College Physics 2013

Eugenia Etkina, Michael Gentle, Alan Van Heuvelen

Chapter 28

Nuclear Physics

Educators

Chapter Questions

Problem 1

Estimate the magnitude of the repulsive electrical force between two protons in a helium nucleus. How are they held together in the nucleus?

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Problem 2

Estimate the magnitude of the repulsive electrical force that the rest of a gold nucleus $(79 \mathrm{Au})$ exerts on a proton near the edge of the nucleus. State any assumptions used in your estimation. This is an order-of-magnitude estimate, so do not become bogged down in details.

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Problem 3

Determine the number of protons, neutrons, and nucleons in the following nuclei;$_{4}^{9} \mathrm{Be},_{8}^{16} \mathrm{O},_{13}^{27} \mathrm{Al}, 26 \mathrm{Fe},^{64} \mathrm{Zn},$ and $^{107} \mathrm{Ag}$

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Problem 4

Suppose the radius of a copper nucleus is 10-14 m and that the radius of the copper atom is about 0.1 nm. Estimate the atom’s size if the nucleus was the size of a tennis ball.

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Problem 5

Estimate the total number of (a) nucleons and (b) electrons in your body. (c) In- dicate roughly the volume in cubic centimeters occupied by these nucleons.

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Problem 6

Measurements with a mass spectrometer indicate the following particle masses: $_{3}^{7} \operatorname{Li}(7.016003 \mathrm{u}),_{1}^{1} \mathrm{H}(1.007825 \mathrm{u}),$ and $_{0}^{1} \mathrm{n}$ $(1.008665 \text { u). Compare the mass of the lithium atom to the }$mass of the particles of which it is made. What do you conclude? Note: $1 \mathrm{u}=1.660539 \times 10^{-27} \mathrm{kg} .$

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Problem 7

A 5.5-g marble initially at rest is dropped in the Earth’s gravitational field. How far must it fall before its decrease in gravitational potential energy is 938 MeV, the same as the rest mass energy of a proton?

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Problem 8

Determine the rest mass energies of an electron, a proton, and a neutron in units of mega-electron volts.

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Problem 9

Use Figure 28.6 to estimate the total binding energy of $_{79}^{197} \mathrm{Au}$

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Problem 10

Determine the total binding energy and the binding energy per nucleon of $_{6}^{12} \mathrm{C}$ .

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Problem 11

Determine the binding energies per nucleon for $^{238}_{92} \mathrm{U}$ and $^{120}_{50} \mathrm{Sn}$. Based on these numbers, which nucleus is more stable? Explain.

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Problem 12

Determine the energy that is needed to remove a neutron from $_{3}^{7} \mathrm{Li}$ to produce $_{3}^{6}$ Li plus a free neutron. [Hint: Compare the mass of $_{3}^{7}$ Li and that of $_{3}^{6} \mathrm{Li}+^{1}_{0} \mathrm{n} ].$

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Problem 13

Insert the missing symbol in the following reactions.
(a) $_{2}^{4} \mathrm{He}+_{6}^{12} \mathrm{C} \rightarrow_{7}^{15} \mathrm{N}+?$
(b) $_{1}^{2} \mathrm{H}+_{1}^{3} \mathrm{H} \rightarrow_{2}^{4} \mathrm{He}+?$
(c) $^{1}_{0} \mathrm{n}+235 \mathrm{U} \rightarrow_{54}^{140} \mathrm{Xe}+?+2_{0}^{1} \mathrm{n}$
(d) $_{1}^{3} \mathrm{H} \rightarrow ?+_{-1}^{0} e$

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Problem 14

Explain why the following reactions violate one or more of the rules for nuclear reactions.
(a) $_{2}^{4} \mathrm{He}+_{13}^{27} \mathrm{Al} \rightarrow_{15}^{32 \mathrm{P}}+_{0}^{1} \mathrm{n}$
(b) $_{1}^{2} \mathrm{H}+_{1}^{3} \mathrm{H} \rightarrow_{2}^{4} \mathrm{He}+_{1}^{1} \mathrm{H}$
(c) $^l{1}_{0} \mathrm{n}+_{94}^{238} \mathrm{Pu} \rightarrow_{54}^{140} \mathrm{Xe}+_{40}^{96} \mathrm{Zr}+2_{0}^{1} \mathrm{n}$
(d) $^{14}_{6} \mathrm{C} \rightarrow \stackrel{14}{7 \mathrm{N}}+_{\mathrm{i}}^{0} e$

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Problem 15

Explain why the following reaction does not occur spontaneously: $_{2}^{4} \mathrm{He} \rightarrow_{1}^{3} \mathrm{H}+_{1}^{1} \mathrm{H}$

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Problem 16

The following reaction occurs in the Sun: $_{2}^{3} \mathrm{He}+_{2}^{4} \mathrm{He} \rightarrow_{4}^{7} \mathrm{Be}$ How much energy is released?

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Problem 17

Nuclear reaction in the Sun Oxygen is produced in stars by the following reaction: $^{12}_6 \mathrm{C}+_{2}^{4} \mathrm{He} \rightarrow_{8}^{16} \mathrm{O}$ . How much energy is absorbed or released by the reaction in units of mega-elec- tron volts?

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Problem 18

Another reaction in the Sun One part of the carbon-nitrogen cycle that provides energy for the Sun is the reaction $_{6}^{12} \mathrm{C}+_{1}^{1} \mathrm{H} \rightarrow_{7}^{13} \mathrm{N}+1.943 \mathrm{MeV}$ Using the known masses of $^{12} \mathrm{C}$ and $^{1} \mathrm{H}$ and the results of this reaction, determine the mass of $^{13} \mathrm{N}$ .

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Problem 19

Determine the missing nucleus in the following reaction and calculate how much energy is released: $^{232}_{92} \mathrm{U} \rightarrow ?+_{2}^{4} \mathrm{He}+$ energy

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Problem 20

Determine the missing nucleus in the following reaction and calculate its mass: ? $\rightarrow^{211}_{83} \mathrm{Bi}+_{2}^{4} \mathrm{He}+8.20 \mathrm{MeV}$

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Problem 21

Determine (a) the number of protons and neutrons in the missing fragment of the reaction shown below and (b) the mass of that fragment:
$$^{1}_{0} \mathrm{n}+^{255}_{92} \mathrm{U} \rightarrow ?+_{54}^{136} \mathrm{Xe}+12_{0}^{1} \mathrm{n}+126.5 \mathrm{MeV}$$

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Problem 22

A series of reactions in the Sun leads to the fusion of three helium nuclei $\left(_{2}^{4} \mathrm{He}\right)$ to form one carbon nucleus $\left(^{12}_{6} \mathrm{C}\right)$. (a) Determine the net energy released by the reactions. (b) What fraction of the total mass of the three helium nuclei is converted to energy?

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Problem 23

A series of reactions that provides energy for the Sun and stars is summarized by the following equation: $6_{1}^{2} \mathrm{H} \rightarrow 2_{1}^{1} \mathrm{H}+2 ^{1}_{0} \mathrm{n}+2_{2}^{4} \mathrm{He}$ ( a) Determine the net energy released by the reaction. (b) Convert this answer to units of joules per kilogram of deuterium $\left(_{1}^{2} \mathrm{H}\right)$.

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Problem 24

Equations for determining the mass defect for two nuclear reactions are shown below. Represent each reaction in symbolic form (as in the previous problems). Decide whether each reaction results in energy release or energy absorption.
(a) $235.0439 \mathrm{u}+1.0087 \mathrm{u} \rightarrow 95.9343 \mathrm{u}+137.9110 \mathrm{u}$ $+2(1.0087 \mathrm{u})+$ energy
(b) $3(4.002602 \mathrm{u}) \rightarrow 12.000000 \mathrm{u}+$ energy

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Problem 25

In 1913, Frederick Soddy collected the following data related to the radioactive transformation of uranium (Figure P28.25). The first product that appears in the sample is thorium, then protactinium, then another isotope of uranium, and so on. Examine the series of the transformation found by Soddy and explain using your knowledge of alpha and beta decays. Discuss what quantities are conserved in each
process.

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Problem 26

In the 1930s, Meitner, Hahn, and Strassmann did experiments irradiating uranium with neutrons. They predicted three possible outcomes:
Production of a new element (if the neutron undergoes beta decay in the nucleus).
Production of a heavier isotope of uranium (if the neutron stays in the nucleus).
Production of a slightly lighter nucleus (if the neutron captured by the nucleus causes one or two alpha particles to leave).
Instead, the nuclei produced in the reaction included isotopes of barium and other nuclei with about half the mass of uranium. How could they explain their findings?

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Problem 27

The following nuclei produced in a nuclear reactor each undergo radioactive decay. Determine the daughter nucleus formed by each decay reaction: (a) $^{239}_{94} \mathrm{Pu}$ alpha decay; (b) $^{144}_{58} \mathrm{Ce}$ beta-minus decay; (c) $^{129}_{53} \mathrm{I}$ beta-minus decay; and (d) $_{30}^{60} \mathrm{Zn}$ beta-plus decay.

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Problem 28

Potassium-40 $^{19}_{40} \mathrm{K}$ undergoes both beta-plus and beta-minus decay. Determine the daughter nucleus in each case.

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Problem 29

Radon- 222 $\left(\frac{222}{86} \mathrm{Rn}\right)$ is released into the air during uranium
mining and undergoes alpha decay to form ${ }_{84}^{218} \mathrm{Po}$ of mass 218.0090 u. Determine the energy released by the decay reaction. Most of this energy is in the form of alpha particle kinetic energy.

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Problem 30

Carbon- 11$(^{11}_{6} \mathrm{C})$ undergoes beta-plus decay. Determine the product of the decay and the energy released.

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Problem 31

(a) Determine the total binding energy of radium-226 $\left(^{226} _{88} \mathrm{Ra}\right) .$ (b) Determine and add together the binding energies of a radon- 222$(_{86}^{222} \mathrm{Rn})$ and an alpha particle. (c) Determine the difference of these numbers, which equals the energy released during alpha decay of $^{226} \mathrm{Ra}$

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Problem 32

The body contains about 7 $\mathrm{mg}$ of radioactive $_{19}^{40} \mathrm{K}$ that is absorbed with the foods we eat. Each second, about $2.0 \times 10^{3}$ of these potassium nuclei undergo beta decay (either beta-minus or beta-plus). About how many potassium nuclei are in the average body and what fraction decay each second?

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Problem 33

A radioactive $^{60}$ Co nucleus emits a gamma ray of wavelength $0.93 \times 10^{-12} \mathrm{m}$ . If the cobalt was initially at rest, use the conservation of momentum equation to determine its speed following the gamma ray emission.

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Problem 34

Design an experiment to determine whether $\mathrm{O}_{2}$ emitted from plants comes from $\mathrm{H}_{2} \mathrm{O}$ or from $\mathrm{CO}_{2},$ the basic input molecules that lead to plant growth.

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Problem 35

A radioactive sample initially undergoes $4.8 \times 10^{4}$ decays/s. Twenty-four hours later, its activity is $1.2 \times 10^{4}$ decays/s. Determine the half-life of the radioactive species in the sample.

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Problem 36

Cesium-137, a waste product of nuclear reactors, has a half- life of 30 years. Use two different methods to determine the fraction of $^{137}$ Cs remaining in a reactor fuel rod: (a) 120 years after it is removed from the reactor, (b) 240 years after, and (c) 1000 years after.

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Problem 37

A sample of radioactive technetium-99 of half-life 6 h is to be used in a clinical examination. The sample is delayed 15 h before arriving at the lab for use. Use two methods to determine the fraction of radioactive technetium that remains.

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Problem 38

If 120 mg of radioactive gold-198 with half-life 2.7 days is administered to a patient for radiation therapy, what is the gold-198 activity 3 weeks later if none is eliminated from the body by biological means?

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Problem 39

How many years are required for the amount of krypton-85 $(^{35}_{36} \mathrm{Kr})$ in a spent nuclear reactor fuel rod to be reduced by a factor of 1$/ 8 ? 1 / 32 ? 1 / 128 ?$ The half-life of $^{85} \mathrm{Kr}$ is 10.8 years.

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Problem 40

A tree sample was uprooted and buried 60,000 years ago during part of the Wisconsin glaciation. How many years after it was buried was the radioactive carbon-14 $\left(^{14} \mathrm{C}\right)$ in the root reduced by a factor of (a) $1 / 2,$ (b) $1 / 4,$ and $(\mathrm{c}) 1 / 8 ?$ (d) What fraction remained after 60,000 years? The carbon-14 was not replenished after the tree stopped growing.

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Problem 41

How many years are required for the amount of strontium-90 $\left(^{90} \mathrm{Sr}\right)$ released from a nuclear explosion in the atmosphere to be reduced by a factor of $(\mathrm{a}) 1 / 16,(\mathrm{b}) 1 / 64,$ and $(\mathrm{c}) 0.010 ?$

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Problem 42

A student accidentally swallows 0.10 $\mu g$ of iodine-131 while pipetting the radioactive material. (a) Determine the number of $^{131}\mathrm{I}$ atoms swallowed (the atomic mass of $^{131}\mathrm{I}$ is approximately 131 u). (b) Determine the activity of this material. The half-life of $^{131}\mathrm{I}$ is 8.02 days. (c) What is the mass of radioactive iodine-131 that remains 21 days later if none leaves the body?

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Problem 43

An unlabeled container of radioactive material has an activity of 90 decays/min. Four days later the activity is 72 decays/ min. Determine the half-life of the material. When will its activity reach 9 decays/min?

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Problem 44

To estimate the number of ants in a nest, 100 ants are removed and fed sugar made from radioactive carbon of a long half-life. The ants are returned to the nest. Several days later, of 200 ants taken from the nest, only 5 are radioactive. Roughly how many ants are in the nest? Explain your calculation technique.

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Problem 45

One gram of pure, radioactive radium produces 130 J of energy per hour due to radium decay only and has an activity of 1.0 Ci. Determine the average energy in electron volts released by each radioactive decay.

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Problem 46

A mallet found at an archeological excavation site has 1/16 the normal carbon-14 decay rate. Determine the mallet’s age.

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Problem 47

The $^{235} \mathrm{U}$ in a rock decays with a half-life of $7.04 \times 10^{8}$ years. A geologist finds that for each $^{235} \mathrm{U}$ now remaining in the rock, $2.6^{235} \mathrm{U}$ have already decayed to form daughter nuclei. Determine the age of the rock.

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Problem 48

A sample of water from a deep, isolated well contains only 30% as much tritium as fresh rainwater. How old is the water in the well?

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Problem 49

The decay rate of $^{14} \mathrm{C}$ from a bone uncovered at a burial site is 12.6 decays/min, whereas the decay rate from a fresh bone of the same mass is 1610 decays/min. Approximately how old is the uncovered bone?

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Problem 50

The tree sample described in Problem 40 contains 50 g of carbon when it is discovered. (a) If 1 in $10^{12}$ carbon atoms in a fresh tree sample are carbon-14, how many carbon-14 atoms. would be in 50 g of carbon from a fresh tree? (b) Calculate the carbon-14 activity of the sample. (c) Determine the age of the buried tree if its 50 g of carbon has an activity of $-2.2 \mathrm{s}^{-1}$

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Problem 51

A radioactive series different from those shown in Figures 28.14 and P28.25 begins with $^{90}_{232} \mathrm{Th}$ and undergoes the following series of decays: $\alpha \beta^{-} \beta^{-} \alpha \alpha \alpha \alpha \beta^{-} \alpha \beta^{-} .$ Determine each nucleus in the series.

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Problem 52

A 70-kg person receives a 250-mrad whole-body absorbed dose of radiation. (a) How much energy does the person absorb? (b) Is it better to absorb 250 mrad of X-rays or 250 mrad of beta rays? Explain. (c) What is the dose in each case?

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Problem 53

Determine the dose equivalent of a 70-mrad absorbed dose of the following types of radiation: (a) X-rays, (b) beta rays, (c) protons, (d) alpha particles, and (e) heavy ions.

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Problem 54

The yearly whole-body dose caused by radioactive $^{40} \mathrm{K}$ absorbed in our tissues is 17 mrem.(a) Assuming that $^{40} \mathrm{K}$ undergoes beta decay with an $\mathrm{RBE}$ of $1.4,$ determine the absorbed dose in rads. (b) How much beta ray energy does an 80 -kg person absorb in one year? [Note: $^{40} \mathrm{K}$ also emits gamma rays, many of which leave the body before being absorbed. Because fatty tissue has low potassium concentration and muscle has a higher concentration, gamma ray emissions indicate indirectly a person's fat content.]

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Problem 55

Determine the number of $^{40} \mathrm{K}$ nuclei in the body of an 80 -kg person using the information provided in the previous problem and the fact that $^{40} \mathrm{K}$ has a radioactive half-life of $1.28 \times 10^{9}$ years. Assume that each $^{40} \mathrm{K}$ beta decay results in 1.4 $\mathrm{MeV}$ of energy that is deposited in the person's tissue.

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Problem 56

During an X-ray examination a person receives a dose of 80 mrem in 4.0 kg of tissue. The RBE of X- rays is 1.0. (a) Determine the total energy absorbed by the 4.0kg of body tissue. (b) Determine the energy in joules of each 40,000-eV X-ray photon. (c) Determine the number of photons absorbed by that tissue.

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Problem 57

Estimate the temperature at which two protons can come close enough together to form an isotope of a helium nucleus.

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Problem 58

The mass of a helium nucleus is less than the mass of the nucleons inside it. (a) Explain how this observation led scientists to the idea that it is possible to convert hydrogen into helium to produce thermal energy. (b) Will this process mean that energy is not conserved? Explain. (c) Why do you think scientists need very high temperatures and high pressures for this reaction? (d) Estimate the temperature at which two protons will join together due to their nuclear attraction. Remember that nuclear forces are effective at distances less than or equal to $10^{-15} \mathrm{m} .[\text { Hint: Use an energy approach, not a force approach. }]$ (e) Suggest possible ways of containing ionized hydrogen to make the reaction possible (remember that all solid containers will melt at this temperature).

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Problem 59

World energy consumption in 2005 was about $4 \times 10^{20} \mathrm{J} .$ (a) Determine the number of deuterium nuclei $\left(_{1}^{2} \mathrm{H}\right)$ that would be needed to produce this energy. The fusion of two deuterium nuclei releases about 4 MeV of energy. (b) Determine the volume of water of density 1000 $\mathrm{kg} / \mathrm{m}^{3}$ needed to supply the energy. One mole of water has a mass of $18 \mathrm{g},$ and about one in every 6700 $_{1}^{1} \mathrm{H}$ atoms in water is a deuterium $$
{ }_{1}^{2} \mathrm{H.}
$

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Problem 60

Convert the 200 MeV per nucleus energy that is released by $^{235}\mathrm{U}$ fission to units of joules per kilogram. By comparison, the energy release by burning coal is $3.3 \times 10^{7} \mathrm{J} / \mathrm{kg}$ .

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Problem 61

(a) Determine the energy used by a 1200-W hair dryer in 10 $\min .$ (b) Approximately how many $^{235}$ U fissions must occur in a nuclear power plant to provide this energy if 35% of the energy released by fission produces electrical energy? The energy released per fission is about 200 MeV.

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Problem 62

Estimate the number of ${ }^{235} \mathrm{U}$ nuclei that must undergo fission to provide the energy to lift your body $1 \mathrm{~m} .$ Indicate any assumptions made.

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Problem 63

Suppose that a nuclear fission power plant and a coal-fired power plant both operate at 40% efficiency. Determine the ratio of the mass of coal that must be burned in 1 day of operating a 1000 -MW plant compared to the mass of $^{235} \mathrm{U}$ that must undergo fission in the same plant. The energy released by burning coal is $3.3 \times 10^{7} \mathrm{J} / \mathrm{kg}$ . The energy released per $^{235} \mathrm{U}$ fission is 200 $\mathrm{MeV} .$

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Problem 64

The world’s uranium supply is approximately $10^{9} \mathrm{kg}\left(10^{6} \text { tons), } 0.7 \% \text { of which is }^{235} \mathrm{U}\right.$ (a) How much energy is available from the fission of this $^{235}\mathrm{U}$? (b) The world’s energy consumption rate for production of electricity is about $10^{20} \mathrm{J} /$ year. At this rate, how many years would the uranium last if it were used to provide all our electrical energy?

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Problem 65

A radioactive sample contains two different types of radio-active nuclei: A, with half-life 5.0 days, and B, with half-life 30.0 days. Initially, the decay rate of the A-type nucleus is 64 times that of the B-type nucleus. When will their decay rates be equal?

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Problem 66

After a series of alpha and beta-minus decays, plutonium $-239$ $^{239}_{94}\mathrm{Pu}$ becomes lead- 207$(^{207}_{82} \mathrm{Pb}) .$ Determine the number of alpha and beta-minus particles emitted in the complete decay process. Explain your method for determining these numbers.

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Problem 67

Ion production due to ionizing radiation A body cell of $1.0 \times 10^{-5} \mathrm{m}$ radius and density 1000 $\mathrm{kg} / \mathrm{m}^{3}$ receives a radiation absorbed dose of 1 rad. (a) Determine the mass of the cell (assume it is shaped like a sphere). (b) Determine the energy absorbed by the cell. (c) If the average energy needed to produce one positively charged ion is 100 eV, determine the number of positive ions produced in the cell. (d) Repeat the procedure for the $3.0 \times 10^{-6}-$ -m-radius nucleus of the cell, where its genetic information is stored.

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Problem 68

Which answer below is closest to the Fukushima total person-rem exposure?
(a) $2 \times 10^{4}$ person $\cdot$ rem
(b) $2 \times 10^{6}$ person $\cdot$ rem
(c) $2 \times 10^{8}$ person $\cdot$ rem
(d) $2 \times 10^{10}$ person $\cdot$ rem

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Problem 69

Which answer below is closest to the number of statistically likely cancer deaths to be caused by this event?
(a) 10
(b) 100
(c) 1000
(d) $10,000$
(e) $100,000$

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Problem 70

About 4000 of the 2 million residents die each year from cancer caused by other factors, such as smoking. If those Fukushima accident cancer deaths occurred during the 5 years following the accident, which number below is closest to the ratio of Fukushima-caused cancer deaths to normal cancer deaths for the exposed population?
(a) 1/2000
(b) 1/20
(c) 1
(d) 20/1
(e) 200/1

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Problem 71

Which answer below is closest to the number of cancer deaths in the United States due to an estimated 20 mrad/year average radon exposure to each person?
(a) 50
(b) 500
(c) 5000
(d) 50,000
(e) 500,000

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Problem 72

Which is the most important reason why it is difficult statistically to identify cancer deaths due to radiation exposure from the Fukushima accident?
(a) An unknown number of people stayed in their homes following the accident.
(b) The radiation detection devices were inaccurate.
(c) The fraction of people who left the area was unknown.
(d) The normal cancer rate significantly exceeded the cancer caused by the accident.
(e) Authorities often distort information they provide about radiation exposure.

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Problem 73

In neutron activation analysis (NAA) a particular elemental concentration in a sample is determined by measuring which of the following?
(a) Alpha particles
(b) Beta particles
(c) Gamma rays
(d) Neutrons
(e) Two of these choices

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Problem 74

Which answer below is closest to the ratio of the relative amount of lutetium (Lu) in stone from Notre Dame compared to stone from Amiens Cathedral?
(a) 0.5
(b) 1
(c) 2
(d) 10
(e) 1000

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Problem 75

What does neutron absorption by sodium ( $_{11}^{23} \mathrm{Na} )$ produce?
(a) $_{11}^{22} \mathrm{Na}$
(b) $_{11}^{24} \mathrm{Na}$
(c) $_{20}^{40} \mathrm{Ca}$
(d) $_{9 \mathrm{K}}^{41} \mathrm{K}$
(e) $_{19}^{39} \mathrm{K}$

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Problem 76

Neutron absorption by potassium produces an excited $^{40}_{9} \mathrm{K}$ nucleus, which emits a beta-minus and a gamma ray. Which nucleus below is a product of this decay?
(a) $_{19}^{40} \mathrm{K}$
(b) $_{18}^{40} A r$
(c) $^{40}_{20} \mathrm{Ca}$
(d) $_{ 19 }^{41} \mathrm{K}$
(e) $_{19 \mathrm{K}}^{39}$

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Problem 77

Rank the elements Cr, Mn, and Na in stone samples from Notre Dame compared to stone from Amiens Cathedral as the best indicators for distinguishing the two types of stone (rated from best to worst).
(a) Cr, Mn, Na
(b) Mn, Cr, Na
(c) Na, Mn, Cr
(d) Mn, Na, Cr

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