Half-Life & Radioactive Decay Calculator (incl. Carbon Dating)

About the Half-Life & Radioactive Decay Calculator (incl. Carbon Dating)

A radioactive isotope loses half of what is left in every half-life: N = N₀ × (½)^(t / T½). This calculator solves that law for any one unknown — the amount left, the starting amount, the elapsed time or the half-life — for amounts in per cent, grams, atoms, moles, becquerels or curies. It gives the decay constant λ = ln 2 / T½, the mean life, the number of half-lives, a decay curve and a table of the first ten half-lives, and with a molar mass the activity in Bq and Ci and the specific activity. Presets fill in the current half-lives of 27 well-known isotopes from the NNDC NuDat 3 database — carbon-14, tritium, iodine-131, caesium-137, cobalt-60, uranium-238 and more.

A carbon dating mode turns a measured fraction of modern carbon (F¹⁴C, percent modern or two activities) into the conventional radiocarbon age laboratories report — with the Libby half-life of 5,568 years, as the convention requires — and shows the ages the 5,730 and 5,700 year half-lives would give. A rock age mode works out uranium–lead, thorium–lead, rubidium–strontium and potassium–argon ages from a daughter-to-parent ratio or from measured masses, using the IUGS decay constants.

Common uses

• Chemistry and physics homework on half-lives, decay constants and activity.
• Planning how long a laboratory source or waste must be stored before its activity falls to a given level.
• Understanding the numbers behind radiocarbon dates in archaeology reports (ages BP, F¹⁴C, pMC).
• Geology coursework on U–Pb, Rb–Sr and K–Ar ages.
• Converting activities between becquerels and curies.

The formulas

• Decay law: N = N₀ (½)^(t/T½) = N₀ e^(−λt), so t = T½ log₂(N₀/N) and T½ = t ÷ log₂(N₀/N).
• Decay constant and mean life: λ = ln 2 / T½ = 0.693 / T½, τ = 1/λ = 1.4427 T½.
• Activity: A = λN with N the number of nuclei; it halves every half-life too. 1 Bq = 1 decay per second; 1 Ci = 3.7 × 10¹⁰ Bq exactly.
• Specific activity: λ N_A / M decays per second per gram, with M the molar mass of the isotope.
• Years are taken as 365.2422 days, the convention of the ENSDF/NuDat nuclear data.

Carbon-14 conventions

Radiocarbon laboratories report the conventional radiocarbon age, t = −8033 × ln(F¹⁴C) years BP, defined by Stuiver and Polach: it uses the original Libby half-life of 5,568 years (mean life 8,033 years) on purpose, so that ages from different decades stay comparable, and counts “before present” from AD 1950. F¹⁴C, the fraction of modern carbon, is the sample’s ¹⁴C/¹²C activity ratio relative to the 1950 standard after correcting for isotopic fractionation (Reimer et al.); percent modern carbon is 100 × F¹⁴C.

The physical half-life is longer — 5,730 ± 40 years in many textbooks and 5,700 ± 30 years in NuDat 3 today — so the calculator also shows the ages those values give. None of these is a calendar date: the amount of ¹⁴C in the air has varied, so reported ages are converted to calendar years with a calibration curve (IntCal20). Radiocarbon works up to about 50,000 years; beyond that too little ¹⁴C is left (OpenStax Chemistry 2e).

Rock ages

If a mineral started with no daughter atoms and has been a closed system since, the daughter-to-parent ratio fixes its age: t = (1/λ) ln(1 + D/P). For potassium–argon only the electron-capture branch makes argon-40, so t = (1/λ) ln(1 + (λ/λ_e)(⁴⁰Ar/⁴⁰K)). The decay constants are those recommended by the IUGS Subcommission on Geochronology (Steiger and Jäger), still the convention for many published ages: λ(²³⁸U) = 1.55125 × 10⁻¹⁰/y, λ(²³⁵U) = 9.8485 × 10⁻¹⁰/y, λ(²³²Th) = 4.9475 × 10⁻¹¹/y, λ(⁸⁷Rb) = 1.42 × 10⁻¹¹/y and, for ⁴⁰K, λ_e = 0.581 × 10⁻¹⁰/y and λ_β = 4.962 × 10⁻¹⁰/y. For any other system, choose Another isotope and type its half-life.

Sources

• Brookhaven National Laboratory, NNDC, NuDat 3: half-lives and decay modes of the presets (each preset names its evaluation in Nuclear Data Sheets).
• IAEA Nuclear Data Section, LiveChart of Nuclides: atomic masses (AME2020).
• M. Stuiver and H. A. Polach, “Discussion: Reporting of ¹⁴C Data”, Radiocarbon 19, 355–363, doi:10.1017/S0033822200003672.
• P. J. Reimer, T. A. Brown and R. W. Reimer, “Discussion: Reporting and Calibration of Post-Bomb ¹⁴C Data”, Radiocarbon 46, 1299–1304.
• R. H. Steiger and E. Jäger, “Subcommission on geochronology: Convention on the use of decay constants in geo- and cosmochronology”, Earth and Planetary Science Letters 36, 359–362.
• OpenStax, Chemistry 2e (CC BY 4.0), §21.3 Radioactive Decay (Examples 21.5–21.7).
• NIST, SP 811 Appendix B.8: 1 Ci = 3.7 × 10¹⁰ Bq (exact).

Limitations

• Not for medication or radiopharmaceutical dosing: the body also removes substances (biological half-life), and doses need clinical protocols and a qualified professional.
• Decay is random: the formula gives the average, and with only a few atoms left the real number scatters widely.
• Decay chains (a daughter that is itself radioactive, as in the uranium series) are not modelled; each calculation is for one isotope.
• Radiocarbon ages here are uncalibrated and assume the F¹⁴C you enter is already corrected for isotopic fractionation (δ¹³C = −25 ‰), as laboratories report it.
• Rock ages assume a closed system with no initial daughter; real geochronology tests that with isochrons or concordant ages.

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