# Half-life

The time in which one half of the atoms of a particular radioactive substance disintegrates into another nuclear form. Measured halflives vary from millionths of a second to billions of years. Also called physical or radiological halflife.

Radioactive decay is a random process. As such, one cannot state with certainty when an unstable nuclide will decay. The probability that an atom will decay during the time $dt$ is given by $kdt$ where $k$ is the constant of proportionality known as the decay constant. In a system where there are $N(0)$ atoms present initially, the number of atoms decaying in time $dt$ is given by $â�� dN = -k Ndt$ . In the limit of very small time intervals, this can be expressed as

$dN/dt = -k N$

Integration with respect to time gives the number of atoms present at any time t , i.e.

$N(t) = N(0)e^{-kt}$

The half-life, $\tau$, is used to denote the time at which the number of atoms has decreased to half the initial value, i.e. $1/2 = e^{-k\tau}$ . Hence the half-life is related to the decay constant through the relation

$\tau = ln2/k$$0.693/k$

References:

Wikipedia on halflife