Solved: A sample contains radioactive atoms of two types,

What is the structure of an atomic nucleus?

Can quantum mechanics explain the nucleus?

Which isotopes are stable?

What is radioactivity?

What is a half-life?

How is nuclear physics useful?

Three electrons orbit a neutral 6 Li atom. How many electrons orbit a neutral 7 Li atom?

What is the binding energy of the 56Fe nucleus?

The isobars corresponding to one specific value of A are found on the plot of Figure 42.4 along a. A vertical line. b. A horizontal line. c. A diagonal line that goes up and to the left. d. A diagonal line that goes up and to the right.

The iodine isotope 131 I, which has an eight-day half-life, is used in nuclear medicine. A sample of 131 I containing 2.00 * 1012 atoms is created in a nuclear reactor. a. How many 131 I atoms remain 36 hours later when the sample is delivered to a hospital? b. The sample is constantly getting weaker, but it remains usable as long as the number of 131 I atoms exceeds 5.0 * 1011 . What is the maximum delay before the sample is no longer usable?

A very bright spotlight shines on a Geiger counter. Does it click?

The isotope 137 Cs is a standard laboratory source of gamma rays. The half-life of 137 Cs is 30 years. a. How many 137 Cs atoms are in a 5.0 mCi source? b. What is the activity of the source 10 years later?

Archeologists excavating an ancient hunters camp have recovered a 5.0 g piece of charcoal from a fireplace. Measurements on the sample find that the 14 C activity is 0.35 Bq. What is the approximate age of the camp?

A sample starts with 1000 radioactive atoms. How many halflives have elapsed when 750 atoms have decayed? a. 0.25 b. 1.5 c. 2.0 d. 2.5

The uranium isotope 238 U undergoes alpha decay to 234 Th. The atomic masses are 238.0508 u for 238 U and 234.0436 u for 234 Th. What is the kinetic energy, in MeV, of the alpha particle?

The cobalt isotope 60 Co 1Z = 272 decays to the nickel isotope 60 Ni 1Z = 282. The decay process is a. Alpha decay. b. Beta-minus decay. c. Beta-plus decay. d. Electron capture. e. Gamma decay.

How much energy is released in the beta-minus decay of 14 C?

A 75 kg laboratory technician working with the radioactive isotope 137 Cs receives an accidental 1.0 mSv exposure. 137 Cs emits 0.66 MeV gamma-ray photons. How many gamma-ray photons are absorbed in the technicians body?

An 85 kg patient swallows a 30 mCi beta emitter that is to be used as a tracer. The isotopes half-life is 5.0 days. The average energy of the beta particles is 0.35 MeV, and they have an RBE (relative biological effectiveness) of 1.5. Ninety percent of the beta particles are stopped inside the patients body and 10% escape. What total dose equivalent does this patient receive?

Consider the atoms 16 O, 18 O, 18 F, 18 Ne, and 20 Ne. Some of the following questions may have more than one answer. Give all answers that apply. a. Which are isotopes? b. Which are isobars? c. Which have the same chemical properties? d. Which have the same number of neutrons?

a. Is the binding energy of a nucleus with A = 200 more than, less than, or equal to the binding energy of a nucleus with A = 60? Explain. b. Is a nucleus with A = 200 more tightly bound, less tightly bound, or bound equally tightly as a nucleus with A = 60? Explain

a. How do we know the strong force exists? b. How do we know the strong force is short range?

Does each nuclear energy-level diagram in FIGURE Q42.4 represent a nuclear ground state, an excited nuclear state, or an impossible nucleus? Explain. (a) Neutrons Protons Neutrons Protons Neutrons Protons (b) (c)

Are the following decays possible? If not, why not? a. 232 Th 1Z = 902 S 236 U1Z = 922 + a b. 238 Pu 1Z = 942 S 236 U1Z = 922 + a c. 11 B1Z = 52 S 11 B1Z = 52 + g d. 33 P1Z = 152 S 32 S1Z = 162 + e

Nucleus A decays into nucleus B with a half-life of 10 s. At t = 0 s, there are 1000 A nuclei and no B nuclei. At what time will there be 750 B nuclei?

What kind of decay, if any, can occur for the nuclei in FIGURE Q42.7?

Apple A in FIGURE Q42.8 is strongly irradiated by nuclear radiation for 1 hour. Apple B is not irradiated. Afterward, in what ways are apples A and B different?

The three isotopes 212 Po, 137 Cs, and 90 Sr decay as 212 Po S 208 Pb + a, 137 Cs S 137 Ba + e- + g, and 90 Sr S 90 Y + e-. Which of these isotopes would be most useful as a biological tracer? Why?

How many protons and how many neutrons are in (a) 6 Li, (b) 54 Cr, (c) 54 Fe, and (d) 220 Rn?

How many protons and how many neutrons are in (a) 3 He, (b) 32 P, (c) 32 S, and (d) 238 U?

Calculate the nuclear diameters of (a) 4 He, (b) 56 Fe, and (c) 238 U.

Calculate the mass, radius, and density of the nucleus of (a) 7 Li and (b) 207 Pb. Give all answers in SI units

Which stable nuclei have a diameter of 7.46 fm?

Use data in Appendix C to make your own chart of stable and unstable nuclei, similar to Figure 42.4, for all nuclei with Z 8. Use a blue or black dot to represent stable isotopes, a red dot to represent isotopes that undergo beta-minus decay, and a green dot to represent isotopes that undergo beta-plus decay or electron-capture decay.

a. What is the smallest value of A for which there are two stable nuclei? What are they? b. For which values of A less than this are there no stable nuclei?

Calculate (in MeV) the total binding energy and the binding energy per nucleon for 3 H and for 3 He

Calculate (in MeV) the total binding energy and the binding energy per nucleon for 129 I and for 129 Xe

Calculate (in MeV) the binding energy per nucleon for 3 He and 4 He. Which is more tightly bound?

Calculate (in MeV) the binding energy per nucleon for 14 O and 16 O. Which is more tightly bound?

Calculate the chemical atomic mass of silicon.

Use the potential-energy diagram in Figure 42.8 to estimate the strength of the strong force between two nucleons separated by 1.5 fm.

Use the potential-energy diagram in Figure 42.8 to sketch an approximate graph of the strong force between two nucleons versus the distance r between their centers

Use the potential-energy diagram in Figure 42.8 to estimate the ratio of the gravitational potential energy to the nuclear potential energy for two neutrons separated by 1.0 fm.

a. Draw energy-level diagrams, similar to Figure 42.11, for all A = 10 nuclei listed in Appendix C. Show all the occupied b. Which of these nuclei are stable? What is the decay mode of any that are radioactive?neutron and proton levels.

a. Draw energy-level diagrams, similar to Figure 42.11, for all A = 14 nuclei listed in Appendix C. Show all the occupied neutron and proton levels. b. Which of these nuclei are stable? What is the decay mode of any that are radioactive?

The barium isotope 131 Ba has a half-life of 12 days. A 250 mg sample of 131 Ba is prepared. What is the mass of 131 Ba after (a) 1 day, (b) 10 days, and (c) 100 days?

The radium isotope 226 Ra has a half-life of 1600 years. A sample begins with 1.00 * 1010 226 Ra atoms. How many are left after (a) 200 years, (b) 2000 years, and (c) 20,000 years?

A sample of 1.0 * 1010 atoms that decay by alpha emission has a half-life of 100 min. How many alpha particles are emitted between t = 50 min and t = 200 min?

The radioactive hydrogen isotope 3 H, called tritium, has a half-life of 12 years. a. What are the decay mode and the daughter nucleus of tritium? b. What are the lifetime and the decay rate of tritium?

What is the half-life in days of a radioactive sample with 5.0 * 1015 atoms and an activity of 5.0 * 108 Bq?

The half-life of 60 Co is 5.27 years. The activity of a 60 Co sample is 3.50 * 109 Bq. What is the mass of the sample?

Identify the unknown isotope X in the following decays. a. X S 224 Ra + a b. X S 207 Pb + e- + n c. 7 Be + e- S X + n d. X S 60 Ni + g

Identify the unknown isotope X in the following decays. a. 230 Th S X + a b. 35 S S X + e- + n c. X S 40 K + e+ + n d. 24 Na S 24 Mg + e- + n S X + g

a. What are the isotopic symbols of all A = 17 isobars? b. Which of these are stable nuclei? c. For those that are not stable, identify both the decay mode and the daughter nucleus.

a. What are the isotopic symbols of all A = 19 isobars? b. Which of these are stable nuclei? c. For those that are not stable, identify both the decay mode and the daughter nucleus.

An unstable nucleus undergoes alpha decay with the release of 5.52 MeV of energy. The combined mass of the parent and daughter nuclei is 452 u. What was the parent nucleus?

What is the energy (in MeV) released in the alpha decay of 239 Pu?

What is the total energy (in MeV) released in the beta-minus decay of 24 Na?

What is the total energy (in MeV) released in the beta-minus decay of 3 H?

What is the total energy (in MeV) released in the beta decay of a neutron?

The doctors planning a radiation therapy treatment have determined that a 100 g tumor needs to receive 0.20 J of gamma radiation. What is the dose in grays?

1.5 Gy of gamma radiation are directed into a 150 g tumor during radiation therapy. How much energy does the tumor absorb?

How many grays of gamma-ray photons cause the same biological damage as 30 Gy of alpha radiation?

A 50 kg laboratory worker is exposed to 20 mJ of beta radiation with RBE = 1.5. What is the dose equivalent in mrem?answer both in atomic mass units and as a percentage of the mass of the helium nucleus. c. Compare your answers to parts a and b. Why do you hear it said that mass is lost in nuclear reactions but not in chemical reactions?

Use the graph of binding energy to estimate the total energy released if three 4 He nuclei fuse together to form a 12 C nucleus.

Use the graph of binding energy to estimate the total energy released if a nucleus with mass number 240 fissions into two nuclei with mass number 120.

Could a 56 Fe nucleus fission into two 28 Al nuclei? Your answer, which should include some calculations, should be based on the curve of binding energy.

What energy (in MeV) alpha particle has a de Broglie wavelength equal to the diameter of a 238 U nucleus?

The activity of a sample of the cesium isotope 137 Cs, with a half-life of 30 years, is 2.0 * 108 Bq. Many years later, after the sample has fully decayed, how many beta particles will have been emitted?

What is the age in years of a bone in which the 14 C/ 12 C ratio is measured to be 1.65 * 10-13 ?

A 1 Ci source of radiation is a significant source. 238 U is an alpha emitter. What mass of 238 U has an activity of 1 Ci?

137 Cs is a common product of nuclear fission. Suppose an accident spills 550 mCi of 137 Cs in a lab room. a. What mass of 137 Cs is spilled? b. If the spill is not cleaned up, how long will it take until the radiation level drops to an acceptable level, for a room this size, of 25 mCi?

A 115 mCi radioactive tracer is made in a nuclear reactor. When it is delivered to a hospital 16 hours later its activity is 95 mCi. The lowest usable level of activity is 10 mCi. a. What is the tracers half-life? b. For how long after delivery is the sample usable?

The radium isotope 223 Ra, an alpha emitter, has a half-life of 11.43 days. You happen to have a 1.0 g cube of 223 Ra, so you decide to use it to boil water for tea. You fill a well-insulated container with 100 mL of water at 18C and drop in the cube of radium. How long will it take the water to boil?

How many half-lives must elapse until (a) 90% and (b) 99% of a radioactive sample of atoms has decayed?

A sample contains radioactive atoms of two types, A and B. Initially there are five times as many A atoms as there are B atoms. Two hours later, the numbers of the two atoms are equal. The half-life of A is 0.50 hour. What is the half-life of B?

Radioactive isotopes often occur together in mixtures. Suppose a 100 g sample contains 131 Ba, with a half-life of 12 days, and 47 Ca, with a half-life of 4.5 days. If there are initially twice as many calcium atoms as there are barium atoms, what will be the ratio of calcium atoms to barium atoms 2.5 weeks later?

The half-life of the uranium isotope 235 U is 700 million years. The earth is approximately 4.5 billion years old. How much more 235 U was there when the earth formed than there is today? Give your answer as the then-to-now ratio

A chest x ray uses 10 keV photons with an RBE of 0.85. A 60 kg person receives a 0.30 mSv dose from one chest x ray that exposes 25% of the patients body. How many x ray photons are absorbed in the patients body?

The rate at which a radioactive tracer is lost from a patients body is the rate at which the isotope decays plus the rate at which the element is excreted from the body. Medical experiments have shown that stable isotopes of a particular element are excreted with a 6.0 day half-life. A radioactive isotope of the same element has a half-life of 9.0 days. What is the effective half-life of the isotope, in days, in a patients body?

The plutonium isotope 239 Pu has a half-life of 24,000 years and decays by the emission of a 5.2 MeV alpha particle. Plutonium is not especially dangerous if handled because the activity is low and the alpha radiation doesnt penetrate the skin. However, there are serious health concerns if even the tiniest particles of plutonium are inhaled and lodge deep in the lungs. This could happen following any kind of fire or explosion that disperses plutonium as dust. Lets determine the level of danger. a. Soot particles are roughly 1 mm in diameter, and it is known that these particles can go deep into the lungs. How many atoms are in a 1.0@mm@diameter particle of 239 Pu? The density of plutonium is 19,800 kg/m3 . b. What is the activity, in Bq, of a 1.0@mm@diameter particle? c. The activity of the particle is very small, but the penetrating power of alpha particles is also very small. The alpha particles are all stopped, and each deposits its energy in a 50@mm@diameter sphere around the particle. What is the dose, in mSv/year, to this small sphere of tissue in the lungs? Assume that the tissue density is that of water. d. Is this exposure likely to be significant? How does it compare to the natural background of radiation exposure?

The uranium isotope 238 U is naturally present at low levels in many soils. One of the nuclei in the decay series of 238 U is the radon isotope 222 Rn, which decays by emitting a 5.50 MeV alpha particle with t1/2 = 3.82 days. Radon is a gas, and it tends to seep from soil into basements. The Environmental Protection Agency recommends that homeowners take steps to remove radon, by pumping in fresh air, if the radon activity exceeds 4 pCi per liter of air. a. How many 222 Rn atoms are there in 1 m3 of air if the activity is 4 pCi/L? b. The range of alpha particles in air is 3 cm. Suppose we model a person as a 180-cm-tall, 25-cm-diameter cylinder with a mass of 65 kg. Only decays within 3 cm of the cylinder can cause exposure, and only 50, of the decays direct the alpha particle toward the person. Determine the dose in mSv per year for a person who spends the entire year in a room where the activity is 4 pCi/L. c. Does the EPA recommendation is trapped inside and cannot escape. A geologist brings you a piece of solidified lava in which you find the 40 Ar/ 40 K ratio to be 0.013. What is the age of the rock? seem appropriate? Why?

Beta-plus decay is A XZ S A YZ-1 + e+ + n. a. Determine the mass threshold for beta-plus decay. That is, what is the minimum atomic mass mX for which this decay is energetically possible? Your answer will be in terms of the atomic mass mY and the electron mass me. b. Can 13 N undergo beta-plus decay into 13 C? If so, how much energy is released in the decay?

All the very heavy atoms found in the earth were created long ago by nuclear fusion reactions in a supernova, an exploding star. The debris spewed out by the supernova later coalesced into the gases from which the sun and the planets of our solar system were formed. Nuclear physics suggests that the uranium isotopes 235 U and 238 U should have been created in roughly equal numbers. Today, 99.28% of uranium is 238 U and only 0.72% is 235 U. How long ago did the supernova occur?

Alpha decay occurs when an alpha particle tunnels through the Coulomb barrier. FIGURE CP42.63 shows a simple one-dimensional model of the potential-energy well of an alpha particle in a nucleus with A 235. The 15 fm width of this one-dimensional potential-energy well is the diameter of the nucleus. Further, to keep the model simple, the Coulomb barrier has been modeled as a 20-fm-wide, 30-MeV-high rectangular potential-energy barrier. The goal of this problem is to calculate the half-life of an alpha particle in the energy level E = 5.0 MeV. a. What is the kinetic energy of the alpha particle while inside the nucleus? What is its kinetic energy after it escapes from the nucleus? b. Consider the alpha particle within the nucleus to be a point particle bouncing back and forth with the kinetic energy you found in part a. What is the particles collision rate, the number of times per second it collides with a wall of the potential? c. What is the tunneling probability Ptunnel ? d. Ptunnel is the probability that on any one collision with a wall the alpha particle tunnels through instead of reflecting. The probability of not tunneling is 1 - Ptunnel . Hence the probability that the alpha particle is still inside the nucleus after N collisions is 11 - Ptunnel 2N 1 - NPtunnel , where weve used the binomial approximation because Ptunnel V 1. The half-life is the time at which half the nuclei have not yet decayed. Use this to determine (in years) the half-life of the nucleus.

It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, lets analyze the decay of a free neutron that is at rest in the laboratory. Well ignore the antineutrino and consider the decay n S p+ + e-. The analysis requires the use of relativistic energy and momentum, from Chapter 36. a. What is the total kinetic energy, in MeV, of the proton and electron? b. Write the equation that expresses the conservation of relativistic energy for this decay. Your equation will be in terms of the three masses mn,mp, and me and the relativistic factors gp and ge.c. Write the equation that expresses the conservation of relativistic momentum for this decay. Let v represent speed, rather than velocity, then write any minus signs explicitly. d. You have two simultaneous equations in the two unknowns vp and ve. To help in solving these, first prove that gv = 1g2 - 121/2 c. e. Solve for vp and ve. (Its easiest to solve for gp and ge, then find v from g.) First get an algebraic expression for each, in terms of the masses. Then evaluate each, giving v as a fraction of c. f. Calculate the kinetic energy in MeV of the proton and the electron. Verify that their sum matches your answer to part a. g. Now explain why the electron is ejected in beta decay while the proton remains in the nucleus.

What are the overarching findings of Part VIII?

What laws and principles of physics govern relativity and quantum physics?

What are the most important models used in quantum physics?

What are the most important tools introduced in Part VIII?