Summary 1017

SUMMARY The goals of Chapter 30 have been to understand the physics of the nucleus and some of the applications of nuclear physics.

GENERAL PRINCIPLES

The Nucleus The nucleus is a s)TIaJl , dense, positive core at the center of an atom.

Z protons, charge +e. spin ~ N neulrons, charge 0, spin r

The mass number is

A =Z+N

~PW'O"

~Neutron

Isotopes of an element have the same value of Z but different values of N.

The strong force holds nuclei together:

It acts between any two nucleons.

It is short range.

Adding neutrons to a nucleus allows the strong force to overcome the repUlsive Coulomb force between protons.

The binding energy B of a nucleus depends on the mass difference between an atom and its const ituenLS:

IMPORTANT CONCEPTS

Energy levels

Nucleons fill nuclear energy levels, similar to filling electron energy levels in atoms. Nucleons can often jump to lower energy levels by emitt ing beta particles or gamma photons.

APPLICATIONS

Radioactive decay

The number of undecayed nuclei decreases exponen­ti ally with time t:

N = Nue 1/.

N-Nu -

Energy

N

Neutrons

-'- -­,

Protons

_ (I)""" 2

The half-life

0+---;"---;-­o

11/2 = 71n2 = 0.6937

is the time in which hal f of any sample decays.

Nuclear Stability N Low·Z nuclei Line of stabili ty move c loser to the line of stabil ity by beta decay.

\. Alpha decay is energetically favorJb le for high-Z nuclei.

Most nuclei are not stable. Unstable nucle i undergo radioactive decay. Stable nuclei cluster along the line of stability in a plOl of the isotopes. ~------z

Mechanisms by which unstable nucle i decay:

Decay Particle Penetration

alpha 4He nucleus low

beta-minus e medium

beta-plus e+ medium

gamma photon high

Alpha and beta decays change the nucleus; the daughter nucleus is a different element.

Alpha decay: Beta-minus decay:

1x --+ ~ jY + a + energy ~X --+ l+1 y + f3 + energy

The quark model

Nucleons (and other particles) are made of quarks. Quarks and leptons are fundamental particles.

Measuring radiation

Proton Neutron

Up quark Down quark

The activity of a rad ioactive sample is the number of decays per sec­ond. Acti vity is re lated to the half-life as

O.693N N R = --= -

I t1'2 7

The radiation dose is measured in grays, where

I Gy = 1.00 J/kg of absorbed energy

The relative biological effectiveness (RBE) is the biological e ffect of a dose relati ve to the biological effects of x rays. The dose equivalent is measured in sieverts, where

dose equivalent in Sv = dose in Gy X RBE

1018 CHAPTER 30 Nuc lear Phys ics

U:;;.TM For homewo~k assig~ed on MasteringPhysics, go to

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Problem difficulty is labeled as I (straightforward) to 11111 (challenging).

QUESTIONS

Conceptual Questions

1. Atom A has a large r atomic mass than atom B. Does thi s mean that atom A also has a larger atomic number? Explain.

2. Given that ifill = 1.007825 U, is the mass ofa hydrogen atom IH greater than, less than. or equal 10 1112 the mass of a 12C atom? Explain.

3. a. Is there a stable ~Li nucleus? Explain how YOll made your determi nation.

b. Is there a stable 1~~U nucleus? Explai n how you made yo ur determination.

4. Rounding slightly, the nucleus 3He has a bi nd ing energy of 2.5 MeV/nucleon and the nuc leus 6Li has a bind ing energy of 5 MeV/nucleon. a . What is the binding e nergy of 3He? b. What is the binding energy of 6Li ? c . Is it energeticall y poss ible for two 3He nuclei to jo in or fu se

together into a 6Li nucleus? Expla in. d. Is it energeticall y possible fo r a 6Li nucleus to split or fi s­

sion in to two 3He nucle i? Explain. 5. A sample contains a mix of isotopes of an elemen t. Us ing a

spectrometer to measure the spec trum of e mitted light will not reveal the mix of isotopes; analyz ing the sample wi th a mass spectrometer will . Explain.

6. For each nuclear energy- leve l d iagram in Figure Q30.6, state whether it re presen ts a nuclear ground state, an exc ited nuclear state, or an imposs ible nucleus.

Energy (, ) Energy (b)

---22--- ---22---

--#-" 4#- fftt 44tttt 4#-22 --#- * ' '---tt-Neutrons Protons Neutrons Protons

FIGURE 030.6

7. Figure Q30.7 shows how N

the nu mber of nucle i of 1.000.000

one parti cular isotope

Energy «1

-+-22+ 4#-" ## --#- '2--#-Neutrons Protons

va ri es with t ime. What is the half- life of the nucleus?

250,000 o+---------~--­

o FIGURE 030.7

8. A rad ioactive sample has a half- life of 10 s. 10,000 nucle i are present at t = 20 s. a. How many nucle i were there at I = 0 s? b. How many nuclei will there be at t = 40 s?

Prob lems labeled INT integrate significant material from earlier

chapters; BID are of biological or medical interest.

9 . Nucleus A decays in to the stable nucleus B with a hal f-li fe o f 10 s. At' = 0 s there are 1000 A nucle i and no B nuclei. At what time will there be 750 B nuclei?

10. A radioac ti ve sample's half-l ife is 1.0 mi n, so each nucleus in the sample has a 50% chance of undergo ing a decay somet ime between t = a and t = I min. O ne part icula r nucleus has not decayed at I = I S min. What is the probability thi s nucleus wi ll decay between t = 15 and t = 16 min?

II . Four samples each contain a single rad ioact ive isotope. Sample A has 1 mol of matter and an acti vity of 100 Bq. Sample B has 10 mol and 100 Bq, sample C has 100 mol and 100 Bq, and sample 0 has 100 mol and 1000 Bq. Rank in o rder, from largest to smallest, the hal f-l ives of these four isotopes. Explain .

12. O il and coal generall y contain no measurable 14c. What does thi s tell us about how long they have been buried?

13. Rad iocarbon dating assumes that the abundance of 14C in the environ ment has been constant. Suppose 14C was less abundant 10,000 years ago than it is today. Would th is cause a lab using rad iocarbon dating to overestimate or underest imate the age of a 10,000-year-old artifac t? (Tn fact~ the abundance of 14C in the e nviro nment does vary slightly with time. But the issue has been weU studied, and the ages of arti fac ts are adjusted to com­pensate fo r thi s variation.)

14. Identify the unknown X in the following decays: a. 2~~Rn ~ 2~~PO + X b. 2~~Ra ~ 2~SAc + X c. 1;~Xe ~ I~CS + X d. ~Cu ~ ~Ni + X

15. Are the follow ing decays poss ible? If not, why not? a. 2~~Th ~ 2~~U + a b. 2~~PU ~ 2~~U + a c. ~~P ~nS+e-

16. What ki nd of decay, if any, would you expect for the nucle i with the energy-level diagrams shown in Figure Q30. 16?

Energy (a )

-22+ ##'4--#­--#-" --#-Neutrons PrOlOns

FIGURE 030.16

Energy (h) Energy (c)

Neutrons PrOlOns Neutrons Protons

17. The nuclei of 4He and 160 are very stab le and are o fte n referred to as ;;doubly mag ic" nucle i. Use what you know about e nergy levels to ex plain what is spec ial aboul these particul ar nuclei.

18. A and B are fresh apples. Apple A is strongly irradiated by BID nuclear rad iat ion for I hour. Apple B is not irradi ated. After­

ward, in what ways are apples A and B d ifferent? 19. A pat ient's tumor is irradi ated with gamma rays from an ex ter­BID nal source. Afterward, is hi s body rad ioacti ve? Explain.

20. It's poss ible that a bone tumor will not show up on an x-ray BID image bu t will show up in a gamma scan. Ex plain why this is so. 2 1. Four rad iation doses are as fo llows: Dose A is 10 rad with an BID RBE of I, dose B is 20 rad with an RBE of I, dose C is 10 rad

with an RBE of 2, and dose D is 20 rad with an RBE of 2. a . Rank in order, from larges t to smallest, the amount of

energy de li vered by these four doses. b. Rank in order, from largest to smallest, the biological dam­

age caused by these four doses . 22. Two di fferent sources of radiation give the same dose equ iva­BID lent in Sv. Does thi s mean that the radiat ion from each source

has the same RBE? Ex plain . 23. Some types of MRl can produce images of resolution and detail BID similar to PET. Though the images are simi lar, MRI is gene rally

prefe rred over PET for studies o f brain function in volv ing healthy subjects. Why?

24. Sulfur colloid particles tagged wi th 9~C are taken up and BID retained by ce ll s in the liver and sp leen. A patient is suspected

of hav ing a liver tumor that would des troy these cells. Ex plain how a gamma camera scan could be used to confi rm or rule out the ex istence of a tumor.

25. The first two ietlers in the acronym SPECT, which describes a BID nuclear imaging technique, stand for "s ingle photon." Is a

SPECT done with a gamma emitler or a positron emitler?

The following two questions concern an uncommon nuclear decay mode known as electro" captllre. Certain nuc lei that are proton-ri ch but energet ically prohibi ted from unde rgo ing beta-plus decay can capture an electron from the Is she ll , which then combi nes with a proton to make a neutron. The basic react ion is

p + e ~ n + "e 26. Give a description of the e lectron capture process in terms of

quarks. 27. Electron capture is usually fo llowed by the emiss ion of an x ray.

Why?

Multiple-Choice Ouestions

28. j A signi fican t fract ion of the rad iation dose you will rece ive BID dur ing your life comes from radioacti ve materi als in your body.

The most importan t source of thi s radiation is the potassium isotope 4OK, which decays to the stab le calcium isotope ,wCa. What particle is emitted in the decay? A. A he lium nucleus B. A neutron C. An e lectron D. A positron

PROBLEMS

Section 30.1 Nuclear Structure

I. I How many protons and how many neutro ns are in (a) 3H, (b) "'Ar, (e) "'Ca, and (d) "'Pu?

2. 1 How many protons and how many neutrons are in (a) 3He, (b) '"Ne, (e) "'Co, and (d) 226Ra?

3. II Use the data in Appendix D to calculate the chemical ato mic mass of lithium, to two dec imal places .

4 . 1 Use the data in Appendix D to ca lcu late the che mica l atomic mass of neon, to two dec imal places.

29. I A ce rtain watch's luminous glow is due to zinc sulfide paint thaI is energized by beta part icles given off by triTium, the rad ioacti ve hydrogen iso tope 3H, which has a ha lf-l ife of 12,3 years. This glow has about 1110 of its initial brightness . J-Iow many years old is the watch?

Problems

A. 20yr B. 30yr C. 40yr D. 50 yr

1019

30. I W hat is the unknow n isotope in the following fi ss ion reac­tion: n + 235 U ~ 131 1 +? + 3n A. HfiR b B. 102Rb C. H9y D. lIJ2y

3 1. II The uran ium in the earth's crust is 0.7% 235 U and 99.3% 23~ U, Two bill ion years ago, 235 U comprised approx imately 3% of the uran ium in the earth's crust. Thi s teBs you something about the relati ve half-l ives of the two isotopes. Suppose you have a sam­ple of 235 U and a sample of 231\U, each with exactly the same number of atoms. A. The sample of 235U has a higher act ivity. B. The sample of 238U has a higher activity. C. The two samples have the same act ivity.

32. I Suppose you have a I g sample of 226Ra, half·l ife 1600 years. How long will it be un til only 0.1 g of rad ium is le ft ? A. 1600 yr B. 3200 yr C. 5300 yr D. 16,000 yr

33. 1 A sample of 131 1, hal f-li fe 8.0 days, is reg isteri ng 100 counts per second on a Geiger counter, How long wiiJ it be before the sample reg isters only I coun t per second? A. 8 days B. 53 days C. 80 days D. 800 days

34. II The..complete expression for the decay of the radioactive hydro­gen isotope tritillm may be wri tten as 3H ~ 3He + X + Y. The symbols X and Y represent A. X = e+, Y= ve B. X = e- , Y = ve C. X = e+, Y = VC D, X = e-, Y = lie

35. I The quark compos ition of the proton and neutron are, respec-. . ,

tlvely, uud and udd, where U IS an up quark (charge +je) and d

is a down quark (charge -1e) . There are also anti-up u (charge

-~e) and anti -down d (charge +~e) quarks. The combinat ion of a quark and an antiquark is ca ll ed a mesol/. The mesons known as pions have the compos ition 1T ~ = ud and 1T- = lid, Suppose a proton collides with an antineutron. During such col­l.i s ions, the vari ous quarks and antiquarks annihi late whenever possible. When the remaining quarks combine to form a sing le particle, it is a A. Proton

Section 30.2 Nuclear Stability

B. Ne Ulron D. 7T

5. I Calculate (in MeV) the to tal bind ing energy and the binding energy per nucleon (a) for 3H and (b) for 3He,

6. I Calculate (in MeV) the total bind ing energy and the binding energy per nucleon (a) for 40Ar and (b) fo r 4OK.

7. I Calcu late (in MeV) the binding energy pe r nucleon for 3J-1e and 4J-1e. Which is more ti ghtly bound?

8. II Calcu late (in MeV) the binding energy per nucleon for 12C and Dc. Which is more tightly bound?

1020 CHAPTER 30 Nuclear Physics

9. I Calculate (in MeV) the binding energy per nucleon for (a) 14N , (b) 56Fe, and (c) 21l7Pb.

10. 11 When a nucleus Of 235U undergoes fi ss ion, it breaks in to two smaller, more tightly bound fragments. Calculate the bindi ng energy per nucleon for 235U and for the fi ssion product 137es.

11. II When a nucleus of 24UpU undergoes f iss ion, it breaks into two smaller, more tightly bound fragments. Calculate the binding energy per nucleon for 24UpU and for the fi ss ion product 133Xe.

Section 30.3 Forces and Energy in the Nucleus

12. 11 Draw an energy-level di agram, similar to Figure 30.9 for the protons and neutrons in II Be. Do YO LI expect thi s nucleus to be stable?

13. II Draw energy-leve l diagrams. similar to Figure 30.9, for all A = 10 nuclei li sted in Appendix D. Show aJJ the occupied neu­tron and proton levels. Which of these nuclei do you expect to be stable?

14. II Draw energy-leve l diagrams, similar to Figure 30.9, for all A = 14 nuclei li sted in Appendix D. Show all the occupied neu­tron and proton levels. Which of these nuclei do you expect to be stable?

15. III You have seen that filled electron energy levels correspond to chemically stable nuclei. A similar princi ple holds for nuclear energy levels; nucle i with equally filled proton and neutron energy levels are especiall y stab le. What are the three lightest isotopes whose prolon and neutron energy levels are both filled, and fill ed equally?

Section 30.4 Radiation and Radioactivity

16. I 150 and 131 1 are isotopes used in medical imag ing. 150 is a BIO beta-pl us emitter, 131 1 a beta-minus emitter. What are the daugh-

ter nuclei of the two decays? 17. I Spacecraft have been powered with energy from the alpha

decay of 23g pu. What is the daughter nuc leus? 18. I Identify the unknown isotope X in the following decays.

a. 2J4U -+ X + (}' b. 32 p _ X + e-

c. X - 3uSi + e+ d. 24Mg - X + l' 19. I Identify the unknown isotope X in the following decays.

a. X _ 224 Ra+a b. X_ 2u7Pb +e-c. 7Be + e - X d. X _ 6UNi +1'

20. I What is the energy ( in MeV) released in the al pha decay of 239PU?

2 1. I What is the energy ( in MeV) released in the alpha decay of WITh?

22. 11 What is the total energy (in MeV) released in the beta decay of a neutron?

23 . I Medical gamma imag ing is generaLl y done with the tech­BIO netium isotope ~c·, which decays by emitti ng a gamma-ray

photon with energy 140 keY. What is the mass loss of the nucleus, in u, upon emiss ion of thi s gamma ray?

Section 30.5 Nuclear Decay and Half-Lives

24. I The rad ioacti ve hydrogen isotope 3H is called tritium. It decays by beta-m inus decay with a half-life of 12.3 years. a. What is the daughter nucleus of tritium? b. A watch uses the decay of tritium to energize its glowing

dial. What fraction of the tritium remains 20 years after the wa tch was created?

25. I The barium isotope m Ba has a hal f- life of 10.5 years. A sam­ple beg ins with 1.0 X lOW 133 Ba atoms. How many are left after (a) 2 years, (b) 20 years, and (c) 200 years?

26. I The cadmium isotope I09Cd has a half-l ife of 462 days . A sample begins with 1.0 X 1012 I09Cd atoms. How many are left after (a) 50 days, (b) 500 days, and (c) 5000 days?

27 . II How many half-li ves must elapse until (a) 90% and (b) 99% of a radioact ive sample of atoms has decayed?

28. II The Chernobyl reac tor acc ident in what is now Ukraine was BlO the worst nuclear di saster of all time. Fission products from the

reactor core spread over a wide area. The primary radiation exposure to people in western Europe was due to the short-li ved (half-l ife 8.0 days) isotope 131 1, which fell across the landscape and was ingested by grazing cows that concen trated the isotope in their milk . Farmers couldn't se ll the contaminated milk , so many opted to use the milk to make cheese. aging it until the radioactivity decayed to acceptable levels. How much time must elapse for the act ivity of a block of cheese containing 131 ( to drop to 1.0% of its initial value?

29. 11 11 What is the age in years of a bone in which the 14C/ 12C ratio is measured to be 1.65 X 1O- 13?

30. II 1I5Sr is a short- li ved (half- life 65 days) isotope used in bone BIO scans. A typical pati ent receives a dose of lISSr with an activity

of 0.10 mC i. If all of the .'lSSr is retained by the body, what wil l be its aC li vity in the patient 's body after one year has passed?

3 1. II What is tbe halJ-life in days of a radioactive sample with 5.0 X 10 15 atoms and an acti vity of 5.0 X 108 Bq?

32. III What is the activity, in Bq and Ci, of 1.0 g of 226Ra? Mari e INT Curie was the discoverer of radium ; can you see where the unit

of acti vity named after her came from? 33 . II Many med ical PET scans use the isotope IgF, which has a BIO half-life of 1.8 hr. A sample prepared at 10:00 A.M. has an acti v­

ity of 20 mCi. What is the ac ti vity at 1 :00 P.M., when the patient is injec ted?

34. III An inves ti gator collects a sample of a radioact ive isotope with an acti vity of 370,000 Bq. 48 hours later, the acti vity is 120,000 Bq. What is the half-l ife of the sample?

Section 30.6 Medical Applications of Nuclear Physics

35 . II A 50 kg nuclear plant worker is exposed to 20 mJ of neutron BIO radiation with an RBE of 10. Whal is the dose in mSv? 36. II A gamma scan showing the BlO act ive vo lume of a patient' s

lungs can be created by having a patient breathe the radioacti ve isotope 133 Xe, which undergoes beta-minus decay with a subse­quent gamma emiss ion from the daughter nucleus. A typical procedure gives a dose of 0.30 rem to the lungs. How much energy is deposited in the 1.2 kg mass of a patient's lungs?

37. II How many rad of gamma-ray photons cause the same biolog-ica l damage as 30 rad of alpha rad iation?

38. I 150 rad of gamma rad iat ion are directed into a 150 g tumor. BlO How much energy does the tumor absorb? 39. II During the 1950s, nuclear bombs were tested on islands in

the South Pac ific. In one test, personnel on a nearby island received 10 mGy per hour of beta and gamma radiation. At thi s rate, how long would it take to rece ive a potentially lethal dose equi valent of 4.5 Sv?

c 40. III 131 1 undergoes beta-minus BID decay wilh a subsequent gamma

emiss ion from the daughter nucleus. Iodine in the body is almost entirely taken up by the thyroid g land. so a gamma scan using th is isotope will show a bright area corresponding to the thyroid g land with the sur­rounding ti ssue appearing dark. Because the isotope is concen­trated in the gland , so is the radiation dose , most of which results from (he bela emission . In a typical procedure, a patient receives 0.050 mCi of 131 1 . Assume that all of the iodine is absorbed by the 0.15 kg thyro id g land. Each 131 ( decay produces

a 0.97 MeV beta particle. Assume (hal half the energy of each bela particle is deposited in the gland. What dose eq ui va lent in Sv will the gland recei ve io the firs t hour?

41. I The doctors planning a radiat ion therapy treatment have BID determined that a 100 g tumor needs to receive 0.20 J of gamma

radiat ion. What is the dose in Gy? 42. III \IOSr decays with the emission of a 2.8 Me V beta panicle. BKl Strontium is chem ically similar to calc ium and is laken up by

bone. A 75 kg person exposed to waste from a nuclear accident abso rbs \IOSr with an activity of 370,000 8q. Assume thal all of thi s 90Sr ends up in the skeleton. The ske leton form s 17% of the person's body mass. If 50% of the decay energy is absorbed by the skeleton , what dose equivale nt in Sv wi ll be received by the person 's skeleton in the first month?

Section 30.7 The Ultimate Building Blocks of Matter

43. U What are the minimum energies of the two o ppositely BID directed gamma rays in a PET procedure? 44. 1111 Positive and negative pion s, denoted 1;+ and 1; , are ant i·

particles of each o ther. Each has a res t mass of 140 MeV/c2•

Suppose a collis ion between an electron and positron , each with kinetic energy K, produces a 1;+ , 1;- pair. What is the smallest possible value fo r K?

45. II In a particular beta·minus decay of a free neutron (that is. one no t part o f an atomic nucleus), the emilled electron has exactly the same kinetic energy as the emitted e lectron antineutrino. What is the value, in MeV, of thal kinetic e nergy? Assume that the recoiling proton has negligible kinetic energy.

46. 1111 The masses of the ne utrinos are still not prec ise ly deter· INT mined , but let us assume for the purpose of thi s problem that the

mass of an e lectron neutrino is one millionth the mass of an electron. What is the kinetic energy, in eV, of an electron ne u· trino moving at 0.999c'!

General Problems

47. I The chemical atomic mass of hydrogen, with the two stable isotopes lH and 2H (deuterium). is 1.00798 u. Use thi s va lue to determine the natural abundance of these two isotopes.

48. n You learned in Chapter 29 that the b ind ing e nergy of the elec­tron in a hydrogen atom is 13.6 ev' a. By how much does the mass decrease when a hyd rogen

atom is formed from a proton and an e lectron ? Give your answer both in atomic mass units and as a percentage of the mass of the hydrogen atom.

b. By how much does the mass decrease when a helium nucleus is formed from two protons and two nelltrons? Give

Problems 1021

your answer both in alOmic mass units and as a percentage of the mass of the hel ium nuclells.

c. Compare your answers to parts a and b. Why do you hear it said that mass is "'05( ' in nuclear react ions but not in chem­ical react ions?

49. III Use the graph of binding energy of Figure 30.6 to es timate the total energy released if a nucleus with mass number 240 fi s­sions into two nuclei with mass number 120.

50. III Could a 56Fe nucleus fiss ion into two 28 AI nucle i? Your

answer, which should include some calculations, should be based on the cu rve of binding energy of Figure 30.6.

5 1. III a. What arc the isotopic symbols of all isotopes in Appendix D wi th A = 17?

b. Which ofthese are stab le nuclei? c. For those that are not stable, identify both the decay mode

and the daughter nucleus. 52. III What is the activity in 8q and in C i o f a 2.0 mg sample of 3H? 53. The activi ty of a sample of the ces ium isotope 137Cs is

2.0 X 108 8q. Many years later, after the sample has full y decayed, how many beta particles will have been emilled?

54. III A il S mCi sample of a rad ioactive isotope is made in a reac­tor. When delivered to a hospita l 16 hours later, its act iv ity is 95 mCi. The lowest usable activi ty level is 10 mCi. a. What is the isotope's half-life ? b. For how long a fter de li very is the sample usable?

55. III You are ass isting in an anthropology lab over the summe r by BID carry ing out 14C dating. A graduate slUdent found a bone he

believes to be 20,000 years o ld . You ex tract the carbon from the bone and prepare an equal-mass sample o f carbon from modern organic material. To determine the activity of a sample with the accuracy your supervi sor demands. you need to measure the time it takes for 10,000 decays 10 occur. a. The act ivity of the modern sample is 1.06 Bq. How long

does that measurement lake? b. It turns out that the graduate student 's estimate of the bone's

age was accu rate . How long docs it take to measure the act ivity of the ancient carbon?

56. 1111 A sample o f wood from an archaeo log ical excavation is BID dated by using a mass spectrometer to measure the fraction of

14C atoms. Suppose 100 atoms of 14C are found for every 1.0 X 10 15 atoms of 12C in the sample. What is the wood 's age?

57. II A sample of 1.0 X 1010 atoms that decay by alpha emi ss ion has a half-life of 100 min. How many alpha particles are emit­ted between I = 50 min and I = 200 min?

58. II A sample conta ins radioactive atoms of two types, A and B. InitialJy there are five times as many A atoms as there me B atoms. Two hours later. the numbers of the lwo atoms are equal. The half-l ife of A is 0.50 hOllrs. What is the half-l ife of B?

59. 1111 The technique known as potass ium­argon dating is used to date old lava nows and thus any fossili zed skeletons found in them, like thi s 1.8-million-year o ld hominid sku ll. The potassium iso­tope --10K has a 1.28-bi llion-year ha lf- life and is naturall y present at ve ry low lev­e ls. 40K decays by beta emiss ion into the stable isotope 4OAr. Argon is a gas. and the re is no argon in flowin g lava because the gas escapes. Once the lava so lidifies. any argon produced in the decay of 40K is trapped inside and cannot escape. What is the age of a piece of solidified lava with a 40 ArfOK rat io of 0. 12?

1022 CHAPTER 30 Nuclear Physics

60. 1111 Corals take lip certain elements from seawater, includ ing ura­BID nium but not thorium. After the corals die, the uranium isotopes

slowly decay into thorium isotopes. A measurement of the rela­tive frac tion of certain isotopes therefore provides a determina­tion of the coral's age. A compl icating factor is that the thorium isotopes decay as we ll . One scheme uses the alpha decay of 234U to 2300yh. After a long lime, the two species reach an equ i­librium in which the number of 234U decays per second (each producing an alom o f 2300yh) is exactl y eq ual to the num ber o f n u.yh decays per second. What is the relative concentrat ion of the two isotopes- the ratio of 2J4U to 2311"h_ when thi s equ ili b­

rium is reached? 61. III All the very heavy atoms found in the earth were created

long ago by nuclear fusion reactions in a supernova, an explod­ing star. The debris spewed out by the supernova later coa­lesced to form the sun and the planets of our solar system. Nuclear physics suggests that the uranium isotopes 235 U (t l l2 =

7.04 X 108 yr) and 238U (/1/2 = 4.47 X 109 yr) should have been created in roughl y equal amounts. Today, 99.2S% of uranium is 238U and 0.72% is 235U. How long ago did the supernova occur?

62. III 235 U decays to 207Pb via the decay seri es shown in Figure 30.16. The first decay in the cha in , that of 235 U, has a half-l ife of 7.0 X 108 years. The subsequent decays are much more rapid, so we can take thi s as the half-l ife for the decay o f 235 U to 207Pb. 238U decays, with a half- life of 4.5 X 109 years, via a s imilar decay series that e nds in a different lead isotope, 206Pb . Aga in , the subsequent decays are much more rapid, so we can take thi s as the half- life for the decay o f 238U to 2U6Pb. The two uranium

decay chains can be used to precisely detennine the age of celta in minerals that exclude lead from their crystal structure but easily incorporate uranium. When these minerals form , they conta in both 238U and 235U, but no lead. As time goes o n, the isotopes of

uranium decay, producing isotopes of lead. A measurement of the 238UP06Pb ratio allows a determination of the age- which can be checked using a measuremen t of the 235Upo7Pb ratio. If 75% of the 235U present in a particular rock has decayed to lead, what percent of Lhe 23IiU has decayed to lead?

63. II A 75 kg pat ie nt swallows a 30 ,uCi beta emitter with a half­BID life of 5.0 days. The beta particles are emitted with an average

energy of 0.35 MeV. Ninety percent o f the beta particles are absorbed within the pati ent's body and 10% escape. What dose equi valent does the patien t receive?

64. III About 12% of your body mass is carbon; some of thi s is BID radioactive 14C, a beta-emitter. If you absorb 100% of Lhe 49 keV INT energy of each 14C decay, what dose equi valem in Sv do you

receive each year from the 14C in your body? 65. 11111 Ground beef may be irrad iated with high-energy e lectrons BID from a linear accelerator to kill pathogens. In a standard treat­INT men t, 1.0 kg of beef rece ives 4.5 kGy of radiarion in 40 s.

a. How much energy is deposited in the beef? b. What is the average rate (in W) of energy deposition? c. Est imate the temperature increase of the beef due to this

procedure . The specific heat o f beef is approximately 3/4 of that of water.

66. III A 70 kg human body typ ical.l y contains 140 g of potass ium. BID Potass ium has a chemical atomic mass of 39.1 u and has three

natu ra Uy occurring isotopes. O ne of those isotopes, 4O K, is rad ioact ive with a half- life of 1.3 billio n years and a natural abunda nce o f 0.012% . Each 4°K decay deposits, on average, 1.0 MeV of ene rgy into the body. What yearly dose in Gy does the typ ical person rece ive from the decay of 40K in the body?

67. 1111 What dose in rads of gamma radiation must be absorbed by a INT block of ice at O°C to transform the entire block to li quid wate r

at O°C? 6S. II A chest x ray uses 10 keV photons. A 60 kg person receives a BID 30 mrem dose from one x ray that exposes 25% of the patient 's

body. How many x-ray photons are absorbed in the patient 's body?

69. 11 11 The pluto nium isoto pe 239pU has a half-life of 24,000 years BID and decays by the emiss io n of a 5.2 MeV alpha part icle. Pluto­

nium is not espec ially dangerous if handled because the act ivity is low and the alpha rad ia ti on doesn ' t penetrate the skin . But the

t iniest speck o f plutonium can cause problems if it is inhaled and lodges deep in the lungs. Let' s see why. a. Soot particles are roughly I ,urn in diameter, and it is kn own

that these part icles can go deep into the lungs. How man y 239pU atoms are in a 1.0-,um-diameter particle of 239 pU? The

de nsity Ofp[ulo nium is 19,5OO kg/m3. b. What is the activity, in Bq, of this [.O-,um-diameter particle? c. The activity of the particle is very small , but the pe netrat ing

power o f alpha part icles is also very small , so the damage is concentrated . The al pha particles deposit their e nergy in a 50-,um-diameter sphere around the pluto nium part icle. In one year, what is the dose equi valent in mS v to thi s smaiJ sphere o f ti ssue in the lungs? Assume Lhat the ti ssue density is that of water.

d. How does the exposure to thi s ti ssue compare to the natural background exposure?

70. III Uranium is naturally present at low levels in many so ils and BID rocks. The 238 U decay series includes the short- li ved radon iso­

tope 222 Rn, with '112 = 3.82 days. Radon is a gas, and it can seep into basements. The Environ menta[ Protection Agency recom­me nds that homeowners take steps to remove radon if the radon acti vity exceeds 4 pC i per [iter of air. The daughter nucle i from rado n decay are of s ignifi cant concern , but the radon itse lf does prov ide some exposure. a. How many m Rn atoms are there in 1.0 m] of air if the

activ ity is 4.0 pCi/L? b. The range of a lpha pan icles in air is 3 cm . Let's model a

person as a ISO-cm-tall , 25-cm-diamete r cyli nde r wi th a mass of 65 kg. Only decays within 3 cm of the cyl inder cause exposure, and onl y 50% of the decays direct the al pha particle toward the person. What is the dose equi va len t ( in mSv) for a person who spends an ent ire year in a room where the activity is 4 pC i/L?

Passage Problems

Nuclear Fission

The uranium isotope mU can fissio n-break into two smaUer-mass components and free neutrons-if it is struck by a free ncutron. A typical reaction is

As YOLI can see, the subsc ripts (the number of protons) and the super­scripts (the number of nucleons) "balance" before and after the fis­sion event ; there is no change in the number of protons or neutrons. Significant energy is released in thi s reaclion. IF a fi ssion event hap­pens in a large chunk of ~mU~ the neutrons released may induce the fi ss ion of other m U aloms, resulting in a chain react ion. This is how a nuclear reactor works.

The number of neutrons required to create a stable nucleus increases with atomic number. When the heavy 235U nucleus fi s­s ions, the lighter react ion products are thus neutron rich and are likely unstable. Man y of the short- li ved radioacti ve nuclei used in medic ine are produced in fi ss ion reactions in nuclear reactors.

Stop to Think 30.1: Three. Different isotopes of an element have different numbers of neutrons but the same number o f proton s. The number of electrons in a neutral ato m matc hes the number of protons.

Stop to Think 30.2: Yes. 12C has fill ed levels of protons and ne u­trons; the neutron we add to make I1C will be in a higher energy level, but there is no " hole" in a lower level for it to move to, so we expect thi s nucleus to be stable.

Problems 1023

7 1. I What statement can be made about the masses of atoms in the above react ion?

A. me~~U) > ",C ~~Ba) + m(~Kr) + 211/(0n) B. me~~U) < l1Ie~~Ba) + m(j~Kr) + 2mCbn) C. l1/e~~ U) = mC~~Ba) + m(~Kr) + 2m(6n) D. me~~U) = m(' ;~Ba) + m(~~Kr) + 3mCbn)

72. I Because the decay products in the above fission reaction are neutron rich , they will likely decay by what process? A. Alpha decay B. Beta decay C. Gamma decay

73. I 235 U is radioactive , with a long haJr-l ife of 704 million years. The decay products o f a 235U fi ss io n reaction typicaLly have half-lives of a few minutes. This means that the decay products o f a fi ssion react ion have A. Much higher activity than the original uranium. B. Much lower activity than the original uranium. C. The same acti vity as the original uranium.

74. I If a 2~~U nucleus is struck by a neutron , it Illay absorb the neutron. The resulting nucle us then rapidl y undergoes beta­minus decay. The daughter nucleus of that decay is A. 2ijjPa B. 2ij~U C. 2~~Np D. 2ij~ Pu

Stop to Think 30.3: B. An increase o f Z with no change in A occurs when a neutron changes to a proton and an electron, ejecting the electron .

Stop to Think 30.4: C. One-quarter of the ato ms are left. This is o ne-half of one-half, or C 112)2 , so two half-lives have elapsed.

Stop to Think 30.5: A. Dose equivalent is the product o f dose in Gy (the same for each) and RBE (highest for alpha particles).