Chapter 25

What nucleus is obtained in each process? (a) \(\frac{234}{94}\) Pu decays by \(\alpha\) emission. \(\text { (b) } \begin{array}{l}248 \\\97\end{array}\)Bk decays by \(\beta^{-}\) emission. \(\text { (c) } \begin{array}{r}196 \\\82\end{array}\) Pb goes through two successive EC processes.

Based on a favorable \(N-Z\) ratio for the product nucleus, write the most plausible equation for the decay of \(\frac{14}{6} \mathrm{C}\).

Write a plausible equation for the decay of tritium, 3 \(\mathrm{H}\), the radioactive isotope of hydrogen. 1 \(\textrm{ }\).

Just as the uranium series is called the "4n \(+2^{\prime \prime}\) series, the thorium series can be called the "4n" series and the actinium series the "4n \(+3 "\) series. A \(4 n+1 "\) series has also been established, with \(^{241} \mathrm{Pu}\) as the parent nuclide. To which series does each of the following belong: (a) \(\frac{214}{83} \mathrm{Bi} ;\) (b) \(\frac{216}{84} \mathrm{Po} ;\) (c) \(\frac{215}{85} \mathrm{At}\) (d) \(\frac{235}{92} \mathrm{U} ?\)

Supply the missing information in each of the following nuclear equations representing a radioactive decay process.(a) \(160_?\mathrm{W} \longrightarrow\\{\mathrm{Hf}+?\) (b) \(38_? \mathrm{Cl} \longrightarrow_{?}^{?} \mathrm{Ar}+?\) (c) \(^{214} ? \longrightarrow_{?}^{?} \mathrm{Po}+_{-1}^{0} \boldsymbol{\beta}\) (d) \(_{17}^{32} \mathrm{Cl} \longrightarrow_{1}^{?} ?+?\)

Complete the following nuclear equations. (a) \(\frac{23}{11} \mathrm{Na}+? \longrightarrow_{11}^{24} \mathrm{Na}+_{1}^{1} \mathrm{H}\) (b) \(_{27}^{59} \mathrm{Co}+_{0}^{1} \mathrm{n} \longrightarrow_{25}^{56} \mathrm{Mn}+?\) (c) \(?+_{1}^{2} \mathrm{H} \longrightarrow_{94}^{240} \mathrm{Pu}+_{-1}^{0} \beta\) (d) \(^{246} \mathrm{Cm}+? \longrightarrow_{102}^{254} \mathrm{No}+5_{0}^{1} \mathrm{n}\) (e) \(^{238} \mathrm{U}+? \longrightarrow_{99}^{246} \mathrm{Es}+6 \frac{1}{0} \mathrm{n}\)

Write equations for the following nuclear reactions. (a) bombardment of \(^{7} \mathrm{Li}\) with protons to produce \(^{8} \mathrm{Be}\) and \(\gamma\) rays (b) bombardment of \(^{9} \mathrm{B}\) with \(_{1}^{2} \mathrm{H}\) to produce \(^{10} \mathrm{B}\) (c) bombardment of \(^{14} \mathrm{N}\) with neutrons to produce \(^{14} \mathrm{C}\)

Write equations for the following nuclear reactions. (a) bombardment of \(^{238} \mathrm{U}\) with \(\alpha\) particles to produce \(^{239} \mathrm{Pu}\) (b) bombardment of tritium ( \(^{3} \mathrm{H}\) ) with \(_{1}^{2} \mathrm{H}\) to produce \(^{4} \mathrm{He}\) (c) bombardment of \(^{33} \mathrm{S}\) with neutrons to produce \(^{33} \mathrm{P}\).

Write nuclear equations to represent the formation of an isotope of element 111 with a mass number of 272 by the bombardment of bismuth-209 by nickel-64 nuclei, followed by a succession of five \(\alpha\) -particle emissions.

Write nuclear equations to represent the formation of a hypothetical isotope of element 118 with a mass number of 293 by the bombardment of lead- 208 by krypton-86 nuclei, followed by a chain of \(\alpha\) -particle emissions to the element seaborgium.

Scientists from Dubna, Russia, observed the existence of elements 118 and 116 at the Joint Institute for Nuclear Research U400 cyclotron in 2005. This was the result of bombarding calcium- 48 ions on a californium-249 target. Write a complete nuclear equation for this reaction.

The immediate decay product of element 118 is thought to be element \(116 .\) Write a complete nuclear equation for this reaction.

Element-120 is located in a region of the neutron versus proton map known as the island of stability. Write a nuclear equation for the generation of element- 120 by bombarding iron isotopes on a plutonium target.

Another possible nuclear reaction leading to the formation of element-120 is between uranium-238 and nickel-64. Write a nuclear equation for this nuclear reaction.

In a comparison of two radioisotopes, isotope \(A\) requires 18.0 hours for its decay rate to fall to \(\frac{1}{16}\) its initial value, while isotope B has a half-life that is 2.5 times that of A. How long does it take for the decay rate of isotope \(B\) to decrease to \(\frac{1}{32}\) of its initial value?

The disintegration rate for a sample containing \(_{27}^{60} \mathrm{Co}\) as the only radioactive nuclide is 6740 dis \(\mathrm{h}^{-1}\). The half-life of 20 Co is 5.2 years. Estimate the number of atoms of \(_{27}^{60}\) Co in the sample.

A sample containing \(_{88}^{224} \mathrm{Ra},\) which decays by \(\alpha\) -particle emission, disintegrates at the following rate, expressed as disintegrations per minute or counts per minute \((\mathrm{cpm}): t=0,1000 \mathrm{cpm} ; t=1 \mathrm{h}\) \(992 \mathrm{cpm} ; t=10 \mathrm{h}, 924 \mathrm{cpm} ; t=100 \mathrm{h}, 452 \mathrm{cpm}\) \(t=250 \mathrm{h}, 138 \mathrm{cpm} .\) What is the half-life of this nuclide?

Iodine-129 is a product of nuclear fission, whether from an atomic bomb or a nuclear power plant. It is a \(\beta^{-}\) emitter with a half-life of \(1.7 \times 10^{7}\) years. How many disintegrations per second would occur in a sample containing \(1.00 \mathrm{mg}^{129} \mathrm{I} ?\)

What should be the mass ratio \(^{208} \mathrm{Pb} /^{232} \mathrm{Th}\) in a meteorite that is approximately \(2.7 \times 10^{9}\) years old? The half-life of \(^{232} \mathrm{Th}\) is \(1.39 \times 10^{10}\) years. [Hint: One \(^{208} \mathrm{Pb}\) atom is the final decay product of one \(^{232}\) Th atom.

A lunar rock was analyzed for argon by mass spectrometry and for potassium by atomic absorption. The results of these analyses showed that the sample contained \(3.02 \times 10^{-5} \mathrm{mL} \mathrm{g}^{-1}\) of argon and \(0.083 \%\) of potassium. The half-life of potassium- 40 is \(1.248 \times\) \(10^{9} \mathrm{y} \cdot\) Calculate the age of the lunar rock.

What is the age of a piece of volcanic rock that has a mass ratio of argon- 40 to potassium- 40 of \(2.9 ?\) The half-life of potassium-40 by \(\beta\) decay is \(1.248 \times 10^{9} \mathrm{y}\) and by electron capture \(t_{1 / 2}=1.4 \times 10^{9} \mathrm{y}\).

Which member of the following pairs of nuclides would you expect to be most abundant in natural sources: (a) \(_{10}^{20} \mathrm{Ne}\) or \(_{10}^{22} \mathrm{Ne} ;\) (b) \(_{8}^{17} \mathrm{O}\) or \(_{8}^{18} \mathrm{O} ;\) (c) \(_{3}^{6} \mathrm{Li}\) or \(_{3}^{7} \mathrm{Li} ?\) Explain your reasoning.

Which member of the following pairs of nuclides would you expect to be most abundant in natural sources: (a) \(_{20}^{40} \mathrm{Ca}\) or \(_{20}^{42} \mathrm{Ca} ;\) (b) \(_{15}^{31} \mathrm{P}\) or \(_{15}^{32} \mathrm{P} ;\) (c) \(_{30}^{63} \mathrm{Zn}\) or 64 \(\mathrm{Zn} ?\) Explain your reasoning?

One member each of the following pairs of radioisotopes decays by \(\beta^{-}\) emission, and the other by positron \(\left(\beta^{+}\right)\) emission: \((\mathrm{a})_{15}^{29} \mathrm{P}\) and \(_{15}^{33} \mathrm{P} ;(\mathrm{b}) \stackrel{120}{53} \mathrm{I}\) and \(_{53}^{134} \mathrm{I} .\) Which is which? Explain your reasoning.

Each of the following isotopes is radioactive: (a) \(\frac{28}{15} \mathrm{P}\) (b) \(45 \mathrm{K} ;\) (c) \(^{73} \mathrm{Zn}\). Which would you expect to decay by 30 . \(\beta^{+}\) emission?

Both \(\beta^{-}\) and \(\beta^{+}\) emissions are observed for artificially produced radioisotopes of low atomic numbers, but only \(\beta^{-}\) emission is observed with naturally occurring radioisotopes of high atomic number. Why do you suppose this is so?

Explain why more energy is released in a fusion process than in a fission process.

Explain why the rem is more satisfactory than the rad as a unit for measuring radiation dosage.

Discuss briefly the basic difficulties in establishing the physiological effects of low-level radiation.

"Sr is both a product of radioactive fallout and a radioactive waste in a nuclear reactor. This radioisotope is a \(\beta^{-}\) emitter with a half-life of 27.7 years. Suggest reasons why \(^{90} \mathrm{Sr}\) is such a potentially hazardous substance.

\(^{222} \mathrm{Rn}\) is an \(\alpha\) -particle emitter with a half-life of 3.82 days. Is it hazardous to be near a flask containing this isotope? Under what conditions might \(^{222} \mathrm{Rn}\) be hazardous?

Describe how you might use radioactive materials to find a leak in the \(\mathrm{H}_{2}(\mathrm{g})\) supply line in an ammonia synthesis plant.

Explain why neutron activation analysis is so useful in identifying trace elements in a sample, in contrast to ordinary methods of quantitative analysis, such as precipitation or titration.

A small quantity of \(\mathrm{NaCl}\) containing radioactive \(_{11}^{24} \mathrm{Na}\) is added to an aqueous solution of \(\mathrm{NaNO}_{3}\). The solution is cooled, and \(\mathrm{NaNO}_{3}\) is crystallized from the solution. Would you expect the \(\mathrm{NaNO}_{3}(\mathrm{s})\) to be radioactive? Explain.

The following reactions are carried out with HCl(aq) containing some tritium ( \(_{1}^{3} \mathrm{H}\) ) as a tracer. Would you expect any of the tritium radioactivity to appear in the \(\mathrm{NH}_{3}(\mathrm{g}) ?\) In the \(\mathrm{H}_{2} \mathrm{O} ?\) Explain. $$\begin{array}{c} \mathrm{NH}_{3}(\mathrm{aq})+\mathrm{HCl}(\mathrm{aq}) \longrightarrow \mathrm{NH}_{4} \mathrm{Cl}(\mathrm{aq}) \\ \mathrm{NH}_{4} \mathrm{Cl}(\mathrm{aq})+\mathrm{NaOH}(\mathrm{aq}) \longrightarrow\\\\\mathrm{NaCl}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(1)+\mathrm{NH}_{3}(\mathrm{g})\end{array}$$

In some cases, the most abundant isotope of an element can be established by rounding off the atomic mass to the nearest whole number, as in \(^{39} \mathrm{K},^{85} \mathrm{Rb}\), and \(^{88} \mathrm{Sr}\). But in other cases, the isotope corresponding to the rounded-off atomic mass does not even occur naturally, as in \(^{64} \mathrm{Cu}\). Explain the basis of this observation.

The overall change in the radioactive decay of \({238}_{92} \mathrm{U}\)to 206 \(\mathrm{Pb}\) is the emission of eight \(\alpha\) particles. Show that if \(_{82}^{206} \mathrm{Pb}\)this loss of eight \(\alpha\) particles were not also accompanied by six \(\beta^{-}\) emissions, the product nucleus would still be radioactive.

One method of dating rocks is based on their \(^{87} \mathrm{Sr} /^{87} \mathrm{Rb}\) ratio. \(^{87} \mathrm{Rb}\) is a \(\beta^{-}\) emitter with a half- life of \(5 \times 10^{11}\) years. A certain rock has a mass ratio \(^{87} \mathrm{Sr} /^{87} \mathrm{Rb}\) of \(0.004 / 1.00 .\) What is the age of the rock?

The carbon-14 dating method is based on the assumption that the rate of production of \(^{14} \mathrm{C}\) by cosmic ray bombardment has remained constant for thousands of years and that the ratio of \(^{14} \mathrm{C}\) to \(^{12} \mathrm{C}\) has also remained constant. Can you think of any effects of human activities that could invalidate this assumption in the future?

Calculate the minimum kinetic energy (in megaelectronvolts) that \(\alpha\) particles must possess to produce the nuclear reaction $$_{2}^{4} \mathrm{He}+^{14}_{7} \mathrm{N} \longrightarrow^{17}_{8} \mathrm{O}+_{1}^{1} \mathrm{H}.$$ The nuclidic masses are \(_{2}^{4} \mathrm{He}=4.00260 \mathrm{u}\); \(_{7}^{14} \mathrm{He}=14.00307\mathrm{u}\);\(_{1}^{1} \mathrm{H}=1.00783 \mathrm{u}\);\(_{8}^{17} \mathrm{H}=16.99913 \mathrm{u}\);

Hydrogen gas is spiked with tritium to the extent of \(5.00 \%\) by mass. What is the activity in curies of a \(4.65 \mathrm{L}\) sample of this gas at \(25.0^{\circ} \mathrm{C}\) and 1.05 atm pressure? [Hint: Use 3.02 u as the atomic mass of tritium and data from elsewhere in the text, as necessary.]

A certain shale deposit containing \(0.006 \%\) U by mass is being considered for use as a potential fuel in a breeder reactor. Assuming a density of \(2.5 \mathrm{g} / \mathrm{cm}^{3},\) how much energy could be released from \(1.00 \times 10^{3} \mathrm{cm}^{3}\) of this material? Assume a fission energy of \(3.20 \times 10^{-11} \mathrm{J}\) per fission event (that is, per U atom).

An ester forms from a carboxylic acid and an alcohol. $$\mathrm{RCO}_{2} \mathrm{H}+\mathrm{HOR}^{\prime} \longrightarrow \mathrm{RCO}_{2} \mathrm{R}^{\prime}+\mathrm{H}_{2} \mathrm{O}.$$ This reaction is superficially similar to the reaction of an acid with a base such as sodium hydroxide. The mechanism of the reaction can be followed by using the tracer \(^{18} \mathrm{O}\). This isotope is not radioactive, but other physical measurements can be used to detect its presence. When the esterifcation reaction is carried out with the alcohol containing oxygen-18 atoms, no oxygen-18 beyond its naturally occurring abundance is found in the water produced. How does this result affect the perception that this reaction is like an acid-base reaction?

The conversion of \(\mathrm{CO}_{2}\) into carbohydrates by plants via photosynthesis can be represented by the reaction $$6 \mathrm{CO}_{2}(\mathrm{g})+6 \mathrm{H}_{2} \mathrm{O} \stackrel{\text { light }}{\longrightarrow} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+6 \mathrm{O}_{2}(\mathrm{g}).$$ To study the mechanism of photosynthesis, algae were grown in water containing \(^{18}\) O, that is, \(\mathrm{H}_{2}^{18} \mathrm{O}\) The oxygen evolved contained oxygen-18 in the same ratio to the other oxygen isotopes as the water in which the reaction was carried out. In another experiment, algae were grown in water containing only \(^{16} \mathrm{O}\),but with oxygen-18 present in the \(\mathrm{CO}_{2}\). The oxygen evolved in this experiment contained no oxygen-18. What conclusion can you draw about the mechanism of photosynthesis from these experiments?

Assume that when Earth formed, uranium-238 and uranium-235 were equally abundant. Their current percent natural abundances are \(99.28 \%\) uranium- 238 and \(0.72 \%\) uranium- \(235 .\) Given half-lives of \(4.5 \times 10^{9}\) years for uranium-238 and \(7.1 \times 10^{8}\) years for uranium-235, determine the age of Earth corresponding to this assumption.

The packing fraction of a nuclide is related to the fraction of the total mass of a nuclide that is converted to nuclear binding energy. It is defined as the fraction \((M-A) / A,\) where \(M\) is the actual nuclidic mass and \(A\) is the mass number. Use data from a handbook (such as the Handbook of Chemistry and Physics, published by the CRC Press) to determine the packing fractions of some representative nuclides. Plot a graph of packing fraction versus mass number, and compare it with Figure \(25-6 .\) Explain the relationship between the two.

For medical uses, radon-222 formed in the radioactive decay of radium-226 is allowed to collect over the radium metal. Then, the gas is withdrawn and sealed into a glass vial. Following this, the radium is allowed to disintegrate for another period, when a new sample of radon- 222 can be withdrawn. The procedure can be continued indefinitely. The process is somewhat complicated by the fact that radon-222 itself undergoes radioactive decay to polonium- 218 , and so on. The half-lives of radium-226 and radon-222 are \(1.60 \times 10^{3}\) years and 3.82 days, respectively.(a) Beginning with pure radium- \(226,\) the number of radon-222 atoms present starts at zero, increases for a time, and then falls off again. Explain this behavior. That is, because the half-life of radon-222 is so much shorter than that of radium- \(226,\) why doesn't the radon-222 simply decay as fast as it is produced, without ever building up to a maximum concentration?(b) Write an expression for the rate of change \((d \mathrm{D} / d t)\) in the number of atoms (D) of the radon- 222 daughter in terms of the number of radium- 226 atoms present initially ( \(\mathrm{P}_{0}\) ) and the decay constants of the parent \(\left(\lambda_{\mathrm{p}}\right)\) and daughter \(\left(\lambda_{\mathrm{d}}\right)\) (c) Integration of the expression obtained in part (b) yields the following expression for the number of atoms of the radon-222 daughter (D) present at a time \(t\).$$\mathrm{D}=\frac{\mathrm{P}_{0} \lambda_{\mathrm{p}}\left(\mathrm{e}^{-\lambda_{\mathrm{p}} \times t}-\mathrm{e}^{-\lambda_{\mathrm{d}} \times t}\right)}{\lambda_{\mathrm{d}}-\lambda_{\mathrm{p}}}$$,Starting with \(1.00 \mathrm{g}\) of pure radium- \(226,\) approximately how long will it take for the amount of radon222 to reach its maximum value: one day, one week, one year, one century, or one millennium?

Radioactive decay and mass spectrometry are often used to date rocks after they have cooled from a magma. \(^{87} \mathrm{Rb}\) has a half-life of \(4.8 \times 10^{10}\) years and follows the radioactive decay $$^{87} \mathrm{Rb} \longrightarrow^{87} \mathrm{Sr}+\beta^{-}$$ A rock was dated by assaying the product of this decay. The mass spectrum of a homogenized sample of rock showed the \(^{87} \mathrm{Sr} /^{86} \mathrm{Sr}\) ratio to be \(2.25 .\) Assume that the original \(^{87} \mathrm{Sr} /^{86} \mathrm{Sr}\) ratio was 0.700 when the rock cooled. Chemical analysis of the rock gave \(15.5 \mathrm{ppm}\) Sr and 265.4 ppm \(\mathrm{Rb},\) using the average atomic masses from a periodic table. The other isotope ratios were \(^{86} \mathrm{Sr} /^{88} \mathrm{Sr}=\) 0.119 and \(^{84} \mathrm{Sr} /^{88} \mathrm{Sr}=0.007 .\) The isotopic ratio for \(^{87} \mathrm{Rb} /^{85} \mathrm{Rb}\) is 0.330. The isotopic masses are as follows:Calculate the following: (a) the average atomic mass of Sr in the rock (b) the original concentration of \(\mathrm{Rb}\) in the rock in \(\mathrm{ppm}\) (c) the percentage of rubidium- 87 decayed in the rock (d) the time since the rock cooled.

5\. In your own words, define the following symbols: (a) \(\alpha ;\) (b) \(\beta^{-} ;\) (c) \(\beta^{+} ;\) (d) \(\gamma ;\) (e) \(t_{1 / 2}\).

Briefly describe each of the following ideas, phenomena, or methods: (a) radioactive decay series;(b) charged-particle accelerator; (c) neutron-to- proton ratio; (d) mass-energy relationship; (e) background radiation.