Secular equilibrium

In radioactive decay

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:

:\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 t1/2 by \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 dN_B/dt = 0, or

: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

: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, \lambda_A t \ll 1 and the exponential can be approximated as 1.

References

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