How to Calculate Half-Life and the Amount Remaining

The half-life is the time a radioactive substance takes to fall to half its amount. From the half-life and the time elapsed, this finds how much is left. Enter the elapsed time and the half-life in the same unit.

The half-life is the time a radioactive substance takes to fall to half its amount. Each half-life that passes leaves a half, then a quarter, then an eighth.

N = N_0 \left(\dfrac{1}{2}\right)^{t/T}

Here $N_0$ is the initial amount, $t$ the time elapsed, $T$ the half-life and $N$ what remains. The exponent $t/T$ counts how many half-lives have gone by, and it need not be a whole number.

The decay constant

The same decay can be written with the base of natural logarithms.

N = N_0 e^{-\lambda t}, \qquad \lambda = \dfrac{\ln 2}{T}

The quantity $\lambda$ is the decay constant, the probability that any one atom decays in unit time. Fix either the half-life or the decay constant and the other follows.

Worked example

Starting from 100, with a half-life of 12.3 and 30 units of time elapsed, about 18.44 remains.

Since 30 ÷ 12.3 = 2.439, some 2.44 half-lives have passed. Raising 0.5 to the power 2.439 gives 0.1844, and multiplying by 100 gives 18.44. The decay constant is 0.693 ÷ 12.3 = 0.0564.

Points to watch

Enter the elapsed time and the half-life in the same unit, years with years or days with days. The answer then carries the unit of the initial amount.

A half-life says nothing about when any individual atom will decay. No atom has a scheduled moment; the half-life is the time in which a large collection of them falls by half. Once only a handful are left, the formula no longer describes what happens.