21.E: Exercises

21.1: Radioactivity

Conceptual Problems

1. Why are many radioactive substances warm to the touch? Why do many radioactive substances glow?
2. Describe the differences between nonionizing and ionizing radiation in terms of the intensity of energy emitted and the effect each has on an atom or molecule after collision. Which nuclear decay reactions are more likely to produce ionizing radiation? nonionizing radiation?
3. Would you expect nonionizing or ionizing radiation to be more effective at treating cancer? Why?
4. Historically, concrete shelters have been used to protect people from nuclear blasts. Comment on the effectiveness of such shelters.
5. Gamma rays are a very high-energy radiation, yet α particles inflict more damage on biological tissue. Why?
6. List the three primary sources of naturally occurring radiation. Explain the factors that influence the dose that one receives throughout the year. Which is the largest contributor to overall exposure? Which is the most hazardous?
7. Because radon is a noble gas, it is inert and generally unreactive. Despite this, exposure to even low concentrations of radon in air is quite dangerous. Describe the physical consequences of exposure to radon gas. Why are people who smoke more susceptible to these effects?
8. Most medical imaging uses isotopes that have extremely short half-lives. These isotopes usually undergo only one kind of nuclear decay reaction. Which kind of decay reaction is usually used? Why? Why would a short half-life be preferred in these cases?
9. Which would you prefer: one exposure of 100 rem, or 10 exposures of 10 rem each? Explain your rationale.

Conceptual Answers

3. Ionizing radiation is higher in energy and causes greater tissue damage, so it is more likely to destroy cancerous cells.

9. Ten exposures of 10 rem are less likely to cause major damage.

Numerical Problems

1. A 2.14 kg sample of rock contains 0.0985 g of uranium. How much energy is emitted over 25 yr if 99.27% of the uranium is ²³⁸U, which has a half-life of 4.46 × 10⁹ yr, if each decay event is accompanied by the release of 4.039 MeV? If a 180 lb individual absorbs all of the emitted radiation, how much radiation has been absorbed in rads?
2. There is a story about a “radioactive boy scout” who attempted to convert thorium-232, which he isolated from about 1000 gas lantern mantles, to uranium-233 by bombarding the thorium with neutrons. The neutrons were generated via bombarding an aluminum target with α particles from the decay of americium-241, which was isolated from 100 smoke detectors. Write balanced nuclear reactions for these processes. The “radioactive boy scout” spent approximately 2 h/day with his experiment for 2 yr. Assuming that the alpha emission of americium has an energy of 5.24 MeV/particle and that the americium-241 was undergoing 3.5 × 10⁶ decays/s, what was the exposure of the 60.0 kg scout in rads? The intrepid scientist apparently showed no ill effects from this exposure. Why?

21.2: Patterns of Nuclear Stability

Conceptual Problems

1. How do chemical reactions compare with nuclear reactions with respect to mass changes? Does either type of reaction violate the law of conservation of mass? Explain your answers.
2. Why is the amount of energy released by a nuclear reaction so much greater than the amount of energy released by a chemical reaction?
3. Explain why the mass of an atom is less than the sum of the masses of its component particles.
4. The stability of a nucleus can be described using two values. What are they, and how do they differ from each other?
5. In the days before true chemistry, ancient scholars (alchemists) attempted to find the philosopher’s stone, a material that would enable them to turn lead into gold. Is the conversion of Pb → Au energetically favorable? Explain why or why not.
6. Describe the energy barrier to nuclear fusion reactions and explain how it can be overcome.
7. Imagine that the universe is dying, the stars have burned out, and all the elements have undergone fusion or radioactive decay. What would be the most abundant element in this future universe? Why?
8. Numerous elements can undergo fission, but only a few can be used as fuels in a reactor. What aspect of nuclear fission allows a nuclear chain reaction to occur?
9. How are transmutation reactions and fusion reactions related? Describe the main impediment to fusion reactions and suggest one or two ways to surmount this difficulty.

Numerical Problems

1. Using the information provided in Chapter 33, complete each reaction and calculate the amount of energy released from each in kilojoules.
   1. ²³⁸Pa → ? + β⁻
   2. ²¹⁶Fr → ? + α
   3. ¹⁹⁹Bi → ? + β⁺

2. Using the information provided in Chapter 33, complete each reaction and calculate the amount of energy released from each in kilojoules.

1. ¹⁹⁴Tl → ? + β⁺
2. ¹⁷¹Pt → ? + α
3. ²¹⁴Pb → ? + β⁻

3. Using the information provided in Chapter 33, complete each reaction and calculate the amount of energy released from each in kilojoules per mole.

1. \(_{91}^{234}\textrm{Pa}\rightarrow \,?+\,_{-1}^0\beta\)
2. \(_{88}^{226}\textrm{Ra}\rightarrow \,?+\,_2^4\alpha\)

4. Using the information provided in Chapter 33, complete each reaction and then calculate the amount of energy released from each in kilojoules per mole.

1. \(_{27}^{60}\textrm{Co}\rightarrow\,?+\,_{-1}^0\beta\) (The mass of cobalt-60 is 59.933817 amu.)
2. technicium-94 (mass = 93.909657 amu) undergoing fission to produce chromium-52 and potassium-40

5. Using the information provided in Chapter 33, predict whether each reaction is favorable and the amount of energy released or required in megaelectronvolts and kilojoules per mole.

1. the beta decay of bismuth-208 (mass = 207.979742 amu)
2. the formation of lead-206 by alpha decay

6. Using the information provided, predict whether each reaction is favorable and the amount of energy released or required in megaelectronvolts and kilojoules per mole.

1. alpha decay of oxygen-16
2. alpha decay to produce chromium-52

7. Calculate the total nuclear binding energy (in megaelectronvolts) and the binding energy per nucleon for ⁸⁷Sr if the measured mass of ⁸⁷Sr is 86.908877 amu.
8. Calculate the total nuclear binding energy (in megaelectronvolts) and the binding energy per nucleon for ⁶⁰Ni.
9. The experimentally determined mass of ⁵³Mn is 52.941290 amu. Find each of the following.

1. the calculated mass
2. the mass defect
3. the nuclear binding energy
4. the nuclear binding energy per nucleon

10. The experimentally determined mass of ²⁹S is 28.996610 amu. Find each of the following.

1. the calculated mass
2. the mass defect
3. the nuclear binding energy
4. the nuclear binding energy per nucleon

11. Calculate the amount of energy that is released by the neutron-induced fission of ²³⁵U to give ¹⁴¹Ba, ⁹²Kr (mass = 91.926156 amu), and three neutrons. Report your answer in electronvolts per atom and kilojoules per mole.
12. Calculate the amount of energy that is released by the neutron-induced fission of ²³⁵U to give ⁹⁰Sr, ¹⁴³Xe, and three neutrons. Report your answer in electronvolts per atom and kilojoules per mole.
13. Calculate the amount of energy released or required by the fusion of helium-4 to produce the unstable beryllium-8 (mass = 8.00530510 amu). Report your answer in kilojoules per mole. Do you expect this to be a spontaneous reaction?
14. Calculate the amount of energy released by the fusion of ⁶Li and deuterium to give two helium-4 nuclei. Express your answer in electronvolts per atom and kilojoules per mole.
15. How much energy is released by the fusion of two deuterium nuclei to give one tritium nucleus and one proton? How does this amount compare with the energy released by the fusion of a deuterium nucleus and a tritium nucleus, which is accompanied by ejection of a neutron? Express your answer in megaelectronvolts and kilojoules per mole. Pound for pound, which is a better choice for a fusion reactor fuel mixture?

Numerical Answers

1

1. \(_{91}^{238}\textrm{Pa}\rightarrow\,_{92}^{238}\textrm{U}+\,_{-1}^0\beta\); −5.540 × 10⁻¹⁶ kJ
2. \(_{87}^{216}\textrm{Fr}\rightarrow\,_{85}^{212}\textrm{At}+\,_{2}^4\alpha\); −1.470 × 10⁻¹⁵ kJ
3. \(_{83}^{199}\textrm{Bi}\rightarrow\,_{82}^{199}\textrm{Pb}+\,_{+1}^0\beta\); −5.458 × 10⁻¹⁶ kJ

1. \(_{91}^{234}\textrm{Pa}\rightarrow\,_{92}^{234}\textrm{U}+\,_{-1}^0\beta\); 2.118 × 10⁸ kJ/mol
2. \(_{88}^{226}\textrm{Ra}\rightarrow\,_{86}^{222}\textrm{Rn}+\,_{2}^4\alpha\); 4.700 × 10⁸ kJ/mol

1. The beta decay of bismuth-208 to polonium is endothermic (ΔE = 1.400 MeV/atom, 1.352 × 10⁸ kJ/mol).
2. The formation of lead-206 by alpha decay of polonium-210 is exothermic (ΔE = −5.405 MeV/atom, −5.218 × 10⁸ kJ/mol).

7. 757 MeV/atom, 8.70 MeV/nucleon

1. 53.438245 amu
2. 0.496955 amu
3. 463 MeV/atom
4. 8.74 MeV/nucleon

11. −173 MeV/atom; 1.67 × 10¹⁰ kJ/mol

13. ΔE = + 9.0 × 10⁶ kJ/mol beryllium-8; no

15. D–D fusion: ΔE = −4.03 MeV/tritium nucleus formed = −3.89 × 10⁸ kJ/mol tritium; D–T fusion: ΔE = −17.6 MeV/tritium nucleus = −1.70 × 10⁹ kJ/mol; D–T fusion

21.3: Nuclear Transmutations

Conceptual Problems

1. Why do scientists believe that hydrogen is the building block of all other elements? Why do scientists believe that helium-4 is the building block of the heavier elements?
2. How does a star produce such enormous amounts of heat and light? How are elements heavier than Ni formed?
3. Propose an explanation for the observation that elements with even atomic numbers are more abundant than elements with odd atomic numbers.
4. During the lifetime of a star, different reactions that form different elements are used to power the fusion furnace that keeps a star “lit.” Explain the different reactions that dominate in the different stages of a star’s life cycle and their effect on the temperature of the star.
5. A line in a popular song from the 1960s by Joni Mitchell stated, “We are stardust….” Does this statement have any merit or is it just poetic? Justify your answer.
6. If the laws of physics were different and the primary element in the universe were boron-11 (Z = 5), what would be the next four most abundant elements? Propose nuclear reactions for their formation.

Conceptual Answers

3. The raw material for all elements with Z > 2 is helium (Z = 2), and fusion of helium nuclei will always produce nuclei with an even number of protons.

Numerical Problems

1. Write a balanced nuclear reaction for the formation of each isotope.

1. ²⁷Al from two ¹²C nuclei
2. ⁹Be from two ⁴He nuclei

2. At the end of a star’s life cycle, it can collapse, resulting in a supernova explosion that leads to the formation of heavy elements by multiple neutron-capture events. Write a balanced nuclear reaction for the formation of each isotope during such an explosion.

1. ¹⁰⁶Pd from nickel-58
2. selenium-79 from iron-56

3. When a star reaches middle age, helium-4 is converted to short-lived beryllium-8 (mass = 8.00530510 amu), which reacts with another helium-4 to produce carbon-12. How much energy is released in each reaction (in megaelectronvolts)? How many atoms of helium must be “burned” in this process to produce the same amount of energy obtained from the fusion of 1 mol of hydrogen atoms to give deuterium?

21.4: Rates of Radioactive Decay

Conceptual Problems

1. What do chemists mean by the half-life of a reaction?
2. If a sample of one isotope undergoes more disintegrations per second than the same number of atoms of another isotope, how do their half-lives compare?

Conceptual Answers

Numerical Problems

1. Half-lives for the reaction A + B → C were calculated at three values of [A]₀, and [B] was the same in all cases. The data are listed in the following table:

[A]0 (M)	t½ (s)
0.50	420
0.75	280
1.0	210

Does this reaction follow first-order kinetics? On what do you base your answer?

2. Ethyl-2-nitrobenzoate (NO₂C₆H₄CO₂C₂H₅) hydrolyzes under basic conditions. A plot of [NO₂C₆H₄CO₂C₂H₅] versus t was used to calculate t½, with the following results:

[NO2C6H4CO2C2H5] (M/cm^3)	t½ (s)
0.050	240
0.040	300
0.030	400

Is this a first-order reaction? Explain your reasoning.

3. Azomethane (CH₃N₂CH₃) decomposes at 600 K to C₂H₆ and N₂. The decomposition is first order in azomethane. Calculate t½ from the data in the following table:

Time (s)	\(P_{\large{\mathrm{CH_3N_2CH_3}}}\) (atm)
0	8.2 × 10^−2
2000	3.99 × 10^−2
4000	1.94 × 10^−2

How long will it take for the decomposition to be 99.9% complete?

4. The first-order decomposition of hydrogen peroxide has a half-life of 10.7 h at 20°C. What is the rate constant (expressed in s⁻¹) for this reaction? If you started with a solution that was 7.5 × 10⁻³ M H₂O₂, what would be the initial rate of decomposition (M/s)? What would be the concentration of H₂O₂ after 3.3 h?

Numerical Answer

1. No; the reaction is second order in A because the half-life decreases with increasing reactant concentration according to t_1/2 = 1/k[A₀].

3. t_1/2 = 1.92 × 10³ s or 1920 s; 19100 s or 5.32 hrs.

21.6: Energy Changes in Nuclear Reactions

Conceptual Problems

Conceptual Answers

Numerical Problems

21.7: Nuclear Fission

Conceptual Problems

Conceptual Answers

Numerical Problems

21.8: Nuclear Fusion

Conceptual Problems

Conceptual Answers

Numerical Problems

21.9: Biological Effects of Radiation

Conceptual Problems

Conceptual Answers

Numerical Problems