Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Transient equilibrium
open-in-new
<h2 id="activity-in-transient-equilibrium">Activity in transient equilibrium</h2> <p>The activity of the daughter is given by the Bateman equation: </p> A d = A P ( 0 ) λ d λ d − λ P × ( e − λ P t − e − λ d t ) × B R + A d ( 0 ) e − λ d t , {\displaystyle A_{d}=A_{P}(0){\frac {\lambda _{d}}{\lambda _{d}-\lambda _{P}}}\times (e^{-\lambda _{P}t}-e^{-\lambda _{d}t})\times BR+A_{d}(0)e^{-\lambda _{d}t},} <p>where A P {\displaystyle A_{P}} and A d {\displaystyle A_{d}} are the activity of the parent and daughter, respectively. T P {\displaystyle T_{P}} and T d {\displaystyle T_{d}} are the half-lives (inverses of reaction rates λ {\displaystyle \lambda } in the above equation modulo ln(2)) of the parent and daughter, respectively, and BR is the <a href="/facts/Branching_ratio/QrbbS9PI">branching ratio</a>. </p><p>In transient equilibrium, the Bateman equation cannot be simplified by assuming the daughter's half-life is negligible compared to the parent's half-life. The ratio of daughter-to-parent activity is given by: </p> A d A P = T P T P − T d × B R . {\displaystyle {\frac {A_{d}}{A_{P}}}={\frac {T_{P}}{T_{P}-T_{d}}}\times BR.} <h2 id="time-of-maximum-daughter-activity">Time of maximum daughter activity</h2> <p>In transient equilibrium, the daughter activity increases and eventually reaches a maximum value that can exceed the parent activity. The time of maximum activity is given by: </p> t max = 1.44 × T P T d T P − T d × ln T P T d , {\displaystyle t_{\max }={\frac {1.44\times T_{P}T_{d}}{T_{P}-T_{d}}}\times \ln {\frac {T_{P}}{T_{d}}},} <p>where T P {\displaystyle T_{P}} and T d {\displaystyle T_{d}} are the half-lives of the parent and daughter, respectively. In the case of Tc 99 m − 99 Mo {\displaystyle {\ce {^{99\!m}Tc-^{99}Mo}}} generator, the time of maximum activity ( t max {\displaystyle t_{\max }} ) is approximately 24 hours, which makes it convenient for medical use.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bateman_equation/fTIktMak">Bateman equation</a></li> <li><a href="/facts/Secular_equilibrium/sshnQTf8">Secular equilibrium</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>transient equilibrium Archived June 6, 2011, at the Wayback Machine <a href="http://web.ead.anl.gov/marssim/acrogloss/dsp_wordpopup.cfm?word_id=504" target="_blank">http://web.ead.anl.gov/marssim/acrogloss/dsp_wordpopup.cfm?word_id=504</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Johnson, Thomas E.; Birky, Brian K.; Shleien, Bernard (2012). Health physics and radiological health (4th ed.). Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins. p. 1205. ISBN 9781609134198. <a href="9781609134198" target="_blank">9781609134198</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>S.R. Cherry; J.A. Sorenson; M.E. Phelps (2003). Physics in Nuclear Medicine. A Saunders Title; 3 edition. ISBN 0-7216-8341-X. <a href="0-7216-8341-X" target="_blank">0-7216-8341-X</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>