Radioactive Decay Calculator

Calculate remaining isotope quantity, decayed amount, and decay constant from an initial amount, half-life, and elapsed time.

Radioactive decay calculation

Use consistent time units to model exponential nuclear decay.

Initial Amount

Amount Unit

Half-Life

Elapsed Time

Time Unit

About radioactive decay

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable state. A decay event may release an alpha particle, a beta particle, gamma radiation, or another product depending on the isotope. Although the moment when any individual nucleus changes cannot be predicted, a large population follows a highly regular exponential pattern. That statistical regularity makes half-life useful in science, medicine, engineering, archaeology, and radiation safety. Half-life is the interval required for half of the radioactive nuclei in a sample to decay. After one half-life, 50 percent remains; after two, 25 percent remains; and after three, 12.5 percent remains. The process never reaches exactly zero in the mathematical model. The calculator evaluates the remaining amount as the initial amount multiplied by two raised to the negative ratio of elapsed time to half-life. It then subtracts that result from the initial amount to find how much has decayed. The decay constant expresses the probability of decay per unit time and is related to half-life by lambda equals the natural logarithm of two divided by half-life. This tool converts the selected half-life unit to seconds before displaying the constant in inverse seconds. The exponential ratio itself is unit independent as long as half-life and elapsed time use the same unit. Mixing a half-life in years with elapsed time in days without conversion would produce a meaningless answer, which is why one shared time selector controls both fields. Quantity units do not alter the fraction remaining. The same model can describe a count of atoms, an amount in moles, or a mass in grams or kilograms. A result in atoms may be fractional when the model represents an expected population, even though an actual observation cannot contain part of an atom. Mass and mole calculations assume that decay products are not being counted as the original isotope. Applications range from carbon-14 dating of once-living material to iodine-131 treatment planning and cobalt-60 source management. Geologists use long-lived isotopes to date rocks, while hospitals track short-lived tracers to balance image quality and patient dose. Nuclear facilities estimate activity changes during storage. In every case, uncertainty in the measured starting amount, isotope half-life, contamination, or system history affects the interpretation. This calculator models a single isotope with a constant half-life and no additional production. It does not solve multi-step decay chains, branching pathways, daughter-product ingrowth, biological clearance, or changing irradiation. Those situations require coupled differential equations and domain-specific safety controls. Treat the output as a transparent mathematical estimate and use validated nuclear data and professional procedures for clinical, regulatory, or radiation-protection decisions.

Radioactive decay examples

The same exponential law applies across very different isotope time scales.

Starting conditions	Remaining amount	Application
1,000 atoms, half-life 5,730 years, elapsed 11,460 years	250 atoms	Carbon-14 has passed through two half-lives.
5 g, half-life 8 days, elapsed 16 days	1.25 g	Iodine-131 has passed through two half-lives.
1 mol, half-life 4.468 billion years, elapsed 4.5 billion years	About 0.498 mol	Uranium-238 changes slowly over geological time.
2 kg, half-life 5.27 years, elapsed 10 years	About 0.537 kg	Cobalt-60 source strength falls substantially over a decade.

How to calculate radioactive decay

Enter the initial isotope quantity and select atoms, moles, grams, or kilograms.
Enter the isotope half-life and the elapsed time using the same time scale.
Select the matching time unit from seconds through years.
Choose Calculate Decay to see the remaining amount, decayed amount, and decay constant.
Check the isotope and unit assumptions before using the estimate in further work.

Radioactive decay FAQ

Does all radioactive material disappear after one half-life?

No. One half-life leaves half of the original radioactive nuclei. Each additional half-life removes half of what remained, so the modeled quantity approaches zero gradually.

Why is the decay constant shown in inverse seconds?

Inverse seconds provide a standard unit that makes constants easier to compare. The calculator converts the selected half-life unit to seconds before applying the natural-logarithm relationship.

Can I enter mass instead of a number of atoms?

Yes. Exponential decay preserves the same remaining fraction for atoms, moles, and mass. Select the amount unit you want the result to retain.

What assumptions does this calculation make?

It assumes one isotope, a constant half-life, and no new production of that isotope. It does not include decay chains, branching, contamination, or biological removal.

Is half-life affected by sample size?

Ordinary nuclear half-life does not depend on how much material is present. Larger samples contain more nuclei and therefore more total decays, but the expected fraction lost over time is unchanged.