Half-Life Calculator
To compute radioactive decay: N(t) = N₀ × (½)^(t/t½). Enter initial amount, half-life, and elapsed time; the tool returns remaining quantity, decayed fraction, decay constant λ, and mean lifetime τ. Includes 12 real isotopes from F-18 (109 min) to U-238 (4.5 Gyr).
Quick Conversion
Formula: N(t) = N₀ × (1/2)^(t/t½)
Exponential Decay Curve
Red marker = current N(t); amber ticks = 1·t½ … 7·t½
After 11460.0 years (2.00 half-lives), 25.00% of the original N₀ = 100.0 remains. The decay constant λ = 1.2097e-4 years⁻¹ means each nucleus has a 0.0121% probability of decay per years on average.
Real isotope presets
Fraction remaining vs half-lives elapsed
| Half-lives | Fraction remaining | Percent | For 100 mg C-14 |
|---|---|---|---|
| 0 | 1.0000e+0 | 100.0000% | 100.0 mg |
| 0.5 | 7.0711e-1 | 70.7107% | 70.7107 mg |
| 1 | 5.0000e-1 | 50.0000% | 50.0000 mg |
| 1.5 | 3.5355e-1 | 35.3553% | 35.3553 mg |
| 2 | 2.5000e-1 | 25.0000% | 25.0000 mg |
| 3 | 1.2500e-1 | 12.5000% | 12.5000 mg |
| 4 | 6.2500e-2 | 6.2500% | 6.2500 mg |
| 5 | 3.1250e-2 | 3.1250% | 3.1250 mg |
| 6 | 1.5625e-2 | 1.5625% | 1.5625 mg |
| 7 | 7.8125e-3 | 0.7813% | 0.7813 mg |
| 8 | 3.9063e-3 | 0.3906% | 0.3906 mg |
| 10 | 9.7656e-4 | 0.0977% | 0.0977 mg |
| 15 | 3.0518e-5 | 0.0031% | 0.0031 mg |
| 20 | 9.5367e-7 | 0.0001% | 9.5367e-5 mg |
10 half-lives ≈ 0.1% remaining; commonly used as the "effectively zero" threshold for waste-storage planning.
Formulas
N(t) = N₀ × (1/2)^(t / t½) = N₀ × e^(−λt)λ = ln(2) / t½ τ = 1/λ = t½ / ln(2)Worked example (radiocarbon dating): A bone fragment has 25% of modern ¹⁴C. Time elapsed = t½ × log₂(1/0.25) = 5730 × 2 = 11,460 years. The bone is from the early Holocene.
How to read the decay curve in 5 steps
- Pick or enter the half-life. Use one of the 12 isotope presets (C-14, Tc-99m, U-238...) or type a custom value with a chosen time unit.
- Enter N₀ (initial amount). Any positive quantity - milligrams of sample, Bq of activity, or arbitrary "units". The calculator scales linearly.
- Set elapsed time t. The curve redraws with a red vertical marker dropping onto the exponential.
- Read N(t), λ, and τ. Output panels show remaining quantity, decay constant, and mean lifetime in your chosen unit system.
- Compare to half-life ticks. Amber dashed lines mark 1·t½ through 7·t½ - after 7 half-lives less than 1% remains, a useful rule of thumb.
A century of decay constants: from Becquerel to BIPM
Why this calculator exists: In 2026, a nuclear-medicine technologist preparing a morning F-18 FDG schedule must plan injection times around the 109.77-minute physical half-life and the biological clearance of the patient. A clean curve + λ readout removes the spreadsheet errors that plague Monday-morning dose calibration.
Radioactivity was discovered by Henri Becquerel in March 1896 - uranium salts fogged a photographic plate in a closed drawer. Marie and Pierre Curie isolated polonium (1898) and radium (1898) and coined "radioactivity". The exponential decay law was articulated by Ernest Rutherford and Frederick Soddy in 1903: dN/dt = -λN, integrating to N(t) = N₀ e^(-λt). Rutherford's 1911 gold-foil experiment then identified the nucleus as the seat of α emission, finishing the model.
Dmitri Mendeleev's 1869 periodic table was the framework that radioactivity disrupted then strengthened - the discovery of isotopes by Soddy (1913) meant that "atomic weight" was an isotope-weighted average, and that elements could have multiple half-lives. Henry Moseley's 1913 X-ray work fixed atomic number Z as the true periodic-table organiser, and the Z-N chart of all known nuclides (now ~3,300 entries at the IAEA Nuclear Data Section) is the master catalogue of half-lives.
The half-life concept underwrites four enormous practical industries. Radiocarbon dating (Willard Libby 1949, Nobel 1960) revolutionised archaeology - the Libby half-life of 5,568 yr was later refined to the Cambridge half-life of 5,730 ± 40 yr in 1962; modern AMS labs achieve ±20 yr per sample. U-Pb geochronology uses U-238 (4.468 Gyr) and U-235 (704 Myr) to date Earth's oldest zircons at 4.404 Gyr - very nearly the age of the planet itself. Nuclear medicine centres on Tc-99m (6 h) and F-18 (110 min) - short half-lives let imaging be done without long radiation burden. Nuclear waste planning uses Pu-239 (24,110 yr) and Cs-137 (30.05 yr) to define storage timeframes.
IUPAC and the BIPM standardised the units. The becquerel (Bq, 1 decay/s) replaced the curie (Ci, 3.7 × 10¹⁰ Bq, defined from one gram of radium) in 1975 SI work. The 2019 SI redefinition fixed the second by the cesium-133 transition and the kelvin by Boltzmann's constant; half-life values are quoted in SI seconds, ensuring that decay-constant tables remain consistent across decades. The IAEA Nuclear Data Section publishes the canonical t½ for every isotope ever measured - this calculator's 12 presets are pulled from that database.
Twenty-first-century improvements continue. The 2020 IntCal20 calibration curve corrects atmospheric ¹⁴C variations across the past 55,000 years using tree rings, speleothems, and varved lake sediments. The 2019 LIGO/Virgo gravitational-wave detection of a binary neutron-star merger (GW170817) was matched to a kilonova whose light-curve fitted r-process nuclide decays - an entirely new arena for half-life physics. Lab-scale precision keeps improving: NIST's 2023 Po-209 half-life is now known to ±0.04 yr out of 102 yr.
This calculator implements the bare formula in five-line clarity, with twelve named isotopes covering the medical-geological span. For atom-count conversions see atoms to moles; for stable-isotope chemistry see molar mass.
Trusted by nuclear medicine, geochronology labs, and educators
“I use the F-18 and Tc-99m presets every morning for dose calibration. The exponential-decay curve next to a live N(t) readout is exactly what I need to plan injection times - saves five clicks compared with our LIS.”
“The U-238 and K-40 presets, plus the ability to handle Gyr-scale times without overflow, make this the cleanest pedagogical tool I have linked from a course syllabus this year. Decay-constant readout matches my lecture notes.”
“When I onboard new techs, the 12 named isotopes covering nuclear medicine, fallout, and geochronology gives them the entire isotope landscape in one screen. The Po-210 footnote about Litvinenko sticks with everyone.”
“Students drag the time slider and watch N(t) march down past 7 half-lives. The visual click - 'one half, one half again, one half again' - is what they need before the algebra. Best half-life teaching tool I have used.”
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Last reviewed: 2026-05
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