Half-Life, Activity & Decay Chain Calculator
Radioactive decay: remaining amount and activity, time to a level, half-life fits and chains.
How it works
Work out radioactive decay the way textbooks and labs do: how much of a nuclide is left after a time and what its activity is (Bq or Ci, from a percentage, an activity, a mass or a number of atoms), how long it takes to fall to a given level, which half-life fits your measured count rates, and how the members of a decay chain such as Mo-99 → Tc-99m or radon and its progeny grow in and decay (Bateman equations with branching). Nineteen preset nuclides carry evaluated half-lives from ENSDF/NUBASE2020. Part of the Proofline math notebook.
- N(t) = N₀·2^(−t/T½) = N₀·e^(−λt) with λ = ln 2 / T½; activity A = λ·N in becquerel (1 Ci = 3.7 × 10¹⁰ Bq); atoms from a mass via the molar mass of the nuclide (AME2020 where Proofline’s chemistry engine has it, otherwise the mass number) and the exact Avogadro constant. Time to a level: t = ln(N₀/N)/λ.
- Half-life fits: least squares of ln(value) against time with Proofline’s QR fitter gives λ and its standard error; two points give the half-life directly.
- Chains: the Bateman equations Nₖ(t) = N₁(0)·Πbᵢλᵢ·Σ e^(−λᵢt)/Π(λⱼ − λᵢ) with branching fractions bᵢ; the time of the daughter maximum is ln(λ₂/λ₁)/(λ₂ − λ₁).
- Verified: 1 g of Ra-226 has 0.989 Ci (the historical curie), C-14 has a specific activity of 166 GBq/g, the Tc-99m activity of a Mo-99 generator peaks after 22.85 h, and radon progeny reach secular equilibrium within hours.
Decay is a statistical law: small samples fluctuate around these averages. No dose, shielding or exposure calculation is made. Nothing here is advice on handling chemicals or radioactive material safely.
Sources
- NuDat 3 — evaluated nuclear structure data (ENSDF): half-lives and decay modes (National Nuclear Data Center, Brookhaven National Laboratory)
- F. G. Kondev, M. Wang, W. J. Huang, S. Naimi, G. Audi: The NUBASE2020 evaluation of nuclear physics properties, Chinese Physics C 45 (2021) 030001 (IOP Publishing)
- H. Bateman: The solution of a system of differential equations occurring in the theory of radioactive transformations, Proc. Cambridge Phil. Soc. 15 (1910) 423–427 (Cambridge Philosophical Society)
- CODATA 2018: molar gas constant R and Avogadro constant (exact SI values) (NIST)
Part of Proofline
-
Proofline Math notebook
Solve, graph, test and visualise in one notebook — with data, findings and a report.
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Calculate radioactive decay, half-lives and decay chains
Work out how much of a radioactive nuclide remains after a time, how long it takes to reach a level, fit a half-life to your measurements and follow decay chains.
What it is for
Radioactive decay follows one simple law: after every half-life, half of the nuclei are left. The arithmetic around it is still easy to get wrong, with units like MBq, curie, days and years mixed together, activity versus mass, and chains where a daughter grows in and then decays.
The tool has four modes. Decay gives the amount, activity and mass left after a time. Time tells you how long it takes to fall to a target level. Fit finds the half-life from your own measurements with a log-linear least-squares fit. Chain follows a parent and its daughters with the exact Bateman equations, including branching.
Pick one of 19 preset nuclides, whose half-lives come from ENSDF/NuDat and NUBASE2020, or type any half-life. Amounts can be a percentage, an activity (Bq, Ci), a mass or a number of atoms. The tool does physics only. It does not give dose or safety advice.
Worked example
Iodine-131, starting at 100 MBq, after 30 d: 7.493 % remains, which is 3.738 half-lives of 8.0252 d. The activity is now 7.4935 MBq. That corresponds to only 1.63 ng of iodine, about 7.5 × 10¹² atoms.
Carbon-14 in time mode, from 100 % down to 25 %: 11400 years, exactly two half-lives of 5700 years. The specific activity of pure C-14 is 165.72 GBq per gram.
One gram of radium-226 has an activity of 36.581 GBq, 0.9887 Ci. The curie was originally defined as the activity of one gram of radium, and the small difference comes from the modern half-life of 1600 years.
In chain mode, the Mo-99 → Tc-99m generator starting at 10 GBq of Mo-99: the Tc-99m activity peaks 22.847 h later. That is why hospitals elute the generator about once a day. After 7 days the two are in transient equilibrium.
How to read the result
- Remaining and half-lives elapsed are independent of the units. Activity and mass appear when the starting amount allows converting between them.
- Decay constant λ = ln 2 / T½ and mean life τ = 1/λ.
- In fit mode, the standard error of the half-life and R² show how well the data follow a single exponential. The residual column shows which points stray.
- In chain mode, the equilibrium type (secular, transient or none) and the time of the daughter's peak are given.
Tips and common mistakes
- Write units in full:
30 d,5700 y,164.3 µs,10 MBq,2 mCi. One year is 365.2422 days. - For the fit, enter one measurement per line as time and value, for example
6 h 501. Subtract the background first. - Mass and atom counts of a preset use the nuclide's atomic mass, or its mass number with a note when that is all that is available.
- Very short-lived daughters (Po-214, 164 µs) follow their parent within microseconds. The curves show them in equilibrium.
FAQ
What does the half-life fit need?
At least three measurements at different times, all above zero. The more half-lives the data cover, the more precise the result.
Can I enter my own decay chain?
Yes. Write one member per line with its half-life, add a branching fraction where only part of the parent decays to that member, and end with stable if you like.
Why is the Tc-99m activity lower than Mo-99 in equilibrium?
Only 87.6 % of Mo-99 decays go to Tc-99m. The rest bypass the metastable state.
Does the tool count random fluctuations?
No. It gives expected values; real counts scatter around them.
Guide last updated:
For scripts and AI agents
- JSON API:
POST https://prooflinehalflife.usefulweb.app/api/v1/calculate(same calculation as this page) - Interface description: OpenAPI 3.1 · AI guide: llms.txt
- MCP server (Streamable HTTP):
https://prooflinehalflife.usefulweb.app/mcp· tool nameproofline_half_life
Example request
curl -s -X POST https://prooflinehalflife.usefulweb.app/api/v1/calculate \
-H 'content-type: application/json' \
-d '{"mode":"decay","nuclide":"i131","initial":"100 MBq","elapsed":"30 d"}'