Chapter 43

How many protons and how many neutrons are there in a nucleus of the most common isotope of (a) silicon, \(_{14}^{28} \mathrm{Si} ;\) (b) rubidium, \(\frac{85}{37} \mathrm{Rb} ;\) (c) thallium, \(_{81}^{205} \mathrm{Tl} ?\)

Hydrogen atoms are placed in an external \(1.65-\mathrm{T}\) magnetic field. (a) The protons can make transitions between states where the nuclear spin component is parallel and antiparallel to the field by absorbing or emitting a photon. Which state has lower energy: the state with the nuclear spin component parallel or antiparallel to the field? What are the frequency and wavelength of the photon? In which region of the electromagnetic spectrum does it lie? (b) The electrons can make transitions between states where the electron spin component is parallel and antiparallel to the field by absorbing or emitting a photon. Which state has lower energy: the state with the electron spin component parallel or antiparallel to the field? What are the frequency and wavelength of the photon? In which region of the electromagnetic spectrum does it lie?

Hydrogen atoms are placed in an external magnetic field. The protons can make transitions between states in which the nuclear spin component is parallel and antiparallel to the field by absorbing or emitting a photon. What magnetic- field magnitude is required for this transition to be induced by photons with frequency 22.7 \(\mathrm{MHz} ?\)

What nuclide is produced in the following radioactive decays? (a) \(\alpha\) decay of \(^{239} \mathrm{Pu} ;\) (b) \(\beta^{-}\) decay of \(_{11}^{24} \mathrm{Na} ;(\mathrm{c}) \beta^{+}\) decay of \(_{8}^{15} \mathrm{O}\)

What particle \((\alpha\) particle, electron, or positron) is emitted in the following radioactive decays? (a) \(_{14}^{27} \mathrm{Si} \rightarrow_{13}^{27} \mathrm{Al}\) (b) \(^{238} \mathrm{U} \rightarrow_{90}^{234} \mathrm{Th} ;\) (c) \(_{33}^{74} \mathrm{As} \rightarrow_{34}^{74} \mathrm{Se}\)

Tritium (\(^{3}_{1}\)H) is an unstable isotope of hydrogen; its mass, including one electron, is 3.016049 u. (a) Show that tritium must be unstable with respect to beta decay because the decay products ( \(^{3}_{2}\)He plus an emitted electron ) have less total mass than the tritium. \right. (b) Determine the total kinetic energy (in MeV) of the decay products, taking care to account for the electron masses correctly.

BIO 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 \(\mathrm{Ci} .\) When its activity falls below \(3500 \mathrm{Ci},\) 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 \(6,2004\) . It is now April \(6,2007\) . Is the source still usable? The half-life of \(^{60} \mathrm{Co}\) is 5.271 years.

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?

BIO Radioactive Tracers. Radioactive isotopes are often introduced into the body through the bloodstream. Their spread through the body can then be monitored by detecting the appearance of radiation in different organs. \(^{131} \mathrm{I},\) 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 d (about 3\(\frac{1}{2}\) weeks) later? (b) If the decay rate in the thyroid 24 d later is actually measured to be 17.0 Bq, what percentage of the tracer went to that gland? (c) What isotope remains after the I-131 decays?

As a health physicist, you are being consulted about a spill in a radiochemistry lab. The isotope spilled was 500\(\mu C\) of \(^{131} \mathrm{Ba}\), which has a half-life of 12 days. (a) What mass of \(^{131} \mathrm{Ba}\) was spilled? (b) Your recommendation is to clear the lab until the radiation level has fallen 1.00\(\mu\) Ci. How long will the lab have to be closed?

Measurements on a certain isotope tell you that the decay rate decreases from 8318 decays/min to 3091 decays/min in 4.00 days. What is the half-life of this isotope?

The isotope \(^{226} \mathrm{Ra}\) undergoes \(\alpha\) decay with a half-life of 1620 years. What is the activity of 1.00 \(\mathrm{g}\) of \(^{226} \mathrm{Ra}\) ? Express your answer in \(\mathrm{Bq}\) and in \(\mathrm{Ci}\) .

The radioactive nuclide \(^{199} \mathrm{Pt}\) has a half-life of 30.8 minutes. A sample is prepared that has an initial activity of \(7.56 \times 10^{11} \mathrm{Bq}\). (a) How many \(^{199} \mathrm{Pt}\) nuclei are initially present in the sample? (b) How many are present after 30.8 minutes? What is the activity at this time? (c) Repeat part (b) for a time 92.4 minutes after the sample is first prepared.

BIO (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 \(\mathrm{RBE}\) 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 xays 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)?

BIO A person exposed to fast neutrons receives a radiation dose of 200 rem on part of his hand, affecting 25 g of tissue. The RBE of these neutrons is 10. (a) How many rad did he receive? (b) How many joules of energy did this person receive? (c) Suppose the person received the same rad dosage, but from beta rays with an RBE of 1.0 instead of neutrons. How many rem would he have received?

BIO A nuclear chemist receives an accidental radiation dose of 5.0 Gy from slow neutrons (RBE = 4.0). What does she receive in rad, rem, and J/kg?

BIO 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 mSv to only 5.0 kg of tissue. How many chest \(x\) rays would deliver the same total amount of energy to the body of a 75 -kg person as one whole-body scan?

BIO In an industrial accident a 65-kg person receives a lethal whole-body equivalent dose of 5.4 \(\mathrm{Sv}\) 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 kg of water 0.010 \(\mathrm{C}^{\circ}\) ?

BIO A 67 -kg person accidentally ingests 0.35 Ci of tritium. (a) Assume that the tritium spreads uniformly throughout the body and that each decay leads on the average to the absorption of 5.0 \(\mathrm{keV}\) of energy from the electrons emitted in the decay. The half-life of tritium is \(12.3 \mathrm{y},\) and the RBE of the electrons is \(1.0 .\) Calculate the absorbed dose in rad and the equivalent dose in rem during one week. (b) The \(\beta^{-}\) decay of tritium releases more than 5.0 keV of energy. Why is the average energy absorbed less than the total energy released in the decay?

The United States uses \(1.0 \times 10^{20} \mathrm{J}\) 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 and (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 \(^{235} \mathrm{U} .\) )

Consider the nuclear reaction $$ _{14}^{28} \mathrm{Si}+\gamma \rightarrow_{12}^{24} \mathrm{Mg}+\mathrm{X} $$ where \(X\) is a nuclide. (a) What are \(Z\) and \(A\) for the nuclide \(X\) ? (b) Ignoring the effects of recoil, what minimum energy must the photon have for this reaction to occur? The mass of a \(_{14}^{28}\) Si atom is 27.976927 \( \mathrm{u},\) and the mass of a \(^{24}_{12} \mathrm{Mg}\) atom is 23.985042 \(\mathrm{u}\)

Comparison of Energy Released per Gram of Fuel. (a) When gasoline is burned, it releases \(1.3 \times 10^{8} \mathrm{J}\) of energy per gallon \((3.788 \mathrm{L}) .\) 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 \(\mathrm{J} / \mathrm{g}\) 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 measured rate of \(3.86 \times 10^{26} \mathrm{W}\) . 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.)

Use conservation of mass-energy to show that the energy released in alpha decay is positive whenever the mass of the original neutral atom is greater than the sum of the masses of the final neutral atom and the neutral \(^{4}\) He atom. (Hint: Let the parent nucleus have atomic number \(Z\) and nucleon number \(A .\) First write the reaction in terms of the nuclei and particles involved, and then add \(Z\) electron masses to both sides of the reaction and allot them as needed to arrive at neutral atoms.)

Use conservation of mass-energy to show that the energy released in \(\beta^{+}\) decay is positive whenever the neutral atomic mass of the original atom is at least two electron masses greater than that of the final atom. (See the hint in Problem \(43.49 .\))

BIO 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 percentage of the original level of \(^{90}\) 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 \(_{12}^{25} \mathrm{Mg}\) is 24.985837 \(\mathrm{u},\) and the atomic mass of \(^{25}_{13}\) Al is 24.990429 u. (a) Which of these nuclei will decay into the other? (b) What type of decay will occur? Explain how you determined this. (c) How much energy (in MeV) is released in the decay?

Gold, \(_{79}^{198} \mathrm{Au}\), undergoes \(\beta^{-}\) decay to an excited state of \(^{198}_{80} \mathrm{Hg}\). If the excited state decays by emission of a photon with energy 0.412 MeV, what is the maximum kinetic energy of the electron emitted in the decay? This maximum occurs when the antineutrino has negligible energy. (The recoil energy of the \(^{198}_{80} \mathrm{Hg}\) nucleus can be ignored. The masses of the neutral atoms in their ground states are 197.968225 u for \(^{198}_{80} \mathrm {Au}\) and 197.966752 u for \(\frac{198}{80} \mathrm{Hg}_{\cdot} \))

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 \(no\) 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 \(^{99} \mathrm{Tc}\) 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."

BIO A 70.0-kg person experiences a whole-body exposure to \(\alpha\) radiation with energy 4.77 MeV. A total of \(6.25 \times 10^{12} \alpha\) particles are absorbed. (a) What is the absorbed dose in rad? (b) What is the equivalent dose in rem? (c) If the source is 0.0320 \(\mathrm{g}\) of \(^{226} \mathrm{Ra}\) (half-life 1600 \(\mathrm{y}\)) somewhere in the body, what is the activity of this source? (d) If all the alpha particles produced are absorbed, what time is required for this dose to be delivered?

Measurements indicate that 27.83\(\%\) of all rubidium atoms currently on the earth are the radioactive \(^{87} \mathrm{Rb}\) isotope. The rest are the stable \(^{87} \mathrm{Rb}\) isotope. The half-life of \(^{87} \mathrm{Rb}\) is \(4.75 \times 10^{10} \mathrm{y}\) . Assuming that no rubidium atoms have been formed since, what percentage of rubidium atoms were \(^{87} \mathrm{Rb}\) when our solar system was formed \(4.6 \times 10^{9} \mathrm{y}\) ago?

A \(^{186}_{76} \mathrm{Os}\) nucleus at rest decays by the emission of a 2.76 -MeV \(\alpha\) particle. Calculate the atomic mass of the daughter nuclide produced by this decay, assuming that it is produced in its ground state. The atomic mass of \(^{186}_{76} \mathrm{Os}\) is 185.953838 \(\mathrm{u}\)

\(\mathrm{A}^{60} \mathrm{Co}\) source with activity \(2.6 \times 10^{-4} \mathrm{Ci}\) is embedded in a tumor that has mass 0.200 \(\mathrm{kg} .\) The source emits \(\gamma\) photons with average energy 1.25 \(\mathrm{MeV} .\) 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\(?\) (d) What exposure time is required for an equivalent dose of 200 rem?

The nucleus \(^{15}_{8} \mathrm{O}\) has a half-life of 122.2 \(\mathrm{s}\); \(^{19}_{8} \mathrm{O}\) has a half-life of 26.9 s. If at some time a sample contains equal amounts of \(^{15}_{8} \mathrm{O}\) and \(^{19}_{8} \mathrm{O}\), what is the ratio of \(^{15}_{8} \mathrm{O}\) to \(^{19}_{8} \mathrm{O}\) (a) after 4.0 minutes and (b) after 15.0 minutes?

An Oceanographic Tracer. Nuclear weapons tests in the 1950 s and 1960 s released significant amounts of radioactive tritium \((^{3}_{1} \mathrm{H},\) 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, \(^{3}_{2} \mathrm{He},\) to the remaining tritium in the water. For example, if the ratio of \(_{2}^{3} \mathrm{He}\) 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 \(_{2}^{3}\) He to \(_{1}^{3} \mathrm{H}\) is 4.3 to 1.0. How many years ago did this water sink below the surface?

In the 1986 disaster at the Chernobyl reactor in the Soviet Union (now Ukraine), about \(\frac{1}{8}\) of the \(^{137} \mathrm{Cs}\) present in the reactor was released. The isotope \(^{137} \mathrm{Cs}\) has a half-life for \(\beta\) decay of 30.07 \(\mathrm{y}\) and decays with the emission of a total of 1.17 \(\mathrm{MeV}\) of energy per decay. Of this, 0.51 \(\mathrm{MeV}\) goes to the emitted electron and the remaining 0.66 \(\mathrm{MeV}\) to a \(\gamma\) ray. The radioactive \(^{137} \mathrm{Cs}\) is absorbed by plants, which are eaten by livestock and humans. How many \(^{137} \mathrm{Cs}\) atoms would need to be present in each kilogram of body tissue if an equivalent dose for one week is 3.5 \(\mathrm{Sv}\) ? Assume that all of the energy from the decay is deposited in that 1.0 \(\mathrm{kg}\) of tissue and that the RBE of the electrons is 1.5.

(a) Prove that when a particle with mass \(m\) and kinetic energy \(K\) collides with a stationary particle with mass \(M,\) the total kinetic energy \(K_{\mathrm{cm}}\) in the center-of-mass coordinate system (the energy available to cause reactions) is $$K_{\mathrm{cm}}=\frac{M}{M+m} K$$ Assume that the kinetic energies of the particles and nuclei are much lower than their rest energies. (b) If \(K_{\text { th }}\) is the minimum, or threshold, kinetic energy to cause an endoergic reaction to occur in the situation of part (a), show that $$K_{\mathrm{th}}=-\frac{M+m}{M} Q$$

Calculate the energy released in the fission reaction \(_{92}^{235} \mathrm{U}+_{0}^{1} \mathrm{n} \rightarrow ^{140}_{54} \mathrm{Xe}+ ^{94}_{38} \mathrm{Sr}+2_{0}^{1} \mathrm{n}\) You can ignore the initial kinetic energy of the absorbed neutron. The atomic masses are \(_{92}^{235} \mathrm{U},\) 235.04392 \(\mathrm{u};\) \(^{140}_{54} \mathrm{Xe},\) 139.921636 \(\mathrm{u};\) and \(^{94}_{38} \mathrm{Sr},\) 93.915360 \(\mathrm{u}.\)

Industrial Radioactivity. Radioisotopes are used in a variety of manufacturing and testing techniques. Wear measurements can be made using the following method. An automobile engine is produced using piston rings with a total mass of \(100 \mathrm{g},\) which includes 9.4\(\mu \mathrm{Ci}\) of \(^{59} \mathrm{Fe}\) whose half-life is 45 days. The engine is test-run for 1000 hours, after which the oil is drained and its activity is measured. If the activity of the engine oil is 84 decays/s, how much mass worn from the piston rings per hour of operation?