The half-life of a nuclide is a statistical property and is a valid concept only because of the very large number of atoms involved. Any individual atom of a radionuclide may be transformed at any time, from zero to infinity. For some calculations, it is convenient to use the average life or mean life of a radionuclide. The average or mean life is defined as the sum of the lifetimes of all the infdividual atoms divided by the total number of atoms present originally.
During a time interval from t to t+dt, the total number of transformations is kdt. Each atom that decayed during this time interval had existed for a total lifetime t. The sum of the lifetimes of all atoms that were transformed during the time interval dt, havinfg survived from t=0 is tkNdt. The average lifetime is then given by
It is then straightforward to show that the relationship between the average energy or mean lifetime l and the half-life t is given by.