text: Secular equilibrium can occur in a radioactive decay chain only if the half-life of the daughter radionuclide B is much shorter than the half-life of the parent radionuclide A. In such a case, the decay rate of A and hence the production rate of B is approximately constant, because the half-life of A is very long compared to the time scales considered. The quantity of radionuclide B builds up until the number of B atoms decaying per unit time becomes equal to the number being produced per unit time. The quantity of radionuclide B then reaches a constant, equilibrium value. Assuming the initial concentration of radionuclide B is zero, full equilibrium usually takes several half-lives of radionuclide B to establish.
  The quantity of radionuclide B when secular equilibrium is reached is determined by the quantity of its parent A and the half-lives of the two radionuclide. That can be seen from the time rate of change of the number of atoms of radionuclide B:
  where λ and λ are the decay constants of radionuclide A and B, related to their half-lives t by , and N and N are the number of atoms of A and B at a given time.
  Secular equilibrium occurs when , or
  Over long enough times, comparable to the half-life of radionuclide A, the secular equilibrium is only approximate; N decays away according to
  and the "equilibrium" quantity of radionuclide B declines in turn.  For times short compared to the half-life of A,  and the exponential can be approximated as 1.