Nuclear Physics Worksheet

Transcription

1 Nuclear Physics Worksheet The ucleus [lural: uclei] is the core of the atom ad is comosed of articles called ucleos, of which there are two tyes: rotos (ositively charged); the umber of rotos i a ucleus is called the Atomic Number [symbol: Z], ad idetifies which elemet corresods to that articular ucleus. eutros (o electric charge); the umber of eutros i a ucleus is called the eutro umber. The ucleos are held together by the strog force, also called the uclear force. Differet isotoes of a sigle elemet all have the same Z, but differet umbers of eutros. Radioactive Decay: Certai tyes of uclei are said to udergo radioactive decay whe they trasform ito aother tye of ucleus. (Nuclei that decay are said to be ustable.) The trasformatio, or decay, is accomaied by emissio of certai tyes of articles ( alha ad beta articles), or of a gamma ray. It is ot ossible to determie whe oe articular ustable ucleus will decay. However, it is ossible to determie to a very high degree of accuracy what roortio of a very large samle of ustable uclei of ay give tye will decay each secod. For istace, if we have 1000 uclei of tye A, we might fid that 4% of them will decay each secod. (We will assume here that they each decay ito a sigle oradioactive ucleus.) I this examle, we would fid that of the origial samle of 1000 uclei, 40 of them will decay i the first secod. For ay other samle cosistig of 1000 uclei of tye A, we ca be cofidet that very close to 40 of them will decay i oe secod. We would say that the decay costat of this articular ucleus is 4%/s [four ercet er secod]. Usig the symbol λ for decay costat, we would say that λ = 0.04/s for this ucleus. Some symbols we will use: λ [ decay costat ]: fractio of uclei reset which decay er uit time. Note that i this cotext, λ does ot rereset a wavelegth! λ is usually exressed i uits of s 1. N: umber of radioactive uclei of a give tye reset at a articular momet i time. N is a fuctio of time: N = N(t). [Note that the umber of decays er secod is equal to λn.] N: chage i value of N. A decrease i N would corresod to a egative value of N. If there are decays i time, the value of N will decrease by. Therefore, we ca see that: N = umber of decays er secod Chater 14: Nuclear Physics Worksheet 1

2 1. Cosider a samle of N 0 uclei (where N 0 is the origial umber, that is, the umber at t = 0 s). As time goes o, we cotiue to make observatios o this samle of uclei. a) Will the value of N (i.e., the umber of radioactive uclei reset) icrease, decrease, or remai the same as time goes o? b) Will the umber of decays er secod icrease, decrease, or remai the same as time goes o? 2. Cosider a radioactive ucleus of a tye that has decay costat equal to λ = 0.10/s [we ca also write this as λ = 0.10 s 1 ]. a) If at a give momet you have a umber A of these uclei, how may of them will decay i oe secod? b) If at aother momet you have 0.7 A of these uclei, how may of them will decay i oe secod? c) Is your aswer to 1(b) cosistet with your aswers to (a) ad (b) here? 3. a) Write a equatio relatig decays er secod to λ ad N. decays er secod = b) Write a equatio relatig age oe. N = N to λ ad N. Hit: You eed a ( ) sig; refer to otes o 4. Suose a samle of A radioactive uclei has B decays er secod at a give momet, where B reresets some secific umber such as 10 or 20. Whe the umber of these uclei has decreased to 0.5 A, how may decays er secod will be there be? Exress your aswer i terms of the umber B. Hit: Review your aswer to #2 above. 5. Give two differet samles each cotaiig 1000 uclei, which oe would be the first to have its value of N hit 500: A samle with λ = 0.10/s, or oe with λ = 0.20/s? Why? Chater 14: Nuclear Physics Worksheet 2

3 The time required for a samle of N radioactive uclei of a certai tye to decay to half of its origial value is called the half-life [symbol: T 1/2 ] of that tye of ucleus. 6. Which would have the loger half-life: a ucleus with large λ, or oe with small λ? Does your aswer suggest that T 1/2 λ, or istead that T 1/2 1/λ? (It s ot exactly equal to either oe.) 7. Let s cosider a samle of 1000 uclei with λ = 0.20/s. a) How may uclei would decay i the first secod? b) If this same umber of decays er secod were to cotiue, how log would it take for half of the origial umber of uclei to decay away? c) Is the actual half-life of this material larger tha, smaller tha, or exactly equal to the your aswer for [b]? Hit: How may uclei decay betwee t = 0s ad t = 1s? How does this comare to the umber that decay betwee t = 1 s ad t = 2s? d) Work out a more recise estimate of the actual half-life of this material. 2 1 e) Which would be a better estimate of the half-life of this material: or? Hit: Fid λ 2λ the umerical values of these two exressios. 8. A isotoe of kryto has a half-life of 3 miutes. A samle of this isotoe roduces 1280 couts er secod i a Geiger couter at 3:00 PM. Each cout corresods to the radioactive decay of oe ucleus. a) As time goes o, will the umber of couts er secod icrease, decrease, or remai the same? b) After oe half-life has elased, how may couts er secod would you exect to observe? Hit: Refer to your aswer to #4. c) Determie the umber of couts er secod roduced by this samle at 3:15 PM. Chater 14: Nuclear Physics Worksheet 3

4 Radioactive Datig: Radioactive materials are ofte used to rovide estimates of the ages of substaces that cotai those radioactive materials. A articularly imortat examle is radiocarbo datig, which makes use of the radioactive ucleus 14 C that has a half-life of 5730 years. (The umber 14 refers to the total umber of ucleos rotos [6] lus eutros [8] i this ucleus. The oradioactive form of carbo is 12 C, ad has 6 rotos ad 6 eutros.) The relative roortios of 14 C ad 12 C i the atmoshere have bee fairly stable for the ast 50,000 years or so. I a samle of uclei, aroximately oe will be 14 C, ad the rest will be 12 C. Plats absorb carbo (i the form of carbo dioxide) from the air ad aimals eat lats. As log as the orgaisms are alive, the ratio of 14 C to 12 C i their tissues remais at the atmosheric ratio of 1 i However, whe the orgaism dies, the radioactive carbo is o loger releished from the atmoshere ad so its roortio decreases. By measurig the relative roortios of the two tyes of carbo i a samle of biological material, the age of the samle ca be determied with great accuracy (u to aroximately 50,000 years). Let s cosider a samle of 120 g of carbo extracted from some material of ukow age. This is equal to te moles, each of which cotais atoms. The decay costat of 14 C is s 1. Suose we determie (usig some form of Geiger couter, for istace) that the samle roduces 10 radioactive decays every secod. We wat to fid out the aroximate age of the material, usig this iformatio. How may we roceed? 9. Suose we had a samle of 120 g of carbo from a livig orgaism (this samle cotais both tyes of uclei). How may decays er secod would we exect to detect? Hit: Determie the umber of radioactive uclei i this origial samle (this is much less tha the total umber of uclei reset); use the decay costat to fid the umber of decays er secod. 10. How may decays er secod would we exect to detect from that samle 5730 years after the death of the orgaism? Hit: Fid the umber of radioactive uclei reset at this time, ad agai use the decay costat. 11. Fid the aroximate age of the material. Determie two differet ages which bracket the actual age: material defiitely older tha: yrs material defiitely youger tha: yrs Chater 14: Nuclear Physics Worksheet 4

5 12. Diagrams of several differet uclei are show. How may differet elemets are rereseted? For each elemet, state how may isotoes of that elemet are rereseted. 13. I the followig diagrams, the white circles rereset oradioactive uclei. The black circles rereset 14 C, which is radioactive ad has a half-life of 5730 years. Five differet samles are show. Rak order them accordig to how may uclei will decay each secod, o the average (most to fewest): B C D E A Most decays er secod: Fewest decays er secod Exlai your aswer. 14. These five samles were take from trees that have bee dead for differet amouts of time. Agai, the black circles rereset 14 C; whe they decay they become white circles. (The roortios of radioactive uclei are ot realistic.) Fid the aroximate ages of samles B, C, D ad E. A [oe year sice death] B C D E Hit: Aroximately how may decays are likely to have occurred i samle A, give its age? B: C: D: E: Exlai your aswers. Chater 14: Nuclear Physics Worksheet 5