Chapter 21

The steps below show three of the steps in the radioactive decay chain for \(_{90}^{232}\) Th. The half-life of each isotope is shown below the symbol of the isotope. (a) Identify the type of radioactive decay for each of the steps (i), (ii), and (iii). (b) Which of the isotopes shown has the highest activity? (c) Which of the isotopes shown has the lowest activity? (d) The next step in the decay chain is an alpha emission. What is the next isotope in the chain? [Sections 21.2 and 21.4]

The accompanying graph illustrates the decay of \(_{42}^{88} \mathrm{Mo}\) which decays via positron emission. (a) What is the half-life of the decay? (b) What is the rate constant for the decay?(c) What fraction of the original sample of \(_{42}^{88} \mathrm{Mo}\) remains after 12 \(\mathrm{min}\) ? (d) What is the product of the decay process? [Section 21.4\(]\)

The diagram shown here illustrates a fission process. \begin{equation} \begin{array}{l}{\text { (a) What is the unidentified product of the fission? }} \\ {\text { (b) Use Figure } 21.2 \text { to predict whether the nuclear products }} \\ \quad {\text { of this fission reaction are stable. [ Section } 21.7 ]}\end{array} \end{equation}

Indicate the number of protons and neutrons in the following nuclei: \((\mathbf{a}) _{24}^{56} \mathrm{Cr},(\mathbf{b})^{193} \mathrm{Tl},(\mathbf{c})\) argon-\(38.\)

Indicate the number of protons and neutrons in the following nuclei: \((\mathbf{a}) _{53}^{129} I,(\mathbf{b})^{138} \mathrm{Ba},(\mathbf{c})\) neptunium\(-237 .\)

Give the symbol for \((\mathbf{a})\) a neutron, \((\mathbf{b})\) an alpha particle, \((\mathbf{c})\) gamma radiation.

Give the symbol for \((\mathbf{a})\) a proton, \((\mathbf{b})\) a beta particle, \((\mathbf{c})\) a positron.

Write balanced nuclear equations for the following processes: \((\mathbf{a})\) rubidium-90 undergoes beta emission; \((\mathbf{b})\) selenium- 72 undergoes electron capture; \((\mathbf{c})\) krypton-76 undergoes positron emission; \((\mathbf{d})\) radium-226 emits alpha radiation.

Write balanced nuclear equations for the following transformations: \((\mathbf{a})\) bismuth-213 undergoes alpha decay; \((\mathbf{b})\) nitrogen-13 undergoes electron capture; \((\mathbf{c})\) technicium-98 undergoes electron capture; \((\mathbf{d})\) gold-188 decays by positron emission.

Decay of which nucleus will lead to the following products: \((\mathbf{a})\) bismuth-211 by beta decay; \((\mathbf{b})\) chromium-so by positron emission; \((\mathbf{c})\) tantalum-179 by electron capture; \((\mathbf{d})\) radium-226 by alpha decay?

What particle is produced during the following decay processes: \((\mathbf{a})\) sodium-24 decays to magnesium-24; \((\mathbf{b})\) mercury-188 decays to gold-188; \((\mathbf{c})\)iodine-122 decays to xenon-122; \((\mathbf{d})\) plutonium-242 decays to uranium-238?

The naturally occurring radioactive decay series that begins with \(_{92}^{235} \mathrm{U}\) stops with formation of the stable \(_{82}^{207} \mathrm{Pb}\) nucleus. The decays proceed through a series of alpha-particle and beta- particle emissions. How many of each type of emission are involved in this series?

A radioactive decay series that begins with \(_{90}^{232}\) \(\mathrm{Th}\) ends with formation of the stable nuclide \(_{82}^{208}\) \(\mathrm{Pb}\) . How many alpha-particle emissions and how many beta-particle emissions are involved in the sequence of radioactive decays?

Predict the type of radioactive decay process for the following radionuclides: \((\mathbf{a})_{5}^{8} \mathrm{B}\) \((\mathbf{b})_{29}^{68} \mathrm{Cu},(\mathbf{c})\) phosphorus-32 \((\mathbf{d}),\)chlorine-39.

Each of the following nuclei undergoes either beta decay or positron emission. Predict the type of emission for each: \((\mathbf{a})\) tritium, \(_{1}^{3} \mathrm{H},(\mathbf{b})_{38}^{89} \mathrm{Sr},(\mathbf{c})\) iodine-120, \((\mathbf{d})\) (d) silver-102.

One of the nuclides in each of the following pairs is radioactive. Predict which is radioactive and which is stable: \((\mathbf{a})_{19}^{39} \mathrm{K}\) and \(_{19}^{40} \mathrm{K},\) \((\mathbf{b})^{209} \mathrm{Bi}\) and \(^{208} \mathrm{Bi}\) \((\mathbf{c})\) nickel-58 and nickel-65.

One nuclide in each of these pairs is radioactive. Predict which is radioactive and which is stable: \((\mathbf{a})_{20}^{40} \mathrm{Ca}\) and \(_{20}^{45} \mathrm{Ca}\) , \((\mathbf{b})^{12} \mathrm{C}\) and \(^{14} \mathrm{C},\) \((\mathbf{c})\) lead-206 and thorium-230. Explain your choice in each case.

Which of the following statements best explains why alpha emission is relatively common, but proton emission is extremely rare? \begin{equation}\begin{array}{l}{\text { (a) Alpha particles are very stable because of magic numbers }} \\ \quad {\text { of protons and neutrons. }} \\\ {\text { (b) Alpha particles occur in the nucleus. }} \\ {\text { (c) Alpha particles are the nuclei of an inert gas. }} \\ {\text { (d) An alpha particle has a higher charge than a proton. }}\end{array}\end{equation}

Which of the following nuclides would you expect to be radioactive: \(_{26}^{58} \mathrm{Fe}, _{27}^{60} \mathrm{Co},\) \(_{41}^{92} \mathrm{Nb},\) mercury-202, radium-226? Justify your choices.

Which statement best explains why nuclear transmutations involving neutrons are generally easier to accomplish than those involving protons or alpha particles? \begin{equation} \begin{array}{l}{\text { (a) Neutrons are not a magic number particle. }} \\\ {\text { (b) Neutrons do not have an electrical charge. }} \\ {\text { (c) Neutrons are smaller than protons or alpha particles. }} \\ {\text { (d) Neutrons are attracted to the nucleus even at long distances,}} \\ \quad {\text { whereas protons and alpha particles are repelled. }}\end{array} \end{equation}

In 1930 the American physicist Ernest Lawrence designed the first cyclotron in Berkeley, California. In 1937 Lawrence bombarded a molybdenum target with deuterium ions, producing for the first time an element not found in nature. What was this element? Starting with molybdenum-96 as your reactant, write a nuclear equation to represent this process.

Complete and balance the following nuclear equations by supplying the missing particle: \begin{equation}\begin{array}{l}{\text { (a) }_{98}^{252} \mathrm{Cf}+_{5}^{10} \mathrm{B} \longrightarrow 3_{0}^{1} \mathrm{n}+?} \\\ {\text { (b) }_{1}^{2} \mathrm{H}+_{2}^{3} \mathrm{He} \longrightarrow_{2}^{4} \mathrm{He}+?}\\\ {\text { (c) }_{1}^{1} \mathrm{H}+_{5}^{11} \mathrm{B} \longrightarrow 3?} \\ {\text { (d) }_{53}^{122} \mathrm{I}\longrightarrow_{54}^{122} \mathrm{Xe}+?}\\\ {\text { (e) }_{26}^{59} \mathrm{Fe}\longrightarrow_{-1}^{0} \mathrm{e}+?}\end{array}\end{equation}

Complete and balance the following nuclear equations by supplying the missing particle: \begin{equation}\begin{array}{l}{\text { (a) }_{17}^{14} \mathrm{N}+_{2}^{4} \mathrm{He} \longrightarrow ? +_{1}^{1} \mathrm{H}} \\ {\text { (b) }_{19}^{40} \mathrm{K}+_{1}^{0} \mathrm{e} \ \mathrm{(orbital \ electron) \longrightarrow ?}}\\\ {\text { (c) }_{}{\underline{\phantom{xx}}} \mathrm{?}+_{2}^{4} \mathrm{He} \longrightarrow_{14}^{30} \mathrm{Si} +_{1}^{1} \mathrm{H}}\\\ {\text { (d) }_{26}^{58} \mathrm{Fe} +2 _{0}^{1} \mathrm{n} \longrightarrow_{27}^{60} \mathrm{Co}+?}\\\ {\text { (e) }_{92}^{235} \mathrm{U}\longrightarrow+_{0}^{1} n \longrightarrow_{54}^{135} \mathrm{Xe}+2_{0}^{1} \mathrm{n}+?} \end{array}\end{equation}

Write balanced equations for each of the following nuclear reactions: \((\mathbf{a}) _{92}^{238} \mathrm{U}(\mathrm{n}, \gamma)_{92}^{239} \mathrm{U},\) \((\mathbf{b})_{8}^{16} \mathrm{O}(\mathrm{p}, \alpha)_{7}^{13} \mathrm{N},\) \((\mathbf{c})_{8}^{18} \mathrm{O}\left(\mathrm{n}, \beta^{-}\right)_{9}^{19} \mathrm{F}.\)

It has been suggested that strontium-90 (generated by nuclear testing deposited in the hot desert will undergo radioactive decay more rapidly because it will be exposed to much higher average temperatures. (a) Is this a reasonable suggestion? (b) Does the process of radioactive decay have an activation energy, like the Arrhenius behavior of many chemical reactions (Section 14.5\()?\)

Some watch dials are coated with a phosphor, like ZnS, and a polymer in which some of the \(^{1} \mathrm{H}\) atoms have been replaced by \(^{3} \mathrm{H}\) atoms, tritium. The phosphor emits light when struck by the beta particle from the tritium decay, causing the dials to glow in the dark. The half-life of tritium is 12.3 yr. If the light given off is assumed to be directly proportional to the amount of tritium, by how much will a dial be dimmed in a watch that is 50 yr old?

It takes 4 \(\mathrm{h} 39\) min for a 2.00 - mg sample of radium-230 to decay to 0.25 \(\mathrm{mg.}\) What is the half-life of radium-230?

Cobalt-so is a strong gamma emitter that has a half-life of 5.26 yr. The cobalt-60 in a radiotherapy unit must be replaced when its radioactivity falls to 75\(\%\) of the original sample. If an original sample was purchased in June \(2016,\) when will it be necessary to replace the cobalt-60?

Radium-226, which undergoes alpha decay, has a half-life of 1600 yr. (a) How many alpha particles are emitted in 5.0 min by a 10.0 -mg sample of \(^{226} \mathrm{Ra}\) ? (b) What is the activity of the sample in mCi?

Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. (a) How many beta particles are emitted in 600 s by a 3.75 -mg sample of \(^{60} \mathrm{Co} ?(\mathbf{b})\) What is the activity of the sample in \(\mathrm{Bq}\) ?

The cloth shroud from around a mummy is found to have \(\mathrm{a}^{14} \mathrm{C}\) activity of 9.7 disintegrations per minute per gram of carbon as compared with living organisms that undergo 16.3 disintegrations per minute per gram of carbon. From the half-life for \(^{14} \mathrm{C}\) decay, 5715 yr, calculate the age of the shroud.

A wooden artifact from a Chinese temple has a \(^{14} \mathrm{C}\) cactivity of 38.0 counts per minute as compared with an activity of 58.2 counts per minute for a standard of zero age. From the half-life for \(^{14} \mathrm{C}\) decay, 5715 yr, determine the age of the artifact.

Potassium-40 decays to argon-40 with a half-life of \(1.27 \times 10^{9}\) yr. What is the age of a rock in which the mass ratio of \(^{40} \mathrm{Ar}\) to \(^{40} \mathrm{K}\) is 4.2?

The half-life for the process \(^{238} \mathrm{U} \longrightarrow^{206} \mathrm{Pb}\) is \(4.5 \times 10^{9} \mathrm{yr}.\) A mineral sample contains 75.0 \(\mathrm{mg}\) of \(^{238} \mathrm{U}\) and 18.0 \(\mathrm{mg}\) of \(^{206} \mathrm{pb} .\) What is the age of the minineral?

The thermite reaction, \(\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+2 \mathrm{Al}(s) \longrightarrow 2 \mathrm{Fe}(s)+\) \(\mathrm{Al}_{2} \mathrm{O}_{3}(s), \Delta H^{\circ}=-851.5 \mathrm{kJ} / \mathrm{mol},\) is one of the most exothermic reactions known. Because the heat released is sufficient to melt the iron product, the reaction is used to weld metal under the ocean. How much heat is released per mole of \(\mathrm{Al}_{2} \mathrm{O}_{3}\) produced? How does this amount of thermal energy compare with the energy released when 2 mol of protons and 2 mol of neutrons combine to form 1 mol of alpha particles?

How much energy must be supplied to break a single \(^{21} \mathrm{Ne}\) nucleus into separated protons and neutrons if the nucleus has a mass of 20.98846 amu? What is the nuclear binding energy for 1 mol of \(^{21} \mathrm{Ne} ?\)

The atomic masses of hydrogen-2 (deuterium), helium-4, and lithium-6 are 2.014102 amu, 4.002602 amu, and 6.0151228 amu, respectively. For each isotope, calculate (a) the nuclear mass, (b) the nuclear binding energy, (c) the nuclear binding energy per nucleon. (d) Which of these three isotopes has the largest nuclear binding energy per nucleon? Does this agree with the trends plotted in Figure 21.12\(?\)

The atomic masses of nitrogen-14, titanium-48, and xenon-129 are 13.999234 amu, 47.935878 amu, and 128.90479 amu, respectively. For each isotope, calculate (a) the nuclear mass, (b) the nuclear binding energy, (c) the nuclear binding energy per nucleon.

The energy from solar radiation falling on Earth is \(1.07 \times 10^{16} \mathrm{kJ} / \mathrm{min.}\) (a) How much loss of mass from the Sun occurs in one day from just the energy falling on Earth? (\mathbf{b} )If the energy released in the reaction \begin{equation}^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \longrightarrow_{56}^{141} \mathrm{Ba}+_{36}^{92} \mathrm{Kr}+3_{0}^{1} \mathrm{n} \end{equation}\(\left(^{235} \mathrm{U}\right.\)nuclear mass,234.9935 amu; \(^{235} \mathrm{Ba}.\) nuclear mass, 140.8833 amu; \(^{92} \mathrm{Kr}\) nuclear mass, 91.9021 amu) is taken as typical of that occurring in a nuclear reactor, what mass of uranium-235 is required to equal 0.10\(\%\) of the solar energy that falls on Earth in 1.0 day?

Based on the following atomic mass values \(-^{1} \mathrm{H}, 1.00782\) \(\mathrm{amu} ;^{2} \mathrm{H}, 2.01410 \mathrm{amu}\); \(^{3} \mathrm{H}, 3.01605 \mathrm{amu} ;^{3} \mathrm{He}, 3.01603\) \(\mathrm{amu} ;^{4} \mathrm{He}, 4.00260 \mathrm{amu}-\) amu—and the mass of the neutron given in the text, calculate the energy released per mole in each of the following nuclear reactions, all of which are possibilities for a controlled fusion process: \begin{equation}(\mathbf{a})\quad_{1}^{2} \mathrm{H}+_{1}^{3} \mathrm{H} \longrightarrow _{4}^{2} \mathrm{He}+_{1}^{0} \mathrm{n}\end{equation} \begin{equation}(\mathbf{b})\quad_{1}^{2} \mathrm{H}+_{1}^{2} \mathrm{H} \longrightarrow_{2}^{3} \mathrm{He}+_{0}^{1} \mathrm{n}\end{equation} \begin{equation}(\mathbf{c})\quad_{1}^{2} \mathrm{H}+_{2}^{3} \mathrm{He} \longrightarrow_{2}^{4} \mathrm{He}+_{1}^{1} \mathrm{H}\end{equation}

Iodine-131 is a convenient radioisotope to monitor thyroid activity in humans. It is a beta emitter with a half-life of 8.02 days. The thyroid is the only gland in the body that uses iodine. A person undergoing a test of thyroid activity drinks a solution of Nal, in which only a small fraction of the iodide is radioactive. (a) Why is Nal a good choice for the source of iodine? (b) If a Geiger counter is placed near the person's thyroid (which is near the neck) right after the sodium iodide solution is taken, what will the data look like as a function of time? (c) A normal thyroid will take up about 12\(\%\) of the ingested iodide in a few hours. How long will it take for the radioactive iodide taken up and held by the thyroid to decay to 0.01\(\%\) of the original amount?

Why is it important that radioisotopes used as diagnostic tools in nuclear medicine produce gamma radiation when they decay? Why are alpha emitters not used as diagnostic tools?

(a) Which of the following are required characteristics of an isotope to be used as a fuel in a nuclear power reactor? (i) It must emit gamma radiation. (ii) On decay, it must release two or more neutrons. (iii) It must have a half-life less than one hour. (iv) It must undergo fission upon the absorption of a neutron. (b) What is the most common fissionable isotope in a commercial nuclear power reactor?

Which of the following statements about the uranium used in nuclear reactors is or are true? (i) Natural uranium has too little \(^{235} U\) to be used as a fuel. (ii) \(^{238} U\) cannot be used as a fuel because it forms a supercritical mass too easily. (iii) To be used as fuel, uranium must be enriched so that it is more than 50\(\%^{235} \mathrm{U}\) in composition. (iv) The neutron-induced fission of \(^{235} \mathrm{U}\) releases more neutrons per nucleus than fission of \(^{238} \mathrm{U}\).

What is the function of the control rods in a nuclear reactor? What substances are used to construct control rods?Why are these substances chosen?

(a) What is the function of the moderator in a nuclear reactor? (b) What substance acts as the moderator in a pressurized water generator? (c) What other substances are used as a moderator in nuclear reactor designs?

Complete and balance the nuclear equations for the following fission reactions: \begin{equation}(a) ^{2235} \mathrm{U}+_{0}^{1} \mathrm{n} \longrightarrow _{62}^{160} \mathrm{Sm}+ _{30}^{72} \mathrm{Zn}+_{0}^{1} \mathrm{n}\end{equation}\begin{equation} (b)^{239} \mathrm{Pu}+_{0}^{1} \mathrm{n} \longrightarrow _{58}^{144} \mathrm{Ce}+ 2 _{0}^{1} \mathrm{n} \end{equation}

A portion of the Sun's energy comes from the reaction \begin{equation}4_{1}^{1} \mathrm{H} \longrightarrow_{2}^{4} \mathrm{He}+2_{1}^{0} \mathrm{e} \end{equation}which requires a temperature of \(10^{6}\) to \(10^{7} \mathrm{K}\) . Use the mass of the helium-4 nucleus given in Table 21.7 to determine how much energy is released per mol of hydrogen atoms.

The spent fuel elements from a fission reactor are much more intensely radioactive than the original fuel elements. (a) What does this tell you about the products of the fission process in relationship to the belt of stability, Figure 21.2? (b) Given that only two or three neutrons are released per fission event and knowing that the nucleus undergoing fission has a neutron- to-proton ratio characteristic of a heavy nucleus, what sorts of decay would you expect to be dominant among the fission products?

Which type or types of nuclear reactors have these characteristics? \(\begin{array}{l}{\text { (a) Does not use a secondary coolant }} \\ {\text { (b) Creates more fissionable material than it consumes }} \\ {\text { (c) Uses a gas, such as He or } \mathrm{CO}_{2}, \text { as the primary coolant }}\end{array}\)