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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 30

Nuclear Energy and Elementary Particles

Educators


Problem 1

Natural uranium ore contains about 0.720$\%$ of the fissile uranium $-235$ isotope. Suppose a sample of uranium ore contains $2.50 \times 10^{28}$ uranium nuclei. Determine the number of uranium-235 nuclei in the sample.

Ren Jie T.
Numerade Educator

Problem 2

A typical uranium-234 fission event releases 208 $\mathrm{MeV}$ of energy. Determine (a) the energy released per event in joules and (b) the change in mass during the event.

Ren Jie T.
Numerade Educator

Problem 3

If the average energy released in a fission event is 208 MeV, find the total number of fission events required to operate a 100. - W lightbulb for 1.0 h.

Ren Jie T.
Numerade Educator

Problem 4

Find the energy released in the fission reaction
$$
\mathrm{n}+_{92}^{235} \mathrm{U} \quad \rightarrow \quad_{40}^{98} \mathrm{Zr}+\frac{135}{52} \mathrm{Te}+3 \mathrm{n}
$$
The atomic masses of the fission products are 97.9120 u for 98 Zr and 134.9087 u for $_{52}^{135} \mathrm{Te}$ .

Ren Jie T.
Numerade Educator

Problem 5

Find the energy released in the fission reaction
$$
_{0}^{1} \mathrm{n}+_{92}^{295} \mathrm{U} \rightarrow_{38}^{88} \mathrm{Sr}+\frac{136}{54} \mathrm{Xe}+12_{0}^{1} \mathrm{n}
$$

Ren Jie T.
Numerade Educator

Problem 6

According to one estimate, the first atomic bomb released an energy equivalent to $20 .$ kilotons of TNT. If 1.0 ton of TNT releases about $4.0 \times 10^{9} \mathrm{J}$ , how much uranium was lost through fission in this bomb? (Assume 208 MeV released per fission.)

Ren Jie T.
Numerade Educator

Problem 7

Assume ordinary soil contains natural uranium in amounts of 1 part per million by mass. (a) How much uranium is in the top 1.00 $\mathrm{m}$ of soil on a 1 -acre $\left(43560-\mathrm{ft}^{2}\right)$ plot of ground, assuming the specific gravity of soil is 4.00$?(\mathrm{b})$ How much of the isotope 235 $\mathrm{U}$ , appropriate for nuclear reactor fuel, is in this soil? Hint: See Appendix $\mathrm{B}$ for the percent abundance of $^{235} \mathrm{U}$

Ren Jie T.
Numerade Educator

Problem 8

A typical nuclear fission power plant produces about 1.00 GW of electrical power. Assume the plant has an overall efficiency of 40.0% and each fission produces 200. MeV of thermal energy. Calculate the mass of $^{235} \mathrm{U}$ consumed each day.

Ren Jie T.
Numerade Educator

Problem 9

In order to minimize neutron leakage from a reactor, the ratio of the surface area to the volume must be as small as possible. Assume that a sphere of radius a and a cube both have the same volume. Find the surface - to - volume ratio for (a) the sphere and (b) the cube. (c) Which of these reactor shapes would have the minimum leakage?

Ren Jie T.
Numerade Educator

Problem 10

According to one estimate, there are $4.4 \times 10^{6}$ metric tons of world uranium reserves extractable at $\$ 130 / \mathrm{kg}$ or less. About 0.70$\%$ of naturally occurring uranium is the fissionable isotope 235 $\mathrm{U}$ . (a) Calculate the mass of $^{2355} \mathrm{U}$ in this reserve in grams. (b) Find the number of moles of 235 $\mathrm{U}$ and convert to a number of atoms. (c) Assuming 208 MeV is obtained from each reaction and all this energy is captured, calculate the total energy that can be extracted from the reserve in joules. (d) Assuming world power consumption to be constant at $1.5 \times 10^{13} \mathrm{J} / \mathrm{s},$ how many years could the uranium reserves provide for all the world's energy needs using conventional reactors that don't generate nuclear fuel? (e) What conclusions can be drawn?

Ren Jie T.
Numerade Educator

Problem 11

An all-electric home uses approximately $2.00 \times 10^{3} \mathrm{kWh}$ of electric energy per month. How much uranium- 235 would be required to provide this house with its energy needs for one year? Assume 100$\%$ conversion efficiency and 208 $\mathrm{MeV}$ released per fission.

Ren Jie T.
Numerade Educator

Problem 12

Seawater contains 3 mg of uranium per cubic meter. (a) Given that the average ocean depth is about 4 km and water covers two - thirds of Earth’s surface, estimate the amount of uranium dissolved in the ocean. (b) Estimate how long this uranium could supply the world’s energy needs at the current usage of $1.5 \times 10^{13} \mathrm{J} / \mathrm{s}$ . (c) Where does the dissolved uranium come from? Is it a renewable energy source? Can uranium from the ocean satisfy our energy requirements? Discuss. Note: Breeder reactors increase the efficiency of nuclear fuel use by approximately two orders of magnitude.

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Numerade Educator

Problem 13

Suppose a deuterium-deuterium fusion reactor is designed to have a plasma confinement time of 1.50 s. Determine the minimum ion density per cubic cm required to obtain a net power output from the reactor.

Ren Jie T.
Numerade Educator

Problem 14

The proton-proton cycle responsible for the Sun's $3.84 \times 10^{26} \mathrm{W}$ power output yields about 26.7 $\mathrm{MeV}$ of energy for every four protons that are fused into a helium nucleus. Determine (a) the energy in joules released during each proton-proton cycle fusion reaction, (b) the number of proton-proton cycles occurring per second in the sun, and (c) the change in the Sun's mass each second due to this energy release.

Ren Jie T.
Numerade Educator

Problem 15

When a star has exhausted its hydrogen fuel, it may fuse other nuclear fuels. At temperatures above $1.0 \times 10^{8} \mathrm{K}$ , helium fusion can occur. Write the equations for the following processes. (a) Two alpha particles fuse to produce a nucleus $A$ and a gamma ray. What is nucleus $A ?$ (b) Nucleus
$A$ absorbs an alpha particle to produce a nucleus $B$ and a gamma ray. What is nucleus $B ?(c)$ Find the total energy released in the reactions given in parts (a) and (b). Note: The mass of $_{4}^{8} \mathrm{Be}=8.005305 \mathrm{u} .$

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Numerade Educator

Problem 16

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

Ren Jie T.
Numerade Educator

Problem 17

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

Ren Jie T.
Numerade Educator

Problem 18

Another series of nuclear reactions that can produce energy in the interior of stars is the cycle described below. 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. What is nucleus $F ?$ Note: If nucleus $F$ is not $1^{12} \mathrm{C}-$ that is, the nucleus you started with - you have made an error and should review the sequence of events.

Ren Jie T.
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Problem 19

Assume a deuteron and a triton are at rest when they fuse according to the reaction
$$
_{1}^{2} \mathrm{H}+_{1}^{3} \mathrm{H} \rightarrow_{2}^{4} \mathrm{He}+_{0}^{1} \mathrm{n}+17.6 \mathrm{MeV}
$$
Neglecting relativistic corrections, determine the kinetic energy acquired by the neutron.

Ren Jie T.
Numerade Educator

Problem 20

A 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:
$$
_{1}^{1} \mathrm{H}+\frac{11}{5} \mathrm{B} \rightarrow 3\left(_{2}^{4} \mathrm{He}\right)
$$
This reaction is an attractive possibility because boron is easily obtained from 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 to the initiation of uranium fission by slow neutrons. (a) How much energy is released in each reaction? (b) Why must the reactant particles have high kinetic energies?

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Numerade Educator

Problem 21

A photon produces a proton-antiproton pair according to the reaction $\gamma \rightarrow \mathrm{p}+\overline{\mathrm{p}} .$ What is the minimum possible frequency of the photon? What is its wavelength?

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Numerade Educator

Problem 22

A photon with an energy of 2.09 GeV creates a proton –antiproton pair in which the proton has a kinetic energy of 95.0 MeV. What is the kinetic energy of the antiproton?

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Problem 23

A neutral pion at rest decays into two photons according to
$$
\pi^{0} \quad \rightarrow \quad \gamma+\gamma
$$
Find the energy, momentum, and frequency of each photon.

Ren Jie T.
Numerade Educator

Problem 24

(a) Determine the baryon number of the reaction $\mathrm{p}+\overline{\mathrm{p}} \rightarrow 2 \gamma .$ Determine (b) the baryon number and (c) the electron-lepton number of the reaction $\Omega^{-} \rightarrow \Lambda^{0}+\mathrm{K}^{-}$

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Numerade Educator

Problem 25

(a) Determine the muon-lepton number in the reaction $\mu^{-} \rightarrow \mathrm{e}^{-}+\overline{\nu}_{\mathrm{e}}+\nu_{\mu} \cdot(\mathrm{b})$ Determine the value of strangeness in the reaction $\pi^{-}+\mathrm{p} \rightarrow \Lambda^{0}+\mathrm{K}^{0}$ .

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Numerade Educator

Problem 26

For the following two reactions, the first may occur but the second cannot. Explain.
$$\begin{aligned} \mathrm{K}^{0} & \rightarrow \pi^{+}+\pi^{-}(\text { can occur) }\\ \Lambda^{0} & \rightarrow \pi^{+}+\pi^{-} \text { (cannot occur) } \end{aligned}$$

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Problem 27

Each of the following reactions is forbidden. Determine a conservation law that is violated for each reaction.
(a) $\quad \mathrm{p}+\overline{\mathrm{p}} \rightarrow \mu^{+}+\mathrm{e}^{-}$
(b) $\quad \pi^{-}+\mathrm{p} \rightarrow \mathrm{p}+\pi^{+}$
(c) $\quad \mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\pi^{+}$
(d) $p+p \rightarrow p+p+n$
(e) $\quad \gamma+p \rightarrow n+\pi^{0}$

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Numerade Educator

Problem 28

Determine which of the reactions below can occur. For those that cannot occur, determine the conservation law (or laws) that each violates.
$$
\text {(a)} \mathrm{p} \rightarrow \pi^{+}+\pi^{0} \quad \text { (d) } \mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{-}+\overline{\nu}_{\mathrm{e}}
$$
$$
\text {(b)} \mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\pi^{0} \quad \text { (e) } \pi^{+} \rightarrow \mu^{+}+\mathrm{n}
$$
$$
\pi^{+} \rightarrow \mu^{+}+\nu_{\mu}
$$

Ren Jie T.
Numerade Educator

Problem 29

Which of the following processes are allowed by the strong interaction, the electromagnetic interaction, the weak interaction, or no interaction at all?
$$
\text {(a)} \pi^{-}+p \rightarrow 2 \eta^{0} \quad \text { (d) } \Omega^{-} \rightarrow \Xi^{-}+\pi^{0}
$$
$$
\text {(b)} \mathrm{K}^{-}+\mathrm{n} \rightarrow \Lambda^{0}+\pi^{-} \quad \text { (e) } \eta^{0} \rightarrow 2 \gamma
$$
$$
\text {(c)} \mathbf{K} \rightarrow \pi^{-}+\pi^{0}
$$

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Numerade Educator

Problem 30

(a) Show that baryon number and charge are conserved in the following reactions of a pion with a proton:
$$
\begin{array}{l}{\text { (1) } \pi^{+}+\mathrm{p} \rightarrow \mathrm{K}^{+}+\Sigma^{+}} \\ {\text { (2) } \pi^{+}+\mathrm{p} \rightarrow \pi^{+}+\Sigma^{+}}\end{array}
$$
(b) The first reaction is observed, but the second never occurs. Explain these observations. (c) Could the second reaction happen if it created a third particle? If so, which particles in Table 30.2 might make it possible? Would the reaction require less energy or more energy than the reaction of Equation (1)? Why ?

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Problem 31

Determine whether or not strangeness is conserved in the following decays and reactions.
$$
\text {(A)} \quad \Lambda^{0} \rightarrow \mathrm{p}+\pi^{-} \quad \text { (d) } \pi^{-}+\mathrm{p} \rightarrow \pi^{-}+\Sigma^{+}
$$
$$
\text {(b)} \pi^{-}+p \rightarrow \Lambda^{0}+K^{0} \quad(e) \equiv \rightarrow \Lambda^{0}+\pi^{-}
$$
$$
\text {(c)} \overline{\mathrm{p}}+\mathrm{p} \rightarrow \overline{\Lambda^{0}}+\Lambda^{0} \quad \text { (f) } \Xi^{0} \rightarrow \mathrm{p}+\pi^{-}
$$

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Problem 32

The quark composition of the proton is uud, whereas that of the neutron is udd. Show that the charge, baryon number, and strangeness of these particles equal the sums of these numbers for their quark constituents.

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Problem 33

Find the number of electrons, and of each species of quark, in 1.00 L of water.

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Problem 34

The quark compositions of the $\mathrm{K}^{0}$ and $\Lambda^{0}$ particles are $\mathrm{d} \overline{s}$ and uds, respectively. Show that the charge, baryon number, and strangeness of these particles equal the sums of these numbers for their quark constituents.

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Problem 35

Identify the particles corresponding to the quark states (a) suu, (b) \overlined, (c) sd, and (d) ssd.

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Problem 36

What is the electrical charge of the baryons with the quark compositions (a) $\overline{\mathrm{uu} \mathrm{d}}$ and (b) $\overline{\mathrm{u}} \mathrm{dd}$ ? What are these baryons called?

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Problem 37

A $\Sigma^{0}$ particle traveling through matter strikes a proton. A $\Sigma^{+},$ a gamma ray, as well as a third particle, emerge. Use the quark model of each to determine the identity of the third particle.

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Numerade Educator

Problem 38

Name at least one conservation law that prevents each of the following reactions from occurring.
$$
\begin{array}{l}{\text { (a) } \pi^{-}+\mathrm{p} \rightarrow \Sigma^{+}+\pi^{0}} \\ {\text { (b) } \mu^{-} \quad \rightarrow \quad \pi^{-}+\nu_{\mathrm{c}}} \\ {\text { (c) } \mathrm{p} \rightarrow \pi^{+}+\pi^{+}+\pi^{-}}\end{array}
$$

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Problem 39

Find the energy released in the fusion reaction
$$
_{1}^{1} \mathrm{H}+_{2}^{3} \mathrm{He} \rightarrow_{2}^{4} \mathrm{He}+\mathrm{e}^{+}+\nu
$$

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Numerade Educator

Problem 40

Occasionally, high-energy muons collide with electrons and produce two neutrinos according to the reaction $\mu^{+}+\mathrm{e}^{-} \rightarrow$ 2$\nu .$ What kind of neutrinos are they?

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Problem 41

Fill in the missing particle. Assume that (a) occurs via the strong interaction while (b) and (c) involve the weak interaction.
$$
\begin{array}{l}{\text { (a) } \mathrm{K}^{+}+\mathrm{p} \rightarrow ?+\mathrm{p}} \\ {\text { (b) } \Omega^{-} \rightarrow ?^{2}+\pi^{-}} \\ {\text { (c) } \mathrm{K}^{+} \rightarrow ?+\mu^{+}+\nu_{\mu}}\end{array}
$$

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Problem 42

Two protons approach each other with 70.4 MeV of kinetic energy and engage in a reaction in which a proton and a positive pion emerge at rest. What third particle, obviously uncharged and therefore difficult to detect, must have been created?

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Problem 43

A 2.0 -MeV neutron is emitted in a fission reactor. If it loses one-half its kinetic energy in each collision with a moderator atom, how many collisions must it undergo to reach an energy associated with a gas at a room temperature of $20.0^{\circ} \mathrm{C}$ ?

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Numerade Educator

Problem 44

The fusion reaction $\frac{2}{1} \mathrm{D}+\frac{2}{1} \mathrm{D} \rightarrow_{2}^{3} \mathrm{He}+_{0}^{1} \mathrm{n}$ releases 3.27 MeV of energy. If a fusion reactor operates strictly on the basis of this reaction, (a) how much energy could it produce by completely reacting 1.00 $\mathrm{kg}$ of deuterium? (b) At eight cents a kilowatt-hour, how much would the produced energy be worth? (c) Heavy water $\left(\mathrm{D}_{2} \mathrm{O}\right)$ costs about $\$ 300$ . per kilogram. Neglecting the cost of separating the deuterium from the oxygen via electrolysis, how much does 1.00 $\mathrm{kg}$ of deuterium cost, if derived from $\mathrm{D}_{2} \mathrm{O}$ ? (d) Would it be cost effective to use deuterium as a source of energy? Discuss, assuming the cost of energy production is nine-tenths the value of energy produced.

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Problem 45

(a) Show that about $1.0 \times 10^{10} \mathrm{J}$ would be released by the fusion of the deuterons in 1.0 gal of water. Note that 1 of every 6500 hydrogen atoms is a deuteron. (b) The average energy consumption rate of a person living in the United States is about $1.0 \times 10^{4} \mathrm{J} / \mathrm{s}(\text { an average power of } 10 . \mathrm{kW}) .$ At this rate, how long would the energy needs of one person be supplied by the fusion of the deuterons in 1.0 gal of water? Assume the energy released per deuteron is 1.64 $\mathrm{MeV}$

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Problem 46

The oceans have a volume of 317 million cubic miles and con- $\operatorname{tain} 1.32 \times 10^{21} \mathrm{kg}$ of water. Of all the hydrogen nuclei in this water, 0.0156$\%$ are deuterium. (a) If all of these deuterium nuclei were fused to helium via the first reaction in Equation 30.4 , determine the total amount of energy that could be released. (b) Present world electric power consumption is about $7.00 \times 10^{12} \mathrm{W}$ . If consumption were 100 times greater, how many years would the energy supply calculated in (a) last?

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Numerade Educator

Problem 47

A $\pi$ -meson at rest decays according to
$$
\pi^{-} \rightarrow \mu^{-}+\overline{\nu}_{\mu}
$$
What is the energy carried off by the neutrino? Assume the neutrino has no mass and moves off with the speed of light. Take $m_{\pi} c^{2}=139.6 \mathrm{MeV}$ and $m_{\mu} c^{2}=105.7 \mathrm{MeV.}$ Note: Use relativity; see Equation 26.13 .

Ren Jie T.
Numerade Educator

Problem 48

The reaction $\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{0}+\Lambda^{0}$ occurs with high probability, whereas the reaction $\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{0}+\mathrm{n}$ never occurs. Analyze these reactions at the quark level. Show that the first reaction conserves the total number of each type of quark and the second reaction does not.

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Problem 49

The sun radiates energy at the rate of $3.85 \times 10^{26} \mathrm{W} .$ Suppose the net reaction
$$
4 \mathrm{p}+2 \mathrm{e}^{-} \rightarrow \alpha+2 \nu_{\mathrm{e}}+6 \gamma
$$
accounts for all the energy released. Calculate the number of
protons fused per second. Note: Recall that an alpha particle is
a helium-4 nucleus.

Ren Jie T.
Numerade Educator

Problem 50

A $\mathrm{K}^{0}$ particle at rest decays into a $\pi^{+}$ and a $\pi^{-} .$ The mass of the $\mathrm{K}^{0}$ is 497.7 $\mathrm{MeV} / c^{2}$ and the mass of each pion is 139.6 $\mathrm{MeV} / c^{2} .$ What will be the speed of each of the pions?

Ren Jie T.
Numerade Educator