Half-Life Calculator

Pick an isotope or type your own half-life, then leave one box blank and it gets solved: what is left, how long it took, what you started with, or the half-life itself. Every result comes with the decay constant, the mean lifetime, and the ladder of what remains after each of the first ten half-lives.

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How this half-life calculator works

Pick an isotope or type your own half-life, fill in what you know, and leave one box blank. You can solve for the amount remaining, the time that has passed, the amount you started with, or the half-life itself if you have two measurements and the gap between them. Every result comes with the decay constant, the mean lifetime, and a ladder showing what is left after each of the first ten half-lives, because "how many half-lives" is the intuition this is really about.

The second mode handles radiocarbon dating, and it exists because that calculation carries a genuinely strange piece of history: the half-life the world reports dates with is not the half-life carbon-14 actually has, and that is deliberate.

The formula

N = N0 × (1/2)t/T
t = T × log2(N0 ÷ N)
λ = ln2 ÷ T
mean lifetime τ = T ÷ ln2 = 1.4427 × T
radiocarbon age = −8033 × ln(F)

N0 is what you started with, N what is left, t the time elapsed and T the half-life. The units of N never matter, because only the ratio appears: grams, counts per minute and percent all give the same answer. λ is the decay constant, the fraction lost per unit time, and τ is the mean lifetime of a single atom. In the radiocarbon line, F is the fraction of modern carbon-14 remaining and 8033 is the mean lifetime that follows from a 5,568 year half-life.

Worked example

A technetium-99m dose, prepared at 8am and imaged at noon. Its half-life is 6.0067 hours, so four hours is 0.666 half-lives and 63.03% of the activity is still there. That short half-life is the entire reason the isotope is used: long enough to inject, scan and interpret, short enough that the patient is not meaningfully radioactive by the next morning. After 24 hours, 6.27% remains; after two days, 0.39%.

The mean lifetime surprise. Carbon-14's half-life is 5,700 years, but the average carbon-14 atom lives 8,223 years, which is 1.4427 times longer. Both are true. Half the atoms are gone by 5,700 years, but the ones that survive can survive a very long time, and they drag the average up. If you have ever wondered why decay is quoted in half-lives rather than lifetimes, this is why: the half-life is the number that behaves.

And the rule of thumb worth memorising. After 5 half-lives, 3.1% is left. After 10, 0.098%. After 20, one part in a million. Nothing ever reaches zero, which is why "how long until it is gone" is a question with no answer and "how long until 99% of it is gone" is a question with a very precise one.

The radiocarbon half-life that everyone knows is wrong

In 1949 Willard Libby measured the half-life of carbon-14 as 5,568 years. In 1962 a redetermination at Cambridge put it at 5,730 years, and the current evaluated value is about 5,700. Libby was off by roughly 2.4%.

Radiocarbon laboratories still report ages using 5,568. Not by oversight, and not because they disagree: by international convention, formalised by Stuiver and Polach in 1977. The reason is that seventy years of published dates were calculated on Libby's value, and quietly switching would make every old date incomparable with every new one without a single measurement having changed. So the convention holds, and a number reported as a "conventional radiocarbon age" is understood by everyone in the field to be calculated on a half-life nobody believes.

What rescues this from being absurd is calibration, and it is the real lesson of the second mode above. A radiocarbon age is not a calendar date, because the amount of carbon-14 in the atmosphere has never been constant: it moves with solar activity, with the magnetic field, with ocean circulation, and lately with fossil fuel burning and atomic bomb testing. So every raw age must be run through a calibration curve built from tree rings and other independently dated material before it becomes a range of real years. And because the same half-life is used to build the curve as to date the sample, the error cancels exactly. The convention is not a mistake being preserved; it is a shared unit of account that the calibration step converts into truth.

The calculator shows all three ages side by side so the size of the difference is visible rather than theoretical. On a sample at 55% modern carbon it is about 114 years, and it grows with age. It is also why you should never mix a "years BP" figure with a calendar year without saying which you mean: BP counts from 1950, the ages are conventional, and the two only reconcile after calibration.

What decay does not depend on

Radioactive decay is close to unique in nature for what it ignores. Heat it, freeze it, compress it, dissolve it, put it in a chemical compound: the half-life does not move. This is because decay happens in the nucleus, and chemistry happens in the electrons, and the two barely speak. It is why radiometric dating works at all, and why a rock that has been through mountain building and metamorphism still carries an honest clock.

There are two footnotes an examiner might enjoy. Decay modes that involve the electrons, such as electron capture, can be shifted by a few tenths of a percent by extreme chemical or pressure conditions, because they depend on electron density at the nucleus. And the half-life of a fully ionised atom moving at relativistic speed genuinely changes, which is time dilation rather than chemistry. Neither of those matters for anything you will do with this page, but "nothing affects it" is a slightly stronger claim than the physics actually makes, and it is worth knowing where the edges are.

The other thing worth stating plainly is that a half-life is a statistical property, not a promise about any one atom. An individual carbon-14 atom has no memory and no schedule; it has a fixed probability of decaying in the next second, and that probability is the same whether it was made yesterday or twenty thousand years ago. The tidy exponential curve above is what emerges from an enormous number of independent coin flips, which is exactly why it stops being tidy when very few atoms remain, and why counting statistics, rather than the arithmetic, sets the real limit on how old a sample can be dated.

Sources

Where the numbers on this page come from. We go to the body that publishes the figure, not to another calculator. See how we verify.

Frequently asked questions

How do I calculate how much is left after a given time?

Multiply the starting amount by one half raised to the power of time divided by half-life. Four hours of a 6.0067 hour half-life is 0.666 half-lives, so 0.5 to the power 0.666 leaves 63.03% of what you had. The units of the amount never matter because only the ratio appears in the equation, so grams, counts per minute and percent all give the same answer.

How do I find the half-life from two measurements?

Half-life equals the elapsed time multiplied by ln2 and divided by the natural log of the ratio between the two amounts. If a sample falls from 100 to 25 over 40 minutes, that is two half-lives, so the half-life is 20 minutes. Leave the half-life box blank above and enter both amounts and the time, and it solves that way round.

What is the difference between half-life and mean lifetime?

The half-life is when half the atoms are gone; the mean lifetime is the average an individual atom actually survives, and it is longer by a factor of 1 divided by ln2, which is 1.4427. Carbon-14's half-life is 5,700 years but the average carbon-14 atom lives 8,223 years, because the minority that survive a long time drag the average up. Half-lives are quoted rather than lifetimes because the half-life is the number that behaves predictably.

How many half-lives until something is gone?

Never, strictly, which is why the question is always asked as a percentage. After 5 half-lives 3.1% is left, after 7 about 0.8%, after 10 about 0.098%, and after 20 roughly one part in a million. The 5 half-life figure is the one worth carrying around: it is where most things stop mattering, and it is the same arithmetic that governs how long a drug takes to clear the bloodstream and how long a spill stays a problem.

Why do radiocarbon labs use the wrong half-life?

On purpose, and by international agreement. Libby measured carbon-14 at 5,568 years in 1949; a 1962 redetermination gave 5,730 and the current value is about 5,700, so Libby was low by roughly 2.4%. Laboratories still report conventional radiocarbon ages on 5,568 because seventy years of published dates were calculated that way and switching would silently make them all incomparable. Since calibration uses the same half-life on both sides, the discrepancy cancels and no date is actually wrong.

Is a radiocarbon age the same as a calendar date?

No, and this is the most common misreading of a radiocarbon result. The carbon-14 content of the atmosphere has never been constant, so a raw age has to be calibrated against a curve built from tree rings and other independently dated material before it becomes a range of real years. A radiocarbon age is a measurement, quoted in years BP counting back from 1950; a calendar date is that measurement plus a curve, and it usually comes out as a range rather than a single year.

Can anything change a half-life?

Essentially nothing you can do to a sample. Heat, pressure, chemical state and dissolution leave it alone, because decay happens in the nucleus while chemistry happens in the electrons. There are two genuine edge cases worth knowing: decay modes that involve the electrons, such as electron capture, can shift by a few tenths of a percent under extreme conditions, and a fully ionised nucleus travelling at relativistic speed really does decay more slowly, which is time dilation rather than chemistry. Neither matters for any ordinary calculation.

What can carbon dating actually date, and how far back?

Anything that was once alive and took carbon from the atmosphere: wood, charcoal, bone, textiles, seeds. It cannot date rock, metal or pottery, though it can date charcoal trapped in them. The practical limit is around 50,000 years, after which so little carbon-14 remains that contamination and background dominate the measurement. It is also poor for the last few centuries, where natural variation in atmospheric carbon-14 swamps the decay signal and tree rings do the job better.

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