Half-Life Calculator (Decay & Remaining Amount)

The Decay

Half-life and elapsed time can use different units; they are converted before use.

Starting quantityAny unit — grams, counts, becquerels

Half-life

Half-life unit

Time elapsed

Elapsed time unit

Target remaining fractionHow long until this much is left

AMOUNT REMAINING

25 remaining

Fraction remaining25% of the original

Half-lives elapsed2 half-lives

Decay constant (λ)3.833e-12 per second (0.000121 per year)

Mean lifetime (1/λ)8,266.643 years (1.443 × the half-life)

Time to reach the target19,034.648 years to reach 10%

After 11,460 years — that is 2 half-lives — 25% remains, or 25 of the original 100. The decay constant is ln2 divided by the half-life — 3.833e-12 per second, not 0.5 divided by it, which is the most common slip here because the wrong answer lands in the right ballpark. The mean lifetime, 8,266.643 years, is longer than the half-life by a factor of 1.443: most nuclei decay early but a long tail drags the average up. Half-life does not change as the sample shrinks — each nucleus has a fixed probability per unit time and nuclei do not age.

The formula and the constant

The amount left after a time t is N = N₀ × (½)^t/T, where T is the half-life. Equivalently it is N₀e^-λt, with the decay constant λ = ln2 / T — note the natural logarithm of two, roughly 0.693, and not 0.5. Getting that wrong is the single most common slip here, and it produces an answer in the right ballpark, which is precisely what makes it hard to notice.

Why the half-life never changes

Decay is a random process at the level of individual nuclei: each has a fixed probability of decaying in any given interval, regardless of how long it has already existed and regardless of what its neighbors are doing. Nuclei do not age. The consequence is that the time for half of a sample to go is the same whether you start with a kilogram or a microgram, which is what makes the process usable as a clock. It also means the sample never quite reaches zero mathematically, though in practice it reaches a single atom and then that atom decays.

Mean lifetime is longer than half-life

The mean lifetime, 1/λ, is the average time an individual nucleus survives. It is about 1.44 times the half-life, not equal to it, because the distribution has a long tail — most nuclei decay early but a few last a very long time, and they drag the average up. Physics papers usually quote mean lifetime while chemistry and medicine quote half-life, so a factor of 1.44 discrepancy between two sources is often this and not an error.

Carbon dating and its limits

Carbon-14 has a half-life of about 5730 years, which sets the useful range of radiocarbon dating at roughly ten half-lives — around 50,000 years, beyond which too little remains to measure reliably. The method also assumes the atmospheric carbon-14 level was constant, which it was not; calibration curves built from tree rings and other records correct for that. Different isotopes suit different timescales: potassium-argon for millions of years, tritium for a few decades.

Beyond radioactivity

The same mathematics describes any process where the rate of loss is proportional to the amount present. Drug elimination from the bloodstream is quoted as a biological half-life, and dosing intervals are set from it — after about five half-lives roughly 97% has gone, which is the usual rule of thumb for a drug being cleared. Capacitor discharge, sound absorption and atmospheric pressure with altitude all follow the same exponential form with a different constant.

Frequently Asked Questions

Is the decay constant 0.5 divided by the half-life?

No — it is the natural logarithm of 2 divided by the half-life, about 0.693/T. Using 0.5 gives an answer in roughly the right range, which makes the error easy to miss.

Why is mean lifetime longer than half-life?

Because the distribution has a long tail. Most nuclei decay early but a few last much longer and pull the average up. The mean lifetime is about 1.44 times the half-life.

Does a half-life change as the sample shrinks?

No. Each nucleus has a fixed probability of decaying per unit time regardless of age or of how many others remain, so the half-life is constant. That is what makes it usable as a clock.

How long until essentially nothing is left?

After ten half-lives about 0.1% remains, which is the usual practical limit. Mathematically the amount never reaches zero, but in practice you run out of atoms.

Where these numbers come from

Sources

Official publications only. Links open the original document in a new tab.

National Institute of Standards and Technology CODATA fundamental physical constants Fundamental constants used in decay calculations
National Institute of Standards and Technology Time and Frequency Division US civil time standard

Half-Life Calculator