Radioactive Decay Calculator

What the radioactive decay calculator does

Radioactive material decays away over time at a steady, predictable rate. This calculator handles the whole relationship: enter any three of the initial amount, the remaining amount, the elapsed time, and the half-life, and it solves for the fourth. It also accepts the decay constant or mean lifetime in place of the half-life.

Below is what radioactive decay is, the equation behind it, how half-life relates to its cousins, and a worked example.

How to use it

1. Choose what to solve for: remaining amount, initial amount, elapsed time, or half-life.
2. Enter the other three values, giving the timing as a half-life, mean lifetime, or decay constant.
3. Press Calculate for the result, or Reset to clear it.

What radioactive decay is

Radioactive decay is the process by which unstable atomic nuclei break down over time, transforming into other nuclei and releasing energy and particles as they go. Each unstable atom decays at a random moment that cannot be predicted individually, yet when you have a great many of them, their collective behaviour is extraordinarily regular. A fixed fraction decays in any given period, so the amount of material left falls in a smooth, predictable curve.

This blend of individual randomness and collective certainty is what makes radioactive decay so useful. You cannot say when any one atom will decay, but you can say with precision how much of a large sample will remain after a given time. The decay never quite reaches zero; it keeps halving, leaving an ever-smaller remnant. This calculator works with that decay law, relating how much material you start with, how much is left, how much time has passed, and how quickly the substance decays.

The equation it uses

The amount remaining follows an exponential decay governed by the half-life:

N = N₀ × (1/2)^{t / t½}

Here N is the amount remaining, N₀ is the initial amount, t is the elapsed time, and t½ is the half-life, the time for half the material to decay. The exponent is the number of half-lives that have passed, and each one halves what is left. After one half-life, half remains; after two, a quarter; after three, an eighth, and so on. The calculator rearranges this same relationship to solve for whichever of the four quantities you leave unknown.

Half-life, decay constant, and mean lifetime

The pace of decay can be described in three equivalent ways, and the calculator accepts any of them. The half-life is the most intuitive: the time for half the material to disappear. The decay constant is the probability per unit time that any given atom decays, a measure of how eager the substance is to break down. The mean lifetime is the average time an individual atom survives before decaying.

These three are simply different expressions of the same decay rate, linked by fixed relationships. A short half-life corresponds to a large decay constant and a short mean lifetime, all describing a substance that decays quickly. A long half-life means a small decay constant and a long mean lifetime, a substance that lingers. The calculator converts whichever you provide into the others internally, so you can work in the terms most natural to your problem, whether you have a half-life from a reference table or a decay constant from a measurement.

Carbon dating and other uses

The steady clock of radioactive decay underlies some of science's most powerful dating and measurement techniques. Radiocarbon dating uses the decay of a radioactive form of carbon, with a half-life of about 5,730 years, to determine the age of once-living material. By measuring how much of the carbon remains, archaeologists can date wood, bone, and cloth tens of thousands of years old. Other isotopes with far longer half-lives date rocks and meteorites, revealing the age of the Earth and the Solar System in billions of years.

The same decay law governs many other fields. In medicine, radioactive isotopes used for imaging and treatment have carefully chosen half-lives, long enough to do their job but short enough to clear from the body, and the decay law sets the doses and timing. Nuclear power and waste management depend on knowing how long materials remain radioactive. From archaeology to medicine to geology, this single exponential relationship is at work, and the calculator lets you apply it to any of these situations.

Units and precision

The calculator works with amounts in any consistent unit, since only their ratio matters, and timing as a half-life, mean lifetime, or decay constant. The elapsed time and the half-life should be given in the same time unit, and the result comes out in that unit. It uses the exact exponential decay law, so the results are accurate across spans from fractions of a second to billions of years, covering everything from fast-decaying laboratory isotopes to the ancient clocks used in geology.

A worked example

Suppose you start with 100 units of a substance whose half-life is 10 years, and you want to know how much remains after 30 years.

Thirty years is three half-lives, so the amount remaining is N = N₀ × (1/2)³ = 100 × 1/8 = 12.5 units. Going the other way, to find how long it takes to fall from 100 to 25 units, that is two half-lives, or 20 years. For radiocarbon dating, a sample with a quarter of its original carbon left has aged two half-lives of about 5,730 years, giving an age of roughly 11,460 years.

Questions people ask

How do you calculate radioactive decay?

Use N = N₀ × (1/2)^t/t½, where t½ is the half-life. The exponent is the number of half-lives elapsed, and each one halves the amount remaining.

What is half-life?

The time for half of a radioactive substance to decay. After one half-life, half remains; after two, a quarter; after three, an eighth, and so on, never quite reaching zero.

What is the decay constant?

The probability per unit time that an atom decays, an alternative to the half-life. A large decay constant means fast decay and a short half-life; the two are directly linked.

How does carbon dating work?

By measuring how much radioactive carbon, with a half-life of about 5,730 years, remains in once-living material. The amount left reveals how long ago the organism died.

References

A quick note on where the physics comes from. The radioactive decay law, half-life, and decay constant are standard nuclear physics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. NIST maintains reference data on isotopes and half-lives. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, University Physics Volume 3, Section 10.4, Half-Life and Activity. https://openstax.org/books/university-physics-volume-3/pages/10-4-half-life-and-activity
2. HyperPhysics, Radioactive Decay and Half-Life. http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/radact.html
3. National Institute of Standards and Technology (NIST), Atomic and Nuclear Data. https://www.nist.gov/pml/radiation-physics