Radioactive Decay Calculator
Radioactive decay shrinks a sample exponentially: every half-life, exactly half of what remains decays away. Choose a common isotope or enter your own half-life, set the elapsed time, and get the remaining amount, the decay activity in becquerels or curies, the mean lifetime, and a full show-your-work breakdown. You can also reverse-solve for the elapsed time or the initial amount.
Formula
Worked example
Carbon-14 half-life: 5730 years. After 11 460 years (two half-lives): λ = ln2/5730 ≈ 0.000121 per year, N = 100 × e^(-0.000121 × 11460) ≈ 25. A 1 g sample has activity A = 6.022×10^23 × (1/14.003) × (ln2/1.8046×10^11 s) ≈ 0.165 Bq/g.
How radioactive decay works
Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration, releasing energy as alpha particles, beta particles, or gamma rays. For any single nucleus the moment of decay is entirely random, but across a large ensemble of atoms the process is statistically perfect: a fixed fraction of the remaining nuclei decays in each unit of time. This produces exponential decay, described by N(t) = N₀ × e^(-λt), where N₀ is the starting quantity, N(t) is the amount surviving after time t, and λ is the decay constant. The rate is highest at the start when the most unstable atoms are present, and it tails off continuously as the sample depletes toward zero.
Half-life, decay constant and mean lifetime
The half-life (t½) is the time it takes for exactly half the sample to decay. It connects to the decay constant through λ = ln2 / t½. A third quantity, the mean lifetime τ = 1/λ = t½/ln2, is the average time an individual nucleus exists before it decays. After one half-life 50% remains; after two, 25%; after ten, less than 0.1%. Half-lives span an extraordinary range: Technetium-99m (used in medical imaging) has a six-hour half-life; Carbon-14 takes 5730 years; Uranium-238 persists for 4.47 billion years. The same exponential law governs every isotope.
Activity: becquerels and curies
Activity A is the number of disintegrations occurring per second, measured in becquerels (Bq, one decay per second) or curies (Ci, 3.7 × 10^10 Bq, the original standard based on 1 g of radium-226). Activity is A = λ × N, where N is the number of radioactive atoms present. Because N = N_A × (mass in grams / molar mass), the full formula is A = N_A × (m/M) × (ln2 / t½_seconds). Specific activity (Bq per gram) is the activity per unit mass and depends only on the isotope, not the sample size. Short-lived isotopes have very high specific activity.
Reverse-solving: dating and forensics
The decay law can be rearranged to find the elapsed time t = -(1/λ) × ln(N/N₀) given the original and remaining amounts, or to find N₀ = N / e^(-λt) given the current amount and the time elapsed. Radiocarbon dating uses the first form: by measuring the fraction of carbon-14 remaining in an organic sample relative to the atmospheric ratio (N₀), archaeologists calculate the age to within decades over tens of thousands of years. The same principle applies to uranium-lead, potassium-argon, and other decay pairs for geological dating. In nuclear medicine the reverse calculation lets pharmacists work out how much of a short-lived radiopharmaceutical to prepare hours ahead of a patient scan.
Common isotopes, half-lives and specific activity
| Isotope | Half-life | Molar mass (g/mol) | Specific activity |
|---|---|---|---|
| Tritium (H-3) | 12.32 years | 3.016 | 3.57 × 10^14 Bq/g |
| Carbon-14 (C-14) | 5730 years | 14.003 | 1.65 × 10^11 Bq/g |
| Strontium-90 (Sr-90) | 28.8 years | 89.908 | 5.11 × 10^12 Bq/g |
| Iodine-131 (I-131) | 8.03 days | 130.906 | 4.59 × 10^15 Bq/g |
| Cesium-137 (Cs-137) | 30.1 years | 136.907 | 3.22 × 10^12 Bq/g |
| Cobalt-60 (Co-60) | 5.27 years | 59.934 | 4.19 × 10^13 Bq/g |
| Tc-99m | 6.01 hours | 98.906 | 1.93 × 10^17 Bq/g |
| Radium-226 (Ra-226) | 1600 years | 226.025 | 3.66 × 10^10 Bq/g |
| Uranium-238 (U-238) | 4.47 × 10^9 years | 238.051 | 1.24 × 10^4 Bq/g |
| Potassium-40 (K-40) | 1.25 × 10^9 years | 39.964 | 2.63 × 10^5 Bq/g |
Specific activity calculated for a pure 1 g sample of the isotope. Actual material may be diluted.
Frequently asked questions
What units should I use for the initial amount?
Any consistent unit works: grams, moles, atoms, or becquerels. The result for "amount remaining" and "amount decayed" comes out in that same unit. For activity calculations, enable the activity toggle and enter the sample mass in grams and the molar mass of the isotope.
Why does the sample never reach exactly zero?
Exponential decay removes a fixed fraction of whatever is left in each interval rather than a fixed amount, so the curve halves and halves again without touching zero. In practice, once only a handful of atoms remain, the statistical model breaks down and the last nuclei decay at individually unpredictable moments.
How do I find the age of a sample (reverse-solve)?
Select "Elapsed time" from the "Solve for" dropdown. Enter the initial amount and the amount currently remaining; the calculator rearranges t = -(t½ / ln2) × ln(N / N₀) to give you the age. This is exactly how radiocarbon dating works.
What is the difference between Bq and Ci?
Both measure radioactive activity (decays per second). One becquerel (Bq) equals one decay per second. One curie (Ci) equals 3.7 × 10^10 decays per second, defined to match the activity of one gram of radium-226. The curie is still common in medical and older industrial contexts; the becquerel is the SI unit.
What is specific activity and why does it matter?
Specific activity is the activity per unit mass of an isotope, usually expressed in Bq/g or Ci/g. It is a fixed property of each isotope and depends only on its half-life and molar mass. Short-lived isotopes have very high specific activity. Knowing it lets you calculate how much radioactive material you actually have in a sample diluted with stable carrier.
Why do the presets not match a calculator I find elsewhere?
Half-life values are continuously refined by measurement. This calculator uses data from the 2020 Atomic Mass Evaluation (AME2020 / NUBASE2020), which is the current international standard. Small differences from older references are normal and do not affect the practical result for most calculations.