Chapter 30

How many protons and how many neutrons are there in a nucleus of (a) neon, \(_{10}^{21} \mathrm{Ne}(\mathrm{b})\) zinc, \(_{30}^{65} \mathrm{Zn}(\mathrm{c})\) silver, \(_{47}^{108} \mathrm{Ag}\)

Density of the nucleus. (a) Using the empirical formula for the radius of a nucleus, show that the volume of a nucleus is directly proportional to its nucleon number \(A .\) (b) Give a reasonable argument concluding that the mass \(m\) of a nucleus of nucleon number \(A\) is approximately \(m=m_{\mathrm{p}} A,\) where \(m_{\mathrm{p}}\) is the mass of a proton. (c) Use the results of parts (a) and (b) to show that all nuclei should have about the same density. Then calculate this density in \(\mathrm{kg} / \mathrm{m}^{3},\) and compare it with the density of lead (which is 11.4 \(\mathrm{g} / \mathrm{cm}^{3} )\) and a neutron star (about \(10^{17} \mathrm{kg} / \mathrm{m}^{3} ) .\)

For the common isotope of nitrogen, \(^{14} \mathrm{N},\) calculate (a) the mass defect, \((\mathrm{b})\) the binding energy, and \((\mathrm{c})\) the binding energy per nucleon.

(a) Calculate the total binding energy (in MeV) of the nuclei of \(^{56} \mathrm{Fe}\) (of atomic mass 55.934937 \(\mathrm{u} )\) and of \(^{207} \mathrm{Pb}\) (of atomic mass 206.975897 u). (b) Calculate the binding energy per nucleon for each of these atoms.(c) How much energy would be needed to totally take apart each of these nuclei? (d) For which of these atoms are the individual nucleons more tightly bound? Explain your reasoning.

Thorium series. The following decays make up the thorium decay series (the \(X\) 's are unknowns for you to identify): $$^{232} \mathrm{Th} \stackrel{\alpha}{\longrightarrow} X_{1}, \quad^{228} \mathrm{Ra} \stackrel{\beta^{-}}{\longrightarrow}^{228} \mathrm{Ac}, \quad X_{2} \stackrel{\beta^{-}}{\longrightarrow}^{228} \mathrm{Th}$$ $$^{228} \mathrm{Th} \stackrel{x_{3}}{\longrightarrow}^{224} \mathrm{Ra}, \quad^{224} \mathrm{Ra} \stackrel{\alpha}{\longrightarrow}^{220} \mathrm{Rn}, \quad^{220} \mathrm{Rn} \stackrel{\alpha}{\longrightarrow} X_{4}$$ \(X_{5} \stackrel{\alpha}{\longrightarrow}^{212} \mathrm{Pb},\) and \(^{212 \mathrm{Pb}} \stackrel{x_{6}}{\longrightarrow}^{212} \mathrm{Bi} .\) The \(^{212} \mathrm{Bi}\) then decays by an \(\alpha\) decay and a \(\beta^{-}\) decay, which can occur in either order \((\alpha\) followed by \(\beta\) or \(\beta\) followed by \(\alpha)\) . (a) Identify each of the six unknowns \((X_{1}, X_{2},\) etc. \()\) by nucleon number, atomic number, neutron number, and name. (b) Write out the decays of \(^{212} \mathrm{Bi}\) and indicate the end product of this series. (For some guidance, see the discussion under "Decay Series" in Section \(30.3 . )(\mathrm{c})\) Draw a Segre chart for the thorium series, similar to the one shown in Figure \(30.5 .\)

Suppose that 8.50 g of a nuclide of mass number 105 decays at a rate of \(6.24 \times 10^{11}\) Bq. What is its half-life? (Hint: Use the fact that \(\Delta N / \Delta t=-\lambda N .\) You are given \(\Delta N / \Delta t\) and can figure out \(N\) knowing the mass number and mass of your sample.)

A radioisotope has a half-life of 5.00 min and an initial decay rate of \(6.00 \times 10^{3}\) Bq. (a) What is the decay constant? (b) What will be the decay rate at the end of (i) 5.00 min, (ii) 10.0 min, (iii) 25.0 min?

A radioactive isotope has a half-life of 76.0 min. A sample is prepared that has an initial activity of \(16.0 \times 10^{10}\) Bq. (a) How many radioactive nuclei are initially present in the sample? (b) How many are present after 76.0 min? What is the activity at this time? (c) Repeat part (b) for a time of 152 min after the sample is first prepared.

Radioactive tracers. Radioactive isotopes are often intro- duced into the body through the bloodstream. Their spread through the body can then be monitored by detecting the appearance of radiation in different organs. \(^{131} \mathrm{I}, \mathrm{a} \beta^{-}\) emitter with a half-life of 8.0 \(\mathrm{d}\) is one such tracer. Suppose a scientist introduces a sample with an activity of 375 \(\mathrm{Bq}\) and watches it spread to the organs. (a) Assuming that the sample all went to the thyroid gland, what will be the decay rate in that gland 24 \(\mathrm{d}\) (about 2\(\frac{1}{2}\) weeks) later? (b) If the decay rate in the thyroid 24 d later is actually measured to be 17.0 \(\mathrm{Bq}\) , what percent of the tracer went to that gland? (c) What isotope remains after the I-131 decays?

The common isotope of uranium, \(^{238} \mathrm{U},\) has a half-life of \(4.47 \times 10^{9}\) years, decaying to \(^{234} \mathrm{Th}\) by alpha emission. (a) What is the decay constant? (b) What mass of uranium is required for an activity of 1.00 curie? (c) How many alpha particles are emitted per second by 10.0 g of uranium?

A sample of the radioactive nuclide \(^{199} \mathrm{Pt}\) is prepared that has an initial activity of \(7.56 \times 10^{11}\) Bq. (a) 92.4 min after the sample is prepared, the activity has fallen to \(9.45 \times 10^{10}\) Bq. What is the half-life of this nuclide? (b) How many radioactive nuclei were initially present in the sample?

We are stardust. In \(1952,\) spectral lines of the element technetium-99 \(\left(^{99} \mathrm{Tc}\right)\) were discovered in a red-giant star. Red giants are very old stars, often around 10 billion years old, and near the end of their lives. Technetium has \(n o\) stable isotopes, and the half life of \(^{99} \mathrm{Tc}\) is \(200,000\) years. (a) For how many half-lives has the \(^{99} \mathrm{Tc}\) been in the red-giant star if its age is 10 billion years? (b) What fraction of the original \(^{99} \mathrm{Tc}\) would be left at the end of that time? This discovery was extremely important because it provided convincing evidence for the theory (now essentially known to be true) that most of the atoms heavier than hydrogen and helium were made inside of stars by thermonuclear fusion and other nuclear processes. If the Tc had been part of the star since it was born, the amount remaining after 10 billion years would have been so minute that it would not have been detectable. This knowledge is what led the late astronomer Carl Sagan to proclaim that "we are stardust."

Radioactive isotopes used in cancer therapy have a "shelf- life," like pharmaceuticals used in chemotherapy. Just after it has been manufactured in a nuclear reactor, the activity of a sample of \(^{60} \mathrm{Co}\) is 5000 Ci. When its activity falls below 3500 Ci, it is considered too weak a source to use in treatment. You work in the radiology department of a large hospital. One of these \(^{60} \mathrm{Co}\) sources in your inventory was manufactured on October \(6,2008 .\) It is now April \(6,2011 .\) Is the source still usable? The half-life of \(^{60} \mathrm{Co}\) is 5.271 years.

A nuclear chemist receives an accidental radiation dose of 5.0 Gy from slow neutrons \((R B E=4.0) .\) What does she receive in rad, rem, and \(J / k g ?\)

(a) If a chest \(x\) ray delivers 0.25 \(\mathrm{mSv}\) to 5.0 \(\mathrm{kg}\) of tissue, how many total joules of energy does this tissue receive? (b) Natural radiation and cosmic rays deliver about 0.10 \(\mathrm{mSv}\) per year at sea level. Assuming an RBE of \(1,\) how many rem and rads is this dose, and how many joules of energy does a 75 kg person receive in a year? (c) How many chest x rays like the one in part (a) would it take to deliver the same total amount of energy to a 75 kg person as she receives from natural radiation in a year at sea level, as described in part (b)?

To scan or not to scan? It has become popular for some people to have yearly whole-body scans (CT scans, formerly called CAT scans), using x rays, just to see if they detect anything suspicious. A number of medical people have recently questioned the advisability of such scans, due in part to the radiation they impart. Typically, one such scan gives a dose of 12 \(\mathrm{mSv}\) , applied to the whole body. By contrast, a chest \(x\) ray typically administers 0.20 \(\mathrm{mSv}\) to only 5.0 \(\mathrm{kg}\) of tissue. How many chest \(\mathrm{x}\) rays would deliver the same total amount of energy to the body of a 75 \(\mathrm{kg}\) person as one whole-body scan?

In an industrial accident a \(65-\mathrm{kg}\) person receives a lethal whole- body equivalent dose of 5.4 Sv from \(\mathrm{x}\) rays. (a) What is the equivalent dose in rem? (b) What is the absorbed dose in rad? (c) What is the total energy absorbed by the person's body? How does this amount of energy compare to the amount of energy required to raise the temperature of 65 \(\mathrm{kg}\) of water \(0.010^{\circ} \mathrm{C} ?\)

In a diagnostic x-ray procedure, \(5.00 \times 10^{10}\) photons are absorbed by tissue with a mass of 0.600 \(\mathrm{kg}\) . The x-ray wavelength is 0.0200 \(\mathrm{nm}\) . (a) What is the total energy absorbed by the tissue? (b) What is the equivalent dose in rem?

Assuming that 200 \(\mathrm{MeV}\) is released per fission, how many fissions per second take place in a 100 \(\mathrm{MW}\) reactor?

The United States uses \(1.0 \times 10^{19} \mathrm{J}\) of electrical energy per year. If all this energy came from the fission of \(^{235} \mathrm{U},\) which releases 200 MeV per fission event, (a) how many kilograms of \(^{235}\mathrm{U}\) would be used per year; (b) how many kilograms of uranium would have to be mined per year to provide that much \(^{235} \mathrm{U} ?\) (Recall that only 0.70\(\%\) of naturally occurring uranium is \(^{235} \mathrm{U} . )\)

At the beginning of Section \(30.6,\) a fission process is illustrated in which \(^{235} \mathrm{U}\) is struck by a neutron and undergoes fission to produce \(^{144} \mathrm{Ba}, 89 \mathrm{Kr}\) , and three neutrons. The measured masses of these isotopes are 235.043930 u \(\left(^{235} \mathrm{U}\right)\) \(143.922953 \mathrm{u}\left(^{144} \mathrm{Ba}\right), 88.917630 \mathrm{u}\left(^{89} \mathrm{Kr}\right),\) and 1.0086649 \(\mathrm{u}\) (neutron). (a) Calculate the energy (in MeV) released by each fission reaction. (b) Calculate the energy released per gram of \(^{235} \mathrm{U},\) in \(\mathrm{MeV} / \mathrm{g} .\)

Comparison of energy released per gram of fuel. (a) When gasoline is burned, it releases \(1.3 \times 10^{8}\) J per gallon \((3.788\) L) of energy. Given that the density of gasoline is \(737 \mathrm{kg} / \mathrm{m}^{3},\) express the quantity of energy released in \(\mathrm{J} / \mathrm{g}\) of fuel. (b) During fission, when a neutron is absorbed by a \(^{235} \mathrm{U}\) nucleus, about 200 \(\mathrm{MeV}\) of energy is released for each nucleus that undergoes fission. Express this quantity in \(\mathrm{J} / \mathrm{g}\) of fuel. (c) In the proton-proton chain that takes place in stars like our sun, the overall fusion reaction can be summarized as six protons fusing to form one 4 He nucleus with two leftover protons and the liberation of 26.7 \(\mathrm{MeV}\) of energy. The fuel is the six protons. Express the energy produced here in units of J/g of fuel. Notice the huge difference between the two forms of nuclear energy, on the one hand, and the chemical energy from gasoline, on the other. (d) Our sun produces energy at a meas- ured rate of \(3.92 \times 10^{26} \mathrm{W}\) . If its mass of \(1.99 \times 10^{30} \mathrm{kg}\) were all gasoline, how long could it last before consuming all its fuel? (Historical note: Before the discovery of nuclear fusion and the vast amounts of energy it releases, scientists were confused. They knew that the earth was at least many millions of years old, but could not explain how the sun could survive that long if its energy came from chemical burning.)

Show that the net result of the proton-proton fusion chain that occurs inside our sun can be summarized as $$6 \mathrm{p}^{+} \rightarrow_{2}^{4} \mathrm{He}+2 \mathrm{p}^{+}+2 \beta^{+}+2 \gamma+2 \nu_{\mathrm{e}}$$

Pair annilation. Consider the case where an electron \(\mathrm{e}^{-}\) and a positron \(\mathrm{e}^{+}\) annihilate each other and produce photons. Assume that these two particles collide head-on with equal, but small, speeds. (a) Show that it is not possible for only one photon to be produced. (Hint: Consider the conservation law that must be true in any collision.) (b) Show that if only two photons are produced, they must travel in opposite directions and have equal energy. (c) Calculate the wavelength of each of the photons in part (b). In what part of the electromagnetic spectrum do they lie?

A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon emitted (a) if the initial kinetic energies of the proton and antiproton are negligible and (b) if each particle has an initial kinetic energy of 830 \(\mathrm{MeV} .\)

Which of the following reactions obey the conservation of baryon number? (a) \(\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{e}^{+},(\mathrm{b}) \mathrm{p}+\mathrm{n} \rightarrow 2 \mathrm{e}^{+}+\mathrm{e}^{-}\)(c) \(\mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{-}+\overline{\nu}_{\mathrm{c}},(\mathrm{d}) \mathrm{p}+\overline{\mathrm{p}} \rightarrow 2 \gamma\)

Determine the electric charge, baryon number, strangeness quantum number, and charm quantum number for the following quark combinations: (a) uus, (b) \(\overline{c s},(\mathrm{c}) \overline{d d u},\) and \((\mathrm{d}) \overline{c} \boldsymbol{b}\)

The critical density of the universe is \(5.8 \times 10^{-27} \mathrm{kg} / \mathrm{m}^{3}\) . (a) Assuming that the universe is all hydrogen, express the critical density in the number of \(\mathrm{H}\) atoms per cubic meter. (b) If the density of the universe is equal to the critical density, how many atoms, on the average, would you expect to find in a room of dimensions 4 \(\mathrm{m} \times 7 \mathrm{m} \times 3 \mathrm{m} ?\) (c) Compare your answer in part (b) with the number of atoms you would find in this room under normal conditions on the earth.

The starship Enterprise, of television and movie fame, is powered by the controlled combination of matter and antimatter. If the entire 400 \(\mathrm{kg}\) antimatter fuel supply of the Enterprise combines with matter, how much energy is released?

A 70.0 kg person experiences a whole-body exposure to alpha radiation with energy of 1.50 MeV. A total of \(5.00 \times 10^{12}\) alpha particles is absorbed. (a) What is the absorbed dose in rad? (b) What is the equivalent dose in rem? (c) If the source is 0.0100 g of \(^{226}\) Ra (half-life 1600 years) somewhere in the body, what is the activity of the source? (d) If all the alpha particles produced are absorbed, what time is required for this dose to be delivered?

\(\mathrm{A}^{60} \mathrm{Co}\) source with activity 15.0 Ci is imbedded in a tumor that has a mass of 0.500 \(\mathrm{kg}\) . The Co source emits gamma-ray photons with average energy of 1.25 MeV. Half the photons are absorbed in the tumor, and half escape. (a) What energy is delivered to the tumor per second? (b) What absorbed dose (in rad) is delivered per second? (c) What equivalent dose (in rem) is delivered per second if the RBE for these gamma rays is 0.70\(?(\mathrm{d})\) What exposure time is required for an equivalent dose of 200 \(\mathrm{rem} ?\)

The nucleus \(_{8}^{15} \mathrm{O}\) has a half-life of 2.0 min. \(_{8}^{19}\mathrm{O}\) has a half-life of about 0.5 min. (a) If, at some, time, a sample contains equal amounts of \(_{8}^{15} \mathrm{O}\) and \(_{8}^{19} \mathrm{O},\) what is the ratio of \(_{8}^{15} \mathrm{O}\) to \(_{8}^{19} \mathrm{O}\) after 2.0 min? (b) After 10.0 min?

Radiation treatment of prostate cancer. In many cases, prostate cancer is treated by implanting 60 to 100 small seeds of radioactive material into the tumor. The energy released from the decays kills the tumor. One isotope that is used (there are others is palladium \((103 \mathrm{Pd}),\) with a half-life of 17 days. If a typical grain contains 0.250 \(\mathrm{g}\) of \(^{103} \mathrm{Pd},\) (a) what is its initial activity rate in \(\mathrm{Bq},\) and (b) what is the rate 68 days later?

An unstable isotope of cobalt, \(^{60} \mathrm{Co}\) , has one more neutron in its nucleus than the stable \(^{59}\mathrm{Co}\) and is a beta emitter with a half-life of 5.3 years. This isotope is widely used in medicine. A certain radiation source in a hospital contains 0.0400 \(\mathrm{g}\) of \(^{60} \mathrm{Co.}\) (a) What is the decay constant for that iso- tope? (b) How many atoms are in the source? (c) How many decays occur per second? (d) What is the activity of the source, in curies?

An oceanographic tracer. Nuclear weapons tests in the 1950 s and 1960 s released significant tritium (\(_{ 1 }^{ 3 }{H},\) half-life 12.3 years) into the atmosphere. The tritium atoms were quickly bound into water molecules and rained out of the air, most of them ending up in the ocean. For any of this tritium-tagged water that sinks below the surface, the amount of time during which it has been isolated from the surface can be calculated by measuring the ratio of the decay product, \(_{2}^{3} \mathrm{He},\) to the remaining tritium in the water. For example, if the ratio of \(\frac{3}{2} \mathrm{He}\) to \(_{1}^{3} \mathrm{H}\) in a sample of water is \(1 : 1,\) the water has been below the surface for one half-life, or approximately 12 years. This method has provided oceanographers with a convenient way to trace the movements of subsurface currents in parts of the ocean. Suppose that in a particular sample of water, the ratio of \(^{3} \mathrm{He}\) to \(^{3} \mathrm{H}\) is 4.3 to \(1.0 .\) How many years ago did this water sink below the surface?

Radioactive fallout. One of the problems of in-air testing of nuclear weapons (or, even worse, the use of such weapons!) is the danger of radioactive fallout. One of the most problematic nuclides in such fallout is strontium-90 \(\left(^{90} \mathrm{Sr}\right),\) which breaks down by \(\beta^{-}\) decay with a half-life of 28 years. It is chemically similar to calcium and therefore can be incorporated into bones and teeth, where, due to its rather long half- life, it remains for years as an internal source of radiation. (a) What is the daughter nucleus of the \(^{90}\) Sr decay? (b) What percent of the original level of 90 \(\mathrm{Sr}\) is left after 56 years? (c) How long would you have to wait for the original level to be reduced to 6.25\(\%\) of its original value?

The atomic mass of \(\frac{56}{26} \mathrm{Fe}\) is 55.934939 \(\mathrm{u}\) , and the atomic mass of mass of \(\frac{56}{27}\) Co is 55.939847 \(\mathrm{u}\) (a) Which of these nuclei will decay into the other? (b) What type of decay will occur? (c) How much kinetic energy will the products of the decay have?

The measured energy width of the \(\phi\) meson is 4.0 \(\mathrm{MeV}\) and its mass is 1020 \(\mathrm{MeV} / c^{2} .\) Using the uncertainty principle (in the form \(\Delta E \Delta t \geq h / 2 \pi ),\) estimate the lifetime of the \(\phi\) meson.

Given that each particle contains only combinations of \(u, d,\) \(s, \overline{u}, \overline{d},\) and \(\overline{s},\) deduce the quark content of (a) a particle with charge \(+e,\) baryon number \(0,\) and strangeness \(+1 ;\) (b) a particle with charge \(+e,\) baryon number \(-1,\) and strangeness \(+1 ;\) (c) a particle with charge \(0,\) baryon number \(+1,\) and strangeness \(-2 .\)

What is the energy of each of the photons resulting from an annihilation event? A. \(\frac{1}{2} m_{\mathrm{e}} v^{2},\) where \(v\) is the speed of the positron. B. \(m_{\mathrm{e}} v^{2}\) C. \(\frac{1}{2} m_{\mathrm{e}} c^{2}\) D. \(m_{\mathrm{e}} c^{2}\)