Half-Life Calculator
Calculate radioactive decay for any half-life: solve for the amount remaining, the time elapsed or the half-life itself, convert between half-life, decay constant and mean lifetime, date samples by carbon-14, and watch the decay curve.
Exponential decay
Enter any three values and the fourth is computed from N = N₀·(½)^(t/T).
Result
Decay curve
About the Half-Life Calculator
Half-life is the time it takes for half of a radioactive sample to decay. Decay is exponential: after one half-life half of the atoms remain, after two half-lives a quarter remain, after three an eighth, and so on — the amount never quite reaches zero. This calculator works in every direction. Give it any three of the four quantities — the initial amount N₀, the amount remaining N, the elapsed time t and the half-life T — and it solves for the fourth, then shows the fraction and percentage remaining and how many half-lives have passed. It converts freely between the three equivalent measures of decay rate: the half-life T, the decay constant λ = ln2/T (the probability a nucleus decays per unit time) and the mean lifetime τ = 1/λ (the average life of a single atom). A built-in carbon-14 dating mode turns the percentage of ¹⁴C left in a sample into an age in years, the same method used to date archaeological finds. Twelve real isotope presets — from Carbon-14 and Iodine-131 to Uranium-238 and Plutonium-239 — fill in accurate half-lives with one tap, and the decay curve plots N/N₀ against time so you can see the exponential fall and mark exactly where your sample sits.
How to use this calculator
- 1 Pick a mode: "Decay solver" to find an amount, time or half-life; "Constants" to convert between T, λ and τ; or "Carbon-14 dating" to date a sample.
- 2 Choose the time unit, then choose what to solve for and enter the other three values — or tap an isotope preset to fill in its half-life automatically.
- 3 Read the answer along with the fraction and percentage remaining, the number of half-lives elapsed, the decay constant and mean lifetime, and the decay-curve plot.
How radioactive decay works
Radioactive decay follows an exponential law. The amount remaining after time t is: N = N₀·(½)^(t/T) = N₀·e^(−λt) where N₀ is the starting amount, T is the half-life and λ is the decay constant. Setting t = T gives N = N₀/2 — exactly half — which is what "half-life" means. The three rate measures are linked by: • Decay constant: λ = ln2 / T ≈ 0.693 / T (per unit time) • Mean lifetime: τ = 1 / λ = T / ln2 ≈ 1.443·T To solve for the elapsed time, invert the law: t = T·log₂(N₀/N). Carbon-14 dating is this same equation with T = 5730 years applied to the fraction f of ¹⁴C remaining: age = 5730·log₂(1/f). Living things keep a fixed ¹⁴C level by exchanging carbon with the atmosphere; once they die the ¹⁴C decays with no replacement, so the fraction left is a clock. The method is reliable to roughly 50,000 years, after which too little ¹⁴C remains to measure.
Frequently asked questions
What is half-life?
Half-life is the time it takes for half of the atoms in a radioactive sample to decay. It is a constant for each isotope, independent of how much you start with: after one half-life 50% remains, after two 25%, after three 12.5%, and so on. Half-lives range from fractions of a second to billions of years — Carbon-14 is 5730 years, while Uranium-238 is about 4.5 billion years.
How do you calculate the amount remaining after a given time?
Use N = N₀·(½)^(t/T), where N₀ is the initial amount, t is the elapsed time and T is the half-life (t and T in the same unit). For example, with N₀ = 80 g, t = 5730 years and T = 5730 years, you get N = 80·(½)¹ = 40 g. After two half-lives (11460 years) it would be 80·(½)² = 20 g. This calculator does the arithmetic and also shows the fraction and percentage left.
What is carbon-14 dating?
Carbon-14 dating estimates the age of once-living material from how much ¹⁴C it still contains. Living organisms maintain a steady ¹⁴C level; after death the ¹⁴C decays with a half-life of 5730 years and is not replaced, so the fraction remaining gives the age: age = 5730·log₂(1/fraction). If 25% of the original ¹⁴C is left, that is two half-lives, so the sample is about 11,460 years old. The technique works up to roughly 50,000 years.
What is the difference between half-life and decay constant?
They describe the same decay rate in different ways. The half-life T is the time for half the sample to decay; the decay constant λ is the probability that any one nucleus decays per unit time. They are inversely related: λ = ln2 / T ≈ 0.693 / T. A short half-life means a large decay constant (fast decay). The mean lifetime τ = 1/λ = T/ln2 ≈ 1.443·T is the average lifetime of a single atom, always a bit longer than the half-life.
Related tools
Keep exploring science on UnitConv. Compare the energy of radiation in the radiation dose experience, write very large and very small numbers cleanly with the scientific notation converter, and try the molar mass and other science calculators where decay and atomic quantities meet.