# Half-Life, Activity & Decay Chain Calculator

> Radioactive decay: remaining amount and activity, time to a level, half-life fits and chains. 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.

Part of usefulweb.app (https://usefulweb.app/). Part of [Proofline](https://proofline.usefulweb.app/) (ownerApp: proofline). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://prooflinehalflife.usefulweb.app/ (interactive tool at https://prooflinehalflife.usefulweb.app/tool)
- JSON API: POST https://prooflinehalflife.usefulweb.app/api/v1/calculate with a JSON object of inputs (schema: https://prooflinehalflife.usefulweb.app/api/v1/tool)
- OpenAPI: https://prooflinehalflife.usefulweb.app/openapi.json
- MCP (Streamable HTTP): https://prooflinehalflife.usefulweb.app/mcp - tool `proofline_half_life`

## Inputs
- mode (select): Calculation
- nuclide (select): Nuclide
- halfLife (text): Half-life (own nuclide) - A number with a time unit: s, min, h, d or y (365.2422 days).
- initial (text): Starting amount - A percentage, an activity (Bq, kBq, MBq, GBq, Ci, mCi, µCi), a mass (g, mg; preset nuclides only) or a number of atoms. Empty: 100 % (chains: 1 MBq).
- elapsed (text): Elapsed time
- target (text): Remaining level (time mode) - A percentage, or an activity or mass in the same kind of unit as the start.
- measurements (textarea): Measurements (fit mode) - One measurement per line: the time with its unit and the measured count rate or activity.
- chainPreset (select): Decay chain
- chain (textarea): Chain members (own chain) - One member per line: name, half-life (or “stable” for the last) and optionally the share of the previous member’s decays that feeds it.
- duration (text): Follow the chain for - Empty: the preset chain’s own period.

## Method
- 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.

## Limits
- 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.