How much energy is released when a 238U nucleus decays by

A 7 Li nucleus with a kinetic energy of 3.00 MeV is sent toward a 232Th nucleus. What is the least center-to-center separation between the two nuclei, assuming that the (more massive) 232Th nucleus does not move?

Calculate the distance of closest approach for a head-on collision between a 5.30 MeV alpha particle and a copper nucleus

A 10.2 MeV Li nucleus is shot directly at the center of a Ds nucleus. At what center-to-center distance does the Li momentarily stop, assuming the Ds does not move?

In a Rutherford scattering experiment, assume that an incident alpha particle (radius 1.80 fm) is headed directly toward a target gold nucleus (radius 6.23 fm). What energy must the alpha particle have to just barely touch the gold nucleus?

When an alpha particle collides elastically with a nucleus, the nucleus recoils. Suppose a 5.00 MeV alpha particle has a headon elastic collision with a gold nucleus that is initially at rest. What is the kinetic energy of (a) the recoiling nucleus and (b) the rebounding alpha particle?

The strong neutron excess (defined as N # Z) of high-mass nuclei is illustrated by noting that most high-mass nuclides could never fission into two stable nuclei without neutrons being left over. For example, consider the spontaneous fission of a 235U nucleus into two stable daughter nuclei with atomic numbers 39 and 53. From Appendix F, determine the name of the (a) first and (b) second daughter nucleus. From Fig. 42-5, approximately how many neutrons are in the (c) first and (d) second? (e) Approximately how many neutrons are left over?

What is the nuclear mass density rm of (a) the fairly low-mass nuclide 55Mn and (b) the fairly high- mass nuclide 209Bi? (c) Compare the two answers, with an explanation.What is the nuclear charge density rq of (d) 55Mn and (e) 209Bi? (f) Compare the two answers, with an explanation

(a) Show that the mass M of an atom is given approximately by Mapp ! Amp, where A is the mass number and mp is the proton mass. For (b) 1 H, (c) 31P, (d) 120Sn, (e) 197Au, and (f) 239Pu, use Table 42-1 to find the percentage deviation between Mapp and M: (g) Is a value of Mapp accurate enough to be used in a calculation of a nuclear binding energy?

The nuclide 14C contains (a) how many protons and (b) how many neutrons?

What is the mass excess )1 of 1 H (actual mass is 1.007 825 u) in (a) atomic mass units and (b) MeV/c 2 ? What is the mass excess )n of a neutron (actual mass is 1.008 665 u) in (c) atomic mass units and (d) MeV/c 2 ? What is the mass excess )120 of 120Sn (actual mass is 119.902 197 u) in (e) atomic mass units and (f) MeV/c 2 ?

Nuclear radii may be measured by scattering highenergy (high speed) electrons from nuclei. (a) What is the de Broglie wavelength for 200 MeV electrons? (b) Are these electrons suitable probes for this purpose?

The electric potential energy of a uniform sphere of charge q and radius r is given by (a) Does the energy represent a tendency for the sphere to bind together or blow apart? The nuclide 239Pu is spherical with radius 6.64 fm. For this nuclide, what are (b) the electric potential energy U according to the equation, (c) the electric potential energy per proton, and (d) the electric potential energy per nucleon? The binding energy per nucleon is 7.56 MeV. (e) Why is the nuclide bound so well when the answers to (c) and (d) are large and positive?

A neutron star is a stellar object whose density is about that of nuclear matter, 2 " 1017 kg/m3 . Suppose that the Sun were to collapse and become such a star without losing any of its present mass. What would be its radius?

What is the binding energy per nucleon of the americium isotope Am? Here are some atomic masses and the neutron mass. 244.064 279 u 1 H 1.007 825 u n 1.008 665 u

(a) Show that the energy associated with the strong force between nucleons in a nucleus is proportional to A, the mass number of the nucleus in question. (b) Show that the energy associated with the Coulomb force between protons in a nucleus is proportional to Z(Z # 1). (c) Show that, as we move to larger and larger nuclei (see Fig. 42-5), the importance of the Coulomb force increases more rapidly than does that of the strong force

What is the binding energy per nucleon of the europium isotope Eu? Here are some atomic masses and the neutron mass. 151.921 742 u 1 H 1.007 825 u n 1.008 665 u

Because the neutron has no charge, its mass must be found in some way other than by using a mass spectrometer.When a neutron and a proton meet (assume both to be almost stationary), they combine and form a deuteron, emitting a gamma ray whose energy is 2.2233 MeV. The masses of the proton and the deuteron are 1.007 276 467 u and 2.013 553 212 u, respectively. Find the mass of the neutron from these data.

What is the binding energy per nucleon of the rutherfordium isotope Rf? Here are some atomic masses and the neutron mass. 259.105 63 u 1.008 665 u

A periodic table might list the average atomic mass of magnesium as being 24.312 u, which is the result of weighting the atomic masses of the magnesium isotopes according to their natural abundances on Earth. The three isotopes and their masses are 24Mg (23.985 04 u), 25Mg (24.985 84 u), and 26Mg (25.982 59 u). The natural abundance of 24Mg is 78.99% by mass (that is, 78.99% of the mass of a naturally occurring sample of magnesium is due to the presence of 24Mg). What is the abundance of (a) 25Mg and (b) 26Mg?

What is the binding energy per nucleon of 262Bh? The mass of the atom is 262.1231 u.

(a) Show that the total binding energy Ebe of a given nuclide is Ebe ! Z)H ' N)n # ), where )H is the mass excess of 1 H,)n is the mass excess of a neutron, and ) is the mass excess of the given nuclide. (b) Using this method, calculate the binding energy per nucleon for 197Au. Compare your result with the value listed in Table 42-1. The needed mass excesses, rounded to three significant figures, are )H ! '7.29 MeV, )n ! '8.07 MeV, and )197 ! #31.2 MeV. Note the economy of calculation that results when mass excesses are used in place of the actual masses

An a particle (4 He nucleus) is to be taken apart in the following steps. Give the energy (work) required for each step: (a) remove a proton, (b) remove a neutron, and (c) separate the remaining proton and neutron. For an a particle, what are (d) the total binding energy and (e) the binding energy per nucleon? (f) Does either match an answer to (a), (b), or (c)? Here are some atomic masses and the neutron mass. 4 He 4.002 60 u 2 H 2.014 10 u 3 H 3.016 05 u 1 H 1.007 83 u n 1.008 67 u

Verify the binding energy per nucleon given in Table 42-1 for the plutonium isotope 239Pu. The mass of the neutral atom is 239.052 16 u.

A penny has a mass of 3.0 g. Calculate the energy that would be required to separate all the neutrons and protons in this coin from one another. For simplicity, assume that the penny is made entirely of 63Cu atoms (of mass 62.929 60 u).The masses of the protonplus-electron and the neutron are 1.007 83 u and 1.008 66 u, respectively.

Cancer cells are more vulnerable to x and gamma radiation than are healthy cells. In the past, the standard source for radiation therapy was radioactive 60Co, which decays, with a half-life of 5.27 y, into an excited nuclear state of 60Ni. That nickel isotope then immediately emits two gamma-ray photons, each with an approximate energy of 1.2 MeV. How many radioactive 60Co nuclei are present in a 6000 Ci source of the type used in hospitals? (Energetic particles from linear accelerators are now used in radiation therapy.)

The half-life of a radioactive isotope is 140 d. How many days would it take for the decay rate of a sample of this isotope to fall to one-fourth of its initial value?

A radioactive nuclide has a half-life of 30.0 y. What fraction of an initially pure sample of this nuclide will remain undecayed at the end of (a) 60.0 y and (b) 90.0 y?

The plutonium isotope 239Pu is produced as a by-product in nuclear reactors and hence is accumulating in our environment. It is radioactive,decaying with a half-life of 2.41 " 104 y.(a) How many nuclei of Pu constitute a chemically lethal dose of 2.00 mg? (b) What is the decay rate of this amount?

A radioactive isotope of mercury, 197Hg, decays to gold, 197Au, with a disintegration constant of 0.0108 h#1 . (a) Calculate the half-life of the 197Hg.What fraction of a sample will remain at the end of (b) three half-lives and (c) 10.0 days?

The half-life of a particular radioactive isotope is 6.5 h. If there are initially 48 " 1019 atoms of this isotope, how many remain at the end of 26 h?

Consider an initially pure 3.4 g sample of 67Ga, an isotope that has a half-life of 78 h. (a) What is its initial decay rate? (b) What is its decay rate 48 h later?

When aboveground nuclear tests were conducted, the explosions shot radioactive dust into the upper atmosphere. Global air circulations then spread the dust worldwide before it settled out on ground and water. One such test was conducted in October 1976. What fraction of the 90Sr produced by that explosion still existed in October 2006? The half-life of 90Sr is 29 y

The air in some caves includes a significant amount of radon gas, which can lead to lung cancer if breathed over a prolonged time. In British caves, the air in the cave with the greatest amount of the gas has an activity per volume of 1.55 " 105 Bq/m3 . Suppose that you spend two full days exploring (and sleeping in) that cave. Approximately how many 222Rn atoms would you take in and out of your lungs during your two-day stay? The radionuclide 222Rn in radon gas has a half-life of 3.82 days. You need to estimate your lung capacity and average breathing rate

Calculate the mass of a sample of (initially pure) 40K that has an initial decay rate of 1.70 " 105 disintegrations/s.The isotope has a half-life of 1.28 " 109 y.

A certain radionuclide is being manufactured in a cyclotron at a constant rate R. It is also decaying with disintegration constant l. Assume that the production process has been going on for a time that is much longer than the half-life of the radionuclide. (a) Show that the number of radioactive nuclei present after such time remains constant and is given by N ! R/l. (b) Now show that this result holds no matter how many radioactive nuclei were present initially. The nuclide is said to be in secular equilibrium with its source; in this state its decay rate is just equal to its production rate.

Plutonium isotope 239Pu decays by alpha decay with a halflife of 24 100 y. How many milligrams of helium are produced by an initially pure 12.0 g sample of 239Pu at the end of 20 000 y? (Consider only the helium produced directly by the plutonium and not by any by-products of the decay process.)

The radionuclide 64Cu has a half-life of 12.7 h. If a sample contains 5.50 g of initially pure 64Cu at t ! 0, how much of it will decay between t ! 14.0 h and t ! 16.0 h?

A dose of 8.60 mCi of a radioactive isotope is injected into a patient. The isotope has a half-life of 3.0 h. How many of the isotope parents are injected?

The radionuclide 56Mn has a half-life of 2.58 h and is produced in a cyclotron by bombarding a manganese target with deuterons. The target contains only the stable manganese isotope 55Mn, and the manganesedeuteron reaction that produces 56Mn is If the bombardment lasts much longer than the half-life of 56Mn, the activity of the 56Mn produced in the target reaches a final value of 8.88 " 1010 Bq. (a) At what rate is 56Mn being produced? (b) How many 56Mn nuclei are then in the target? (c) What is their total mass?

A source contains two phosphorus radionuclides, 32P (T1/2 ! 14.3 d) and 33P (T1/2 ! 25.3 d). Initially, 10.0% of the decays come from 33P.How long must one wait until 90.0% do so?

A 1.00 g sample of samarium emits alpha particles at a rate of 120 particles/s. The responsible isotope is 147Sm, whose natural abundance in bulk samarium is 15.0%. Calculate the half-life

What is the activity of a 20 ng sample of 92Kr, which has a half-life of 1.84 s?

A radioactive sample intended for irradiation of a hospital patient is prepared at a nearby laboratory. The sample has a halflife of 83.61 h.What should its initial activity be if its activity is to be 7.4 " 108 Bq when it is used to irradiate the patient 24 h later?

Figure 42-19 shows the decay of parents in a radioactive sample. The axes are scaled by Ns ! 2.00 " 106 and ts ! 10.0 s. What is the activity of the sample at t ! 27.0 s?

In 1992, Swiss police arrested two men who were attempting to smuggle osmium out of Eastern Europe for a clandestine sale. However, by error, the smugglers had picked up 137Cs. Reportedly, each smuggler was carrying a 1.0 g sample of 137Cs in a pocket! In (a) bequerels and (b) curies, what was the activity of each sample? The isotope 137Cs has a half-life of 30.2 y. (The activities of radioisotopes commonly used in hospitals range up to a few millicuries.)

The radioactive nuclide 99Tc can be injected into a patients bloodstream in order to monitor the blood flow, measure the blood volume, or find a tumor, among other goals. The nuclide is produced in a hospital by a cow containing 99Mo, a radioactive nuclide that decays to 99Tc with a half-life of 67 h. Once a day, the cow is milked for its 99Tc, which is produced in an excited state by the 99Mo; the 99Tc de-excites to its lowest energy state by emitting a gamma-ray photon, which is recorded by detectors placed around the patient. The de-excitation has a half-life of 6.0 h. (a) By what process does 99Mo decay to 99Tc? (b) If a patient is injected with an 8.2 " 107 Bq sample of 99Tc, how many gamma-ray photons are initially produced within the patient each second? (c) If the emission rate of gamma-ray photons from a small tumor that has collected 99Tc is 38 per second at a certain time, how many excitedstate 99Tc are located in the tumor at that time?

After long effort, in 1902 Marie and Pierre Curie succeeded in separating from uranium ore the first substantial quantity of radium, one decigram of pure RaCl2. The radium was the radioactive isotope 226Ra, which has a half-life of 1600 y. (a) How many radium nuclei had the Curies isolated? (b) What was the decay rate of their sample, in disintegrations per second?

How much energy is released when a 238U nucleus decays by emitting (a) an alpha particle and (b) a sequence of neutron, proton, neutron, proton? (c) Convince yourself both by reasoned argument and by direct calculation that the difference between these two numbers is just the total binding energy of the alpha particle. (d) Find that binding energy. Some needed atomic and particle masses are 238U 238.050 79 u 234Th 234.043 63 u 237U 237.048 73 u 4 He 4.002 60 u 236Pa 236.048 91 u 1 H 1.007 83 u 235Pa 235.045 44 u n 1.008 66 u

Generally, more massive nuclides tend to be more unstable to alpha decay. For example, the most stable isotope of uranium, 238U, has an alpha decay half-life of 4.5 " 109 y.The most stable isotope of plutonium is 244Pu with an 8.0 " 107 y half-life, and for curium we have 248Cm and 3.4 " 105 y. When half of an original sample of 238U has decayed,what fraction of the original sample of (a) plutonium and (b) curium is left?

Large radionuclides emit an alpha particle rather than other combinations of nucleons because the alpha particle has such a stable, tightly bound structure. To confirm this statement, calculate the disintegration energies for these hypothetical decay processes and discuss the meaning of your findings: (a) (b) (c) The needed atomic masses are 232Th 232.0381 u 3 He 3.0160 u 231Th 231.0363 u 4 He 4.0026 u 230Th 230.0331 u 5 He 5.0122 u 235U 235.0429 u

A 238U nucleus emits a 4.196 MeV alpha particle. Calculate the disintegration energy Q for this process, taking the recoil energy of the residual 234Th nucleus into account.

Under certain rare circumstances, a nucleus can decay by emitting a particle more massive than an alpha particle. Consider the decays and Calculate the Q value for the (a) first and (b) second decay and determine that both are energetically possible. (c) The Coulomb barrier height for alpha-particle emission is 30.0 MeV. What is the barrier height for 14C emission? (Be careful about the nuclear radii.) The needed atomic masses are 223Ra 223.018 50 u 14C 14.003 24 u 209Pb 208.981 07 u 4 He 4.002 60 u 219Rn 219.009 48 u

The cesium isotope 137Cs is present in the fallout from aboveground detonations of nuclear bombs. Because it decays with a slow (30.2 y) half-life into 137Ba, releasing considerable energy in the process, it is of environmental concern. The atomic masses of the Cs and Ba are 136.9071 and 136.9058 u, respectively; calculate the total energy released in such a decay.

Some radionuclides decay by capturing one of their own atomic electrons, a K-shell electron, say.An example is 49V ' e# : 49Ti ' <, T1/2 ! 331 d Show that the disintegration energy Q for this process is given by Q ! (mV # mTi)c 2 # EK, where mV and mTi are the atomic masses of 49V and 49Ti, respectively, and EK is the binding energy of the vanadium K-shell electron. (Hint: Put mV and mTi as the corresponding nuclear masses and then add in enough electrons to use the atomic masses.)

A free neutron decays according to Eq. 42-26. If the neutronhydrogen atom mass difference is 840 mu, what is the maximum kinetic energy Kmax possible for the electron produced in a neutron decay?

An electron is emitted from a middle-mass nuclide (A ! 150, say) with a kinetic energy of 1.0 MeV. (a) What is its de Broglie wavelength? (b) Calculate the radius of the emitting nucleus. (c) Can such an electron be confined as a standing wave in a box of such dimensions? (d) Can you use these numbers to disprove the (abandoned) argument that electrons actually exist in nuclei?

The radionuclide 11C decays according to The maximum energy of the emitted positrons is 0.960 MeV. (a) Show that the disintegration energy Q for this process is given by Q ! (mC # mB # 2me)c 2 , where mC and mB are the atomic masses of 11C and 11B, respectively, and me is the mass of a positron. (b) Given the mass values mC ! 11.011 434 u, mB ! 11.009 305 u, and me ! 0.000 548 6 u, calculate Q and compare it with the maximum energy of the emitted positron given above. (Hint: Let mC and mB be the nuclear masses and then add in enough electrons to use the atomic masses.)

Two radioactive materials that alpha decay, 238U and 232Th, and one that beta decays, 40K, are sufficiently abundant in granite to contribute significantly to the heating of Earth through the decay energy produced. The alpha-decay isotopes give rise to decay chains that stop when stable lead isotopes are formed. The isotope 40K has a single beta decay. (Assume this is the only possible decay of that isotope.) Here is the information: Stable Decay Half-Life End Q f Parent Mode (y) Point (MeV) (ppm) 238U a 4.47 " 109 206Pb 51.7 4 232Th a 1.41 " 1010 208Pb 42.7 13 40K b 1.28 " 109 40Ca 1.31 4 In the table Q is the total energy released in the decay of one parent nucleus to the final stable end point and f is the abundance of the isotope in kilograms per kilogram of granite; ppm means parts per million. (a) Show that these materials produce energy as heat at the rate of 1.0 " 10#9 W for each kilogram of granite. (b) Assuming that there is 2.7 " 1022 kg of granite in a 20-km- thick spherical shell at the surface of Earth, estimate the power of this decay process over all of Earth. Compare this power with the total solar power intercepted by Earth, 1.7 " 1017 W.

The radionuclide 32P decays to 32S as described by Eq. 42-24. In a particular decay event, a 1.71 MeV electron is emitted, the maximum possible value. What is the kinetic energy of the recoiling 32S atom in this event? (Hint: For the electron it is necessary to use the relativistic expressions for kinetic energy and linear momentum.The 32S atom is nonrelativistic.)

A 5.00 g charcoal sample from an ancient fire pit has a 14C activity of 63.0 disintegrations/min. A living tree has a 14C activity of 15.3 disintegrations/min per 1.00 g. The half-life of 14C is 5730 y. How old is the charcoal sample?

The isotope 238U decays to 206Pb with a half-life of 4.47 " 109 y.Although the decay occurs in many individual steps, the first step has by far the longest half-life; therefore, one can often consider the decay to go directly to lead.That is, A rock is found to contain 4.20 mg of 238U and 2.135 mg of 206Pb. Assume that the rock contained no lead at formation, so all the lead now present arose from the decay of uranium. How many atoms of (a) 238U and (b) 206Pb does the rock now contain? (c) How many atoms of 238U did the rock contain at formation? (d) What is the age of the rock?

A particular rock is thought to be 260 million years old. If it contains 3.70 mg of 238U, how much 206Pb should it contain? See Problem 61.

A rock recovered from far underground is found to contain 0.86 mg of 238U, 0.15 mg of 206Pb, and 1.6 mg of 40Ar. How much 40K will it likely contain? Assume that 40K decays to only 40Ar with a half- life of 1.25 " 109 y. Also assume that 238U has a half-life of 4.47 " 109 y.

The isotope 40K can decay to either 40Ca or 40Ar; assume both decays have a half-life of 1.26 " 109 y. The ratio of the Ca produced to the Ar produced is 8.54/1 ! 8.54. A sample originally had only 40K. It now has equal amounts of 40K and 40Ar; that is, the ratio of K to Ar is 1/1 ! 1. How old is the sample? (Hint: Work this like other radioactive-dating problems, except that this decay has two products.)

The nuclide 198Au, with a half-life of 2.70 d, is used in cancer therapy. What mass of this nuclide is required to produce an activity of 250 Ci?

A radiation detector records 8700 counts in 1.00 min. Assuming that the detector records all decays, what is the activity of the radiation source in (a) becquerels and (b) curies?

An organic sample of mass 4.00 kg absorbs 2.00 mJ via slow neutron radiation (RBE ! 5). What is the dose equivalent (mSv)?

A 75 kg person receives a whole-body radiation dose of 2.4 " 10#4 Gy, delivered by alpha particles for which the RBE factor is 12. Calculate (a) the absorbed energy in joules and the dose equivalent in (b) sieverts and (c) rem.

An 85 kg worker at a breeder reactor plant accidentally ingests 2.5 mg of 239Pu dust. This isotope has a half-life of 24 100 y, decaying by alpha decay. The energy of the emitted alpha particles is 5.2 MeV, with an RBE factor of 13. Assume that the plutonium resides in the workers body for 12 h (it is eliminated naturally by the digestive system rather than being absorbed by any of the internal organs) and that 95% of the emitted alpha particles are stopped within the body. Calculate (a) the number of plutonium atoms ingested, (b) the number that decay during the 12 h, (c) the energy absorbed by the body, (d) the resulting physical dose in grays, and (e) the dose equivalent in sieverts.

A typical kinetic energy for a nucleon in a middle-mass nucleus may be taken as 5.00 MeV. To what effective nuclear temperature does this correspond, based on the assumptions of the collective model of nuclear structure?

A measurement of the energy E of an intermediate nucleus must be made within the mean lifetime )t of the nucleus and necessarily carries an uncertainty )E according to the uncertainty principle )E . )t . (a) What is the uncertainty )E in the energy for an intermediate nucleus if the nucleus has a mean lifetime of 10#22 s? (b) Is the nucleus a compound nucleus?

In the following list of nuclides, identify (a) those with filled nucleon shells, (b) those with one nucleon outside a filled shell, and (c) those with one vacancy in an otherwise filled shell: 13C, 18O, 40K, 49Ti, 60Ni, 91Zr, 92Mo, 121Sb, 143Nd, 144Sm, 205Tl, and 207Pb.

Consider the three formation processes shown for the compound nucleus 20Ne in Fig. 42-14. Here are some of the atomic and particle masses: 20Ne 19.992 44 u a 4.002 60 u 19F 18.998 40 u p 1.007 83 u 16O 15.994 91 u What energy must (a) the alpha particle, (b) the proton, and (c) the g-ray photon have to provide 25.0 MeV of excitation energy to the compound nucleus?

In a certain rock, the ratio of lead atoms to uranium atoms is 0.300.Assume that uranium has a half- life of 4.47 109 y and that the rock had no lead atoms when it formed. How old is the rock?

A certain stable nuclide, after absorbing a neutron, emits an electron, and the new nuclide splits spontaneously into two alpha particles. Identify the nuclide

A typical chest x-ray radiation dose is 250 mSv, delivered by x rays with an RBE factor of 0.85. Assuming that the mass of the exposed tissue is one-half the patients mass of 88 kg, calculate the energy absorbed in joules.

How many years are needed to reduce the activity of 14C to 0.020 of its original activity? The half-life of 14C is 5730 y.

Radioactive element AA can decay to either element BB or element CC.The decay depends on chance, but the ratio of the resulting number of BB atoms to the resulting number of CC atoms is always 2/1.The decay has a half-life of 8.00 days.We start with a sample of pure AA. How long must we wait until the number of CC atoms is 1.50 times the number of AA atoms?

One of the dangers of radioactive fallout from a nuclear bomb is its 90Sr, which decays with a 29-year half-life. Because it has chemical properties much like those of calcium, the strontium, if ingested by a cow, becomes concentrated in the cows milk. Some of the 90Sr ends up in the bones of whoever drinks the milk. The energetic electrons emitted in the beta decay of 90Sr damage the bone marrow and thus impair the production of red blood cells. A 1 megaton bomb produces approximately 400 g of 90Sr. If the fallout spreads uniformly over a 2000 km2 area, what ground area would hold an amount of radioactivity equal to the allowed limit for one person, which is 74 000 counts/s?

Because of the 1986 explosion and fire in a reactor at the Chernobyl nuclear power plant in northern Ukraine, part of Ukraine is contaminated with 137Cs, which undergoes beta-minus decay with a half- life of 30.2 y. In 1996, the total activity of this contamination over an area of 2.6 " 105 km2 was estimated to be 1 " 1016 Bq. Assume that the 137Cs is uniformly spread over that area and that the beta-decay electrons travel either directly upward or directly downward. How many beta-decay electrons would you intercept were you to lie on the ground in that area for 1 h (a) in 1996 and (b) today? (You need to estimate your cross-sectional area that intercepts those electrons.)

Figure 42-20 shows part of the decay scheme of 237Np on a plot of mass number A versus proton number Z; five lines that represent either alpha decay or beta-minus decay connect dots that represent isotopes.What is the isotope at the end of the five decays (as marked with a question mark in Fig. 42-20)?

After a brief neutron irradiation of silver, two isotopes are present: 108Ag (T1/2 ! 2.42 min) with an initial decay rate of 3.1 " 105 /s, and 110Ag (T1/2 ! 24.6 s) with an initial decay rate of 4.1 " 106 /s. Make a semilog plot similar to Fig. 42-9 showing the total combined decay rate of the two isotopes as a function of time from t ! 0 until t ! 10 min. We used Fig. 42-9 to illustrate the extraction of the half-life for simple (one isotope) decays. Given only your plot of total decay rate for the two-isotope system here, suggest a way to analyze it in order to find the half-lives of both isotopes.

Because a nucleon is confined to a nucleus, we can take the uncertainty in its position to be approximately the nuclear radius r. Use the uncertainty principle to determine the uncertainty )p in the linear momentum of the nucleon. Using the approximation p % )p and the fact that the nucleon is nonrelativistic, calculate the kinetic energy of the nucleon in a nucleus with A ! 100.

A radium source contains 1.00 mg of 226Ra, which decays with a half-life of 1600 y to produce 222Rn, a noble gas. This radon isotope in turn decays by alpha emission with a half-life of 3.82 d. If this process continues for a time much longer than the half-life of 222Rn, the 222Rn decay rate reaches a limiting value that matches the rate at which 222Rn is being produced, which is approximately constant because of the relatively long half-life of 226Ra. For the source under this limiting condition, what are (a) the activity of 226Ra, (b) the activity of 222Rn, and (c) the total mass of 222Rn?

Make a nuclidic chart similar to Fig. 42-6 for the 25 nuclides 118122Te, 117121Sb, 116120Sn, 115119In, and 114118Cd. Draw in and la-bel (a) all isobaric (constant A) lines and (b) all lines of constant neutron excess, defined as N # Z.

A projectile alpha particle is headed directly toward a target aluminum nucleus. Both objects are assumed to be spheres. What energy is required of the alpha particle if it is to momentarily stop just as its surface touches the surface of the aluminum nucleus? Assume that the target nucleus remains stationary.

Consider a 238U nucleus to be made up of an alpha particle (4 He) and a residual nucleus (234Th). Plot the electrostatic potential energy U(r), where r is the distance between these particles. Cover the approximate range 10 fm / r / 100 fm and compare your plot with that of Fig. 42-10.

Characteristic nuclear time is a useful but loosely defined quantity, taken to be the time required for a nucleon with a few million electron-volts of kinetic energy to travel a distance equal to the diameter of a middle-mass nuclide. What is the order of magnitude of this quantity? Consider 5 MeV neutrons traversing a nuclear diameter of 197Au; use Eq. 42-3.

What is the likely mass number of a spherical nucleus with a radius of 3.6 fm as measured by electron-scattering methods?

Using a nuclidic chart, write the symbols for (a) all stable isotopes with Z ! 60, (b) all radioactive nuclides with N ! 60, and (c) all nuclides with A ! 60.

If the unit for atomic mass were defined so that the mass of 1 H were exactly 1.000 000 u, what would be the mass of (a) 12C (actual mass 12.000 000 u) and (b) 238U (actual mass 238.050 785 u)?

High-mass radionuclides, which may be either alpha or beta emitters, belong to one of four decay chains, depending on whether their mass number A is of the form 4n, 4n ' 1, 4n ' 2, or 4n ' 3, where n is a positive integer. (a) Justify this statement and show that if a nuclide belongs to one of these families, all its decay products belong to the same family. Classify the following nuclides as to family: (b) 235U, (c) 236U, (d) 238U, (e) 239Pu, (f) 240Pu, (g) 245Cm, (h) 246Cm, (i) 249Cf, and (j) 253Fm.

From data presented in the first few paragraphs of Module 42-3, find (a) the disintegration constant l and (b) the half-life of 238U.