Chapter 40

Radium- 226 decays by emitting an alpha particle. What is the daughter nucleus? a) \(\mathrm{Rd}\) b) \(\mathrm{Rn}\) c) Bi d) \(\mathrm{Pb}\)

Which of the following decay modes is due to a transition between states of the same nucleus? a) alpha decay c) gamma decay b) beta decay d) none of the above

In neutron stars, which are roughly \(90 \%\) neutrons and supported almost entirely by nuclear forces, which of the following binding-energy terms becomes relatively dominant compared to ordinary nuclei? a) the Coulomb term b) the asymmetry term c) the pairing term d) all of the above e) none of the above

When a target nucleus is bombarded by an appropriate beam of particles, it is possible to produce a) a less massive nucleus, but not a more massive one. b) a more massive nucleus, but not a less massive one. c) a nucleus with smaller charge number, but not one with a greater charge number. d) a nucleus with greater charge number, but not one with a smaller charge number. e) a nucleus with either greater or smaller charge number.

The strong force (select all that apply.) a) is only attractive. b) does not act on electrons. c) only acts over a few \(\mathrm{fm}\). d) All of the above are true. e) None of the above are true.

Cobalt has a stable isotope, \({ }^{59} \mathrm{Co},\) and 22 radioactive isotopes. The most stable radioactive isotope is \({ }^{60} \mathrm{Co} .\) What is the dominant decay mode of this \({ }^{60}\) Co isotope? a) \(\beta^{+}\) b) \(\beta^{-}\) c) \(\mathrm{P}\) d) \(n\)

The mass of an atom (atomic mass) is equal to a) the sum of the masses of the protons. b) the sum of the masses of protons and neutrons. c) the sum of the masses of protons, neutrons and electrons. d) the sum of the masses of protons, neutrons, and electrons minus the atom's binding energy.

What is more dangerous, a radioactive material with a short half-life or a long one?

Apart from fatigue, what is another reason the Federal Aviation Administration limits the number of hours intercontinental pilots can travel annually?

The binding energy of \({ }_{2}^{3}\) He is lower than that of \({ }_{1}^{3} \mathrm{H} .\) Provide a plausible explanation, considering the Coulomb interaction between two protons in \({ }_{2}^{3}\) He.

Which of the following quantities is conserved during a nuclear reaction, and how? a) charge d) linear momentum b) the number of nucleons, A e) angular momentum c) mass-energy

Some food is treated with gamma radiation to kill bacteria. Why is there not a concern that people who eat such food might be consuming food containing gamma radiation?

Refer to the subsection "Terrestrial Fusion" in Section 40.4 to see how achieving controlled fusion would be the solution to mankind's energy problems, and how difficult it is to do. Why is it so hard? The Sun does it all the time (see the previous subsection, "Stellar Fusion"). Do we need to understand better how the Sun works to build a useful nuclear fusion reactor?

Why are atomic nuclei more or less limited in size and neutron-proton ratios? That is, why are there no stable nuclei with 10 times as many neutrons as protons, and why are there no atomic nuclei the size of marbles?

A nuclear reaction of the kind \({ }_{2}^{3} \mathrm{He}+{ }_{6}^{12} \mathrm{C} \rightarrow \mathrm{X}+\alpha\) is called a pick-up nuclear reaction. a) Why is it called a pick-up reaction, that is, what is picked up, what picked it up, and where did it come from? b) What is the resulting nucleus X? c) What is the \(\mathrm{Q}\) -value of this reaction? d) Is this reaction endothermic or exothermic?

Before you look it up, make a prediction of the spin (intrinsic spin, i.e., actual angular momentum) of the deuteron, \({ }^{2} \mathrm{H}\). Explain your reasoning.

\(^{39} \mathrm{Ar}\) is an isotope with a half-life of \(269 \mathrm{yr}\). If it decays through beta-minus emission, what isotope will result?

A neutron star is essentially a gigantic nucleus with mass 1.35 times that of the Sun, or mass number of order \(10^{57} .\) It consists of approximately \(99 \%\) neutrons, the rest being protons and an equal number of electrons. Explain the physics that determines these features.

What is the nuclear configuration of the daughter nucleus associated with the alpha decay of \(\mathrm{Hf}(A=157,\) \(Z=72\) )?

Using the table of isotopes in Appendix B, calculate the binding energies of the following nuclei. a) \({ }^{7} \mathrm{Li}\) b) \({ }^{12} \mathrm{C}\) c) \({ }^{56} \mathrm{Fe}\) d) \({ }^{85} \mathrm{Rb}\)

According to the standard notation, find the number of protons, nucleons, neutrons, and electrons of \({ }_{54}^{134}\) Xe.

Calculate the binding energy for the following two uranium isotopes: a) \({ }_{92}^{238} \mathrm{U},\) which consists of 92 protons, 92 electrons, and 146 neutrons, with a total mass of \(238.0507826 \mathrm{u}\). b) \({ }^{235} \mathrm{U},\) which consists of 92 protons, 92 electrons, and 143 neutrons, with a total mass of \(235.0439299 \mathrm{u} .\) The atomic mass unit \(\mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg} .\) Which isotope is more stable (or less unstable)?

Write down equations to describe the \(\beta^{-}\) -decay of the following atoms: a) \({ }^{60} \mathrm{Co}\) b) \({ }^{3} \mathrm{H}\) c) \({ }^{14} \mathrm{C}\)

Write down equations to describe the alpha decay of the following atoms: a) \(^{212} \mathrm{Rn}\) b) \({ }^{241} \mathrm{Am}\)

A certain radioactive isotope decays to one-eighth its original amount in \(5.0 \mathrm{~h}\). a) What is its half-life? b) What is its mean lifetime?

A certain radioactive isotope decays to one-eighth its original amount in \(5.00 \mathrm{~h} .\) How long would it take for \(10.0 \%\) of it to decay?

Determine the decay constant of radium- 226 , which has a half-life of \(1600 \mathrm{yr}\).

The half-life of a sample of \(10^{11}\) atoms that decay by alpha emission is \(10 \mathrm{~min} .\) How many alpha particles are emitted between the time interval 100 min and 200 min?

The specific activity of a radioactive material is the number of disintegrations per second per gram of radioactive atoms. a) Given the half-life of \({ }^{14} \mathrm{C}\) of \(5730 \mathrm{yr}\), calculate the specific activity of \({ }^{14} \mathrm{C}\). Express your result in disintegrations per second per gram, becquerel per gram, and curie per gram. b) Calculate the initial activity of a \(5.00-\mathrm{g}\) piece of wood. c) How many \({ }^{14} \mathrm{C}\) disintegrations have occurred in a \(5.00-\mathrm{g}\) piece of wood that was cut from a tree January \(1,1700 ?\)

In a simple case of chain radioactive decay, a parent radioactive species of nuclei, A, decays with a decay constant \(\lambda_{1}\) into a daughter radioactive species of nuclei, B, which then decays with a decay constant \(\lambda_{2}\) to a stable element C. a) Write the equations describing the number of nuclei in each of the three species as a function of time, and derive an expression for the number of daughter nuclei, \(N_{2}\), as a function of time, and for the activity of the daughter nuclei, \(A_{2},\) as a function of time. b) Discuss the results in the case when \(\lambda_{2}>\lambda_{1}\left(\lambda_{2} \approx 10 \lambda_{1}\right)\) and when \(\lambda_{2}>>\lambda_{1}\left(\lambda_{2} \approx 100 \lambda_{1}\right)\).

Consider the Bethe-Weizsäcker formula for the case of odd \(A\) nuclei. Show that the formula can be written as a quadratic in \(Z\) -and thus, that for any given \(A\), the binding energies of the isotopes having that \(A\) take a quadratic form, \(B=a+b Z+c Z^{2} .\) Find the most deeply bound isotope (the most stable one) having \(A=117\) using your result.

a) What is the energy released in the fusion reaction \({ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \rightarrow{ }_{2}^{4} \mathrm{He}+Q ?\) b) The oceans have a total mass of water of \(1.50 \cdot 10^{16} \mathrm{~kg}\), and \(0.0300 \%\) of this quantity is deuterium, \({ }_{1}^{2} \mathrm{H} .\) If all the deuterium in the oceans were fused by controlled fusion into \({ }_{2}^{4} \mathrm{He},\) how many joules of energy would be released? c) World power consumption is about \(1.00 \cdot 10^{13} \mathrm{~W}\). If consumption were to stay constant and all problems arising from ocean water consumption (including those of political, meteorological, and ecological nature) could be avoided, how many years would the energy calculated in part (b) last?

The Sun radiates energy at the rate of \(3.85 \cdot 10^{26} \mathrm{~W}\) a) At what rate, in \(\mathrm{kg} / \mathrm{s}\), is the Sun's mass converted into energy? b) Why is this result different from the rate calculated in Example \(40.6,6.02 \cdot 10^{11}\) kg protons being converted into helium each second? c) Assuming that the current mass of the Sun is \(1.99 \cdot 10^{30} \mathrm{~kg}\) and that it radiated at the same rate for its entire lifetime of \(4.50 \cdot 10^{9} \mathrm{yr}\), what percentage of the Sun's mass was converted into energy during its entire lifetime?

Consider the following fusion reaction, which allows stars to produce progressively heavier elements: \({ }_{2}^{3} \mathrm{He}+{ }_{2}^{4} \mathrm{He} \rightarrow{ }_{4}^{7} \mathrm{Be}+\gamma\). The mass of \({ }_{2}^{3} \mathrm{He}\) is \(3.016029 \mathrm{u}\), the mass of \({ }_{2}^{4}\) He is \(4.002603 \mathrm{u}\), and the mass of \({ }_{4}^{7} \mathrm{Be}\) is \(7.0169298 \mathrm{u}\). The atomic mass unit is \(u=1.66 \cdot 10^{-27} \mathrm{~kg} .\) Assuming the Be atom is at rest after the reaction and neglecting any potential energy between the atoms and kinetic energy of the He nuclei, calculate the minimum possible energy and maximum possible wavelength of the photon \(\gamma\) that is released in this reaction.

What is the average kinetic energy of protons at the center of a star where the temperature is \(1.00 \cdot 10^{7} \mathrm{~K} ?\) What is the average velocity of those protons?

Billions of years ago, our Solar System was created out of the remnants of exploding stars. Nuclear scientists believe that two isotopes of uranium, \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U},\) were created in equal amounts at the time of a stellar explosion. However, today \(99.28 \%\) of uranium is in the form of \({ }^{238} \mathrm{U}\) and only \(0.72 \%\) is in the form of \({ }^{235} \mathrm{U}\). Assuming a simplified model in which all of the matter in the Solar System originated in a single exploding star, estimate the approximate time of this explosion.

A drug containing \({ }_{43}^{99} \mathrm{Tc}\left(t_{1 / 2}=6.05 \mathrm{~h}\right)\) with an activity of \(1.50 \mu \mathrm{Ci}\) is to be injected into a patient at \(9.30 \mathrm{a} . \mathrm{m} .\) You are to prepare the sample \(2.50 \mathrm{~h}\) before the injection (at 7: 00 a.m.). What activity should the drug have at the preparation time (7:00 a.m.)?

Consider a 42.58 -MHz photon needed to produce NMR transition in free protons in a magnetic field of \(1.000 \mathrm{~T}\). What is the wavelength of the photon, its energy, and the region of the spectrum in which it lies? Could it be harmful to the human body?

The radon isotope \({ }^{222} \mathrm{Rn}\), which has a half-life of 3.825 days, is used for medical purposes such as radiotherapy. How long does it take until \({ }^{222} \mathrm{Rn}\) decays to \(10.00 \%\) of its initial quantity?

Radiation therapy is one of the modalities for cancer treatment. Based on the approximate mass of a tumor, oncologists can calculate the radiation dose necessary to treat their patients. Suppose a patient has a 50.0 -g tumor and needs to receive 0.180 J of energy to kill the cancer cells. What rad (radiation absorbed dose) should the patient receive?

How close can a \(5.00-\mathrm{MeV}\) alpha particle get to a uranium- 238 nucleus, assuming the only interaction is Coulomb?

Pu decays with a half-life of 24,100 yr via a 5.25 \(\mathrm{MeV}\) alpha particle. If you have a \(1.00 \mathrm{~kg}\) spherical sample of \({ }^{239} \mathrm{Pu},\) find the initial activity in \(\mathrm{Bq} .\)

\(^{8} \mathrm{Li}\) is an isotope that has a lifetime of less than one second. Its mass is \(8.022485 \mathrm{u} .\) Calculate its binding energy in \(\mathrm{MeV}\).

What is the total energy released in the decay \(n \rightarrow p+e^{-}+\bar{\nu}_{e} ?\)

\(10^{30}\) Atoms of a radioactive sample remain after 10 half-lives. How many atoms remain after 20 half-lives?

Calculate the binding energy per nucleon of a) \({ }_{2}^{4} \mathrm{He}(4.002603 \mathrm{u})\). c) \({ }_{1}^{3} \mathrm{H}(3.016050 \mathrm{u})\) b) \({ }_{2}^{3} \mathrm{He}(3.016030 \mathrm{u}) .\) d) \({ }_{1}^{2} \mathrm{H}(2.014102 \mathrm{u})\).

The mean lifetime for a radioactive nucleus is \(4300 \mathrm{~s}\) What is its half-life?

\(^{214} \mathrm{Pb}\) has a half-life of \(26.8 \mathrm{~min}\). How many minutes must elapse for \(90.0 \%\) of a given sample of \({ }^{214} \mathrm{~Pb}\) atoms to decay?

You have developed a grand unified theory which predicts the following things about the decay of the proton: (1) protons never get any older, in the sense that their probability of decay per unit time never changes, and (2) half the protons in any given collection of protons will have decayed in \(1.80 \cdot 10^{29}\) yr. You are given experimental facilities to test your theory: A tank containing \(1.00 \cdot 10^{4}\) tons of water and sensors to record proton decays. You will be allowed access to this facility for two years. How many proton decays will occur in this period if your theory is correct?

The precession frequency of the protons in a laboratory NMR spectrometer is \(15.35850 \mathrm{MHz}\). The magnetic moment of the proton is \(1.410608 \cdot 10^{-26} \mathrm{~J} / \mathrm{T}\), while its spin angular momentum is \(0.5272863 \cdot 10^{-34} \mathrm{~J}\) s. Calculate the magnitude of the magnetic field in which the protons are immersed.