Half-Life and Radioactive Decay Calculator
Determine how much of a radioactive substance remains after any period of time, or back-calculate the half-life from measured decay data. Ideal for radiometric dating, nuclear physics coursework, and radiation safety assessments.
About this calculator
Radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. The fundamental law states that the remaining quantity N(t) follows an exponential relationship: N(t) = N₀ × (0.5)^(t / t½), where N₀ is the initial amount, t is the elapsed time, and t½ is the half-life. Every full half-life period reduces the remaining quantity by exactly 50%. The decay constant λ is related to the half-life by λ = ln(2) / t½. Different decay modes (alpha, beta, gamma) do not change this mathematical relationship — only the half-life value differs between isotopes. This calculator supports a decay-type multiplier so you can model effective or combined decay scenarios.
How to use
Suppose you start with 1,000 atoms of Carbon-14 (half-life = 5,730 years) and want to know how many remain after 11,460 years. Enter N₀ = 1000, half-life = 5730 years, time elapsed = 11460 years, and decay type = 1. The formula gives: N = 1000 × (0.5)^(11460 / 5730 × 1) = 1000 × (0.5)² = 1000 × 0.25 = 250 atoms. After exactly two half-lives, 25% of the original sample remains. This result is consistent with the well-known rule that each half-life halves the remaining quantity.
Frequently asked questions
What is the half-life of a radioactive element and why does it matter?
The half-life is the time required for exactly half of a given radioactive sample to decay into its daughter products. It is a fixed, characteristic property of each radioactive isotope that does not change with temperature, pressure, or chemical state. Half-life values span an enormous range — from microseconds for highly unstable nuclei to billions of years for stable isotopes like Uranium-238. Knowing the half-life is essential for applications ranging from carbon dating archaeological artifacts to determining safe storage timelines for nuclear waste.
How do I calculate the amount of radioactive material remaining after multiple half-lives?
After each complete half-life, the remaining quantity is halved. After n half-lives, the fraction remaining is (0.5)ⁿ, or equivalently N(t) = N₀ × (0.5)^(t / t½). For example, after 3 half-lives only 12.5% of the original material remains. This exponential relationship means that theoretically a substance never fully disappears, but it becomes negligibly small after about 10 half-lives (less than 0.1% remaining). This calculator automates that exponentiation for any combination of initial amount and elapsed time.
When is radioactive decay used in real-world dating and medical applications?
Radiocarbon dating uses the known half-life of Carbon-14 (5,730 years) to estimate the age of organic materials up to about 50,000 years old. Uranium-lead dating extends this principle to rocks billions of years old. In nuclear medicine, short-lived isotopes like Technetium-99m (half-life ~6 hours) are used as tracers because they decay quickly, minimizing patient radiation exposure. Radiation therapy also exploits the predictable decay of isotopes to deliver targeted doses to tumors. Understanding decay rates is therefore fundamental to both Earth science and clinical practice.