RANDOM PROCESS: Identical to Unimolecular Decomposition. ΔE a

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1 7 Lecure 13: Radioacive Decay Kineics I. Kineics of Firs-Order Processes. Mechanism: 1. ucleus has Inernal Srucure Z X Z Decay involves inernal + Y + Q ; Q =+ rearrangemen of sysem. RDOM PROCESS: Idenical o Unimolecular Decomposiion 2. Chemical nalogy V(r) 2 O 4 O 2 ΔE a ΔH Rae = d O d 2 4 = k [ 2O 4 ] rx k = e Δ E/RTxf(J a, π...) 2 O 4 2 O 2 OTE: Since aoms and molecules are neural, if T >, hey will have (3/2) kt kineic energy; Collisions may also induce reacions. Collisions are SECOD ORDER; mus disinguish order for chemical reacions For nuclei, T and Coulomb barrier prevens collisions; OLY FIRST ORDER DECY 3. uclear Sysem a. omenclaure Change [ ] o o, he number of nuclei iniially (o vogadro's number) k, he rae consan = f(q, Iπ) Firs-order Rae Law is d Rae = = d Insananeous decay rae

2 71 b. Example 137 Cs β m a + ν (1/2 = 3y) Mahemaics of Firs Order Decay Rae consan Probabiliy 137m a γ a ( = T) 56 1/2 1. Soluion: d - d = - d dt = ln or = e =, where is differen for every nuclide 2. For a pure sample e = ln ln = slope 3. 1/2 : The Half-Life Expresses probabiliy in erms of a characerisic ime i.e., high probabiliy, shor decay ime and vice versa DEFIITIO: The half-life ( 1/2 ) of a nucleus is he ime required for one-half he nuclei in a sample o decay. i.e., afer = 1/2, = /2 /2 = e = 1 = = e /2 1/2 ln 2 = 1/2 1/2 = 1n 2/ =.693/ = 1/2

3 72 4. Mahemaical Shorcus (bu = e always works). a. If < < 1/2 / = e lim(e x ) x 1 x + / = 1 = (1 ) = = OR Δ = Δ & Δ Δ = LWYS TRY TO SEE IF THIS WORKS; RULE: If <.1 1/2, good o 3 sig. figs. b. Problem: How many 238 U nuclei will decay in 1. y from a sample ha conains 2.38 mg of uranium? remain? abundance = %, 1/2 = y Tes Rule: 1. << y, OK o use Δ = Δ g = ( ) g/mole 23 aoms mole = aoms 238 U =.693/ y = y 1 Δ = ( y 1 ) ( aoms) (1.y) = aoms decay (remaining) = Δ = = aoms remain i.e., no change a 3 sig fig level c. If > > 1/2, rivial resul e = e.693 / 1/2 = e = i.e., = and Δ = ; i.e., all (or mos of) sample has decayed RULE: IF > > 1 1/2,

4 73 d. Inegral half-lives le n = / 1/2 = ineger, 1, 2, 3 ec. 1 = 2 n (same equaion, differen form). e. verage Lifeime τ τ = = 1 / i = i d Changing variables as τ 1 d = 1 (- d) = - (- e- = d = - e d MTH: ax xe ax e dx = ( ax -1) 2 a RESULT: τ = 1/ = /2 i.e., average is longer because long imes skew disribuion. 5. Rae of Energy Loss Imporance: Heaing of earh's crus ( 4 K, 232 Th, 235,238 U) Miniaure power sources ( 238 Pu) Spurious heaing in hermochemisry of radioacive elemens a. Definiions: Rae = de d = d d de d = Q =Q e- E Δ = Δ Q Insananeous Inegral

5 74 b. Problem: Calculae he rae of energy loss for 21 Po in kj/mole-min. α-decay ; 1/2 = d ; Q α = 5.35 MeV Δ < < 1/2 ΔE/Δ = Q α Q α = 5.35 MeV ( kj/mev) = kj/aom =.693/(138.4 d)(144 min/d) = /min = 1 mole = aoms ΔE/Δ = ( kj/aom)( /min)( aoms) ΔE/Δ = 187 kj/min-mole Comparable o chemical reacions (ΔH), bu a end of day, sill have same source. fer d ΔE/Δ = 935 kj/min-mole. II. civiy Pracical specs of Radioaciviy Usually measure emied paricles, no (paren or daugher nuclei).. Definiion: civiy is he radiaion ha is measured from a radioacive source. = civiy = c d d = c ~ c Δ Δ number of paricles emied emission ime inerval 1. d/d = SOLUTE DISITEGRTIO RTE (dps, dpm, dph, ec.) c = deecion coefficien = Gε. when G is a geomery facor and ε is he deecion efficiency 2. Schemaic Source radius. r deecor Deecor area = πr 2 Spherical area = 4πR 2 R G = πr 2 /4πR 2 = r 2 /4R 2 SIC PRICIPLE OF RDITIO SFETY aciviy decreases a square of disance R

6 75 ε = Fracion of paricles ha srike he deecor and give signal ccurae deerminaion of c is criical o absolue measuremens. Firs-Order Decay Law in Terms of civiy = e c = c e = e ; uncerainy is ± Δ where Δ is number of couns Δ = = emied paricles C. Pracical pplicaion 1. Plo is on semilog paper 2. Deermine and 1/2 graphically 3. If Δ 1/2, mus correc for decay 4. Deermine c wih sandard source Δ(1) > Δ(2) (1) (2) Δ(oal) = e 1 e < > = D. Unis of Reaciviy 1. Curie: Ci 1 Ci = dps 2. equerel: q 1 q = 1 dps E. Minigeneraor Experimen m a γ a 2. Measure as a f() 3. ackground Couns in deecor ha originae in surroundings insead of source ; due o 4 K, cosmic rays, ec. sample = source bkg Error = ± number of couns / ime To firs approximaion: = umber of couns/ime

7 76 III. Mixure of Independen civiies PREVIOUS: Paren OW: Paren 1 Paren 2 Paren 3 1/2 (1) 1/2 (2) 1/2 (3) Sable Daugher ll Daughers Sable [umber of componens = umber of slope changes + 1]. Example I: Two Componens. General Case oal bkg = (1) + (2) + (3) + = (1) e + (2) e + (3) e + RULE: s, SHORTEST-LIVED COMPOETS DISPPER c i, i, i can all differ C. Rules for Decomposing a 3-Componen Decay Curve (or greaer) (or why I'm happy he compuer can do his now). PLOT O SEMILOGPPER 1. Sar: LYZE LOG-TIME PORTIO OF DECY CURVE FIRST. ecause shorer-lived species will have decayed away, only a single pure (linear) componen will remain. 2. Draw a sraigh line (bes fi) hrough his componen his line describes decay of his componen al all imes. slope = ; = inercep = 3. SUTRCT long lived componen (seps 1 & 2) from he oal aciviy. oal bkg 3 = Repea procedure for second longes componen, ec. 5. Schemaic (3) (2) (2) ln (3)

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9 78 IV. Paren-Daugher Relaionships (The Rae-Deermining Sep). Case of Radioacive Daugher C e.g. 21 i 21 Po 26 Pb 5.1 d β d α Sable Same problem as a sepwise chemical reacion. Mahemaics of he Problem 1. Paren: Decay: d/d = ; = e (nohing new) 2. Daugher: Formaion: Decay: ET: d /d = d /d = d = d = e - - Linear firs-order differenial equaion 3. Soluion = e - e - + e - If pure iniially, = and second erm vanishes 4. Daugher: C = + + C

10 79 5. Special Cases: Long ime soluions classificaion ime relaionship Rae-deermining sep a. o Equilibrium > > C b. Transien Equilibrium > > c. Secular Equilibrium >> > (c. is case of U-Th decay series) C. o Equilibrium Daugher is rae-deermining sep 1/2 () > 1/2 () Operaive Word 1. Fig. 3-4 a. Curve a: Toal aciviy of iniially PURE sample of OTE: resembles wo-componen independen decay curve. OT!!! (oal) = () + () b. Curve b: Paren aciviy, () d. Curve d: Daugher aciviy, () ; noe growh curve 2. Curve c: Long-erm behavior: > > 1/2 () / 1/2 () = and e OR = - ( ) e oe: < ; ( )( ) = + 3. Long erm curve: ll of has disappeared ; his defines 1/2 () D. Transien Equilibrium Paren decay is rae-deermining sep 1/2 () > 1/2 () 1. Fig. 3-2 a. Curve a: Toal aciviy of iniially pure sample of (oal) = () + () decay curve ha iniially increases wih ime is a signaure of ransien or secular equilibrium.

11 8 b. Curve b: Paren civiy: () c. Curve d: Daugher civiy: () d. Iniial growh of () and hen decay according o he half-life of he paren. 2. Maximum Daugher civiy a. Maximum occurs when d /d = differeniae equaion for and se his equal o zero; his defines max as max = ln( / ) ; i.e., is maximum when = max b. Imporance Medical isoopes Milking a cow how long mus one wai before exracing he daugher aciviy again? 137 1β c. Example: 55 Cs 137m γ 5 a 3 y 26. m 137 a (sable) max = ln( / ) = ln[ = ln [ (3y/2.6min) /2 ()/ 1/2 ()] min/y] y (.693 y/2.6m) 2.6 m 3 y 2.6 m max = 58 min 3. Long-Time Soluion: Curve e a. For iniially pure, > > 1/2 () e = e or = ( ) e, OR = i.e., a long ime, raio / is COSTT wih ime SYSTEM PPERS TO E I EQUILIRIUM

12 81 4. Consequences a. Long erm decay is governed by paren b. civiy: muliply equaion in 3a. above by c c c = - = = c. Half-life of can be deermined by combining: long erm behavior 1/2 () aciviy raio above