When the radioactive substance is only disintegrating: Radioactive decay is a random process. Each decay is an independent event, and one cannot tell when a particular nucleus will decay. When a given nucleus decays, it is transformed into another nuclide which may or may not be radioactive. When there is a very large number of nuclei in a sample, the rate of decay (i.e - dN/dt) is proportional to the number of nuclei, N, that are present i.e - dN/dt ∝ N ⇒ - dN/dt = λ/N where λ is called the decay constant.
This equation may be expressed in the form dN/dt = -λdt and integrated to get
where N0 is the initial number of parent nuclei at t = 0. The number that survive at time t is therefore
N = N0 e–λt
