Physics

John D. Cutnell, Kenneth W. Johnson

Chapter 31

Nuclear Physics and Radioactivity - all with Video Answers

Educators

Chapter Questions

Problem 1

By what factor does the nucleon number of a nucleus have to increase in order for the nuclear radius to double?

Mike Gaerlan

Mike Gaerlan

Numerade Educator

Problem 2

For ${ }_{82}^{208} \mathrm{~Pb}$ find (a) the net electrical charge of the nucleus, (b) the number of neutrons, (c) the number of nucleons, (d) the approximate radius of the nucleus, and (e) the nuclear density.

Declan Nell

Numerade Educator

Problem 3

In each of the following cases, what element does the symbol X represent and how many neutrons are in the nucleus: (a) $\frac{195}{78} \mathrm{X},$ (b) $\frac{32}{16} \mathrm{X},$ (c) $\frac{63}{29} \mathrm{X}$, (d) $\frac{11}{5} X,$ and (e) $\frac{239}{94} \mathrm{X} ?$ Use the periodic table on the in side of the back cover as needed.

Declan Nell

Numerade Educator

Problem 4

What is the radius of a nucleus of titanium $\frac{48}{22} \mathrm{~T}$

Narayan Hari

Numerade Educator

Problem 5

The largest stable nucleus has a nucleon number of 209 , and the smallest has a nucleon number of $1 .$ If each nucleus is assumed to be a sphere, what is the ratio (largest/ smallest) of the surface areas of these spheres?

Mike Gaerlan

Numerade Educator

Problem 6

The ratio $r_{X} / r_{T}$ of the radius of an unknown nucleus $A_{Z}^{A}$ to a tritium nucleus $\frac{3}{1} \mathrm{~T}$ is $\frac{r_{X}}{r_{T}}=1.10 .$ Both nuclei contain the same number of neutrons. Identify the unknown nucleus in the form $\underset{z}{A}_{X}$. Use the periodic table on the inside of the back cover as needed.

Narayan Hari

Numerade Educator

Problem 7

An unknown nucleus contains 70 neutrons and has twice the volume of the nickel $60 \mathrm{Ni}$ nucleus. Identify the unknown nucleus in the form $A_{X}$. Use the periodic table on the inside of the back cover as needed.

Narayan Hari

Numerade Educator

Problem 8

Conceptual Example 1 provides some useful background for this problem. (a) Determine an approximate value for the density (in $\mathrm{kg} / \mathrm{m}^{3}$ ) of the nucleus. (b) If a BB (radius $=2.3$ $\mathrm{mm}$ ) from an air rifle had a density equal to the nuclear density, what mass would the $\mathrm{BB}$ have? (c) Assuming the mass of a supertanker is about $1.5 \times 10^{8} \mathrm{~kg}$, how many "supertankers" of mass would this hypothetical BB have?

Narayan Hari

Numerade Educator

Problem 9

Refer to Conceptual Example 1 for a discussion of nuclear densities. A neutron star is composed of neutrons and has a density that is approximately the same as that of a nucleus. What is the radius of a neutron star whose mass is 0.40 times the mass of the sun?

Mike Gaerlan

Numerade Educator

Problem 10

Determine the mass defect (in atomic mass units) for (a) helium ${ }_{2}^{3} \mathrm{He},$ which has an atomic mass of $3.016030 \mathrm{u},$ and $(\mathrm{b})$ the isotope of hydrogen known as tritium $\frac{3}{1} \mathrm{~T}$ which has an atomic mass of $3.016050 \mathrm{u}$. (c) On the basis of your answers to parts (a) and (b), state which nucleus requires more energy to disassemble it into its separate and stationary constituent nucleons. Give your reasoning.

Narayan Hari

Numerade Educator

Problem 11

The earth revolves around the sun, and the two represent a bound system that has a binding energy of $2.6 \times 10^{33} \mathrm{~J}$. Suppose the earth and sun were completely separated, so that they were infinitely far apart and at rest. What would be the difference between the mass of the separated system and that of the bound system?

Narayan Hari

Numerade Educator

Problem 12

Find the binding energy (in $\mathrm{MeV}$ ) for lithium $\frac{7}{3} \mathrm{Li}$ (atomic mass $=7.016003 \mathrm{u}$ ).

Mike Gaerlan

Numerade Educator

Problem 13

For lead ${ }_{82}^{206} \mathrm{~Pb}$ (atomic mass $=205.974440 \mathrm{u}$ ) obtain (a) the mass defect in atomic mass units, (b) the binding energy (in $\mathrm{MeV}$ ), and (c) the binding energy per nucleon (in $\mathrm{MeV} /$ nucleon ).

Yaqub Khan

Numerade Educator

Problem 14

Use the plot of binding energy per nucleon in Figure $31-5$ to determine the mass defect for the oxygen ${ }_{8}^{16} \mathrm{O}$ nucleus. Express your answer in kilograms.

Narayan Hari

Numerade Educator

Problem 15

Two isotopes of a certain element have binding energies that differ by $5.03 \mathrm{MeV}$. The isotope with the larger binding energy contains one more neutron than the other isotope. Find the difference in atomic mass between the two isotopes.

Prashant Bana

Numerade Educator

Problem 16

(a) Energy is required to separate a nucleus into its constituent nucleons, as Figure $31-3$ indicates; this energy is the total binding energy of the nucleus. In a similar way one can speak of the energy that binds a single nucleon to the remainder of the nucleus. For example, separating nitrogen ${ }_{7}^{14} N$ into nitrogen ${ }_{7}^{13} \mathrm{~N}$ and a neutron takes energy equal to the binding energy of the neutron, as shown below: $$ \frac{14}{7} N+\text { Energy } \rightarrow \frac{13}{7} N+\frac{1}{0} n $$ Find the energy (in MeV) that binds the neutron to the $\frac{14}{7} N$ nucleus by considering the mass of $13 \mathrm{~N}$ (atomic mass $=13.005738 \mathrm{u}$ ) and the mass of $\frac{1}{0^{\mathrm{n}}}$ (atomic mass $=1.008$ $665 \mathrm{u}$ ), as compared to the mass of ${ }_{7}^{14} \mathrm{~N}$ ( atomic mass $=14.003074 \mathrm{u}$ ). (b) Similarly, one can speak of the energy that binds a single proton to the$\frac{14}{7} \mathrm{~N}$ nucleus: $$ { }_{7}^{14} \mathrm{~N}+\text { Energy } \rightarrow{ }_{6}^{13} \mathrm{C}+{ }_{1}^{1} \mathrm{H} $$ Following the procedure outlined in part (a), determine the energy (in MeV) that binds the proton (atomic mass $=1.007825 \mathrm{u}$ ) to the $\frac{14}{7} \mathrm{~N}$ nucleus. The atomic mass of carbon $\frac{13}{6} \mathrm{C}$ is $13.003335 \mathrm{u}$ (c) Which nucleon is more tightly bound, the neutron or the proton?

Declan Nell

Numerade Educator

Problem 17

Write the $\beta^{-}$ decay process for $\frac{35}{16} \mathrm{~S}$, including the chemical symbol and values for $Z$ and $A$

Narayan Hari

Numerade Educator

Problem 18

Write the $\beta^{-}$ decay process for $\frac{35}{16} \mathrm{~S}$, including the chemical symbol and values for $Z$ and $A$

Vishal Gupta

Numerade Educator

Problem 19

Write the $\beta^{+}$ decay process for each of the following nuclei, being careful to include $Z$ and $A$ and the proper chemical symbol for each daughter nucleus: (a) 18 9 $\mathrm{~F}$ and (b) 15 8.

Narayan Hari

Numerade Educator

Problem 20

Find the energy (in MeV) released when $\alpha$ decay converts radium 226 Ra (atomic mass $=226.02540 \mathrm{u}$ ) into radon $\frac{222}{86} \mathrm{Rn}($ atomic mass $=222.01757 \mathrm{u}) .$ The atomic mass of an $\alpha$ particle is $4.002603 \mathrm{u}$.

Narayan Hari

Numerade Educator

Problem 21

decay occurs for each of the following nuclei. Write the decay process for each, including the chemical symbols nd values for $Z$ and $A$ for the daughter nuclei: (a) ${ }_{84}^{212} \mathrm{P} \circ$ and $(\mathrm{b}){ }_{92}^{232} \mathrm{U}$.

Declan Nell

Numerade Educator

Problem 22

Multiple-Concept Example 7 reviews the concepts needed to solve this problem. When uranium 235 U decays, it emits (among other things) a $\gamma$ ray that has a wavelength of $1.14 \times 10^{-11} \mathrm{~m} .$ Determine the energy (in $\mathrm{MeV}$ ) of this $\gamma$ ray.

Narayan Hari

Numerade Educator

Problem 23

Find the energy released when lead $\underset{82}^{211} \mathrm{~Pb}$ (atomic mass $=210.988735 \mathrm{u}$ ) undergoes $\beta^{-}$ decay to become bismuth $\underset{83}{211} \mathrm{Bi}$ (atomic mass $\left.=210.987255 \mathrm{u}\right)$.

Narayan Hari

Numerade Educator

Problem 24

Review Conceptual Example 5 as background for this problem. The $\alpha$ decay of uranium $238 \mathrm{U}$ produces thorium $234 \mathrm{Th}$ (atomic mass $=234.0436 \mathrm{u}$ ). In Example 4 , the energy released in this decay is determined to be $4.3 \mathrm{MeV}$. Determine how much of this energy is carried away by the recoiling $\frac{234}{90}$ Th daughter nucleus and how much by the $\alpha$ particle (atomic mass $=4.002603 \mathrm{u}$ ). Assume that the energy of each particle is kinetic energy, and ignore the small amount of energy carried away by the $\gamma$ ray that is also emitted. In addition, ignore relativistic effects.

Declan Nell

Numerade Educator

Problem 25

Refer to Interactive Solution 31.25 at to review a model for solving this type of problem. Polonium ${ }_{84}^{210} \mathrm{P} \circ($ atomic mass $=209.982848 \mathrm{u})$ undergoes $\alpha$ decay. Assuming that all the released energy is in the form of kinetic energy of the $\alpha$ particle (atomic mass $=4.002603 \mathrm{u}$ ) and ignoring the recoil of the daughter nucleus (lead $826 \mathrm{p}, 205.974440 \mathrm{u}),$ find the speed of the $\alpha$ particle. Ignore relativistic effects.

Declan Nell

Numerade Educator

Problem 26

Rado $220 \mathrm{Rn}$ produces a daughter nucleus that is radioactive. The daughter, in turn, produces its own radioactive daughter, and so on. This process continues until lead $83 \underline{208} \mathrm{~b}$ is reached. What are the total number $N_{\alpha}$ of $\alpha$ particles and the total number $N_{\beta}$ of $\beta$ particles that are generated in this series of radioactive decays?

Narayan Hari

Numerade Educator

Problem 27

Find the energy (in MeV) released when $\beta^{+}$ decay converts sodium $\underset{11}{22} \mathrm{Na}$ (atomic mass $=21.994434 \mathrm{u}$ ) into neon $\underset{10}{22} \mathrm{Ne}$ (atomic mass $=21.991383 \mathrm{u}$ ). Notice that the atomic mass for $22 \mathrm{Na}$ includes the mass of 11 electrons, whereas the atomic mass for 22 Ne includes the mass of only 10 electrons.

Narayan Hari

Numerade Educator

Problem 28

Interactive LearningWare 31.1 at reviews the concepts that are involved in this problem. An isotope of beryllium (atomic mass $=7.017 \mathrm{u}$ ) emits a $\gamma$ ray and recoils with a speed of $2.19 \times 10^{4} \mathrm{~m} / \mathrm{s}$. Assuming that the beryllium nucleus is stationary to begin with, find the wavelength of the $\gamma$ ray.

Narayan Hari

Numerade Educator

Problem 29

The isotope ${ }_{88}^{224}$ Ra of radium has a decay constant of $2.19 \times 10^{-6} \mathrm{~s}^{-1}$. What is the halflife (in days) of this isotope?

Narayan Hari

Numerade Educator

Problem 30

The half-lives in two different samples, $A$ and $B$, of radioactive nuclei are related according to $T_{1 / 2, B}=\frac{1}{2} T_{1 / 2, A}$ In a certain period the number of radioactive nuclei in sample A decreases to one-fourth the number present initially. In this same period the number of radioactive nuclei in sample $\mathrm{B}$ decreases to a fraction $f$ of the number present initially. Find $f$

Mike Gaerlan

Numerade Educator

Problem 31

How many half-lives are required for the number of radioactive nuclei to decrease to one-millionth of the initial number?

Declan Nell

Numerade Educator

Problem 32

Strontium ${ }_{38} \mathrm{Sr}$ has a half-life of $29.1 \mathrm{yr} .$ It is chemically similar to calcium,
enters the body through the food chain, and collects in the bones. Consequently, ${ }_{38} \mathrm{Sr}$ is a particularly serious health hazard. How long (in years) will it take for $99.9900 \%$ of the 38 Sr released in a nuclear reactor accident to disappear?

Narayan Hari

Numerade Educator

Problem 33

A device used in radiation therapy for cancer contains $0.50 \mathrm{~g}$ of cobalt $\frac{60}{27} \mathrm{Co}$ $(59.933819 \mathrm{u}) .$ The half-life of $\underset{27}{60} \mathrm{C} \circ$ is $5.27 \mathrm{yr}$. Determine the activity of the radioactive material.

Narayan Hari

Numerade Educator

Problem 34

Iodine $\frac{131}{53} \mathrm{I}$ is used in diagnostic and therapeutic techniques in the treatment of
thyroid disorders. This isotope has a half-life of 8.04 days. What percentage of an initial sample of $\frac{131}{53}$ I remains after 30.0 days?

Narayan Hari

Numerade Educator

Problem 35

The number of radioactive nuclei present at the start of an experiment is $4.60 \times 10^{15}$. The number present twenty days later is $8.14 \times 10^{14}$. What is the half-life (in days) of the nuclei?

Mike Gaerlan

Numerade Educator

Problem 36

To make the dial of a watch glow in the dark, $1.000 \times 10^{-9} \mathrm{~kg}$ of radium $\frac{226}{88} \mathrm{Ra}$ is used. The half-life of this isotope has a value of $1.60 \times 10^{3}$ yr. How many kilograms of radium disappear while the watch is in use for fifty years?

Narayan Hari

Numerade Educator

Problem 37

Refer to Interactive Solution 31.37 at for one approach to solving this problem. To see why one curie of activity was chosen to be $3.7 \times 10^{10} \mathrm{~Bq},$ determine the activity (in disintegrations per second) of one gram of radium $\frac{226}{88} \mathrm{Ra}\left(T_{1 / 2}=1.6 \times 10^{3} \mathrm{yr}\right)$

Declan Nell

Numerade Educator

Problem 38

The isotope $\frac{198}{79} \mathrm{Au}$ (atomic mass $=197.968 \mathrm{u}$ ) of gold has a half-life of 2.69
days and is used in cancer therapy. What mass (in grams) of this isotope is required to produce an activity of $315 \mathrm{Ci} ?$

Declan Nell

Numerade Educator

Problem 39

Two radioactive nuclei $A$ and $B$ are present in equal numbers to begin with. Three days later, there are three times as many A nuclei as there are $\mathrm{B}$ nuclei. The half-life of species $\mathrm{B}$ is 1.50 days. Find the half-life of species $\mathrm{A}$.

Mike Gaerlan

Numerade Educator

Problem 40

Multiple-Concept Example 11 reviews most of the concepts that are needed to solve this problem. Material found with a mummy in the arid highlands of southern Peru has a $14^{4} \mathrm{C}$ activity per gram of carbon that is $78.5 \%$ of the activity present initially. How long ago (in years) did this individual die?

Narayan Hari

Numerade Educator

Problem 41

The practical limit to ages that can be determined by radio carbon dating is about 41 000 yr. In a 41 000-yr-old sample, what percentage of the original ${ }_{6}^{14} \mathrm{C}$ atoms remains?

Narayan Hari

Numerade Educator

Problem 42

The half-life for the $\alpha$ decay of uranium $\frac{238}{92} \mathrm{U}$ is $4.47 \times 10^{9} \mathrm{yr} .$ Determine the age (in years) of a rock specimen that contains sixty percent of its original number of $\frac{238}{92} \mathrm{U}$ atoms.

Narayan Hari

Numerade Educator

Problem 43

The shroud of Turin is a religious artifact known since the Middle Ages. In 1988 its age was measured using the radiocarbon dating technique, which revealed that the shroud could not have been made before $1200 \mathrm{AD}$. Of the $\frac{14}{6} \mathrm{C}$ nuclei that were present in the living matter from which the shroud was made, what percentage remained in $1988 ?$

Declan Nell

Numerade Educator

Problem 44

Review Conceptual Example 12 before starting to solve this problem. The number of unstable nuclei remaining after a time $t=5.00 \mathrm{yr}$ is $N,$ and the number present initially is $N_{0}$. Find the ratio $N / N_{0}$ for (a) ${ }_{6}{ }^{14} \mathrm{C}$ (half-life $\left.=5730 \mathrm{yr}\right),$ (b) ${ }_{8}^{15} \mathrm{O}$ (half-life $=122.2$ s; use $t=1.00 \mathrm{~h}$, since otherwise the answer is out of the range of your calculator), and (c) ${ }_{1}^{3} \mathrm{H}$ (half-life $=12.33 \mathrm{yr}$ ). Verify that your answers are consistent with the reasoning in Conceptual Example 12 .

Declan Nell

Numerade Educator

Problem 45

When any radioactive dating method is used, experimental error in the measurement of the sample's activity leads to error in the estimated age. In an application of the radiocarbon dating technique to certain fossils, an activity of $0.100 \mathrm{~Bq}$ per gram of carbon is measured to within an accuracy of $\pm 10.0 \%$. Find the age of the fossils and the maximum error (in years) in the value obtained. Assume that there is no error in the 5730-year half-life of $\frac{14}{6} \mathrm{C}$ nor in the value of 0.23 Bq per gram of carbon in a living organism.

Declan Nell

Numerade Educator

Problem 46

A sample is being dated by the radiocarbon technique. If the sample were uncontaminated, its activity would be 0.011 Bq per gram of carbon. Find the true age (in years) of the sample. (b) Suppose the sample is contaminated, so that only $98.0 \%$ of its carbon is ancient carbon. The remaining $2.0 \%$ is fresh carbon, in the sense that the $\frac{14}{6} \mathrm{C}$ it contains has not had any time to decay. Assuming that the lab technician is unaware of the contamination, what apparent age (in years) would be determined for the sample?

Cyra Jelle Calleja

Cyra Jelle Calleja

Numerade Educator

Problem 47

Mercury ${ }_{80}^{202} \mathrm{Hg}$ has an atomic mass of $201.970617 \mathrm{u}$. Obtain the binding energy per nucleon (in MeV/nucleon).

Narayan Hari

Numerade Educator

Problem 48

In electrically neutral atoms, how many (a) protons are in the uranium $\frac{238}{92} \mathrm{U}$ nucleus,
(b) neutrons are in the mercury $202 \mathrm{Hg}$ nucleus, and (c) electrons are in orbit about the niobium 93 $\mathrm{Nb}$ nucleus? $41^{?} ?$

Narayan Hari

Numerade Educator

Problem 49

If the activity of a radioactive substance is initially 398 disintegrations/min and two days later it is 285 disintegrations/min, what is the activity four days later still, or six days after the start? Give your answer in disintegrations/min.

Declan Nell

Numerade Educator

Problem 50

Osmium ${ }_{76}^{191} \mathrm{Os}$ (atomic mass $=190.960920 \mathrm{u}$ ) is converted into Iridium $\frac{191}{77} \mathrm{Ir}$ (atomic mass $=190.960584 \mathrm{u}$ ) via $\beta^{-}$ decay. What is the energy (in $\mathrm{MeV}$ ) released in this process?

Narayan Hari

Numerade Educator

Problem 51

Complete the following decay processes by stating what the symbol "X" represents $\left(\mathrm{X}=\alpha, \beta^{-}, \beta^{+}\right.$ or $\left.\gamma\right)$
a. $\quad{ }_{82}^{211} \mathrm{~Pb} \rightarrow{ }_{83}^{211} \mathrm{Bi}+\mathrm{X}$
b. $\quad{ }_{6}^{11} \mathrm{C} \rightarrow{ }_{5}^{11} \mathrm{~B}+\mathrm{X}$
c. $\quad{ }_{90}^{231} \mathrm{Th}^{*} \rightarrow{ }_{90}^{231} \mathrm{Th}+\mathrm{X}$
d. $\quad{ }_{84}^{210} \mathrm{P} \circ \rightarrow{ }_{82}^{206} \mathrm{~Pb}+\mathrm{X}$

Declan Nell

Numerade Educator

Problem 52

Review Multiple-Concept Example 11 for help in approaching this problem. An archeological specimen containing $9.2 \mathrm{~g}$ of carbon has an activity of $1.6 \mathrm{~Bq} .$ How old (in years) is the specimen?

Declan Nell

Numerade Educator

Problem 53

The photomultiplier tube in a commercial scintillation counter contains 15 of the special electrodes, or dynodes. Each dynode produces 3 electrons for every electron that strikes it. One photoelectron strikes the first dynode. What is the maximum number of electrons that strike the 15 th dynode?

Mike Gaerlan

Numerade Educator

Problem 54

A sample of ore containing radioactive strontium ${ }_{38}^{90} \mathrm{Sr}$ has an activity of $6.0 \times 10^{5} \mathrm{~Bq}$ The atomic mass of strontium is $89.908 \mathrm{u},$ and its half-life is 29.1 yr. How many grams of strontium are in the sample?

Yaqub Khan

Numerade Educator

Problem 55

Determine the symbol $A_{X}$ for the parent nucleus whose $\alpha$ decay produces the same daughter as the $\beta^{-}$ decay of thallium $\frac{208}{81} \mathrm{~T} 1$

Narayan Hari

Numerade Educator

Problem 56

Both gold $\frac{198}{79} \mathrm{Au}\left(T_{1 / 2}=2.69\right.$ days $)$ and iodine $\frac{131}{53} \mathrm{I}\left(T_{1 / 2}=8.04\right.$ days $)$ are used in diagnostic medicine related to the liver. At the time laboratory supplies are monitored, the activity of the gold is observed to be five times greater than the activity of the iodine. How many days later will the two activities be equal?

Declan Nell

Numerade Educator

Problem 57

Concept Questions (a) In a nucleus, each proton experirences a repulsive
electrostatic force from each of the other protons. Write an expression for the magnitude of the force that one proton (charge $=+e$ ) applies to another proton that is located a distance $r$ away. (b) The force that acts on either of two particular protons in the nucleus has the smallest possible magnitude. Relative to one another, where in the nucleus must these two protons be located? Explain.

Declan Nell

Numerade Educator

Problem 58

Concept Questions(a) How is the total released energy related to the decrease in mass that accompanies $\beta^{-}$ decay? (b) When a parent nucleus undergoes $\beta^{-}$ decay, a daughter nucleus, a $\beta^{-}$ particle, and an antineutrino $\bar{\gamma}$ are produced. The released energy is shared among these three particles. Assume that the kinetic energy of the $\beta^{-}$ particle is known and that the antineutrino carries away the maximum possible energy. What then can you say about the kinetic energy of the recoiling daughter nucleus?

Cyra Jelle Calleja

Cyra Jelle Calleja

Numerade Educator

Problem 59

Concept Questions (a) Two radioactive nuclei A and B have half-lives of $T_{1 / 2, A}$ and $T_{1 / 2, \mathrm{~B}},$ where $T_{1 / 2, \mathrm{~A}}$ is greater than $T_{1 / 2, \mathrm{~B}}$. During the same time period, is the fraction of nuclei A that decay greater than, smaller than, or the same as the fraction of nuclei B that decay? (b) The numbers of these nuclei present initially are $N_{0, A}$ and $N_{0, B}$, the ratio of the two being $N_{0, A} / N_{0, B}$. Is the ratio $N_{A} / N_{B}$ of the number of nuclei present at a later time greater than, smaller than, or the same as $N_{0, \mathrm{~A}} / N_{0, \mathrm{~B}} ?$ Justify your answers.
Problem Two waste products from nuclear reactors are strontium $38 \mathrm{Sr}$ $\left(T_{1 / 2}=29.1 \mathrm{yr}\right)$ and cesium $\frac{134}{55} \mathrm{Cs}\left(T_{1 / 2}=2.06 \mathrm{yr}\right) .$ These two species are present
initially in a ratio of $N_{0, \mathrm{Sr}} / \mathrm{N}_{0, \mathrm{Cs}}=7.80 \times 10^{-3} .$ What is the ratio $N_{\mathrm{Sr}} / N_{\mathrm{Cs}}$ fifteen
years later? Verify that your answer is consistent with your answers to the Concept Questions.

Declan Nell

Numerade Educator

Problem 60

Concept Questions(a) Write an expression for the heat $Q$ needed to melt a mass $m$ of water. (b) When a nucleus disintegrates via $\alpha$ decay, how is the released energy related to the decrease in mass that accompanies the decay? (c) During $\alpha$ decay, the energy released by each disintegration is $E$. What is the total energy $E_{\text {Total }}$ released by a
number $n$ of disintegrations? (d) If the number of radioactive nuclei present initially is $N_{0}$, what is the number of disintegrations that occur in one half-life?
Problem A one-gram sample of radium $\frac{224}{88} \mathrm{Ra}$ (atomic mass $=224.020186 \mathrm{u}$ $T_{1 / 2}=3.66$ days ) contains $2.69 \times 10^{21}$ nuclei and undergoes $\alpha$ decay to produce radon $220 \mathrm{Rn}($ atomic mass $=220.011368 \mathrm{u}) .$ The atomic mass of an $\alpha$ particle is $4.002603 \mathrm{u}$ With the energy released in 3.66 days, how many kilograms of ice could be melted at $0^{\circ}$ $\mathrm{C} ?$

Declan Nell

Numerade Educator

Problem 61

Concept Questions Outside the nucleus, the neutron itself is radioactive and decays into a proton, an electron, and an antineutrino. (a) Suppose that, originally, a number $N_{0}$ of neutrons are outside the nucleus. What is meant by the statement "The half- life of the neutron is X minutes"? (b) How is the number of neutrons remaining at any time $t$ related to the original number and the half-life of the neutron? (c) Suppose each of the neutrons is moving with the same kinetic energy. How is the speed of each neutron related to its kinetic energy? Ignore relativistic effects.
Problem The half-life of a neutron (mass $=1.675 \times 10^{-27} \mathrm{~kg}$ ) outside the nucleus is $10.4 \mathrm{~min}$. On average, over what distance (in meters) would a beam of $5.00-\mathrm{eV}$ neutrons travel before the number of neutrons decreases to $75.0 \%$ of its initial value? Ignore

Declan Nell

Numerade Educator

Problem 62

Concept Questions (a) Physically, what does the binding energy of a nucleus
represent? (b) What is the relationship between the binding energy of a nucleus and its mass defect? (c) How is the mass defect of a nucleus related to the mass of the intact nucleus and the masses of the individual nucleons? (d) A piece of metal, such as a coin, contains a certain number of atoms, and, hence, nuclei. How much energy would be required to break all the nuclei into their constituent protons and neutrons? Give your answer in terms of the number of atoms in the coin and the binding energy of each nucleus. (e) How is the number of atoms in a piece of metal related to its mass? Express your answer in terms of the atomic mass of the atoms and Avogadro's number.
Problem A copper penny has a mass of $3.0 \mathrm{~g}$. Determine the energy (in $\mathrm{MeV}$ ) that would be required to break all the copper nuclei into their constituent protons and neutrons. Ignore the energy that binds the electrons to the nucleus and the energy that binds one atom to another in the structure of the metal. For simplicity, assume that all the copper nuclei are $\frac{63}{29} \mathrm{Cu}$ (atomic mass $=62.939598 \mathrm{u}$ ).

Cyra Jelle Calleja

Cyra Jelle Calleja

Numerade Educator