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Half-Life Calculator

Calculate half-life, radioactive decay, remaining quantity, decay constant, and mean lifetime with step-by-step solutions, scientific visualizations, and exponential decay analysis.

Half-Life Solver

any units (g, atoms, %)
same units as N₀

Formula Explorer

Three equivalent ways to describe the same exponential decay. They're mathematically identical — pick whichever frame fits the problem.

Half-Life form

N(t) = N₀ · (½)^(t / t½)

Most intuitive: each t½ halves the amount remaining.

Mean Lifetime form

N(t) = N₀ · e^(−t / τ)

Used when discussing average atom survival time τ.

Decay Constant form

N(t) = N₀ · e^(−λt)

Standard physics form. λ is the per-atom decay rate.

Why they're equivalent

t½ = τ · ln 2 = ln 2 / λ

Setting (½)^(t / t½) = e^(−λt) gives λ · t½ = ln 2. From there, τ = 1/λ closes the triangle.

Isotope Half-Life Library

Published half-lives for isotopes used in archaeology, medicine, geology, and nuclear engineering.

IsotopeHalf-Life
C-14Carbon-145,730 years
U-238Uranium-2384.468 billion years
U-235Uranium-235703.8 million years
K-40Potassium-401.25 billion years
Th-232Thorium-23214.05 billion years
Tc-99mTechnetium-99m6.0 hours
I-131Iodine-1318.02 days
F-18Fluorine-18109.77 minutes
Co-60Cobalt-605.27 years
Cs-137Caesium-13730.17 years
Sr-90Strontium-9028.79 years
Pu-239Plutonium-23924,110 years
Ra-226Radium-2261,600 years
H-3Tritium12.32 years
P-32Phosphorus-3214.27 days

What Is Half-Life?

Half-life is the time required for half the atoms in a radioactive sample to decay — or, more generally, the time required for any exponentially decaying quantity to fall to half its starting value. The half-life is intrinsic to the substance: it doesn't depend on temperature, pressure, chemical bonding, or how much material you start with. One gram of Carbon-14 and one tonne of Carbon-14 both shed exactly half their atoms in 5,730 years.

The same maths describes radioactive isotopes, drug elimination from the bloodstream, capacitor discharge, light intensity through an absorbing medium, atmospheric tracers, and many population processes. Anywhere a quantity decreases at a rate proportional to its current size, the half-life concept applies.

How Half-Life Calculation Works

The Core Equation

N(t) = N₀ · (½)^(t / t½). After one half-life, half remains. After two half-lives, a quarter. After three, an eighth. The decay is multiplicative — never linear — which is why a 100-gram sample with a 10-year half-life still has 6.25 g after 40 years, not zero.

Three Equivalent Forms

Physicists write the same law three ways. Half-life form uses t½. Mean-lifetime form uses τ = t½ / ln 2. Decay-constant form uses λ = ln 2 / t½. Each is identical maths; the choice depends on which constant you happen to know.

Solving for Any Variable

Given any three of {N₀, N(t), t, t½}, the fourth is fixed. Rearranging with natural logarithms recovers time t = t½ · ln(N/N₀) / ln(½), or the half-life itself from observed decay over a measured interval.

Why Exponential?

Each atom has an independent, fixed probability of decaying per unit time. Multiplied across billions of atoms, the macroscopic result is a perfectly smooth exponential curve — a direct consequence of the central limit theorem applied to a Poisson process.

6 Ways to Use the Half-Life Calculator

1

Radiocarbon dating

Date organic samples up to ~50,000 years old from the ratio of remaining Carbon-14.

2

Medical dose planning

Plan radiopharmaceutical injection times so the tracer activity matches the imaging window.

3

Pharmacokinetics

Estimate when a drug falls below therapeutic concentration to schedule repeat dosing.

4

Geological age estimation

Use Uranium-238, Potassium-40, or Thorium-232 ratios to date rocks and meteorites.

5

Nuclear safety planning

Project how long contaminated zones remain hazardous after a release of Cs-137 or Sr-90.

6

Classroom physics

Verify exponential decay homework, visualise decay curves, and check formula rearrangements.

Best Practices When Working With Half-Lives

Match time units consistently

Keep half-life and elapsed time in the same units. Mixing years and days is the single most common source of error in homework and lab reports.

Use published half-lives

Half-lives are measured to high precision and tabulated by IAEA, NIST, and ENSDF. Don't guess — published values include uncertainty bounds you can propagate.

Report significant figures honestly

Most measured isotope half-lives are known to 3–4 significant figures. Don't carry six decimal places into a final answer unless the measurements actually support it.

Consider decay chains

U-238 doesn't decay directly to lead — it passes through 14 daughter isotopes. For accurate age-dating, model the full chain rather than the parent alone.

Account for contamination

Real Carbon-14 measurements correct for fossil-fuel CO₂, nuclear-bomb spike, and lab handling. Educational calculators give a clean theoretical age, not a publication-quality date.

Use the appropriate form

Half-life form is intuitive for whole half-lives. Decay-constant form is cleaner for calculus and physics derivations. Mean-lifetime form is natural for atomic physics and quantum mechanics.

Why Half-Life Matters

Archaeology & Anthropology

Carbon-14 dating revolutionised the chronology of human history. Egyptian dynasties, Mesoamerican cities, and Neolithic settlements all carry independent radiometric dates.

Medicine & Pharmacology

Drug half-life decides dosing intervals. Tc-99m (6.0 hours) and F-18 (110 minutes) are short enough to image patients without long-term radiation exposure.

Energy & Nuclear Engineering

Fuel design, reactor operation, and waste storage all hinge on the half-lives of the actinides and fission products involved.

Tricky Cases & Misconceptions

Half-life ≠ how long until it's gone

After ten half-lives only ~0.1% remains, but mathematically the curve never hits zero. Saying "Plutonium-239 lasts 240,000 years" means the next 0.1% is still around, not that it's harmless at that point.

Mean lifetime is longer than half-life

τ = t½ / ln 2 ≈ 1.44 × t½. After τ, only e⁻¹ ≈ 36.8% remains — not half. Confusing the two doubles or halves your answer.

Carbon-14 starts the clock at death

Living things continually exchange carbon with the atmosphere. The ¹⁴C ratio only begins falling when the organism dies and exchange stops.

Activity vs amount

Decay rate (becquerels) = λ · N. A small mass of a very short-lived isotope can have more activity than a much larger mass of a long-lived one.

Half-life isn't affected by chemistry

Chemical reactions affect electrons; nuclear decay involves the nucleus. Heating, cooling, or dissolving a sample does not change its half-life (with rare exceptions in electron-capture isotopes at extreme pressures).

Initial atoms is not always known

For most dating problems, 'original' N₀ is inferred from a reference sample (a living organism, a young rock). The age is only as good as that reference.

Core Half-Life Formulas

Half-life form

N(t) = N₀ · (½)^(t / t½)

Most intuitive. Each elapsed t½ halves what's left.

Decay-constant form

N(t) = N₀ · e^(−λt)

Continuous form used in physics derivations.

Mean-lifetime form

N(t) = N₀ · e^(−t / τ)

τ is the average lifetime of an individual atom.

Solve for elapsed time

t = t½ · ln(Nₜ / N₀) ÷ ln(½)

Foundation of radiometric dating.

Solve for the half-life

t½ = t · ln(½) ÷ ln(Nₜ / N₀)

Determined experimentally from a measured decay curve.

Constants identity

t½ = τ · ln 2 = ln 2 / λ

Single relation linking all three constants.

Activity

A(t) = λ · N(t) = A₀ · e^(−λt)

Decays per second, measured in becquerels.

Number of half-lives elapsed

n = t / t½ ⇒ fraction remaining = (½)^n

Quick mental check: n=1 → 50%, n=2 → 25%, n=3 → 12.5%.

Common Mistakes

Using log base 10 instead of ln

The decay equation uses the natural log, ln. Substituting log₁₀ gives an answer off by a factor of ln 10 ≈ 2.303.

Forgetting the negative sign

Decay constant λ is positive, but it appears with a minus sign in the exponent: N(t) = N₀ e^(−λt). A missing minus sign turns decay into growth.

Adding half-lives

Three half-lives leaves (½)³ = 12.5% — not 50% − 25% − 12.5% = 12.5% by subtraction. Half-lives multiply, they don't subtract.

Confusing t with t½

Elapsed time t and half-life t½ are different variables. The ratio t / t½ tells you how many half-lives have passed.

Forgetting unit conversion

If λ is in "per year" but t is in days, divide t by 365.25 first — or the answer is off by three orders of magnitude.

Treating the answer as exact

Real measurements have uncertainty in N₀, Nₜ, and even t½. Single-point answers are estimates, not certainties.

How Carbon-14 Dating Works

Cosmic rays bombard nitrogen atoms in the upper atmosphere, producing a steady trickle of radioactive Carbon-14. Plants absorb ¹⁴C through photosynthesis; animals absorb it by eating plants; living tissue equilibrates with the atmospheric ¹⁴C / ¹²C ratio. The moment an organism dies, the exchange stops — and the ¹⁴C in its remains begins decaying with a half-life of 5,730 years.

By measuring the surviving ¹⁴C fraction in a sample (bone, wood, charcoal, shell, parchment) and comparing to the living-tissue reference, archaeologists can back-calculate years since death. The technique was developed by Willard Libby (1949) and earned him the Nobel Prize in Chemistry in 1960.

Modern accelerator mass spectrometry pushes the upper limit to ~55,000 years on samples as small as a few milligrams. Calibration curves correct for past variations in atmospheric ¹⁴C — IntCal20 is the current international standard.

Real-World Half-Life Examples

Tritium glow signs

Tritium (¹H-3) keeps exit signs and watch dials glowing for ~12 years. After 24.6 years, only a quarter of the original ³H remains and brightness drops accordingly.

PET scan radiotracers

Fluorine-18 (110 min half-life) is produced in a cyclotron, injected within minutes, imaged within an hour, and decayed away by the next day — leaving no residual radioactivity.

Caesium-137 in soil

Released from nuclear weapons tests and Chernobyl, ¹³⁷Cs is still detectable in soils worldwide. With a 30-year half-life, contamination zones near reactor accidents will remain measurable for centuries.

Uranium-238 in granite

Trace U-238 in granite countertops decays through 14 daughter isotopes to stable lead, releasing radon gas — a measurable but usually low household exposure.

Drug half-life in pharmacology

Caffeine has a biological half-life of ~5 hours; ibuprofen ~2 hours; methadone ~24 hours. Dosing schedules are designed around these values to maintain therapeutic levels.

Potassium-40 in bananas

Natural K-40 (t½ = 1.25 billion years) means every banana is mildly radioactive. The dose is trivial (~0.1 µSv per banana), but it's a useful classroom demonstration.

Derivation of the Half-Life Constants

Setting the half-life form equal to the decay-constant form lets you derive the relationships between t½, τ, and λ in a few lines.

  1. Step 1 — Equate the two forms

    (½)^(t / t½) = e^(−λt)

  2. Step 2 — Take the natural log of both sides

    (t / t½) · ln(½) = −λt

  3. Step 3 — Cancel t, use ln(½) = −ln 2

    −ln 2 / t½ = −λ ⇒ λ · t½ = ln 2

  4. Step 4 — Use τ = 1 / λ to close the triangle

    t½ = τ · ln 2 = ln 2 / λ

Drug Half-Life in Pharmacokinetics

In pharmacology, the "biological half-life" describes how long it takes for the plasma concentration of a drug to fall to half its peak value. The body clears most drugs via first-order kinetics — meaning a constant fraction is eliminated per unit time, just like radioactive decay.

After five half-lives, the drug concentration falls to ~3% of its peak — usually below the therapeutic threshold. This is why short-half-life drugs (e.g., ibuprofen ~2 hr) need repeated dosing, while long-half-life drugs (e.g., fluoxetine ~1–4 days) reach steady state more slowly but persist longer between doses.

The decay-constant form is identical to radioactive decay: C(t) = C₀ · e^(−kₑ · t), where kₑ is the elimination rate constant — the pharmacokinetic equivalent of λ.

Radioactivity Activity vs. Quantity

The activity A of a sample is the number of decays per second, measured in becquerels (Bq). It equals λ · N — the decay constant times the current number of atoms. Because λ is fixed and N decreases exponentially, A also decays exponentially with the same half-life: A(t) = A₀ · e^(−λt).

A short-lived isotope (small t½, large λ) packs high activity per atom but burns out quickly. A long-lived isotope has low activity per atom but persists for ages. Both behaviours fall out of the same equation — choose your reporting units (Bq, Ci, atoms, percent) to match the context.

About This Half-Life Calculator

Scientifically Accurate

Uses the canonical exponential decay model N(t) = N₀ e^(−λt). Isotope half-lives sourced from IAEA Live Chart of Nuclides and NIST.

Free & Private

All math runs in your browser. No measurements, samples, or inputs are sent to a server.

Multi-Mode

One toolkit for half-life solving, constants conversion, radioactive decay, decay milestones, and Carbon-14 dating.

Related tools: Scientific Calculator, Percentage Calculator, Time Calculator, Molarity Calculator, Molecular Weight Calculator.

Frequently Asked Questions

Half-life is the time required for half the atoms in a radioactive sample — or half of any exponentially decaying quantity — to disappear or transform. It is intrinsic to the substance and does not depend on temperature, pressure, chemical bonding, or starting amount. One gram of Carbon-14 and one tonne of Carbon-14 both lose exactly half their atoms in 5,730 years. The same idea also describes drug elimination, capacitor discharge, and light intensity through absorbing media.

The half-life equation is N(t) = N₀ · (½)^(t / t½). Given any three of {N₀, Nₜ, t, t½}, the fourth follows by algebra and natural logarithms. To solve for elapsed time: t = t½ · ln(Nₜ / N₀) ÷ ln(½). To solve for the half-life from a measured decay: t½ = t · ln(½) ÷ ln(Nₜ / N₀). Use the Half-Life Solver tab at the top of this page to compute any of the four values automatically.

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into one or more daughter nuclei, releasing alpha, beta, or gamma radiation in the process. Each individual atom has a fixed probability of decaying per unit time (the decay constant λ). Multiplied across billions of atoms, this gives a perfectly smooth exponential decrease in the parent population — N(t) = N₀ · e^(−λt).

The decay constant λ (lambda) is the instantaneous probability per atom of decay, with units of inverse time. It connects to the half-life through λ = ln 2 / t½ ≈ 0.693 / t½. A large λ means rapid decay (short half-life); a small λ means slow decay (long half-life). The product λ · N gives the activity — the number of decays per second, measured in becquerels.

Mean lifetime τ (tau) is the average time an individual atom survives before decaying. It equals τ = 1 / λ = t½ / ln 2 ≈ 1.4427 × t½. After τ has elapsed, e⁻¹ ≈ 36.8% of the original quantity remains — not 50%. Half-life and mean lifetime are different but related: confusing the two is one of the most common errors in nuclear physics homework.

Living organisms continually exchange carbon with the atmosphere, maintaining the atmospheric ratio of radioactive Carbon-14 to stable Carbon-12. When the organism dies, the exchange stops and the ¹⁴C in its remains begins to decay with a half-life of 5,730 years. Measuring the surviving ¹⁴C / ¹²C ratio and comparing it to a living reference gives the years since death: t = 5,730 · ln(Nₜ / N₀) ÷ ln(½). Use the Carbon Dating tab above for guided calculations.

Carbon-14 is a radioactive isotope of carbon, formed when cosmic-ray neutrons collide with nitrogen-14 in the upper atmosphere. It has six protons and eight neutrons, decays by beta emission back to nitrogen-14, and has a half-life of 5,730 years. Despite being radioactive, Carbon-14 makes up only ~1 part in 10¹² of natural carbon — yet that trace level is precisely measurable, enabling radiocarbon dating up to roughly 50,000 years.

Each atom has an independent, fixed probability of decaying per unit time. The number of decays per second is therefore proportional to the number of remaining atoms — dN/dt = −λN. Solving that differential equation gives the exponential function N(t) = N₀ · e^(−λt). It is exactly the same maths that governs first-order chemical kinetics, drug elimination, and capacitor discharge.

After n half-lives, the fraction remaining is (½)ⁿ. One half-life leaves 50%, two leaves 25%, three leaves 12.5%, four leaves 6.25%, and so on. After ten half-lives only ~0.1% remains — usually the practical threshold for declaring a sample 'decayed away.' Mathematically, the curve never truly reaches zero, which is why the Exponential Decay Analyzer tab projects out to 0.1% and beyond.

The maths is exact — the only floating-point error comes from the double-precision IEEE 754 representation used by JavaScript, which keeps roughly 15 significant digits. Real-world accuracy is bounded by your inputs: published isotope half-lives carry their own measurement uncertainty (typically 0.1–1%), and Carbon-14 dating in particular requires calibration against tree-ring chronologies (IntCal20) for publication-quality dates. Treat this calculator as a teaching tool and first-cut estimator, not a substitute for laboratory radiometric dating.

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