# Because the tritium in water is being replenished, we assume that the amount is constant until the wine

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1 Physics 111 Fall 007 Radioactive Decay Problems Solutions 1 The 3 1 H isotope of hydrogen, which is called tritium (because it contains three nucleons), has a half-life of 133 yr It can be used to measure the age of objects up to about 100 yr It is produced in the upper atmosphere by cosmic rays and brought to Earth by rain As an application, determine approximately the age of a bottle of wine 3 1 whose H radiation is about that present in new wine 1 10 Because the tritium in water is being replenished, we assume that the amount is constant until the wine is made, and then it decays We find the number of half-lives from N () n ; () 1 n, or n log log 10 (), which gives n 33 Thus the time is t nt 1 ( 33)133 ( yr) 41 yr Strontium-90 is produced as a nuclear fission product of uranium in both reactors and atomic bombs Look at its location in the periodic table to see what other elements it might be similar to chemically, and tell why you think it might be dangerous to ingest It has too many neutrons, and it decays with a half-life of about 9 yr How 90 long will we have to wait for the amount of 38 Sr on the Earth s surface to reach 1% of its current level, assuming no new material is scattered about? Write down the decay reaction, including the daughter nucleus The daughter is radioactive: write down its decay We see from the periodic chart that Sr is in the same column as calcium If strontium is ingested, the body will treat it chemically as if it were calcium, which means it will be stored by the body in bones We find the number of half-lives to reach a 1% level from N () n ; () 1 n, or n log log(100), which gives n 664

2 Thus the time is t nt 1 ( 664)9 ( yr) 193 yr The decay reactions are 90 Sr Y + 0 e + v, Y is radioactive; 90 Y Zr + 0 e + v, Zr is stable 14 3 An old wooden tool is found to contain only 60% of 6 C that a sample of fresh wood would How old is the tool? Since the carbon is being replenished in living trees, we assume that the amount of until the wood is cut, and then it decays We find the number of half-lives from N () 1 n ; 0060 () 1 n, or n log log 167 ( ), which gives n 406 Thus the time is 14 C is constant t nt 1 ( 406)5730 ( yr) yr 4 An amateur archeologist finds a bone that she believes to be from a dinosaur and she sends a chip off to a laboratory for 14 C dating The lab finds that the chip contains 5g of carbon and has an activity of 05 Bq How old is the bone and could it be from a dinosaur? The activity when the bone chip is measured is 05 decays/sec The initial activity when the animal died needs to be determined In the bone there is found 5g of carbon Since 1 mole of carbon contains 60x10 3 atoms and 1 mole of carbon has a mass of 1 g, there are 51x10 3 carbon nuclei Further the ration of 14 C/ 1 C has remained relatively constant and has a value of 13x10-1 Thus the number of 14 C nuclei is given as (13x10-1 )(51x10 3 nuclei) 36x C nuclei when the animal died The initial activity is a product of the decay constant and the number of 14 C nuclei present when the animal died The decay constant is found from the half-life of carbon (5730yrs) λ yr sec 7 t yrs 3 10 sec The initial activity is λn (378x10-1 sec -1 )(36x C nuclei) 13 Bq To calculate the age of the bone we use

3 the radioactive decay law λt ( sec ) t 05 A Aoe 05Bq 13Bqe ln( ) ( sec ) t ( sec ) t t sec 7456yrs Since the bone is only about 7500 years old and knowing that the dinosaurs disappeared over 65 million years ago, it is probably not the bone of a dinosaur 5 Calculate the decay energy for the β - decay of 4 Na given the following data: m( 4 Na) u, m( 4 Mg) u, m( 4 Ne) u, m(β - ) 549x10-4 u What is the range of the possible energies of the emitted beta particle? For the β-decay reaction of 4 Na, Q MeV ( M 4 M 4 M ) c ( ) uc 55MeV 3 uc 0 11 Na 1 Mg 1 e 6 Show that in alpha decay from a stationary parent nuclide that the conservation of energy and momentum lead to a relation between the decay energy for the nuclear reaction and the kinetic energy gained by the alpha particle, KE, given 4 m( ) by + He Q KE 1 m( daughter ) What is the kinetic energy of the alpha particle emitted in the decay of 38 U? For alpha decay: Q ( M M M ) c If the parent is at rest when it decays, parent daughter then from conservation of momentum, the daughter gets a recoil velocity in the direction opposite direction to the velocity of the alpha particle Conservation of momentum gives: mα 0 m daughter vdaughter + mα vα vdaughter vα Next we apply conservation of m He daughter

4 energy and we find: 1 1 mparentc mdaughtervdaughter + mhevhe + mdaughterc + mhec Next we can replace the velocity of the recoiling daughter atom in terms of the velocity of the alpha particle which can be measured m parent c 1 Bringing all of mα 1 m daughter v α m HevHe mdaughterc mhec mdaughter the rest energy terms to one side and combinging the terms involving the velocity of the alpha particle, we find that 1 m 1 α parent daughter He Q + mα vα mdaughter Factoring out the mass of the alpha particle on the LHS we have: ( m c m c m c ) 1 m Q 1 + α mα vα, which is the desired result mdaughter Looking up the decay energy, Q 48 MeV, m α 40015u, m daughter m thorium u, and we find the velocity of the alpha particle to be 14x10 7 m/s 7 Calculate the binding energies of radium-6 (m u), radium-8 (m u), and thorium-3 (m u) The nuclear binding energy is given through: NBE Zmp c + NmNc matomc ;where mp 10077u and mn u For radium-6: 9315MeV NBE 88 1uc NBE 17318MeV MeV The 766 nucleon nucleon 6 [ ( 10077u) + 138( u) u] c 17318MeV For radium-8: 9315MeV NBE 88 1uc NBE 1747MeV MeV The 764 nucleon nucleon 8 [ ( 10077u) + 140( u) 79875u] c 1747MeV

5 For thorium-3: 9315MeV NBE 90 1uc NBE 17669MeV MeV The 76 nucleon nucleon 3 [ ( 10077u) + 14( u) u] c 17669MeV 8 What is the activity of 1gram of radium-6? The activity is given as the product λn, where λ is the decay constant and N is the number of nuclei that decay Given that are sample is radium-6, with a half-life of 1600 years, we can yr 1 calculate the decay constant λ sec 7 t yrs 3 10 sec To calculate the number of nuclei present we use the mass given, 1g There are 6 g of radium per mole and in 1 mole there are 60x10 3 nuclei Thus in 1 g there are 66x10 1 nuclei The activity is therefore λn (135x10-11 sec -1 )(66x10 1 nuclei) 36x10 10 decays/sec 36x10 10 Bq 9 Strontium is chemically similar to calcium and can replace calcium in bones The radiation from 90 Sr can damage bone marrow where blood cells are produced, and lead to serious health problems How long would it take for all but 001% of a sample of 90 Sr to decay? Here we take the initial mass M 0 1 and the final mass is 001% of the initial mass Thus M f 1x10-4 M 0 The decay constant for 90 Sr is λ 004 yr -1 From the radioactive decay law: λt ( 004yr ) t M M e M M e t 3834yrs f o 0 o 10 After the sudden release of radioactivity from the Chernobyl nuclear reactor accident in 1986, the radioactivity of milk in Poland rose to 000 Bq/L due to iodine-131 present in the grass eaten by dairy cattle Radioactive iodine, with half-life 804 days, is particularly hazardous because the thyroid gland concentrates iodine The Chernobyl accident caused a measurable increase in thyroid cancers among children in Belarus (a) For comparison, find the activity of milk due to potassium Assume that one liter of milk contains 00 g of potassium, of which % is the isotope 40 K with half-life yr (b) After what time interval would the activity due to iodine fall below that due to potassium? (a) One liter of milk contains this many 40 K nuclei:

6 N 00 g ( ) nuclei mol 391 g mol λ ln ln 1 yr T yr s s R λn ( s )360 ( ) 618 Bq (b) For the iodine, R R 0 e λ t with λ ln 804 d t 1 λ ln R 0 R 804 d 000 ln ln d nuclei 11 A small building has become accidentally contaminated with radioactivity The 90 longest-lived material in the building is strontium-90 ( 38 Sr has an atomic mass u, and its half-life is 91 yr It is particularly dangerous because it substitutes for calcium in bones) Assume that the building initially contained 500 kg of this substance uniformly distributed throughout the building and that the safe level is defined as less than 100 decays/min (to be small in comparison to background radiation) How long will the building be unsafe? mass present mass of nucleus 500 kg ( u ) kg u ( ) nuclei λ ln ln T 1 91 yr yr min R 0 λ ( min )335 ( 10 5 ) counts min R e λ t 100 counts min 659 R counts min and λ t ln( ) 396 giving t 396 λ yr yr 1 Natural uranium must be processed to produce uranium enriched in 35 U for bombs and power plants The processing yields a large quantity of nearly pure 38 U as a byproduct, called depleted uranium Because of its high mass density, it is used in armor-piercing artillery shells (a) Find the edge dimension of a 700-kg cube of 38 U The density of uranium is kg/m 3 (b) The isotope 38 U has a long half-life of yr As soon as one nucleus decays, it begins a relatively rapid series of 14 steps that together constitute the net reaction

7 4 9 8( He ) + 6( e ) + 8 Pb + 6 v + Q net 38 U Find the net decay energy (Refer to Table A3) (c) Argue that a radioactive sample with decay rate R and decay energy Q has power output QR (d) Consider an artillery shell with a jacket of 700 kg of 38 U Find its power output due to the radioactivity of the uranium and its daughters Assume that the shell is old enough that the daughters have reached steady-state amounts Express the power in joules per year (e) A 17-year-old soldier of mass 700 kg works in an arsenal where many such artillery shells are stored Assume that his radiation exposure is limited to absorbing 455 mj per year per kilogram of body mass Find the net rate at which he can absorb energy of radiation, in joules per year (a) V l 3 m ρ, so l m ρ kg kg m m (b) Add 9 electrons to both sides of the given nuclear reaction Then it becomes 38 9U atom 8 4 He atom Pb atom + Q net Q net M 8 M M 38 9 U 4 He 8 06 Pb c [ ( ) ]u 9315 MeV u ( ) Q net 517 MeV (c) (d) If there is a single step of decay, the number of decays per time is the decay rate R and the energy released in each decay is Q Then the energy released per time is P QR If there is a series of decays in steady state, the equation is still true, with Q representing the net decay energy The decay rate for all steps in the radioactive series in steady state is set by the parent uranium: N g 38 g mol ( nuclei mol) nuclei λ ln ln T yr yr R λn yr ( nuclei) decays yr, so ( ) P QR 517 MeV yr ( JMeV) Jyr (e) The allowed whole-body dose is then 700 kg ( ) J kg yr 318 J yr

8 3 13 A sealed capsule containing the radiopharmaceutical phosphorus-3 ( 15 P ), an e emitter, is implanted into a patient s tumor The average kinetic energy of the beta particles is 700 kev The initial activity is 5 MBq Determine the energy absorbed during a 100-day period Assume that the beta particles are completely absorbed within the tumor The half-life of 3 P is 146 d Thus, the decay constant is λ ln ln T d d s R 0 λ At decay s nuclei s t 100 days, the number remaining is N e λt ( d )100 ( d) ( nuclei)e nuclei so the number of decays has been N and the energy released is E ( )700 ( kev) ( JkeV) 0400 J 14 To destroy a cancerous tumor, a dose of gamma radiation totaling an energy of 1 J is to be delivered in 300 days from implanted sealed capsules containing palladium- 103 Assume that this isotope has half-life 170 d and emits gamma rays of energy 10 kev, which are entirely absorbed within the tumor (a) Find the initial activity of the set of capsules (b) Find the total mass of radioactive palladium that these seeds should contain The decay constant is λ ln ln d The number of nuclei remaining after 30 d T 1 17 d is N e λt e ( 30 ) 094 The number decayed is N ( 1 094) 0706 Then the energy release is 1 J ev ( ) J 1 J J (a) R 0 λ d 1 ev ( d ) s Bq original sample mass m N original m one atom u (b) kg g 153 ng ( ) kg 1 u

9 15 A living specimen in equilibrium with the atmosphere contains one atom of 14 C (halflife yr) for every stable carbon atoms An archeological sample of wood (cellulose, C 1 H O 11 ) contains 10 mg of carbon When the sample is placed inside a shielded beta counter with 880% counting efficiency, 837 counts are accumulated in one week Assuming that the cosmic-ray flux and the Earth s atmosphere have not changed appreciably since the sample was formed, find the age of the sample ( ) g N C 10 g mol molecules mol ( N C carbon atoms) of which 1 in is a 14 C atom ( ) C , λ C-14 ln 5730 yr yr s R λn λ e λ t At t 0, R 0 λ ( s ) ( ) ( ) s 1 week At time t, R decays week 088 Taking logarithms, ln R λt so t R 0 λ ln R decays week t yr ln yr R 0 16 In an experiment on the transport of nutrients in the root structure of a plant, two radioactive nuclides X and Y are used Initially 50 times more nuclei of type X are present than of type Y Just three days later there are 40 times more nuclei of type X than of type Y Isotope Y has a half-life of 160 d What is the half-life of isotope X? We have all this information: N x ( 0) 50N y ( 0)

10 ( 3d) 40N y ( 3d) N x ()e 0 λ x 3d 40N y ()e 0 λ y 3d 40 N x () 0 N x e 3dλ x 5 4 e3dλ y 3dλ x ln dλ y 3d 0693 ln 5 T 1x 4 + 3d d 0781 T 1 x 66 d 50 e λ y 3d

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