1. Burning one metric ton (1 000 kg) of coal can yield an energy of 3.30 × 10^{10} J. Fission of one nucleus of uranium-235 yields an average of about 208 MeV. What mass of uranium produces the same energy as a ton of coal?
2. Find the energy released in the fission reaction

The atomic masses of the fission products are: , 97.912 7 u; , 134.916 5 u.
3. Strontium-90 is a particularly dangerous fission product of ^{235}U because it is radioactive and it substitutes for calcium in bones. What other direct fission products would accompany it in the neutron-induced fission of ^{235}U? (Note: This reaction may release two, three, or four free neutrons.)
4. List the nuclear reactions required to produce ^{239}Pu from ^{238}U under fast neutron bombardment.
5. List the nuclear reactions required to produce ^{233}U from ^{232}Th under fast neutron bombardment.
6 Note: Problem 53 in Chapter 25 and Problems 21, 51, and 68 in Chapter 44 can be assigned with this section.

. (a) The following fission reaction is typical of those occurring in a nuclear electric generating station:

Find the energy released. The required masses are
= 1.008 665 u

= 235.043 923 u

= 140.914 4 u

= 91.926 2 u
(b) What fraction of the initial mass of the system is transformed?
7. A reaction that has been considered as a source of energy is the absorption of a proton by a boron-11 nucleus to produce three alpha particles:

This is an attractive possibility because boron is easily obtained from the Earth’s crust. A disadvantage is that the protons and boron nuclei must have large kinetic energies in order for the reaction to take place. This is in contrast to the initiation of uranium fission by slow neutrons. (a) How much energy is released in each reaction? (b) Why must the reactant particles have high kinetic energies?
8. A typical nuclear fission power plant produces about 1.00 GW of electrical power. Assume that the plant has an overall efficiency of 40.0% and that each fission produces 200 MeV of energy. Calculate the mass of ^{235}U consumed each day.
9.Review problem. Suppose enriched uranium containing 3.40% of the fissionable isotope is used as fuel for a ship. The water exerts an average friction force of magnitude 1.00 × 10^{5} N on the ship. How far can the ship travel per kilogram of fuel? Assume that the energy released per fission event is 208 MeV and that the ship’s engine has an efficiency of 20.0%.
Section 45.3 Nuclear Reactors 10. To minimize neutron leakage from a reactor, the surface area-to-volume ratio should be a minimum. For a given volume V, calculate this ratio for (a) a sphere, (b) a cube, and (c) a parallelepiped of dimensions a × a × 2a. (d) Which of these shapes would have minimum leakage? Which would have maximum leakage?
11. It has been estimated that on the order of 10^{9} tons of natural uranium is available at concentrations exceeding 100 parts per million, of which 0.7% is the fissionable isotope ^{235}U. Assume that all the world’s energy use (7 × 10^{12} J/s) were supplied by ^{235}U fission in conventional nuclear reactors, releasing 208 MeV for each reaction. How long would the supply last? The estimate of uranium supply is taken from K. S. Deffeyes and I. D. MacGregor, “World Uranium Resources,” Scientific American 242(1):66, 1980.
12. If the reproduction constant is 1.000 25 for a chain reaction in a fission reactor and the average time interval between successive fissions is 1.20 ms, by what factor will the reaction rate increase in one minute?
13. A large nuclear power reactor produces about 3 000 MW of power in its core. Three months after a reactor is shut down, the core power from radioactive byproducts is 10.0 MW. Assuming that each emission delivers 1.00 MeV of energy to the power, find the activity in becquerels three months after the reactor is shut down.
Section 45.4 Nuclear Fusion 14. (a) Consider a fusion generator built to create 3.00 GW of power. Determine the rate of fuel burning in grams per hour if the D–T reaction is used. (b) Do the same for the D–D reaction assuming that the reaction products are split evenly between (n, ^{3}He) and (p, ^{3}H).
15. Two nuclei having atomic numbers Z_{1} and Z_{2} approach each other with a total energy E. (a) Suppose they will spontaneously fuse if they approach within a distance of 1.00 × 10^{–14} m. Find the minimum value of E required to produce fusion, in terms of Z_{1} and Z_{2}. (b) Evaluate the minimum energy for fusion for the D–D and D–T reactions (the first and third reactions in Eq. 45.4).
16. Review problem. Consider the deuterium–tritium fusion reaction with the tritium nucleus at rest:

(a) Suppose that the reactant nuclei will spontaneously fuse if their surfaces touch. From Equation 44.1, determine the required distance of closest approach between their centers. (b) What is the electric potential energy (in eV) at this distance? (c) Suppose the deuteron is fired straight at an originally stationary tritium nucleus with just enough energy to reach the required distance of closest approach. What is the common speed of the deuterium and tritium nuclei as they touch, in terms of the initial deuteron speed v_{i}? (Suggestion: At this point, the two nuclei have a common velocity equal to the center-of-mass velocity.) (d) Use energy methods to find the minimum initial deuteron energy required to achieve fusion. (e) Why does the fusion reaction actually occur at much lower deuteron energies than that calculated in (d)?
17. To understand why plasma containment is necessary, consider the rate at which an unconfined plasma would be lost. (a) Estimate the rms speed of deuterons in a plasma at 4.00 × 10^{8} K. (b) What If? Estimate the order of magnitude of the time interval during which such a plasma would remain in a 10-cm cube if no steps were taken to contain it.
18. Of all the hydrogen in the oceans, 0.030 0% of the mass is deuterium. The oceans have a volume of 317 million mi^{3}. (a) If nuclear fusion were controlled and all the deuterium in the oceans were fused to , how many joules of energy would be released? (b) What If? World power consumption is about 7.00 × 10^{12} W. If consumption were 100 times greater, how many years would the energy calculated in part (a) last?
19. It has been suggested that fusion reactors are safe from explosion because there is never enough energy in the plasma to do much damage. (a) In 1992, the TFTR reactor achieved an ion temperature of 4.0 × 10^{8} K, an ion density of 2.0 × 10^{13} cm^{–3}, and a confinement time of 1.4 s. Calculate the amount of energy stored in the plasma of the TFTR reactor. (b) How many kilograms of water could be boiled away by this much energy? (The plasma volume of the TFTR reactor is about 50 m^{3}.)
20. Review problem. To confine a stable plasma, the magnetic energy density in the magnetic field (Eq. 32.14) must exceed the pressure 2nk_{B}T of the plasma by a factor of at least 10. In the following, assume a confinement time τ = 1.00 s. (a) Using Lawson’s criterion, determine the ion density required for the D–T reaction. (b) From the ignition-temperature criterion, determine the required plasma pressure. (c) Determine the magnitude of the magnetic field required to contain the plasma.
21. Find the number of ^{6}Li and the number of ^{7}Li nuclei present in 2.00 kg of lithium. (The natural abundance of ^{6}Li is 7.5%; the remainder is ^{7}Li.)
22. One old prediction for the future was to have a fusion reactor supply energy to dissociate the molecules in garbage into separate atoms and then to ionize the atoms. This material could be put through a giant mass spectrometer, so that trash would be a new source of isotopically pure elements—the mine of the future. Assuming an average atomic mass of 56 and an average charge of 26 (a high estimate, considering all the organic materials), at a beam current of 1.00 MA, how long would it take to process 1.00 metric ton of trash?
Section 45.5 Radiation Damage 23. A building has become accidentally contaminated with radioactivity. The longest-lived material in the building is strontium-90. ( has an atomic mass 89.907 7 u, and its half-life is 29.1 yr. It is particularly dangerous because it substitutes for calcium in bones.) Assume that the building initially contained 5.00 kg of this substance uniformly distributed throughout the building (a very unlikely situation) and that the safe level is defined as less than 10.0 decays/min (to be small in comparison to background radiation). How long will the building be unsafe?
24. Review problem. A particular radioactive source produces 100 mrad of 2-MeV gamma rays per hour at a distance of 1.00 m. (a) How long could a person stand at this distance before accumulating an intolerable dose of 1 rem? (b) What If? Assuming the radioactive source is a point source, at what distance would a person receive a dose of 10.0 mrad/h?
25. Assume that an x-ray technician takes an average of eight x-rays per day and receives a dose of 5 rem/yr as a result. (a) Estimate the dose in rem per photograph taken. (b) How does the technician’s exposure compare with lowlevel background radiation?
26. When gamma rays are incident on matter, the intensity of the gamma rays passing through the material varies with depth x as I(x) = I_{0}e^{–μ}^{x}, where μ is the absorption coefficient and I_{0} is the intensity of the radiation at the surface of the material. For 0.400-MeV gamma rays in lead, the absorption coefficient is 1.59 cm^{–1}. (a) Determine the “half-thickness” for lead—that is, the thickness of lead that would absorb half the incident gamma rays. (b) What thickness will reduce the radiation by a factor of 10^{4}?
27. A “clever” technician decides to warm some water for his coffee with an x-ray machine. If the machine produces 10.0 rad/s, how long will it take to raise the temperature of a cup of water by 50.0°C?
28. Review problem. The danger to the body from a high dose of gamma rays is not due to the amount of energy absorbed but occurs because of the ionizing nature of the radiation. To illustrate this, calculate the rise in body temperature that would result if a “lethal” dose of 1 000 rad were absorbed strictly as internal energy. Take the specific heat of living tissue as 4 186 J/kg · °C.
29. Technetium-99 is used in certain medical diagnostic procedures. Assume 1.00 × 10^{–8} g of ^{99}Tc is injected into a 60.0-kg patient and half of the 0.140-MeV gamma rays are absorbed in the body. Determine the total radiation dose received by the patient.
30. Strontium-90 from the testing of atomic bombs can still be found in the atmosphere. Each decay of ^{90}Sr releases 1.1 MeV of energy into the bones of a person who has had strontium replace the calcium. Assume a 70.0-kg person receives 1.00 μg of ^{90}Sr from contaminated milk. Calculate the absorbed dose rate (in J/kg) in one year. Take the half-life of ^{90}Sr to be 29.1 yr.
Section 45.6 Radiation Detectors 31.In a Geiger tube, the voltage between the electrodes is typically 1.00 kV and the current pulse discharges a 5.00-pF capacitor. (a) What is the energy amplification of this device for a 0.500-MeV electron? (b) How many electrons participate in the avalanche caused by the single initial electron?
32. Assume a photomultiplier tube (Figure 40.12) has seven dynodes with potentials of 100, 200, 300, . . . , 700 V. The average energy required to free an electron from the dynode surface is 10.0 eV. Assume that just one electron is incident and that the tube functions with 100% efficiency. (a) How many electrons are freed at the first dynode? (b) How many electrons are collected at the last dynode? (c) What is the energy available to the counter for each electron?
33. (a) Your grandmother recounts to you how, as young children, your father, aunts, and uncles made the screen door slam continually as they ran between the house and the back yard. The time interval between one slam and the next varied randomly, but the average slamming rate stayed constant at 38.0/h from dawn to dusk every summer day. If the slamming rate suddenly dropped to zero, the children would have found a nest of baby field mice or gotten into some other mischief requiring adult intervention. How long after the last screen-door slam would a prudent and attentive parent wait before leaving her or his work to see about the children? Explain your reasoning. (b) A student wishes to measure the half-life of a radioactive substance, using a small sample. Consecutive clicks of her Geiger counter are randomly spaced in time. The counter registers 372 counts during one 5.00-min interval, and 337 counts during the next 5.00 min. The average background rate is 15 counts per minute. Find the most probable value for the half-life. (c) Estimate the uncertainty in the half-life determination. Explain your reasoning.
Section 45.7 Uses of Radiation 34. During the manufacture of a steel engine component, radioactive iron (^{59}Fe) is included in the total mass of 0.200 kg. The component is placed in a test engine when the activity due to this isotope is 20.0 μCi. After a 1 000-h test period, some of the lubricating oil is removed from the engine and found to contain enough ^{59}Fe to produce 800 disintegrations/min/L of oil. The total volume of oil in the engine is 6.50 L. Calculate the total mass worn from the engine component per hour of operation. (The half-life of ^{59}Fe is 45.1 d.)
35. At some time in your past or future, you may find yourself in a hospital to have a PET scan. The acronym stands for positron-emission tomography. In the procedure, a radioactive element that undergoes e^{+} decay is introduced into your body. The equipment detects the gamma rays that result from pair annihilation when the emitted positron encounters an electron in your body’s tissue. Suppose you receive an injection of glucose that contains on the order of 10^{10}atoms of ^{14}O. Assume that the oxygen is uniformly distributed through 2 L of blood after 5 min. What will be the order of magnitude of the activity of the oxygen atoms in 1 cm^{3} of the blood?
36. You want to find out how many atoms of the isotope ^{65}Cu are in a small sample of material. You bombard the sample with neutrons to ensure that on the order of 1% of these copper nuclei absorb a neutron. After activation you turn off the neutron flux, and then use a highly efficient detector to monitor the gamma radiation that comes out of the sample. Assume that half of the ^{66}Cu nuclei emit a 1.04-MeV gamma ray in their decay. (The other half of the activated nuclei decay directly to the ground state of ^{66}Ni.) If after 10 min (two half-lives) you have detected 10^{4} MeV of photon energy at 1.04 MeV, (a) about how many ^{65}Cu atoms are in the sample? (b) Assume the sample contains natural copper. Refer to the isotopic abundances listed in Table A.3 and estimate the total mass of copper in the sample.
37. Neutron activation analysis is a method for chemical analysis at the level of isotopes. When a sample is irradiated by neutrons, radioactive atoms are produced continuously and then decay according to their characteristic half-lives. (a) Assume that one species of radioactive nuclei is produced at a constant rate R and that its decay is described by the conventional radioactive decay law. If irradiation begins at time t = 0, show that the number of radioactive atoms accumulated at time t is

(b) What is the maximum number of radioactive atoms that can be produced?
Additional Problems 38. The nuclear bomb dropped on Hiroshima on August 6, 1945, released 5 × 10^{13} J of energy, equivalent to that from 12 000 tons of TNT. The fission of one nucleus releases an average of 208 MeV. Estimate (a) the number of nuclei fissioned, and (b) the mass of this .
39. Carbon detonations are powerful nuclear reactions that temporarily tear apart the cores inside massive stars late in their lives. These blasts are produced by carbon fusion, which requires a temperature of about 6 × 10^{8} K to overcome the strong Coulomb repulsion between carbon nuclei. (a) Estimate the repulsive energy barrier to fusion, using the temperature required for carbon fusion. (In other words, what is the average kinetic energy of a carbon nucleus at 6 × 10^{8} K?) (b) Calculate the energy (in MeV) released in each of these “carbon-burning” reactions:

(c) Calculate the energy (in kWh) given off when 2.00 kg of carbon completely fuses according to the first reaction.
40. Review problem. Consider a nucleus at rest, which then spontaneously splits into two fragments of masses m_{1} and m_{2}. Show that the fraction of the total kinetic energy that is carried by fragment m_{1} is

and the fraction carried by m_{2} is

assuming relativistic corrections can be ignored. (Note: If the parent nucleus was moving before the decay, then the fission products still divide the kinetic energy as shown, as long as all velocities are measured in the center-of-mass frame of reference, in which the total momentum of the system is zero.)
41. A stationary nucleus fissions spontaneously into two primary fragments, and . (a) Calculate the disintegration energy. The required atomic masses are 86.920 711 u for , 148.934 370 u for , and 236.045 562 u for . (b) How is the disintegration energy split between the two primary fragments? You may use the result of Problem 40. (c) Calculate the speed of each fragment immediately after the fission.
42. Compare the fractional energy loss in a typical ^{235}U fission reaction with the fractional energy loss in D–T fusion.
43. The half-life of tritium is 12.3 yr. If the TFTR fusion reactor contained 50.0 m^{3} of tritium at a density equal to 2.00 × 10^{14} ions/cm^{3}, how many curies of tritium were in the plasma? Compare this value with a fission inventory (the estimated supply of fissionable material) of 4 × 10^{10} Ci.
44. Review problem. A very slow neutron (with speed approximately equal to zero) can initiate the reaction

The alpha particle moves away with speed 9.25 × 10^{6} m/s. Calculate the kinetic energy of the lithium nucleus. Use nonrelativistic equations.
45. Review problem. A nuclear power plant operates by using the energy released in nuclear fission to convert 20°C water into 400°C steam. How much water could theoretically be converted to steam by the complete fissioning of 1.00 g of ^{235}U at 200 MeV/fission? 46. Review problem. A nuclear power plant operates by using the energy released in nuclear fission to convert liquid water at T_{c}into steam at T_{h}. How much water could theoretically be converted to steam by the complete fissioning of a mass m of ^{235}U at 200 MeV/fission? 47. About 1 of every 3 300 water molecules contains one deuterium atom. (a) If all the deuterium nuclei in 1 L of water are fused in pairs according to the D–D fusion reaction ^{2}H + ^{2}H ^{3}He + n + 3.27 MeV, how much energy in joules is liberated? (b) What If? Burning gasoline produces about 3.40 × 10^{7} J/L. Compare the energy obtainable from the fusion of the deuterium in a liter of water with the energy liberated from the burning of a liter of gasoline.
48. Review problem. The first nuclear bomb was a fissioning mass of plutonium-239 exploded in the Trinity test, before dawn on July 16, 1945, at Alamogordo, New Mexico. Enrico Fermi was 14 km away, lying on the ground facing away from the bomb. After the whole sky had flashed with unbelievable brightness, Fermi stood up and began dropping bits of paper to the ground. They first fell at his feet in the calm and silent air. As the shock wave passed, about 40 s after the explosion, the paper then in flight jumped about 5 cm away from ground zero. (a) Assume that the shock wave in air propagated equally in all directions without absorption. Find the change in volume of a sphere of radius 14 km as it expands by 5 cm. (b) Find the work PΔV done by the air in this sphere on the next layer of air farther from the center. (c) Assume the shock wave carried on the order of one tenth of the energy of the explosion. Make an order-of-magnitude estimate of the bomb yield. (d) One ton of exploding trinitrotoluene (TNT) releases 4.2 GJ of energy. What was the order of magnitude of the energy of the Trinity test in equivalent tons of TNT? The dawn revealed the mushroom cloud. Fermi’s immediate knowledge of the bomb yield agreed with that determined days later by analysis of elaborate measurements.
49. A certain nuclear plant generates internal energy at a rate of 3.065 GW and transfers energy out of the plant by electrical transmission at a rate of 1.000 GW. Of the wasted energy, 3.0% is ejected to the atmosphere and the remainder is passed into a river. A state law requires that the river water be warmed by no more than 3.50°C when it is returned to the river. (a) Determine the amount of cooling water necessary (in kg/h and m^{3}/h) to cool the plant. (b) Assume fission generates 7.80 × 10^{10} J/g of ^{235}U. Determine the rate of fuel burning (in kg/h) of ^{235}U.
50. The alpha-emitter polonium-210 () is used in a nuclear energy source on a spacecraft (Fig. P45.50). Determine the initial power output of the source. Assume that it contains 0.155 kg of ^{210}Po and that the efficiency for conversion of radioactive decay energy to energy transferred by electrical transmission is 1.00%.

Courtesy of NASA Ames Figure P45.50: The Pioneer 10 spacecraft leaves the Solar System. It carries radioactive power supplies at the ends of two booms. Solar panels would not work in this region far from the Sun.
51. Natural uranium must be processed to produce uranium enriched in ^{235}U for bombs and power plants. The processing yields a large quantity of nearly pure ^{238}U as a byproduct, called “depleted uranium.” Because of its high mass density, it is used in armor-piercing artillery shells. (a) Find the edge dimension of a 70.0-kg cube of ^{238}U. (Refer to Table 1.5.) (b) The isotope ^{238}U has a long half-life of 4.47 × 10^{9} yr. As soon as one nucleus decays, it begins a relatively rapid series of 14 steps that together constitute the net reaction

Find the net decay energy. (Refer to Table A.3.) (c) Argue that a radioactive sample with decay rate R and decay energy Q has power output = QR. (d) Consider an artillery shell with a jacket of 70.0 kg of ^{238}U. Find its power output due to the radioactivity of the uranium and its daughters. Assume the shell is old enough that the daughters have reached steady-state amounts. Express the power in joules per year. (e) What If? A 17-year-old soldier of mass 70.0 kg works in an arsenal where many such artillery shells are stored. Assume his radiation exposure is limited to 5.00 rem per year. Find the rate at which he can absorb energy of radiation, in joules per year. Assume an average RBE factor of 1.10.
52. A 2.0-MeV neutron is emitted in a fission reactor. If it loses half its kinetic energy in each collision with a moderator atom, how many collisions must it undergo in order to become a thermal neutron, with energy 0.039 eV?
53. Assuming that a deuteron and a triton are at rest when they fuse according to the reaction , determine the kinetic energy acquired by the neutron.
54. A sealed capsule containing the radiopharmaceutical phosphorus-32 (), an e^{–} emitter, is implanted into a patient’s tumor. The average kinetic energy of the beta particles is 700 keV. The initial activity is 5.22 MBq. Determine the absorbed dose during a 10.0-day period. Assume the beta particles are completely absorbed in 100 g of tissue. (Suggestion: Find the number of beta particles emitted.)
55. (a) Calculate the energy (in kilowatt-hours) released if 1.00 kg of ^{239}Pu undergoes complete fission and the energy released per fission event is 200 MeV. (b) Calculate the energy (in electron volts) released in the deuterium–tritium fusion reaction

(c) Calculate the energy (in kilowatt-hours) released if 1.00 kg of deuterium undergoes fusion according to this reaction. (d) What If? Calculate the energy (in kilowatthours) released by the combustion of 1.00 kg of coal if each reaction yields 4.20 eV. (e) List advantages and disadvantages of each of these methods of energy generation.
56. The Sun radiates energy at the rate of 3.77 × 10^{26} W. Suppose that the net reaction

accounts for all the energy released. Calculate the number of protons fused per second.
57. Consider the two nuclear reactions
(I)

(II)

(a) Show that the net disintegration energy for these two reactions (Q_{net} = Q_{I} + Q_{II}) is identical to the disintegration energy for the net reaction

(b) One chain of reactions in the proton–proton cycle in the Sun’s core is

Based on part (a), what is Q_{net} for this sequence?
58. Suppose the target in a laser fusion reactor is a sphere of solid hydrogen that has a diameter of 1.50 × 10^{–4} m and a density of 0.200 g/cm^{3}. Also assume that half of the nuclei are ^{2}H and half are ^{3}H. (a) If 1.00% of a 200-kJ laser pulse is delivered to this sphere, what temperature does the sphere reach? (b) If all of the hydrogen “burns” according to the D–T reaction, how many joules of energy are released? In addition to the proton–proton cycle described in the chapter text, the carbon cycle, first proposed by Hans Bethe in 1939, is another cycle by which energy is released in stars as hydrogen is converted to helium. The carbon cycle requires higher temperatures than the proton–proton cycle. The series of reactions is
^{12}C + ^{1}H ^{13}N + γ

^{13}N ^{13}C + e^{+} + v

e^{+} + e^{–} 2γ

^{13}C + ^{1}H ^{14}N + γ

^{14}N + ^{1}H ^{15}O + γ

^{15}O ^{15}N + e^{+} + v

e^{+} + e^{–} 2γ

^{15}N + ^{1}H ^{12}C + ^{4}He
(a) If the proton–proton cycle requires a temperature of 1.5 × 10^{7} K, estimate by proportion the temperature required for the carbon cycle. (b) Calculate the Q value for each step in the carbon cycle and the overall energy released. (c) Do you think the energy carried off by the neutrinos is deposited in the star? Explain.
60. When photons pass through matter, the intensity I of the beam (measured in watts per square meter) decreases exponentially according to
I = I_{0}e^{– μx} where I_{0} is the intensity of the incident beam and I is the intensity of the beam that just passed through a thickness x of material. The constant μ is known as the linear absorption coefficient, and its value depends on the absorbing material and the wavelength of the photon beam. This wavelength (or energy) dependence allows us to filter out unwanted wavelengths from a broad-spectrum x-ray beam. (a) Two x-ray beams of wavelengths λ_{1} and λ_{2} and equal incident intensities pass through the same metal plate. Show that the ratio of the emergent beam intensities is

(b) Compute the ratio of intensities emerging from an aluminum plate 1.00 mm thick if the incident beam contains equal intensities of 50 pm and 100 pm x-rays. The values of μ for aluminum at these two wavelengths are μ_{1} = 5.4 cm^{–1} at 50 pm and μ_{2} = 41.0 cm^{–1} at 100 pm. (c) Repeat for an aluminum plate 10.0 mm thick.
61. To build a bomb. (a) At time t = 0 a sample of uranium is exposed to a neutron source that causes N_{0} nuclei to undergo fission. The sample is in a supercritical state, with a reproduction constant K > 1. A chain reaction occurs which proliferates fission throughout the mass of uranium. The chain reaction can be thought of as a succession of generations. The N_{0} fissions produced initially are the zeroth generation of fissions. From this generation, N_{0}K neutrons go off to produce fission of new uranium nuclei. The N_{0}K fissions that occur subsequently are the first generation of fissions, and from this generation, N_{0}K^{2} neutrons go in search of uranium nuclei in which to cause fission. The subsequent N_{0}K^{2} fissions are the second generation of fissions. This process can continue until all the uranium nuclei have fissioned. Show that the cumulative total of fissions N that have occurred up to and including the nth generation after the zeroth generation, is given by

(b) Consider a hypothetical uranium bomb made from 5.50 kg of isotopically pure ^{235}U. The chain reaction has a reproduction constant of 1.10, and starts with a zeroth generation of 1.00 × 10^{20} fissions. The average time interval between one fission generation and the next is 10.0 ns. How long after the zeroth generation does it take the uranium in this bomb to fission completely? (c) Assume that the bulk modulus of uranium is 150 GPa. Find the speed of sound in uranium. You may ignore the density difference between ^{235}U and natural uranium. (d) Find the time interval required for a compressional wave to cross the radius of a 5.50-kg sphere of uranium. This time interval indicates how quickly the motion of explosion begins. (e) Fission must occur in a time interval that is short compared to that in part (d), for otherwise most of the uranium will disperse in small chunks without having fissioned. Can the bomb considered in part (b) release the explosive energy of all of its uranium? If so, how much energy does it release, in equivalent tons of TNT? Assume that one ton of TNT releases 4.20 GJ and that each uranium fission releases 200 MeV of energy.