Last updated: 2026-07-01

Radioactive Decay

Radioactive decay is the spontaneous breakdown of unstable atomic nuclei. Each nucleus decays at a random moment, but with a fixed probability per unit time, so a large collection of nuclei thins out in a smooth, predictable way. That collective behaviour is captured by the exponential decay law:

This calculator takes the initial quantity $N_0$, the half-life $t_{1/2}$ and the elapsed time $t$, and returns the remaining quantity $N$, the decay constant $\lambda$, the fraction remaining and the number of half-lives that have passed.

How the model works

Two constants describe the same decay in different language. The half-life $t_{1/2}$ is the time for half of the sample to disappear. The decay constant $\lambda$ is the probability per unit time that any one nucleus decays. Setting $N/N_0 = \tfrac{1}{2}$ in the exponential law gives the link between them:

Because a half-life always halves what remains, the law can be rewritten in a form that makes the halving explicit. With $n = t / t_{1/2}$ half-lives elapsed:

The two forms are identical. The exponential $e^{-\lambda t}$ handles any elapsed time smoothly, while the $(\tfrac{1}{2})^{n}$ form is easier to read off when $n$ is a whole number.

Formula

The fraction remaining depends only on the number of half-lives $n$, never on the starting amount, which is why a percentage is the most portable way to state it.

Worked example

A sample starts with $N_0 = 1000$ atoms of an isotope whose half-life is $t_{1/2} = 10$ years. How much remains after $t = 25$ years?

Step 1 — Decay constant:

Step 2 — Half-lives elapsed:

Step 3 — Remaining quantity:

Step 4 — Fraction remaining:

After 25 years, roughly 177 of the original 1000 atoms have not yet decayed.

Fraction remaining after whole half-lives

Activity: the rate of decay

The activity of a sample is the number of decays per unit time, $A = \lambda N$. It is measured in becquerels (1 Bq = 1 decay per second) or curies (1 Ci = 3.7 × 10¹⁰ Bq). Because activity is proportional to the remaining number of nuclei, it follows the same exponential curve with the same half-life: a source at half its original activity has aged exactly one half-life, regardless of the isotope.

Real-world relevance

The decay law underpins radiometric dating. In radiocarbon dating, a once-living sample stops absorbing carbon-14 at death; the carbon-14 then decays with a 5730-year half-life. Measuring the surviving fraction and solving $N = N_0\,e^{-\lambda t}$ for $t$ gives the age — a sample with 25% of its carbon-14 has passed two half-lives, about 11,460 years. The same mathematics sets medical-isotope dosing schedules, fixes the storage times for nuclear waste, and calibrates the deep-time clocks of geology using uranium and potassium isotopes.

Limitation: single decay path, large samples

The exponential law assumes a single, constant decay constant and a sample large enough for statistical averaging to hold. It does not by itself track a decay chain, where the product of one decay is itself radioactive and partly replenishes what is lost; those cases need the coupled Bateman equations. For a single isotope decaying to a stable product, and for sample sizes far above a handful of atoms, the formula above is exact.