Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 42

Nuclear Physics - all with Video Answers

Educators

Chapter Questions

Problem 1

A ${ }^{7} \mathrm{Li}$ nucleus with a kinetic energy of $3.00 \mathrm{MeV}$ is sent toward a ${ }^{232}$ Th nucleus. What is the least center-to-center separation between the two nuclei, assuming that the (more massive) ${ }^{232}$ Th nucleus does not move?

Keshav Singh

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Problem 2

Calculate the distance of closest approach for a head-on collision between a 5.30 MeV alpha particle and a copper nucleus.

Ben Nicholson

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Problem 3

A $10.2 \mathrm{MeV}$ Li nucleus is shot directly at the center of a Ds nucleus. At what center-to-center distance does the Li momentarily stop. assuming the Ds does not move?

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Numerade Educator

Problem 4

Go In a Rutherford scattering experiment, assume that an incident alpha particle (radius $1.80 \mathrm{fm}$ ) is headed directly toward a target gold nucleus (radius $6.23 \mathrm{fm}) .$ What energy must the alpha particle have to just barely "touch" the gold nucleus?

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Problem 5

(co) When an alpha particle collides elastically with a nucleus, the nucleus recoils. Suppose a $5.00 \mathrm{MeV}$ alpha particle has a headon elastic collision with a gold nucleus that is initially at rest. What is the kinetic energy of (a) the recoiling nucleus and (b) the rebounding alpha particle?

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Problem 6

The strong neutron excess (defined as $N-Z$ ) of high-mass nuclei is illustrated by noting that most high-mass nuclides could never fission into two stable nuclei without neutrons being left over. For example, consider the spontaneous fission of a ${ }^{235} \mathrm{U}$ nucleus into two stable daughter nuclei with atomic numbers 39 and
53. From Appendix $\mathrm{F}$, determine the name of the (a) first and (b) second daughter nucleus. From Fig. $42-5$, approximately how many

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Problem 7

What is the nuclear mass density $\rho_{m}$ of (a) the fairly low-mass nuclide ${ }^{55} \mathrm{Mn}$ and $(\mathrm{b})$ the fairly high-mass nuclide ${ }^{209} \mathrm{Bi} ?(\mathrm{c})$ Compare the two answers, with an explanation. What is the nuclear charge density $\rho_{q}$ of (d) ${ }^{55} \mathrm{Mn}$ and (e) ${ }^{209} \mathrm{Bi}$ ? (f) Compare the two answers, with an explanation.

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Problem 8

(a) Show that the mass $M$ of an atom is given approximately by $M_{\mathrm{app}}=A m_{\mathrm{p}}$, where $A$ is the mass number and $m_{\mathrm{p}}$ is the proton mass. For (b) ${ }^{1} \mathrm{H},(\mathrm{c})^{31} \mathrm{P},(\mathrm{d}){ }^{120} \mathrm{Sn},(\mathrm{e}){ }^{197} \mathrm{Au}$, and $(\mathrm{f})^{239} \mathrm{Pu}$, use Table
$42-1$ to find the percentage deviation between $M_{\text {app }}$ and $M$ :

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Problem 9

The nuclide ${ }^{14} \mathrm{C}$ contains (a) how many protons and (b) how many neutrons?

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Problem 10

What is the mass excess $\Delta_{1}$ of ${ }^{1} \mathrm{H}$ (actual mass is $\left.1.007825 \mathrm{u}\right)$ in
(a) atomic mass units and (b) $\mathrm{MeV} / \mathrm{c}^{2}$ ? What is the mass excess $\Delta_{\mathrm{n}}$ of a neutron (actual mass is $1.008665 \mathrm{u}$ ) in (c) atomic mass units and
(d) $\mathrm{MeV} / \mathrm{c}^{2} ?$ What is the mass excess $\Delta_{120}$ of ${ }^{120} \mathrm{Sn}$ (actual mass is $119.902197 \mathrm{u})$ in (e) atomic mass units and (f) $\mathrm{MeV} / \mathrm{c}^{2}$ ?

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Problem 11

Nuclear radii may be measured by scattering highenergy (high speed) electrons from nuclei.
(a) What is the de Broglie wavelength for $200 \mathrm{MeV}$ electrons? (b) Are these electrons suitable probes for this purpose?

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Problem 12

The electric potential energy of a uniform sphere of charge $q$ and radius $r$ is given by
$$
U=\frac{3 q^{2}}{20 \pi \varepsilon_{0} r}
$$
(a) Does the energy represent a tendency for the sphere to bind together or blow apart? The nuclide ${ }^{239} \mathrm{Pu}$ is spherical with radius $6.64$ $\mathrm{fm}$. For this nuclide, what are (b) the electric potential energy $U$ according to the equation, (c) the electric potential energy per proton, and (d) the electric potential energy per nucleon? The binding energy per nucleon is $7.56 \mathrm{MeV}$. (e) Why is the nuclide bound so well when the answers to $(\mathrm{c})$ and $(\mathrm{d})$ are large and positive?

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Problem 13

A neutron star is a stellar object whose density is about that of nuclear matter, $2 \times 10^{17} \mathrm{~kg} / \mathrm{m}^{3}$. Suppose that the Sun were to collapse and become such a star without losing any of its present mass. What would be its radius?

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Problem 14

co What is the binding energy per nucleon of the americium isotope ${ }_{95}^{244} \mathrm{Am}$ ? Here are some atomic masses and the neutron mass.

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Problem 15

(a) Show that the energy associated with the strong force between nucleons in a nucleus is proportional to $A$, the mass number of the nucleus in question. (b) Show that the energy associated with the Coulomb force between protons in a nucleus is proportional to $Z(Z-1)$. (c) Show that, as we move to larger and larger nuclei (see Fig. 42-5), the importance of the Coulomb force increases more rapidly than does that of the strong force.

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Problem 16

(co) What is the binding energy per nucleon of the europium isotope ${ }_{63}^{152}$ Eu? Here are some atomic masses and the neutron mass.
$$
\begin{array}{llll}
51.921742 \mathrm{u} & { }^{1} \mathrm{H} & 1.007825 \\
1.008665 \mathrm{u} & &
\end{array}
$$

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Problem 17

Because the neutron has no charge, its mass must be found in some way other than by using a mass spectrometer. When a neutron and a proton meet (assume both to be almost stationary), they combine and form a deuteron, emitting a gamma ray whose energy is $2.2233 \mathrm{MeV} .$ The masses of the proton and the deuteron are $1.007276467 \mathrm{u}$ and $2.013553212 \mathrm{u}$, respectively. Find the mass of the neutron from these data.

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Problem 18

(co) What is the binding energy per nucleon of the rutherfordium isotope ${ }_{104}^{259} \mathrm{Rf}$ ? Here are some atomic masses and the neutron mass.
${ }_{104}^{259} \mathrm{Rf}$
$259.10563 \mathrm{u} \quad{ }^{1} \mathrm{H} \quad 1.007825 \mathrm{u}$
$1.008665 \mathrm{u}$ $\mathrm{l}_{104} \mathrm{KI}$

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Problem 19

A periodic table might list the average atomic mass of magnesium as being $24.312 \mathrm{u}$, which is the result of weighting the atomic masses of the magnesium isotopes according to their natural abundances on Earth. The three isotopes and their masses are ${ }^{24} \mathrm{Mg}(23.98504 \mathrm{u}),{ }^{25} \mathrm{Mg}(24.98584 \mathrm{u})$, and ${ }^{26} \mathrm{Mg}(25.98259 \mathrm{u})$
The natural abundance of ${ }^{24} \mathrm{Mg}$ is $78.99 \%$ by mass (that is, $78.99 \%$ of the mass of a naturally occurring sample of magnesium is due to the presence of ${ }^{24} \mathrm{Mg}$ ). What is the abundance of (a) ${ }^{25} \mathrm{Mg}$ and
(b) ${ }^{26} \mathrm{Mg}$ ?

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Problem 20

What is the binding energy per nucleon of ${ }^{262} \mathrm{Bh}$ ? The mass of the atom is $262.1231 \mathrm{u}$.

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Problem 21

SSM WWW (a) Show that the total binding energy $E_{\mathrm{be}}$ of a given nuclide is

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Problem 22

$.$ co An $\alpha$ particle $\left({ }^{4} \mathrm{He}\right.$ nucleus $)$ is to be taken apart in the following steps. Give the energy (work) required for each step: (a) remove a proton, (b) remove a neutron, and (c) separate the remaining proton and neutron. For an $\alpha$ particle, what are (d) the total binding energy and (e) the binding energy per nucleon? (f) Does either match an answer to (a), (b), or (c)? Here are some atomic masses and the neutron mass.
$\begin{array}{llll}{ }^{4} \mathrm{He} & 4.00260 \mathrm{u} & { }^{2} \mathrm{H} & 2.01410 \mathrm{u} \\ { }^{3} \mathrm{H} & 3.01605 \mathrm{u} & { }^{1} \mathrm{H} & 1.00783 \mathrm{u} \\ \mathrm{n} & 1.00867 \mathrm{u} & & \end{array}$

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Problem 23

Verify the binding energy per nucleon given in Table $42-1$ for the plutonium isotope ${ }^{239} \mathrm{Pu}$. The mass of the neutral atom is $239.05216 \mathrm{u}$.

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Problem 24

A penny has a mass of $3.0 \mathrm{~g} .$ Calculate the energy that would be required to separate all the neutrons and protons in this coin from one another. For simplicity, assume that the penny is made entirely of ${ }^{63} \mathrm{Cu}$ atoms (of mass $62.92960 \mathrm{u}$ ). The masses of the protonplus-electron and the neutron are $1.00783 \mathrm{u}$ and $1.00866 \mathrm{u}$, respectively.

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Problem 25

Cancer cells are more vulnerable to $x$ and gamma radiation than are healthy cells. In the past, the standard source for radiation therapy was radioactive ${ }^{60} \mathrm{Co}$, which decays, with a half-life of $5.27$ $\mathrm{y}$, into an excited nuclear state of ${ }^{60} \mathrm{Ni}$. That nickel isotope then immediately emits two gamma-ray photons, each with an approximate energy of $1.2 \mathrm{MeV}$. How many radioactive ${ }^{60} \mathrm{Co}$ nuclei are present in a $6000 \mathrm{Ci}$ source of the type used in hospitals? (Energetic particles from linear accelerators are now used in radiation therapy.)

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Problem 26

The half-life of a radioactive isotope is $140 \mathrm{~d}$. How many days would it take for the decay rate of a sample of this isotope to fall to one-fourth of its initial value?

Ben Nicholson

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Problem 27

A radioactive nuclide has a half-life of $30.0 \mathrm{y} .$ What fraction of an initially pure sample of this nuclide will remain undecayed at the end of (a) $60.0 \mathrm{y}$ and (b) $90.0 \mathrm{y}$ ?

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Problem 28

The plutonium isotope ${ }^{239} \mathrm{Pu}$ is produced as a by-product in nuclear reactors and hence is accumulating in our environment. It is radioactive, decaying with a half-life of $2.41 \times 10^{4} \mathrm{y} .$ (a) How many nuclei of Pu constitute a chemically lethal dose of $2.00 \mathrm{mg} ?(\mathrm{~b})$ What is the decay rate of this amount?

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Problem 29

A radioactive isotope of mercury, ${ }^{197} \mathrm{Hg}$, decays to gold, ${ }^{197} \mathrm{Au}$, with a disintegration constant of $0.0108 \mathrm{~h}^{-1}$. (a) Calculate the half-life of the ${ }^{197} \mathrm{Hg}$. What fraction of a sample will remain at the end of (b) three half-lives and (c) $10.0$ days?

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Problem 30

The half-life of a particular radioactive isotope is $6.5 \mathrm{~h}$. If there are initially $48 \times 10^{19}$ atoms of this isotope, how many remain at the end of $26 \mathrm{~h}$ ?

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Problem 31

Consider an initially pure $3.4 \mathrm{~g}$ sample of ${ }^{67} \mathrm{Ga}$, an isotope that has a half-life of $78 \mathrm{~h}$. (a) What is its initial decay rate? (b) What is its decay rate $48 \mathrm{~h}$ later?

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Problem 32

When aboveground nuclear tests were conducted, the explosions shot radioactive dust into the upper atmosphere. Global air circulations then spread the dust worldwide before it settled out on ground and water. One such test was conducted in October $1976 .$ What fraction of the ${ }^{90}$ Sr produced by that explosion still existed in October 2006? The half-life of ${ }^{90} \mathrm{Sr}$ is $29 \mathrm{y}$.

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Problem 33

The air in some caves includes a significant amount of radon gas, which can lead to lung cancer if breathed over a prolonged time. In British caves, the air in the cave with the greatest amount of the gas has an activity per volume of $1.55 \times 10^{5} \mathrm{~Bq} / \mathrm{m}^{3}$. Suppose that you spend two full days exploring (and sleeping in) that cave. Approximately how many ${ }^{222} \mathrm{Rn}$ atoms would you take in and out of your lungs during your two-day stay? The radionuclide ${ }^{222} \mathrm{Rn}$ in radon gas has a half-life of $3.82$ days. You need to estimate your lung capacity and average breathing rate.

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Problem 34

Calculate the mass of a sample of (initially pure) ${ }^{40} \mathrm{~K}$ that has an initial decay rate of $1.70 \times 10^{5}$ disintegrations/s. The isotope has a half-life of $1.28 \times 10^{9} \mathrm{y}$.

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Problem 35

A certain radionuclide is being manufactured in a cyclotron at a constant rate $R$. It is also decaying with disintegration constant $\lambda$. Assume that the production process has been going on for a time that is much longer than the half-life of the radionuclide.
(a) Show that the number of radioactive nuclei present after such time remains constant and is given by $N=R / \lambda .$ (b) Now show that this result holds no matter how many radioactive nuclei were present initially. The nuclide is said to be in secular equilibrium with its source; in this state its decay rate is just equal to its production rate.

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Problem 36

Plutonium isotope ${ }^{239}$ Pu decays by alpha decay with a halflife of $24100 \mathrm{y}$. How many milligrams of helium are produced by an initially pure $12.0 \mathrm{~g}$ sample of ${ }^{239} \mathrm{Pu}$ at the end of $20000 \mathrm{y}$ ? (Consider only the helium produced directly by the plutonium and not by any by-products of the decay process.)

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Problem 37

The radionuclide ${ }^{64} \mathrm{Cu}$ has a half-life of $12.7 \mathrm{~h} .$ If a sample contains $5.50 \mathrm{~g}$ of initially pure ${ }^{64} \mathrm{Cu}$ at $t=0$, how much of it will decay between $t=14.0 \mathrm{~h}$ and $t=16.0 \mathrm{~h} ?$

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Problem 38

A dose of $8.60 \mu \mathrm{Ci}$ of a radioactive isotope is injected into a patient. The isotope has a half-life of $3.0 \mathrm{~h}$. How many of the isotope parents are injected?

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Problem 39

The radionuclide ${ }^{56} \mathrm{Mn}$ has a half-life of $2.58 \mathrm{~h}$ and is produced in a cyclotron by bombarding a manganese target with deuterons. The target contains only the stable manganese isotope ${ }^{55} \mathrm{Mn}$, and the manganese-deuteron reaction that produces ${ }^{56} \mathrm{Mn}$ is
$$
{ }^{55} \mathrm{Mn}+\mathrm{d} \rightarrow{ }^{56} \mathrm{Mn}+\mathrm{p}
$$
If the bombardment lasts much longer than the half-life of ${ }^{56} \mathrm{Mn}$, the activity of the ${ }^{56} \mathrm{Mn}$ produced in the target reaches a final value of $8.88 \times 10^{10} \mathrm{~Bq} .$ (a) At what rate is ${ }^{56} \mathrm{Mn}$ being produced? (b) How many ${ }^{56} \mathrm{Mn}$ nuclei are then in the target? (c) What is their total mass?

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Problem 40

A source contains two phosphorus radionuclides, ${ }^{32} \mathrm{P}\left(T_{1 / 2}=\right.$ $14.3 \mathrm{~d}$ ) and ${ }^{33} \mathrm{P}\left(T_{1 / 2}=25.3 \mathrm{~d}\right)$. Initially, $10.0 \%$ of the decays come from ${ }^{33} \mathrm{P}$. How long must one wait until $90.0 \%$ do so?

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Problem 41

A $1.00 \mathrm{~g}$ sample of samarium emits alpha particles at a rate of 120 particles/s. The responsible isotope is ${ }^{147} \mathrm{Sm}$, whose natural abundance in bulk samarium is $15.0 \%$. Calculate the half-life.

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Problem 42

What is the activity of a 20 ng sample of ${ }^{92} \mathrm{Kr}$, which has a half-life of $1.84 \mathrm{~s}$ ?

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Problem 43

$.$ co A radioactive sample intended for irradiation of a hospital patient is prepared at a nearby laboratory. The sample has a halflife of $83.61$ h. What should its initial activity be if its activity is to be $7.4 \times 10^{8} \mathrm{~Bq}$ when it is used to irradiate the patient $24 \mathrm{~h}$ later?

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Problem 44

(co) Figure $42-19$ shows the decay of parents in a radioactive sample. The axes are scaled by $N_{s}=2.00 \times 10^{6}$ and $t_{s}=10.0 \mathrm{~s}$. What is the activity of the sample at $t=27.0 \mathrm{~s}$ ?

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Problem 45

(co) Figure $42-19$ shows the decay of parents in a radioactive sample. The axes are scaled by $N_{s}=2.00 \times 10^{6}$ and $t_{s}=10.0 \mathrm{~s}$. What is the activity of the sample at $t=27.0 \mathrm{~s}$ ?

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Problem 46

The radioactive nuclide ${ }^{99} \mathrm{Tc}$ can be injected into a patient's bloodstream in order to monitor the blood flow, measure the blood volume, or find a tumor, among other goals. The nuclide is produced in a hospital by a "cow" containing ${ }^{99}$ Mo, a radioactive nuclide that decays to ${ }^{99} \mathrm{Tc}$ with a half-life of $67 \mathrm{~h} .$ Once a day, the cow is "milked" for its ${ }^{99} \mathrm{Tc}$, which is produced in an excited state by the ${ }^{99} \mathrm{Mo} ;$ the ${ }^{99} \mathrm{Tc}$ de-excites to its lowest energy state by emitting a gamma-ray photon, which is recorded by detectors placed around the patient. The de-excitation has a half-life of $6.0 \mathrm{~h}$. (a) By what process does ${ }^{99}$ Mo decay to ${ }^{99} \mathrm{Tc}$ ? (b) If a patient is injected with an $8.2 \times 10^{7}$ Bq sample of ${ }^{99} \mathrm{Tc}$, how many gamma-ray photons are initially produced within the patient each second? (c) If the emission rate of gamma-ray photons from a small tumor that has collected ${ }^{99} \mathrm{Tc}$ is 38 per second at a certain time, how many excitedstate ${ }^{99} \mathrm{Tc}$ are located in the tumor at that time?

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Problem 47

SSM After long effort, in 1902 Marie and Pierre Curie succeeded in separating from uranium ore the first substantial quantity of radium, one decigram of pure $\mathrm{RaCl}_{2}$. The radium was the radioactive isotope ${ }^{226} \mathrm{Ra}$, which has a half-life of $1600 \mathrm{y}$. (a) How many radium nuclei had the Curies isolated? (b) What was the decay rate of their sample, in disintegrations per second?

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Problem 48

How much energy is released when a ${ }^{238} \mathrm{U}$ nucleus decays by emitting (a) an alpha particle and (b) a sequence of neutron, proton, neutron, proton? (c) Convince yourself both by rea-soned argument and by direct calculation that the difference between these two numbers is just the total binding energy of the alpha particle. (d) Find that binding energy. Some needed atomic and particle masses are

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Problem 49

Generally, more massive nuclides tend to be more unstable to alpha decay. For example, the most stable isotope of uranium, ${ }^{238} \mathrm{U}$, has an alpha decay half-life of $4.5 \times 10^{9} \mathrm{y} .$ The most stable isotope of plutonium is ${ }^{244} \mathrm{Pu}$ with an $8.0 \times 10^{7} \mathrm{y}$ half-life, and for curium we have ${ }^{248} \mathrm{Cm}$ and $3.4 \times 10^{5} \mathrm{y}$. When half of an original sample of ${ }^{238} \mathrm{U}$ has decayed, what fraction of the original sample of (a) plutonium and (b) curium is left?

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Problem 50

Large radionuclides emit an alpha particle rather than other combinations of nucleons because the alpha particle has such a stable, tightly bound structure. To confirm this statement, calculate the disintegration energies for these hypothetical decay processes and discuss the meaning of your findings:
(a) ${ }^{235} \mathrm{U} \rightarrow{ }^{232} \mathrm{Th}+{ }^{3} \mathrm{He}$
(b) ${ }^{235} \mathrm{U} \rightarrow{ }^{231} \mathrm{Th}+{ }^{4} \mathrm{He}$
(c) ${ }^{235} \mathrm{U} \rightarrow{ }^{230} \mathrm{Th}+{ }^{5} \mathrm{He}$.
The needed atomic masses are
$\begin{array}{llll}{ }^{232} \mathrm{Th} & 232.0381 \mathrm{u} & { }^{3} \mathrm{He} & 3.0160 \mathrm{u} \\ { }^{231} \mathrm{Th} & 231.0363 \mathrm{u} & { }^{4} \mathrm{He} & 4.0026 \mathrm{u} \\ { }^{230} \mathrm{Th} & 230.0331 \mathrm{u} & { }^{5} \mathrm{He} & 5.0122 \mathrm{u} \\ { }^{235} \mathrm{U} & 235.0429 \mathrm{u} & & \end{array}$

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Problem 51

$\mathrm{A}^{238} \mathrm{U}$ nucleus emits a $4.196 \mathrm{MeV}$ alpha particle. Calculate the disintegration energy $Q$ for this process, taking the recoil energy of the residual ${ }^{234} \mathrm{Th}$ nucleus into account.

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Problem 52

Under certain rare circumstances, a nucleus can decay by emitting a particle more massive than an alpha particle. Consider the decays
$$
{ }^{223} \mathrm{Ra} \rightarrow{ }^{209} \mathrm{~Pb}+{ }^{14} \mathrm{C} \quad \text { and } \quad{ }^{223} \mathrm{Ra} \rightarrow{ }^{219} \mathrm{Rn}+{ }^{4} \mathrm{He}
$$
Calculate the $Q$ value for the (a) first and (b) second decay and determine that both are energetically possible. (c) The Coulomb barrier height for alpha-particle emission is $30.0 \mathrm{MeV}$. What is the barrier height for ${ }^{14} \mathrm{C}$ emission? (Be careful about the nuclear radii.) The needed atomic masses are
$\begin{array}{ll}{ }^{223} \mathrm{Ra} & 223.01850 \mathrm{u} \\ { }^{209} \mathrm{~Pb} & 208.98107 \mathrm{u}\end{array}$
$$
{ }^{14} \mathrm{C} \quad 14.00324 \mathrm{u}
$$
$14.00324 \mathrm{u}$
$4.00260 \mathrm{u}$ ${ }^{4} \mathrm{He}$
$$
{ }^{219} \mathrm{Rn} \quad 219.00948 \mathrm{u}
$$

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Problem 53

The cesium isotope ${ }^{137} \mathrm{Cs}$ is present in the fallout from aboveground detonations of nuclear bombs. Because it decays with a slow ( $30.2 \mathrm{y}$ ) half-life into ${ }^{137} \mathrm{Ba}$, releasing considerable energy in the process, it is of environmental concern. The atomic masses of the $\mathrm{Cs}$ and $\mathrm{Ba}$ are $136.9071$ and $136.9058 \mathrm{u}$, respectively; calculate the total energy released in such a decay.

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Problem 54

Some radionuclides decay by capturing one of their own atomic electrons, a $K$ -shell electron, say. An example is
$$
{ }^{49} \mathrm{~V}+\mathrm{e}^{-} \rightarrow{ }^{49} \mathrm{Ti}+\nu, \quad T_{1 / 2}=331 \mathrm{~d}
$$Show that the disintegration energy $Q$ for this process is given by
$$
Q=\left(m_{\mathrm{V}}-m_{\mathrm{Ti}}\right) c^{2}-E_{K}
$$
where $m_{\mathrm{V}}$ and $m_{\mathrm{Ti}}$ are the atomic masses of ${ }^{49} \mathrm{~V}$ and ${ }^{49} \mathrm{Ti}$, respectively, and $E_{K}$ is the binding energy of the vanadium $K$ -shell electron. (Hint: Put $\mathbf{m}_{\mathrm{V}}$ and $\mathbf{m}_{\mathrm{Ti}}$ as the corresponding nuclear masses and then add in enough electrons to use the atomic masses.)

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Problem 55

A free neutron decays according to Eq. 42-26. If the neutron-hydrogen atom mass difference is $840 \mu \mathrm{u}$, what is the maximum kinetic energy $K_{\max }$ possible for the electron produced in a neutron decay?

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Problem 56

An electron is emitted from a middle-mass nuclide $(A=150$, say) with a kinetic energy of $1.0 \mathrm{MeV}$. (a) What is its de Broglie wavelength? (b) Calculate the radius of the emitting nucleus. (c) Can such an electron be confined as a standing wave in a "box" of such dimensions? (d) Can you use these numbers to disprove the (abandoned) argument that electrons actually exist in nuclei?

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Problem 57

$.$ co The radionuclide ${ }^{11} \mathrm{C}$ decays according to
$$
{ }^{11} \mathrm{C} \rightarrow{ }^{11} \mathrm{~B}+\mathrm{e}^{+}+\nu, \quad T_{1 / 2}=20.3 \mathrm{~min}
$$
The maximum energy of the emitted positrons is $0.960 \mathrm{MeV}$. (a) Show that the disintegration energy $Q$ for this process is given by
$$
Q=\left(m_{\mathrm{C}}-m_{\mathrm{B}}-2 m_{\mathrm{e}}\right) c^{2}
$$
where $m_{\mathrm{C}}$ and $m_{\mathrm{B}}$ are the atomic masses of ${ }^{11} \mathrm{C}$ and ${ }^{11} \mathrm{~B}$, respectively, and $m_{\mathrm{e}}$ is the mass of a positron. (b) Given the mass values $m_{\mathrm{C}}=11.011434 \mathrm{u}, m_{\mathrm{B}}=11.009305 \mathrm{u}$, and $m_{\mathrm{e}}=0.0005486 \mathrm{u}$,
calculate $Q$ and compare it with the maximum energy of the emitted positron given above. (Hint: Let $\mathbf{m}_{\mathrm{C}}$ and $\mathbf{m}_{\mathrm{B}}$ be the nuclear masses and then add in enough electrons to use the atomic masses.)

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Problem 58

Two radioactive materials that alpha decay, ${ }^{238} \mathrm{U}$ and ${ }^{232}$ Th, and one that beta decays, ${ }^{40} \mathrm{~K}$, are sufficiently abundant in granite to contribute significantly to the heating of Earth through the decay energy produced. The alpha-decay isotopes give rise to decay chains that stop when stable lead isotopes are formed. The isotope ${ }^{40} \mathrm{~K}$ has a single beta decay. (Assume this is the only possible decay of that isotope.) Here is the information:

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Problem 59

The radionuclide ${ }^{32} \mathrm{P}$ decays to ${ }^{32} \mathrm{~S}$ as described by Eq. 42-24. In a particular decay event, a $1.71 \mathrm{MeV}$ electron is emitted, the maximum possible value. What is the kinetic energy of the recoiling ${ }^{32} \mathrm{~S}$ atom in this event? (Hint: For the electron it is necessary to use the relativistic expressions for kinetic energy and linear momentum. The ${ }^{32} \mathrm{~S}$ atom is nonrelativistic.)

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Problem 60

A $5.00 \mathrm{~g}$ charcoal sample from an ancient fire pit has a ${ }^{14} \mathrm{C}$ activity of $63.0$ disintegrations $/ \mathrm{min}$. A living tree has a ${ }^{14} \mathrm{C}$ activity of $15.3$ disintegrations $/$ min per $1.00 \mathrm{~g}$. The half-life of ${ }^{14} \mathrm{C}$ is $5730 \mathrm{y}$. How old is the charcoal sample?

Ben Nicholson

Numerade Educator

Problem 61

The isotope ${ }^{238} \mathrm{U}$ decays to ${ }^{206} \mathrm{~Pb}$ with a half-life of $4.47 \times 10^{9}$
y. Although the decay occurs in many individual steps, the first step has by far the longest half-life; therefore, one can often consider the decay to go directly to lead. That is,
$$
{ }^{238} \mathrm{U} \rightarrow{ }^{206} \mathrm{~Pb}+\text { various decay products. }
$$
A rock is found to contain $4.20 \mathrm{mg}$ of ${ }^{238} \mathrm{U}$ and $2.135 \mathrm{mg}$ of ${ }^{206} \mathrm{~Pb}$. Assume that the rock contained no lead at formation, so all the lead now present arose from the decay of uranium. How many atoms of (a) ${ }^{238} \mathrm{U}$ and $(\mathrm{b})^{206} \mathrm{~Pb}$ does the rock now contain? (c) How many atoms of ${ }^{238} \mathrm{U}$ did the rock contain at formation? (d) What is the age of the rock?

Keshav Singh

Numerade Educator

Problem 62

A particular rock is thought to be 260 million years old. If it contains $3.70 \mathrm{mg}$ of ${ }^{238} \mathrm{U}$, how much ${ }^{206} \mathrm{~Pb}$ should it contain? See Problem 61 .

Keshav Singh

Numerade Educator

Problem 63

(co A rock recovered from far underground is found to contain $0.86 \mathrm{mg}$ of ${ }^{238} \mathrm{U}, 0.15 \mathrm{mg}$ of ${ }^{206} \mathrm{~Pb}$, and $1.6 \mathrm{mg}$ of ${ }^{40}$ Ar. How much ${ }^{40} \mathrm{~K}$ will it likely contain? Assume that ${ }^{40} \mathrm{~K}$ decays to only ${ }^{40}$ Ar with a half-life of $1.25 \times 10^{9} \mathrm{y}$. Also assume that ${ }^{238} \mathrm{U}$ has a half-life of $4.47 \times 10^{9} \mathrm{y}$

Keshav Singh

Numerade Educator

Problem 64

co The isotope ${ }^{40} \mathrm{~K}$ can decay to either ${ }^{40} \mathrm{Ca}$ or ${ }^{40} \mathrm{Ar}$; assume both decays have a half-life of $1.26 \times 10^{9} \mathrm{y}$. The ratio of the Ca produced to the Ar produced is $8.54 / 1=8.54$. A sample originally had only ${ }^{40} \mathrm{~K}$. It now has equal amounts of ${ }^{40} \mathrm{~K}$ and ${ }^{40} \mathrm{Ar} ;$ that is, the ratio of $\mathrm{K}$ to Ar is $1 / 1=1$. How old is the sample? (Hint: Work this like other radioactive-dating problems, except that this decay has two products.)

Keshav Singh

Numerade Educator

Problem 65

The nuclide ${ }^{198} \mathrm{Au}$, with a half-life of $2.70 \mathrm{~d}$, is used in cancer therapy. What mass of this nuclide is required to produce an activity of $250 \mathrm{Ci}$ ?

Keshav Singh

Numerade Educator

Problem 66

A radiation detector records 8700 counts in $1.00 \mathrm{~min}$. Assuming that the detector records all decays, what is the activity of the radiation source in (a) becquerels and (b) curies?

Ben Nicholson

Numerade Educator

Problem 67

An organic sample of mass $4.00 \mathrm{~kg}$ absorbs $2.00 \mathrm{~mJ}$ via slow neutron radiation $(\mathrm{RBE}=5)$. What is the dose equivalent (mSv)?

Keshav Singh

Numerade Educator

Problem 68

A $75 \mathrm{~kg}$ person receives a whole-body radiation dose of $2.4 \times 10^{-4}$ Gy, delivered by alpha particles for which the RBE factor is 12. Calculate (a) the absorbed energy in joules and the dose equivalent in (b) sieverts and (c) rem.

Ben Nicholson

Numerade Educator

Problem 69

An $85 \mathrm{~kg}$ worker at a breeder reactor plant accidentally ingests $2.5 \mathrm{mg}$ of ${ }^{239} \mathrm{Pu}$ dust. This isotope has a half-life of $24100 \mathrm{y}$, decaying by alpha decay. The energy of the emitted alpha particles is $5.2 \mathrm{MeV}$, with an RBE factor of $13 .$ Assume that the plutonium resides in the worker's body for $12 \mathrm{~h}$ (it is eliminated naturally by the digestive system rather than being absorbed by any of the internal organs) and that $95 \%$ of the emitted alpha particles are stopped within the body. Calculate (a) the number of plutonium atoms ingested, (b) the number that decay during the $12 \mathrm{~h},(\mathrm{c})$ the

Keshav Singh

Numerade Educator

Problem 70

A typical kinetic energy for a nucleon in a middle-mass nucleus may be taken as $5.00 \mathrm{MeV}$. To what effective nuclear temperature does this correspond, based on the assumptions of the collective model of nuclear structure?

Ben Nicholson

Numerade Educator

Problem 71

A measurement of the energy $E$ of an intermediate nucleus must be made within the mean lifetime $\Delta t$ of the nucleus and necessarily carries an uncertainty $\Delta E$ according to the uncertainty principle
$$
\Delta E \cdot \Delta t=\hbar .
$$
(a) What is the uncertainty $\Delta E$ in the energy for an intermediate nucleus if the nucleus has a mean lifetime of $10^{-22} \mathrm{~s} ?(\mathrm{~b})$ Is the nucleus a compound nucleus?

Keshav Singh

Numerade Educator

Problem 72

In the following list of nuclides, identify (a) those with filled nucleon shells, (b) those with one nucleon outside a filled shell, and
(c) those with one vacancy in an otherwise filled shell: ${ }^{13} \mathrm{C},{ }^{18} \mathrm{O},{ }^{40} \mathrm{~K}$, ${ }^{49} \mathrm{Ti},{ }^{60} \mathrm{Ni},{ }^{91} \mathrm{Zr},{ }^{92} \mathrm{Mo},{ }^{121} \mathrm{Sb},{ }^{143} \mathrm{Nd},{ }^{144} \mathrm{Sm},{ }^{205} \mathrm{Tl}$, and ${ }^{207} \mathrm{~Pb}$.

Keshav Singh

Numerade Educator

Problem 73

Consider the three formation processes shown for the compound nucleus ${ }^{20}$ Ne in Fig. 42-14. Here are some of the atomic and particle masses:
$$
\begin{array}{llll}
{ }^{20} \mathrm{Ne} & 19.99244 \mathrm{u} & \alpha & 4.00260 \mathrm{u} \\
{ }^{19} \mathrm{~F} & 18.99840 \mathrm{u} & \mathrm{p} & 1.00783 \mathrm{u} \\
{ }^{16} \mathrm{O} & 15.99491 \mathrm{u} & &
\end{array}
$$
What energy must (a) the alpha particle, (b) the proton, and (c) the $\gamma$ -ray photon have to provide $25.0 \mathrm{MeV}$ of excitation energy to the compound nucleus?

Keshav Singh

Numerade Educator

Problem 74

(co) In a certain rock, the ratio of lead atoms to uranium atoms is $0.300$. Assume that uranium has a half-life of $4.47 \times 10^{9} \mathrm{y}$ and that the rock had no lead atoms when it formed. How old is the rock?

Keshav Singh

Numerade Educator

Problem 75

A certain stable nuclide, after absorbing a neutron, emits an electron, and the new nuclide splits spontaneously into two alpha particles. Identify the nuclide.

Keshav Singh

Numerade Educator

Problem 76

A typical chest x-ray radiation dose is $250 \mu \mathrm{Sv}$, delivered by $\mathrm{x}$ rays with an $\mathrm{RBE}$ factor of $0.85$. Assuming that the mass of the exposed tissue is one-half the patient's mass of $88 \mathrm{~kg}$, calculate the energy absorbed in joules.

Ben Nicholson

Numerade Educator

Problem 77

How many years are needed to reduce the activity of ${ }^{14} \mathrm{C}$ to $0.020$ of its original activity? The half-life of ${ }^{14} \mathrm{C}$ is $5730 \mathrm{y}$.

Keshav Singh

Numerade Educator

Problem 78

co Radioactive element $A A$ can decay to either element $B B$ or element $C C$. The decay depends on chance, but the ratio of the resulting number of $B B$ atoms to the resulting number of $C C$ atoms is always $2 / 1$. The decay has a half-life of $8.00$ days. We start with a sample of pure $A A$. How long must we wait until the number of $C C$ atoms is $1.50$ times the number of $A A$ atoms?

Keshav Singh

Numerade Educator

Problem 79

(co) SSM One of the dangers of radioactive fallout from a nuclear bomb is its ${ }^{90} \mathrm{Sr}$, which decays with a 29 -year half-life. Because it has chemical properties much like those of calcium, the strontium, if ingested by a cow, becomes concentrated in the cow's milk. Some of the ${ }^{90}$ Sr ends up in the bones of whoever drinks the milk. The energetic electrons emitted in the beta decay of ${ }^{90}$ Sr damage the bone marrow and thus impair the production of red blood cells. A 1 megaton bomb produces approximately $400 \mathrm{~g}$ of ${ }^{90} \mathrm{Sr}$. If the fallout spreads uniformly over a $2000 \mathrm{~km}^{2}$ area, what ground area

Keshav Singh

Numerade Educator

Problem 80

Because of the 1986 explosion and fire in a reactor at the Chernobyl nuclear power plant in northern Ukraine, part of Ukraine is contaminated with ${ }^{137} \mathrm{Cs}$, which undergoes beta-minus decay with a half-life of $30.2 \mathrm{y} .$ In 1996, the total activity of this contamination over an area of $2.6 \times 10^{5} \mathrm{~km}^{2}$ was estimated to be $1 \times$ $10^{16} \mathrm{~Bq} .$ Assume that the ${ }^{137} \mathrm{Cs}$ is uniformly spread over that area and that the beta-decay electrons travel either directly upward or directly downward. How many beta-decay electrons would you intercept were you to lie on the ground in that area for $1 \mathrm{~h}$ (a) in 1996 and (b) today? (You need to estimate your cross-sectional area that intercepts those electrons.)

Keshav Singh

Numerade Educator

Problem 81

Figure 42-20 shows part of the decay scheme of ${ }^{237} \mathrm{~Np}$ on a plot of mass number $A$ versus proton number $Z$; five lines that represent either alpha decay or beta-minus decay connect dots that represent isotopes. What is the isotope at the end of the five decays (as marked with a question mark in Fig. $42-20$ ) ?

Keshav Singh

Numerade Educator

Problem 82

After a brief neutron irradiation of silver, two isotopes are present: ${ }^{108} \mathrm{Ag}\left(T_{1 / 2}=2.42 \mathrm{~min}\right)$ with an initial decay rate of $3.1 \times$ $10^{5} / \mathrm{s}$, and ${ }^{110} \mathrm{Ag}\left(T_{1 / 2}=24.6 \mathrm{~s}\right)$ with an initial decay rate of $4.1 \times$
$10^{6} / \mathrm{s}$. Make a semilog plot similar to Fig. 42-9 showing the total combined decay rate of the two isotopes as a function of time from $t=0$ until $t=10 \mathrm{~min}$. We used Fig. 42-9 to illustrate the extraction of the half-life for simple (one isotope) decays. Given only your plot of total decay rate for the two-isotope system here, suggest a way to analyze it in order to find the half-lives of both isotopes.

Timothy Black

Numerade Educator

Problem 83

Because a nucleon is confined to a nucleus, we can take the uncertainty in its position to be approximately the nuclear radius $r$ Use the uncertainty principle to determine the uncertainty $\Delta p$ in the linear momentum of the nucleon. Using the approximation $p \approx \Delta p$ and the fact that the nucleon is nonrelativistic, calculate the kinetic energy of the nucleon in a nucleus with $A=100$.

Salamat Ali

Numerade Educator

Problem 84

A radium source contains $1.00 \mathrm{mg}$ of ${ }^{226} \mathrm{Ra}$, which decays with a half-life of $1600 \mathrm{y}$ to produce ${ }^{222} \mathrm{Rn}$, a noble gas. This radon isotope in turn decays by alpha emission with a half-life of $3.82 \mathrm{~d}$. If this process continues for a time much longer than the half-life of ${ }^{222} \mathrm{Rn}$, the ${ }^{222}$ Rn decay rate reaches a limiting value that matches the rate at which ${ }^{222} \mathrm{Rn}$ is being produced, which is approximately constant because of the relatively long half-life of ${ }^{226} \mathrm{Ra}$. For the source under this limiting condition, what are (a) the activity of ${ }^{226} \mathrm{Ra},(\mathrm{b})$ the activity of ${ }^{222} \mathrm{Rn}$, and $(\mathrm{c})$ the total mass of ${ }^{222} \mathrm{Rn}$ ?

Keshav Singh

Numerade Educator

Problem 85

Make a nuclidic chart similar to Fig. 42-6 for the 25 nuclides ${ }^{118-122} \mathrm{Te},{ }^{117-121} \mathrm{Sb},{ }^{116-120} \mathrm{Sn},{ }^{115-119} \mathrm{In}$, and ${ }^{114-118} \mathrm{Cd}$. Draw in and la-bel (a) all isobaric (constant $A$ ) lines and (b) all lines of constant neutron excess, defined as $N-Z$.

Dominador Tan

Numerade Educator

Problem 86

co A projectile alpha particle is headed directly toward a target aluminum nucleus. Both objects are assumed to be spheres. What energy is required of the alpha particle if it is to momentarily stop just as its "surface" touches the "surface" of the aluminum nucleus? Assume that the target nucleus remains stationary.

Keshav Singh

Numerade Educator

Problem 87

Consider a ${ }^{238} \mathrm{U}$ nucleus to be made up of an alpha particle $\left({ }^{4} \mathrm{He}\right)$ and a residual nucleus $\left({ }^{234} \mathrm{Th}\right)$. Plot the electrostatic potential energy $U(r)$, where $r$ is the distance between these particles. Cover the approximate range $10 \mathrm{fm}<r<100 \mathrm{fm}$ and compare your plot with that of Fig. $42-10$.

Keshav Singh

Numerade Educator

Problem 88

Characteristic nuclear time is a useful but loosely defined quantity, taken to be the time required for a nucleon with a few million electron-volts of kinetic energy to travel a distance equal to the diameter of a middle-mass nuclide. What is the order of magnitude of this quantity? Consider $5 \mathrm{MeV}$ neutrons traversing a nuclear diameter of ${ }^{197} \mathrm{Au} ;$ use Eq. $42-3$.

Keshav Singh

Numerade Educator

Problem 89

What is the likely mass number of a spherical nucleus with a radius of $3.6 \mathrm{fm}$ as measured by electron-scattering methods?

Salamat Ali

Numerade Educator

Problem 90

Using a nuclidic chart, write the symbols for (a) all stable isotopes with $Z=60,(\mathrm{~b})$ all radioactive nuclides with $N=60$, and $(\mathrm{c})$ all nuclides with $A=60$.

Ben Nicholson

Numerade Educator

Problem 91

If the unit for atomic mass were defined so that the mass of ${ }^{1} \mathrm{H}$ were exactly $1.000000 \mathrm{u}$, what would be the mass of $(\mathrm{a}){ }^{12} \mathrm{C}$ (actual mass $12.000000 \mathrm{u}$ ) and $(\mathrm{b})^{238} \mathrm{U}($ actual mass $238.050785 \mathrm{u})$ ?

Salamat Ali

Numerade Educator

Problem 92

High-mass radionuclides, which may be either alpha or beta emitters, belong to one of four decay chains, depending on whether their mass numher $A$ is of the form $4 n, 4 n+1,4 n+2$, or $4 n+3$ where $n$ is a positive integer. (a) Justify this statement and show that if a nuclide belongs to one of these families, all its decay products belong to the same family. Classify the following nuclides as to family:
(b) ${ }^{235} \mathrm{U},(\mathrm{c})^{236} \mathrm{U}$
(d) $^{238} \mathrm{U},(\mathrm{e})^{239} \mathrm{Pu},(\mathrm{f})^{240} \mathrm{Pu},(\mathrm{g})^{245} \mathrm{Cm}$,
(h) $^{246} \mathrm{Cm}$, (i)
${ }^{249} \mathrm{Cf}$, and $(\mathrm{j})^{253} \mathrm{Fm}$.

Keshav Singh

Numerade Educator

Problem 93

Find the disintegration energy $Q$ for the decay of ${ }^{49} \mathrm{~V}$ by $K$. electron capture (see Problem 54). The needed data are $m_{\mathrm{V}}=$ $48.94852 \mathrm{u}, m_{\mathrm{Ti}}=48.94787 \mathrm{u}$, and $E_{K}=5.47 \mathrm{keV}$

Keshav Singh

Numerade Educator

Problem 94

Locate the nuclides displayed in Table $42-1$ on the nuclidic chart of Fig. $42-5$. Verify that they lie in the stability zone.

Ben Nicholson

Numerade Educator

Problem 95

The radionuclide ${ }^{32} \mathrm{P}\left(T_{1 / 2}=14.28 \mathrm{~d}\right)$ is often used as a tracer to follow the course of biochemical reactions involving phosphorus. (a) If the counting rate in a particular experimental setup is initially 3050 counts/s, how much time will the rate take to fall to 170 counts/s? (b) A solution containing ${ }^{32} \mathrm{P}$ is fed to the root system of an experimental tomato plant, and the ${ }^{32} \mathrm{P}$ activity in a leaf is measured $3.48$ days later. By what factor must this reading be multiplied to correct for the decay that has occurred since the experiment began?

Keshav Singh

Numerade Educator

Problem 96

At the end of World War II, Dutch authorities arrested Dutch artist Hans van Meegeren for treason because, during the war, he had sold a masterpiece painting to the Nazi Hermann Goering. The painting, Christ and His Disciples at Emmaus by Dutch master Johannes Vermeer (1632-1675), had been discovered in 1937 by van Meegeren, after it had been lost for almost 300 years. Soon after the discovery, art experts proclaimed that Emmaus was possibly the best Vermeer ever seen. Selling such a Dutch national treasure to the enemy was unthinkable treason.

However, shortly after being imprisoned, van Meegeren suddenly announced that he, not Vermeer, had painted Emmaus. He

Sheryl Ezze

Numerade Educator

Problem 97

From data presented in the first few paragraphs of Module 42-3, find (a) the disintegration constant $\lambda$ and (b) the half-life of ${ }^{238} \mathrm{U}$.

Salamat Ali

Numerade Educator