Half-Life Calculator

Article — Half-Life Calculator

How the half-life calculator works

Half-life is the time required for half of a radioactive sample (or any first-order decaying quantity) to disappear. The formula N(t) = N₀ · (1/2)^t/t_½ describes the exponential decay. Carbon-14 has a half-life of 5,730 years (used for archaeological dating), technetium-99m has 6.01 hours (the most common medical radioisotope), and uranium-238 has 4.47 billion years (used to date the Earth itself).

The same first-order kinetics apply to drug elimination in the body: ibuprofen has a half-life of about 2 hours, caffeine 5 hours, and fluoxetine 1–3 days. After 5 half-lives less than 3% remains, which is why doctors space doses by half-life multiples.

What is half-life

Half-life is the most intuitive way to describe exponential decay. Start with 100 atoms; after one half-life, 50 remain; after two, 25; after three, 12.5; and so on. The pattern is geometric, halving each interval. Half-life is the property of an individual nucleus or molecule; the macroscopic decay curve smooths out the statistical noise across many trillions of particles.

Crucially, half-life does not depend on temperature, pressure, chemical environment, or how many other atoms are present (with one tiny exception for electron-capture isotopes, where chemical state shifts the rate by parts per million). The decay is a quantum-mechanical property of the nucleus.

The half-life formula

The exponential decay form is:

N(t) = N0 · e−λt

The half-life form is mathematically identical but more intuitive:

N(t) = N0 · (1/2)t/t½

Where N₀ is the initial amount, t is elapsed time, and t_½ is the half-life. After n = t/t_½ half-lives, the fraction remaining is (1/2)ⁿ. Twelve half-lives leave less than 0.025% — one part in 4096.

Half-life and the decay constant

The decay constant λ is the per-unit-time probability that any single nucleus or molecule decays. It links to half-life through:

λ = ln 2 / t½ ≈ 0.693 / t½

The mean lifetime τ = 1/λ is the average time a single particle survives. It is 1.443 times longer than the half-life, because of the way the exponential weights tail values. For carbon-14 with t_½ = 5730 years, τ works out to 8266 years.

Activity A = λN is the rate of decay in becquerels (decays per second). One gram of pure radium-226 has an activity of about 3.7 × 10¹⁰ Bq — the historical definition of one curie.

Radioactive half-lives of common isotopes

Half-lives span an extraordinary range, from microseconds (most short-lived nuclear decay products) to billions of years (primordial uranium and thorium):

• Technetium-99m — 6.007 hours, the workhorse of nuclear medicine imaging
• Iodine-131 — 8.025 days, used for thyroid imaging and ablation
• Radon-222 — 3.823 days, the indoor air contaminant from uranium decay
• Cobalt-60 — 5.271 years, used in industrial sterilization and historic teletherapy
• Caesium-137 — 30.17 years, the dominant Chernobyl and Fukushima fallout product
• Carbon-14 — 5,730 years, radiocarbon dating standard
• Plutonium-239 — 24,110 years, fuel and weapons material
• Uranium-238 — 4.468 × 10⁹ years, the most abundant uranium isotope

Drug half-life in pharmacology

Most drugs follow first-order elimination kinetics in the body, so half-life is the same statistic that describes radioactive decay. The mechanisms differ — for drugs it is liver enzymes and kidney excretion — but the math is identical.

Dosing intervals are usually one half-life or shorter. After 5 half-lives the residual is below 3% — the practical "washout" window. For caffeine at a 5-hour half-life, that is roughly a full day from last cup to cleared system. For Prozac at 1–3 days plus a much longer-lived active metabolite, complete washout takes weeks.

Carbon-14 dating with half-life

Living organisms maintain the atmospheric C-14/C-12 ratio because they continuously exchange carbon with the environment. At death this exchange stops and C-14 decays exponentially with a 5,730-year half-life.

Measure the surviving C-14 fraction in a sample, take the negative log, multiply by t_½ / ln 2, and you get the age. A sample with 50% of the modern C-14 is one half-life old (5,730 years). A sample with 25% is two half-lives (11,460 years). The practical limit is about 10 half-lives or 50,000–60,000 years, beyond which the remaining C-14 signal is too weak.

Half-life in uranium-lead dating

For ages beyond the carbon-14 limit, geologists turn to longer-lived isotopes. U-238 decays to Pb-206 with a half-life of 4.468 × 10⁹ years, and U-235 decays to Pb-207 with 7.04 × 10⁸ years. By measuring the Pb/U ratios in zircon crystals, geologists can date rocks back to the formation of Earth itself (4.54 billion years).

Zircon (ZrSiO₄) is the preferred mineral because it incorporates uranium during crystallization but rejects lead almost completely. Any Pb found inside a zircon grain therefore came from in situ U decay, making the U-Pb ratio a clean clock. The oldest dated zircon, from Jack Hills in Western Australia, returned an age of 4.4 billion years — only about 150 million years younger than the formation of the planet.

Common half-life mistakes

• Confusing half-life with full lifetime — after one half-life the sample is not gone; it is halved
• Linear thinking — two half-lives leave 25%, not 0%; four half-lives still leave 6.25%
• Wrong units — half-life and elapsed time must be in the same unit before the exponent makes sense
• Treating half-life as a deadline — some atoms decay much earlier and others much later; half-life is a statistical median, not a fixed countdown
• Mixing physical and biological half-life — for radiopharmaceuticals the effective clearance combines both via 1/T_eff = 1/T_phys + 1/T_biol

Half-life shows up beyond physics and pharmacology too. The interest of a continuously-compounded loan has a halving time given by 0.693/r where r is the rate. Population studies use doubling time (the negative-rate twin of half-life) to describe exponential growth. Bacterial generation times, viral load curves, and even the disappearance rate of a meme on social media all follow the same mathematics that governs radioactive decay.