Radioactive decay is a random process at the single-atom level; is impossible to predict exactly when a particular atom will decay. However, the chance that a given atom will decay is constant over time. And for a large number of atoms, the decay rate for the collection as a whole can be computed from the measured decay constants of the nuclides (or equivalently from the half-lives) .
Given a sample of a particular radionuclide, the half-life is the time taken for half of its atoms to decay . The following equation is used to predict the number of atoms (N) of a a given radioactive sample that remain after a given time (t):
Where: τ, "tau" is the average lifetime of a radioactive particle before decay. λ, the decay constant "lambda" is the inverse of the mean lifetime, and N0 is the value of N at t=0. The equation indicates that the decay constant λ has units of t-1, and can thus also be represented as 1/τ, where τ is a characteristic time of the process called the time constant.
The half-life is related to the decay constant as follows: set N = N0/2 and t = T1/2 to obtain:
This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer. Half-lives vary widely: that of 209Bi is 1019 years, while unstable nuclides can have half-lives that have been measured as short as 10−23 seconds.
Rewriting the initial equation after converting logarithmic bases, the following equation can be shown to be valid:
Since radioactive decay is exponential with a constant probability, each process could as easily be described with a different constant time period that gave its "(1/3)-life" (how long until only 1/3 is left) or "(1/10)-life" (a time period until only 10% is left), and so on. Thus, the choice of τ and t1/2 for marker-times, are only for convenience, and from convention. Mathematically, the nth life for the above situation would be found in the same way as above—by setting N = N0/n, and substituting into the decay solution to obtain: