Chapter 41

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

(I) What is the approximate radius of an alpha particle \(\left({ }_{2}^{4} \mathrm{He}\right) ?\)

(II) (a) What is the approximate radius of a \(148 \mathrm{~d}\) nucleus? (b) Approximately what is the value of \(A\) for a nucleus whose radius is \(3.7 \times 10-15 \mathrm{~m} ?\)

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

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

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

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

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

(I) The \({ }_{3}^{7}\) Li nucleus has an excited state \(0.48 \mathrm{MeV}\) above the ground state. What wavelength gamma photon is emitted when the nucleus decays from the excited state to the ground state?

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

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

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

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

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

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

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

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

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

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

(I) \((a)\) What is the decay constant of \({ }_{92}^{238} \mathrm{U}\) whose half-life is \(4.5 \times 10^{9}\) yr? \((b)\) The decay constant of a given nucleus is \(3.2 \times 10^{-5} \mathrm{~s}^{-1} .\) What is its half-life?

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

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

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

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

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

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

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

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

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

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

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

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

(II) Calculate the activity of a pure \(8.7-\mu \mathrm{g}\) sample of \({ }_{15}^{32} \mathrm{P}\left(T_{\frac{1}{2}}=1.23 \times 10^{6} \mathrm{~s}\right)\)

(II) A sample of \({ }_{92}^{233} \mathrm{U} \quad\left(T_{1}=1.59 \times 10^{5} \mathrm{yr}\right)\) contains \(5.50 \times 10^{18}\) nuclei. \(\quad(a)\) What \({ }^{2}\) is the decay constant? (b) Approximately how many disintegrations will occur per minute?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Strontium-90 is produced as a nuclear fission product of uranium in both reactors and atomic bombs. Look at its location in the Periodic Table to see what other elements it might be similar to chemically, and tell why you think it might be dangerous to ingest. It has too many neutrons, and it decays with a half-life of about 29 yr. How long will we have to wait for the amount of \({ }_{38}^{90} \mathrm{Sr}\) on the Earth's surface to reach \(1 \%\) of its current level, assuming no new material is scattered about? Write down the decay reaction, including the daughter nucleus. The daughter is radioactive: write down its decay.