Editing Radioactive decay (section)

====Time constant and mean-life====

For the one-decay solution ''{{math|A → B}}'':

:<math>N = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/ \tau}, \,\!</math>

the equation indicates that the decay constant {{math|''λ''}} has units of {{math|''t''<sup>−1</sup>}}, and can thus also be represented as 1/{{math|''τ''}}, where {{math|''τ''}} is a characteristic time of the process called the ''[[time constant]]''.

In a radioactive decay process, this time constant is also the [[mean lifetime]] for decaying atoms. Each atom "lives" for a finite amount of time before it decays, and it may be shown that this mean lifetime is the [[arithmetic mean]] of all the atoms' lifetimes, and that it is {{math|''τ''}}, which again is related to the decay constant as follows:

:<math>\tau = \frac{1}{\lambda}.</math>

This form is also true for two-decay processes simultaneously ''{{math|A → B + C}}'', inserting the equivalent values of decay constants (as given above)

:<math> \lambda = \lambda_B + \lambda_C \,</math>

into the decay solution leads to:

:<math>\frac{1}{\tau} = \lambda = \lambda_B + \lambda_C = \frac{1}{\tau_B} + \frac{1}{\tau_C}\,</math>

[[File:Halflife-sim.gif|thumb|right|Simulation of many identical atoms undergoing radioactive decay, starting with either 4 atoms (left) or 400 (right). The number at the top indicates how many half-lives have elapsed.]]