Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 45

Applications of Nuclear Physics - all with Video Answers

Educators

Chapter Questions

Problem 1

Find the energy released in the fission reaction
$$_{0}^{1} n+\frac{235}{92} U\rightarrow \frac{98}{40} \mathrm{Zr}+\frac{135}{52} \mathrm{Te}+3\left(\begin{array}{l}{1} \\ {0}\end{array}\right)$$
The atomic masses of the fission products are 97.912735 $\mathrm{u}$
for $_{40}^{98} \mathrm{Zr}$ and 134.916450 u for $_{52}^{135} \mathrm{Te}$

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Ren Jie Tuieng

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Problem 2

Burning one metric ton $(1000 \mathrm{kg})$ of coal can yield an energy of $3.30 \times 10^{10} \mathrm{J}$ . Fission of one nucleus of uranium-235 yields an average of approximately 200 $\mathrm{MeV}$ . What mass of uranium produces the same energy in fission as burning one metric ton of coal?

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Problem 3

Strontium- 90 is a particularly dangerous fission product of $^{235} \mathrm{U}$ because it is radioactive and it substitutes for calcium in bones. What other direct fission products would accompany it in the neutron-induced fission of $^{255} \mathrm{U}$ ? Note: This reaction may release two, three, or four free neutrons.

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Problem 4

A typical nuclear fission power plant produces approximately 1.00 $\mathrm{GW}$ of electrical power. Assume the plant has an overall efficiency of 40.0$\%$ and each fission reaction produces 200 $\mathrm{MeV}$ of energy. Calculate the mass of $^{255} \mathrm{U}$ consumed each day.

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Problem 5

List the nuclear reactions required to produce $^{233} \mathrm{U}$ from $^{232} \mathrm{Th}$ under fast neutron bombardment.

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Problem 6

The following fission reaction is typical of those occurring in a nuclear electric generating station:
$$_{0}^{1} n+\frac{255}{92} U\rightarrow \quad 141 \mathrm{Ba}+\underset{36}{92} \mathrm{Kr}+3\left(\begin{array}{l}{1} \\ {0}\end{array}\right)$$
a) Find the energy released in the reaction. The masses of the products are 140.914 411 u for $^{141} \mathrm{Ba}$ and 91.926 156 u for 92 36 $\mathbf{K r}$ (b) What fraction of the initial rest energy of the system is transformed to other forms?

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Problem 7

Find the energy released in the fission reaction
$$_{0}^{1} \mathbf{n}+2 \underset{92}{92} \mathrm{U}\rightarrow_{38}^{88 \mathrm{Sr}}+\underset{186}{54} \mathrm{Xe}+12\left(\frac{1}{0} n\right)$$

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Problem 8

A 2.00 -MeV neutron is emitted in a fission reactor. If it loses half its kinetic energy in each collision with a moderator atom, how many collisions does it undergo as it becomes a thermal neutron, with energy $0.039 \mathrm{eV} ?$

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Problem 9

Review. Suppose seawater exerts an average frictional drag force of $1.00 \times 10^{5} \mathrm{N}$ on a nuclear-powered ship. The fuel consists of enriched uranium containing 3.40$\%$ of the fissionable isotope $\underset{92}{235} \mathrm{U},and the ship’s reactor has an efficiency of 20.0%. Assuming 200 MeV is released per fission event, how far can the ship travel per kilogram of fuel?

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Problem 10

Seawater contains 3.00 mg of uranium per cubic meter. (a) Given that the average ocean depth is about 4.00 km and water covers two-thirds of the Earth’s surface, estimate the amount of uranium dissolved in the ocean. (b) About 0.700% of naturally occurring uranium is the fissionable isotope 285 $\mathrm{U}$ Estimate how long the uranium in the oceans could supply the world’s energy needs at the current usage of $1.50 \times 10^{13} \mathrm{J} / \mathrm{s}$ . (c) Where does the dissolved uranium come from? (d) Is it a renewable energy source?

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Problem 11

If the reproduction constant is 1.00025 for a chain reaction in a fission reactor and the average time interval between successive fissions is 1.20 $\mathrm{ms}$ , by what factor does the reaction rate increase in one minute?

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Problem 12

A large nuclear power reactor produces approximately $3000 \mathrm{MW}$ of power in its core. Three months after a reactor is shut down, the core power from radioactive by-products is $10.0 \mathrm{MW}$. Assuming each emission delivers $1.00 \mathrm{MeV}$ of energy to the power, find the activity in becquerels three months after the reactor is shut down.

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Problem 13

The probability of a nuclear reaction increases dramatically when the incident particle is given energy above the “Coulomb barrier,” which is the electric potential energy of the two nuclei when their surfaces barely touch. Compute the Coulomb barrier for the absorption of an alpha particle by a gold nucleus.

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Problem 14

To minimize neutron leakage from a reactor, the ratio of the surface area to the volume should be a minimum. For a given volume $V,$ calculate this ratio for (a) a sphere, (b) a cube, and (c) a parallelepiped of dimensions $a \times a \times 2 a$ . (d) Which of these shapes would have minimum leakage? Which would have maximum leakage? Explain your answers.

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Problem 15

A particle cannot generally be localized to distances much smaller than its de Broglie wavelength. This fact can be taken to mean that a slow neutron appears to be larger to a target particle than does a fast neutron in the sense that the slow neutron has probabilities of being found over a larger volume of space. For a thermal neutron at room temperature of 300 K, find (a) the linear momentum and (b) the de Broglie wavelength. (c) State how this effective size compares with both nuclear and atomic dimensions.

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Problem 16

Why is the following situation impossible? An engineer working on nuclear power makes a breakthrough so that he is able to control what daughter nuclei are created in a fission reaction. By carefully controlling the process, he is able to restrict the fission reactions to just this single possibility: the uranium-235 nucleus absorbs a slow neutron and splits into lanthanum-141 and bromine-94. Using this break-
through, he is able to design and build a successful nuclear reactor in which only this single process occurs.

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Problem 17

According to one estimate, there are $4.40 \times$ $10^{6}$ metric tons of world uranium reserves extractable at $\$ 130 / \mathrm{kg}$ or less. We wish to determine if these reserves are sufficient to supply all the world's energy needs. About 0.700$\%$ of naturally occurring uranium is the fissionable isotope $^{235} \mathrm{U} .$ (a) Calculate the mass of $^{235} \mathrm{U}$ in the reserve in grams. (b) Find the number of moles of $^{235} \mathrm{U}$ in the reserve. (c) Find the number of $^{235} \mathrm{U}$ nuclei in the reserve. (d) Assuming 200 MeV is obtained from each fission reaction and all this energy is captured, calculate the total energy in joules that can be extracted from the reserve. (e) Assuming the rate of world power consumption remains constant at $1.5 \times 10^{13} \mathrm{J} / \mathrm{s}$ , how many years could the uranium reserve provide for all the world's energy needs? (f) What conclusion can be drawn?

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Problem 18

An all-electric home uses 2000 kWh of electric energy per month. Assuming all energy released from fusion could be captured, how many fusion events described by the reaction $_{1}^{2} \mathrm{H}+_{1}^{3} \mathrm{H} \rightarrow_{2}^{4} \mathrm{He}+_{0}^{1} \mathrm{n}$ would be required to keep this home running for one year?

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Problem 19

When a star has exhausted its hydrogen fuel, it may fuse other nuclear fuels. At temperatures above $1.00 \times 10^{8} \mathrm{K}$ , helium fusion can occur. Consider the following processes. (a) Two alpha particles fuse to produce a nucleus $A$ and a gamma ray. What is nucleus $A$ ? (b) Nucleus $A$ from part (a) absorbs an alpha particle to produce nucleus $B$ and a gamma ray. What is nucleus $B$ ? (c) Find the total energy released in the sequence of reactions given in parts (a) and (b).

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Problem 20

Find the energy released in the fusion reaction
$$_{1}^{1} \mathrm{H}+_{1}^{2} \mathrm{H} \rightarrow_{2}^{3} \mathrm{He}+\gamma$$

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Problem 21

(a) Consider a fusion generator built to create 3.00 GW of power. Determine the rate of fuel burning in grams per hour if the D–T reaction is used. (b) Do the same for the D–D reaction, assuming the reaction products are split evenly between $\left(\mathrm{n},^{3} \mathrm{He}\right)$ and $\left(\mathrm{p},^{3} \mathrm{H}\right)$

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Problem 22

Two nuclei having atomic numbers $Z_{1}$ and $Z_{2}$ approach each other with a total energy $E .$ (a) When they are far apart, they interact only by electric repulsion. If they approach to a distance of $1.00 \times 10^{-14} \mathrm{m},$ the nuclear force suddenly takes over to make them fuse. Find the minimum value of $E,$ in terms of $Z_{1}$ and $Z_{2},$ required to produce fusion. (b) State how $E$ depends on the atomic numbers. (c) If $Z_{1}+Z_{2}$ is to have a certain target value such as $60,$ would it be energetically favorable to take $Z_{1}=1$ and $Z_{2}=59,$ or $Z_{1}=Z_{2}=30,$ or some other choice? Explain your answer. (d) Evaluate from your expression the minimum energy for fusion for the $\mathrm{D}-\mathrm{D}$ and $\mathrm{D}-$ T reactions (the first and third reactions in Eq. $45.4 ) .$

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Problem 23

Of all the hydrogen in the oceans, 0.0300$\%$ of the mass is deuterium. The oceans have a volume of 317 million $\mathrm{mi}^{3} .$ (a) If nuclear fusion were controlled and all the deuterium in the oceans were fused to $_{2}^{4} \mathrm{He}$ , how many joules of energy would be released? (b) What If? World power consumption is approximately 1.50 $\times 10^{13} \mathrm{W}$ . If consumption were 100 times greater, how many years would the energy calculated in part (a) last?

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Problem 24

Consider the deuterium–tritium fusion reaction with the tritium nucleus at rest:
$$_{1}^{2} \mathrm{H}+_{1}^{3} \mathrm{H} \rightarrow_{2}^{4} \mathrm{He}+_{0}^{1} \mathrm{n}$$
(a) Suppose the reactant nuclei will spontaneously fuse if their surfaces touch. From Equation 44.1, determine the required distance of closest approach between their centers. (b) What is the electric potential energy (in electron volts) at this distance? (c) Suppose the deuteron is fired straight at an originally stationary tritium nucleus with just enough energy to reach the required distance of closest approach. What is the common speed of the deuterium and tritium nuclei, in terms of the initial deuteron speed $v_{i},$ as they touch? (d) Use energy methods to find the minimum initial deuteron energy required to achieve fusion. (e) Why does the fusion reaction actually occur at much lower deuteron energies than the energy calculated in part (d)?

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Problem 25

To understand why plasma containment is necessary, consider the rate at which an unconfined plasma would be lost. (a) Estimate the rms speed of deuterons in a plasma at a temperature of $4.00 \times 10^{8} \mathrm{K} .$ (b) What If? Estimate the order of magnitude of the time interval during which such a plasma would remain in a 10.0 -cm cube if no steps were taken to contain it.

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Problem 26

It has been suggested that fusion reactors are safe from explosion because the plasma never contains enough energy to do much damage. (a) In $1992,$ the TFTR reactor, with a plasma volume of approximately $50.0 \mathrm{m}^{3},$ achieved an ion temperature of $4.00 \times 10^{8} \mathrm{K}$ , an ion density of $2.00 \times 10^{13} \mathrm{cm}^{-3},$ and a confinement time of 1.40 s. Calculate the amount of energy stored in the plasma of the TFTR reactor. (b) How many kilograms of water at $27.0^{\circ} \mathrm{C}$ could be boiled away by this much energy?

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Problem 27

To confine a stable plasma, the magnetic energy density in the magnetic field (Eq. 32.14) must exceed the pressure 2$n k_{\mathrm{B}} T$ of the plasma by a factor of at least $10 .$ In this problem, assume a confinement time $\tau=1.00 \mathrm{s}$ . (a) Using Lawson's criterion, determine the ion density required for the D-T reaction. (b) From the ignition-temperature criterion, determine the required plasma pressure. (c) Determine the magnitude of the magnetic field required to contain the plasma.

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Problem 28

Another series of nuclear reactions that can produce energy in the interior of stars is the carbon cycle first proposed by Hans Bethe in 1939, leading to his Nobel Prize in Physics in 1967. This cycle is most efficient when the central temperature in a star is above $1.6 \times 10^{7} \mathrm{K}$ . Because the temperature at the center of the Sun is only $1.5 \times 10^{7} \mathrm{K}$ , the following cycle produces less than 10$\%$ of the Sun's energy. (a) A high-energy proton is absorbed by $^{12} \mathrm{C}$ . Another nucleus, $A,$ is produced in the reaction, along with a gamma ray. Identify nucleus $A .$ (b) Nucleus $A$ decays through positron emission to form nucleus $B$ . Identify nucleus $B$ (c) Nucleus $B$ absorbs a proton to produce nucleus C and a gamma ray. Identify nucleus $C$ . (d) Nucleus C absorbs a proton to produce nucleus $D$ and a gamma ray. Identify nucleus $D .$ (e) Nucleus $D$ decays through positron emission to produce nucleus $E .$ Identify nucleus $E .$ (f) Nucleus $E$ absorbs a proton to produce nucleus $F$ plus an alpha particle. Identify nucleus $F$ . (g) What is the significance of the final nucleus in the last step of the cycle outlined in part (f)?

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Problem 29

A particular radioactive source produces 100 mrad of 2.00-MeV gamma rays per hour at a distance of 1.00 m from the source. (a) How long could a person stand at this distance before accumulating an intolerable dose of 1.00 rem? (b) What If? Assuming the radioactive source is a point source, at what distance would a person receive a dose of 10.0 mrad/h?

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Problem 30

Assume an x-ray technician takes an average of eight x-rays per workday and receives a dose of 5.0 rem/yr as a result. (a) Estimate the dose in rem per x-ray taken. (b) Explain how the technician’s exposure compares with low-level background radiation.

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Problem 31

When gamma rays are incident on matter, the intensity of the gamma rays passing through the material varies with depth $x$ as $I(x)=I_{0} e^{-\mu x}$ , where $I_{0}$ is the intensity of the radiation at the surface of the material (at $x=0 )$ and $\mu$ is the linear absorption coefficient. For 0.400 -MeV gamma rays in lead, the linear absorption coefficient is 1.59 $\mathrm{cm}^{-1}$. (a) Determine the “half-thickness” for lead, that is, the thickness of lead that would absorb half the incident gamma rays. (b) What thickness reduces the radiation by a factor of $10^{4} ?$

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Problem 32

When gamma rays are incident on matter, the intensity of the gamma rays passing through the material varies with depth $x$ as $I(x)=I_{0} e^{-\mu x}$ , where $I_{0}$ is the intensity of the radiation at the surface of the material (at $x=0 )$ and $\mu$ is the linear absorption coefficient. (a) Determine the "half-thickness" for a material with linear absorption coefficient $\mu,$ that is, the thickness of the material that would absorb half the incident gamma rays. (b) What thickness changes the radiation by a factor of $f ?$

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Problem 33

The danger to the body from a high dose of gamma rays is not due to the amount of energy absorbed; rather, it is due to the ionizing nature of the radiation. As an illustration, calculate the rise in body temperature that results if a “lethal” dose of 1 000 rad is absorbed strictly as internal energy. Take the specific heat of living tissue as 4186 $\mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}$.

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Problem 34

Why is the following situation impossible? A “clever” technician takes his 20-min coffee break and boils some water for his coffee with an x-ray machine. The machine produces 10.0 rad/s, and the temperature of the water in an insulated cup is initially 50.0°C.

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Problem 35

A small building has become accidentally contaminated with radioactivity. The longest-lived material in the building is strontium- 90 ($_{38}^{90} \mathbf{Sr}$ has an atomic mass 89.9077 $\mathrm{u}$ and its half-life is 29.1 yr. It is particularly dangerous because it substitutes for calcium in bones.) Assume the building initially contained 5.00 kg of this substance uniformly distributed throughout the building and the safe level is defined as less than 10.0 decays/min (which is small compared with background radiation). How long will the building be unsafe?

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Problem 36

Technetium-99 is used in certain medical diagnostic procedures. Assume $1.00 \times 10^{-8} \mathrm{g}$ of $^{99} \mathrm{Tc}$ is injected into a 60.0-kg patient and half of the 0.140-MeV gamma rays are absorbed in the body. Determine the total radiation dose received by the patient.

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Problem 37

To destroy a cancerous tumor, a dose of gamma radiation with a total energy of 2.12 J is to be delivered in 30.0 days from implanted sealed capsules containing palladium-103. Assume this isotope has a half-life of 17.0 d and emits gamma rays of energy 21.0 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 these “seeds” should contain.

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Problem 38

Strontium- 90 from the testing of nuclear bombs can still be found in the atmosphere. Each decay of 90 $\mathrm{Sr}$ releases 1.10 $\mathrm{MeV}$ of energy into the bones of a person who has had strontium replace his or her body's calcium. Assume a 70.0 -kg person receives 1.00 $\mathrm{ng}$ of 90 $\mathrm{Sr}$ from contaminated milk. Take the half-life of 90 $\mathrm{Sr}$ to be 29.1 yr. Calculate the
absorbed dose rate (in joules per kilogram) in one year.

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Problem 39

In a Geiger tube, the voltage between the electrodes is typically 1.00 $\mathrm{kV}$ and the current pulse discharges a $5.00-\mathrm{pF}$ capacitor. (a) What is the energy amplification of this device for a 0.500 -MeV electron? (b) How many electrons participate in the avalanche caused by the single initial electron?

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Problem 40

(a) A student wishes to measure the half-life of a radioactive substance using a small sample. Consecutive clicks of her Geiger counter are randomly spaced in time. The counter registers 372 counts during one 5.00 -min interval and 337 counts during the next 5.00 min. The average background rate is 15 counts per minute. Find the most probable value for the half-life. (b) Estimate the uncertainty in the half-life determination in part (a). Explain your reasoning.

Victor Salazar

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Problem 41

When gamma rays are incident on matter, the intensity of the gamma rays passing through the material varies with depth $x$ as $I(x)=I_{0} e^{-\mu x},$ where $I_{0}$ is the intensity of the radiation at the surface of the material (at $x=0 )$ and $\mu$ is the linear absorption coefficient. For low-energy gamma rays in steel, take the absorption coefficient to be 0.720 $\mathrm{mm}^{-1}$ . (a) Determine the "half-thickness" for steel, that is, the thickness of steel that would absorb half the incident gamma rays. (b) In a steel mill, the thickness of sheet steel passing into a roller is measured by monitoring the intensity of gamma radiation reaching a detector below the rapidly moving metal from a small source immediately above the metal. If the thickness of the sheet changes from 0.800 $\mathrm{mm}$ to $0.700 \mathrm{mm},$ by what percentage does the gamma-ray intensity change?

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Problem 42

A method called neutron activation analysis can be used for chemical analysis at the level of isotopes. When a sample is irradiated by neutrons, radioactive atoms are produced continuously and then decay according to their characteristic half-lives. (a) Assume one species of radioactive nuclei is produced at a constant rate $R$ and its decay is described by the conventional radioactive decay law. Assuming irradiation begins at time $t=0$ , show that the number of radio- active atoms accumulated at time $t$ is
$$
N=\frac{R}{\lambda}\left(1-e^{-\lambda t}\right)
$$
(b) What is the maximum number of radioactive atoms that can be produced?

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Problem 43

You want to find out how many atoms of the isotope 65 $\mathrm{Cu}$ are in a small sample of material. You bombard the sample with neutrons to ensure that on the order of 1$\%$ of these copper nuclei absorb a neutron. After activation, you turn off the neutron flux and then use a highly efficient detector to monitor the gamma radiation that comes out of the sample. Assume half of the 66 $\mathrm{Cu}$ nuclei emit a 1.04 -MeV gamma ray in their decay. (The other half of the activated nuclei decay directly to the ground state of $66 \mathrm{Ni} . )$ If after 10 $\mathrm{min}$ (two half-lives) you have detected $1.00 \times 10^{4} \mathrm{MeV}$ gamma ray in their decay. (The other half of the activated nuclei decay directly to the ground state of 66 $\mathrm{Ni}$ . If after 10 min (two half-lives) you have detected $1.00 \times 10^{4} \mathrm{MeV}$ of photon energy at $1.04 \mathrm{MeV},(\mathrm{a})$ approximately how many 65 $\mathrm{Cu}$ atoms are in the sample? (b) Assume the sample contains natural copper. Refer to the isotopic abundances listed in Table 44.2 and estimate the total mass of copper in the sample.

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Problem 44

A fusion reaction that has been considered as a source of energy is the absorption of a proton by a boron- 11 nucleus to produce three alpha particles:
$$
_{1}^{1} \mathrm{H}+\frac{11}{5} \mathrm{B} \rightarrow 3\left(\frac{4}{2} \mathrm{He}\right)
$$
This reaction is an attractive possibility because boron is easily obtained from the Earth's crust. A disadvantage is that the protons and boron nuclei must have large kinetic energies for the reaction to take place. This requirement contrasts with the initiation of uranium fission by slow neutrons. (a) How much energy is released in each reac-
tion? (b) Why must the reactant particles have high kinetic
energies?

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Problem 45

Review. A very slow neutron (with speed approximately equal to zero) can initiate the reaction
$$
\frac{1}{0} \mathbf{n}+\frac{10}{5} \mathbf{B} \rightarrow_{3}^{7} \mathbf{L} \mathbf{i}+_{2}^{4} \mathbf{H} \mathbf{e}
$$
The alpha particle moves away with speed $9.25 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Calculate the kinetic energy of the lithium nucleus. Use nonrelativistic equations.

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Problem 46

Review. The first nuclear bomb was a fissioning mass of plutonium- 239 that exploded in the Trinity test before dawn on July $16,1945,$ at Alamogordo, New Mexico. Enrico Fermi was 14 $\mathrm{km}$ away, lying on the ground facing away from the bomb. After the whole sky had flashed with unbelievable brightness, Fermi stood up and began dropping bits of paper to the ground. They first fell at his feet in the calm and silent air. As the shock wave passed, about 40 $\mathrm{s}$ after the explosion, the paper then in flight jumped approximately 2.5 $\mathrm{m}$ away from ground zero. (a) Equation 17.10 describes the relationship between the pressure amplitude $\Delta P_{\max }$ of a sinusoidal air compression wave and its displacement amplitude $s_{\max }$ . The compression pulse produced by the bomb explosion was not a sinusoidal wave, but let's use the same equation to compute an estimate for the pressure amplitude, taking $\omega \sim 1 \mathrm{s}^{-1}$ as an estimate for the angular frequency at which the pulse ramps up and down. (b) Find the change in volume $\Delta V$ of a sphere of radius 14 $\mathrm{km}$ when its radius increases by $2.5 \mathrm{m} .$ (c) The energy carried by the blast wave is the work done by one layer of air on the next as the wave crest passes. An extension of the logic used to derive Equation 20.8 shows that this work is given by $\left(\Delta P_{\max }\right)(\Delta V) .$ Compute an estimate for this energy. (d) Assume the blast wave carried on the order of one-tenth of the explosion's energy. Make an order-of-magnitude estimate of the bomb yield. (e) One ton of exploding TNT releases 4.2 $\mathrm{G} \mathrm{J}$ of energy. What was the order of magnitude of the energy of the Trinity test in equivalent tons of TNT? Fermi's immediate knowledge of the bomb yield agreed with that determined days later by analysis of elaborate measurements.

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Problem 47

Review. Consider a nucleus at rest, which then spontaneously splits into two fragments of masses $m_{1}$ and $m_{2}$ .
(a) Show that the fraction of the total kinetic energy carried by fragment $m_{1}$ is
$$
\frac{K_{1}}{K_{\mathrm{tot}}}=\frac{m_{2}}{m_{1}+m_{2}}
$$
and the fraction carried by $m_{2}$ is
$$
\frac{K_{2}}{K_{\mathrm{tot}}}=\frac{m_{1}}{m_{1}+m_{2}}
$$
assuming relativistic corrections can be ignored. A stationary 236 $\mathrm{U}$ nucleus fissions spontaneously into two primary fragments, $\frac{87}{33} \mathrm{Br}$ and $\frac{149}{57} \mathrm{La}$ . (b) Calculate the disintegration energy. The required atomic masses are 86.920711 u for $_{35}^{87} \mathrm{Br}, 148.934370 \mathrm{ufor}$ $_{57}^{149} \mathrm{La},$ and 236.045562 u for $_{92}^{236} \mathrm{U}$ (c) How is the disintegration energy split between the two primary fragments? (d) Calculate the speed of each fragment immediately after the fission.

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Problem 48

A fission reactor is hit by a missile, and $5.00 \times 10^{6} \mathrm{Ci}$ of $90 \mathrm{Sr},$
with half-life $29.1 \mathrm{yr},$ evaporates into the air. The strontium falls out over an area of $10^{4} \mathrm{km}^{2}$ . After what time interval will the activity of the "So Sr reach the agriculturally "safe"
level of 2.00$\mu \mathrm{Ci} / \mathrm{m}^{27}$

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Problem 49

The alpha-emitter plutonium- 238 $\left(_{94}^{238} \mathrm{Pu}\right.$ atomic mass $238.049560 \mathrm{u},$ half-life 87.7 $\mathrm{yr} )$ was used in a nuclear energy source on the Apollo Lunar Surface Experiments Package (Fig. P45.49). The energy source, called the Radioisotope Thermoelectric Generator, is the small gray object to the left of the gold-shrouded Central Station in the photograph. Assume the source contains 3.80 $\mathrm{kg}$ of $^{ 238 \mathrm{Pu}$ and the efficiency for conversion of radioactive decay energy to energy transferred by electrical transmission is 3.20$\%$ . Determine the initial power output of the source.

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Problem 50

The half-life of tritium is 12.3 yr. (a) If the TFTR fusion reactor contained 50.0 $\mathrm{m}^{3}$ of tritium at a density equal to $2.00 \times 10^{14}$ ions/cm ^{3} , \text { how many curies of tritium } were in the plasma? (b) State how this value compares with a fission inventory (the estimated supply of fissionable
material) of $4.00 \times 10^{10} \mathrm{Ci} .$

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Problem 51

Review. A nuclear power plant operates by using the energy released in nuclear fission to convert $20^{\circ} \mathrm{C}$ water into $400^{\circ} \mathrm{C}$ steam. How much water could theoretically be converted to steam by the complete fissioning of 1.00 $\mathrm{g}$ of $^{35} \mathrm{U}$ at 200 $\mathrm{MeV} /$ fission?

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Problem 52

A nuclear power plant operates by using the energy released in nuclear fission to convert liquid water at $T_{c}$ into steam at $T_{h} .$ How much water could theoretically be converted to steam by the complete fissioning of a mass $m$ of ${ }^{235} \mathrm{U}$ if the energy released per fission event is $E ?$

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Problem 53

Consider a 1.00 -kg sample of natural uranium composed primarily of $^{238} \mathrm{U},$ a smaller amount $(0.720 \% \text { by }$ mass) of $^{235} \mathrm{U},$ and a trace $(0.00500 \% \text ) $of $^{234} \mathrm{U}, \text { which has a }$ half-life of $2.44 \times 10^{5}$ yr. (a) Find the activity in curies due to each of the isotopes. (b) What fraction of the total activity is due to each isotope? (c) Explain whether the activity of this sample is dangerous.

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Problem 54

Approximately 1 of every 3 300 water molecules contains one deuterium atom. (a) If all the deuterium nuclei in 1 $\mathrm{L}$ of water are fused in pairs according to the $\mathrm{D}-\mathrm{D}$ fusion reaction $^{2} \mathrm{H}+^{2} \mathrm{H} \rightarrow^{3} \mathrm{He}+\mathrm{n}+3.27 \mathrm{MeV}$ , how much energy in joules is liberated? (b) What If? Burning gasoline produces approximately $3.40 \times 10^{7} \mathrm{J} / \mathrm{L}$ . State how the energy obtainable from the fusion of the deuterium in 1 $\mathrm{L}$ of water compares with the energy liberated from the burning of 1 $\mathrm{L}$ of gasoline.

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Numerade Educator

Problem 55

Carbon detonations are powerful nuclear reactions that temporarily tear apart the cores inside massive stars late in their lives. These blasts are produced by carbon fusion, which requires a temperature of approximately $6 \times 10^{8} \mathrm{K}$ to overcome the strong Coulomb repulsion between carbon nuclei. (a) Estimate the repulsive energy barrier to fusion, using the temperature required for carbon fusion. (In other words, what is the average kinetic energy of a carbon
nucleus at $6 \times 10^{8} \mathrm{K}^{2}$ (b) Calculate the energy (in MeV) released in each of these "carbon-burning" reactions:
$$
\begin{array}{l}{^{12} \mathrm{C}+^{12} \mathrm{C} \rightarrow^{20} \mathrm{Ne}+^{4} \mathrm{He}} \\ {^{12} \mathrm{C}+^{12} \mathrm{C} \rightarrow^{24} \mathrm{Mg}+\gamma}\end{array}
$$
(c) Calculate the energy in kilowatt-hours given off when 2.00 $\mathrm{kg}$ of carbon completely fuse according to the first reaction.

Ren Jie Tuieng

Numerade Educator

Problem 56

A sealed capsule containing the radiopharmaceutical phosphorus- $32,$ an $\mathrm{e}^{-}$ emitter, is implanted into a patient's tumor. The average kinetic energy of the beta particles is 700 keV. The initial activity is 5.22 $\mathrm{MBq}$ . Assume the beta particles are completely absorbed in 100 $\mathrm{g}$ of tissue. Determine the absorbed dose during a 10.0 -day period.

Ren Jie Tuieng

Numerade Educator

Problem 57

A certain nuclear plant generates internal energy at a rate of 3.065 $\mathrm{GW}$ and transfers energy out of the plant by electrical transmission at a rate of 1.000 $\mathrm{GW}$ . Of the waste energy,
3.0$\%$ is ejected to the atmosphere and the remainder is passed into a river. A state law requires that the river water be warmed by no more than $3.50^{\circ} \mathrm{C}$ when it is returned to the river. (a) Determine the amount of cooling water necessary (in kilograms per hour and cubic meters per hour) to cool the plant. (b) Assume fission generates $7.80 \times 10^{10} \mathrm{J} / \mathrm{g}$ of $^{235} \mathrm{U}$ . Determine the rate of fuel burning (in kilograms per hour) of $^{255} \mathrm{U} .$

Ren Jie Tuieng

Numerade Educator

Problem 58

The Sun radiates energy at the rate of $3.85 \times 10^{26} \mathrm{W}$ . Suppose the net reaction $4(1 \mathrm{H})+2\left(0^{\circ} \mathrm{e}\right) \rightarrow_{2}^{4} \mathrm{He}+2 v+\gamma$
accounts for all the energy released. Calculate the number of protons fused per second.

Ren Jie Tuieng

Numerade Educator

Problem 59

Consider the two nuclear reactions
$$
\begin{array}{l}{\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}+\mathrm{E}} \\ {\mathrm{C}+\mathrm{D} \rightarrow \mathrm{F}+\mathrm{G}}\end{array}
$$
(a) Show that the net disintegration energy for these two reactions $\left(Q_{\text { net }}=Q_{1}+Q_{\mathrm{I}}\right)$ is identical to the disintegration energy for the net reaction
$$
\mathrm{A}+\mathrm{B}+\mathrm{D} \rightarrow \mathrm{E}+\mathrm{F}+\mathrm{G}
$$
(b) One chain of reactions in the proton-proton cycle in the Sun's core is
$$
\begin{array}{l}{_{1}^{1} \mathbf{H}+_{1}^{1} \mathbf{H} \quad \rightarrow_{1}^{2} \mathbf{H}+_{1}^{0} \mathbf{e}+\nu} \\ {_{1}^{0} \mathbf{e}+_{-1}^{0} \mathbf{e} \rightarrow 2 \gamma}\end{array}
$$
$$
_{1}^{1} \mathrm{H}+_{1}^{2} \mathrm{H} \rightarrow_{2}^{3} \mathrm{He}+\gamma
$$
$$
_{1}^{1} \mathbf{H}+_{2}^{3} \mathbf{H} \mathbf{e} \rightarrow_{2}^{4} \mathbf{H} \mathbf{e}+_{1}^{0} \mathbf{e}+\boldsymbol{\nu}
$$
$$
_{1}^{0} \mathrm{e}+_{-1}^{0} \mathrm{e} \rightarrow 2 \gamma
$$
Based on part (a), what is $Q_{\mathrm{net}}$ for this sequence?

Ren Jie Tuieng

Numerade Educator

Problem 60

Natural uranium must be processed to produce uranium enriched in $^{235} \mathrm{U}$ for weapons and power plants. The processing yields a large quantity of nearly pure $^{238} \mathrm{U}$ as a by-product, called "depleted uranium." Because of its high mass density, 238 $\mathrm{U}$ is used in armor-piercing artillery shells.
(a) Find the edge dimension of a $70.0-\mathrm{kg}$ cube of $^{238}\mathrm{U}$ $\left(\rho=19.1 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}\right) .$ (b) The isotope $^{238}\mathrm{U}$ has a long half-life of $4.47 \times 10^{9}$ y. As soon as one nucleus decays, a relatively rapid series of 14 steps begins that together constitute the net reaction
$$
_{92}^{238} \mathrm{U} \rightarrow 8\left(_{2}^{4} \mathrm{He}\right)+6\left(_{-1}^{0} \mathrm{e}\right)+_{82}^{206} \mathrm{Pb}+6 \overline{\nu}+Q_{\mathrm{net}}
$$
Find the net decay energy. (Refer to Table $44.2 . )(\mathrm{c})$ Argue that a radioactive sample with decay rate $R$ and decay energy $Q$ has power output $P=Q R .$ (d) Consider an artillery shell with a jacket of 70.0 $\mathrm{kg}$ of $^{238}\mathrm{U}$ . Find its power output due to the radioactivity of the uranium and its daughters. Assume the shell is old enough that the daughters have reached steady-state amounts. Express the power in joules per year. (e) What If? A 17 -year-old soldier of mass
70.0 $\mathrm{kg}$ works in an arsenal where many such artillery shells are stored. Assume his radiation exposure is limited to 5.00 rem per year. Find the rate in joules per year at which he can absorb energy of radiation. Assume an average $\mathrm{RBE}$ factor of $1.10 .$

Ren Jie Tuieng

Numerade Educator

Problem 61

Suppose the target in a laser fusion reactor is a sphere of solid hydrogen that has a diameter of $1.50 \times 10^{-4} \mathrm{m}$ and a density of $0.200 \mathrm{g} / \mathrm{cm}^{3} .$ Assume half of the nuclei are 2 $\mathrm{H}$ and half are $^{3} \mathrm{H}$ . (a) If 1.00$\%$ of a $200-\mathrm{kJ}$ laser pulse is delivered to this sphere, what temperature does the sphere reach? (b) If all the hydrogen fuses according to the $\mathrm{D}-\mathrm{T}$ reaction, how many joules of energy are released?

Ren Jie Tuieng

Numerade Educator

Problem 62

When photons pass through matter, the intensity $I$ of the beam (measured in watts per square meter) decreases exponentially according to
$$
I=I_{0} e^{-\mu x}
$$
where $I$ is the intensity of the beam that just passed through a thickness $x$ of material and $I_{0}$ is the intensity of the incident beam. The constant $\mu$ is known as the linear absorption coefficient, and its value depends on the absorbing material and the wavelength of the photon beam. This wavelength (or energy) dependence allows us to filter out unwanted wavelengths from a broad-spectrum x-ray beam.
(a) Two x-ray beams of wavelengths $\lambda_{1}$ and $\lambda_{2}$ and equal incident intensities pass through the same metal plate. Show that the ratio of the emergent beam intensities is
$$
\frac{I_{2}}{I_{1}}=e^{-\left(\mu_{2}-\mu_{1}\right) x}
$$
(b) Compute the ratio of intensities emerging from an aluminum plate 1.00 $\mathrm{mm}$ thick if the incident beam contains equal intensities of 50 $\mathrm{pm}$ and 100 $\mathrm{pm}$ x-rays. The values of $\mu$ for aluminum at these two wavelengths are $\mu_{1}=5.40 \mathrm{cm}^{-1}$ at 50 $\mathrm{pm}$ and $\mu_{2}=41.0 \mathrm{cm}^{-1}$ at 100 $\mathrm{pm}$ .
(c) Repeat part (b) for an aluminum plate 10.0 $\mathrm{mm}$ thick.

Ren Jie Tuieng

Numerade Educator

Problem 63

Assume a deuteron and a triton are at rest when they fuse according to the reaction
$$
_{1}^{2} \mathrm{H}+_{1}^{3} \mathrm{H} \rightarrow_{2}^{4} \mathrm{He}+_{0}^{1} \mathrm{n}
$$
Determine the kinetic energy acquired by the neutron.

Ren Jie Tuieng

Numerade Educator

Problem 64

(a) Calculate the energy (in kilowatt-hours) released if 1.00 $\mathrm{kg}$ of 239 $\mathrm{Pu}$ undergoes complete fission and the energy released per fission event is 200 $\mathrm{MeV}$ . (b) Calculate the energy (in electron volts) released in the deuterium-tritium fusion reaction
$$
\frac{2}{1} \mathrm{H}+_{1}^{3} \mathrm{H} \rightarrow_{2}^{4} \mathrm{He}+_{0}^{1} \mathrm{n}
$$
(c) Calculate the energy (in kilowatt-hours) released if 1.00 $\mathrm{kg}$ of deuterium undergoes fusion according to this reaction. (d) What If? Calculate the energy (in kilowatt-hours) released by the combustion of 1.00 $\mathrm{kg}$ of carbon in coal if each $\mathrm{C}+\mathrm{O}_{2} \rightarrow \mathrm{CO}_{2}$ reaction yields 4.20 $\mathrm{eV}$ . (e) List advantages and disadvantages of each of these methods of energy generation.

Ren Jie Tuieng

Numerade Educator

Problem 65

During the manufacture of a steel engine component, radioactive iron $(59 \text { Fe) with a half-life of } 45.1 \mathrm{d} \text { is included }$ in the total mass of 0.200 $\mathrm{kg}$ . The component is placed in a test engine when the activity due to this isotope is 20.0$\mu \mathrm{Ci}$ . After a 1000 -h test period, some of the lubricating oil is removed from the engine and found to contain enough $^{59}\mathrm{Fe}$ to produce 800 disintegrations/min/L of oil. The total volume of oil in the engine is 6.50 L. Calculate the total mass worn from the engine component per hour of operation.

Ren Jie Tuieng

Numerade Educator

Problem 66

Assume a photomultiplier tube has seven dynodes with potentials of $100,200,300, \ldots, 700 \mathrm{V}$ as shown in Figure $\mathrm{P} 45.66$ . The average energy required to free an electron from the dynode surface is 10.0 eV. Assume only one electron is incident and the tube functions with 100$\%$ efficiency. (a) How many electrons are freed at the first dynode at 100 $\mathrm{V}$ ? (b) How many electrons are collected at the last dynode? (c) What is the energy available to the counter for all the electrons arriving at the last dynode?

Ren Jie Tuieng

Numerade Educator

Problem 67

(a) At time $t=0$, a sample of uranium is exposed to a neutron source that causes $N_{0}$ nuclei to undergo fission. The sample is in a supercritical state, with a reproduction constant $K>1$. A chain reaction occurs that proliferates fission throughout the mass of uranium. The chain reaction can be thought of as a succession of generations. The $N_{0}$ fissions produced initially are the zeroth generation of fissions. From this generation, $N_{0} K$ neutrons go off to produce fission of new uranium nuclei. The $N_{0} K$ fissions that occur subsequently are the first generation of fissions, and from this generation $N_{0} K^{2}$ neutrons go in search of uranium nuclei in which to cause fission. The subsequent $N_{0} K^{2}$ fissions are the second generation of fissions. This process can continue until all the uranium nuclei have fissioned. Show that the cumulative total of fissions $N$ that have occurred up to and including the $n$ th generation after the zeroth generation is given by $$N=N_{0}\left(\frac{K^{n+1}-1}{K-1}\right).$$
(b) Consider a hypothetical uranium weapon made from $5.50 \mathrm{~kg}$ of isotopically pure ${ }^{235} \mathrm{U}$. The chain reaction has a reproduction constant of $1.10$ and starts with a zeroth generation of $1.00 \times 10^{20}$ fissions. The average time interval between one fission generation and the next is $10.0 \mathrm{~ns}$. How long after the zeroth generation does it take the uranium in this weapon to fission completely?
(c) Assume the bulk modulus of uranium is 150 GPa. Find the speed of sound in uranium. You may ignore the density difference between ${ }^{235} \mathrm{U}$ and natural uranium. (d) Find the time interval required for a compressional wave to cross the radius of a $5.50$ -kg sphere of uranium. This time interval indicates how quickly the motion of explosion begins.
(e) Fission must occur in a time interval that is short compared with that in part (d); otherwise, most of the uranium will disperse in small chunks without having fissioned. Can the weapon considered in part (b) release the explosive energy of all its uranium? If so, how much energy does it release in equivalent tons of TNT? Assume one ton of TNT releases $4.20$ GJ and each uranium fission releases $200 \mathrm{MeV}$ of energy.

Ren Jie Tuieng

Numerade Educator