# End-of-Chapter Exercises

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1 The equation that describes the exponential decay in the number of nuclei of a particular radioactive isotope as a function of time t is, (Equation 29.12: The exponential decay of radioactive nuclei) where Ni is a measure of the initial number of radioactive nuclei (the number at t = 0). The decay constant,!, is related to the half-life, T1/2, (the time for half of the nuclei to decay) by. (Eq : The connection between decay constant and half-life) Nuclear fusion and nuclear fission The most stable nuclei (those with the highest average binding energy per nuclei) are nickel-62, iron-58 and iron-56. Nuclei that are lighter than these most stable nuclei can generally become more stable (increasing the average binding energy per nuclei) by joining together with other light nuclei this process is known as nuclear fusion. Very heavy nuclei, in contrast, can generally become more stable by splitting apart, usually into two medium-sized nuclei and a few neutrons. This process is known as nuclear fission. The fission of uranium-235, driven by the bombardment of the uranium-235 atoms with neutrons, is exploited in a nuclear reactor to produce nuclear energy, while the fission of plutonium-239 is what drives the explosion of a nuclear bomb. End-of-Chapter Exercises Exercises 1 12 are mainly conceptual questions that are designed to see if you have understood the main concepts of the chapter. 1. The symbol for the isotope iron-56 is. A neutral iron-56 atom has how many (a) protons? (b) neutrons? (c) nucleons? 2. Which of these numbers is larger, the mass of an iron-56 atom, or the total mass of the individual constituents (neutrons, protons, and electrons) of an iron-56 atom? Briefly explain your answer. 3. If you could convert 1 kg of matter entirely to energy, how much energy would you get? 4. (a) Fill in the blank to complete this decay process: Ra! He. (b) What kind of decay is this? (c) Based on the fact that this decay process happens spontaneously, which side of the equation do you expect to have more mass? Why? 5. (a) What kind of radioactive decay process gives rise to a positron? (b) What is the electric charge of a positron? (c) Complete this sentence: The mass of a positron is the same as the mass of. 6. Fill in the blanks to complete the following decay processes: (a)! 15 7 N e + + " e. (b) Sc! + 0 "1 e " + # e. (c) Ni *! + ". Chapter 29 The Nucleus Page 29-19

3 15. In almost all nuclei, there are more than 2 nucleons, and thus there is more than one pair of nucleons interacting via the nuclear force. The situation is much simpler in a deuterium ( ) atom, in which the binding energy is associated with the attractive nuclear force between the single neutron and the single proton. For the deuterium atom, calculate (a) the mass defect, in atomic mass units, (b) the binding energy, in MeV, which is almost entirely associated with the nuclear force between the proton and neutron. 16. In general, for a nucleus to be stable, the attractive forces between nuclei must balance the repulsive forces associated with the charged protons. A helium-4 atom is particularly stable. To help understand why this is, consider how many pairs of (a) interacting protons, and (b) interacting nucleons there are in a helium-4 atom. (c) Briefly explain why your answers to (a) and (b) support the idea that the helium-4 atom is particularly stable. 17. Lead-208 is stable, which is relatively rare for a high-mass nucleus. For, calculate (a) the mass defect, in atomic mass units, (b) the total binding energy, in MeV, and (c) the average binding energy per nucleon. 18. Nickel-62 has the largest average binding energy per nucleon of any isotope. Calculate the average binding energy per nucleon for nickel-62. Exercises involve radioactive decay processes. Make use of the data in Table 29-3 in Section Plutonium-239 ( ) decays via alpha decay. (a) Write out the decay equation for plutonium-239. (b) Calculate the energy released in the decay of one Pu-239 atom. 20. Scandium-46 ( ) decays via the beta-minus process. (a) Write out the complete decay equation for scandium-46. (b) Calculate the energy released in the decay of one scandium-46 atom. 21. A certain isotope decays via the beta-plus process to neon-22 ( ). (a) Write out the complete decay equation for this situation. (b) The mass of a neon-22 atom is u. Calculate the energy released in this beta-plus decay process. 22. (a) According to Table 29.3, what does krypton-85 decay into? (b) Write out the decay equation for krypton-85. (c) Calculate the energy released in this decay process. 23. Silver-111,, which has an atomic mass of u, is radioactive, spontaneously decaying via either alpha decay, beta-plus decay, or beta-minus decay. The masses of the possible decay products are u for, u for, and u for. (a) Explain how you can use these numbers to determine the spontaneous decay process for silver-111. (b) Write out the complete decay reaction for the spontaneous decay of silver-111. Chapter 29 The Nucleus Page 29-21

4 Exercises involve radioactivity. 24. Oxygen-15 has a half-life of 2 minutes. Nuclear activity is often measured in units of becquerels (Bq), which are the number of nuclear decays per second. If the activity of a sample of oxygen containing some oxygen-15 is 64! 10 6 Bq at t = 0, what is the activity level of the sample at (a) t = 4 minutes, (b) t = 16 minutes, and (c) t = 20 minutes? 25. Fluorine-20 decays to neon-20 with a half-life of 11 seconds. At t = 0, a sample contains 120 grams of fluorine-20 atoms. How many grams of fluorine-20 atoms remain in the sample at (a) t = 5.0 seconds, (b) t = 30 seconds, and (c) t = 1.0 minutes? 26. A supply of fluorodeoxyglucose (FDG) arrives at a positron emission tomography clinic at 8 am. At 5 pm, at the end of the working day, the activity level of the FDG has dropped significantly because of the 110 minute half-life of the radioactive fluorine. Considering equal masses of FDG at 8 am and 5 pm, by what factor has the activity level been reduced by 5 pm? 27. Four hours after being injected with a radioactive isotope for a positron emission tomography scan, the activity level of the material injected into the patient has been reduced by a factor of 8 compared to its initial value (at the time of the injection). After another four hours elapses, will the activity level be reduced by another factor of 8? Briefly justify your answer. 28. After 3 hours, the activity level of a sample of a particular radioactive isotope, which decays into a stable isotope, has fallen to 20% of its initial value. Calculate the half-life of this isotope. Exercises involve nuclear fusion and nuclear fission. Make use of the data in Table 29-3 in Section (a) What is the second object produced in the following fusion reaction? 1 2 H Li! 3 7 Li +. (b) How much energy is produced in this reaction? 30. In Exercise 10, we examined one of the fusion reactions in the CNO (carbon-nitrogen-oxygen) cycle, which is a primary method of fusing hydrogen into helium in massive stars. The primary reactions in the CNO cycle are shown in Figure Two more of the fusion reactions in the CNO cycle are 1 1H C! 14 7 N and 1 1 H N! 15 8 O. Calculate the energy released in each of these fusion reactions. Figure 29.7: The CNO cycle of fusion reactions that produces much of the energy in massive stars. The CNO cycle has the net effect of fusing hydrogen into helium. Chapter 29 The Nucleus Page 29-22

5 31. In relatively low-mass stars, such as our Sun, much of the energy generated by the star comes from the proton-proton chain, which has the net effect of fusing hydrogen into helium. The step that produces the helium-4 atom is: 2 3 He He! 2 4 He H H. The helium-3 atoms are produced from the fusion of hydrogen isotopes in earlier steps in the process. How much energy is released in the step that produced helium-4? Note that the mass of is u. 32. Complete the following fission reaction: 1 0 n U! U! Ba + + 3( 1 0 n). 33. How much energy is released in the following fission reaction? 1 0n U! U! Ba Kr + 3( 1 0 n). Exercises involve applications of nuclear physics. 34. A common application of nuclear radiation is food irradiation, which is the exposure of food products (such as meat, poultry, and fruit) to ionizing radiation to kill bacteria, prolong shelf life, and delay the ripening of fruit. One example of this treatment is the exposure of a meat product to a dose of 3 kilograys of gamma radiation from a cobalt-60 source. One gray (Gy) represents 1 joule of energy exposure per kilogram of food. Cobalt-60 decays by the beta-minus process to an excited form of nickel-60, which then decays to its ground state through the emission of two gamma rays, with energies of 1.17 MeV and 1.33 MeV. (a) How many nuclei are there in 1 gram of cobalt-60, which has an atomic mass of u? (b) Cobalt-60 has a half-life of 5.27 years. How many nuclei, in a 1-gram sample of cobalt-60, would decay in 1 second? (c) If the two gamma rays associated with each decaying cobalt-60 nuclei deposit all their energy in a 1 kilogram package of meat, how long would the meat have to be exposed to receive a dose of 3 kgy? 35. When radiocarbon dating was carried out on the Shroud of Turin in 1988, the three labs doing the measurements agreed that the shroud dated to approximately the year 1300 AD, providing evidence against the idea that the shroud (a picture of which is shown in Figure 29.8) was the burial cloth of Jesus Christ. If the ratio of carbon-14 to carbon-12 in the atmosphere has remained constant over time at 1.2! 10-12, (a) approximately what ratio did the researchers find for the sample of shroud they measured?, and (b) what ratio would they have found if the cloth was 2000 years old? Figure 29.8: A photograph of part of the Shroud of Turin, on the left, and the corresponding negative image, on the right, for Exercise 35. Image credit: Wikimedia Commons. Chapter 29 The Nucleus Page 29-23

8 50. You have two samples of radioactive nuclei. At t = 0, sample A contains N nuclei of a particular radioactive isotope, while sample B contains twice as many nuclei of a different radioactive isotope. Also at t = 0, the decay rate, measured in nuclei per second, of sample A is three times larger than the decay rate in sample B. How do the half-lives of the two different types of nuclei compare? 51. The graph in Figure 29.9 shows the ratio of the number of undecayed nuclei to the initial number of undecayed nuclei, as a function of time. Use the data in the graph to estimate the half-life of the nuclei. 52. Instead of defining the half-life for a particular isotope, we could define the onetenth-life (or the life for any other fraction less than 1). After a time of a single onetenth-life has passed, only 1/10 th of the initial number of radioactive nuclei remain. (a) Determine the ratio of a one-tenth-life to a half-life. (b) The half-life of cesium-137 is 30 years. What is the one-tenth-life for cesium-137? (c) After how many one-tenthlives does at activity level of a sample of radioactive material drop to 1% of its initial value? (d) How much time elapses until the activity level of a sample of cesium-137 drops to 1% of its initial value? Figure 29.9: A graph of the number of undecayed nuclei to the initial number of undecayed nuclei, as a function of time, for Exercise After 6 days, the activity level of a sample of radioactive material has fallen to 40% of its initial value. After an additional 6 days elapses, what is the activity level of the sample, compared to its initial value? (Hint: there is a very easy way to do this calculation it is possible to work out the answer without a calculator.) 54. Right now, a sample contains 1.0! undecayed nuclei, and a much larger number of decayed nuclei. If the half-life of the undecayed nuclei is 5 months, how many undecayed nuclei did the sample contain a year ago? 55. At t = 0 s, a sample of radioactive nuclei contains 1.0! undecayed nuclei, which have a half-life of 10 seconds. First, use equation 29.10, with N = 1.0! nuclei, to estimate the number of undecayed nuclei remaining at (a) t = 1 s, (b) t = 10 s, and (c) t = 20 s. Second, use equation to estimate the number of undecayed nuclei remaining at (d) t = 1 s, (e) t = 10 s, and (f) t = 20 s. (g) Explain which method is the correct method to use to accurately determine the number of undecayed nuclei remaining, and why the other method gives inaccurate results. Chapter 29 The Nucleus Page 29-26

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