Secular equilibrium

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:

${\displaystyle {\frac {dN_{B}}{dt}}=\lambda _{A}N_{A}-\lambda _{B}N_{B},}$

where λ_A and λ_B are the decay constants of radionuclide A and B, related to their half-lives t_1/2 by ${\displaystyle \lambda =\ln(2)/t_{1/2}}$, and N_A and N_B are the number of atoms of A and B at a given time.

Secular equilibrium occurs when ${\displaystyle dN_{B}/dt=0}$, or

${\displaystyle N_{B}={\frac {\lambda _{A}}{\lambda _{B}}}N_{A}.}$

Over long enough times, comparable to the half-life of radionuclide A, the secular equilibrium is only approximate; N_A decays away according to

${\displaystyle N_{A}(t)=N_{A}(0)e^{-\lambda _{A}t},}$

and the "equilibrium" quantity of radionuclide B declines in turn. For times short compared to the half-life of A, ${\displaystyle \lambda _{A}t\ll 1}$ and the exponential can be approximated as 1.

• Bateman equation
• Transient equilibrium

• IUPAC definition
  • "secular equilibrium", IUPAC definition (IUPAC Compendium of Chemical Terminology 2nd Edition, 1997) (in English)
• Radioactive Equilibrium, EPA definition
• Radioactive Equilibrium. An equilibrium as old as the Earth, radioactivity.eu.com, IN2P3, EDP Science