Chapter 29

Determine the number of protons, neutrons, and electrons in a neutral atom with the following nuclei: (a) \({ }^{90} \mathrm{Zr}\) and (b) \(^{208} \mathrm{~Pb}\)

Magnesium has three stable isotopes. Write these isotopes in nuclear notation including nucleon, proton, and neutron number on the elemental symbol.

An isotope of potassium has the same number of neutrons as argon- \(40 .\) Write this potassium isotope in nuclear notation.

\({ }^{35} \mathrm{Cl}\) and \({ }^{37} \mathrm{Cl}\) are two isotopes of chlorine. What are the numbers of protons, neutrons, and electrons in each if (a) the atom is electrically neutral, (b) the ion has a -2 charge, and (c) the ion has a +1 charge?

One isotope of uranium has a mass number of 235 , and another has a mass number of \(238 .\) What are the numbers of protons, neutrons, and electrons in a neutral atom of each isotope?

(a) Isotopes of an element must have the same (1) atomic number, (2) neutron number, (3) mass number. (b) Write two possible isotopes for gold-197.

An approximate experimental expression for the radius \((R)\) of a nucleus is \(R=R_{\mathrm{o}} A^{1 / 3},\) where \(R_{\mathrm{o}}=1.2 \times 10^{-15} \mathrm{~m}\) and \(A\) is the mass number of the nucleus. (a) Find the nuclear radii of atoms of the noble gases: \(\mathrm{He}, \mathrm{Ne}, \mathrm{Ar}, \mathrm{Kr}, \mathrm{Xe},\) and \(\mathrm{Rn}\) (b) Determine the density of the nuclei associated with each of these species and compare them. Does your answer surprise you?

Tritium is radioactive. (a) Would you expect it to (1) \(\beta^{+},(2) \beta^{-},\) or (3) alpha decay? Why? (b) Write the nuclear equation for the correct decay and identify the daughter nucleus. Is it stable?

Write the nuclear equations expressing (a) the beta decay of \({ }^{60} \mathrm{Co}\) and \((\mathrm{b})\) the alpha decay of \({ }^{222} \mathrm{Rn}\)

Write the nuclear equations for (a) the alpha decay of \(237 \mathrm{~Np},\) (b) the \(\beta^{-}\) decay of \(32 \mathrm{P}\), (c) the \(\beta^{+}\) decay of \({ }^{56} \mathrm{Co}\) (d) electron capture in \({ }^{56} \mathrm{Co},\) and \((\mathrm{e})\) the \(\gamma\) decay of \({ }^{42} \mathrm{~K}\) from an excited nuclear state to the ground state (not excited).

Polonium-214 can decay by alpha decay. (a) The product of its decay has how many fewer protons than polonium- 214:(1) zero, (2) one, (3) two, or (4) four? (b) Write the nuclear equation for this decay and determine the daughter nucleus.

A lead-209 nucleus results from both alpha-beta sequential decays and beta- alpha sequential decays. What was the grandparent nucleus? Show this result for both decay routes by writing the nuclear equations for both sequential decay processes.

A particular radioactive sample undergoes \(2.50 \times 10^{6}\) decays/s. What is the activity of the sample in (a) curies and (b) becquerels?

At present, a radioactive beta source with a long halflife has an activity of \(20 \mathrm{mCi}\). (a) What is the present decay rate in decays per second? (b) Assuming that one beta particle is emitted per decay, how many are currently emitted per minute?

The half-life of a radioactive isotope is known to be exactly \(1 \mathrm{~h}\). (a) What fraction of a sample would be left after exactly \(3 \mathrm{~h}:(1)\) one-third, (2) one-eighth, or (3) oneninth? (b) What fraction of a sample would be left after exactly 1 day?

A 1.25-\muCi alpha source gives off alpha particles each with a kinetic energy of \(2.78 \mathrm{MeV}\). At what rate (in watts) is kinetic energy being produced?

A sample of technetium- 104 , which has a half-lfe of \(18.0 \mathrm{~min}\), has an initial activity of \(10.0 \mathrm{mCi}\). Determine the activity of the sample after exactly 1 h has elapsed.

Calculate the time required for a sample of radioactive tritium to lose \(80.0 \%\) of its activity. (Tritium has a half-life of 12.3 years.)

Carbon- 14 dating is used to determine the age of some unearthed bones. (a) If the activity of bone \(\mathrm{A}\) is higher than that of bone \(\mathrm{B}\), then bone \(\mathrm{A}\) is (1) older than, (2) younger than, (3) the same age as bone B. Explain your reasoning. (b) A sample of bone \(\mathrm{A}\) is found to have 4.0 beta decays/min per gram of carbon, while bone \(\mathrm{B}^{\prime} \mathrm{s}\) activity is only 1.0 beta decay \(/\) min for each gram of carbon. What is the age difference between the two bones?

Prove that the number \(N\) of radioactive nuclei remaining in a sample after an integer number \((n)\) of half-lives has elapsed is \(N=\frac{N_{o}}{2^{n}}=\left(\frac{1}{2}\right)^{n} N_{0} .\) Here \(N_{0}\) stands for the initial number of nuclei.

(a) What is the decay constant of fluorine- 17 if its half-life is known to be \(66.0 \mathrm{~s} ?\) (b) How long will it take for the activity of a sample of \(17 \mathrm{~F}\) to decrease to \(80 \%\) of its initial value? (c) Repeat part (b), but instead determine the time to decrease to an additional \(20 \%\) to \(60 \%\) of its initial value. Does it take twice as long to decay to \(60 \%\) compared to \(80 \%\) of its initial activity? Explain.

Francium- \(223\left(\begin{array}{c}223 \\ 87\end{array} \mathrm{Fr}\right)\) has a half-life of \(21.8 \mathrm{~min}\). (a) How many nuclei are initially present in a \(25.0-\mathrm{mg}\) sample of \({ }^{223} \mathrm{Fr} ?\) (b) What is its initial activity? (c) How many nuclei will be present \(1 \mathrm{~h}\) and 49 min later? (d) What will be the sample's activity at this later time?

A basement room containing radon gas \(\left(t_{1 / 2}=3.82\right.\) days \()\) is sealed to be airtight. (a) If \(7.50 \times 10^{10}\) radon atoms are trapped in the room, estimate how many radon atoms remain in the room after one week. (b) Radon undergoes alpha decay. After 30 days, is the number of its daughter nuclei equal to, or less than, the number of radon parents that have decayed? Explain your reasoning.

Nitrogen-13, with a half-life of \(10.0 \mathrm{~min}\), decays by beta emission. (a) Write down the decay equation to determine the daughter product and whether the beta particle is a positron or electron. (b) If a sample of pure \({ }^{13} \mathrm{~N}\) has a mass of \(1.50 \mathrm{~g}\) at a certain time, what is the activity 35.0 min later? (c) What percentage of the sample is \({ }^{13} \mathrm{~N}\) at this time? (d) What alternative process could have happened to the nitrogen-13? Write down its decay equation and determine the daughter product for this process.

Which one of each of the following pairs of nuclei would you expect it to be easier to remove a neutron from: (a) \({ }^{16} \mathrm{O}\) or \({ }^{17} \mathrm{O} ;\) (b) \({ }_{20}^{40} \mathrm{Ca}\) or \({ }_{20}^{42} \mathrm{Ca}\) ; (c) \({ }^{10} \mathrm{~B}\) or \(\frac{11}{5} \mathrm{~B}\) (d) \({ }^{208} \mathrm{~Pb}\) or \({ }_{83}^{209}\) Bi? State your reasoning for your choice in each case.

Only two isotopes of \(\mathrm{Sb}\) (antimony, \(Z=51\) ) are stable. Pick the two most likely stable isotopes from the follow- (b) \({ }^{121} \mathrm{Sb}\) ing list and explain your rationale: (a) \({ }^{120} \mathrm{Sb}\), (c) \(^{122} \mathrm{Sb} ;\) (d) \({ }^{123} \mathrm{Sb} ;\) (e) \({ }^{124} \mathrm{Sb}\)

Suppose an alpha particle could be removed intact from an aluminum- 27 nucleus \((m=26.981541 \mathrm{u})\) (a) Write the equation that represents this process and determine the daughter nuclide. (b) If the daughter nuclide has mass of \(22.989770 \mathrm{u}\), how much energy would be required to perform this operation?

In a diagnostic procedure, a patient in a hospital ingests \(80 \mathrm{mCi}\) of gold- \(198\left(t_{1 / 2}=2.7\right.\) days \() .\) What is the activity at the end of one month, assuming none of the gold is eliminated from the body by biological functions?

Neutron activation analysis was performed on small pieces of hair that had been taken from the exiled Napoleon after he died on the island of St. Helena in 1821\. This procedure involves exposing the samples to a source of neutrons. Some (stable) arsenic nuclei, if present in the sample, will absorb a neutron. In Napoleon's case the samples did contain abnormally high levels of arsenic, which supported the theory that his death was not due to natural causes. (a) These results came from studying beta emissions of the resulting \({ }^{76} \mathrm{As},\) nucleus. Write the nuclear equation for the neutron absorption and use it to determine the arsenic isotope initially present in the hair. (b) Write the nuclear equation for the subsequent beta decay of \({ }^{76} \mathrm{As}\). Use it to determine the nucleus after this decay.

A cancer treatment called the gamma knife (see Insight 29.1, Biological and Medical Application of Radiation) uses focused \({ }^{60}\) Co sources to treat tumors. Each \({ }^{60}\) Co nucleus emits two gamma rays, of energy \(1.33 \mathrm{MeV}\) and \(1.17 \mathrm{MeV}\), in quick succession. Assume that \(50.0 \%\) of the total gamma-ray energy is absorbed by a tumor. Further assume that the total activity of the \({ }^{60}\) Co sources is \(1.00 \mathrm{mCi}\), the tumor's mass is \(0.100 \mathrm{~kg}\), and the patient is exposed to the gamma radiation for an hour. Determine the effective radiation dose received by the tumor. (Since the \({ }^{60}\) Co half-life is 5.3 years, changes in its activity during treatment are negligible.)

The radioactive source in most smoke detectors is \(241 \mathrm{Am},\) which has a half-life of 432 years. In a typical detector, only about \(0.100 \mathrm{mg}\) of this material is needed. (a) Write down its alpha decay equation and predict the product nucleus. (b) What is the initial source activity? (c) What would be the source's activity after 40 years in operation? (d) How many \({ }^{241} \mathrm{Am},\) nuclei would have decaved in this 40 -vear period?

A sample of \({ }^{215} \mathrm{Bi}\), which beta decays \(\left(t_{1 / 2}=2.4 \mathrm{~min}\right)\) initially contains one-hundreth of Avogadro's number of nuclei. (a) What is the sample's mass? (b) Write down the beta decay equation and predict the product nucleus. (c) How many bismuth nuclei are present after \(10 \mathrm{~min} ?\) (d) After \(1.0 \mathrm{~h} ?\) (d) What are the activities, in curies and becquerels, at these times?

High-energy gamma ray photons can remove nucleons from nuclei in a process called the photonuclear effect. This is a process analogous to the photoelectric effect in atoms (see Section 27.2). (a) If you wanted to remove a single neutron from a nucleus, which of the following nuclei would likely require the higher energy photon: (1) \({ }^{12} \mathrm{C} ;\) (2) \({ }^{13} \mathrm{C} ;\) or (3) the energies would be about the same? Explain your reasoning. (b) Calculate the minimum energy of a photon required to eject a neutron from each of the two isotopes in part (a), neglecting any kinetic energy of the neutron or resulting nucleus. (c) Determine the wavelengths of the light associated (d) Explain why the actual with the photons in part (b). minimum energy in part (b) is higher than your result. [Hint: The initial photon contains linear momentum that must be conserved.]

The experimental expression for the (approximate) radius \((R)\) of a nucleus is \(R=R_{\mathrm{o}} A^{1 / 3},\) where \(R_{\mathrm{o}}=1.2 \times 10^{-15} \mathrm{~m}\) and \(A\) is the mass number of the nucleus. Assuming that nuclei are spherical (they are approximately so in many cases), (a) determine the average nucleon density in a nucleus in units of nucleons \(/ \mathrm{m}^{3}\) and (b) estimate the nuclear density in \(\mathrm{kg} / \mathrm{m}^{3}\). Are you surprised at the magnitude of your answer? (c) A neutron star is the last phase of evolution for some types of stars. Typically, a neutron star has a diameter of \(15 \mathrm{~km}\) and a mass twice that of our Sun. Determine the average density of a typical neutron star and compare it to your answer to part (b). What can you conclude about the structure of the neutron star and how it got its name?

\({ }_{1}^{3} \mathrm{H}\) (tritium) can be produced in water surrounding a strong source of neutrons, such as that occuring in nuclear reactors. One of the ways tritium can form is via neutron capture by deuterium. (a) Write down the equation for this capture reaction. (b) Tritium has a half-life of 12.33 years. What percentage of a sample containing \({ }_{1}^{3} \mathrm{H}\) will remain after exactly 6 years? (c) Determine the gamma-ray energy emitted during the capture (assuming the tritium ends up in its ground state and the incoming neutron kinetic energy is negligible). (d) Write down the reaction for the subsequent beta decay of the tritium and determine the stable daughter identity. (e) If all the energy released in the beta decay went into the beta particle, determine its energy.