Physics

James S. Walker

Chapter 32

Nuclear Physics and Nuclear Radiation - all with Video Answers

Educators

Chapter Questions

Problem 1

ldentify $Z, N,$ and $A$ for the following isotopes: (a) ${ }^{238} \mathrm{U}$, (b) ${ }_{94}^{239} \mathrm{Pu}$, (c) ${ }^{144} \mathrm{Nd}$.

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

Identify $Z, N,$ and $A$ for the following isotopes: (a) ${ }^{20} 80 \mathrm{Hg}$, (b) ${ }^{2}{ }_{86}^{20} \mathrm{Rn},$ (c) ${ }_{4}^{93} \mathrm{Nb}$.

Narayan Hari

Numerade Educator

Problem 3

What are the nuclear radii of (a) 197 Au and (b) $\frac{60}{27}$ Co?

Narayan Hari

Numerade Educator

Problem 4

A certain chlorine nucleus has a radius of approximately $4.0 \times 10^{-15} \mathrm{m} .$ How many neutrons are in this nucleus?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 5

(a) What is the nuclear density of ${ }^{228}$ Th? (b) Do you expect the nuclear density of an alpha particle to be greater than, less than, or the same as that of ${ }^{228}$Th? Explain. (c) Calculate the nuclear density of an alpha particle.

Narayan Hari

Numerade Educator

Problem 6

(a) What initial kinetic energy must an alpha particle have if it is to approach a stationary gold nucleus to within a distance of $22.5 \mathrm{fm} ?$ (b) If the initial speed of the alpha particle is reduced by a factor of $2,$ by what factor is the distance of closest approach changed? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

(a) What initial kinetic energy must an alpha particle have if it is to approach a stationary gold nucleus to within a distance of $22.5 \mathrm{fm} ?$ (b) If the initial speed of the alpha particle is reduced by a factor of $2,$ by what factor is the distance of closest approach changed? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

An $\alpha$ particle with a kinetic energy of 0.85 MeV approaches a stationary gold nucleus. (a) What is the speed of the $\alpha$ particle? (b) What is the distance of closest approach between the $\alpha$ particle and the gold nucleus? $(c)$ If this same $\alpha$ particle were fired at a copper nucleus instead, would its distance of closest approach be greater than, less than, or the same as that found in part (b)? Explain. (To obtain the mass of an alpha particle, refer to Appendix $F$ and subtract the mass of two electrons from the mass of $^{4}_{2}$ He.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

(a) What initial kinetic energy must an alpha particle have if it is to approach a stationary gold nucleus to within a distance of $22.5 \mathrm{fm} ?$ (b) If the initial speed of the alpha particle is reduced by a factor of $2,$ by what factor is the distance of closest approach changed? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

Suppose a marble with a radius of $1.5 \mathrm{cm}$ has the density of a nucleus, as given in Example $32-2$. (a) What is the mass of this marble? (b) How many of these marbles would be required to have a mass equal to the mass of Earth?

Narayan Hari

Numerade Educator

Problem 9

(a) Find the nuclear radius of $_{ 15}^{30}$ P. (b) What mass number would be required for a nucleus to have twice the radius found in part (a)? (c) Verify your answer to part (b) with an explicit calculation.

Narayan Hari

Numerade Educator

Problem 10

An alpha particle is the nucleus of a $^{4}_{2}$ He atom. (a) How many nucleons are in a nucleus with twice the radius of an alpha particle? Explain. (b) Write the symbol for a phosphorus nucleus that has twice the radius of an alpha particle.

Narayan Hari

Numerade Educator

Problem 11

Suppose a uranium-236 nucleus undergoes fission by splitting into two smaller nuclei of equal size. (a) Is the radius of each of the smaller nuclei one-half, more than one-half, or less than one-half the radius of the uranium-236 nucleus? Explain. (b) Calculate the radius of the uranium-236 nucleus. (c) Calculate the radii of the two smaller nuclei.

Narayan Hari

Numerade Educator

Problem 12

A hypothetical nucleus weighs 1 lb. (a) How many nucleons are in this nucleus? (b) What is the radius of this nucleus?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 13

Consider a nucleus that undergoes $\alpha$ decay. (a) Is the radius of the resulting daughter nucleus greater than, less than, or equal to the radius of the original nucleus? (b) Choose the best explanation from among the following:
I. The decay adds an alpha particle to the nucleus, causing its radius to increase.
II. When the nucleus undergoes decay it ejects two neutrons and two protons. This decreases the number of nucleons in the nucleus, and therefore its radius will decrease.
III. An $\alpha$ decay leaves the number of nucleons unchanged. As a result, the radius of the nucleus stays the same.

Narayan Hari

Numerade Educator

Problem 14

Consider a nucleus that undergoes $\beta$ decay. (a) Is the radius of the resulting daughter nucleus greater than, less than, or the same as that of the original nucleus? (b) Choose the best explanation from among the following:
I. Capturing a $\beta$ particle will cause the radius of a nucleus to increase. Therefore, the daughter nucleus has the greater radius.
II. The original nucleus emits a $\beta$ particle, and anytime a particle is emitted from a nucleus the result is a smaller radius. Therefore, the radius of the daughter nucleus is less than the radius of the original nucleus.
III. When a nucleus emits a $\beta$ particle a neutron is converted to a proton, but the number of nucleons is unchanged. As a result, the radius of the daughter nucleus is the same as that of the original nucleus.

Narayan Hari

Numerade Educator

Problem 15

Which of the three decay processes $(\alpha, \beta$ or $\gamma$ ) results in a new element? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

Complete the following nuclear reaction:
$$^{7}_{3} \mathrm{Li}+{ }_{1}^{1} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+?$$

Narayan Hari

Numerade Educator

Problem 17

Complete the following nuclear reaction:
$${ }_{90}^{234} \mathrm{Th} \rightarrow{ }_{88}^{28} \mathrm{Ra}+?$$

Narayan Hari

Numerade Educator

Problem 18

Complete the following nuclear reaction:
$? \rightarrow_{7 } ^{14} \mathrm{N}+\mathrm{e}^{-}+\bar{v}$

Narayan Hari

Numerade Educator

Problem 19

One possible decay series for ${ }_{92}^{238} \mathrm{U}$ is ${ }_{90}^{234} \mathrm{Th},{ }_{91}^{234} \mathrm{Pa},{ }_{92}^{234} \mathrm{U}$ $^{230}_{90} \mathrm{Th}, \quad ^{226}_{88} \mathrm{Ra}, \quad ^{222}_{86} \mathrm{Rn}$ ${ }_{84}^{218} \mathrm{Po}, \quad{ }_{85}^{218} \mathrm{At}, \quad{ }_{86}^{218} \mathrm{Rn}$ ${ }_{84}^{214} \mathrm{Po}, \quad{ }_{82}^{210} \mathrm{Pb}, \quad{ }_{83}^{21} \mathrm{Bi}$ $^{206}_{81} \mathrm{T}$, and ${ }_{82}^{206} \mathrm{Pb}$. Identify, in the order given, each of the $14 \mathrm{de}$ cavs that occur in this series.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

Complete the following nuclear reaction and determine the amount of energy it releases:
$${ }_{1}^{3} \mathrm{H} \rightarrow{ }_{2}^{3} \mathrm{He}+?+?$$
Be sure to take into account the mass of the electrons associated with the neutral atoms.

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 21

The following nuclei are observed to decay by emitting an $\alpha$ particle: (a) ${ }_{84}^{212} \mathrm{Po}$ and (b) ${ }_{94}^{234} \mathrm{Pu}$. Write out the decay process for each of these nuclei, and determine the energy released in each reaction. Be sure to take into account the mass of the electrons associated with the neutral atoms.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

The following nuclei are observed to decay by emitting a $\beta^{-}$ particle: (a) ${ }_{16}^{35} \mathrm{S}$ and (b) $_{82}^{212} \mathrm{Pb}$. Write out the decay process for each of these nuclei, and determine the energy released in each reaction. Be sure to take into account the mass of the electrons associated with the neutral atoms.

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 23

The following nuclei are observed to decay by emitting a $\beta^{+}$ particle: (a) ${ }_{9}^{18} \mathrm{F}$ and (b) ${ }_{11}^{22} \mathrm{Na}$. Write out the decay process for each of these nuclei, and determine the energy released in each reaction. Be sure to take into account the mass of the electrons associated with the neutral atoms.

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 24

Find the energy released when $_{82 }^{211}$ Pb undergoes $\beta^{-}$ decay to become $_{ 83}^{211} \mathrm{Bi}$. Be sure to take into account the mass of the electrons associated with the neutral atoms.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 25

It is observed that $^{66}_{28} \mathrm{Ni},$ with an atomic mass of $65.9291 \mathrm{u}$ decays by $\beta^{-}$ emission. (a) Identify the nucleus that results from this decay. (b) If the nucleus found in part (a) has an a tomic mass of $65.9289 \mathrm{u}$, what is the maximum kinetic energy of the emitted electron?

Narayan Hari

Numerade Educator

Problem 26

The half-life of carbon-14 is 5730 y. (a) Is it possible for a particular nucleus in a sample of carbon-14 to decay after only 1s has passed? Explain. (b) Is it possible for a particular nucleus to decay after 10,000 y? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

Suppose we were to discover that the ratio of carbon-14 to carbon-12 in the atmosphere was significantly smaller 10,000 years ago than it is today. How would this affect the ages we have assigned to objects on the basis of carbon-14 dating? In particular, would the true age of an object be greater than or less than the age we had previously assigned to it? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

A radioactive sample is placed in a closed container. Two days later only one-quarter of the sample is still radioactive. What is the half-life of this sample?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

Radon gas has a half-life of 3.82 d. What is the decay constant for radon?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 30

A radioactive substance has a decay constant equal to $8.9 \times 10^{-3} \mathrm{s}^{-1} .$ What is the half-life of this substance?

Narayan Hari

Numerade Educator

Problem 31

The number of radioactive nuclei in a particular sample decreases over a period of $18 \mathrm{d}$ to one-sixteenth the original number. What is the half-life of these nuclei?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 32

The half-life of ${ }_{8}^{15} \mathrm{O}$ is 122 s. How long does it take for the number of ${ }^{15} \mathrm{O}$ nuclei in a given sample to decrease by a factor of $10^{-4} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

A Radioactive Tag A drug prepared for a patient is tagged with $^{99}_{43} \mathrm{Tc},$ which has a half-life of $6.05 \mathrm{h}$. (a) What is the decay constant of this isotope? (b) How many $\$ 3$ Tc nuclei are required to give an activity of $1.50 \mu$ Ci?

Narayan Hari

Numerade Educator

Problem 34

Referring to Problem $33,$ suppose the drug containing $^{99}_{43} \mathrm{Tc}$ with an activity of $1.50 \mu \mathrm{Ci}$ is injected into the patient $2.05 \mathrm{h}$ after it is prepared. What is its activity at the time it is injected?

Narayan Hari

Numerade Educator

Problem 35

An archeologist on a dig finds a fragment of an ancient basket woven from grass. Later, it is determined that the carbon-14 content of the grass in the basket is $9.25 \%$ that of an equal carbon sample from present-day grass. What is the age of the basket?

Narayan Hari

Numerade Educator

Problem 36

The bones of a saber-toothed tiger are found to have an activity per gram of carbon that is $15.0 \%$ of what would be found in a similar live animal. How old are these bones?

Narayan Hari

Numerade Educator

Problem 37

Charcoal from an ancient fire pit is found to have a carbon14 content that is only $17.5 \%$ that of an equivalent sample of carbon from a living tree. What is the age of the fire pit?

Narayan Hari

Numerade Educator

Problem 38

One of the many isotopes used in cancer treatment is $_{79 }^{198} \mathrm{Au}$, with a half-life of 2.70 d. Determine the mass of this isotope that is required to give an activity of $225 \mathrm{Ci}$.

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 39

Smoke Detectors The radioactive isotope $_{95 }^{241}\mathrm{Am},$ with a half-life of $432 \mathrm{y},$ is the active element in many smoke detectors. Suppose such a detector will no longer function if the activity of the ${95 }^{241}$, $\mathrm{Am}$ it contains drops below $\frac{1}{525}$ of its initial activity. How long will this smoke detector work?

Narayan Hari

Numerade Educator

Problem 40

Radioactivity in the Bones Because of its chemical similarity to calcium, ${ }_{38}^{90} \mathrm{Sr}$ health risk. What percentage of ${ }_{38}^{90} \mathrm{Sr}$ present initially still exists after a period of (a) $50.0 y$ (b) $60.0 \mathrm{y}$, and (c) $70.0 \mathrm{y} ?$

Narayan Hari

Numerade Educator

Problem 41

The atomic mass of gold-197 is 196.96654 u. How much energy is required to completely separate the nucleons in a a gold-197 nucleus?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 42

The atomic mass of lithium-7 is 7.016003 u. How much energy is required to completely separate the nucleons in a lithium-7 nucleus?

Narayan Hari

Numerade Educator

Problem 43

Calculate the average binding energy per nucleon of (a) $^{56}_{26} \mathrm{Fe}$ and (b) $^{238}_{92} U.$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

Calculate the average binding energy per nucleon of (a) $^{4}_{2} \mathrm{He}$ and (b) $^{64}_{30} \mathrm{zn}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

Find the energy required to remove one neutron from ${ }_{8}^{16} \mathrm{O}$.

Narayan Hari

Numerade Educator

Problem 46

(a) Consider the following nuclear process, in which a proton is removed from an oxygen nucleus:
$$
_{ 8}^{16} \mathrm{O}+\text { energy } \rightarrow{ }_{7}^{15} \mathrm{N}+{ }_{1}^{1} \mathrm{H}
$$
Find the energy required for this process to occur. (b) Now consider a process in which a neutron is removed from an oxygen nucleus:
$$
_{ 8}^{16} \mathrm{O}+\text { energy } \rightarrow_{7 }^{15} \mathrm{O}+{ }_{0}^{1} \mathrm{n}
$$
Find the energy required for this process to occur. (c) Which particle, the proton or the neutron, do you expect to be more tightly bound in the oxygen nucleus? Verify your answer.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 47

Find the number of neutrons released by the following fission reaction:
$$_{0}^{1} \mathrm{n}+{ }_{92}^{235} \mathrm{U} \rightarrow_{50 }^{132} \mathrm{Sn}+{ }_{42}^{101} \mathrm{Mo}+(?) \text { neutrons }$$

Narayan Hari

Numerade Educator

Problem 48

Complete the following fission reaction and determine the amount of energy it releases:
$$_{0}^{1}\mathrm{n}+{ }_{92}^{235} \mathrm{U} \rightarrow{ }_{51}^{133} \mathrm{Sb}+?+5 _{0}^{1}\mathrm{n}$$

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 49

Complete the following fission reaction and determine the amount of energy it releases:
$$
^{1}_{0} n+^{235}_{92} U \rightarrow ^{88}_{35} \mathrm{Sr}+{ }_{54}^{136} \mathrm{Xe}+(?) \text { neutrons }
$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

A gallon of gasoline releases about $2.0 \times 10^{8}$ J of energy when it is burned. How many gallons of gas must be burned to release the same amount of energy as is released when 1.0 lb of ${ }_{92}^{23} \mathrm{U}$ undergoes fission. (Assume that each fission reaction in 235 U releases 173 MeV.)

Narayan Hari

Numerade Educator

Problem 51

Assuming a release of 173 MeV per fission reaction, determine the minimum mass of ${ }_{92}^{235} \mathrm{U}$ that must undergo fission to supply the annual energy needs of the United States. (The amount of energy consumed in the United States each year is $.8.4 \times 10^{19} \mathrm{J} .$)

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 52

Assuming a release of 173 MeV per fission reaction, calculate how many reactions must occur per second to produce a power output of $150 \mathrm{MW}$.

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 53

Consider a fusion reaction in which two deuterium nuclei fuse to form a tritium nucleus and a proton. How much energy is released in this reaction?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 54

Consider a fusion reaction in which a proton fuses with a neutron to form a deuterium nucleus. How much energy is released in this reaction?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 55

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

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 56

(a) Complete the following fusion reaction and determine the energy it releases:
$${ }_{1}^{2} \mathrm{H}+{ }_{1}^{3} \mathrm{H} \rightarrow ?+^{1}_{0} \mathrm{n}$$
(b) How many of these reactions must occur per second to produce a power output of $25 \mathrm{MW}$ ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

The Evaporating Sun The Sun radiates energy at the prodigious rate of $3.90 \times 10^{26} \mathrm{W} .$ (a) At what rate, in kilograms per second, does the Sun convert mass into energy? (b) Assuming that the Sun has radiated at this same rate for its entire lifetime of $4.50 \times 10^{9} \mathrm{y},$ and that the current mass of the Sun is $2.00 \times 10^{30} \mathrm{kg},$ what percentage of its original mass has been converted to energy?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 58

Radiation Damage A sample of tissue absorbs a 55-rad dose of $\alpha$ particles $(\mathrm{RBE}=20)$. How many rad of protons $(\mathrm{RBE}=10)$ cause the same amount of damage to the tissue?

Narayan Hari

Numerade Educator

Problem 59

$X$-ray Damage How many rad of $200-$ keV $X$-rays cause the same amount of biological damage as 50 rad of heavy ions?

Narayan Hari

Numerade Educator

Problem 60

(a) Find the energy absorbed by a 78-kg person who is exposed to 52 mrem of $\alpha$ particles with an RBE of $15 .$ (b) If the RBE of the $\alpha$ particles is increased, does the energy absorbed increase, decrease, or stay the same? Explain.

Narayan Hari

Numerade Educator

Problem 61

A patient undergoing radiation therapy for cancer receives a 225-rad dose of radiation. (a) Assuming the cancerous growth has a mass of $0.17 \mathrm{kg}$, calculate how much energy it absorbs. (b) Assuming the growth to have the specific heat of water, determine its increase in temperature.

Narayan Hari

Numerade Educator

Problem 62

Alpha particles with an RBE of 13 deliver a 32-mrad whole-body radiation dose to a $72-k g$ patient. (a) What dosage, in rem, does the patient receive? (b) How much energy is absorbed by the patient?

Narayan Hari

Numerade Educator

Problem 63

A Radioactive Pharmaceutical As part of a treatment program, a patient ingests a radioactive pharmaceutical containing ${ }_{15}^{32} \mathrm{P},$ which emits $\beta$ rays with an $\mathrm{RBE}$ of $1.50 .$ The halflife of ${ }_{15}^{32} \mathrm{P}$ is $14.28 \mathrm{d},$ and the initial activity of the medication is $1.34 \mathrm{MBq}$ (a) How many electrons are emitted over the period of $7.00 \mathrm{d} ?$ (b) If the $\beta$ rays have an energy of $705 \mathrm{keV}$, what is the total amount of energy absorbed by the patient's body in $7.00 \mathrm{d} ?$ (c) Find the absorbed dosage in rem, assuming the radiation is absorbed by $125 \mathrm{g}$ of tissue.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 64

An $\alpha$ particle (charge $+2 e$ ) and a $\beta$ particle (charge $-e$ ) deflect in opposite directions when they pass through a magnetic field. Which particle deflects by a greater amount, given that both particles have the same speed? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 65

Radioactive samples A and B have equal half-lives. The initial activity of sample A is twice that of sample B. What is the ratio of the activity of sample $A$ to that of sample $B$ after two half-lives have elapsed?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

The initial activity of sample $A$ is twice that of sample B. After two half-lives of sample A have elapsed, the two samples have the same activity. What is the ratio of the half-life of $B$ to the half-life of A?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

To produce a given amount of electrical energy, is the amount of coal burned in a coal-burning power plant greater than, less than, or the same as the amount of $_{92 }^{235} \mathrm{U}$ consumed in a nuclear power plant? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 68

Determine the number of neutrons and protons in
(a) $_{90 }^{232} \mathrm{Th}$
(b) $_{82 }^{211} \mathrm{Pb},$ and
(c) $^{60}_{27} \mathrm{Co}$

Narayan Hari

Numerade Educator

Problem 69

Identify the daughter nucleus that results when (a) ${ }_{82}^{210} \mathrm{Pb}$ undergoes $\alpha$ decay, (b) $_{92 }^{239} \mathrm{U}$ undergoes $\beta^{-}$ decay, and (c) ${ }_{6}^{11} \mathrm{C}$ undergoes $\beta^{+}$ decay.

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 70

Suppose it is desired to give a cancerous tumor a dose of 3800 rem. How many rads are needed if the tumor is exposed to alpha radiation?

Narayan Hari

Numerade Educator

Problem 71

A patient is exposed to 260 rad of gamma rays. What is the dose the patient receives in rem?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 72

The two radioactive decay series that begin with $_{ 90}^{232}$ Th and end with ${ }_{82}^{208} \mathrm{Pb}$ are shown in Figure $32-14$. Identify the ten intermediary nuclei that appear in these series.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

Moon Rocks In one of the rocks brought back from the Moon, it is found that $80.5 \%$ of the initial potassium-40 in the rock has decayed to argon-40. (a) If the half-life for this decay is $1.20 \times 10^{9}$ years, how old is the rock? (b) How much longer will it take before only $10.0 \%$ of the original potassium-40 is still present in the rock?

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 74

Mantles in Gas Lanterns Gas lanterns used on camping trips have mantles (small lacy bags that give off light) made from a rayon mesh impregnated with thorium and other materials. Thorium is used, even though it is radioactive, because it forms an oxide that can withstand being incandescent for long periods of time. Almost all natural thorium is $_{90 }^{232} \mathrm{Th},$ which has a half-life of $1.405 \times 10^{10} y .$ (a) A typical mantle contains $325 \mathrm{mg}$ of thorium. What is the activity of the mantle? (b) If the half-life of thorium had been double its actual value, by what factor would the activity of a mantle be changed? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 75

Identify the nucleus whose $\beta^{-}$ decay produces the same nucleus as that produced by the $\alpha$ decay of $_{84 }^{214}$ Po.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 76

An $\alpha$ particle fired head-on at a stationary nickel nucleus approaches to a radius of $15 \mathrm{fm}$ before being turned around. (a) What is the maximum Coulomb force exerted on the $\alpha$ particle? (b) What is the electric potential energy of the $\alpha$ particle at its point of closest approach? (c) Find the initial kinetic energy of the $\alpha$ particle.

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 77

Calculate the number of disintegrations per second that one would expect from a $1.7-g$ sample of $_{88 }^{226}$ Ra. What is the activity of this sample in curies?

Rory Naguib

Numerade Educator

Problem 78

Initially, a sample of radioactive nuclei of type A contains four times as many nuclei as a sample of radioactive nuclei of type B. Two days later $(2.00 \mathrm{d})$ the two samples contain the same number of nuclei. (a) Which type of nucleus has the longer half-life? Explain. (b) Determine the half-life of type B nuclei if the half-life of type A nuclei is known to be $0.500 \mathrm{d}$.

Rory Naguib

Numerade Educator

Problem 79

Stable nuclei have mass numbers that range from a minimum of 1 to a maximum of 209 . (a) Find the corresponding range in nuclear radii. (b) Assuming all nuclei to be spherical, determine the ratio of the surface area of the largest stable nucleus to the surface area of the smallest nucleus. (c) Repeat part (b), only this time find the ratio of the volumes.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 80

Radius of a Neutron Star Neutron stars are so named because they are composed of neutrons and have a density the same as that of a nucleus. Referring to Example $32-2$ for the nuclear density, find the radius of a neutron star whose mass is 0.50 that of the Sun.

Narayan Hari

Numerade Educator

Problem 81

A specimen taken from the wrappings of a mummy contains $7.82 \mathrm{g}$ of carbon and has an activity of $1.38 \mathrm{Bq} .$ How old is the mummy? (Refer to pages 1132 and 1133 for relevant information regarding the isotopes of carbon.)

Rory Naguib

Numerade Educator

Problem 82

(a) How many fission reactions are required to light a 120 -W lightbulb for 2.5 d? Assume an energy release of 212 MeV per fission reaction and a $32 \%$ conversion efficiency. (b) What mass of $_{ 92}^{235} \mathrm{SU}$ corresponds to the number of fission reactions found in part (a)?

Rory Naguib

Numerade Educator

Problem 83

Energy is released when three $\alpha$ particles fuse to form carbon-12. (a) Is the mass of carbon-12 greater than, less than, or the same as the mass of three $\alpha$ particles? Explain. (b) Calculate the energy given off in this fusion reaction.

Rory Naguib

Numerade Educator

Problem 84

Find the dose of $\gamma$ rays that must be absorbed by a block of ice at $0^{\circ} \mathrm{C}$ to convert it to water at $0^{\circ} \mathrm{C}$. Give the dosage in rad.

Narayan Hari

Numerade Educator

Problem 85

Find the dose of $\gamma$ rays that must be absorbed by a block of ice at $0^{\circ} \mathrm{C}$ to convert it to water at $0^{\circ} \mathrm{C}$. Give the dosage in rad.

Narayan Hari

Numerade Educator

Problem 86

Chest X-rays A typical chest X-ray uses X-rays with an RBE of $0.85 .$ If the radiation dosage is 35 mrem, find the energy absorbed by a $72-\mathrm{kg}$ patient, assuming one-quarter of the patient's body is exposed to the X-ravs.

Narayan Hari

Numerade Educator

Problem 87

A ray photon emitted by $^{226}_{88}$ Ra has an energy of 0.186 MeV. Use conservation of linear momentum to calculate the recoil speed of a $_{88}^{226} \mathrm{Ra}$ nucleus after such a $\gamma$ ray is emitted. Assume that the nucleus is at rest initially, and that relativistic effects can be ignored.

Narayan Hari

Numerade Educator

Problem 88

The energy released by $\alpha$ decay in a $50.0-\mathrm{g}$ sample of $_{94 }^{239} \mathrm{Pu}$ is to be used to heat $4.75 \mathrm{kg}$ of water. Assuming all the energy released by the radioactive decay goes into heating the water, find how much the temperature of the water increases in $1.00 \mathrm{h}$.

Cyra Jelle Calleja

Cyra Jelle Calleja

Numerade Educator

Problem 89

Consider a solid sphere of $_{ 92}^{235} \mathrm{g} \mathrm{U}$ with a radius of $2.25 \mathrm{cm}$ in a room with a temperature of 293 K. Assume that all the energy released by $\alpha$ decay goes into heating the sphere, and that the sphere radiates heat to its surroundings as a blackbody. What is the change in temperature of the sphere as a result of the $\alpha$ decay? (Note: The density of uranium is $18.95 \mathrm{g} / \mathrm{cm}^{3}$.)

Cyra Jelle Calleja

Cyra Jelle Calleja

Numerade Educator

Problem 90

What is the decay constant, $\lambda$, for iodine-131?
A. $9.98 \times 10^{-7} \mathrm{s}^{-1}$
B. $1.44 \times 10^{-6} \mathrm{s}^{-1}$
C. $2.39 \times 10^{-5} \mathrm{s}^{-1}$
D. $5.99 \times 10^{-5} \mathrm{s}^{-1}$

Narayan Hari

Numerade Educator

Problem 91

If a sample of iodine-131 contains $4.5 \times 10^{16}$ nuclei, what is the activity of the sample? Express your answer in curies.
A. $0.27 \mathrm{Ci}$
B. $1.2 \mathrm{Ci}$
C. $1 .7 \mathrm{Ci}$
D. $4.5 \mathrm{Ci}$

Narayan Hari

Numerade Educator

Problem 92

If the half-life of iodine-131 were only half of its actual value, would the activity of the sample in Problem 91 be increased or decreased?

Narayan Hari

Numerade Educator