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Master the Concepts
Conceptual Questions
  1. How could Henri Becquerel and other scientists determine that there were three different kinds of radiation before having determined the electric charges or masses of the α, β, and γ rays?

  2. What technique could Becquerel and others have used to determine that α rays are positively charged, β rays negatively charged, and γ rays uncharged? Explain how they could find that α rays have a charge-to-mass ratio half that of the H+ ion, and β rays have the same charge-to-mass ratio as “cathode rays” (electrons). (See Chapter 19 for some ideas.)

  3. Why is a slow neutron more likely to induce a nuclear reaction (as in neutron activation and induced fission) than a proton with the same kinetic energy?

  4. Explain why neutron-activated nuclides tend to decay by β rather than β+.

  5. Why can we ignore the binding energies of the atomic electrons in calculations such as Example 29.4? Isn't there a mass defect due to the binding energy of the electrons?

  6. Why would we expect atmospheric testing of nuclear weapons to increase the relative abundance of carbon-14 in the atmosphere? Why would we expect the widespread burning of fossil fuels to decrease the relative abundance of carbon-14 in the atmosphere?

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    Isolated atoms (or atoms in a dilute gas) radiate photons at discrete energies characteristic of that atom. In dense matter, the spectrum radiated is quasi-continuous. Why doesn't the same thing happen with nuclear spectra: why do the γ rays have the same characteristic energies even when emitted from a solid?

  8. Section 29.8 states that the total energy released by the proton-proton cycle is the same as that released by the carbon cycle. Why must the total energy released be the same?

  9. Iodine is eliminated from the body through biological processes with an effective half-life of about 140 days. The radioactive half-life of iodine-131 is 8 days. Suppose some radioactive 131I nuclei are present in the body. Assuming that no new 131I nuclei are introduced into the body, how much time must pass until only half as much 131I is left in the body: less than 8 days, between 8 and 140 days, or more than 140 days? Explain your reasoning.

  10. Why does a fission reaction tend to release one or more neutrons? Why is the release of neutrons necessary in order to sustain a chain reaction?

  11. Radioactive α-emitters are relatively harmless outside the body, but can be dangerous if ingested or inhaled. Explain.

  12. Fission reactors and cyclotrons tend to produce different kinds of isotopes. A reactor produces isotopes primarily through neutron activation; thus, the isotopes tend to be neutron-rich (high neutron-to-proton ratio). A cyclotron can only accelerate charged particles such as protons or deuterons. When stable nuclei are bombarded with protons or deuterons, the resulting radioisotopes are neutron-deficient (low neutron-to-proton ratio). (a) Explain why a cyclotron cannot accelerate neutrons. (b) Suppose a hospital needs a supply of radioisotopes to use in positron-emission tomography (PET). Would the radioisotopes more likely come from a reactor or a cyclotron? Explain.

  13. Why would a fusion reactor produce less radioactive waste than a fission reactor? [Hint: Compare the products of a fission reaction with those from a fusion reaction.]

  14. Radon-222 is created in a series of radioactive decays starting with and ending with The half-life of 222Rn is 3.8 days. (a) If the half-life is so short, why hasn't all the 222Rn gas decayed by now? (b) If the half-life of 222Rn were much shorter, say a few seconds, would it be more dangerous to us or less dangerous? What if the half-life were much longer, say thousands of years?

Multiple-Choice Questions
  1. Radioactive decays into Which of these particles is released in the decay?

    • (a) a proton

    • (b) an electron

    • (c) a positron

    • (d) an α particle

    • (e) a neutron

    • (f) none of the above

  2. For all stable nuclei

    • (a) there are equal numbers of protons and neutrons.

    • (b) there are more protons than neutrons.

    • (c) there are more neutrons than protons.

    • (d) none of the above have to be true.

  3. For all stable nuclei

    • (a) the mass of the nucleus is less than Zmp + (AZ)mn.

    • (b) the mass of the nucleus is greater than Zmp + (AZ)mn.

    • (c) the mass of the nucleus is equal to Zmp + (AZ)mn.

    • (d) none of the above have to be true.

  4. Of the hypothetical nuclear reactions listed here, which would violate conservation of charge?

    • (a)

    • (b)

    • (c)

    • (d)

    • (e) none of them

    • (f) all of them

    • (g) all but (c)

    • (h) (a) and (d)

  5. Of the hypothetical nuclear reactions listed in Multiple-Choice Question 4, which would violate conservation of nucleon number?

  6. In a fusion reaction, two deuterons produce a helium-3 nucleus. What is the other product of the reaction?

    • (a) an electron

    • (b) a proton

    • (c) a neutron

    • (d) an α particle

    • (e) a positron

    • (f) a neutrino

  7. The activity of a radioactive sample (with a single radioactive nuclide) decreases to one eighth its initial value in a time interval of 96 days. What is the half-life of the radioactive nuclide present?

    • (a) 6 days

    • (b) 8 days

    • (c) 12 days

    • (d) 16 days

    • (e) 24 days

    • (f) 32 days

  8. Solid lead has more than four times the mass density of solid aluminum. What is the main reason that lead is so much more dense?

    • (a) The Pb atom is smaller than the Al atom.

    • (b) The Pb nucleus is smaller than the Al nucleus.

    • (c) The Pb nucleus is more massive than the Al nucleus.

    • (d) The Pb nucleus is more dense than the Al nucleus.

    • (e) The Pb atom has many more electrons than the Al atom.

  9. Which of these are appropriate units for the decay constant λ of a radioactive nuclide?

    • (a) s

    • (b) Ci

    • (c) rd

    • (d) s−1

    • (e) rem

    • (f) MeV

  10. Which of the units listed in Multiple-Choice Question 9 are appropriate for the biologically equivalent dose that results when a person is exposed to radiation?

Problems
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29.1 Nuclear Structure
  • 1. Estimate the number of nucleons found in the body of a 75-kg person.

 
  • 2. Calculate the mass density of nuclear matter.

  • 3. A neutron star is a star that has collapsed into a collection of tightly packed neutrons. Thus, it is something like a giant nucleus; but since it is electrically neutral, there is no Coulomb repulsion to break it up. The force holding it together is gravity. Suppose the Sun were to collapse into a neutron star. What would its radius be? Assume that the density is about the same as for a nucleus. Express your answer in kilometers.

 
 
  • 4. Write the symbol (in the form ) for the nuclide with 38 protons and 50 neutrons and identify the element.

  • 5. Write the symbol (in the form ) for the isotope of potassium with 21 neutrons.

 
 
  • 6. How many neutrons are found in a 35Cl nucleus?

  • 7. How many protons are found in a 136Xe nucleus?

 
  • 8. Write the symbol (in the form ) for the nuclide that has 78 neutrons and 53 protons.

  • 9. Find the radius and volume of the nucleus.

29.2 Binding Energy
 
  • 10. What is the binding energy of an α particle (a 4He nucleus)? The mass of an α particle is 4.001 51 u.

  • 11. Find the binding energy of a deuteron (a 2H nucleus). The mass of a deuteron (not the deuterium atom) is 2.013 553 u.

 
  • 12. What is the average binding energy per nucleon for

 
  • 13. (a) Find the binding energy of the 16O nucleus. (b) What is the average binding energy per nucleon? Check your answer using Fig. 29.2.

  • 14. Calculate the binding energy per nucleon of the nucleus.

 
  • 15. What is the mass defect of the 14N nucleus?

  • 16. What is the mass of an 16O atom in units of MeV/c2? (1 MeV/c2 is the mass of a particle with rest energy 1 MeV.)

  • 17. (a) What is the mass defect of the 1H atom due to the binding energy of the electron (in the ground state)? (b) Should we worry about this mass defect when we calculate the mass of the 1H nucleus by subtracting the mass of one electron from the mass of the 1H atom?

  • 18. Show that c2 = 931.494 MeV/u. [Hint: Start with the conversion factors to SI units for MeV and atomic mass units.]

  • 19. Using a mass spectrometer, the mass of the ion is found to be 238.050 24 u. (a) Use this result to calculate the mass of the nucleus. (b) Now find the binding energy of the nucleus.

  • 20. To make an order-of-magnitude estimate of the energy level spacings in the nucleus, assume that a nucleon is confined to a one-dimensional box of width 10 fm (a typical nuclear diameter). Calculate the energy of the ground state.

29.3 Radioactivity
 
  • 21. Identify the daughter nuclide when decays via β decay.

  • 22. Thorium-232 decays via α decay. Write out the reaction and identify the daughter nuclide.

 
 
  • 23. Write out the reaction and identify the daughter nuclide when decays by electron capture.

  • 24. Write out the reaction and identify the daughter nuclide when decays by emitting a positron.

 
  • 25. Radium-226 decays as If the nucleus is at rest before the decay and the nucleus is in its ground state, estimate the kinetic energy of the α particle. (Assume that the nucleus takes away an insignificant fraction of the kinetic energy.)

  • 26. Which decay mode would you expect for radioactive α, β, or β+? Explain. [Hint: Look at the neutron-to-proton ratio.]

 
  • 27. Calculate the maximum kinetic energy of the β particle when decays via β decay.

  • 28. Calculate the energy of the antineutrino when decays via β decay if the β particle has a kinetic energy of 435 keV.

 
  • 29. Show that the spontaneous α decay of 19O is not possible.

  • 30. Calculate the kinetic energy of the α particle in Probelm 25. This time, do not assume that the nucleus is at rest after the reaction. Start by figuring out the ratio of the kinetic energies of the α particle and the Rn nucleus.

  • 31. An isotope of sodium, decays by β+ emission. Estimate the maximum possible kinetic energy of the positron by assuming that the kinetic energy of the daughter nucleus and the total energy of the neutrino emitted are both zero. [Hint: Remember to keep track of the electron masses.]

  • 32. The nucleus in a atom captures one of the atom's electrons, changing the nucleus to and emitting a neutrino. What is the total energy of the emitted neutrino? [Hint:You can use the classical expression for the kinetic energy of the atom and the extremely relativistic expression for the kinetic energy of the neutrino.]

29.4 Radioactive Decay Rates and Half-Lives
  • 33. A certain radioactive nuclide has a half-life of 200.0 s. A sample containing just this one radioactive nuclide has an initial activity of 80,000.0 s−1. (a) What is the activity 600.0 s later? (b) How many nuclei were there initially? (c) What is the probability per second that any one of the nuclei decays?

 
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    34. The half-life of I-131 is 8.0 days. A sample containing I-131 has an activity of 6.4 × 108 Bq. How many days later will the sample have an activity of 2.5 × 106 Bq?

    Tutorial: Radioactive Decay

  • 35. Some bones discovered in a crypt in Guatemala are carbon-dated. The 14C activity of the bones is measured to be 0.242 Bq per gram of carbon. Approximately how old are the bones?

    Tutorial: Radiocarbon Dating

 
 
  • 36. Carbon-14 dating is used to date a bone found at an archaeological excavation. If the ratio of C-14 to C-12 atoms is 3.25 × 10−13, how old is the bone? [Hint: Note that this ratio is one fourth the ratio of 1.3 × 10−12 that is found in a living sample.]

  • 37. A sample of radioactive which has a half-life of 19.9 min, has an activity of 0.058 Ci. What is its activity 1.0 h later?

 
  • 38. The activity of a sample containing radioactive 108Ag is 6.4 × 104 Bq. Exactly 12 min later, the activity is 2.0 × 103 Bq. Calculate the half-life of 108Ag.

 
  • 39. Calculate the activity of 1.0 g of radium-226 in Ci.

  • 40. What is the activity in becquerels of 1.0 kg of 238U?

 
  • 41. In this problem, you will verify the statement (in Section 29.4) that the 14C activity in a living sample is 0.25 Bq per gram of carbon. (a) What is the decay constant λ for 14C? (b) How many 14C atoms are in 1.00 g of carbon? One mole of carbon atoms has a mass of 12.011 g, and the relative abundance of 14C is 1.3 × 10−12. (c) Using your results from parts (a) and (b), calculate the 14C activity per gram of carbon in a living sample.

  • 42. A radioactive sample has equal numbers of 15O and 19O nuclei. Use the half-lives found in Appendix B to determine how long it will take before there are twice as many 15O nuclei as 19O. What percent of the 19O nuclei have decayed during this time?

  • 43. Show mathematically that if and only if T1/2 = τ ln 2. [Hint: Take the natural logarithm of each side.]

  • 44. The Physics at Home in Section 29.4 suggests tossing coins as a model of radioactive decay. An improved version is to toss a large number of dice instead of coins: each die that comes up a “one” represents a nucleus that has decayed. Suppose that N dice are tossed. (a) What is the average number of dice you expect to decay on one toss? (b) What is the average number of dice you expect to remain undecayed after three tosses? (c) What is the average number of dice you expect to remain undecayed after four tosses? (d) What is the half-life in numbers of tosses?

29.5 Biological Effects of Radiation
  • 45. An α particle produced in radioactive α decay has a kinetic energy of typically about 6 MeV. When an α particle passes through matter (e.g., biological tissue), it makes ionizing collisions with molecules, giving up some of its kinetic energy to supply the binding energy of the electron that is removed. If a typical ionization energy for a molecule in the body is around 20 eV, roughly how many molecules can the alpha particle ionize before coming to rest?

  • 46. If meat is irradiated with 2000.0 Gy of x-rays, most of the bacteria are killed and the shelf life of the meat is greatly increased. (a) How many 100.0-keV photons must be absorbed by a 0.30-kg steak so that the absorbed dose is 2000.0 Gy? (b) Assuming steak has the same specific heat as water, what temperature increase is caused by a 2000.0-Gy absorbed dose?

  • 47. Some types of cancer can be effectively treated by bombarding the cancer cells with high energy protons. Suppose 1.16 × 1017 protons, each with an energy of 950 keV, are incident on a tumor of mass 3.82 mg. If the quality factor for these protons is 3.0, what is the biologically equivalent dose?

  • 48. Make an order-of-magnitude estimate of the amount of radon-222 gas, measured in curies, found in the lungs of an average person. Assume that 0.1 rem/yr is due to the alpha particles emitted by radon-222. The half-life is 3.8 days. You will need to calculate the energy of the alpha particles emitted.

29.6 Induced Nuclear Reactions
  • 49. A certain nuclide absorbs a neutron. It then emits an electron, and then breaks up into two α particles. (a) Identify the original nuclide and the two intermediate nuclides (after absorbing the neutron and after emitting the electron). (b) Would any (anti)neutrino(s) be emitted? Explain.

  • 50. A neutron-activated sample emits gamma rays at energies that are consistent with the decay of mercury-198 nuclei from an excited state to the ground state. If the reaction that takes place is what is the nuclide “(?)” that was present in the sample before neutron activation?

    Tutorial: Neutron Detector

  • 51. Irène and Jean Frédéric Joliot-Curie, in an experiment that led to the 1935 Nobel Prize in chemistry, bombarded aluminum with α particles to form a highly unstable isotope of phosphorus, The phosphorus immediately decayed into another isotope of phosphorus, plus another product. Write out these reactions, identifying the other product.

  • 52. The reactions listed in Problem 51 did not stop there. To the surprise of the Curies, the phosphorus decay continued after the α bombardment ended with the phosphorus emitting a β+ to form yet another product. Write out this reaction, identifying the other product.

29.7 Fission
  • 53. A 235U nucleus captures a low-energy neutron to form the compound nucleus 236U*. Find the excitation energy of the compound nucleus. Ignore the small initial kinetic energy of the captured neutron.

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    54. Estimate the energy released in the fission reaction of Eq. (29-31). Look up the binding energy per nucleon of the nuclides in Fig. 29.2.

  • 55. Calculate the energy released in the fission reaction of Eq. (29-30). The atomic masses of and are 140.914 u and 91.926 u, respectively.

  • 56. One possible fission reaction for 235U is 235U + n → 141Cs + 93Rb + ?n, where “?n” represents one or more neutrons. (a) How many neutrons? (b) From the graph in Fig. 29.2, you can read the approximate binding energies per nucleon for the three nuclides involved. Use that information to estimate the total energy released by this fission reaction. (c) Do a precise calculation of the energy released. (d) What fraction of the rest energy of the 235U nucleus is released by this reaction?

29.8 Fusion
  • 57. Consider the fusion reaction of a proton and a deuteron: (a) Identify the reaction product X. (b) The binding energy of the deuteron is about 1.1 MeV per nucleon and the binding energy of “X” is about 2.6 MeV per nucleon. Approximately how much energy (in MeV) is released in this fusion reaction? (c) Why is this reaction unlikely to occur in a room temperature setting?

  • 58. What is the total energy released by the proton-proton cycle [Eq. (29-34)]? (The total energy released is the total energy of the neutrinos and γ rays plus the kinetic energy of the 4He nucleus minus the initial kinetic energies of the protons and electrons.)

  • 59. Estimate the minimum total kinetic energy of the 2H and 3H nuclei necessary to allow the fusion reaction of Eq. (29-32) to take place.

  • 60. Compare the amount of energy released when 1.0 kg of the uranium isotope 235U undergoes the fission reaction of Eq. (29-30) with the energy released when 1.0 kg of hydrogen undergoes the fusion reaction of Eq. (29-32).

Comprehensive Problems
Answers to Practice Problems
Answers to Checkpoints