Radioactive Decay

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 = N0e-λt

where N is the number of unchanged atoms at time t

and N0 is the number of atoms at any chosen t=0

The radioactive half life (T1/2) is defined as the time required for the number of parent atoms to fall from 

N = N0 to N = N0/2

N = N0/2 = N0e-λT1/2 

Integrating both sides gives

T1/2 = ln2/λ = 0.693/λ

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

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

or relatively short half-lives e.g.

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