Chemical Engineering 412

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1 Chemical Engineering 412 Introductory Nuclear Engineering Lecture 7 Nuclear Decay Behaviors

2 Spiritual Thought Sooner or later, I believe that all of us experience times when the very fabric of our world tears at the seams, leaving us feeling alone, frustrated, and adrift. It can happen to anyone. No one is immune. Everyone s situation is different, and the details of each life are unique. Nevertheless, I have learned that there is something that would take away the bitterness that may come into our lives. There is one thing we can do to make life sweeter, more joyful, even glorious. We can be grateful! Dieter F. Uchtdorf

3 Last Time: Decay Mechanisms Gamma (γγ) Alpha (αα) Beta (+/-) (ββ +, ββ ) Electron capture (EC) Proton (P) Neutron (N) Internal conversion (IC)

4 Decay Constant (λ) NN = number of atoms in the radionuclide sample NN = atoms that undergo decay during t λλ lim tt 0 NN NN tt Definition: the probability that a radionuclide will decay over an infinitesimally small unit of time λλ ff(tt, PP, NN ii, tt)

5 Exponential Decay Now we convert to continuous mathematics, and: NN(tt) = average or expected number of radionuclides at time tt dddd = λλnn tt dddd dddd(tt) dddd = λλnn tt N tt = NN oo ee λλλλ Radioactive Decay Law where NN oo = number of radionuclides in the sample at tt = 0

6 Half-Life (TT 1/2 ) Definition: constant representing the time it takes for a radionuclide to decay to ½ of its initial sample size NN TT 1 2 = NN oo 2 = NN ooee λλtt 1 2 TT 1/2 = lllll λλ λλ TT 1 2 ff(tt, PP, NN ii, tt) When solving for number of half-lives nn needed for a radioactive sample to decay to a fraction εε of its initial value: εε = NN nntt 1 2 NN oo = 1 2 nn ln εε nn = ln ln εε

7 Example A sample contains pure TTTT (Tellurium) which undergoes ββ decay and has a TT 1 2 of 9.35 h. (a) What is the decay constant (in h -1 ) for this radionuclide. λλ = ln 2 = ln 2 = h 1 TT 1 2 (9.35 h) (b) How long will it take for the sample to decay to 2 5 of the original number of atoms? ln εε nn = ln 2 = ln (2 5) = ln 2 tt = nn TT 1 2 = h = h

8 Half Life Histogram Age of Earth Age of Universe

9 Decay Probability & Avg. Lifetime Decay Probability for a Finite Time Interval: NN tt PP tt = = exp λλλλ NN 0 PP tt = 1 PP tt = 1 exp λλλλ pp tt = λλ exp λλλλ NN tt = number of atoms at time tt. PP tt = probability of existence during 0-tt. PP tt = decay probability in time 0-tt. pp tt = probability density Average Lifetime: TT aaaaaa = tt pp tt dddd = 1 0 λλ

10 Parallel Decay Routes 64 29CCCC ZZZZ ββ 39.0% = PPPPPPP NNNN ββ % = PPPPPPP 2 64 NNNN EEEE 43.6% = PPPPPPP 3 28 ddnn CCCC dddd 3 = 1 λλ ii NN CCCC (tt) = NN CCCC (tt) 3 1 λλ ii = λλ tttttt NN CCCC (tt)

11 ff ii = λλ ii λλ tttttttttt Parallel Decay Routes λλ tttttttttt λλ tttttttttt = ii λλ ii

12 Example 64 29CCCC ZZZZ ββ 39.0% = PPPPPPP NNNN ββ % = PPPPPPP 2 64 NNNN EEEE 43.6% = PPPPPPP 3 28 If CCCC has an overall TT 1 2 of 12.7 h, calculate the decay constants for the three decay modes in h -1. λλ tttttt = ln 2 = ln 2 = h 1 TT 1 2 (12.7 h) λλ ββ = ff ββ λλ tttttt = λλ tttttt = h 1 λλ ββ + = ff ββ +λλ tttttt = λλ tttttt = h 1 λλ EEEE = ff EEEE λλ tttttt = λλ tttttt = h 1

13 Activity Definition: number of decays or transmutations per unit of time that occur within the sample Units: Becquerel (Bq) = 1 transformation / sec Curie (Ci) = 3.7*10 10 Bq AA tt ddnn ii dddd = λλnn ii tt = AA 0 exp λλλλ Specific Activity = activity per unit mass AA tt = AA tt mm(tt) = λλnn aa MM Nuclides undergoing single decay mechanisms exhibit decreasing activity and constant specific activity with time.

14 Series Decay with Production When the decay of a radionuclide is accompanied by the creation of new atoms of the radionuclide: ddnn ii (tt) dddd = λλnn ii (tt) + QQ ii (tt) tt NN ii tt = NN ii,0 ee λλλλ + QQ ii ttt ee λλ tt ttt ddddd 0 For QQ = constant: NN ii tt = NN ii,0 exp λλλλ + QQ ii,0 λλ 1 exp λλλλ And as tt = : NN ii eeee = QQ ii,0 λλ

15 General Decay Chain Decay chain: when a parent radionuclide decays into a daughter, that then decays into another daughter, as so on, until a stable daughter is formed XX 1 λλ 1 XX 2 λλ 2 XX ii λλ ii λλ nn 1 XX nn (stable) ddnn 1 (tt) dddd = λλ 1 NN 1 (tt) ddnn 2 (tt) = λλ dddd 1 NN 1 (tt) λλ 2 NN 2 (tt) ddnn ii (tt) = λλ dddd ii 1 NN ii 1 (tt) λλ ii NN ii (tt) ddnn nn (tt) = λλ dddd nn 1 NN nn 1 (tt)

16 General Decay Chain For the case when only radionuclides of the parent XX 1 are initially present (NN 1 (0) 0 and NN ii 0 = 0, ii > 1): AA jj tt = λλ jj NN jj tt = NN 1 0 CC 1 ee λλ 1tt + CC 2 ee λλ 2tt + + CC jj ee λλ jjtt AA jj tt = NN 1 (0) CC mm ee λλ mmtt jj mm=1 CC mm = jj Π ii=1 ii mm jj Π ii=1 λλ ii λλ ii λλ mm = λλ 1 λλ 2 λλ 3 λλ jj λλ 1 λλ mm λλ 2 λλ mm λλ 3 λλ mm λλ jj λλ mm

17 Secular Equilibrium At long times compared to the half lives of the daughters (but short compared to the chain head), the activities of all species are the same. NN 1 λλ 1 = NN 2 λλ 2 = = NN jj λλ jj AA oo = AA 1 = AA 2 = = AA jj Species with short half lives (large λλ) have low concentrations, but concentrations can be estimated from species with longer half lives, in particular from the head of the chain. NN 1 λλ 1 = NN kk λλ kk NN 1 = NN kk (TT 1 2 ) 1 (TT 1 2 ) kk

18 Example Secular Equilibrium Example Problem

19 Natural Radionuclides 65 natural isotopes Cosmogenically produced 3 H, 7 Be, 14 C H and C used for dating. All three source of radioactivity in air samples. Primordial Isotopes Singly occurring (17) including 40 K and 87 Rb (parts of humans) Decay Products with Z > 83 form from 232 Th, 235 U, or 238 U via αα and ββ emission. Isolated natural radionuclides

20 Thorium Decay Chain (4n)

21 Uranium Decay Chain (4n+2)

22 Actinium Decay Chain (4n+3)

23 4 Possible decay chains Mass number given by 4n ( 232 Th), 4n+2 ( 238 U or 234 Np), and 4n+3 ( 235 U or 239 Pu) are near secular equilibrium. 4n+1 ( 241 Ac/Pu) has no step slow enough. Radioisotopes from it have long since decayed and are not found in nature.

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