Simulated Radioactive Decay Using Dice Nuclei


 Howard Foster
 2 years ago
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1 Purpos: In a radioactiv sourc containing a vry larg numbr of radioactiv nucli, it is not possibl to prdict whn any on of th nucli will dcay. Although th dcay tim for any on particular nuclus cannot b prdictd, th avrag rat of dcay of a larg sampl of radioactiv nucli is highly prdictabl. This laboratory uss 2sidd dic to simulat th dcay of radioactiv nucli. Whn a 1 or a 2 is facing up aftr a throw of th dic, it rprsnts a dcay of that nuclus. Masurmnts on a collction of ths dic will b usd to accomplish th following objctivs: 1. Dmonstration of th analogy btwn th dcay of radioactiv nucli and th dcay of dic nucli 2. Dmonstration that both th numbr of nucli not yt dcayd (N) and th rat of dcay (dn/dt) both dcras xponntially 3. Dtrmination of xprimntal and thortical valus of th dcay probability constant λ for th dic nucli 4. Dtrmination of th xprimntal and thortical valus for th halflif of th dic nucli Equipmnt: 2sidd dic Thory: On of th most noticabl diffrncs btwn classical physics known prior to 19 and modrn physics sinc that tim is th incrasd rol that probability plays in modrn physical thoris. Th xact bhavior of many physical systms cannot b prdictd in advanc. On th othr hand, thr ar som systms that involv a vry larg numbr of possibl vnt, ach of which is not prdictabl; and yt, th bhavior of th systm as a whol is quit prdictabl. On xampl of such a systm is a collction of radioactiv nucli that mit α, β, or γradiation. It is not possibl to prdict whn any on radioactiv nuclus will dcay and mit a particl. Howvr, sinc any rasonabl sampl of radioactiv matrial contains a larg numbr of nucli (say at last 1 12 nucli), it is possibl to prdict th avrag rat of dcay with high probability. A basic concpt of radioactiv dcay is that th probability of dcay for ach typ of radioactiv nuclid is constant. In othr words, thr ar a prdictabl numbr of dcays pr scond vn though it is not possibl to prdict which nucli among th sampl will dcay. A quantity calld th dcay constant, or λ, charactrizs this concpt. It is th probability of dcay pr unit tim for on radioactiv nuclus. Th fundamntal concpt is that bcaus λ is constant, it is possibl to prdict th rat of dcay for a radioactiv sampl. Th valu of th constant λ is, of cours, diffrnt for ach radioactiv nuclid. Considr a sampl of N radioactiv nucli with a dcay constant of λ. Th rat of dcay of ths nucli dn dt is rlatd to λ and N by th quation 1 of 7
2 dn dt = λn Eq. 1 Th symbol dn dt stands for th rat of chang of N with tim t. Th minus sign in th quation mans that dn dt must b ngativ bcaus th numbr of radioactiv nucli is dcrasing. Th numbr of radioactiv nucli at tim t = is dsignatd as N. Th qustion of intrst is how many radioactiv nucli N ar lft at som latr tim t. Th answr to that qustion is found by rarranging Eq. 1 and intgrating it subjct to th condition that N = N at t =. Th rsult of that procdur is N = N Eq. 2 Equation 2 stats that th numbr of nucli N at som latr tim t dcrass xponntially from th original numbr N that ar prsnt. A scond qustion of intrst is th valu of th rat of dcay dn dt of th radioactiv sampl. That can b found by substituting th xprssion for N from Eq. 2 back into Eq. 1. Th rsult is Furthrmor, th xprssion for lading to dn dt = λn Eq. 3 dn dt in Eq. 1 can thn b substitutd into Eq. 3, λn = λn Eq. 4 Th quantity λn is th activity of th radioactiv sampl. Sinc λ is th probability of dcay for on nuclus, th quantity λn is th numbr of dcays pr unit tim for N nucli. Typically, λ is xprssd as th probability of dcay pr scond; so in that cas, λn is th numbr of dcays pr scond from a sampl of N nucli. Th symbol A is usd for activity ( A = λn ); thus, Eq. 4 bcoms: A = A Eq. 5 Equations 2 and 5 thus stat that both th numbr of nucli N and th activity A dcay xponntially according to th sam xponntial factor. For masurmnts mad on ral radioactiv nucli, th activity A is th quantity that is usually masurd. An important concpt associatd with radioactiv dcay procsss is th concpt of halflif. Th tim for th sampl to go from th initial numbr of nucli N to half that valu N 2 is dfind as th halflif t ½. If Eq. 2 is solvd for th tim t whn N = 2 th rsult is N ln(2) = λ 1 2 =.693 λ t Eq. 6 This sam rsult could also b obtaind by considring th tim for th activity to go from A to A 2. Figur 1 shows graphs of activity of a radioactiv sampl vrsus tim. Figur 1(a) shows a smilog graph with th activity scal logarithmic and th tim scal linar. Th tim scal is simply markd in units of th halflif. Not that th graph is linar on this 2 of 7
3 smilog plot. Th halflif is th tim to go from any givn valu of activity to half that activity. Figur 1(b) shows th shap of th activity vrsus tim graph if linar scals ar chosn for both quantitis. Th laboratory xrcis to b prformd dos not involv th dcay of ral radioactiv nucli. Instad, it is dsignd to illustrat th concpts dscribd abov by a simulatd dcay of dic nucli. In th xrcis, radioactiv nucli ar simulatd by a collction of 2sidd dic. Th dic ar shakn and thrown, and a dic nucli has dcayd if ithr a 1 or a 2 is facup aftr th throw. In this simulation, th dcay constant λ is qual to th probability of 2 out of 2 of a particular fac coming up. Thus, th thortical dcay constant λ is.1. A uniqu aspct of this simulation xprimnt is that masurmnts can b takn on both th rmaining numbr N and th numbr that dcay. Th numbr that dcay is analogous to th activity A. For ral radioactiv nucli, N cannot b masurd dirctly but is infrrd from masurmnts of th activity Activity (counts/s) Activity (counts/s) t ½ 2 t ½ t ½ 2 t ½ Tim Tim Figur 1 Graph of activity vrsus tim on smilog and on linar scals. Exprimnt: 1. Dpnding on how many dic thr ar, ach di may rprsnt 5 or mor nucli. Each tim intrval may hav svral rounds of rolling th dic to crat a larg nough sampl of nucli. You will nd to kp carful track of how many dic dcay in ach round, and b sur to xclud that numbr from th nxt tim intrval. Your instructor will tll you how many nucli to start with. You will dtrmin how many rounds you nd to rach that numbr. 3 of 7
4 Exampl: You instructor tlls you that your radioactiv sampl contains 5 nucli. You count your dic and find thr ar 5. This mans you will roll all 5 dic in 1 rounds to simulat 5 nucli. 2. Plac all of th dic in a cardboard box or othr containr. Shak th dic and gntly pour thm out onto th tabl (or into a largr box). Count and rcord how many dic show a 1 or a 2 on thir topmost fac (this rcord kping may b don on scratch papr and dos not nd to b includd with your final rport, unlss your instructor dircts you othrwis). Whn all th rounds ar thrown, count and rcord th total numbr of dcayd nucli, and how many nucli ar lft. Dtrmin how many rounds will b thrown in th nxt tim intrval, and whthr any rounds will contain fwr than th total numbr of dic. Exampl: You throw tn rounds of 5 dic ach, and find th following numbr dcay in ach round: Round Numbr of Dcays A total of 52 dic dcayd in that tim intrval. Thus, in th nxt tim intrval you start out with 5 52 = 448 dic. You will throw 8 rounds of 5 dic, and on round of 48 dic. 3. Rpat this procss until you hav fwr than 5 dic rmaining for th tim intrval. If you wish, you may go until you hav zro dic rmaining, but this may tak a whil! Analysis: 1. Entr your data into an Excl spradsht. You should hav two columns: Activity (numbr of dcays), and N (numbr at th bginning of ach throw) 2. Plot N on both a Cartsian and a smilog graph. 3. Plot Activity on both a Cartsian and a smilog graph. 4. Calculat th ratio of th numbr of dic rmovd (activity) aftr a givn throw to th numbr shakn for that throw. Ths ratios will giv an xprimntal valu for th dcay rat, λ. Not that numbr shakn is not th sam as th numbr lft aftr th throw it is th numbr lft aftr th prvious throw. 4 of 7
5 5. Calculat th avrag of ths valus, and rcord it as λ xp. 6. Th thortical valu of λ is.1. Calculat th prcnt rror in th valu of λ xp as compard to λ tho. 7. If you hav not don so alrady, activat on of th plots of N vs. throw, and click on Chart > Add Trndlin. Choos an xponntial trndlin, and undr Options, choos Display quation on chart. Compar th valu of λ rturnd by Excl to your calculatd valu of λ xp. 8. Calculat th thortical halflif from Eq. 6, using th valu of λ =.1. Rcord that valu as (t 1/2 ) tho. For th purposs of this calculation, assum that a fractional throw is possibl. 9. From th xponntial curv of N vs. throw, dtrmin th numbr of throws ndd to go from N to ½ N. Rcord that numbr as (t 1/2 ) xp. For purposs of this dtrmination, considr a fractional throw as possibl. 1. Calculat th prcntag rror in th valu (t 1/2 ) xp compard to th valu of (t 1/2 ) tho. Rcord that prcntag rror. 11. Optional (possibl Extra Crdit? Ask your instructor!): Writ an Excl program that simulats this xprimnt. Start with th sam N, and th sam dcay constant, λ. But notic for som of th throws A was gratr than λn, and for som throws A was lss than λn. You will nd to figur out how to mak your program do this. Graph th rsults, and insrt a trndlin. Notic how th curv fit changs with ach itration of th simulation. Chang N what happns to th rturnd valu of λ as N gts largr or smallr? Your instructor may laborat on ths instructions. Rsults: Writ at last on paragraph dscribing th following: what you xpctd to larn about th lab (i.. what was th rason for conducting th xprimnt?) your rsults, and what you larnd from thm Think of at last on othr xprimnt might you prform to vrify ths rsults Think of at last on nw qustion or problm that could b answrd with th physics you hav larnd in this laboratory, or b xtrapolatd from th idas in this laboratory. 5 of 7
6 ClanUp: Bfor you can lav th classroom, you must clan up your quipmnt, and hav your instructor sign blow. How you divid clanup dutis btwn lab mmbrs is up to you. Clanup involvs: Compltly dismantling th xprimntal stup Rmoving tap from anything you put tap on Dryingoff any wt quipmnt Putting away quipmnt in propr boxs (if applicabl) Rturning quipmnt to propr cabints, or to th cart at th front of th room Throwing away pics of string, papr, and othr dtritus (i.. your watr bottls) Shutting down th computr Anything ls that nds to b don to rturn th room to its pristin, pr lab form. I crtify that th quipmnt usd by has bn cland up. (studnt s nam),. (instructor s nam) (dat) 6 of 7
7 PrLab Assignmnt Rad th xprimnt and answr th following qustions bfor coming to class on lab day. 1. A typical sampl of radioactiv matrial would contain as a lowr limit approximatly how many nucli? (a) 1, (b) 1 6, (c) 1 12, or (d) Th thory of radioactiv dcay can prdict whn ach of th radioactiv nucli in a sampl will dcay. (a) tru (b) fals 3. Stat th dfinition of th dcay constant l. What ar its units? A radioactiv dcay procss has a dcay constant λ = s. Thr ar radioactiv nucli in th sampl at t =. How many radioactiv nucli ar prsnt in th sampl 1 hour latr? Show your work. 5. For th radioactiv sampl dscribd in qustion 4, what is th activity A (in dcays pr scond) at t =? What is th activity 1 hour latr? Show your work. 7 of 7
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