Half-Life Calculator — radioactive decay remaining

How it works

The half-life of a substance is the time for half of it to decay. Radioactive decay is exponential: the amount left keeps halving every half-life, so the fraction remaining after a time t is:

N = N₀ × (½)^(t ÷ T½)

where N₀ is the initial amount and T½ is the half-life. This is the same as the standard decay law N = N₀·e^(−λt) with decay constant λ = ln(2)/T½ ≈ 0.693/T½ — the (½)^(t/T½) form just makes the halving explicit.

The number of half-lives elapsed is t ÷ T½. After 1 half-life, 50% remains; after 2, 25%; after 3, 12.5%; after n, exactly (½)ⁿ. The half-life is an input, so this works for any decaying quantity — carbon-14, a medical isotope, or a drug clearing the body. Keep the time and half-life in the same units.

Worked example

Start with 100 units and a half-life of 5730 (carbon-14's, in years). After one half-life (5730) exactly 50 remain; after two (11 460), 25; after three (17 190), 12.5. The units cancel, so a substance with a 5-unit half-life leaves 80 × (½)^(15/5) = 80 × (½)³ = 10 after 15 units of time.

Frequently asked questions

What is a half-life?

A half-life is the time it takes for half the atoms of a radioactive substance (or half of any exponentially decaying quantity) to decay. It is constant for a given substance, no matter how much you start with.

What is the formula for the amount remaining?

N = N₀ × (½)^(t/T½), where N₀ is the starting amount, t is the elapsed time and T½ is the half-life. Equivalently N = N₀·e^(−λt) with λ = 0.693/T½. The result is the amount still present after time t.

How much is left after several half-lives?

The fraction remaining halves each half-life: 50% after one, 25% after two, 12.5% after three, 6.25% after four — in general (½)ⁿ after n half-lives. It never quite reaches zero, but it becomes negligible after about ten half-lives (under 0.1%).

Does the starting amount change the half-life?

No. The half-life is a fixed property of the substance and does not depend on how much you start with or on temperature, pressure or chemical state. Twice as much material still loses half of itself in one half-life.

What units should I use for time and half-life?

Any units, as long as time and half-life use the **same** one — seconds, days, years, whatever suits the substance. Because the formula uses their ratio, the units cancel, so the amount remaining is in the same units as the initial amount.

How is half-life related to exponential decay?

Half-life is one way to describe exponential decay; the decay constant λ is another. They are linked by λ = ln(2)/T½. A short half-life means a large λ and fast decay; a long half-life means slow decay, like carbon-14's 5730 years used in radiocarbon dating.