The rate of radioactive decay is proportional to the number of atoms

where λ is the disintegration constant or decay rate.

Integrating both sides gives

N = N_{0}e^{-λt}

where N is the number of unchanged atoms at time t

and N_{0} is the number of atoms at any chosen t=0

The radioactive half life (T_{1/2}) is defined as the time required for the number of parent atoms to fall from

N = N_{0} to N = N_{0}/2

N = N_{0}/2 = N_{0}e^{-λT1/2}

Integrating both sides gives

T_{1/2} = ln2/λ = 0.693/λ

Isotopes can have very long half-lives e.g.

^{234}U 244 thousand years^{235}U 704 million years^{238}U 4.5 billion years^{40}K 1.25 billion years^{244}Pu 82 million years

or relatively short half-lives e.g.

^{165}Dy 2 h^{42}K 12 h^{75}Se 120 d^{57}Co 272 d^{201}Th 73 h^{131}I 8 d