secular equilibrium

{{short description|Situation in which the quantity of a radioactive isotope remains constant}}

In nuclear physics, secular equilibrium is a situation in which the quantity of a radioactive isotope remains constant because its production rate (e.g., due to decay of a parent isotope) is equal to its decay rate.

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 t_1/2 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

[http://goldbook.iupac.org/S05532.html IUPAC definition]
[https://web.archive.org/web/20070610000118/http://www.iupac.org/goldbook/S05532.pdf "secular equilibrium", IUPAC definition] (IUPAC Compendium of Chemical Terminology 2nd Edition, 1997) {{in lang|en}}
[https://web.archive.org/web/20040321220209/http://www.epa.gov/radiation/understand/equilibrium.htm Radioactive Equilibrium, EPA definition]
[http://www.radioactivity.eu.com/site/pages/Radioactive_Equilibrium.htm Radioactive Equilibrium. An equilibrium as old as the Earth] {{Webarchive|url=https://web.archive.org/web/20190728193324/http://www.radioactivity.eu.com/site/pages/Radioactive_Equilibrium.htm |date=2019-07-28 }}, radioactivity.eu.com, IN2P3, EDP Science

Category:Radioactivity

secular equilibrium

In radioactive decay

See also

References