Half-Life Calculator
Calculate remaining quantity, half-life time, initial amount, or elapsed decay time.
Decay Results
Working out how long a medicine stays in your body? Use the drug half-life calculator.
Constant proportion, not constant amount
Half-life is the time for a quantity to fall to half its value, and the defining feature is thatit does not depend on how much you started with. A kilogram and a milligram of the same isotope both halve over the same interval, because decay is proportional rather than absolute. After n half-lives the fraction remaining is (½)ⁿ.
That produces the sequence everyone half-remembers: 50%, 25%, 12.5%, 6.25%. It never reaches zero mathematically, which is why "ten half-lives" — leaving about 0.1% — is a common practical threshold for treating something as gone. The relationship to the decay constant ist½ = ln(2)/λ, and the ln(2) is where the familiar 0.693 comes from.
The same mathematics governs very different things. Carbon-14's 5,730-year half-life dates organic remains up to roughly ten of them; drug half-lives determine dosing intervals and how long a medication takes to clear; and capacitor discharge follows the identical curve. Note that biological half-life differs from radioactive half-life — the body eliminates a substance through its own processes, at its own rate.
Frequently asked questions
What is half-life?
Half-life is the time required for a quantity of a substance undergoing exponential decay to reduce to half of its initial value.
What is the half-life formula?
N(t) = N0 x (1/2)^(t / t-half), where N0 is the starting quantity and t-half is the half-life. Equivalently N(t) = N0 x e^(-lambda t) with the decay constant lambda = ln(2) / t-half. The two forms are identical; the exponential form is more convenient when you need the decay constant or mean lifetime.
How many half-lives until something is effectively gone?
After 1 half-life 50% remains, after 2 it is 25%, after 3 12.5%, after 5 about 3%, and after 10 roughly 0.1%. In pharmacology a drug is usually considered cleared after 4-5 half-lives, and by the same arithmetic a drug given repeatedly reaches steady state after about the same number.
Does half-life depend on how much you start with?
No, and that is the defining property of first-order decay: the proportion lost per unit time is constant, so the half-life is the same whether you start with a gram or a tonne. Some processes are zero-order instead - alcohol elimination is the classic example - and for those the concept of a half-life does not apply.