Half-Life Calculator

Two half-lives is a quarter left, not nothing — and ten leaves under a thousandth.

Remaining252 half-lives have passed, leaving 25%

Fraction remaining0.25(½) to the power 2

Decayed75 (75%)

Decay constant λ0.0001209681 per unit timeln2 ÷ half-life. The mean lifetime is 1/λ = 8,266.64, which is LONGER than the half-life by a factor of 1/ln2

Mean lifetime8,266.6426the average time an individual atom survives — 1.4427 times the half-life, because the tail is long

After 1 half-life50%1 in 2 remains

After 2 half-lives25%a QUARTER, not nothing — two half-lives does not mean gone

After 3 half-lives12.5%1 in 8 remains

After 5 half-lives3.125%1 in 32 remains

After 10 half-lives0.097656%under a thousandth, which is the usual working definition of gone

After 20 half-lives0.000095%1 in 1,048,576 remains

It never reaches zeroonly halves, foreverthe curve is asymptotic. In practice the sample runs out of atoms long before the mathematics runs out of halvings

The formula

N = N₀ × (½)^(t ÷ t½); λ = ln2 ÷ t½

Two half-lives is a quarter, not nothing

Each half-life removes half of what is left, so the fractions go 1/2, 1/4, 1/8 and never reach zero.

Ten half-lives leaves under a thousandth, which is the usual working definition of gone. Twenty leaves under a millionth. The curve is asymptotic and the sample simply runs out of atoms before the mathematics runs out of halvings.

The mean lifetime is longer than the half-life

Half the atoms are gone by the half-life, but the survivors have a long tail.

The average lifetime of an individual atom is 1/λ, which is the half-life divided by ln 2 — about 1.44 times longer. Physicists quote the mean lifetime and chemists the half-life, and the two differ by that factor.

Decay is memoryless

An atom that has survived nine half-lives is exactly as likely to decay in the next instant as a freshly made one.

Nothing ages. The exponential form is a direct consequence of that, and it is why a half-life is a property of the isotope rather than of the sample's history.

Ideal behaviour is an approximation

The colligative and gas relations here assume dilute solutions and ideal gases, which real systems approach and do not reach.

At high concentration or high pressure the deviations become large, and the corrections are substance-specific. These figures are right where the assumptions hold and approximate where they do not.