Half-Life Calculator

How to use this calculator

For a remaining-amount question, enter the initial quantity, the half-life and the elapsed time. For dating, enter the fraction remaining (or the initial and remaining quantities) and the half-life; the elapsed time is solved. If you measured a sample twice, enter both quantities and the time between measurements to find its half-life.

"Quantity" can be any unit you like — grams, atoms, becquerels of activity, or percent — as long as initial and remaining use the same unit. The calculator also reports the decay constant λ and the mean lifetime τ, which appear in physics and pharmacokinetics texts. The same mathematics governs drug elimination and first-order chemical reactions.

The formula

N(t) = N₀ × (½)^(t / t½)

λ = ln 2 / t½

τ = 1 / λ

N₀: initial quantity
N(t): remaining quantity
N/N₀: fraction remaining
t: elapsed time
t½: half-life
λ: decay constant
τ: mean lifetime

Worked example

Carbon-14 has a half-life of 5,730 years. What share of the original C-14 remains in a bone after 11,460 years?

t / t½ = 11,460 / 5,730 = 2 half-lives.
N / N₀ = (½)² = 25%, so 100 units become 25.
λ = ln 2 / t½ = 1.21 × 10⁻⁴ per year; τ = 1/λ = 8,267 years.

After two half-lives a quarter of the carbon-14 is left.

Assumptions and limits

Decay is first-order: a constant fraction decays per unit time, independent of amount.
No source replenishes the material during the period (for C-14, the organism has died).

Frequently asked questions

Does anything ever fully decay?

Mathematically the amount approaches zero but never reaches it. In practice, after about 10 half-lives less than 0.1% remains.

What is the difference between half-life and mean lifetime?

Mean lifetime τ is the average time a single atom survives. It is longer than the half-life by a factor of 1/ln 2 ≈ 1.443.

How accurate is radiocarbon dating?

The decay math is exact, but atmospheric C-14 levels varied over time, so raw ages are calibrated against tree-ring and other records.