College Physics

Raymond A. Serway, Jerry S. Faughn, Chris Vuille

Chapter 30

Nuclear Energy and Elementary Particles - all with Video Answers

Educators

Chapter Questions

Problem 1

Burning 1 metric ton (1 $000 \mathrm{~kg}$ ) of coal can yield an energy of $3.30 \times 10^{10} \mathrm{~J}$. Fission of one nucleus of uranium235 yields an average energy of approximately $208 \mathrm{MeV}$. What mass of uranium produces the same energy as 1 metric ton of coal?

Declan Nell

Declan Nell

Numerade Educator

Problem 2

Find the energy released in the fission reaction
$$
\mathrm{n}+{ }_{92}^{295} \mathrm{U} \rightarrow{ }_{40}^{98} \mathrm{Zr}+{ }_{52}^{135} \mathrm{Te}+3 \mathrm{n}
$$
The atomic masses of the fission products are $97.9120 \mathrm{u}$ for ${ }_{40}^{98} \mathrm{Zr}$ and $134.9087 \mathrm{u}$ for ${ }^{135} \mathrm{Te}$.

Declan Nell

Numerade Educator

Problem 3

Find the energy released in the fission reaction
$$
{ }_{0}^{1} \mathrm{n}+{ }_{92}^{235} \mathrm{U} \rightarrow{ }_{38}^{88} \mathrm{Sr}+{ }_{54}^{136} \mathrm{Xe}+12_{0}^{1} \mathrm{n}
$$

Declan Nell

Numerade Educator

Problem 4

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

Declan Nell

Numerade Educator

Problem 5

]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 $4.00$ ?
(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 ${ }_{92}^{235} \mathrm{U}$.

Ashwin Banarsee

Ashwin Banarsee

Numerade Educator

Problem 6

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

Declan Nell

Numerade Educator

Problem 7

Suppose the water exerts an average frictional drag of $1.0 \times 10^{5} \mathrm{~N}$ on a nuclear-powered ship. How far can the

Declan Nell

Numerade Educator

Problem 8

GP 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 $0.7 \%$ of naturally occurring uranium is the fissionable isotope ${ }^{235} \mathrm{U} .$
(a) Calculate the mass of ${ }^{235} \mathrm{U}$ in this reserve in grams. (b) Find the number of moles of ${ }^{235} \mathrm{U}$ and convert to a number of atoms. (c) Assuming $208 \mathrm{MeV}$ 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? (e) What conclusion can be drawn?

Declan Nell

Numerade Educator

Problem 9

An all-electric home uses approximately $2000 \mathrm{kWh}$ of electric energy per month. How much uranium-295 would be required to provide this house with its energy needs for one year? Assume $100 \%$ conversion efficiency and 208 MeV released per fission.

Declan Nell

Numerade Educator

Problem 10

ecp Seawater contains $3 \mathrm{mg}$ of uranium per cubic meter.
(a) Given that the average ocean depth is about $4 \mathrm{~km}$ 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.

Ashwin Banarsee

Ashwin Banarsee

Numerade Educator

Problem 11

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 $\mathrm{a}$ nucleus $B$ and a gamma ray. What is nucleus $B ?(\mathrm{c})$ Find the total energy released in the reactions given in parts
(a) and (b). Note: The mass of ${ }_{4}^{8} \mathrm{Be}=8.005305 \mathrm{u}$.

Ren Jie Tuieng

Numerade Educator

Problem 12

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

Declan Nell

Numerade Educator

Problem 13

If an all-electric home uses approximately $2000 \mathrm{kWh}$ of electric energy per month, 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?

Declan Nell

Numerade Educator

Problem 14

If an all-electric home uses approximately $2000 \mathrm{kWh}$ of electric energy per month, 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?

Declan Nell

Numerade Educator

Problem 15

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 Tuieng

Numerade Educator

Problem 16

ecp 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:
$$
{ }_{1}^{1} \mathrm{H}+{ }_{5}^{11} \mathrm{~B} \rightarrow 3\left({ }_{2}^{4} \mathrm{He}\right)
$$
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?

Declan Nell

Numerade Educator

Problem 17

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?

Ren Jie Tuieng

Numerade Educator

Problem 18

. A photon with an energy of $2.09 \mathrm{GeV}$ creates a protonantiproton pair in which the proton has a kinetic energy of $95.0 \mathrm{MeV}$. What is the kinetic energy of the antiproton?

Declan Nell

Numerade Educator

Problem 18

]A photon with an energy of $2.09 \mathrm{GeV}$ creates a protonantiproton pair in which the proton has a kinetic energy of $95.0 \mathrm{MeV}$. What is the kinetic energy of the antiproton?

Declan Nell

Numerade Educator

Problem 19

A neutral pion at rest decays into two photons according 10
$$
\pi^{0} \rightarrow \gamma+\gamma
$$
Find the energy, momentum, and frequency of each photon.

Declan Nell

Numerade Educator

Problem 20

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

Ren Jie Tuieng

Numerade Educator

Problem 21

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

Ren Jie Tuieng

Numerade Educator

Problem 22

Determine the type of neutrino or antineutrino involved in each of the following processes.
(a) $\pi^{+} \rightarrow \pi^{n}+\mathrm{e}^{+}+?$
(b) $?+\mathrm{p} \rightarrow \mu^{-}+\mathrm{p}+\pi^{+}$
(c) $\Lambda^{0} \rightarrow \mathrm{p}+\mu^{-}+?$
(d) $\tau^{+} \rightarrow \mu^{+}+?+?$

Declan Nell

Numerade Educator

Problem 23

Identify the unknown particle on the left side of the reaction
$$
?+p \rightarrow n+\mu^{+}
$$

Farhanul Hasan

Numerade Educator

Problem 24

ecp (a) Show that baryon number and charge are conserved in the following reactions of a pion with a proton:
(1) $\pi^{+}+\mathrm{p} \rightarrow \mathrm{K}^{+}+\Sigma^{+}$
(2) $\pi^{+}+\mathrm{p} \rightarrow \pi^{+}+\Sigma^{+}$
(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?

Ashwin Banarsee

Ashwin Banarsee

Numerade Educator

Problem 25

ecp Identify the conserved quantities in the following processes.
(a) $\Xi^{-} \rightarrow \Lambda^{0}+\mu^{-}+\nu_{\mu}$
(b) $\mathrm{K}^{0} \rightarrow 2 \pi^{0}$
(c) $\mathrm{K}^{-}+\mathrm{p} \rightarrow \Sigma^{0}+\mathrm{n}$
(d) $\Sigma^{0} \rightarrow \Lambda^{0}+\gamma$
(e) $\mathrm{e}^{+}+\mathrm{e}^{-} \rightarrow \mu^{+}+\mu^{-}$
(f) $\overline{\mathrm{p}}+\mathrm{n} \rightarrow \bar{\Lambda}^{0}+\Sigma^{-}$

Declan Nell

Numerade Educator

Problem 26

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.

Ren Jie Tuieng

Numerade Educator

Problem 27

]Find the number of electrons, and of each species of quark, in $\underline{1} \mathrm{~L}$ of water.

Declan Nell

Numerade Educator

Problem 28

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

Ren Jie Tuieng

Numerade Educator

Problem 29

Identify the particles corresponding to the quark states
(a) suu, (b) $\overline{u d},($ c) $\bar{s} d$, and $($ d $)$ ssd.

Declan Nell

Numerade Educator

Problem 30

What is the electrical charge of the baryons with the quark compositions (a) uud and (b) udd? What are these baryons called?

Robert Zaballa

Numerade Educator

Problem 31

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

Ashwin Banarsee

Ashwin Banarsee

Numerade Educator

Problem 32

Name at least one conservation law that prevents each of the following reactions from occurring.
(a) $\pi^{-}+\mathrm{p} \rightarrow \Sigma^{+}+\pi^{0}$
(b) $\mu^{-} \rightarrow \pi^{-}+v_{\mathrm{c}}$
(c) $\mathrm{p} \rightarrow \overrightarrow{\pi^{+}}+\pi^{+}+\pi$

Declan Nell

Numerade Educator

Problem 33

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

Declan Nell

Numerade Educator

Problem 34

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?

Ren Jie Tuieng

Numerade Educator

Problem 35

ecp Each of the following decays is forbidden. For each process, determine a conservation law that is violated.
(a) $\mu^{-} \rightarrow \mathrm{e}^{-}+\gamma$
(b) $\mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{-}+v_{\mathrm{c}}$
(c) $\Lambda^{0} \rightarrow \mathrm{p}+\pi^{0}$
(d) $\mathrm{P} \rightarrow \mathrm{e}^{+}+\pi^{0}$
(e) $\Xi^{0} \rightarrow n+\pi^{0}$

Ashwin Banarsee

Ashwin Banarsee

Numerade Educator

Problem 36

Two protons approach each other with $70.4 \mathrm{MeV}$ 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?

Ren Jie Tuieng

Numerade Educator

Problem 37

- A $2.0-\mathrm{MeV}$ 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 $20.0^{\circ} \mathrm{C}^{?}$

Ren Jie Tuieng

Numerade Educator

Problem 38

ecp The fusion reaction ${ }_{1}^{2} \mathrm{D}+{ }_{1}^{2} \mathrm{D} \rightarrow{ }_{2}^{3} \mathrm{He}+{ }_{0}^{\mathrm{n}} \mathrm{n}$ releases
$3.27 \mathrm{MeV}$ of energy. If a fusion reactor operates strictly on the basis of this reaction, (a) how much energy could it produce by completely reacting $1 \mathrm{~kg}$ of deuterium?

Ren Jie Tuieng

Numerade Educator

Problem 39

(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 $1.0 \times 10^{4} \mathrm{~J} / \mathrm{s}$ (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 $1.64 \mathrm{MeV}$.

Ren Jie Tuieng

Numerade Educator

Problem 40

Calculate the mass of ${ }^{235} \mathrm{U}$ required to provide the total energy requirements of a nuclear submarine during a 100 -day patrol, assuming a constant power demand of $100000 \mathrm{~kW}$, a conversion efficiency of $30 \%$, and an average energy released per fission of $208 \mathrm{MeV}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

A $\pi$ -meson at rest decays according to
$$
\pi^{-} \rightarrow \mu^{-}+\bar{v}_{\mu}
$$
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 $26.10 .$

Ren Jie Tuieng

Numerade Educator

Problem 42

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.

Ren Jie Tuieng

Numerade Educator

Problem 43

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 Tuieng

Numerade Educator