Half Life

Every radioactive substance decays at its own unchanging pace, and that pace is summed up in one number: the half-life, the time needed for half of the atoms in a sample to transform. The Half-Life Calculator works through the arithmetic of that decay, telling you how much of a substance remains after a given time, or how long a sample must sit before it falls to a chosen amount. It serves physics students learning exponential decay, researchers planning work with tracers and isotopes, and anyone checking a radiocarbon date or a storage period for radioactive material.

Quick Answer

Remaining amount: N = N₀ × (1/2)^(t ÷ t½) — start with the initial amount N₀, divide the elapsed time t by the half-life t½ to get the number of half-lives passed, then halve the sample that many times.

The Pattern Behind Exponential Decay

Radioactive decay is a random process for each individual atom, yet across the billions of atoms in a real sample it follows an exact statistical law: in every half-life period, half of whatever remains will decay. A sample never quite reaches zero — it halves, and halves again, shrinking toward nothing in smaller and smaller steps. The formula above captures this by raising one half to the power of the number of elapsed half-lives, and because the exponent does not need to be a whole number, it works just as well for 2.4 half-lives as for exactly 3.

Half-lives span an enormous range, and the isotope's value decides what it is used for. Carbon-14, with a half-life of about 5,730 years, ticks slowly enough to date once-living material thousands of years old. Iodine-131, at 8.02 days, decays fast enough to be useful in short-lived medical tracers and thyroid studies. Cobalt-60 (5.27 years) powers industrial irradiation and radiotherapy sources, while uranium-238, at 4.47 billion years, changes so slowly that its decay chain helps geologists date the Earth itself. A related quantity, the decay constant λ = ln 2 ÷ t½ ≈ 0.693 ÷ t½, expresses the same behaviour as a per-unit-time probability and appears throughout nuclear physics equations.

How to Use the Half-Life Calculator

1. Enter the starting amount of the substance — grams, milligrams, or a percentage where 100 means the full original sample.
2. Enter the half-life of the isotope, keeping its time unit in mind (days, years, minutes).
3. Enter the elapsed time in the same unit as the half-life.
4. Read the remaining amount, and the percentage of the original sample it represents.
5. To solve the reverse problem, enter the start and end amounts and let the calculator work out the time or number of half-lives instead.
6. Sanity-check the result: after one half-life you should see exactly half, after two, a quarter.

Worked Example: Cobalt-60 Over Three Half-Lives

Take a 64 g sample of cobalt-60, whose half-life is 5.27 years, and let 15.81 years pass. The number of half-lives is 15.81 ÷ 5.27 = exactly 3. Now halve the sample three times: 64 g becomes 32 g after the first half-life, 16 g after the second, and 8 g after the third — one eighth of the original, matching the formula: 64 × (1/2)³ = 64 × 0.125 = 8 g.

The same arithmetic powers radiocarbon dating. Living wood keeps a steady ratio of carbon-14; once the tree dies, the carbon-14 decays away with its 5,730-year half-life. If a wooden artefact shows 25 percent of the carbon-14 level of living wood, that is two halvings (100 → 50 → 25), so the wood is about 2 × 5,730 = 11,460 years old. A few reference half-lives worth knowing:

• Carbon-14: 5,730 years
• Cobalt-60: 5.27 years
• Tritium (hydrogen-3): 12.3 years
• Iodine-131: 8.02 days
• Uranium-238: 4.47 billion years

Assumptions and Easy Mistakes

Units are the first trap: if the half-life is in years and the elapsed time is in days, the exponent is meaningless until both match. Second, half-life does not depend on how much material you start with — a kilogram and a milligram of the same isotope halve on the same schedule, a point question-setters love to test. Third, ordinary chemistry cannot speed it up or slow it down: heating, pressurising or bonding the atoms into compounds leaves nuclear decay essentially untouched, which is precisely why it makes a reliable clock. Fourth, mind the precision of the inputs — published half-lives carry their own measurement uncertainty, and carbon dates in particular are calibrated against tree-ring records before being quoted as calendar ages. Finally, after about ten half-lives less than 0.1 percent remains (1/1024 of the original), which is why ten half-lives is a common rule of thumb for storing short-lived waste until it is effectively inactive.

The Same Mathematics in Medicine and Everyday Life

Half-life thinking escapes the physics lab surprisingly often, because any process that removes a fixed fraction per unit of time behaves the same way. Pharmacology leans on it directly: most drugs are eliminated from the bloodstream at a rate proportional to their concentration, so each drug has an elimination half-life — roughly five hours for caffeine in a typical adult. After one half-life half the dose remains active; after five half-lives only (1/2)⁵ = 1/32, about 3 percent, is left, which is why dosing intervals and "how long until it clears my system" questions are both answered in half-lives. Environmental science uses the identical curve for the breakdown of some pollutants and for the cooling of a hot object toward room temperature in simplified models. Recognising the pattern — constant halving time, steep early change, long shallow tail — lets you estimate any of these on the back of an envelope: count the half-lives, halve that many times, and you have the answer to within the accuracy of the inputs.

Keep Exploring Radioactivity and Decay

Half-life questions rarely stand alone for long. Gas and amount calculations from the same course sit nearby in the science calculators section, where you will also find the STP Calculator for converting gas measurements to standard conditions — a handy companion when decay products are collected and measured as gases. For the underlying mathematics of the decay law itself, the half-life reference article collects the standard derivations and isotope tables.