College Physics

lan Giambattista, Betty McCarthy Richardson, Robert C. Richardson

Chapter 29 Nuclear Physics - all with Video Answers

Educators

Chapter Questions

01:00

Problem 1

Estimate the number of nucleons found in the body of a 75-kg person.

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02:59

Problem 2

Calculate the mass density of nuclear matter.

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Numerade Educator

02:14

A neutron star is a star that has collapsed into a collection of tightly packed neutrons. Thus, it is something like a giant nucleus; but since it is electrically neutral, there is no Coulomb repulsion to break it up. The force holding it together is gravity. Suppose the Sun were to collapse into a neutron star. What would its radius be? Assume that the density is about the same as for a nucleus. Express your answer in $\mathrm{km}$.

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02:23

Problem 4

Write the symbol (in the form $\left.{ }_{Z}^{A} \mathrm{X}\right)$ for the nuclide with 38 protons and 50 neutrons and identify the element.

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01:37

Problem 5

Write the symbol (in the form ${ }_{Z}^{A} \mathrm{X}$ ) for the isotope of potassium with 21 neutrons.

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01:16

Problem 6

How many neutrons are found in a ${ }^{35} \mathrm{Cl}$ nucleus?

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00:57

Problem 7

How many protons are found in a ${ }^{136} \mathrm{Xe}$ nucleus?

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02:05

Problem 8

Write the symbol (in the form $\left.{ }_{Z}^{A} \mathrm{X}\right)$ for the nuclide that has 78 neutrons and 53 protons.

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01:47

Problem 9

Find the radius and volume of the ${ }_{43}^{107}$ Tc nucleus.

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03:21

Problem 10

What is the binding energy of an alpha particle (a ${ }^{4} \mathrm{He}$ nucleus)? The mass of an alpha particle is $4.00151 \mathrm{u}$.

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02:54

Problem 11

Find the binding energy of a deuteron (a ${ }^{2} \mathrm{H}$ nucleus). The mass of a deuteron (not the deuterium atom) is $2.013553 \mathrm{u}$.

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05:57

Problem 12

Using a mass spectrometer, the mass of the ${ }_{92}^{238} \mathrm{U}^{+}$ ion is found to be $238.05024$ u. (a) Use this result to calculate the mass of the ${ }_{92}^{238} \mathrm{U}$ nucleus. (b) Now find the binding energy of the ${ }_{92}^{238} \mathrm{U}$ nucleus.

Declan Nell

Numerade Educator

02:19

Problem 13

What is the average binding energy per nucleon for ${ }_{18}^{40} \mathrm{Ar}$ ?

Narayan Hari

Numerade Educator

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Problem 14

(a) Find the binding energy of the ${ }^{16} \mathrm{O}$ nucleus. (b) What is the average binding energy per nucleon? Check your answer using Fig. $29.2$.

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01:51

Problem 15

Calculate the binding energy per nucleon of the ${ }_{15}^{31} \mathrm{P}$ nucleus.

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02:29

Problem 16

Show that $c^{2}=931.494 \mathrm{MeV} / \mathrm{u} .$ [Hint: Start with the conversion factors to SI units for MeV and u.]

Mayank Tripathi

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02:16

Problem 17

What is the mass defect of the ${ }^{14} \mathrm{~N}$ nucleus?

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01:51

Problem 18

What is the mass of an ${ }^{16} \mathrm{O}$ atom in units of $\mathrm{MeV} / \mathrm{c}^{2}$ ? (1 $\mathrm{MeV} / \mathrm{c}^{2}$ is the mass of a particle with rest energy $1 \mathrm{MeV} .$ )

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03:11

Problem 19

(a) What is the mass defect of the ${ }^{1} \mathrm{H}$ atom due to the binding energy of the electron (in the ground state)?
(b) Should we worry about this mass defect when we calculate the mass of the ${ }^{1} \mathrm{H}$ nucleus by subtracting the mass of one electron from the mass of the ${ }^{1} \mathrm{H}$ atom?

Narayan Hari

Numerade Educator

01:07

Problem 20

To make an order-of-magnitude estimate of the energy level spacings in the nucleus, assume that a nucleon is confined to a one-dimensional box of width $10 \mathrm{fm}$ (a
typical nuclear diameter). Calculate the energy of the ground state.

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02:27

Problem 21

Identify the daughter nuclide when ${ }_{19}^{40} \mathrm{~K}$ decays via $\beta^{-}$ decay.

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01:22

Problem 22

${ }_{90}^{232}$ Th decays via $\alpha$ decay. Write out the reaction and identify the daughter nuclide.

Mayank Tripathi

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01:59

Problem 23

Write out the reaction and identify the daughter nuclide when ${ }_{11}^{22}$ Na decays by electron capture.

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02:07

Problem 24

Write out the reaction and identify the daughter nuclide when ${ }_{11}^{22}$ Na decays by emitting a positron.

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02:48

Problem 25

Radium-226 decays as ${ }_{88}^{226} \mathrm{Ra} \rightarrow{ }_{86}^{222} \mathrm{Rn}+{ }_{2}^{4} \mathrm{He}$. If the
${ }_{88}^{226}$ Ra nucleus is at rest before the decay and the ${ }_{86}^{222} \mathrm{Rn}$ nucleus is in its ground state, estimate the kinetic energy of the alpha particle. (Assume that the ${ }_{86}^{222} \mathrm{Rn}$ nucleus takes away an insignificant fraction of the kinetic energy.)

Mayank Tripathi

Numerade Educator

04:45

Problem 26

Calculate the kinetic energy of the alpha particle in Problem 25 . This time, do not assume that the ${ }^{222} \mathrm{Rn}$ nucleus is at rest after the reaction. Start by figuring out the ratio of the kinetic energies of the alpha particle and the ${ }_{86}^{222} \mathrm{Rn}$ nucleus.

Mayank Tripathi

Numerade Educator

02:08

Problem 27

Which decay mode would you expect for radioactive ${ }_{14}^{31}$ Si: $\alpha, \beta^{-}$, or $\beta^{+}$ ? Explain. [Hint: Look at the neutronto-proton ratio.

Narayan Hari

Numerade Educator

03:31

Problem 28

Show that the spontaneous alpha decay of ${ }^{19} \mathrm{O}$ is not possible.

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04:20

Problem 29

Calculate the maximum kinetic energy of the beta particle when ${ }_{19}^{40} \mathrm{~K}$ decays via $\beta^{-}$ decay.

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02:04

Problem 30

Calculate the energy of the antineutrino when ${ }_{38}^{90} \mathrm{Sr}$ decays via $\beta^{-}$ decay if the beta particle has a kinetic energy of $435 \mathrm{keV}$.

Narayan Hari

Numerade Educator

01:55

Problem 31

An isotope of sodium, ${ }_{11}^{22} \mathrm{Na}$, decays by $\beta^{+}$ emission. Estimate the maximum possible kinetic energy of the positron by assuming that the kinetic energy of the daughter nucleus and the total energy of the neutrino emitted are both zero. [Hint: Remember to keep track of the electron masses.

Narayan Hari

Numerade Educator

08:27

Problem 32

The nucleus in a ${ }_{7}^{12} \mathrm{~N}$ atom captures one of the atom's electrons, changing the nucleus to ${ }_{6}^{12} \mathrm{C}$ and emitting a neutrino. What is the total energy of the emitted neutrino? [Hint: You can use the classical expression for the kinetic energy of the ${ }_{6}^{12} \mathrm{C}$ atom and the extremely relativistic expression for the kinetic energy of the neutrino.]

Ren Jie Tuieng

Numerade Educator

02:52

Problem 33

A certain radioactive nuclide has a half-life of $200.0 \mathrm{~s}$. A sample containing just this one radioactive nuclide has an initial activity of $80,000.0 \mathrm{~s}^{-1}$. (a) What is the activity $600.0$ s later? (b) How many nuclei were there initially? (c) What is the probability per second that any one of the nuclei decays?

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04:50

Problem 34

Calculate the activity of $1.0 \mathrm{~g}$ of radium- 226 in $\mathrm{Ci}$.

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02:35

Problem 35

What is the activity in Bq of $1.0 \mathrm{~kg}$ of ${ }^{238} \mathrm{U}$ ?

Mayank Tripathi

Numerade Educator

02:31

Problem 36

The half-life of I- 131 is $8.0$ days. A sample containing I-131 has an activity of $6.4 \times 10^{8}$ Bq. How many days later will the sample have an activity of $2.5 \times 10^{6} \mathrm{~Bq}$ ?

Mayank Tripathi

Numerade Educator

06:11

Problem 37

Some bones discovered in a crypt in Guatemala are carbon dated. The ${ }^{14} \mathrm{C}$ activity of the bones is measured to be $0.242$ Bq per gram of carbon. Approximately how old are the bones?

Declan Nell

Numerade Educator

03:45

Problem 38

In this problem, you will verify the statement (in Section 29.4) that the ${ }^{14} \mathrm{C}$ activity in a living sample is $0.25 \mathrm{~Bq}$ per gram of carbon. (a) What is the decay constant $\lambda$ for ${ }^{\mathrm{i} 4} \mathrm{C} ?$ (b) How many ${ }^{14} \mathrm{C}$ atoms are in $1.00 \mathrm{~g}$ of carbon? One mole of carbon atoms has a mass of $12.011 \mathrm{~g}$, and the relative abundance of ${ }^{14} \mathrm{C}$ is $1.3 \times 10^{-12}$. (c) Using your results from parts (a) and (b), calculate the ${ }^{14} \mathrm{C}$ activity per gram of carbon in a living sample.

Mayank Tripathi

Numerade Educator

02:24

Problem 39

Carbon- 14 dating is used to date a bone found at an archaeological excavation. If the ratio of $\mathrm{C}-14$ to $\mathrm{C}-12$ atoms is $3.25 \times 10^{-13}$, how old is the bone? [Hint: Note that this ratio is $\frac{1}{4}$ the ratio of $1.3 \times 10^{-12}$ that is found in a living sample.]

Mayank Tripathi

Numerade Educator

01:01

Problem 40

A sample of radioactive ${ }_{83}^{214} \mathrm{Bi}$, which has a half-life of $19.9$ min, has an activity of $0.058 \mathrm{Ci}$. What is its activity $1.0$ h later?

Narayan Hari

Numerade Educator

02:08

Problem 41

The activity of a sample containing radioactive ${ }^{108} \mathrm{Ag}$ is $6.4 \times 10^{4}$ Bq. Exactly 12 min later, the activity is $2.0 \times$ $10^{3}$ Bq. Calculate the half-life of ${ }^{108} \mathrm{Ag}$.

Ren Jie Tuieng

Numerade Educator

05:28

Problem 42

A radioactive sample has equal numbers of ${ }^{15} \mathrm{O}$ and ${ }^{19} \mathrm{O}$ nuclei. Use the half-lives found in Appendix $\mathrm{B}$ to determine how long it will take before there are twice as many ${ }^{15} \mathrm{O}$ nuclei as ${ }^{19} \mathrm{O}$. What percent of the ${ }^{19} \mathrm{O}$ nuclei have decayed during this time?

Ren Jie Tuieng

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02:25

Problem 43

Show mathematically that $2^{-t / T_{1 / 2}}=\left(\frac{1}{2}\right)^{t / T_{1 / 2}}=e^{-t / \tau}$ if and only if $T_{1 / 2}=\tau \ln 2 .[$ Hint $:$ Take the natural logarithm of each side.]

Mayank Tripathi

Numerade Educator

04:31

Problem 44

The Physics at Home in Section $29.4$ suggests tossing coins as a model of radioactive decay. An improved version is to toss a large number of dice instead of coins:
each die that comes up a "one" represents a nucleus that has decayed. Suppose that $N$ dice are tossed. (a) What is the average number of dice you expect to decay on one toss? (b) What is the average number of dice you expect to remain undecayed after three tosses? (c) What is the average number of dice you expect to remain undecayed after four tosses? (d) What is the half-life in numbers of tosses?

Ren Jie Tuieng

Numerade Educator

01:50

Problem 45

An alpha particle produced in radioactive alpha decay has a kinetic energy of typically about $6 \mathrm{MeV}$. When an alpha particle passes through matter (such as biological tissue), it makes ionizing collisions with molecules, giving up some of its kinetic energy to supply the binding energy of the electron that is removed. If a typical ionization energy for a molecule in the body is around $20 \mathrm{eV}$, roughly how many molecules can the alpha particle ionize before coming to rest?

Mayank Tripathi

Numerade Educator

03:58

Problem 46

If meat is irradiated with $2000.0$ Gy of $x$ -rays, most of the bacteria are killed and the shelf life of the meat is greatly increased. (a) How many $100.0$ -keV photons must be absorbed by a $0.30-\mathrm{kg}$ steak so that the absorbed dose is $2000.0$ Gy? (b) Assuming steak has the same specific heat as water, what temperature increase is caused by a 2000.0-Gy absorbed dose?

Ren Jie Tuieng

Numerade Educator

08:51

Problem 47

Make an order-of-magnitude estimate of the amount of radon- 222 gas, measured in curies, found in the lungs of an average person. Assume that $0.1$ rem/yr is due to the alpha particles emitted by radon-222. The half-life is $3.8$ days. You will need to calculate the energy of the alpha particles emitted.

Ren Jie Tuieng

Numerade Educator

03:24

Problem 48

Some types of cancer can be effectively treated by bombarding the cancer cells with high energy protons. Suppose $1.16 \times 10^{17}$ protons, each with an energy of $950 \mathrm{keV}$, are incident on a tumor of mass $3.82 \mathrm{mg}$. If the quality factor for these protons is $3.0$, what is the biologically equivalent dose?

Ren Jie Tuieng

Numerade Educator

03:59

Problem 49

A certain nuclide absorbs a neutron. It then emits an electron, and then breaks up into two alpha particles.
(a) Identify the original nuclide and the two intermediate nuclides (after absorbing the neutron and after emitting the electron). (b) Would any (anti)neutrino(s) be emitted? Explain.

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01:42

Problem 50

A neutron-activated sample emits gamma rays at energies that are consistent with the decay of mercury-198 nuclei from an excited state to the ground state. If the reaction that takes place is $\mathrm{n}+(?) \rightarrow{ }^{198} \mathrm{Hg}^{*}+\mathrm{e}^{-}+\bar{v}$
what is the nuclide "(?)" that was present in the sample before neutron activation?

Mayank Tripathi

Numerade Educator

01:35

Problem 51

Irene and Jean Frederick Joliot-Curie, in an experiment that led to a Nobel Prize in Chemistry, bombarded alu$\operatorname{minum}_{13}^{27} \mathrm{Al}$ with alpha particles to form a highly unstable isotope of phosphorus, ${ }_{15}^{31} \mathrm{P}$. The phosphorus immediately decayed into another isotope of phosphorus, ${ }_{15}^{30} \mathrm{P}$, plus another product. Write out these reactions, identifying the other product.

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01:55

Problem 52

The reactions listed in Problem 51 did not stop there. To the surprise of the Curies, the phosphorus decay continued after the alpha bombardment ended with the phosphorus ${ }_{15}^{30} \mathrm{P}$ emitting a $\beta^{+}$ to form yet another product. Write out this reaction, identifying the other product.

Mayank Tripathi

Numerade Educator

04:38

Problem 53

One possible fission reaction for ${ }^{235} \mathrm{U}$ is ${ }^{235} \mathrm{U}+\mathrm{n} \rightarrow$ ${ }^{141} \mathrm{Cs}^{+}{ }^{93} \mathrm{Rb}+? \mathrm{n}$, where "?n" represents one or more neutrons. (a) How many neutrons? (b) From the graph in Fig. 29.2, you can read the approximate binding energies per nucleon for the three nuclides involved. Use that information to estimate the total energy released by this fission reaction. (c) Do a precise calculation of the energy released. (d) What fraction of the rest energy of the ${ }^{235} \mathrm{U}$ nucleus is released by this reaction?

Narayan Hari

Numerade Educator

03:30

Problem 54

$\mathrm{A}^{235} \mathrm{U}$ nucleus captures a low-energy neutron to form the compound nucleus ${ }^{236} \mathrm{U}^{*}$. Find the excitation energy of the compound nucleus. Ignore the small initial kinetic energy of the captured neutron.

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03:59

Problem 55

Estimate the energy released in the fission reaction of Eq. (29-31). Look up the binding energy per nucleon of the nuclides in Fig. $29.2$.

Ren Jie Tuieng

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01:53

Problem 56

Calculate the energy released in the fission reaction of Eq. (29-30). The atomic masses of ${ }_{56}^{141} \mathrm{Ba}$ and ${ }_{36}^{92} \mathrm{Kr}$ are $140.914 \mathrm{u}$ and $91.926 \mathrm{u}$, respectively.

Mayank Tripathi

Numerade Educator

02:52

Problem 57

Consider the fusion reaction of a proton and a deuteron:
${ }_{1}^{1} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \rightarrow \mathrm{X}$. (a) Identify the reaction product $\mathrm{X}$. (b) The binding energy of the deuteron is about $1.1 \mathrm{MeV}$ per nucleon and the binding energy of " $\mathrm{X}$ " is about $2.6 \mathrm{MeV}$ per nucleon. Approximately how much energy (in MeV) is released in this fusion reaction? (c) Why is this reaction unlikely to occur in a room temperature setting?

Narayan Hari

Numerade Educator

02:52

Problem 58

What is the total energy released by the proton-proton cycle [Eq. (29-34)]? (The total energy released is the total energy of the neutrinos and gamma rays plus the kinetic energy of the ${ }^{4}$ He nucleus minus the initial kinetic energies of the protons and electrons.)

Ren Jie Tuieng

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06:41

Problem 59

Estimate the minimum total kinetic energy of the ${ }^{2} \mathrm{H}$ and ${ }^{3} \mathrm{H}$ nuclei necessary to allow the fusion reaction of Eq. (29-32) to take place.

Ren Jie Tuieng

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08:37

Problem 60

Compare the amount of energy released when $1.0 \mathrm{~kg}$ of the uranium isotope ${ }^{235} \mathrm{U}$ undergoes the fission reaction of Eq. ( $29-30$ ) with the energy released when $1.0 \mathrm{~kg}$ of hydrogen undergoes the fusion reaction of Eq. (29-32).

Ren Jie Tuieng

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05:35

Problem 61

(a) Find the approximate number of water molecules in $1.00 \mathrm{~L}$ of water. (b) What fraction of the liter's volume is occupied by water nuclei?

Ren Jie Tuieng

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01:23

Problem 62

Which of these unidentified nuclides are isotopes of each other? ${ }_{71}^{175}(?),{ }_{32}^{71}(?),{ }_{74}^{175}(?),{ }_{71}^{167}(?),{ }_{30}^{71}(?)$, and ${ }_{74}^{180}(?)$.

Narayan Hari

Numerade Educator

02:16

Problem 63

What is the average binding energy per nucleon for ${ }_{11}^{23} \mathrm{Na}$ ?

Narayan Hari

Numerade Educator

01:59

Problem 64

The carbon isotope ${ }^{15} \mathrm{C}$ decays much faster than ${ }^{14} \mathrm{C}$.
(a) Using Appendix B, write a nuclear reaction show-

Mayank Tripathi

Numerade Educator

02:55

Problem 65

A radioactive sample of radon has an activity of $2050 \mathrm{~Bq}$. How many kg of radon are present?

Mayank Tripathi

Numerade Educator

03:50

Problem 66

Radioactive iodine, ${ }^{131} \mathrm{I}$, with a half-life of $8.0207$ days, is used in some forms of medical diagnostics. (a) If the initial activity of a sample is $64.5 \mathrm{mCi}$, what is the mass of ${ }^{131}$ I in the sample? (b) What will the activity be $4.5$ days later?

Mayank Tripathi

Numerade Educator

07:14

Problem 67

An alpha particle with a kinetic energy of $1.0 \mathrm{MeV}$ is headed straight toward a gold nucleus. (a) Find the distance of closest approach between the centers of the alpha particle and gold nucleus. (Assume the gold nucleus remains stationary. Since its mass is much larger than that of the alpha particle, this assumption is a fairly good approximation.) (b) Will the two get close enough to "touch"? (c) What is the minimum initial kinetic energy of an alpha particle that will make contact with the gold nucleus?

Ren Jie Tuieng

Numerade Educator

02:19

Problem 68

Figure $29.7$ is an energy level diagram for ${ }^{208} \mathrm{Tl}$. What are the energies of the photons emitted for the six transitions shown?

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08:14

Problem 69

A space rock contains $3.00 \mathrm{~g}$ of ${ }_{62}^{147} \mathrm{Sm}$ and $0.150 \mathrm{~g}$ of ${ }_{60}^{143}$ Nd. ${ }_{62}^{147} \mathrm{Sm}$ alpha decays to ${ }_{60}^{143} \mathrm{Nd}$ with a half-life of $1.06 \times 10^{11} \mathrm{yr}$. If the rock originally contained no ${ }_{60}^{143} \mathrm{Nd}$, how old is it?

Ren Jie Tuieng

Numerade Educator

04:28

Problem 70

In naturally occurring potassium, $0.0117 \%$ of the nuclei are radioactive ${ }^{40} \mathrm{~K}$. (a) What mass of ${ }^{40} \mathrm{~K}$ is found in a broccoli stalk containing $300 \mathrm{mg}$ of potassium?
(b) What is the activity of this broccoli stalk due to ${ }^{40} \mathrm{~K}$ ?

Ren Jie Tuieng

Numerade Educator

03:16

Problem 71

${ }_{83}^{212} \mathrm{Bi}$ can alpha-decay to the ground state of ${ }_{81}^{208} \mathrm{Tl}$, or to any of the four excited states of ${ }_{81}^{208} \mathrm{Tl}$ shown in Fig. 29.7. The maximum kinetic energy of the alpha particles emitted by ${ }_{83}^{212} \mathrm{Bi}$ is $6.090 \mathrm{MeV}$. What other alpha-particle kinetic energies are possible? [Hint:
Estimate the atomic mass of ${ }_{81}^{208} \mathrm{Tl}$.]

Ren Jie Tuieng

Numerade Educator

01:47

Problem 72

Approximately what is the total energy of the neutrino emitted when ${ }_{11}^{22}$ Na decays by electron capture?

Narayan Hari

Numerade Educator

03:07

Problem 73

${ }_{52}^{106} \mathrm{Te}$ is radioactive; it alpha decays to ${ }_{50}^{102} \mathrm{Sn} \cdot{ }_{50}^{102} \mathrm{Sn}$ is
itself radioactive and has a half-life of $4.5$ s. At $t=0$, a sample contains $4.00 \mathrm{~mol}$ of ${ }_{52}^{106} \mathrm{Te}$ and $1.50 \mathrm{~mol}$ of ${ }_{50}^{102}$ Sn. At $t=25 \mu \mathrm{s}$, the sample contains $3.00 \mathrm{~mol}$ of ${ }_{52}^{106} \mathrm{Te}$ and $2.50 \mathrm{~mol}$ of ${ }_{50}^{102} \mathrm{Sn}$. How much ${ }_{50}^{102} \mathrm{Sn}$ will there
be at $t=50 \mu \mathrm{s}$ ?

Narayan Hari

Numerade Educator

04:37

Problem 74

Suppose that a radioactive sample contains equal numbers of two radioactive nuclides $\mathrm{A}$ and $\mathrm{B}$ at $t=0 .$ A has
a half-life of $3.0 \mathrm{~h}$, while $\mathrm{B}$ has a half-life of $12.0 \mathrm{~h}$. Eind the ratio of the decay rates or activities $R_{\mathrm{A}} / R_{\mathrm{B}}$ at
(a) $t=0$, (b) $t=12.0 \mathrm{~h}$, and
(c) $t=24.0 \mathrm{~h}$.

Mayank Tripathi

Numerade Educator

04:23

Problem 75

In 1988 the shroud of Turin, a piece of cloth that some people believe is the burial cloth of Jesus, was dated using ${ }^{14} \mathrm{C}$. The measured ${ }^{14} \mathrm{C}$ activity of the cloth was about $0.23 \mathrm{~Bq} / \mathrm{g}$. According to this activity, when was the cloth in the shroud made?

Ren Jie Tuieng

Numerade Educator

06:27

Problem 76

The power supply for a pacemaker is a small amount of radioactive ${ }^{238} \mathrm{Pu}$. This nuclide decays by alpha decay with a half-life of 86 yr. The pacemaker is typically replaced every $10.0$ yr. (a) By what percentage does the activity of the ${ }^{238}$ Pu source decrease in $10 \mathrm{yr} ?$ (b) The energy of the alpha particles emitted is $5.6 \mathrm{MeV}$. Assume an efficiency of $100 \%-$ all of the alphaparticle energy is used to run the pacemaker. If the pacemaker starts with $1.0 \mathrm{mg}$ of ${ }^{238} \mathrm{Pu}$, what is the power output initially and after $10.0$ yr?

Ren Jie Tuieng

Numerade Educator

07:58

Problem 77

The first nuclear reaction ever observed (in 1919 by Ernest Rutherford) was $\alpha+{ }_{7}^{14} \mathrm{~N} \rightarrow \mathrm{p}+\mathrm{X} .$ (a) Show that the reaction product "X" must be ${ }_{8}^{17} \mathrm{O}$. (b) For this reaction to take place, the alpha particle must come in contact with the nitrogen nucleus. Calculate the distance $d$ between their centers when they just make contact.
(c) If the alpha particle and the nitrogen nucleus are initially far apart, what is the minimum value of their kinetic energy necessary to bring the two into contact? Express your answer in terms of the elementary charge $e$, the contact distance $d$, and whatever else you need.
(d) Is the total kinetic energy of the reaction products more or less than the initial kinetic energy in part (c)? Why? Calculate this kinetic energy difference.

Ren Jie Tuieng

Numerade Educator

08:18

Problem 78

The last step in the carbon cycle that takes place inside stars is $\mathrm{p}+{ }^{15} \mathrm{~N} \rightarrow{ }^{12} \mathrm{C}+(?) .$ This step releases
5.00 MeV of energy. (a) Show that the reaction product "(?)" must be an alpha particle. (b) Calculate the atomic mass of helium-4 from the information given.
(c) In order for this reaction to occur, the proton must come into contact with the nitrogen nucleus. Calculate the distance $d$ between their centers when they just "touch." (d) If the proton and nitrogen nucleus are initially far apart, what is the minimum value of their total kinetic energy necessary to bring the two into contact?

Ren Jie Tuieng

Numerade Educator

03:17

Problem 79

Radon gas $(\mathrm{Rn})$ is produced by the $\alpha$ decay of radium ${ }_{88}^{226} \mathrm{Ra}$. (a) How many neutrons and how many protons are present in the nucleus of the isotope of $\mathrm{Rn}$ produced by this decay? (b) In the air in an average size room in a student basement apartment in Ithaca, NY, there are about $10^{7}$ Rn nuclei. The Rn nucleus itself is radioactive; it too decays by emitting an $\alpha$ particle. The half-life of Rn is $3.8$ days. How many $\alpha$ particles per second are emitted by decaying Rn nuclei in the room?

Mayank Tripathi

Numerade Educator

03:22

Problem 80

(a) What fraction of the ${ }^{238} \mathrm{U}$ atoms present at the formation of the Earth still exist? Take the age of the Earth to be $4.5 \times 10^{9}$ yr. (b) Answer the same question for ${ }^{235} \mathrm{U}$. Could this explain why there are more than 100 times as many ${ }^{238} \mathrm{U}$ atoms as ${ }^{235} \mathrm{U}$ atoms in the Earth today?

Ren Jie Tuieng

Numerade Educator

09:04

Problem 81

The radioactive decay of ${ }^{238} \mathrm{U}$ produces alpha particles with a kinetic energy of $4.17 \mathrm{MeV}$. (a) At what speed do these alpha particles move? (b) Put yourself in the place of Rutherford and Geiger. You know that alpha particles are positively charged (from the way they are deflected in a magnetic field). You want to measure the
speed of the alpha particles using a velocity selector. If your magnet produces a magnetic field of $0.30 \mathrm{~T}$, what strength electric field would allow the alpha particles to pass through undeflected? (c) Now that you know the speed of the alpha particles, you measure the radius of their trajectory in the same magnetic field (without the electric field) to determine their charge-to-mass ratio. Using the charge and mass of the alpha particle, what would the radius be in a $0.30$ -T field? (d) Why can you determine only the ratio $q / m$ by this experiment, but not the individual values of $q$ and $m$ ?

Ren Jie Tuieng

Numerade Educator

08:28

Problem 82

Once Rutherford and Geiger determined the charge-tomass ratio of the alpha particle (Problem 81 ), they performed another experiment to determine its charge. An alpha source was placed in an evacuated chamber with a fluorescent screen. Through a glass window in the chamber, they could see a flash on the screen every time an alpha particle hit it. They used a magnetic field to deflect beta particles away from the screen so they were sure that every flash represented an alpha particle.
(a) Why is the deflection of a beta particle in a magnetic field much larger than the deflection of an alpha particle moving at the same speed? (b) By counting the flashes, they could determine the number of alphas per second striking the screen $(R)$. Then they replaced the screen with a metal plate connected to an electroscope and measured the charge $Q$ accumulated in a time $\Delta t$. What is the alpha-particle charge in terms of $R, Q$, and $\Delta t ?$

Ren Jie Tuieng

Numerade Educator

05:44

Problem 83

A water sample is found to have $0.016 \%$ deuterium content (that is, $0.016 \%$ of the hydrogen nuclei in the water are ${ }^{2} \mathrm{H}$ ). If the fusion reaction $\left({ }^{2} \mathrm{H}+{ }^{2} \mathrm{H}\right)$ yields $3.65 \mathrm{MeV}$ of energy on average, how much energy could you get from $1.00 \mathrm{~L}$ of the water? (There are two reactions with approximately equal probabilities; one yields $4.03 \mathrm{MeV}$ and the other $3.27 \mathrm{MeV}$.) Assume that you are able to extract and fuse $87.0 \%$ of the deuterium in the water. Give your answer in $\mathrm{kW} \cdot \mathrm{h}$.

Ren Jie Tuieng

Numerade Educator