Half-Life Calculator

This calculator has two modes — choose the one that matches what you know:

Mode 1 — Remaining Amount: Enter the initial quantity (N₀), the elapsed time (t), and the known half-life (t½). The calculator returns the remaining quantity (Nₜ) after that time.

Mode 2 — Calculate Half-Life: Enter the initial quantity (N₀), the remaining quantity (Nₜ), and the elapsed time (t). The calculator solves for the half-life period (t½).

Make sure all time values use the same unit (seconds, years, etc.) before entering them. The calculator assumes ideal first-order decay with no external interference.

This calculator uses two forms of the exponential decay equation:

Mode 1 — Finding remaining quantity: Nₜ = N₀ × (0.5)^(t / t½)

Mode 2 — Finding half-life: t½ = t × ln(0.5) / ln(Nₜ / N₀)

Where: - N₀ = initial quantity of the substance - Nₜ = remaining quantity after time t - t = elapsed time - t½ = half-life period

The second form is derived by rearranging the first equation: since Nₜ/N₀ = (0.5)^(t/t½), taking the natural logarithm of both sides gives ln(Nₜ/N₀) = (t/t½) × ln(0.5), which rearranges to the formula above.

An equivalent form using the decay constant λ: Nₜ = N₀ × e^(−λt), where λ = ln(2) / t½.

Half-life is the time required for exactly half of the atoms in a radioactive sample to undergo radioactive decay. It is a fundamental concept in nuclear physics that describes how quickly an unstable substance transforms into a more stable form by emitting radiation — alpha particles, beta particles, or gamma rays.

Every radioactive isotope has a fixed, characteristic half-life that is entirely independent of external conditions like temperature, pressure, or chemical state. This makes it a highly reliable quantity for measurement. Half-lives span an extraordinary range: carbon-10 has a half-life of just 19 seconds, making it impossible to encounter in nature, while uranium-238 has a half-life of approximately 4.5 billion years — comparable to the age of the Earth. Radium-226 sits in between at about 1,600 years, and carbon-14, used in radiocarbon dating, has a half-life of 5,730 years.

Beyond nuclear physics, the concept of half-life extends to any exponential decay process. In pharmacology, the biological half-life of a drug describes how long it takes for the body to eliminate half of an administered dose — crucial for determining dosing intervals. In environmental science, the half-life of pollutants determines how long they persist in soil or water. Even in population biology, exponential decay models describe how quickly organisms decline.

Half-life is a probabilistic concept: it does not predict when any individual nucleus will decay, but it describes the statistical behavior of a large ensemble of identical nuclei with remarkable precision. The larger the sample size, the more accurately the half-life formula describes the observed decay — a principle that underpins everything from medical imaging with radioactive tracers to nuclear waste storage planning.