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# Lecture14RadioactDecayKinetics-2-rev_001 - Lecture 14...

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Lecture 14 Nuclear Decay kinetics : Transient and Secular Equilibrium

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IV. Parent-Daughter Relationships ( The Rate-Determining Step ) A. Case of Radioactive Daughter A e . g . Same problem as a stepwise chemical reaction B. Mathematics of the Problem 1. Parent: A D e c a y : dN/dt = A N A ; N A = N A 0 e  A t (nothing new) 2. Daughter: B Formation: dN B /dt = A N A D e c a y : d N B /dt = B N B N E T : dN dt A N AB N B = A N A 0 e - A t - B N B B   Linear first-order differential equation 3. Solution N = N e - A t e -t + N e - B AA 0 BA B 0 B B t If pure A initially, N B 0 = 0 and second term vanishes 4. Daughter: C N A 0 = N A + N B + N C B C 210 Bi 210 Po 206 Pb Stable 5.01 d 138.4 d
5. Special Cases: Long time solutions classification time relationship Rate-determining step a. No Equilibrium t > t B > t A B C b. Transient Equilibrium t > t A > t B A B c. Secular Equilibrium t A >> t > t B A B (c. is case of U-Th decay series) C. No Equilibrium Daughter is rate-determining step t 1/2 (B) > t 1/2 (A) 1. Fig. 3-4 a. Curve a: Total activity of initially PURE sample of A NOTE: resembles two-component independent decay curve. NOT !!! A(total) = A(A) + A(B) b. Curve b: Parent activity, A(A) d. Curve d: Daughter activity, A(B) ; note growth curve Operative Word

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2. Curve c: Long-term behavior: t > > t 1/2 (A) t/t 1/2 (A) = and e  A t 0 OR N B =  A N A 0 B - A B t e () Note: B < A ; ( )( ) = + 3. Long term curve: All of A has disappeared ; this defines t 1/2 (B) D. Transient Equilibrium Parent decay is rate-determining step t 1/2 (A) > t 1/2 (B) 1. Fig. 3-2 a. Curve a: Total activity of initially
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Lecture14RadioactDecayKinetics-2-rev_001 - Lecture 14...

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