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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 29

Nuclear Physics

Educators


Problem 1

Determine the number of (a) electrons, (b) protons, and (c) neutrons in iron $\left(\frac{56}{26} \mathrm{Fe}\right)$

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Problem 2

The atomic mass of an oxygen atom is 15.999 u. Convert this mass to units of (a) kilograms and (b) $\mathrm{MeV} / \mathrm{c}^{2}$

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Problem 3

Find the nuclear radii of the following nuclides: (a) $_{1}^{2} \mathrm{H}$ (b) $^{60}_{27} \mathrm{Co}$ (c) $^{197}_{79} \mathrm{Au}$ (d) $_{94}^{239} \mathrm{Pu}$

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Problem 4

Find the radius of a nucleus of (a) $_{2}^{4}$He and (b) $^{238}_{93}$U

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Problem 5

Using $2.3 \times 10^{17} \mathrm{kg} / \mathrm{m}^{3}$ as the density of nuclear matter, find
the radius of a sphere of such matter that would have a mass equal to that of Earth. Earth has a mass equal to $5.98 \times 10^{24} \mathrm{kg}$ and average radius of $6.37 \times 10^{6} \mathrm{m}$

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Problem 6

Consider the $^{65}_{29} \mathrm{Cm}$ nucleus. Find approximate values for its (a) radius, (b) volume, and (c) density.

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Problem 7

An alpha particle $\left(Z=2, \text { mass }=6.64 \times 10^{-27} \mathrm{kg}\right)$ approaches to within $1.00 \times 10^{-14} \mathrm{m}$ of a carbon nucleus (Z = 6). What are (a) the maximum Coulomb force on the alpha particle, (b) the acceleration of the alpha particle at this time, and (c) the potential energy of the alpha particle at the same time?

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Problem 8

Singly ionized carbon atoms are accelerated through $1.00 \times 10^{3} \mathrm{V}$ and passed into a mass spectrometer to determine the isotopes present. (See Topic 19.) The magnetic field strength in the spectrometer is 0.200 T. (a) Determine the orbital radii for the $^{12} \mathrm{C}$ and the $^{13} \mathrm{C}$ isotopes as they pass through the field. (b) Show that the ratio of the radii may be written in the form
$$\frac{r_{1}}{r_{2}}=\sqrt{\frac{m_{1}}{m_{2}}}$$
and verify that your radii in part (a) satisfy this formula.

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Problem 9

(a) Find the speed an alpha particle requires to come within $3.2 \times 10^{-14} \mathrm{m}$ of a gold nucleus. (b) Find the energy of the alpha particle in MeV.

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Problem 10

At the end of its life, a star with a mass of two times the Sun’s mass is expected to collapse, combining its protons and electrons to form a neutron star. Such a star could be thought of as a gigantic atomic nucleus. If a star of mass $2 \times 1.99 \times$ $10^{30} \mathrm{kg}$ collapsed into neutrons $\left(m_{n}=1.67 \times 10^{-27} \mathrm{kg}\right)$ what would its radius be? Assume $r=r_{0} A^{1 / 3}$.

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Problem 11

Find the average binding energy per nucleon of (a) $_{12}^{24} \mathrm{Mg}$ and (b) $^{85}_{37} \mathrm{Rb}$

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Problem 12

Calculate the binding energy per nucleon for (a) $^{2} \mathrm{H}$, (b) $^{4} \mathrm{He}$ (C) $^{56} \mathrm{Fe}$, (d) $^{238} \mathrm{U}$.

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Problem 13

A pair of nuclei for which $Z_{1}=N_{2}$ and $Z_{2}=N_{1}$ are called mirror isobars. (The atomic and neutron numbers are interchangeable.) Binding - energy measurements on such pairs can be used to obtain evidence of the charge independence of nuclear forces. Charge independence means that the proton–proton, proton–neutron, and neutron–neutron forces are approximately equal. Calculate the difference in binding energy for the two mirror nuclei $^{15}_{8} \mathrm{O}$ and $_{7}^{15} \mathrm{N}$.

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Problem 14

The peak of the stability curve occurs at $^{56} \mathrm{Fe},$ which is why iron is prominent in the spectrum of the Sun and stars. Show that $^{56} \mathrm{Fe},$ has a higher binding energy per nucleon has a higher binding energy per nucleon than its neighbors $^{55} \mathrm{Mn}$ and $^{59} \mathrm{Co}$. Compare your results with Figure 29.4.

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Problem 15

Two nuclei having the same mass number are known as isobars. (a) Calculate the difference in binding energy per nucleon for the isobars $^{23}_{11} \mathrm{Na},$ and $^{23}_{12} \mathrm{Mg},$ (b) How do you account for this difference? (The mass of $^{23}_{12} \mathrm{Mg}=22.994127 \mathrm{u} . )$

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Problem 16

Calculate the binding energy of the last neutron in the $^{43}_{20} \mathrm{Ca}$ nucleus. Hint: You should compare the mass of $^{43}_{20} \mathrm{Ca}$ with the mass of $^{43}_{20} \mathrm{Ca}$ plus the mass of a neutron. The mass of $_{20}^{12} \mathrm{Ca}=$ 41.958 622 u, whereas the mass of $\underset{20}{43} \mathrm{Ca}=42.958770 \mathrm{u}$

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Problem 17

Radon gas has a half - life of 3.83 days. If 3.00 g of radon gas is present at time t 5 0, what mass of radon will remain after 1.50 days have passed?

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Problem 18

A drug tagged with $^{99}_{43} $\mathrm{Tc}$ (half - life 5 6.05 h) is prepared for a patient. If the original activity of the sample was 1.1 3 $10^{4} \mathrm{Bq}$ what is its activity after it has been on the shelf for
2.0 h?

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Problem 19

The half-life of $^{131} \mathrm{I}$ is 8.04 days. ( a ) Convert the half-life to seconds. (b) Calculate the decay constant for this isotope. (c) Convert 0.500 $\mu \mathrm{Ci}$ to the SI unit the becquerel. (d) Find the number of $^{131} \mathrm{I}$ nuclei necessary to produce a sample with an activity of 0.500$\mu \mathrm{Ci} .(\mathrm{e})$ Suppose the activity of a certain $^{131} \mathrm{I}$ is 6.40 $\mathrm{mCi}$ at a given time. Find the number of half-lives the sample goes through in 40.2 $\mathrm{d}$ and the activity at the end of that period.

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Problem 20

Tritium has a half - life of 12.33 years. What fraction of the nuclei in a tritium sample will remain (a) after 5.00 yr? (b) After 10.0 yr? (c) After 123.3 yr? (d) According to Equation 29.4a, an infinite amount of time is required for the entire sample to decay. Discuss whether that is realistic.

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Problem 21

After 2.00 days, the activity of a sample of an unknown type radioactive material has decreased to 84.2% of the initial activity. What is the half - life of this material?

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Problem 22

After a plant or animal dies, its $^{14} \mathrm{C}$ content decreases with a half - life of 5 730 yr. If an archaeologist finds an ancient fire pit containing partially consumed firewood and the $^{14} \mathrm{C}$ Content of the wood is only 12.5$\%$ that of an equal carbon sample from a present-day tree, what is the age of the ancient site?

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Problem 23

A freshly prepared sample of a certain radioactive isotope has an activity of 10.0 mCi. After 4.00 h, the activity is 8.00 mCi. (a) Find the decay constant and half - life of the isotope. (b) How many atoms of the isotope were contained in the freshly prepared sample? (c) What is the sample’s activity in mCi 30.0 h after it is prepared?

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Problem 24

A building has become accidentally contaminated with radioactivity. The longest - lived material in the building is strontium - 90. (The atomic mass of $\underset{38}{90} \mathrm{Sr}$ is 89.907 7 u.) If the
building initially contained 5.0 kg of this substance and the safe level is less than 10.0 counts/min, how long will the building be unsafe?

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Problem 25

Chromium’s radioactive isotope $^{51} \mathrm{Cr}$ has a half - life of 27.7 days and is often used in nuclear medicine as a diagnostic tracer in blood studies. Suppose a $^{51} \mathrm{Cr}$ sample
has an activity of 2.00 $\mu \mathrm{Ci}$ when it is placed on a storage shelf. (a) How many $^{51} \mathrm{Cr}$ nuclei does the sample contain? (b) Calculate the sample’s activity in Bq when it is removed from storage one year later.

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Problem 26

On March 11, 2011, a magnitude 9.0 earthquake struck northwest Japan. The tsunami that followed left thousands of people dead and triggered a meltdown at the Fukushima Daiichi Nuclear Power Plant, releasing radioactive isotopes $^{137} \mathrm{Cs}$ and $^{134} \mathrm{Cs}$ among others, into the atmosphere and into the Pacific Ocean. By December 2015 (about 1 730 days after the meltdown), contaminated seawater reached the U.S. west coast with maximum Cs activities (including both isotopes) per cubic meter of seawater reaching 11.0 Bq/ m3 , more than 500 times below the U.S. government safety limits for drinking water. The half - lives of $^{137} \mathrm{Cs}$ and $^{134} \mathrm{Cs}$ are $1.10 \times 10^{4}$ days and 734 days, respectively. Calculate the number of (a) $^{137} \mathrm{Cs}$ and (b) $^{134} \mathrm{Cs}$ nuclei in the 1.00 $\mathrm{m}^{3}$ seawater sample, assuming $^{137} \mathrm{Cs}$ and $^{134} \mathrm{Cs}$ were originally released in equal amounts.

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Problem 27

Identify the missing nuclides in the following decays:
a) $^{212}_{83}\mathrm{Bi} \rightarrow ?+^{4}_{2} \mathrm{He}$
b) $^{95}_{36}\mathrm{Kr} \rightarrow ?+\mathrm{e}^{-}+\overline{\nu}$
c) $? \rightarrow_{2}^{4} \mathrm{He}+_{\mathrm{58}}^{140} \mathrm{Ce}$

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Problem 28

Complete the following radioactive decay formulas:
a) $^{12}_{5} \mathbf{B} \rightarrow ?+\mathbf{e}^{-}+\overline{\nu}$
b) $^{234}_(90) \mathrm{Th} \rightarrow_{88}^{230} \mathrm{Ra}+?$
c) $? \rightarrow^{14}_{7} \mathrm{N}+\mathrm{e}^{-}+\overline{\nu}$

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Problem 29

The mass of $^{56} \mathrm{Fe}$ is $55.9349 \mathrm{u},$ and the mass of $^{56} \mathrm{Co}$ is
55.9399 u. Which isotope decays into the other and by what process?

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Problem 30

Find the energy released in the alpha decay of $^{238}_{92} \mathrm{U}$ . The following mass value will be useful: $^{234}_{90}$ Th has a mass of 234.043 583 u.

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Problem 31

Determine which of the following suggested decays can occur spontaneously:
(a) $_{20}^{40} \mathrm{Ca} \rightarrow \mathrm{c}^{+}+_{19}^{40} \mathrm{K}$ (b) $_{60}^{144} \mathrm{Nd} \rightarrow_{2}^{4} \mathrm{He}+\stackrel{140}{58} \mathrm{Ce}$

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Problem 32

$\underset{28}{66} \mathrm{Ni}(\text { mass }=65.9291 \mathrm{u})$ undergoes beta decay to $^{66}_{29} \mathrm{Cu}$ (mass 5 65.928 9 u). (a) Write the complete decay formula for this process. (b) Find the maximum kinetic energy of the emerging electrons.

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Problem 33

A $^{3} \mathrm{H}$ (tritium) nucleus beta decays into $^{3} \mathrm{He}$ by creating an electron and an antineutrino according to the reaction
$$_{1}^{3} \mathrm{H} \quad \rightarrow \quad_{2}^{3} \mathrm{He}+\mathrm{e}^{-}+\overline{\nu}$$ Use Appendix B to determine the total energy released in this reaction.

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Problem 34

In the decay $^{234}_{90} \mathrm{Th} \rightarrow_{Z}^{A} \mathrm{Ra}+_{2}^{4} \mathrm{He}$ identify (a) the mass number (by balancing mass numbers) and (b) the atomic number (by balancing atomic numbers) of the Ra nucleus.

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Problem 35

A wooden artifact is found in an ancient tomb. Its carbon - 14 $\left(\begin{array}{c}{14} \\ {6}\end{array}\right)$ activity is measured to be 60.0% of that in a fresh sample of wood from the same region. Assuming the same amount of $^{14} \mathrm{C}$ was initially present in the artifact as is
now contained in the fresh sample, determine the age of the artifact.

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Problem 36

A beam of 6.61 -MeV protons is incident on a target of $^{27}_{13} \mathrm{Al}$ Those protons that collide with the target produce the reaction
$$\mathrm{p}+_{13}^{27} \mathrm{Al} \rightarrow_{14}^{27} \mathrm{Si}+\mathrm{n}$$
$\left(_{14}^{27} \mathrm{Si} \text { has a mass of } 26.986721 \mathrm{u} .\right)$ Neglecting any recoil of the product nucleus, determine the kinetic energy of the emerging neutrons.

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Problem 37

Identify the unknown particles X and X' in the following nuclear reactions:
a) $\mathrm{X}+_{2}^{4} \mathrm{Hc} \rightarrow_{12}^{24} \mathrm{Mg}+_{0}^{1} \mathrm{n}$
b) $_{92}^{295} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow_{38}^{90} \mathrm{Sr}+\mathrm{X}+2_{0}^{1} \mathrm{n}$
c) $2_{1}^{1} \mathrm{H} \rightarrow \frac{2}{1} \mathrm{H}+\mathrm{X}+\mathrm{X}^{\prime}$

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Problem 38

One method of producing neutrons for experimental use is to bombard $_{3}^{7}$Li with protons. The neutrons are emitted
$$_{1}^{1} \mathrm{H}+_{3}^{7} \mathrm{Li} \rightarrow_{4}^{7} \mathrm{Be}+_{0}^{1} \mathrm{n}$$
(a) Calculate the mass in atomic mass units of the particles on the left side of the equation. (b) Calculate the mass (in atomic mass units) of the particles on the right side of the equation.
(c) Subtract the answer for part (b) from that for part (a) and convert the result to mega electron volts, obtaining the Q value for this reaction. (d) Assuming lithium is initially at rest, the proton is moving at velocity v, and the resulting beryllium and neutron are both moving at velocity V after the collision, write an expression describing conservation of momentum for this reaction in terms of the masses $m_{p}, m_{B}, m_{n},$ and the velocities. (e) Write an expression relating the kinetic energies of particles before and after together with Q. (f ) What minimum kinetic energy must the incident proton have if this reaction is to occur?

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Problem 39

(a) Suppose $^{10}_{5} \mathrm{B}$ is struck by an alpha particle, releasing a proton and a product nucleus in the reaction. What is the product nucleus? (b) An alpha particle and a product nucleus are
produced when $^{13}_{6} \mathrm{C}$ is struck by a proton. What is the product nucleus?

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Problem 40

Consider two reactions:
$(1) \mathrm{n}+\frac{2}{3} \mathrm{H} \rightarrow_{1}^{3} \mathrm{H}$
(2) $_{1}^{1} \mathrm{H}+_{1}^{2} \mathrm{H} \rightarrow_{2}^{3} \mathrm{He}$
(a) Compute the Q values for these reactions. Identify whether each reaction is exothermic or endothermic. (b) Which reaction results in more released energy? Why? (c) Assuming the difference is primarily due to the work done by the electric force, calculate the distance between the two protons in
helium - 3.

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Problem 41

Natural gold has only one isotope, $^{197}_{79}$ Au. If gold is bombarded with slow neutrons, $e^{-}$ particles are emitted. (a) Write the appropriate reaction equation. (b) Calculate the maximum energy of the emitted beta particles. The mass of $_{80}^{198} \mathrm{Hg}$ is 197.966 75 u.

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Problem 42

Complete the following nuclear reactions:
(a) $?+\frac{14}{7} \mathrm{N} \rightarrow_{1}^{1} \mathrm{H}+_{8}^{17} \mathrm{O} \quad(\mathrm{b})_{3}^{7} \mathrm{L} \mathrm{i}+_{1}^{1} \mathrm{H} \rightarrow_{2}^{4} \mathrm{He}+?$

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Problem 43

(a) Determine the product of the reaction $_{3}^{7} \mathrm{Li}+_{2}^{4} \mathrm{He} \rightarrow ?+\mathrm{n}$
(b) What is the Q value of the reaction?

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Problem 44

In terms of biological damage, how many rad of heavy ions are equivalent to 100 rad of x - rays?

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Problem 45

A person whose mass is 75.0 kg is exposed to a whole - body dose of 25.0 rad. How many joules of energy are deposited in the person’s body?

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Problem 46

A 200. - rad dose of radiation is administered to a patient in an effort to combat a cancerous growth. Assuming all the energy deposited is absorbed by the growth, (a) calculate the amount of energy delivered per unit mass. (b) Assuming the growth has a mass of 0.25 kg and a specific heat equal to that of water, calculate its temperature rise.

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Problem 47

A “clever” technician decides to heat some water for his coffee with an x - ray machine. If the machine produces 10. rad/s, how long will it take to raise the temperature of a cup of water by 50.8 C? Ignore heat losses during this time.

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Problem 48

An x - ray technician works 5 days per week, 50 weeks per year. Assume the technician takes an average of eight x - rays per day and receives a dose of 5.0 rem/yr as a result. (a) Estimate the dose in rem per x - ray taken. (b) How does this result compare with the amount of low - level background radiation the technician is exposed to?

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Problem 49

A patient swallows a radiopharmaceutical tagged with phosphorus - 32 $\left(\begin{array}{c}{32} \\ {15}\end{array} P\right), \quad a \beta^{-}$ emitter with a half - life of 14.3 days. The average kinetic energy of the emitted electrons is $7.00 \times 10^{2} \mathrm{keV}$ If the initial activity of the sample is 1.31 MBq, determine (a) the number of electrons emitted in a 10.0-day period, (b) the total energy deposited in the body during the 10.0 days, and (c) the absorbed dose if the electrons are completely absorbed in $1 \times 10^{2} \mathrm{g}$ of tissue.

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Problem 50

A particular radioactive source produces 100. mrad of 2 - MeV gamma rays per hour at a distance of 1.0 m. (a) How long could a person stand at this distance before accumulating an intolerable dose of 1.0 rem? (b) Assuming the gamma radiation is emitted uniformly in all directions, at what distance
would a person receive a dose of 10. mrad/h from this source?

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Problem 51

A radioactive sample contains 3.50$\mu \mathrm{g}$ of pure $^{11} \mathrm{C},$ which has a half - life of 20.4 min. (a) How many moles of $^{11} \mathrm{C}$ are present initially? (b) Determine the number of nuclei present initially. What is the activity of the sample (c) initially and (d) after 8.00 h?

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Problem 52

Find the threshold energy that the incident neutron must have to produce the reaction: $:_{0}^{1} n+\frac{4}{2} H e \rightarrow_{1}^{2} H+_{1}^{3} H$

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Problem 53

A 200.0 - mCi sample of a radioactive isotope is purchased by a medical supply house. If the sample has a half - life of 14.0 days, how long will it keep before its activity is reduced to 20.0 mCi?

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Problem 54

The $^{14} \mathrm{C}$ isotope undergoes beta decay according to the process given by Equation 29.15. Find the Q value for this process.

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Problem 55

In a piece of rock from the Moon, the $^{87} \mathrm{Rb}$ content is assayed to be $1.82 \times 10^{10}$ atoms per gram of material and the $^{87}$ Sr content is found to be $1.07 \times 10^{9}$ atoms per gram. (The relevant decay is $^{87} \mathrm{Rb} \rightarrow^{87} \mathrm{Sr}+\mathrm{e}^{-} .$ The half-life of the decay is $4.8 \times 10^{10}$ yr.) (a) Determine the age of the rock. (b) Could the material in the rock actually be much older? (c) What assumption is implicit in using the radioactive -dating method?

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Problem 56

Many radioisotopes have important industrial, medical, and research applications. One of these is $^{60} \mathrm{Co},$ which has a half - life of 5.2 years and decays by the emission of a beta particle (energy 0.31 MeV) and two gamma photons (energies 1.17 MeV and 1.33 MeV). A scientist wishes to prepare a
$^{60} \mathrm{Co}$ sealed source that will have an activity of at least 10 Ci after 30 months of use. What is the minimum initial mass of $^{60} \mathrm{Co}$ required?

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Problem 57

A medical laboratory stock solution is prepared with an initial activity due to $^{24} \mathrm{Na}$ of 2.5 $\mathrm{mCi} / \mathrm{mL}$ , and 10.0 $\mathrm{mL}$ of the stock solution is diluted at $t_{0}=0$ to a working solution whose total volume is 250 mL. After 48 h, a 5.0 - mL sample of the working solution is monitored with a counter. What is the measured activity? Note: 1 mL 5 1 milliliter.

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Problem 58

After the sudden release of radioactivity from the Chernobyl nuclear reactor accident in 1986, the radioactivity of milk in Poland rose to $2.00 \times 10^{3} \mathrm{Bq} / \mathrm{L}$ due to iodine-131, with a half - life of 8.04 days. Radioactive iodine is particularly hazardous because the thyroid gland concentrates iodine. The Chernobyl accident caused a measurable increase in thyroid cancers among children in Belarus. (a) For comparison, find the activity of milk due to potassium. Assume 1 liter of milk
contains 2.00 g of potassium, of which 0.011 7% is the isotope $^{40} \mathrm{K},$ which has a half-life of $1.28 \times 10^{9} \mathrm{yr}$ (b) After what length of time would the activity due to iodine fall below that due to potassium?

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Problem 59

The theory of nuclear astrophysics is that all the heavy elements like uranium are formed in the interior of massive stars. These stars eventually explode, releasing the elements into space. If we assume that at the time of explosion there were equal amounts of $^{235} \mathrm{U}$ and $^{238} \mathrm{U},$ how long ago were the elements that formed our Earth released, given that the present $^{235} \mathrm{U} /^{238} \mathrm{U}$ ratio is 0.007? (The half-lives of $^{235} \mathrm{U}$ and $^{238} \mathrm{U}$ are $0.70 \times$ $10^{9}$ yr and $4.47 \times 10^{9}$ yr, respectively.)

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Problem 60

A by-product of some fission reactors is the isotope $_{94}^{299} \mathrm{Pu}$ , which is an alpha emitter with a half-life of 24 000 years:
$$_{94}^{239} \mathrm{Pu} \quad \rightarrow \quad_{92}^{235} \mathrm{U}+_{2}^{4} \mathrm{He}$$
Consider a sample of 1.0 kg of pure $^{239}_{94} \mathrm{Pu}$ at $t=0$ . Calculate (a) the number of $^{239}_{94} \mathrm{Pu}$ nuclei present at $t=0$ and $(b)$ the initial activity of the sample. (c) How long does the sample have to be stored if a “safe” activity level is 0.10 Bq?

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Problem 61

After how many half-lives will (a) 10.0%, (b) 5.00%, and (c) 1.00% of a radioactive sample remain?

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Problem 62

A piece of charcoal used for cooking is found at the remains of an ancient campsite. A 1.00-kg sample of carbon from the wood has an activity equal to $5.00 \times 10^{2}$ decays per minute. Find the age of the charcoal. Hint: Living material has an activity equal to 15.0 decays/min per gram of carbon present.

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