Exponential decay, solved — remaining amount after any number of half-lives.
Some things in nature disappear on a strict schedule. A gram of carbon-14 halves every 5,730 years, with a regularity so dependable that archaeologists date ancient charcoal by it. The iodine-131 used to treat thyroid disease halves every 8 days, which is why a patient is radioactive for weeks, not forever. Caffeine in your bloodstream halves roughly every 5 hours, which is why the 4 p.m. espresso still has you staring at the ceiling at midnight. All of these follow the same mathematics of exponential decay, and the half-life — the time for any quantity to fall to exactly half — is the single number that describes the whole process. The calculator above takes an initial amount, a half-life, and an elapsed time, and returns what remains, with both steps of the arithmetic shown.
The defining feature of half-life decay is that the fraction lost per unit time is constant, while the absolute amount lost keeps shrinking. After one half-life, half remains; after two, a quarter; after three, an eighth. Each half-life multiplies the remainder by one-half, so after n half-lives the surviving fraction is (1/2)n, where n is simply the elapsed time divided by the half-life. This is why the formula needs no calculus to use: remaining = initial × (0.5)t / half-life. The widget computes the number of elapsed half-lives first, then applies the power — and it works for fractional half-lives too, such as 2.5 half-lives leaving (0.5)2.5 ≈ 17.7% behind.
Half-life thinking extends far beyond radioactivity, which is why this calculator earns its place in a chemistry collection. Pharmacologists use biological half-lives to set dosing intervals: a drug with a 6-hour half-life given every 6 hours accumulates to a steady plateau, and missing two doses drops the blood level to a quarter. Environmental chemists track pesticide half-lives in soil to decide when a field is safe to replant. Food scientists use microbial decimal-reduction times — the half-life’s cousin — to design pasteurization. Wherever something fades by a constant fraction per unit time, the same equation, and the same widget, applies.
Radioactive decay is a game of chance played by every unstable nucleus independently. In any given second, each nucleus of a particular isotope has the same tiny probability of decaying — a probability set by quantum mechanics and unaffected by temperature, pressure, chemistry, or the nucleus’s age. With trillions of nuclei present, the law of large numbers turns those individual coin flips into a beautifully smooth exponential: the number decaying per second is always proportional to the number still present. That proportionality — a constant fraction per unit time — is the entire content of the half-life concept, and it falls out of the statistics without any nucleus “knowing” how old it is.
This memoryless property is what makes half-lives so different from human aging. A 90-year-old person is far more likely to die this year than a 20-year-old, but a billion-year-old uranium-238 nucleus is exactly as likely to decay in the next minute as a nucleus created a second ago. The decay constant lambda quantifies the per-second probability, and the half-life is simply ln(2) ÷ lambda — the time for the survival probability to reach one-half. Different isotopes have wildly different lambdas: polonium-212’s half-life is 0.3 microseconds, uranium-238’s is 4.5 billion years, a span of 23 orders of magnitude described by one equation.
The same mathematics governs every first-order process, which is why pharmacokinetics borrows the vocabulary wholesale. When the liver clears a drug at a rate proportional to its blood concentration, the drug’s level follows the identical exponential, and doctors speak of its “elimination half-life” exactly as physicists speak of a radionuclide’s. Newton’s law of cooling, capacitor discharge, and even the depreciation of some assets follow the same curve. Learn the half-life equation once, and you have learned the master template for a sizable fraction of applied science.
Key terms:
Use this form whenever the decay is first-order: one half-life value, constant in time, and a single decaying species. It accepts fractional half-lives without modification, since the exponent is just a real number. A handy mental rule: 10 half-lives reduce any amount by a factor of about 1000, so “ten half-lives” is the practical meaning of “essentially gone” in radiation safety, pharmacology, and environmental cleanup alike.
A thyroid patient receives a dose containing 100 units of iodine-131 (half-life 8 days). How much remains after 24 days? Here initial = 100, half-life = 8 days, elapsed = 24 days.
This is the widget’s default: 100 initial, half-life 5, elapsed 15 gives the same clean 3-half-life structure and the answer 12.5.
An archaeologist measures a charcoal fragment’s carbon-14 activity at 0.25 times the activity of living wood. Carbon-14’s half-life is 5,730 years. How old is the sample? This runs the formula backward: solve for t given the remaining fraction.
Radiocarbon dating works because living things maintain the atmospheric 14C ratio while alive, and the clock starts at death — the same exponential, read in reverse, becomes a time machine with a 50,000-year range.
First scenario: you drink a 200 mg caffeine coffee at 4 p.m., and caffeine’s half-life in your body is about 5 hours. By 9 p.m. (one half-life), 100 mg remains; by 2 a.m. (two half-lives), 50 mg; by 7 a.m. (three), 25 mg. That 50 mg circulating at 2 a.m. is roughly half a cup of coffee’s worth — the quantitative reason sleep researchers advise a caffeine cutoff 8 to 10 hours before bed. The widget does this instantly: initial 200, half-life 5, elapsed 10 gives 50 remaining.
Second scenario: a hospital stocks a radiotracer with a 6-hour half-life for morning PET scans. The 8 a.m. delivery contains 400 MBq, but the first patient is scanned at 11 a.m. (0.5 half-lives) and the last at 4 p.m. (about 1.33 half-lives). Remaining activity at 4 p.m. is 400 × (0.5)1.333 ≈ 400 × 0.397 = 159 MBq. Nuclear medicine departments schedule doses backward from scan time using exactly this calculation, ordering enough initial activity that the decayed remainder still meets the imaging protocol — too little and the scan is unreadable, too much and the patient’s dose is needlessly high.
Third: an environmental consultant evaluates a field sprayed with a pesticide whose soil half-life is 30 days. The application was 2 kg per hectare, and regulations require residues below 0.125 kg/ha before replanting. Solving (0.5)t/30 = 0.0625 gives t/30 = 4, so t = 120 days — four half-lives, a sixteen-fold reduction. The consultant schedules soil testing at day 120 rather than guessing, and the farmer gets a defensible, documented replant date instead of a hopeful one.
Fourth: a museum curator checks whether a luminous 1960s watch dial (radium-226 paint, half-life 1,600 years) is still a meaningful hazard. Sixty years is 0.0375 half-lives, so the remaining fraction is (0.5)0.0375 ≈ 0.974 — still 97% of the original radium, essentially undecayed. The curator’s takeaway is the inverse of the usual intuition: for long-half-life isotopes, human timescales barely move the needle, so old radium artifacts are treated as permanently radioactive and stored accordingly.
Nuclear medicine: technetium-99m, the workhorse of diagnostic imaging with a 6-hour half-life, is milked daily from molybdenum-99 generators in hospitals worldwide. Its short half-life is a feature: it delivers its gamma rays for the scan and mostly decays away within a day, minimizing patient dose — a triumph of matching the isotope’s clock to the clinical need.
Archaeology: radiocarbon dating (carbon-14, 5,730 years) dates organic artifacts to about 50,000 years; potassium-argon dating (1.25 billion years) dates volcanic rock and early hominin fossils; uranium-lead dating (4.5 billion years) dates the oldest Earth rocks and meteorites. Each method is the same half-life equation pointed at a different clock, chosen so the sample’s age spans a few half-lives — the sweet spot of measurement precision.
Pharmacology: dosing intervals are set from elimination half-lives. A drug with a 24-hour half-life taken daily accumulates to a stable plateau in about 5 half-lives; a drug with a 2-hour half-life needs multiple daily doses or a slow-release formulation. Anesthesiologists think in “context-sensitive half-times” for infusions, deciding when a patient will wake from the same exponential logic.
Environmental cleanup: regulators use half-lives to set remediation timelines. Tritium (12.3 years) in groundwater, cesium-137 (30 years) in Chernobyl’s exclusion zone, and pesticide residues in farmland are all managed by projecting the exponential forward: ten half-lives to “gone,” with monitoring to confirm the model against reality.
Food safety: thermal death-time kinetics for pasteurization use the D-value — the time for a 90% (one-log) microbial kill — which is the half-life’s base-10 sibling. A 12-D process (“botulinum cook”) reduces Clostridium botulinum spores by twelve orders of magnitude in canned food, the same exponential math with a different logarithm base.
Finance and technology: the same curve prices asset depreciation, models the decay of news relevance in feeds, and describes how quickly a viral post’s engagement halves. Whenever analysts speak of something’s “half-life” — from memes to machine-learning model freshness — they are borrowing this equation, because constant-fraction decay is everywhere once you look.
One pattern worth internalizing: think in half-lives, not in time. Convert any elapsed time into “how many half-lives” and the answer becomes intuitive: 1 half-life is 1/2, 2 is 1/4, 3 is 1/8, 7 is about 1/128, 10 is about 1/1000. Memorizing the powers of two up to 210 = 1024 gives you a mental half-life calculator accurate enough for triage — the widget then supplies the precision for the report. Experts in every field that uses decay do this reflexively, and it is the fastest way to develop quantitative intuition for exponentials.
The history began with a fogged photographic plate: in 1896 Henri Becquerel found uranium salts darkening plates in a drawer, and Marie and Pierre Curie isolated polonium and radium from tons of pitchblende. Ernest Rutherford, studying thorium emanation around 1900, noticed the activity fell by half in fixed intervals regardless of conditions — the first measured half-life (about a minute for thoron). His student Frederick Soddy coined the term “half-life,” and the Rutherford-Soddy disintegration theory (1902) gave decay its exponential law. Willard Libby weaponized it for archaeology in 1949 with radiocarbon dating, winning the 1960 Nobel Prize — a straight line from a fogged plate to rewriting human prehistory.
A worked scenario with numbers: after a reactor incident, milk is found to contain 800 Bq/L of iodine-131 (half-life 8 days) against a safety limit of 100 Bq/L. The required reduction factor is 8, and (0.5)n = 1/8 gives n = 3 half-lives, so t = 24 days of storage before the milk tests clean — assuming no new contamination. Authorities actually did versions of this calculation after Chernobyl and Fukushima, holding or diverting milk supplies for a few half-lives while the short-lived iodine decayed away, since iodine concentrates in the thyroid and children are the most vulnerable.
Edge cases and pro tips: when a decay chain is in secular equilibrium (a long-lived parent feeding a short-lived daughter, like Mo-99/Tc-99m generators), the daughter’s activity tracks the parent’s, and the effective half-life you observe is the parent’s — the basis of generator design. For biological half-lives, remember the observed value blends physical decay with bodily excretion; the effective half-life is shorter than either alone: 1/teff = 1/tphysical + 1/tbiological. And when counting statistics are low (a handful of atoms, as in superheavy-element research), the smooth exponential breaks into individual random decays — the half-life then describes the probability distribution, not a deterministic schedule.
When the simple formula is not enough: mixtures of isotopes need a sum of exponentials, one term per half-life — a single “effective half-life” misleads. Growing-into-equilibrium problems (daughter accumulating while parent decays) need the Bateman equations, the chain-rule extension of the same law. And carbon dating beyond ~50,000 years, or any dating where the initial ratio is uncertain, needs calibration curves (dendrochronology, atmospheric models) layered on top of the raw exponential. The widget handles the pure single-isotope case perfectly; recognize when your problem has outgrown it.
Half-life decay follows: half-lives elapsed n = t / t1/2; remaining amount = initial × (0.5)n. Each half-life halves the remainder; after 10 half-lives less than 0.1% remains. Example: 100 units with a 5-unit half-life after 15 time units gives n = 3 and 100 × 0.125 = 12.5 remaining. Applies to radioactive decay, drug elimination, and any first-order process with constant fractional loss per unit time. Source: Sigma Calculator, Half-Life Calculator, https://sigmacalculator.net/chemistry-calculators/half-life-calculator-2/
Cite this page as: Half-Life Calculator, Sigma Calculator, https://sigmacalculator.net/chemistry-calculators/half-life-calculator-2/
Last updated: October 12, 2026