Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 41

Nuclear Physics and Radioactivity - all with Video Answers

Educators

Chapter Questions

Problem 1

The Problems in this Section are ranked $1,11,$ or III according to
estimated difficulty, with (I) Problems being easiest. Level (III)
Problems are meant mainly as a challenge for the best students, for
"extra credit. "The Problems are arranged by Sections, meaning that
the reader should have read up to and including that Section, but
this Chapter also have a group of General Problems that are not
arranged by Section and not ranked.
$$\begin{array}{l}{\text { (I) A pi meson has a mass of } 139 \mathrm{MeV} / \mathrm{c}^{2} . \text { What is this in }} \\ {\text { atomic mass units? }}\end{array}$$

Katie Mcalpine

Katie Mcalpine

Numerade Educator

Problem 2

(1) What is the approximate radius of an alpha particle
$\left(\frac{4}{2} \mathrm{He}\right) ?$

Farhanul Hasan

Numerade Educator

Problem 3

(I) By what $\%$ is the radius of ${ }_{92}^{238} \mathrm{U}$ greater than the radius of ${ }_{92}^{232} \mathrm{U} ?$

Katie Mcalpine

Numerade Educator

Problem 4

(II) $(a)$ What is the approximate radius of a $^{112} \mathrm{Cd}$ nucleus?
(b) Approximately what is the value of $A$ for a nucleus?
whose radius is $3.7 \times 10^{-15} \mathrm{m} ?$

Farhanul Hasan

Numerade Educator

Problem 5

(II) What is the mass of a bare $\alpha$ particle (without electrons)
in MeV/$c^{2} ?$

Katie Mcalpine

Numerade Educator

Problem 6

(II) Suppose two alpha particles were held together so they
were just touching. Estimate the electrostatic repulsive force
each would exert on the other. What would be the accelera-
tion of an alpha particle subjected to this force?

Farhanul Hasan

Numerade Educator

Problem 7

(II) $(a)$ Show that the density of nuclear matter is essen-
tially the same for all nuclei. (b) What would be the radius
of the Earth if it had its actual mass but had the density of
nuclei? (c) What would be the radius of a $\frac{238}{90} \mathrm{U}$ nucleus if it
had the density of the Earth?

Katie Mcalpine

Numerade Educator

Problem 8

(1I) What stable nucleus has approximately half the radius
of a uranium nucleus? [Hint: Find $A$ and use Appendix:
Selected Isotopes to get $Z . ]$

Farhanul Hasan

Numerade Educator

Problem 9

(II) If an alpha particle were released from rest near the
surface of a $\frac{257}{100} \mathrm{Fm}$ nucleus, what would its kinetic energy be
when far away?

Katie Mcalpine

Numerade Educator

Problem 10

(II) $(a)$ What is the fraction of the hydrogen atom's mass
that is in the nucleus? (b) What is the fraction of the
hydrogen atom's volume that is occupied by the nucleus?

Farhanul Hasan

Numerade Educator

Problem 11

(II) Approximately how many nucleons are there in a 1.0 -kg
object? Does it matter what the object is made of? Why or
why not?

Katie Mcalpine

Numerade Educator

Problem 12

(II) How much kinetic energy must an $\alpha$ particle have to
just "touch" the surface of a $\frac{238}{92} \mathrm{U}$ nucleus?

Farhanul Hasan

Numerade Educator

Problem 13

(I) Estimate the total binding energy for $\frac{63}{29} \mathrm{Cu},$ using Fig. 1

Katie Mcalpine

Numerade Educator

Problem 14

(II) Use Appendix: Selected Isotopes to calculate the
binding energy of $_{1}^{2} \mathrm{H}$ (deuterium).

Farhanul Hasan

Numerade Educator

Problem 15

(II) Determine the binding energy of the last neutron in a
is $\frac{32}{15} P$ nucleus.

Katie Mcalpine

Numerade Educator

Problem 16

(II) Calculate the total binding energy, and the binding
energy per nucleon, for $(a)_{3}^{7} \mathrm{Li,}$ $(b)_{79}^{197} \mathrm{Au.}$Use Appendix:
Selected Isotopes.

Farhanul Hasan

Numerade Educator

Problem 17

(II) Compare the average binding energy of a nucleon in
$\frac{23}{11}$ Na to that in $\frac{24}{11} \mathrm{Na.}$

Katie Mcalpine

Numerade Educator

Problem 18

(III) How much energy is required to remove (a) a proton,
(b) a neutron, from $_{7}^{15} \mathrm{N}^{}$? Explain the difference in your
answers.

Farhanul Hasan

Numerade Educator

Problem 19

(III) $(a)$ Show that the nucleus $_{4}^{8}$ Be $($ mass $=8.005305 \mathrm{u})$ is
unstable and will decay into two $\alpha$ particles. $(b)$ Is $_{6}^{12} \mathrm{C}$ stable
against decay into three $\alpha$ particles? Show why or why not.

Katie Mcalpine

Numerade Educator

Problem 20

(I) How much energy is released when tritium, $_{1}^{3} \mathrm{H},$ decays
by $\beta^{-}$ emission?

Farhanul Hasan

Numerade Educator

Problem 21

(I) What is the maximum kinetic energy of an electron
emitted in the $\beta$ decay of a free neutron?

Katie Mcalpine

Numerade Educator

Problem 22

(I) Show that the decay $\frac{11}{6} \mathrm{C} \rightarrow \frac{10}{5} \mathrm{B}+\mathrm{p}$ is not possible
because energy would not be conserved.

Farhanul Hasan

Numerade Educator

Problem 23

$(\mathrm{I})$ The $_{3}$ nucleus has an excited state 0.48 $\mathrm{MeV}$ above the
ground state. What wavelength gamma photon is emitted
when the nucleus decays from the excited state to the
ground state?

Katie Mcalpine

Numerade Educator

Problem 24

(II) Give the result of a calculation that shows whether or
not the following decays are possible:
$$(a)_{y 2}^{233} U \rightarrow_{92}^{232;} U+n$$
$$(b)^{14}_{7} \mathrm{N} \rightarrow_{7}^{3} \mathrm{N}+\mathrm{n;}$$
$$(c)^\stackrel{49}{19} \mathrm{K} \rightarrow \stackrel{39}{19} \mathrm{K}+\mathrm{n.}$$

Farhanul Hasan

Numerade Educator

Problem 25

(II) $\frac{24}{11} \mathrm{Na}$ is radioactive. (a) Is it a $\beta$ or $\beta^{+}$ emitter?
(b) Write down the decay reaction, and estimate the
maximum kinetic energy of the emitted $\beta$ .

Katie Mcalpine

Numerade Educator

Problem 26

(II) When $\frac{23}{10} \mathrm{Ne}$ $($ mass $=22.9945 \mathrm{u}) \quad$ decays to $\quad \frac{23}{11} \mathrm{Na}$
$($ mass $=22.9898 \mathrm{u}),$ what is the maximum kinetic energy of
the emitted electron? What is its minimum energy? What is
the energy of the neutrino in each case? Ignore recoil of the
daughter nucleus.

Farhanul Hasan

Numerade Educator

Problem 27

(II) A ${ }_{92}^{238} \mathrm{U}$ nucleus emits an $\alpha$ particle with kinetic energy $=4.20 \mathrm{MeV}$. (a) What is the daughter nucleus, and (b) what is the approximate atomic mass (in u) of the daughter atom? Ignore recoil of the daughter nucleus.

Katie Mcalpine

Numerade Educator

Problem 28

(II) What is the maximum kinetic energy of the emitted
$\beta$ particle during the decay of $_{27}^{60} \mathrm{Co} ?$

Farhanul Hasan

Numerade Educator

Problem 29

(II) A nucleus of mass 256 $\mathrm{u}$ , initially at rest, emits an
$\alpha$ particle with a kinetic energy of 5.0 $\mathrm{MeV} .$ What is the
kinetic energy of the recoiling daughter nucleus?

Christopher Mullarkey

Christopher Mullarkey

Numerade Educator

Problem 30

(II) The isotope $\frac{218}{84} \mathrm{Po}$ can decay by either $\alpha$ or $\beta$ ' emission.
$\frac{218}{84} \mathrm{Po}$ is 218.008965 $\mathrm{u}$

Farhanul Hasan

Numerade Educator

Problem 31

(II) The nuclide $\frac{32}{15} \mathrm{P}$ decays by emitting an electron whose
maximum kinetic energy can be 1.71 $\mathrm{MeV}$ . (a) What is the
daughter nucleus? (b) Calculate the daughter's atomic mass
(in u).

Katie Mcalpine

Numerade Educator

Problem 32

(II) A photon with a wavelength of $1.00 \times 10^{-13} \mathrm{m}$ is
ejected from an atom. Calculate its energy and explain
why it is a $\gamma$ ray from the nucleus or a photon from the
atom.

Farhanul Hasan

Numerade Educator

Problem 33

(II) How much energy is released in electron capture by
beryllium: $_{4}^{7} \mathrm{Be}+\mathrm{e}^{-} \rightarrow_{3}^{7} \mathrm{Li}+v ?$

Katie Mcalpine

Numerade Educator

Problem 34

(II) How much recoil energy does a $\frac{40}{19} \mathrm{K}$ nucleus get when
it emits a 1.46 -MeV gamma ray?

Farhanul Hasan

Numerade Educator

Problem 35

(II) Determine the maximum kinetic energy of $\beta^{+}$ particles
released when $^\frac{11}{6} \mathrm{C}$ decays to $^\frac{11}{5} \mathrm{B}$ . What is the maximum
energy the neutrino can have? What is the minimum energy
of each?

Katie Mcalpine

Numerade Educator

Problem 36

(III) The $\alpha$ particle emitted when $\frac{238}{92} \mathrm{U}$ decays has 4.20 $\mathrm{MeV}$
of kinetic energy. Calculate the recoil kinetic energy of the
daughter nucleus and the $Q$ -value of the decay.

Tara Appleyard

Numerade Educator

Problem 37

(III) What is the energy of the $\alpha$ particle emitted in the
decay $_{84}^{210} \mathrm{Po} \rightarrow_{82}^{206} \mathrm{Pb}+\alpha ?$ Take into account the recoil of
the daughter nucleus.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

(III) Show that when a nucleus decays by $\beta^{+}$ decay, the total
energy released is equal to
$$\left(M_{\mathrm{P}}-M_{\mathrm{D}}-2 m_{\mathrm{e}}\right) c^{2}$$
where $M_{\mathrm{P}}$ and $M_{\mathrm{D}}$ are the masses of the parent and
daughter atoms (neutral), and $m_{\mathrm{e}}$ is the mass of an electron
or positron.

Ben Nicholson

Numerade Educator

Problem 39

(I) $(a)$ What is the decay constant of 282 $\mathrm{U}$ whose half-life is
$4.5 \times 10^{9} \mathrm{yr} ?(b)$ The decay constant of a given nucleus is
$3.2 \times 10^{-5} \mathrm{s}^{-1} .$ What is its half-life?

Katie Mcalpine

Numerade Educator

Problem 40

(I) A radioactive material produces 1280 decays per minute
at one time, and 3.6 h later produces 320 decays per minute.
What is its half-life?

Farhanul Hasan

Numerade Educator

Problem 41

(I) What fraction of a sample of $\frac{68}{32} \mathrm{Ge},$ whose half-life is
about 9 months, will remain after 2.0 yr?

Katie Mcalpine

Numerade Educator

Problem 42

(I) What is the activity of a sample of $\frac{6}{14} \mathrm{C}$ that contains
$8.1 \times 10^{20}$ nuclei?

Farhanul Hasan

Numerade Educator

Problem 43

(1) What fraction of a sample is left after exactly
6 half-lives?

Katie Mcalpine

Numerade Educator

Problem 44

(II) A sample of $\frac{60}{27} \mathrm{Co}$ and a sample of $\frac{131 \mathrm{I}}{53} \mathrm{I}$ both have $N_{0}$
atoms at $t=0 .$ How long will it take until both have
the same activity? (Use Appendix: Selected Isotopes for
half-life data.)

Farhanul Hasan

Numerade Educator

Problem 45

(II) How many nuclei of $\frac{238}{92} \mathrm{U}$ remain in a rock if the activity
registers 340 decays per second?

Katie Mcalpine

Numerade Educator

Problem 46

(II) In a series of decays, the nuclide $\frac{235}{92} \mathrm{U}$ becomes $\frac{207}{82} P b$
How many $\alpha$ and $\beta^{-}$ particles are emitted in this series?

Farhanul Hasan

Numerade Educator

Problem 47

(II) The iodine isotope $\frac{131}{53} \mathrm{I}$ is used in hospitals for diagnosis
of thyroid function. If 782$\mu \mathrm{g}$ are ingested by a patient,
determine the activity $(a)$ immediately, $(b) 1.00 \mathrm{h}$ later when
the thyroid is being tested, and $(c) 4.0$ months later. Use
Appendix: Selected Isotopes.

Katie Mcalpine

Numerade Educator

Problem 48

(II) 1$\frac{124}{55} \mathrm{Cs}$ has a half-life of 30.8 $\mathrm{s}$ . (a) If we have 7.8$\mu \mathrm{g}$
initially, how many $\mathrm{Cs}$ nuclei are present? $(b)$ How many
are present 2.6 min later? (c) What is the activity at this
time? (d) After how much time will the activity drop to less
than about 1 per second?

Farhanul Hasan

Numerade Educator

Problem 49

(II) Calculate the mass of a sample of pure $\frac{40}{19} \mathrm{K}$ , with an
initial decay rate of $2.0 \times 10^{5} \mathrm{s}^{-1} .$ The half-life of $\frac{40}{19} \mathrm{K}$ is
$1.265 \times 10^{9} \mathrm{yr}$

Katie Mcalpine

Numerade Educator

Problem 50

(II) Calculate the activity of a pure 8.7 -\mug sample of
$\frac{32 \mathrm{P}}{12} \mathrm{P}\left(T_{\frac{1}{2}}=1.23 \times 10^{6} \mathrm{s}\right)$

Farhanul Hasan

Numerade Educator

Problem 51

(II) The activity of a sample of $\frac{35}{16} \mathrm{S}\left(T_{\frac{1}{2}}=87.32$ days \right$)$ is
$3.65 \times 10^{4}$ decays per second. What is the mass of the
sample?

Katie Mcalpine

Numerade Educator

Problem 52

(II) A sample of $\frac{233}{92} \mathrm{U} \quad\left(T_{2}=1.59 \times 10^{5} \mathrm{yr}\right)$ contains
$5.50 \times 10^{18}$ nuclei. $(a)$ What is the decay constant? $(b)$
Approximately how many disintegrations will occur per
minute?

Farhanul Hasan

Numerade Educator

Problem 53

(II) The activity of a sample drops by a factor of 4.0 in
8.6 minutes. What is its half-life?

Katie Mcalpine

Numerade Educator

Problem 54

(II) A 385 -g sample of pure carbon contains 1.3 parts in $10^{12}$
(atoms) of $^{14} \mathrm{C}$ . How many disintegrations occur per second?

Farhanul Hasan

Numerade Educator

Problem 55

(II) A sample of $\frac{238}{92} \mathrm{U}$ is decaying at a rate of
$3.70 \times 10^{2}$ decays/s. What is the mass of the sample?

Katie Mcalpine

Numerade Educator

Problem 56

(II) Rubidium-strontium dating. The rubidium isotope
$\frac{87}{37} \mathrm{Rb}$, a $\beta$ emitter with a half-life of $4.75 \times 10^{10}$ yr, is used to
determine the age of rocks and tossils. Rocks containing
fossils of ancient animals contain a ratio of $\frac{87}{38} \mathrm{Sr}$ to $\frac{87}{37} \mathrm{Rb}$ of
$0.0260 .$ Assuming that there was no $\frac{87}{38}$ sr present when the
rocks were formed, estimate the age of these fossils.

Farhanul Hasan

Numerade Educator

Problem 57

(II) The activity of a radioactive source decreases by 2.5$\%$ in
31.0 hours. What is the half-life of this source?

Katie Mcalpine

Numerade Educator

Problem 58

(II) $^{7}_{4}$ Be decays with a half-life of about 53 d. It is produced
in the upper atmosphere, and filters down onto the Earth's
surface. If a plant leaf is detected to have 350 decays/s of
$\frac{7}{4} \mathrm{Be},(a)$ how long do we have to wait for the decay rate to
drop to 15 per second? (b) Estimate the initial mass of $\frac{7}{4} \mathrm{Be}$
on the leaf.

Farhanul Hasan

Numerade Educator

Problem 59

(II) Two of the naturally occurring radioactive decay sequences start with ${ }_{90}^{232} \mathrm{Th}$ and with ${ }_{92}^{235} \mathrm{U}$. The first five decays of these two sequences are:
$$
\alpha, \beta, \beta, \alpha, \alpha
$$
and
$$
\alpha, \beta, \alpha, \beta, \alpha
$$
Determine the resulting intermediate daughter nuclei in each case.

Katie Mcalpine

Numerade Educator

Problem 60

(II) An ancient wooden club is found that contains 85 g of
carbon and has an activity of 7.0 decays per second. Deter-
mine its age assuming that in living trees the ratio of
14 $\mathrm{C} /^{12} \mathrm{C}$ atoms is about $1.3 \times 10^{-12}$ .

Farhanul Hasan

Numerade Educator

Problem 61

(III) At $t=0,$ a pure sample of radioactive nuclei contains
$N_{0}$ nuclei whose decay constant is $\lambda$ . Determine a formula
for the number of daughter nuclei, $N_{D},$ as a function of
time; assume the daughter is stable and that $N_{D}=0$ at
$t=0 .$

Katie Mcalpine

Numerade Educator

Problem 62

Which radioactive isotope of lead is being produced if the
measured activity of a sample drops to 1.050$\%$ of its original
activity in 4.00 $\mathrm{h} ?$

Farhanul Hasan

Numerade Educator

Problem 63

An old wooden tool is found to contain only 6.0$\%$ of the $\frac{14}{6} \mathrm{C}$
that an equal mass of fresh wood would. How old is the tool?

Katie Mcalpine

Numerade Educator

Problem 64

A neutron star consists of neutrons at approximately
nuclear density. Estimate, for a 10 -km-diameter neutron
star, $(a)$ its mass number, $(b)$ its mass $(\mathrm{kg}),$ and $(c)$ the accel-
eration of gravity at its surface.

Farhanul Hasan

Numerade Educator

Problem 65

Tritium dating. The $_{1}^{3} \mathrm{H}$ isotope of hydrogen, which is called
tritium (because it contains three nucleons), has a half-life
of 12.3 yr. It can be used to measure the age of objects up
to about 100 yr. It is produced in the upper atmosphere by
cosmic rays and brought to Earth by rain. As an applica-
tion, determine approximately the age of a bottle of wine
whose $_{1}^{3} \mathrm{H}$ radiation is about $\frac{1}{10}$ that present in new wine.

Katie Mcalpine

Numerade Educator

Problem 66

Some elementary particle theories suggest that the proton
may be unstable, with a half-life $\geq 10^{33}$ yr. How long would
you expect to wait for one proton in your body to decay
(approximate your body as all water)?

Farhanul Hasan

Numerade Educator

Problem 67

Show, using the decays given in Section 5 of "Nuclear
Physics and Radioactivity," that the neutrino has either spin
$\frac{1}{2}$ or $\frac{3}{2} .$

Katie Mcalpine

Numerade Educator

Problem 68

The original experiments which established that an atom
has a heavy, positive nucleus were done by shooting alpha
particles through gold foil. The alpha particles used had a
kinetic energy of 7.7 $\mathrm{MeV} .$ What is the closest they could get
to a gold nucleus? How does this compare with the size of
the nucleus?

Farhanul Hasan

Numerade Educator

Problem 69

How long must you wait (in half-lives) for a radioactive
sample to drop to 1.00$\%$ of its original activity?

Katie Mcalpine

Numerade Educator

Problem 70

If the potassium isotope $_{19}^{40} \mathrm{K}$ gives 45 $\mathrm{decays} / \mathrm{s}$ in a liter of
milk, estimate how much $_{19}^{40} \mathrm{K}$ and regular $\frac{39}{19} \mathrm{K}$ are in a liter
of milk. Use Appendix: Selected Isotopes.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

(a) In $\alpha$ decay of, say, a $\frac{226}{88}Ra$ nucleus, show that the nucleus
carries away a fraction 1$/\left(1+\frac{1}{4} A_{\mathrm{D}}\right)$ of the total energy
available, where $A_{\mathrm{D}}$ is the mass number of the daughter
nucleus. [Hint: Use conservation of momentum as well as
conservation of energy.] (b) Approximately what percentage
of the energy available is thus carried off by the $\alpha$ particle
when $\frac{226}{88}Ra$ decays?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 72

(a) In $\alpha$ decay of, say, a $\frac{226}{88}Ra$ nucleus, show that the nucleus
carries away a fraction 1$/\left(1+\frac{1}{4} A_{\mathrm{D}}\right)$ of the total energy
available, where $A_{\mathrm{D}}$ is the mass number of the daughter
nucleus. [Hint: Use conservation of momentum as well as
conservation of energy.] (b) Approximately what percentage
of the energy available is thus carried off by the $\alpha$ particle
when $\frac{226}{88}Ra$ decays?
it decays with a half-life of about 29 yr. How long will we
have to wait for the amount of $\frac{90}{38}Sr$ on the Earth's surface
to reach 1$\%$ of its current level, assuming no new material is
scattered about? Write down the decay reaction, including
the daughter nucleus. The daughter is radioactive: write
down its decay.

Farhanul Hasan

Numerade Educator

Problem 73

Using the uncertainty principle and the radius of a nucleus,
estimate the minimum possible kinetic energy of a nucleon
in, say, iron. Ignore relativistic corrections. [Hint: A particle
can have a momentum at least as large as its momentum
uncertainty.]

Katie Mcalpine

Numerade Educator

Problem 74

(a) Calculate the kinetic energy of the $\alpha$ particle emitted
when$\frac{236}{92} \mathrm{U}$ decays. $(b)$ Use Eq. 1 to estimate the radius of an
$\alpha$ particle and a $\frac{232}{90}$ Th nucleus. Use this to estimate $(c)$ the
maximum height of the Coulomb barrier, and $(d)$ its width
AB in Fig. $7 .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 75

The nuclide $\frac{191}{76} \mathrm{Os}$ decays with $\beta^{-}$ energy of 0.14 MeV
accompanied by $\gamma$ rays of energy 0.042 MeV and 0.129
MeV. $(a)$ What is the daughter nucleus? $(b)$ Draw an
energy-level diagram showing the ground states of the
parent and daughter and excited states of the daughter.
(c) To which of the daughter states does $\beta^{-}$ decay of $\frac{191}{76} \mathrm{Os}$
occur?

Katie Mcalpine

Numerade Educator

Problem 76

$\begin{array}{l}{\text { Determine the activities of }(a) 1.0 \mathrm{g} \text { of } \frac{131}{53} \mathrm{I}\left(T_{\frac{1}{2}}=8.02 \text { days) }\right.} \\ {\text { and }(b) 1.0 \mathrm{g} \text { of }_{92}^{238} \mathrm{U}\left(T_{1}=4.47 \times 10^{9} \mathrm{yr}\right)}\end{array}
$

Farhanul Hasan

Numerade Educator

Problem 77

Use Fig. 1 to estimate the total binding energy for copper
and then estimate the energy, in joules, needed to break a
3.0 -g copper penny into its constituent nucleons.

Katie Mcalpine

Numerade Educator

Problem 78

Instead of giving atomic masses for nuclides as in Appendix:
Selected Isotopes, some Tables give the mass excess, $\Delta$ ,
defined as $\Delta=M-A,$ where $A$ is the atomic mass
number and $M$ is the mass in u. Determine the mass excess,
in $u$ and in MeV $/ c^{2},$ for: $(a)_{2}^{4} \mathrm{He;}$ $(b)_{13}^{6} \mathrm{C;}$ $c)_{86}^{38} \mathrm{Sr;}$$(d)_{235}^{92} \mathrm{U.}$
(e) From a glance at Appendix: Selected Isotopes, can you
make a generalization about the sign of $\Delta$ as a function
of $Z$ or $A ?$

Farhanul Hasan

Numerade Educator

Problem 79

When water is placed near an intense neutron source,
the neutrons can be slowed down by collisions with the
water molecules and eventually captured by a hydrogen
nucleus to form the stable isotope called deuterium,
$_{1}^{2} \mathrm{H}$ giving off a gamma ray. What is the energy of the
gamma ray?

Katie Mcalpine

Numerade Educator

Problem 80

(a) Show that the mean life of a radioactive nuclide,
defined as
$$\tau=\frac{\int_{0}^{\infty} t N(t) d t}{\int_{0}^{\infty} N(t) d t}$$
is $\tau=1 / \lambda$ . (b) What fraction of the original number of
nuclei remains after one mean life?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 81

(a) A 72 -gram sample of natural carbon contains the usual
fraction of $\frac{14}{6} \mathrm{C}$. Estimate how long it will take before there
is only one $\frac{14}{6} \mathrm{C}$ nucleus left. (b) How does the answer in
(a) change if the sample is 270 grams? What does this tell
you about the limits of carbon dating?

Katie Mcalpine

Numerade Educator

Problem 82

If the mass of the proton were just a little closer to the mass
of the neutron, the following reaction would be possible
even at low collision energies:
$$\mathrm{e}^{-}+\mathrm{p} \rightarrow \mathrm{n}+\nu$$
(a) Why would this situation be catastrophic? $(b)$ By what
percentage would the proton's mass have to be increased to
make this reaction possible?

Farhanul Hasan

Numerade Educator

Problem 83

What is the ratio of the kinetic energies for an alpha particle
and a beta particle if both make tracks with the same radius
of curvature in a magnetic field, oriented perpendicular to
the paths of the particles?

Katie Mcalpine

Numerade Educator

Problem 84

A 1.00 -g sample of natural samarium emits $\alpha$ particles at a
rate of 120 $\mathrm{s}^{-1}$ due to the presence of $_{62}^{147} \mathrm{Sm.}$ The natural
abundance of $_{62}^{147} \mathrm{Sm}$ is 15$\%$ . Calculate the half-life for this
decay process.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 85

Almost all of naturally occurring uranium is $\frac{238}{92} \mathrm{U}$ with a
half-life of $4.468 \times 10^{9}$ yr. Most of the rest of natural
uranium is $\frac{235}{92} \mathrm{U}$ with a half-life of $7.04 \times 10^{8} \mathrm{yr.}$ Today a
sample contains0.720$\%_{0} \frac{235}{92} U$ (a) What was this percentage
1.0 billion years ago? (b) What percentage of$\frac{235}{92} \mathrm{U}$ will
remain 100 million years from now?

Katie Mcalpine

Numerade Educator

Problem 86

A typical banana contains 400 $\mathrm{mg}$ of potassium, of which
a small fraction is the radioactive isotope $\frac{40}{19} \mathrm{K}$ ( see
Appendix: Selected Isotopes). Estimate the activity of an
average banana due to $\frac{40}{19} \mathrm{K.}$

Farhanul Hasan

Numerade Educator

Problem 87

Some radioactive isotopes have half-lives that are larger than the age of the universe (like gadolinium or samarium). The only way to determine these half-lives is to monitor the decay rate of a sample that contains these isotopes. For example, suppose we find an asteroid that currently contains
about $15,000 \mathrm{kg}$ of $\frac{152}{64} \mathrm{Gd}$ (gadolinium) and we detect an activity of 1 decay/s. What is the half-life of gadolinium (in
years)? activity of 1 decay/s. What is the half-life of gadolinium (in years)?

Katie Mcalpine

Numerade Educator

Problem 88

Decay series, such as that shown in Fig. $12,$ can be classified
into four families, depending on whether the mass numbers
have the form $4 n, 4 n+1,4 n+2,$ or $4 n+3,$ where $n$ is an
integer. Justify this statement and show that for a nuclide in
any family, all its daughters will be in the same family.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 89

(I) A laboratory has a $1.80-\mu \mathrm{g}$ sample of radioactive $\frac{13}{7} \mathrm{N}$
whose decay constant $\lambda=1.16 \times 10^{-3} \mathrm{s
}^{-1} .$ Calculate the
initial number of nuclei, $N_{0},$ present in the sample. Use
the radioactive decay law, $N=N_{0} e^{-\lambda t},$ to determine th
number of nuclei $N$ present at time $t$ for $t=0$ th
30 minutes $(1800$ s) in steps of 0.5 $\min (30$ s). Make a grap
of $N$ versus $t$ and from the graph determine the half-life
the sample.

Lainey Roebuck

Numerade Educator

Problem 90

(II) Construct a spreadsheet (or other numerical tool) that
will reproduce Fig. $1,$ the graph of binding energy per nucleon
(in MeV) vs. the mass number $A .$ Using Appendix: Selected
Isotopes, calculate the binding energy per nucleon for the
most stable isotope of each possible mass number $A \geq 2$
[The first few values will be for $_{1}^{2} \mathrm{H}, _{2}^{3} \mathrm{He}$ (it is more stable
than $_{1}^{3} \mathrm{H} ),_{2}^{4} \mathrm{He},_{3}^{6} \mathrm{Li},$ and $_{3}^{7}$ Li (since it is more stable than $_{4}^{7} \mathrm{Be} ) . ]$
To reduce the amount of data, for $A \geq 20$ plot only points for
even values of $A,$ and plot to a maximum of $A=142 .$

Lainey Roebuck

Numerade Educator