Introduction to Real Analysis 4th edition Robert G. Bartle, Donald R. Sherbert - Solutions

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What is the maximum kinetic energy of an electron emitted in the β decay of a free neutron?
Show that the decay 11/6C → 10/5B + p are not possible because energy would not be conserved.
22/11Na is radioactive. (a) Is it a β – or β + emitter? (b) Write down the decay reaction, and estimate the maximum kinetic energy of the emitted β.
Give the result of a calculation that shows whether or not the following decays are possible; (a) 236/92U → 235/92U + n; (b) 16/8O → 15/8O + n; (c) 23/11Na → 22/11Na + n.
A 238/92U nucleus emits an α particle with kinetic energy = 4.20MeV. (a) What is the daughter nucleus, and (b) What is the approximate atomic mass (in u) of the daughter atom? Ignore recoil of the daughter nucleus.
When 23/10Ne (mass = 22.9945u) decays to 23/11Na (mass = 22.9898u), what is the maximum kinetic energy of the emitted electron? What is its minimum energy? What is the energy of the neutrino in each case? Ignore recoil of the daughter nucleus.
A nucleus of mass 238u, initially at rest, emits an α particle with a KE of 5.0MeV. What is the KE of the recoiling daughter nucleus?
What is the maximum KE of the emitted β particle during the decay of 60/27Co?
The nuclide 32/15P decays by emitting an electron whose maximum kinetic energy can be 1.71MeV.(a) What is the daughter nucleus?(b) Calculate the daughter’s atomic mass (in u).
The isotope 218/84Po can decay by either α or β – emission. What is the energy release in each case? The mass of 218/84Po is 218.008965u.
How much energy is released in electron capture by beryllium: 7/4Be + 0/-1e → 7/3Li _ v?
A photon with a wavelength of 1.00 x 10–13 m is ejected from an atom. Calculate its energy and explain why it is a y ray from the nucleus or a photon from the atom.
Determine the maximum kinetic energy of β+ particle released when 11/6C decays to 11/5B. What is the maximum energy the neutrino can have? What is its minimum energy?
How much recoil energy does a 40/19K nucleus get when it emits a 1.46-MeV gamma ray?
What is the energy of the α particle emitted in the decay 210/84Po → 206/82Pb + α? Take into account the recoil of the daughter nucleus.
The α particle emitted when 238/92U decays has 4.20MeV of kinetic energy. Calculate the recoil kinetic energy of the daughter nucleus and the Q-value of the decay.
Show that when a nucleus decays by β+ decay, the total energy released is equal to (Mp – MD – 2me) c2, where Mp and MD are the masses of the parent and daughter atoms (neutral), and Me is the mass of an electron or positron.
A radioactive material produces 1280 decays per minute at one time, and 4.6h later produces 320 decays per minute. What is its half-life?
(a) What is the decay constant of 238/92U whose half-life is 4.5 X 109 yr?(b) The decay constant of a given nucleus is 8.2 X 10 –5 s –1. What is its half-life?
What is the activity of a sample of 14/6C that contains 3.1 X 1020 nuclei?
What fraction of a sample of 68/32Ge, whose half-life is about 9 months, will remain after 3.0yr?
What fraction of a sample is left after exactly 6 half-lives?
How many nuclei of 238/92U remain in a rock if the activity registers 640decays per second?
In a series of decays, the nuclide 235/92U becomes 207/82Pb. How many α and β – particles are emitted in this series?
The iodine isotope 131/531 is u sed in hospitals for diagnosis of thyroid function. If 682μg are ingested by a patient, determine the activity (a) Immediately, (b) 1.0h later when the thyroid is being tested, and (c) 6 months later. Use Appendix b.
124/55Cs has a half-life of 30.8s. (a) If we have 8.8μg initially, how many Cs nuclei are present? (b) How many are present 2.0min later? (c) What is the activity at this time? (d) After how much time will the activity drops to less than about 1 per second?
Calculate the mass of sample of pure 40/19K with an initial decay rate of 2.0 x 105s –1. The half-life of 40/19K is 1.28 X 109yr.
Calculate the activity of a pure 9.7-μg sample of 32/15P (T1/2 = 1.23 X 106s).
The activity of a sample of 35/16S (T½ = 7.55 X 106s) is 2.65 X 105 decays per second. What is the mass of the sample?
A sample of 233/92U (T½ = 1.59 X 105yr) contains 7.50 X 1019 nuclei.(a) What is the decay constant?(b) Approximately how many disintegrations will occur per minute?
The activity of a sample drops by a factor of 10 in 8.6 minutes. What is its half-life?
A 285-g sample pure carbon contains 1.3 parts in 1012 (atoms) of 14/6C. How many disintegrations occur per second?
A sample of 40/19K is decaying at a rate of 6.70 x 102 decays/s, what is the mass of the sample
The rubidium isotope 87/37Rb, a β emitter with a half-life of 4.75 X 1010 yr, is used to determine the age of rocks and fossils. Rocks containing fossils of ancient animals contain a ratio of 87/38Sr to 87/37Rb of 0.0160. Assuming that there was no 87/38Sr present when the rocks
Use Fig. 30-11 and calculate the relative decay rates for α decay of 218/84 Po and 214/84Po.
7/4Be decays with a half-life of about 53d. It is produced in the upper atmosphere, and filters down onto the Earth’s surface. If a plant leaf is detected to have 450 decays/s of 7/4Be, (a) How long do we have to wait for the decay rate to drop to 15 per second?(b) Estimate the initial mass of
Be decays with a half-life of about 53d. It is produced in the upper atmosphere, and filters down onto the Earth’s surface. If a plant leaf is detected to have 450 decays/s of 7/4Be, (a) How long do we have to wait for the decay rate to drop to 15 per second? (b) Estimate the initial mass of
An ancient wooden club is found that contains 290g of carbon and has an activity of 8.0decays per second. Determine its age assuming that in living trees the ratio of 14C/12C atoms is about 1.3 X 10 –12.
At t = 0, a pure sample of radioactive nuclei contains N0 nuclei whose decay constant is λ. Determine a formula for the number of daughter nuclei, ND, as a function of time; assume the daughter is stable and that ND = 0 at t = 0.
Which radioactive isotope of lead is being produced in a reaction where the measured activity of a sample drops to 1.050% of its original activity in 4.00h?
An old wooden tool is found to contain only 6.0% of 14/6C that a sample of fresh wood would. How old is the tool?
A neutron star consists of neutrons at approximately nuclear density. Estimate, for a 10-km-diameter neutron star,(a) Its mass number,(b) Its mass (kg), and (c) The acceleration of gravity at its surface.
The 3/1H isotope of hydrogen, which is called tritium (because it contains three nucleons), has a half-life of 12.33yr, it can be used to measure the age of objects up to about 100yr. It can be used to measure the age of objects up to about 100yr. It is produced in the upper atmosphere by cosmic
Some elementary particle theories (Section 32-11) suggest that the proton may be unstable, with a half-life > 1032 yr. How long would you expect to wait for one proton in your body to decay (consider that your body is all water)?
How long must you wait (in half-lives) for a radioactive sample to drop to 1.00% of its original activity?
If the potassium isotope 40/19K gives 60 decays/s in a liter of milk, estimate how much 40/19K and regular 36/19K are in a liter of milk. Use Appendix B.
(a) In α decay of, say, a 226/88Ra nucleus, show that the nucleus carries away a fraction 1/ (1 + 1/4AD) of the total energy available, where AD is the mass number of the daughter nucleus. (b) Approximately what percentage of the energy available is thus carried off by the α particle in
Strortium-90 is produced as a nuclear fission product of uranium in both reactors and atomic bombs. Look at its location in the periodic table to see what other elements it might be similar to chemically, and tell why you think it might be dangerous to ingest. It has too many neutrons, and it
The nuclide 191/76Os decays with β – energy of 0.14MeV accompanied by y rays of energy 0.042MeV and 0.129MeV. (a) What is the daughter nucleus? (b) Draw an energy-level diagram showing the ground states of the parent and daughter and excited states of the daughter. To which of the daughter
Determine the activates of (a) 1.0g of 131/53I (T ½ = 8.02 days) and (b) 1.0g of 238/92U (T ½ = 4.47 x 109 yr)?
Estimate the total binding energy copper and then estimate the energy, in joules, needed to break a 3.0-g copper penny into its constituent nucleons.
Instead of giving atomic masses for nuclides as in Appendix B some Tables give the mass excess, ∆ defined as ∆ = M – A, where A is the atomic number and M is the mass in u. Determine the mass excess in u and in MeV/c2, for: (a) 4/2He; (b) 12/6C; (c) 107/47Ag; (d) 235/92U. (e)
(a) A 92-gram sample of natural carbon contains the usual fraction of 14/6C Estimate how long it will take before there is only one 14/6C nucleus left.(b) How does the answer in (a) change if the sample is 280 grams? What does this tell you about the limits of carbon dating?
If the mass of the proton were just a little closer to the mass of the neutron, the following reaction would be possible even at low collision energies: e– + p → n + v. why would this situation be catastrophic? By what percentage would the proton’s mass have to be increased to make this
What is the ratio of the kinetic energies for an alpha particle and a beta particle if both make tracks with the same radius of curvature in a magnetic field, oriented perpendicular to the paths of the particles?
A 1.00-g sample of natural samarium emits α particle at a rate of 120s–1 due to the presence of 147/62Sm. The natural abundance of 147/62Sm is 15%. Calculate the half-life for this decay process.
Almost all of naturally occurring uranium is 238/92U with a half-life of 4.468 x 109 yr. Most of the rest of natural uranium is 235/92U with a half-life of 7.038 x 108yr. Today a sample contains 0.72%U. (a) What was this percentage 1.0billion years ago?(b) What percentage of the sample would be
A typical banana contains 400mg of potassium, of which a small fraction is the radioactive isotope 40/19K (see Appendix B). Estimate the activity of an average banana due to 40/19K.
The practical limit for carbon-14 dating is about 60,000 years. If a bone contains 1.0 kg of carbon, and the animal died 60,000 years ago, what is the activity today?
Decay series, such as that shown in Figure, can be classified into four families, depending on whether the mass numbers have the form 4n, 4n + 1, 4n + 2, or 4n + 3, where n is an integer. Justify this statement and show that for a nuclide in any family, all its daughters will be in the same family.
Fill in the missing particles or nuclei; (a) n + 137/56Ba →? + y; (b) n + 137/56Ba → 137/55Cs + ?; (c) d + 2/1H → 4/2 He + ?; (d) α + 197/79Au →? + d where d stands for deuterium.
The isotope: 32/15P is produced by the reaction; n +? → 32/15P + p. what must be the target nucleus?
When 22/11Na is bombarded by deuterons (2/1H) an α particle is emitted. What is the resulting nuclide?
Why are neutrons such good projectiles for producing nuclear reactions?
A proton strikes a 20/10Ne nucleus and an α particle is observed to emerge. What is the residual nucleus? Write down the reaction equation.
Are fission fragments β+ or β– emitters? Explain.
If 235/92U released only 1.5 neutrons per fission on the average, would a chain reaction be possible? If so, what would be different?
238/92U releases an average of 2.5 neutrons per fission compared to 2.9 for 239/94Pu. Pure samples of which of these two nuclei do you think would have the smaller critical mass? Explain.
The energy from nuclear fission appears in the form of thermal energy-but the thermal energy of what?
Why can’t uranium be enriched by chemical means?
How can a neutron, with practically no kinetic energy, excite a nucleus to the extent shown in Fig.31-2?
Why would a porous block of uranium be more likely to explode if kept under water rather than in air?
A reactor that uses highly enriched uranium can use ordinary water (instead of heavy water) as a moderator and still have a self-sustaining chain reaction. Explain.
Why must the fission process release neutrons if it is to be useful?
Discuss the relative merits and disadvantages, including pollution and safety, of power generation by fossil fuels, nuclear fission, and nuclear fusion.
What is the reason for the “secondary system” in a nuclear reactor, Fig. 31-7? That is, why is the water heated by the fuel in a nuclear reactor not used directly to drive the turbines?
Why are neutrons released in a fission reaction?
Why do gamma particles penetrate matter more easily than beta particles do?
A higher temperature is required for deuterium-deuterium ignition than for deuterium-tritium. Explain.
Light energy emitted by the Sun and stars comes from the fusion process. What conditions in the interior of stars make this possible?
How do stars, and our Sun, maintain confinement of the plasma for fusion?
What is the basic difference between fission and fusion?
People who work around metals that emit alpha particles are trained that there is little danger from proximity or even touching the material, but that they must take extreme precautions against ingesting it. Hence, there are strong rules against cating and drinking while working, and against
Why is the recommended maximum radiation dose higher for women beyond the child-bearing age than for younger women?
Radiation is sometimes used to sterilize medical supplies and even food. Explain how it works.
What is the difference between absorbed dose and effective does? What are the SI units for each?
How might radioactive tracers be used to find a leak in a pipe?
Natural aluminum is all 27/13Al. If it absorbs a neutron, what does it become? Does it decay by β+ or β–? What will be the product nucleus?
Determine whether the reaction 2/1H + 2/1H →3/2H + n requires a threshold energy.
Is the reaction n + 238/92U →239/92U + y possible with slow neutrons? Explain.
Does the reaction p + 7/3Li →4/2He +α required energy, or does it release energy? How much energy?
Calculate the energy released (or energy input required) for the reaction α + 9/4Be → 12/6C + n.
(a) Can the reaction n + 24/12Mg → 23/11Na + d occur if the bombarding particles have 10.00MeV of kinetic energy? (d stands for deuterium 2/1H.) (b) If so, how much energy is released?
(a) Can the reaction p + 7/3Li → 4/2He + α occur if the incident proton has kinetic energy = 2500keV? (b) If so, what is the total kinetic energy of the products?
In the reaction α + 14/7N → 17/8O + p, the incident α particles have 7.68MeV of kinetic energy. (a) Can this reaction occur? (b) If so, what is the total kinetic energy of the products? The mass of 17/8O is 16.999131u.
Calculate the Q-value for the “capture” reaction α + 16/8O → 20/10Ne + y.
Radioactive 14/6C is produced in the atmosphere when a neutron is absorbed by 14/7N. Write the reaction and find its Q-value.
Radioactive 14/6C is produced in the atmosphere when a neutron is absorbed by 14/7N. Write the reaction and find its Q-value.
An example of a “stripping” nuclear reaction is d + 6/3Li →X + p. (a) What is X, the resulting nucleus? (b) Why is it called a “stripping” reaction? (c) What is the Q-value of this reaction? Is the reaction endothermic or exothermic?
An example of a “pick-up” nuclear reaction is 3/2He + 12/6C → X + a/ (a) Why is it called a “pickup” reaction? (b) What is the resulting nucleus? (c) What is the Q-value of this reaction? Is the reaction endothermic or exothermic?

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