Half-Life Calculator — Radioactive Decay

A bone fragment holds 35% of the carbon-14 it had when the animal died. How old is it? Questions like that have four moving parts: the starting amount, the amount left, the half-life and the time between them. Give this Half-Life Calculator any three and it returns the fourth.

Pick the unknown under Solve for. Its box turns into a dashed answer field and the other three stay editable. Every answer comes with the decay constant, the mean lifetime, a curve and a table of the first ten half-lives. Because the law is a ratio, the amount can be grams, becquerels, millicuries or a plain percentage.

Worked Example: Radiocarbon Dating a 35% Sample

The page opens solving for Time elapsed, with an initial amount of 100 %, 35 % left and a half-life of 5730 years. It reports:

• Time elapsed: 8,678.5 years, which is 1.51457 half-lives
• Decay constant: 0.000120968 per year (ln 2 ÷ 5,730)
• Mean lifetime: 8,266.64 years, the average time one atom survives
• The timeline row for 2 half-lives reads 11,460 years and 25%, a quick sanity check

Press Load Sample to step through iodine-131 after 30 days, an unknown isotope that falls from 800 g to 50 g in 12 hours (exactly a 3-hour half-life), cobalt-60 worked backwards to its starting mass, and a hospital dose of technetium-99m.

How the Math Rearranges for Each Unknown

Everything comes from N = N₀ × (½)^{t / T½}. Solving for the amount left multiplies; solving for the initial amount divides by the same factor. Time and half-life need a logarithm: the number of halvings is log₂(N₀ ÷ N), and t equals that count times T½.

Both time boxes carry their own unit, so a half-life in days can meet an elapsed time in hours. Internally both become seconds, with a year taken as 365.25 days, the convention nuclear data tables use. Switch Solve for and the answer you just got is written into its box, so the numbers stay consistent instead of going blank. Clicking an isotope chip while solving for the half-life switches the unknown to time.

Precision at Very Many Half-Lives

Results show six significant figures. Very small or very large values move to scientific notation, below 0.0001 and from one billion up, so the 4.468 × 10⁹-year uranium-238 preset stays readable.

Past about 1,075 half-lives the fraction remaining drops below 5 × 10⁻³²⁴, the smallest number JavaScript can hold, and the amount left shows as 0 with a note explaining why. Impossible inputs are refused rather than bent into an answer. Enter more material at the end than at the start and you get: “The amount left can't be larger than the initial amount: decay only ever reduces it.”