Half-Life Calculator
Solve the half-life equation for any unknown — amount remaining, initial amount, half-life or elapsed time — with time-unit dropdowns, isotope presets such as carbon-14, the decay constant and mean lifetime, a decay curve and the worked steps.
How it works
N = N₀ × (1/2)^(t ÷ t½) · N₀ = N × 2^(t ÷ t½) · t½ = t × ln 2 ÷ ln(N₀ ÷ N) · t = t½ × log₂(N₀ ÷ N) · λ = ln 2 ÷ t½ · τ = t½ ÷ ln 2 = 1 ÷ λ
A half-life is the time it takes for half of a decaying quantity to disappear. After one half-life 50% is left, after two 25%, after three 12.5%, so the amount remaining is the starting amount times one-half raised to the number of half-lives that have passed. Because the equation links four quantities — starting amount, amount left, half-life and elapsed time — you can solve for whichever one is missing: pick it under "Solve for". The time fields have their own unit menus (seconds to years) and are converted for you, so a half-life in days and an elapsed time in hours mix correctly. The same decay can be described two other ways: the decay constant λ (the fraction lost per unit time, ln 2 ÷ t½) and the mean lifetime τ (the average time a nucleus survives, 1 ÷ λ, about 1.44 half-lives). Choosing Atoms as the amount unit also gives the activity A = λN in becquerels. Isotope presets use published half-lives from nuclear data tables; carbon-14 uses the conventional 5,730 years.
Worked example
Carbon dating: a bone keeps 25% of the carbon-14 found in living tissue. With "Solve for: Time", carbon-14 selected, an initial amount of 100 and 25 remaining, the ratio is 100 ÷ 25 = 4, log₂ 4 = 2 half-lives, so the age is 2 × 5,730 = 11,460 years. Solving for the half-life instead: 2.5 kg of a sample falls to 2.1 kg in 5 minutes, so t½ = 5 × 0.693147 ÷ ln(2.5 ÷ 2.1) = 5 × 0.693147 ÷ 0.174353 ≈ 19.88 minutes.
Sources: NIST — Radionuclide half-life measurements · NNDC (Brookhaven) — NuDat nuclear data
Frequently asked questions
Does the elapsed time have to be a whole number of half-lives?
No. The formula uses t ÷ t½ as a continuous exponent, so it works for any elapsed time, whole or fractional.
How do I find the half-life from two measurements?
Choose "Solve for: Half-life", enter the starting amount, the amount left and the time between the two measurements. The calculator uses t½ = t × ln 2 ÷ ln(N₀ ÷ N). The two amounts can be masses, counts, activities or percentages, as long as both use the same unit.
What is the difference between half-life, decay constant and mean lifetime?
They describe the same exponential decay. The decay constant λ = ln 2 ÷ t½ is the fractional loss rate, and the mean lifetime τ = 1 ÷ λ ≈ 1.443 × t½ is the average survival time of one nucleus. The calculator shows all three for every answer.
What kinds of things follow half-life decay?
Radioactive isotopes are the classic example, but the same exponential decay describes drug clearance from the body, capacitor discharge, the cooling of a hot object toward room temperature and first-order chemical reactions.
Will the substance ever fully disappear?
Mathematically it approaches zero but never reaches it, halving again with each half-life. After 10 half-lives about 0.1% is left (1/1,024), which is why "Solve for: Time" needs an amount remaining above zero.
Why does the carbon-14 age differ slightly from a lab date?
This calculator uses the plain decay law with the 5,730-year half-life. Radiocarbon laboratories report ages with the older 5,568-year Libby half-life and then calibrate them against tree rings, corals and other records, because atmospheric carbon-14 has not been constant.
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