Master the Concepts

A particular nuclide is characterized by its atomic number

*Z*(the number of protons) and its nucleon number*A*(the total number of protons and neutrons). The isotopes of an element have the same atomic number but different numbers of neutrons.The mass density of all nuclei is approximately the same. The radius of a nucleus is

(29-4)where

(29-5)- Page 1099
The binding energy

*E*_{B}of a nucleus is the energy that must be supplied to separate a nucleus into individual protons and neutrons. Since the nucleus is a bound system, its total energy is*less*than the energy of*Z*protons and*N*neutrons that are far apart and at rest.(29-8) In any nuclear reaction, the total electric charge and the total number of nucleons are conserved.

An unstable or radioactive nuclide decays by emitting radiation.

(29-10)(29-11)(29-12)Each radioactive nuclide has a characteristic decay probability per unit time λ. The activity

*R*of a sample with*N*nuclei is(29-18)Activity is commonly measured in becquerels (1 Bq = 1 decay per second) or curies (1 Ci = 3.7 × 10

^{10}Bq).The number of remaining nuclei

*N*in radioactive decay (the number that have*not*decayed) is an exponential function:(29-19)where the time constant is τ = 1/λ. The half-life is the time during which half of the nuclei decay:

(29-22)The absorbed dose is the amount of radiation energy absorbed per unit mass of tissue, measured in grays (1 Gy = 1 J/kg) or rads (1 rad = 0.01 Gy).

The quality factor (QF) is a relative measure of the biological damage caused by different kinds of radiation. The biologically equivalent dose is

(29-28)A large nucleus can release energy by splitting into two smaller, more tightly bound nuclei in the process called fission. The energy released in a fission reaction is enormous—typically around 200 MeV for the split of a single nucleus.

Nuclear fusion combines two small nuclei to form a larger nucleus. Fusion typically releases significantly more energy per nucleon than fission.

Conceptual Questions

How could Henri Becquerel and other scientists determine that there were three

*different*kinds of radiation*before*having determined the electric charges or masses of the α, β, and γ rays?What technique could Becquerel and others have used to determine that α rays are positively charged, β rays negatively charged, and γ rays uncharged? Explain how they could find that α rays have a charge-to-mass ratio half that of the H

^{+}ion, and β rays have the same charge-to-mass ratio as “cathode rays” (electrons). (See Chapter 19 for some ideas.)Why is a slow neutron more likely to induce a nuclear reaction (as in neutron activation and induced fission) than a proton with the same kinetic energy?

Explain why neutron-activated nuclides tend to decay by β

^{−}rather than β^{+}.Why can we ignore the binding energies of the atomic

*electrons*in calculations such as Example 29.4? Isn't there a mass defect due to the binding energy of the electrons?Why would we expect atmospheric testing of nuclear weapons to increase the relative abundance of carbon-14 in the atmosphere? Why would we expect the widespread burning of fossil fuels to

*decrease*the relative abundance of carbon-14 in the atmosphere?- Page 1100
Isolated atoms (or atoms in a dilute gas) radiate photons at discrete energies characteristic of that atom. In dense matter, the spectrum radiated is quasi-continuous. Why doesn't the same thing happen with nuclear spectra: why do the γ rays have the same characteristic energies even when emitted from a solid?

Section 29.8 states that the total energy released by the proton-proton cycle is the same as that released by the carbon cycle. Why must the total energy released be the same?

Iodine is eliminated from the body through

*biological*processes with an effective half-life of about 140 days. The radioactive half-life of iodine-131 is 8 days. Suppose some radioactive^{131}I nuclei are present in the body. Assuming that no new^{131}I nuclei are introduced into the body, how much time must pass until only half as much^{131}I is left in the body: less than 8 days, between 8 and 140 days, or more than 140 days? Explain your reasoning.Why does a fission reaction tend to release one or more neutrons? Why is the release of neutrons necessary in order to sustain a chain reaction?

Radioactive α-emitters are relatively harmless outside the body, but can be dangerous if ingested or inhaled. Explain.

Fission reactors and cyclotrons tend to produce different kinds of isotopes. A reactor produces isotopes primarily through neutron activation; thus, the isotopes tend to be neutron-rich (high neutron-to-proton ratio). A cyclotron can only accelerate charged particles such as protons or deuterons. When stable nuclei are bombarded with protons or deuterons, the resulting radioisotopes are neutron-deficient (low neutron-to-proton ratio). (a) Explain why a cyclotron cannot accelerate neutrons. (b) Suppose a hospital needs a supply of radioisotopes to use in positron-emission tomography (PET). Would the radioisotopes more likely come from a reactor or a cyclotron? Explain.

Why would a fusion reactor produce less radioactive waste than a fission reactor? [

*Hint:*Compare the products of a fission reaction with those from a fusion reaction.]Radon-222 is created in a series of radioactive decays starting with and ending with The half-life of

^{222}Rn is 3.8 days. (a) If the half-life is so short, why hasn't all the^{222}Rn gas decayed by now? (b) If the half-life of^{222}Rn were much shorter, say a few seconds, would it be more dangerous to us or less dangerous? What if the half-life were much longer, say thousands of years?

Multiple-Choice Questions

Radioactive decays into Which of these particles is released in the decay?

(a) a proton

(b) an electron

(c) a positron

(d) an α particle

(e) a neutron

(f) none of the above

For all stable nuclei

(a) there are equal numbers of protons and neutrons.

(b) there are more protons than neutrons.

(c) there are more neutrons than protons.

(d) none of the above have to be true.

For all stable nuclei

(a) the mass of the nucleus is less than

*Zm*_{p}+ (*A*−*Z*)*m*_{n}.(b) the mass of the nucleus is greater than

*Zm*_{p}+ (*A*−*Z*)*m*_{n}.(c) the mass of the nucleus is equal to

*Zm*_{p}+ (*A*−*Z*)*m*_{n}.(d) none of the above have to be true.

Of the

*hypothetical*nuclear reactions listed here, which would violate conservation of charge?(a)

(b)

(c)

(d)

(e) none of them

(f) all of them

(g) all but (c)

(h) (a) and (d)

Of the

*hypothetical*nuclear reactions listed in Multiple-Choice Question 4, which would violate conservation of nucleon number?In a fusion reaction, two deuterons produce a helium-3 nucleus. What is the other product of the reaction?

(a) an electron

(b) a proton

(c) a neutron

(d) an α particle

(e) a positron

(f) a neutrino

The activity of a radioactive sample (with a single radioactive nuclide) decreases to one eighth its initial value in a time interval of 96 days. What is the half-life of the radioactive nuclide present?

(a) 6 days

(b) 8 days

(c) 12 days

(d) 16 days

(e) 24 days

(f) 32 days

Solid lead has more than four times the mass density of solid aluminum. What is the main reason that lead is so much more dense?

(a) The Pb atom is smaller than the Al atom.

(b) The Pb nucleus is smaller than the Al nucleus.

(c) The Pb nucleus is more massive than the Al nucleus.

(d) The Pb nucleus is more dense than the Al nucleus.

(e) The Pb atom has many more electrons than the Al atom.

Which of these are appropriate units for the decay constant λ of a radioactive nuclide?

(a) s

(b) Ci

(c) rd

(d) s

^{−1}(e) rem

(f) MeV

Which of the units listed in Multiple-Choice Question 9 are appropriate for the biologically equivalent dose that results when a person is exposed to radiation?

Problems

Combination conceptual/quantitative problem

Biological or medical application

Challenging problem

Detailed solution in the Student Solutions Manual

Problems paired by concept

Interactive or tutorial

Page 1101

29.1 Nuclear Structure

**1.**Estimate the number of nucleons found in the body of a 75-kg person.

2. Calculate the mass density of nuclear matter.

**3.**A neutron star is a star that has collapsed into a collection of tightly packed neutrons. Thus, it is something like a giant nucleus; but since it is electrically neutral, there is no Coulomb repulsion to break it up. The force holding it together is gravity. Suppose the Sun were to collapse into a neutron star. What would its radius be? Assume that the density is about the same as for a nucleus. Express your answer in kilometers.

4. Write the symbol (in the form ) for the nuclide with 38 protons and 50 neutrons and identify the element.

**5.**Write the symbol (in the form ) for the isotope of potassium with 21 neutrons.

6. How many neutrons are found in a

^{35}Cl nucleus?**7.**How many protons are found in a^{136}Xe nucleus?

8. Write the symbol (in the form ) for the nuclide that has 78 neutrons and 53 protons.

**9.**Find the radius and volume of the nucleus.

29.2 Binding Energy

10. What is the binding energy of an α particle (a

^{4}He nucleus)? The mass of an α particle is 4.001 51 u.**11.**Find the binding energy of a deuteron (a^{2}H nucleus). The mass of a deuteron (*not*the deuterium atom) is 2.013 553 u.

12. What is the average binding energy per nucleon for

**13.**(a) Find the binding energy of the^{16}O nucleus. (b) What is the average binding energy per nucleon? Check your answer using Fig. 29.2.14. Calculate the binding energy per nucleon of the nucleus.

15. What is the mass defect of the

^{14}N nucleus?16. What is the mass of an

^{16}O atom in units of MeV/*c*^{2}? (1 MeV/*c*^{2}is the mass of a particle with rest energy 1 MeV.)**17.**(a) What is the mass defect of the^{1}H*atom*due to the binding energy of the*electron*(in the ground state)? (b) Should we worry about this mass defect when we calculate the mass of the^{1}H nucleus by subtracting the mass of one electron from the mass of the^{1}H atom?18. Show that

*c*^{2}= 931.494 MeV/u. [*Hint:*Start with the conversion factors to SI units for MeV and atomic mass units.]19. Using a mass spectrometer, the mass of the

*ion*is found to be 238.050 24 u. (a) Use this result to calculate the mass of the*nucleus.*(b) Now find the binding energy of the nucleus.20. To make an order-of-magnitude estimate of the energy level spacings in the nucleus, assume that a nucleon is confined to a one-dimensional box of width 10 fm (a typical nuclear diameter). Calculate the energy of the ground state.

29.3 Radioactivity

**21.**Identify the daughter nuclide when decays via β^{−}decay.22. Thorium-232 decays via α decay. Write out the reaction and identify the daughter nuclide.

**23.**Write out the reaction and identify the daughter nuclide when decays by electron capture.24. Write out the reaction and identify the daughter nuclide when decays by emitting a positron.

**25.**Radium-226 decays as If the nucleus is at rest before the decay and the nucleus is in its ground state,*estimate*the kinetic energy of the α particle. (Assume that the nucleus takes away an insignificant fraction of the kinetic energy.)26. Which decay mode would you expect for radioactive α, β

^{−}, or β^{+}? Explain. [*Hint:*Look at the neutron-to-proton ratio.]

**27.**Calculate the maximum kinetic energy of the β particle when decays via β^{−}decay.28. Calculate the energy of the antineutrino when decays via β

^{−}decay if the β particle has a kinetic energy of 435 keV.

**29.**Show that the spontaneous α decay of^{19}O is not possible.30. Calculate the kinetic energy of the α particle in Probelm 25. This time, do

*not*assume that the nucleus is at rest after the reaction. Start by figuring out the ratio of the kinetic energies of the α particle and the Rn nucleus.31. An isotope of sodium, decays by β

^{+}emission.*Estimate*the maximum possible kinetic energy of the positron by assuming that the kinetic energy of the daughter nucleus and the total energy of the neutrino emitted are both zero. [*Hint:*Remember to keep track of the electron masses.]32. The nucleus in a atom captures one of the atom's electrons, changing the nucleus to and emitting a neutrino. What is the total energy of the emitted neutrino? [

*Hint:*You can use the classical expression for the kinetic energy of the atom and the extremely relativistic expression for the kinetic energy of the neutrino.]

29.4 Radioactive Decay Rates and Half-Lives

**33.**A certain radioactive nuclide has a half-life of 200.0 s. A sample containing just this one radioactive nuclide has an initial activity of 80,000.0 s^{−1}. (a) What is the activity 600.0 s later? (b) How many nuclei were there initially? (c) What is the probability per second that any one of the nuclei decays?

- Page 1102
34. The half-life of I-131 is 8.0 days. A sample containing I-131 has an activity of 6.4 × 10

^{8}Bq. How many days later will the sample have an activity of 2.5 × 10^{6}Bq?Tutorial: Radioactive Decay

**35.**Some bones discovered in a crypt in Guatemala are carbon-dated. The^{14}C activity of the bones is measured to be 0.242 Bq per gram of carbon. Approximately how old are the bones?Tutorial: Radiocarbon Dating

36. Carbon-14 dating is used to date a bone found at an archaeological excavation. If the ratio of C-14 to C-12 atoms is 3.25 × 10

^{−13}, how old is the bone? [*Hint:*Note that this ratio is one fourth the ratio of 1.3 × 10^{−12}that is found in a living sample.]**37.**A sample of radioactive which has a half-life of 19.9 min, has an activity of 0.058 Ci. What is its activity 1.0 h later?

38. The activity of a sample containing radioactive

^{108}Ag is 6.4 × 10^{4}Bq. Exactly 12 min later, the activity is 2.0 × 10^{3}Bq. Calculate the half-life of^{108}Ag.

**39.**Calculate the activity of 1.0 g of radium-226 in Ci.40. What is the activity in becquerels of 1.0 kg of

^{238}U?

**41.**In this problem, you will verify the statement (in Section 29.4) that the^{14}C activity in a living sample is 0.25 Bq per gram of carbon. (a) What is the decay constant λ for^{14}C? (b) How many^{14}C atoms are in 1.00 g of carbon? One mole of carbon atoms has a mass of 12.011 g, and the relative abundance of^{14}C is 1.3 × 10^{−12}. (c) Using your results from parts (a) and (b), calculate the^{14}C activity per gram of carbon in a living sample.42. A radioactive sample has equal numbers of

^{15}O and^{19}O nuclei. Use the half-lives found in Appendix B to determine how long it will take before there are twice as many^{15}O nuclei as^{19}O. What percent of the^{19}O nuclei have decayed during this time?43. Show mathematically that if and only if

*T*_{1/2}= τ ln 2. [*Hint:*Take the natural logarithm of each side.]44. The Physics at Home in Section 29.4 suggests tossing coins as a model of radioactive decay. An improved version is to toss a large number of dice instead of coins: each die that comes up a “one” represents a nucleus that has decayed. Suppose that

*N*dice are tossed. (a) What is the average number of dice you expect to decay on one toss? (b) What is the average number of dice you expect to remain undecayed after three tosses? (c) What is the average number of dice you expect to remain undecayed after four tosses? (d) What is the half-life in numbers of tosses?

29.5 Biological Effects of Radiation

**45.**An α particle produced in radioactive α decay has a kinetic energy of typically about 6 MeV. When an α particle passes through matter (e.g., biological tissue), it makes ionizing collisions with molecules, giving up some of its kinetic energy to supply the binding energy of the electron that is removed. If a typical ionization energy for a molecule in the body is around 20 eV, roughly how many molecules can the alpha particle ionize before coming to rest?46. If meat is irradiated with 2000.0 Gy of x-rays, most of the bacteria are killed and the shelf life of the meat is greatly increased. (a) How many 100.0-keV photons must be absorbed by a 0.30-kg steak so that the absorbed dose is 2000.0 Gy? (b) Assuming steak has the same specific heat as water, what temperature increase is caused by a 2000.0-Gy absorbed dose?

47. Some types of cancer can be effectively treated by bombarding the cancer cells with high energy protons. Suppose 1.16 × 10

^{17}protons, each with an energy of 950 keV, are incident on a tumor of mass 3.82 mg. If the quality factor for these protons is 3.0, what is the biologically equivalent dose?48. Make an order-of-magnitude estimate of the amount of radon-222 gas, measured in curies, found in the lungs of an average person. Assume that 0.1 rem/yr is due to the alpha particles emitted by radon-222. The half-life is 3.8 days. You will need to calculate the energy of the alpha particles emitted.

29.6 Induced Nuclear Reactions

**49.**A certain nuclide absorbs a neutron. It then emits an electron, and then breaks up into two α particles. (a) Identify the original nuclide and the two intermediate nuclides (after absorbing the neutron and after emitting the electron). (b) Would any (anti)neutrino(s) be emitted? Explain.50. A neutron-activated sample emits gamma rays at energies that are consistent with the decay of mercury-198 nuclei from an excited state to the ground state. If the reaction that takes place is what is the nuclide “(?)” that was present in the sample before neutron activation?

Tutorial: Neutron Detector

51. Irène and Jean Frédéric Joliot-Curie, in an experiment that led to the 1935 Nobel Prize in chemistry, bombarded aluminum with α particles to form a highly unstable isotope of phosphorus, The phosphorus immediately decayed into another isotope of phosphorus, plus another product. Write out these reactions, identifying the other product.

52. The reactions listed in Problem 51 did not stop there. To the surprise of the Curies, the phosphorus decay continued after the α bombardment ended with the phosphorus emitting a β

^{+}to form yet another product. Write out this reaction, identifying the other product.

29.7 Fission

**53.**A^{235}U nucleus captures a low-energy neutron to form the compound nucleus^{236}U^{*}. Find the excitation energy of the compound nucleus. Ignore the small initial kinetic energy of the captured neutron.- Page 1103
54.

*Estimate*the energy released in the fission reaction of Eq. (29-31). Look up the binding energy per nucleon of the nuclides in Fig. 29.2. 55. Calculate the energy released in the fission reaction of Eq. (29-30). The atomic masses of and are 140.914 u and 91.926 u, respectively.

56. One possible fission reaction for

^{235}U is^{235}U + n →^{141}Cs +^{93}Rb + ?n, where “?n” represents one or more neutrons. (a) How many neutrons? (b) From the graph in Fig. 29.2, you can read the approximate binding energies per nucleon for the three nuclides involved. Use that information to*estimate*the total energy released by this fission reaction. (c) Do a precise calculation of the energy released. (d) What fraction of the rest energy of the^{235}U nucleus is released by this reaction?

29.8 Fusion

**57.**Consider the fusion reaction of a proton and a deuteron: (a) Identify the reaction product X. (b) The binding energy of the deuteron is about 1.1 MeV*per nucleon*and the binding energy of “X” is about 2.6 MeV*per nucleon.*Approximately how much energy (in MeV) is released in this fusion reaction? (c) Why is this reaction unlikely to occur in a room temperature setting?58. What is the total energy released by the proton-proton cycle [Eq. (29-34)]? (The total energy released is the total energy of the neutrinos and γ rays plus the kinetic energy of the

^{4}He nucleus minus the initial kinetic energies of the protons and electrons.)59. Estimate the minimum total kinetic energy of the

^{2}H and^{3}H nuclei necessary to allow the fusion reaction of Eq. (29-32) to take place.60. Compare the amount of energy released when 1.0 kg of the uranium isotope

^{235}U undergoes the fission reaction of Eq. (29-30) with the energy released when 1.0 kg of hydrogen undergoes the fusion reaction of Eq. (29-32).

Comprehensive Problems

**61.**Which of these unidentified nuclides are isotopes of each other? and62. What is the average binding energy per nucleon for

63. The carbon isotope

^{15}C decays much faster than^{14}C. (a) Using Appendix B, write a nuclear reaction showing the decay of^{15}C. (b) How much energy is released when^{15}C decays?64. A radioactive sample of radon has an activity of 2050 Bq. How many kilograms of radon are present?

**65.**Figure 29.7 is an energy level diagram for^{208}Tl. What are the energies of the photons emitted for the six transitions shown?66. Approximately what is the total energy of the neutrino emitted when decays by electron capture?

67. is radioactive; it α decays to is itself radioactive and has a half-life of 4.5 s. At

*t*= 0, a sample contains 4.00 mol of and 1.50 mol of At*t*= 25 μs, the sample contains 3.00 mol of and 2.50 mol of How much will there be at*t*= 50 μs?68. In 1988 the shroud of Turin, a piece of cloth that some people believe is the burial cloth of Jesus, was dated using

^{14}C. The measured^{14}C activity of the cloth was about 0.23 Bq/g. According to this activity, when was the cloth in the shroud made?**69.**Radon gas (Rn) is produced by the α decay of radium (a) How many neutrons and how many protons are present in the nucleus of the isotope of Rn produced by this decay? (b) In the air in an average size room in a student basement apartment in Ithaca, NY, there are about 10^{7}Rn nuclei. The Rn nucleus itself is radioactive; it too decays by emitting an α particle. The half-life of Rn is 3.8 days. How many α particles per second are emitted by decaying Rn nuclei in the room?70. (a) What fraction of the

^{238}U atoms present at the formation of the Earth still exist? Take the age of the Earth to be 4.5 × 10^{9}yr. (b) Answer the same question for^{235}U. Could this explain why there are more than 100 times as many^{238}U atoms as^{235}U atoms in the Earth today?71. The radioactive decay of

^{238}U produces α particles with a kinetic energy of 4.17 MeV. (a) At what speed do these α particles move? (b) Put yourself in the place of Rutherford and Geiger. You know that α particles are positively charged (from the way they are deflected in a magnetic field). You want to measure the speed of the α particles using a velocity selector. If your magnet produces a magnetic field of 0.30 T, what strength electric field would allow the α particles to pass through undeflected? (c) Now that you know the speed of the α particles, you measure the radius of their trajectory in the same magnetic field (without the electric field) to determine their charge-to-mass ratio. Using the charge and mass of the α particle, what would the radius be in a 0.30-T field? (d) Why can you determine only the charge-to-mass ratio (*q*/*m*) by this experiment, but not the individual values of*q*and*m?*72. Once Rutherford and Geiger determined the charge-to-mass ratio of the α particle (Problem 71), they performed another experiment to determine its charge. An α source was placed in an evacuated chamber with a fluorescent screen. Through a glass window in the chamber, they could see a flash on the screen every time an α particle hit it. They used a magnetic field to deflect β particles away from the screen so they were sure that every flash represented an alpha particle. (a) Why is the deflection of a β particle in a magnetic field much larger than the deflection of an α particle moving at the same speed? (b) By counting the flashes, they could determine the number of α s per second striking the screen (

*R*). Then they replaced the screen with a metal plate connected to an electroscope and measured the charge*Q*accumulated in a time Δ*t.*What is the α-particle charge in terms of*R,**Q,*and Δ*t?*- Page 1104
**73.**A water sample is found to have 0.016% deuterium content (that is, 0.016% of the hydrogen nuclei in the water are^{2}H). If the fusion reaction (^{2}H +^{2}H) yields 3.65 MeV of energy on average, how much energy could you get from 1.00 L of the water? (There are two reactions with approximately equal probabilities; one yields 4.03 MeV and the other 3.27 MeV.) Assume that you are able to extract and fuse 87.0% of the deuterium in the water. Give your answer in kilowatt hours. 74. (a) Find the approximate number of water molecules in 1.00 L of water. (b) What fraction of the liter's volume is occupied by water

*nuclei?*75. Radioactive iodine,

^{131}I, with a half-life of 8.0252 d, is used in some forms of medical diagnostics. (a) If the initial activity of a sample is 64.5 mCi, what is the mass of^{131}I in the sample? (b) What will the activity be 4.5 d later?76. An α particle with a kinetic energy of 1.0 MeV is headed straight toward a gold nucleus. (a) Find the distance of closest approach between the centers of the α particle and gold nucleus. (Assume the gold nucleus remains stationary. Since its mass is much larger than that of the α particle, this assumption is a fairly good approximation.) (b) Will the two get close enough to “touch”? (c) What is the minimum initial kinetic energy of an α particle that will make contact with the gold nucleus?

**77.**A space rock contains 3.00 g of and 0.150 g of decays to with a half-life of 1.06 × 10^{11}yr. If the rock originally contained no how old is it?78. In naturally occurring potassium, 0.0117% of the nuclei are radioactive

^{40}K. (a) What mass of^{40}K is found in a broccoli stalk containing 300 mg of potassium? (b) What is the activity of this broccoli stalk due to^{40}K?79. The power supply for a pacemaker is a small amount of radioactive

^{238}Pu. This nuclide decays by α decay with a half-life of 86 yr. The pacemaker is typically replaced every 10.0 yr. (a) By what percentage does the activity of the^{238}Pu source decrease in 10 yr? (b) The energy of the α particles emitted is 5.6 MeV. Assume an efficiency of 100%—all of the α-particle energy is used to run the pacemaker. If the pacemaker starts with 1.0 mg of^{238}Pu, what is the power output initially and after 10.0 yr?80. can α decay to the ground state of or to any of the four excited states of shown in Fig. 29.7. The maximum kinetic energy of the α particles emitted by is 6.090 MeV. What other α-particle kinetic energies are possible? [

*Hint:*Estimate the atomic mass of ]**81.**Suppose that a radioactive sample contains equal numbers of two radioactive nuclides A and B at*t*= 0. A has a half-life of 3.0 h, while B has a half-life of 12.0 h. Find the ratio of the decay rates or activities*R*_{A}/*R*_{B}at (a)*t*= 0, (b)*t*= 12.0 h, and (c)*t*= 24.0 h.82. The first nuclear reaction ever observed (in 1919 by Ernest Rutherford) was (a) Show that the reaction product “X” must be (b) For this reaction to take place, the α particle must come in contact with the nitrogen nucleus. Calculate the distance

*d*between their centers when they just make contact. (c) If the α particle and the nitrogen nucleus are initially far apart, what is the minimum value of their kinetic energy necessary to bring the two into contact? Express your answer in terms of the elementary charge*e,*the contact distance*d,*and whatever else you need. (d) Is the total kinetic energy of the reaction products more or less than the initial kinetic energy in part (c)? Why? Calculate this kinetic energy difference.83. The last step in the carbon cycle that takes place inside stars is p +

^{15}N →^{12}C + (?). This step releases 5.00 MeV of energy. (a) Show that the reaction product “(?)” must be an α particle. (b) Calculate the atomic mass of helium-4 from the information given. (c) In order for this reaction to occur, the proton must come into contact with the nitrogen nucleus. Calculate the distance*d*between their centers when they just “touch.” (d) If the proton and nitrogen nucleus are initially far apart, what is the minimum value of their total kinetic energy necessary to bring the two into contact?

Answers to Practice Problems

**29.1**(ruthenium)**29.2**17 u**29.3**1.6 × 10^{−42}m^{3}**29.4**115.492 MeV**29.5**(radium-226)**29.6**5.3044 MeV**29.7**1.3111 MeV**29.8**2.26 × 10^{12}**29.9**5300 yr ago**29.10**± 8 yr**29.11**4.4 μg**29.12**exoergic; 0.6259 MeV released**29.13**From Fig. 29.2, nuclides around*A*≈ 60 are the most tightly bound; they have the highest binding energies per nucleon. Fission cannot occur because the total mass of the daughter nuclides and any neutrons released would be*greater*than the mass of the compound nucleus. More likely, would emit an electron and one or more γ rays, leaving a stable nucleus as the final product.**29.14**1.1985 MeV

Answers to Checkpoints

**29.1**has 11 protons and 23 − 11 = 12 neutrons. The mass number is 23.**29.4**After 3 half-lives have passed, (1/2)^{3}= 1/8 of the Mn-54 nuclei remain. Therefore, during 3 half-lives, 7/8 of them decay.**29.6**Balancing the charge and nucleon numbers reveals that the intermediate nucleus is