Chapter 17

When the nucleons rearrange in the following daughter nuclei, the energy changes by the amount shown and a \(\gamma\)-ray is emitted. Determine the frequency and wavelength of the \(\gamma\)-ray in each case: (a) cobalt-60, \(1.33 \mathrm{MeV}\); (b) arsenic-80, \(1.64 \mathrm{MeV}\); (c) iron-59, 1.10 MeV. (1 MeV \(=1.602 \times 10^{-13} \mathrm{~J}\).)

Write the balanced nuclear equation for each of the following decays: (a) \(\alpha\) decay of californium-250; (b) positron decay of rubidium76 ; (c) electron capture by arsenic-73; (d) \(\beta^{-}\)decay of chromium- \(56 .\)

Write the balanced nuclear equation for each of the following decays: (a) \(\beta^{-}\)decay of uranium-233; (b) proton emission of cobalt- 56 ; (c) \(\beta^{+}\)decay of holmium-158; (d) \(\alpha\) decay of polonium-212.

Determine the particle emitted and write the balanced nuclear equation for each of the following nuclear transformations: (a) sodium-24 to magnesium-24; (b) \({ }^{128} \mathrm{Sn}\) to \({ }^{128} \mathrm{Sb}\); (c) lanthanum-140 to barium-140; (d) \({ }^{228} \mathrm{Th}\) to \({ }^{224} \mathrm{Ra}\).

Determine the particle emitted and write the balanced nuclear equation for each of the following nuclear transformations: (a) gadolinium-148 to samarium-144; (b) fluorine-17 to oxygen- 17 ; (c) silver-112 to cadmium-112; (d) plutonium-238 to uranium-234.

The following nuclides lie outside the band of stability. Predict whether each is most likely to undergo \(\beta^{-}\)decay, \(\beta^{+}\)decay, or \(\alpha\) decay, and identify the daughter nucleus: (a) copper-60; (b) xenon-140; (c) americium-246; (d) neptunium-240.

Write a nuclear equation for each of the following processes: (a) oxygen- 17 produced by \(\alpha\)-particle bombardment of nitrogen- 14 ; (b) americium-240 produced by neutron bombardment of plutonium-239.

Write a nuclear equation for each of the following transformations: (a) \({ }^{257} \mathrm{Rf}\) produced by the bombardment of californium- 245 with carbon- 12 nuclei; (b) the first synthesis of \({ }^{266} \mathrm{Mt}\) by the bombardment of bismuth- 209 with iron- 58 nuclei. Given that the first decay of meitnerium is by \(\alpha\) emission, what is the daughter nucleus?

The activity of a certain radioactive source is \(5.3 \times 10^{5} \mathrm{~Bq}\). Express this activity in curies.

The activity of a sample containing carbon-14 is \(54.8 \mathrm{~Bq}\). Express this activity in microcuries.

Determine the number of disintegrations per second for radioactive sources of each of the following activities: (a) \(2.5 \mu \mathrm{Ci}\); (b) \(142 \mathrm{Ci}\); (c) \(7.2 \mathrm{mCi}\)

A certain Geiger counter is known to respond to only 1 of every 1000 radiation events from a sample. Calculate the activity of each radioactive source in curies, given the following data: (a) 591 clicks in \(100 . \mathrm{s}\); (b) \(2.7 \times 10^{4}\) clicks in \(1.5 \mathrm{~h}\); (c) 159 clicks in \(1.0 \mathrm{~min}\).

Someone is exposed to a source of \(\beta\) radiation that results in a dose rate of \(1.0 \mathrm{rad} \cdot \mathrm{d}^{-1}\). Given that nausea begins after a dose equivalent of about 100 rem, after what period will that symptom of radiation sickness be apparent?

Someone is exposed to a source of \(\alpha\) radiation that results in a dose rate of \(2.0 \mathrm{mrad} \cdot \mathrm{d}^{-1}\). If nausea begins after a dose equivalent of about 100 rem, after what period will nausea become apparent?

Determine the decay constant for (a) tritium, \(t_{1 / 2}=12.3 \mathrm{a}\); (b) lithium- \(8, t_{1 / 2}=0.84 \mathrm{~s}\); (c) nitrogen- \(13, t_{1 / 2}=10.0 \mathrm{~min}\).

Determine the half-life of (a) potassium- \(40, k=5.3 \times\) \(10^{-10} \mathrm{a}^{-1}\); (b) cobalt-60, \(k=0.132 \mathrm{a}^{-1}\); (c) nobelium-255, \(k=3.85 \times 10^{-3} \mathrm{~s}^{-1}\).

The activity of a sample of a radioisotope was found to be 2150 disintegrations per minute. After \(6.0 \mathrm{~h}\) the activity was found to be 1324 disintegrations per minute. What is the half-life of the radioisotope?

A current \(1.00\)-g sample of carbon shows 921 disintegrations per hour. If \(1.00 \mathrm{~g}\) of charcoal from an archaeological dig in a limestone cave in Slovenia shows \(5.50 \times 10^{3}\) disintegrations in \(24.0 \mathrm{~h}\), what is the age of the charcoal sample?

Use the law of radioactive decay to determine the activity of (a) a \(1.0\)-mg sample of radium-226 \(\left(t_{1 / 2}=1.60 \mathrm{ka}\right)\); (b) a \(2.0-\mathrm{mg}\) sample of strontium-90 \(\left.t_{1 / 2}=28.1 \mathrm{a}\right) ;\) (c) a \(0.43-\mathrm{mg}\) sample of promethium- \(147\left(t_{1 / 2}=2.6\right.\) a). The mass of each nuclide as a multiple of the atomic mass constant \(\left(m_{\mathrm{u}}\right)\) is equal to its mass number, within two significant figures.

Use the law of radioactive decay to determine the activity of (a) a \(1.0\)-g sample of \({ }^{235} \mathrm{UO}_{2}\left(t_{1 / 2}=7.1 \times 10^{8} \mathrm{a}\right)\); (b) a \(1.0-\mathrm{g}\) sample of cobalt that is \(1.0 \%{ }^{60} \mathrm{Co}\left(t_{1 / 2}=5.26 \mathrm{a}\right)\); (c) a \(5.0-\mathrm{mg}\) sample of thallium-200 \(\left(t_{1 / 2}=26.1 \mathrm{~h}\right)\). The mass of each nuclide as a multiple of the atomic mass constant \(\left(m_{\mathrm{u}}\right)\) is equal to its mass number, within two significant figures.

Deoxyglucose labeled with fluorine- 18 is commonly used in PET scans to locate tumors. Fluorine- 18 has a half-life of \(109 \mathrm{~min}\). How long will it take for the level of fluorine- 18 in the body to drop to \(10 \%\) of its initial value?

Technetium-99 \(\mathrm{m}\) (the \(\mathrm{m}\) signifies a "metastable," or moderately stable, species) is generated in nuclear reactors and shipped to hospitals for use in medical imaging. The radioisotope has a half-life of \(6.01 \mathrm{~h}\). If a 165 -mg sample of technetium- \(99 \mathrm{~m}\) is shipped from a nuclear reactor to a hospital 125 kilometers away in a truck that averages \(50.0 \mathrm{~km} \cdot \mathrm{h}^{-1}\), what mass of technetium\(99 \mathrm{~m}\) will remain when it arrives at the hospital?

A \(1.40\)-g sample containing radioactive cobalt was kept for \(2.50 \mathrm{a}\), at which time it was found to contain \(0.266 \mathrm{~g}\) of \({ }^{67} \mathrm{Co}\). The half-life of \({ }^{67} \mathrm{Co}\) is \(5.27\) a. What percentage (by mass) of the original sample was \({ }^{67} \mathrm{Co}\) ?

A chemist is studying the mechanism of the following hydrolysis reaction of the organic ester methyl acetate: \(\mathrm{CH}_{3} \mathrm{COOCH} \mathrm{CO}_{3}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{CH}_{3} \mathrm{COOH}+\mathrm{CH}_{3} \mathrm{OH}\). The question arises whether the \(\mathrm{O}\) atom in the product methanol comes from the methyl acetate or from the water. Propose an experiment using isotopes that would allow the chemist to determine the origin of the oxygen atom.

What would you expect to happen to the vibrational frequency of the \(\mathrm{C}-\mathrm{H}\) bond of methane if the hydrogen atoms, which are normally present as \({ }^{1} \mathrm{H}\), are replaced by \({ }^{2} \mathrm{H}\) ? See Major Technique 1, Infrared Spectroscopy.

Calculate the energy in joules that is equivalent to (a) \(1.0 \mathrm{~g}\) of matter; (b) one electron; (c) \(1.0 \mathrm{pg}\) of matter; (d) one proton.

Calculate the energy in joules that is equivalent to (a) \(1.00 \mathrm{~kg}\) of matter; (b) \(1.00 \mathrm{lb}\) of matter ( \(1 \mathrm{lb}=454 \mathrm{~g}\) ); (c) one neutron; (d) one hydrogen atom.

(a) For the fusion reaction \(6 \mathrm{D} \rightarrow 2{ }^{4} \mathrm{He}+2{ }^{1} \mathrm{H}+2 \mathrm{n}\), \(3 \times 10^{8} \mathrm{~kJ}\) of energy is released by a certain sample of deuterium. What is the mass loss (in grams) for the reaction? (b) What was the mass of deuterium converted? The molar mass of deuterium is \(2.014 \mathrm{~g} \cdot \mathrm{mol}^{-1}\).

State whether the following statements are true or false. If false, explain why. (a) The dose equivalent is lower than the actual dose of radiation because it takes into account the differential effects of different types of radiation. (b) Exposure to \(1 \times 10^{8} \mathrm{~Bq}\) of radiation would be much more hazardous than exposure to \(10 \mathrm{Ci}\) of radiation. (c) Spontaneous radioactive decay follows firstorder kinetics. (d) Fissile nuclei can undergo fission when struck with slow neutrons, whereas fast neutrons are required to split fissionable nuclei.

State whether the following statements are true or false. If false, explain why. (a) A subcritical mass of fissionable material is unstable and likely to explode. (b) In order for fusion to occur, the colliding particles must have high kinetic energy. (c) Highly reactive fission products are considered to be no longer dangerous after two half-lives. (d) The larger the binding energy per nucleon, the more stable is the nucleus.

Uranium-238 decays through a series of \(\alpha\) and \(\beta\) emissions to lead-206, with an overall half-life for the entire process of \(4.5 \mathrm{Ga}\). How old is a uranium-bearing ore that is found to have a \({ }^{238} \mathrm{U} /{ }^{206} \mathrm{~Pb}\) ratio of (a) \(1.00\) and (b) \(1.25\) ?

Actinium-225 decays by successive emission of three \(\alpha\) particles. (a) Write the nuclear equations for the three decay processes. (b) Compare the neutron-to-proton ratio of the final daughter product with that of actinium-225. Which is closer to the band of stability?

A nuclear waste storage facility is being proposed for your county and you have been asked to prepare a recommendation for how highly radioactive fission products should be processed and stored. In your recommendation discuss the benefits and drawbacks of at least three modes of nuclear waste storage.