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Half-Life Calculator

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Leave one of the four quantities blank — the starting amount, the amount left, the time, or the half-life — and this solves for it in closed form. It also reports the decay constant and the mean lifetime, which are the two numbers most often confused with the half-life and with each other. Twelve well-known half-lives are built in as starting points.

Solve for

Any unit — grams, atoms, becquerels. The answer comes back in the same one.

How to use the half-life calculator

  1. 1Choose which quantity to solve for.
  2. 2Enter the other three. Amounts can be in any unit — grams, atoms, becquerels — as long as both are in the same one.
  3. 3Pick a half-life from the list, or type your own with its time unit.
  4. 4Read the answer and the half-life-by-half-life table underneath.

Examples

Two half-lives of carbon-14

Input
100 units, 11,460 years, half-life 5,730 years
Result
25 units left — a quarter, because two halvings have passed

Dating a sample

Input
Starting 100, remaining 21.7, half-life 5,730 years, solving for time
Result
About 12,630 years

The arithmetic of radiocarbon dating, though a real date also needs calibration for how atmospheric carbon-14 has varied.

Working out an unknown half-life

Input
1,000 counts falling to 125 over 18 hours
Result
6 hours — three halvings in eighteen

About the half-life calculator

The same shape as compound interest

Exponential decay and compound growth are the same equation with the sign of the exponent flipped. Money at 7% doubles in about a decade; a substance with a constant proportional loss halves in a fixed time. Both are consequences of a rate that applies to whatever is currently there rather than to the original amount.

The intuition that transfers is the one about doubling and halving times. Nobody works out compound interest by multiplying by 1.07 repeatedly in their head — they use the doubling time. Half-lives are the same trick for decay: three half-lives is an eighth, ten is about a thousandth, and those two facts answer most questions without touching a logarithm.

Why carbon dating needs calibration

The arithmetic on this page dates a sample by assuming the atmosphere's carbon-14 level was the same when the organism died as it is now. It was not. Cosmic ray flux varies, the ocean's carbon exchange varies, burning fossil fuels diluted atmospheric carbon-14 through the twentieth century, and atmospheric nuclear testing in the 1950s and 60s roughly doubled it.

So a raw radiocarbon age is converted to a calendar date using a calibration curve built from tree rings and other dated material. That is why published dates carry a range and a curve reference rather than a single number. The half-life arithmetic here is the first step of that process, not the whole of it.

Half-life says nothing about danger

A short half-life means a substance decays quickly, which means it is intensely active while it lasts and then effectively gone. A long half-life means the opposite: little activity per second, sustained for a very long time. Neither is straightforwardly safer than the other, and which matters depends entirely on the exposure — a substance that decays in hours is a serious problem inside a body and a non-problem in a store room a year later.

What the half-life cannot tell you at all is what kind of radiation is emitted, how energetic it is, or what happens if the substance is inhaled or ingested rather than kept at a distance. Those are separate properties of each isotope, and they are what dose and risk are actually computed from. This page does arithmetic on a curve; it is not a radiological assessment.

Frequently asked questions

What is the difference between the half-life and the decay constant?
The half-life is the time for half of a sample to decay. The decay constant λ is the probability per unit time that any one atom decays, and they are related by λ = ln2 / T½. The decay constant is the more fundamental of the two — it is what appears in the physics — while the half-life is the one that is easy to state and easy to measure.
Why is the mean lifetime longer than the half-life?
Because the atoms that survive a long time drag the average up. The mean lifetime τ is 1/λ, which works out to about 1.44 half-lives. Half the atoms are gone by one half-life, but the survivors go on decaying for a very long tail, and the average life of a single atom lands beyond the median. Using one where the other belongs makes an answer wrong by 44%, which is why both are reported here.
Why is decay exponential rather than linear?
Because a nucleus has no memory and no age. Its chance of decaying in the next second is the same as it was a thousand years ago, and does not depend on how long it has already survived. A constant chance per atom means a constant proportion of whatever remains goes each second, and a constant proportion is exactly what an exponential is.
Does it work for anything other than radioactivity?
Any process with a constant proportional rate follows the same equation — the discharge of a capacitor, the clearance of a substance from a system, the cooling of an object towards ambient. That is also why compound interest is the same mathematics with the sign reversed. It is not a drug dosing tool: clinical half-lives are population averages that vary with organ function, other medicines, age and genetics, and dose timing is a clinical decision.
How many half-lives until it is all gone?
Mathematically, never — the curve halves forever and never reaches zero. Physically it ends when the last atom decays, which is a random event the smooth curve cannot predict. Ten half-lives leaves about a thousandth and twenty leaves about a millionth of a millionth, and past that point the arithmetic keeps working while the physics has stopped: with a handful of atoms left, decay is a statistical law without enough statistics.

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