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.