Chapter 21

A rock contains \(0.257 \mathrm{mg}\) of lead-206 for every milligram of uranium-238. The half-life for the decay of uranium-238 to lead-206 is \(4.5 \times 10^{5} \mathrm{yr}\). How old is the rock? SOLUTION Analyze We are told that a rock sample has a certain amount of lead206 for every unit mass of uranium-238 and asked to estimate the age of the rock. Plan Lead-206 is the product of the radioactive decay of uranium-238. We will assume that the only source of lead-206 in the rock is from the decay of uranium-238, with a known half-life. To apply firstorder kinetics expressions (Equations \(21.19\) and 21.20) to calculate the time elapsed since the rock was formed, we first need to calculate how much initial uranium-238 there was for every \(1 \mathrm{mg}\) that remains today. Solve Let's assume that the rock currently contains \(1.000 \mathrm{mg}\) of uranium-238 and therefore \(0.257 \mathrm{mg}\) of lead-206. The amount of uranium-238 in the rock when it was first formed therefore equals \(1.000 \mathrm{mg}\) plus the quantity that has decayed to lead-206. Because the mass of lead atoms is not the same as the mass of uranium atoms, we cannot just add \(1.000 \mathrm{mg}\) and \(0.257 \mathrm{mg}\). We have to multiply the present mass of lead-206 \((0.257 \mathrm{mg})\) by the ratio of the mass number of uranium to that of lead, into which it has decayed. Therefore, the original mass of \({ }_{92}^{239} \mathrm{U}\) was $$ \text { Original } \begin{aligned} { }_{98}^{238} \mathrm{U} &=1.000 \mathrm{mg}+\frac{238}{206}(0.257 \mathrm{mg}) \\ &=1.297 \mathrm{mg} \end{aligned} $$ Using Equation 21.20, we can calculate the decay constant for the process from its half-life: $$ k=\frac{0.693}{4.5 \times 10^{9} \mathrm{yr}}=1.5 \times 10^{-10} \mathrm{yr}^{-1} $$ Rearranging Equation \(21.19\) to solve for time, \(t\), and substituting known quantities gives $$ t=-\frac{1}{k} \ln \frac{N_{t}}{N_{0}}=-\frac{1}{1.5 \times 10^{-10} \mathrm{yr}^{-1}} \ln \frac{1.000}{1.297}=1.7 \times 10^{9} \mathrm{yr} $$ Comment To check this result, you could use the fact that the decay of uranium-235 to lead-207 has a half-life of \(7 \times 10^{8} \mathrm{yr}\) and measure the relative amounts of uranium-235 and lead-207 in the rock.

Calculations Involving Radioactive Decay and Time If we start with \(1.000 \mathrm{~g}\) of strontium- \(90,0.953 \mathrm{~g}\) will remain after \(2.00 \mathrm{yr}\). (a) What is the half-life of strontium- 90 ? (b) How much strontium- 90 will remain after \(5.00 \mathrm{yr}\) ? (c) What is the initial activity of the sample in becquerels and curies? SOLUTION (a) Analyze We are asked to calculate a half-life, \(t_{1 / 2}\) based on data that tell us how much of a radioactive nucleus has decayed in a time interval \(t=2.00 \mathrm{yr}\) and the information \(N_{0}=1.000 \mathrm{~g}, N_{t}=0.953 \mathrm{~g}\). Plan We first calculate the rate constant for the decay, \(k\), and then use that to compute \(t_{1 / 2}\). Solve Equation \(21.19\) is solved for the decay constant, \(k\), and then Equation \(21.20\) is used to calculate half-life, \(t_{1 / 2}\) : $$ \begin{aligned} k &=-\frac{1}{t} \ln \frac{N_{\mathrm{f}}}{N_{0}}=-\frac{1}{2.00 \mathrm{yr}} \ln \frac{0.953 \mathrm{~g}}{1.000 \mathrm{~g}} \\ &=-\frac{1}{2.00 \mathrm{yr}}(-0.0481)=0.0241 \mathrm{yr}^{-1} \\ t_{1 / 2} &=\frac{0.693}{k}=\frac{0.693}{0.0241 \mathrm{yr}^{-1}}=28.8 \mathrm{yr} \end{aligned} $$ (b) Analyze We are asked to calculate the amount of a radionuclide remaining after a given period of time. Plan We need to calculate \(N_{b}\) the amount of strontium present at time \(t\), using the initial quantity, \(N_{0}\), and the rate constant for decay, \(k\), calculated in part (a). Solve Again using Equation \(21.19\), with \(k=0.0241 \mathrm{yr}^{-1}\), we have $$ \ln \frac{N_{t}}{N_{0}}=-k t=-\left(0.0241 \mathrm{yr}^{-1}\right)(5.00 \mathrm{yr})=-0.120 $$ \(N_{t} / N_{0}\) is calculated from \(\ln \left(N_{t} / N_{0}\right)=-0.120\) using the \(e^{x}\) or INV LN function of a calculator: $$ \frac{N_{t}}{N_{0}}=e^{-1.120}=0.887 $$ Because \(N_{0}=1.000 \mathrm{~g}\), we have $$ N_{t}=(0.887) N_{0}=(0.887)(1.000 \mathrm{~g})=0.887 \mathrm{~g} $$ (c) Analyze We are asked to calculate the activity of the sample in becquerels and curies. Plan We must calculate the number of disintegrations per atom per second and then multiply by the number of atoms in the sample. Solve The number of disintegrations per atom per second is given by the decay constant, \(k\) : $$ k=\left(\frac{0.0241}{\mathrm{yr}}\right)\left(\frac{1 \mathrm{yr}}{365 \text { days }}\right)\left(\frac{1 \text { day }}{24 \mathrm{~h}}\right)\left(\frac{\mathrm{lh}}{3600 \mathrm{~s}}\right)=7.64 \times 10^{-10} \mathrm{~s}^{-1} $$ To obtain the total number of disintegrations per second, we calculate the number of atoms in the sample. We multiply this quantity by \(k\), where we express \(k\) as the number of disintegrations per atom per second, to obtain the number of disintegrations per second: $$ \begin{aligned} \left(1.000 \mathrm{~g}^{90} \mathrm{Sr}\right)\left(\frac{1 \mathrm{~mol}^{90} \mathrm{Sr}}{90 \mathrm{~g}^{90} \mathrm{Sr}}\right)\left(\frac{6.022 \times 10^{23} \mathrm{atoms} \mathrm{Sr}}{1 \mathrm{~mol}^{90} \mathrm{Sr}}\right)=6.7 \times 10^{21} \text { atoms }^{90} \mathrm{Sr} \\ \text { Total disintegrations/s } &=\left(\frac{7.64 \times 10^{-10} \text { disintegrations }}{\text { atom }{\underline{\phantom{xx}}}^{2} \mathrm{~s}}\right)\left(6.7 \times 10^{21} \text { atoms }\right) \\ &=5.1 \times 10^{12} \text { disintegrations/s } \end{aligned} $$ Because \(1 \mathrm{~Bq}\) is one disintegration per second, the activity is \(5.1 \times 10^{12} \mathrm{~Bq}\). The activity in curies is given by $$ \left(5.1 \times 10^{12} \text { disintegrations/s }\right)\left(\frac{1 \mathrm{Ci}}{3.7 \times 10^{10} \text { disintegrations/s }}\right)=1.4 \times 10^{2} \mathrm{Ci} $$ We have used only two significant figures in products of these calculations because we do not know the atomic weight of \({ }^{90} \mathrm{Sr}\) to more than two significant figures without looking it up in a special source.

Putting Concepts Together Potassium ion is present in foods and is an essential nutrient in the human body. One of the naturally occurring isotopes of potassium, potassium- 40 , is radioactive. Potassium-40 has a natural abundance of \(0.0117 \%\) and a half- life \(t_{1 / 2}=1.28 \times 10^{9} \mathrm{yr}\). It undergoes radioactive decay in three ways: \(98.2 \%\) is by electron capture, \(1.35 \%\) is by beta emission, and \(0.49 \%\) is by positron emission. (a) Why should we expect \({ }^{40} \mathrm{~K}\) to be radioactive? (b) Write the nuclear equations for the three modes by which \({ }^{40} \mathrm{~K}\) decays. (c) How many \({ }^{40} \mathrm{~K}^{+}\)ions are present in \(1.00 \mathrm{~g}\) of \(\mathrm{KCl}\) ? (d) How long does it take for \(1.00 \%\) of the \({ }^{40} \mathrm{~K}\) in a sample to undergo radioactive decay? SOLUTION (a) The \({ }^{40} \mathrm{~K}\) nucleus contains 19 protons and 21 neutrons. There are very few stable nuclei with odd numbers of both protons and neutrons (Section 21.2). (b) Electron capture is capture of an inner-shell electron by the nucleus: $$ { }_{19}^{40} \mathrm{~K}+{ }_{-1}^{0} \mathrm{e} \longrightarrow{ }_{18}^{40} \mathrm{Ar} $$ Beta emission is loss of a beta particle \((-1 \mathrm{e})\) ) by the nucleus: $$ { }_{19}^{40} \mathrm{~K} \longrightarrow{ }_{20}^{40} \mathrm{Ca}+{ }_{-1}^{0} \mathrm{e} $$ Positron emission is loss of a positron \(\left(+{ }_{+}^{0} \mathrm{e}\right)\) by the nucleus: $$ { }_{19}^{40} \mathrm{~K} \longrightarrow{ }_{18}^{40} \mathrm{Ar}+{ }_{+1}^{0} \mathrm{e} $$ (c) The total number of \(\mathrm{K}^{+}\)ions in the sample is $$ (1.00 \mathrm{~g} \mathrm{KCl})\left(\frac{1 \mathrm{~mol} \mathrm{KCl}}{74.55 \mathrm{~g} \mathrm{KCl}}\right)\left(\frac{1 \mathrm{~mol} \mathrm{~K}}{1 \mathrm{~mol} \mathrm{KCl}}\right)\left(\frac{6.022 \times 10^{23} \mathrm{~K}^{+}}{1 \mathrm{~mol} \mathrm{~K}^{+}}\right)=8.08 \times 10^{21} \mathrm{~K}^{+} \text {ions } $$ Of these, \(0.0117 \%\) are \({ }^{40} \mathrm{~K}^{+}\)ions: $$ \left(8.08 \times 10^{21} \mathrm{~K}^{+} \text {ions }\right)\left(\frac{0.0117^{40} \mathrm{~K}^{+} \text {ions }}{100^{+} \text {ions }}\right)=9.45 \times 10^{17} \text { potassium-40 ions } $$ (d) The decay constant (the rate constant) for the radioactive decay can be calculated from the half-life, using Equation 21.20: $$ k=\frac{0.693}{t_{1 / 2}}=\frac{0.693}{1.28 \times 10^{9} \mathrm{yr}}=\left(5.41 \times 10^{-10}\right) / \mathrm{yr} $$ The rate equation, Equation \(21.19\), then allows us to calculate the time required: $$ \begin{aligned} \ln \frac{N_{t}}{N_{0}} &=-k t \\ \ln \frac{99}{100} &=-\left[\left(5.41 \times 10^{-10}\right) / \mathrm{yr}\right] t \\ -0.01005 &=-\left[\left(5.41 \times 10^{-10}\right) / \mathrm{yr}\right] t \\ t &=\frac{-0.01005}{\left(-5.41 \times 10^{-10}\right) / \mathrm{yr}}=1.86 \times 10^{7} \mathrm{yr} \end{aligned} $$ That is, it would take \(18.6\) million years for just \(1.00 \%\) of the \({ }^{40} \mathrm{~K}\) in a sample to decay.

Draw a diagram similar to that shown in Exercise \(21.2\) that illustrates the nuclear reaction \({ }_{83}^{211} \mathrm{Bi} \longrightarrow{ }_{2}^{4} \mathrm{He}+{ }_{81}^{207} \mathrm{Tl}\). [Section 21.2]

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

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

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

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

Decay of which nucleus will lead to the following products: (a) bismuth- 211 by beta decay; (b) chromium 50 by positron emission; (c) tantalum-179 by electron capture; (d) radium-226 by alpha decay?

What particle is produced during the following decay processes: (a) sodium-24 decays to magnesium-24; (b) mercury-188 decays to gold-188; (c) iodine-122 decays to xenon-122; (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 \({ }_{90}^{232}\) Th ends with formation of the stable nuclide \({ }_{82}^{208} \mathrm{~Pb}\). How many alphaparticle 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: (a) \({ }_{5}^{8} \mathrm{~B}\), (b) \({ }_{29}^{68} \mathrm{Cu}\), (c) phosphorus-32, (d) chlorine-39. 21.20 Each of the following nuclei undergoes either beta decay or positron emission. Predict the type of emission for each: (a) tritium, \({ }_{1}^{3} \mathrm{H}\), (b) \({ }_{38}^{89} \mathrm{Sr}\), (c) iodine-120, (d) silver-102.

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

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

Which of the following nuclides have magic numbers of both protons and neutrons: (a) helium-4, (b) oxygen-18, (c) calcium- 40 , (d) zinc-66, (e) lead-208?

Using the concept of magic numbers, explain why alpha emission is relatively common, but proton emission is nonexistent.

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. Nuclear Transmutations (Section 21.3)

Why are nuclear transmutations involving neutrons generally easier to accomplish than those involving protons or alpha particles?

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: (a) \({ }_{58}^{252} \mathrm{Cf}+{ }_{5}^{10} \mathrm{~B} \longrightarrow 3{ }_{0}^{1} \mathrm{n}+\) ? (b) \({ }_{1}^{2} \mathrm{H}+{ }_{2}^{3} \mathrm{He} \longrightarrow{ }_{2}^{4} \mathrm{He}+\) ? (c) \({ }_{1}^{1} \mathrm{H}+{ }_{5}^{11} \mathrm{~B} \longrightarrow 3\) ? (d) \({ }_{53}^{122} \mathrm{I} \longrightarrow{ }_{54}^{122} \mathrm{Xe}+\) ? (e) \({ }_{26}^{59} \mathrm{Fe} \longrightarrow{ }_{-1}^{0} \mathrm{e}+\) ?

Complete and balance the following nuclear equations by supplying the missing particle: (a) \({ }_{7}^{14} \mathrm{~N}+{ }_{2}^{4} \mathrm{He} \longrightarrow\) ? \(+{ }_{1}^{1} \mathrm{H}\) (b) \({ }_{19}^{40} \mathrm{~K}+{ }_{-1}^{0} \mathrm{c}\) (orbital electron) \(\longrightarrow\) ? (c) \(?+{ }_{2}^{4} \mathrm{He} \longrightarrow{ }_{14}^{30} \mathrm{Si}+{ }_{1}^{1} \mathrm{H}\) (d) \({ }_{26}^{58} \mathrm{Fe}+2{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{27}^{60} \mathrm{Co}+\) ? (e) \({ }_{42}^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{54}^{135} \mathrm{Xe}+2{ }_{0}^{1} \mathrm{n}+\) ?

Write balanced equations for (a) \({ }_{92}^{238} \mathrm{U}(\alpha, \mathrm{n})_{{ }_{94}^{24}}^{24} \mathrm{Pu}\), (b) \({ }_{7}^{14} \mathrm{~N}(\alpha, \mathrm{p})_{8}^{17} \mathrm{O},(\mathrm{c}){ }_{26}^{56} \mathrm{Fe}\left(\alpha, \beta^{-}\right)_{29}^{60} \mathrm{Cu}\).

Write balanced equations for each of the following nuclear reactions: (a) \({ }_{92}^{238} \mathrm{U}(\mathrm{n}, \gamma){ }_{92}^{239} \mathrm{U},(\mathrm{b}){ }_{8}^{16} \mathrm{O}(\mathrm{p}, \alpha){ }_{7}^{13} \mathrm{~N}\), (c) \({ }_{8}^{18} \mathrm{O}(\mathrm{n}, \beta){ }_{9}^{19} \mathrm{~F}\). Rates of Radioactive Decay (Section 21.4)

Each statement that follows refers to a comparison between two radioisotopes, \(A\) and \(X\). Indicate whether each of the following statements is true or false, and why. (a) If the half-life for \(\mathrm{A}\) is shorter than the half-life for \(\mathrm{X}, \mathrm{A}\) has a larger decay rate constant. (b) If \(X\) is "not radioactive," its half-life is essentially zero. (c) If A has a half-life of \(10 \mathrm{yr}\), and \(\mathrm{X}\) has a half-life of \(10,000 \mathrm{yr}\), A would be a more suitable radioisotope to measure processes occurring on the 40 -yr time scale.

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 \(12.3 \mathrm{yr}\). 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 \(4 \mathrm{~h} \mathrm{} 39 \mathrm{~min}\) 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-60 is a strong gamma emitter that has a half-life of \(5.26 \mathrm{yr}\). 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 2013, when will it be necessary to replace the cobalt-60?

How much time is required for a \(6.25-\mathrm{mg}\) sample of \({ }^{51} \mathrm{Cr}\) to decay to \(0.75 \mathrm{mg}\) if it has a half-life of \(27.8\) days?

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

The cloth shroud from around a mummy is found to have \(a^{14} \mathrm{C}\) 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 halflife for \({ }^{14} \mathrm{C}\) decay, \(5715 \mathrm{yr}\), calculate the age of the shroud.

A wooden artifact from a Chinese temple has a \({ }^{14} \mathrm{C}\) activity 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 halflife for \({ }^{14} \mathrm{C}\) decay, \(5715 \mathrm{yr}\), determine the age of the artifact.

Potassium-40 decays to argon-40 with a half-life of \(1.27 \times 10^{9} \mathrm{yr}\), 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 \({ }^{239} \mathrm{U}\) and \(18.0 \mathrm{mg}\) of \({ }^{206} \mathrm{~Pb}\). What is the age of the mineral? Energy Changes in Nuclear Reactions (Section 21.6)

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 \(1 \mathrm{~mol}\) of \({ }^{21} \mathrm{Ne}\) ?

The atomic masses of hydrogen-2 (deuterium), helium-4, and lithium-6 are \(2.014102 \mathrm{amu}_{2} 4.002602 \mathrm{amu}\), and \(6.0151228\) amu, respectively. For cach 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 agrec 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 \(128.904779\) 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 encrgy falling on Farth? (b) If the energy released in the reaction $$ { }^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{56}^{141} \mathrm{Ba}+{ }_{36}^{92} \mathrm{Kr}+3{ }_{0}^{1} \mathrm{n} $$ \(\left({ }^{235} \mathrm{U}\right.\) nuclear mass, \(234.9935 \mathrm{amu} ;{ }^{141} \mathrm{Ba}\) nuclear mass, \(140.8833 \mathrm{amu} ;{ }^{92} \mathrm{Kr}\) 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 - \({ }^{1} \mathrm{H}, 1.00782\) amu; \({ }^{2} \mathrm{H}, 2.01410 \mathrm{amu} ;{ }^{3} \mathrm{H}, 3.01605 \mathrm{amu} ;{ }^{3} \mathrm{He}, 3.01603 \mathrm{amu} ;\) \({ }^{4} \mathrm{He}, 4.00260 \mathrm{amu}-\) 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: (a) \({ }_{1}{\underline{\phantom{xx}}}_{1} \mathrm{H}+{ }_{1}^{3} \mathrm{H} \longrightarrow{ }_{2}^{4} \mathrm{He}+{ }_{0}^{1} \mathrm{n}\) (b) \({ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \longrightarrow{ }_{2}^{3} \mathrm{He}+{ }_{0}^{1} \mathrm{n}\) (c) \({ }_{1}^{2} \mathrm{H}+{ }_{2}^{3} \mathrm{He} \longrightarrow{ }_{2}^{4} \mathrm{He}+{ }_{1}^{1} \mathrm{H}\) \(21.53\) Which of the following nuclei is likely to have the largest mass defect per nucleon: (a) \({ }^{59} \mathrm{Co}\), (b) \({ }^{11} \mathrm{~B}\), (c) \({ }^{118} \mathrm{Sn}\), (d) \({ }^{243} \mathrm{Cm}\) ? Explain your answer.

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 NaI 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?

(a) Which of the following statements about the uranium used in nuclear reactors is or are true? (i) Natural uranium has too little \({ }^{295} \mathrm{U}\) to be used as a fuel. (ii) \({ }^{24} \mathrm{U}\) cannot be used as a fucl because it forms a supereritical mass too casily. (iii) To be used as fuel, uranium must be enriched so that it is more than \(50 \%{ }^{2.35} \mathrm{U}\) in composition. (iv) The neutron-induced fission of \({ }^{235} \mathrm{U}\) releases more neutrons per nucleus than fission of \({ }^{2.85} \mathrm{U}\). (b) Which of the following statements about the plutonium shown in the chapter-opening photograph explains why it cannot be used for nuclear power plants or nuclear weapons? (i) None of the isotopes of Pu possess the characteristics needed to support nuclear fission chain reactions. (ii) The orange glow indicates that the only radioactive decay products are heat and visible light. (iii) The particular isotope of plutonium used for RTGs is incapable of sustaining a chain reaction. (iv) Plutonium can be used as a fuel, but only atter it decays to uranium.

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: (a) \({ }_{92}^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{62}^{160} \mathrm{Sm}+{ }_{30}^{72} \mathrm{Zn}+{ }_{0}^{1} \mathrm{n}\) (b) \({ }_{94}^{239} \mathrm{Pu}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{58}^{444} \mathrm{Ce}+\ldots+2{ }_{0}^{1} \mathrm{n}\)

The table to the right gives the number of protons \((p)\) and neutrons \((n)\) for four isotopes. (a) Write the symbol for each of the isotopes. (b) Which of the isotopes is most likely to be unstable? (c) Which of the isotopes involves a magic number of protons and/or neutrons? (d) Which isotope will yield potassium-39 following positron emission? \begin{equation}\begin{array}{|c|c|c|c|}\hline & {\text { (i) }} & {\text { (ii) }} & {\text { (iii) }} & {\text { (iv) }} \\ \hline p & {19} & {19} & {20} & {20} \\ \hline n & {19} & {21} & {19} & {20} \\ \hline\end{array} \end{equation}

Radon-222 decays to a stable nucleus by a series of three alpha emissions and two beta emissions. What is the stable nucleus that is formed?

Chlorine has two stable nuclides, \({ }^{35} \mathrm{Cl}\) and \({ }^{37} \mathrm{Cl}\). In contrast, \({ }^{36} \mathrm{Cl}\) is a radioactive nuclide that decays by beta emission. (a) What is the product of decay of \({ }^{36} \mathrm{Cl}\) ? (b) Based on the empirical rules about nuclear stability, explain why the nucleus of \({ }^{36} \mathrm{Cl}\) is less stable than either \({ }^{35} \mathrm{Cl}\) or \({ }^{37} \mathrm{Cl}\).