# More general mathematical solution: T half T half. = 0.25 This is the fraction left after 25 years.

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1 Physics 07 Problem 2. O. A. Pringle Tritium has a half-life of 2.5 y against beta decay. What fraction of a sample will remain undecayed after 25 y? Simple solution: time (y) # of half-lives fraction left % % % 25% remains after 25 years. More general mathematical solution: ( t ). 0 exp(. t) ( 25) = 0.25 This is the fraction left after 25 years. Physics 07 Problem 2.2 O. A. Pringle The most probable energy of a thermal neutron is ev at room temperature. In what distance will half of a beam of eV neutrons have decayed? The half-life of the neutron is 0.3 min. In a time of 0.3 min, half of the neutrons in the beam will have decayed. We simply need to calculate how far ev neutrons travel in this time ev is a small energy, so a nonrelativistic calculation is sufficient. E neutron energy in Joules m. neutron v. 2 E m neutron v = meters/second distance v distance = meters Physics 07 Problem 2.3 O. A. Pringle Find the probability that any particular nucleus of Cl-38 will undergo beta decay in any.00-s period. The half-life of Cl-38 is 37.2 min δt The probability is δt: probability. δt probability =

2 Physics 07 Problem 2.4 O. A. Pringle The activity of a certain radionuclide decreases to 5 percent of its original value in 0 d. Find its half-life. The decay law is. 0 e (. t ) Because and T./2 are related, we can use the decay law to find the half-life. e (. ) t 0 e. t ln 0 (. t). ln 0 0 t. ln 0 t. 0 ln t After 0 days, is 5% of.0. t ln 0 t = days Physics 07 Problem 2.5 O. A. Pringle The half-life of a-24 is 5.0 h. How long does it take for 80 percent of a sample of this nuclide to decay. ln( 2) 5 0 Start with 00% % is left after 80% decayed. 0 exp(. t) Solving for t as in example 2.2 gives t. 0 ln t = hours Physics 07 Problem 2.6 O. A. Pringle The radionuclide a-24 beta-decays with a half-life of 5.0 h. A solution that contains µci of a-24 is injected into a person's bloodstream. After 4.50 h the activity of a sample of the person's blood is found to be 8.00 pci/cm 3. How many liters of blood does the person's body contain? 5.0 hours The initial activity was The first step is to find the activity after 4.5 hours: t exp(. t) = or 40.6 nci (n means nano) 2

3 The total activity remaining in the person is 40.6 nci, or 40.6x0 3 pci. One cubic centimeter of blood contains 8.00 pci. We assume the a-24 is uniformly dispersed throughout the person's blood after 4.5 hours. A simple ratio gives the total volume of blood. V total _cc V total V total = cubic centimeters or a total volume of 5.08 liters. 8 Physics 07 Problem 2.7 O. A. Pringle One g of a-226 has an activity of nearly Ci. Determine the half-life of a-226. This looks difficult at first, but let's think about what we know. We can calculate how many a-226 atoms there are in a gram of a. The activity and the number of radioactive nuclei present are related by. So if we know (which we do) and (which we can calculate), then we can calculate, and from we can calculate. First, let's calculate. In words, kg ( _gram).. a_226_atom. 000_gram _u In numbers, u kg = looks reasonable ow we use the activity to find That's Ci of activity.. = Because the activity was given in events/s, this is the time in seconds. The half-life in years is =

4 Physics 07 Problem 2.8 O. A. Pringle The mass of a milicurie of Pb-24 is 3.0x0-4 kg. Find the decay constant of Pb-24. This is similar to problem 2.7. The number of atoms in the given mass of lead is = We are given the activity: one millicurie = Physics 07 Problem 2.9 O. A. Pringle The half-life of 92 U 238 against alpha decay is 4.5x0 9 y. Find the activity of.0 g of U-238. The number of atoms in the given mass of uranium is = From the half-life we get : T..... half Don't forget to convert years to seconds. The activity is then. = disintegrations/second (or Bq) Or, in terms of Curies, = About 0.34 µci Physics 07 Problem 2.0 O. A. Pringle The potassium isotope K-40 undergoes beta decay with a half-life of.83x0 9 y. Find the number of beta decays that occur per second in.00 g of pure K-40. T..... half Half-life in seconds, using 365 days per year. 0 The number of atoms in a gram of K-40: = = events/s, or Bq 4

5 Physics 07 Problem 2. O. A. Pringle The half-life of the alpha-emitter Po-20 is 38 d. What mass of Po-20 is needed for a 0 mci source? The equation to use is =. Once we calculate, the number of Po-20 atoms, we can calculate milli Curies T.. half half-life in seconds ln( 2) = This is the number of Po-20 atoms which gives the desired activity. M mass in kg M = kg Physics 07 Problem 2.6 O. A. Pringle In Example 2.5 it is noted that the present radiocarbon activity of living things is 6 disintegrations per minute per gram of carbon content. From this figure find the ratio of C-4 to C-2 atoms in the CO 2 of the atmosphere.. We can use the above equation to find the number of radioactive carbon-4 atoms in a gram of carbon: m m m. m. 6 min. gm years days.. hours.. minutes year day hour. 6 min. gm Working out the numbers: = This is how many radioactive atoms there are in a gram of carbon. There are 6.02x0 23 atoms in mole, or 2 grams of carbon, so the fraction of carbon-4 atoms is f f =

6 Physics 07 Problem 2.7 O. A. Pringle The relative radiocarbon activity in a piece of charcoal from the remains of an ancient campfire is 0.8 that of a contemporary specimen. How long ago did the fire occur? ln( 2) 5760 t. ln.8 t = years Physics 07 Problem 2.8 O. A. Pringle atural thorium consists entirely of the alpha-radioactive isotope Th-232 which has a half-life of.4x0 0 y. If a rock sample known to have solidified 3.5 billion years ago contains 0.00 percent of Th-232 today, what was the percentage of this nuclide it contained when the rock solidified? The equation for geological dating is t. ln ln( 2) T t half Let M=the number of atoms in the rock sample. Then is the present number of Th-232 atoms, and =0.00*0-2 *M (the 0-2 comes from "percentage"). The initial number of Th-232 atoms was =f*m, where f is the initial fraction. t. ln f. x.00. x. t f.00. exp(. t ) f = ln f.00 f.00 exp(. t) Physics 07 Problem 2.2 O. A. Pringle The radionuclide 92 U 238 decays into a lead isotope through the successive emissions of eight alpha particles and six electrons. What is the symbol of the lead isotope? What is the total energy released? Each alpha particle changes A by four and Z by two. Each emission of an electron signifies the decay of a neutron to a proton, increasing Z by one. leaving A constant. Each emission of a positron signifies the decay of a proton to a neutron, decreasing Z by one, leaving A constant. The emission of eight alpha particles thus reduced Z by sixteen and A by thirty two. The emission of six electrons increased Z by six. The final A should be = 206. The final Z should be = 82. A check of the Appendix gives: 82 Pb 206, a stable isotope of 24.% relative abundance. The total energy released is proportional to the mass difference between the original U-238 less eight helium atoms, six electrons, and lead 206. m He m Pb m e m U E 8. m.. He 6 m e m Pb m U ( 93.5) The total energy released: E = MeV 6

7 Physics 07 Problem 2.22 O. A. Pringle The radionuclide U-232 alpha-decays into Th-228. (a) Find the energy released in the decay. m U m Th m He Q m. U m Th m He 93.5 Q = 5.47MeV (b) Is it possible for U-232 to decay into U-23 by emitting a neutron? m n m U Q m. U m U23 m n 93.5 Q = A negative Q means energy must be input; this reaction will not go spontaneously. (c) Is it possible for U-232 to decay into Pa-23 by emitting a proton? m H Use H mass to account for electron. m Pa Q m. U m Pa m H 93.5 Q = A negative Q means energy must be input; this reaction is therefore not possible. Physics 07 Problem 2.24 O. A. Pringle The energy liberated in the alpha decay of a-226 is 4.87 MeV. (a) Identify the daughter nuclide 88 a226 --> 86 n He 4 (b) Find the energy of the alpha particle and the recoil energy of the daughter atom. m a m n m He Q m. a m n m He 93.5 A 222 A 4 K. α A Q K α = MeV K n Q K α K n = MeV (c) If the alpha particle has the energy in (b) within the nucleus, how many of its de Broglie wavelengths fit within the nucleus. The nuclear diameter is D The de Broglie wavelength is =h/mv D =

8 mass m.. He v 2. K... α mass Don't forget 0 6 for MeV v = looks OK About 2.2 wavelengths fit inside the = D = 2.23 mass. v nucleus. (d) How many times per second does the alpha particle strike the nuclear boundary. The number of times per second is just the inverse of the time it takes the alpha particle to travel one nuclear diameter. time D v time = frequency time frequency = The alpha particle strikes 0 2 times per second Physics 07 Problem 2.25 O. A. Pringle Positron emission resembles electron emission in all respects except that the shapes of their respective energy spectra are different: there are many low energy electrons emitted, but few low energy positrons. Thus, the average electron energy in beta decay is about 0.3 K max, whereas the average positron energy is about 0.4 K max. What is the reason for this difference? Because the electron has negative charge, its kinetic energy is slightly reduced by the Coulomb attraction to the nucleus. The positron, on the other hand, is repelled by the like charge of the nucleus and accelerated outward, shifting the energy distribution to higher energies. Physics 07 Problem 2.30 O. A. Pringle Calculate the maximum energy of the electrons emitted in the beta decay of 5 B 2. Because the mass difference between the emitted particle (electron) and the recoiling nuclide is very large, we assume that the total energy released is carried by the electron. In practice appreciable energy is carried by the antineutrino so that the observed energy of the electron is rarely equal to the total energy released. The energy released is the mass difference between 5 B 2 and 6 C 2. This is because the 6 C 2 appears as an IO, with the electron flying free. To find the energy, we take the difference between the ATOMIC reaction products and the ATOMIC parent. The mass of the electron need not be calculated separately. In other words, we can use 5 B2 --> 6 C2 + + e - or 5 B2 --> 6 C 2 whichever is more convenient. m B m C m e E. max m B m C ( 93.5 ) E max = 3.37 MeV 8

9 Physics 07 Problem 2.35 O. A. Pringle The cross section for comparable neutron and proton induced nuclear reactions vary with energy in approximately the manner shown in Fig Why does the neutron cross section decrease with increasing energy whereas the proton cross section increases? The probability of capture of a neutron depends upon the time δt the neutron spends above the potential well of the nucleus. The transit time of the neutron depends inversely upon the velocity and directly upon the width. The velocity depends upon the square root of the kinetic energy. Therefore, increasing kinetic energy leads to decreasing capture probability for neutrons. The probability for proton capture at lower energies is lower because of the coulomb repulsion of the proton by the positively charged nucleus. Physics 07 Problem 2.36 O. A. Pringle A slab of absorber is exactly one mean free path thick for a beam of certain incident particles. What percentage of the particles will emerge from the slab? According to equation 2.2, the mean free path is =/nσ. Plugging into equation 2.20 gives. 0 exp( n. σ. ). 0 exp ( n. σ ) exp( ) n. σ 0 Fraction Fraction exp( ) Fraction = Physics 07 Problem 2.37 O. A. Pringle The capture cross section of Co-59 for thermal neutrons is 37b. (a) What percentage of a beam of thermal neutrons will penetrate a mm sheet of Co-59? The density of Co-59 is 8.9 x 0 3 kg/m 3. (b) What is the mean free path of thermal neutrons in Co-59? The intensity of a beam of particles varies as: ( x ). o e n. σ. x Where n is the number of target nuclei per unit volume, and σ is the interaction cross section. The number of target nuclei per unit volume is: n ρ. A n n = M The cross section is: b 0 28 σ. 37 b The fraction passing through mm (thickness=x) is: x 0 3 f.. n σ x e f = 0.75 about 7.5% The mean free path is /nσ and equals: l. n σ l = meters 3 9

10 Physics 07 Problem 2.38 O. A. Pringle The cross section for the interaction of a neutrino with matter is 0-47 m 2. Find the mean free path of neutrinos in solid iron, whose density is 7.8x0 3 kg/m 3 and whose average atomic mass in 55.9 u. Express the answer in light years, the distance light travels in free space in a year. lyr µ ρ M 55.9 σ n. n σ ρ. M µ n = Just checking n. = 25.8 lyr This distance is beyond any reasonable experimentally achievable = meters value. Any experiments must find atoms with larger neutrino cross sections. Physics 07 Problem 2.39 O. A. Pringle The boron isotope B-0 captures neutrons in an (n,α) - neutron in alpha particle out - reaction whose cross section for thermal neutrons is 4x0 3 b. The density of B-0 is 2.2x0 3 kg/m 3. What thickness of B-0 is needed to absorb 99 percent of an incident beam of thermal neutrons? M A b 0 28 σ b ρ n ρ. A M n = The fraction absorbed in distance x with density n and cross section σ is: f... n σ mm e n. σ = Solving for x as a function of the fraction to be absorbed gives the following (note that if 99 percent is absorbed, 0.0 percent goes through) x( f) ln( f) n. σ σ = x( 0.0) = meters 0

11 Physics 07 Problem 2.40 O. A. Pringle There are approximately 6x0 28 atoms/m 3 in solid aluminum. A beam of 0.5 MeV neutrons is directed at an aluminum foil 0. mm thick. If the capture cross section for neutrons of this energy in aluminum is 2x0-3 m 2, find the fraction of incident neutrons that are captured. A = b 0 28 σ n σ = b The fraction absorbed in distance x with density n and cross section σ is: f... e n σ n. σ = f = Solving for the fraction absorbed gives the following (note that if 99 percent pass through, 0.0 percent is absorbed.) Fraction absorbed: f = Physics 07 Problem 2.43 O. A. Pringle Complete these nuclear reactions: 3Li6 +? --> 4Be7 + 0n Answer: 3Li6 + H2 --> 4Be7 + 0n 7Cl35 +? --> 6S32 + 2He4 Answer: 7Cl35 +? --> 6S32 + 2He4 4Be9 + 2He4 --> 3(2He4) +? Answer: 4Be9 + 2He4 --> 3(2He4) + 0n 35Br79 + H2 -->? + 2(0n) Answer: 35Br79 + H2 --> 36Kr79 + 2(0n) Make sure that the sum of nucleons (the upper numbers) is conserved. eutrons and protons may interchange, but their number is conserved. Charge conservation is the key to the sum of the lower numbers (number of protons). Physics 07 Problem 2.54 O. A. Pringle U-235 loses about 0. percent of its mass when it undergoes fission. c (a) How much energy is released when kg of U-235 undergoes fission? E c 2 The fraction "burned" times the mass times c squared. E = The energy released in Joules.

12 (b) One ton of TT releases about 4GJ when it is detonated. How many tons of TT are equivalent in destructive power to a bomb that contains kg of U-235? E Tons Tons = Twenty two kilotons of TT. Physics 07 Problem 2.60 O. A. Pringle In their old age, heavy stars obtain part of their energy by the reaction 2He4 + 6C2 --> 8O6 How much energy does each such event give off? m He m C m O Q m. He m C m O 93.5 Q = 7.6 MeV Physics 07 Problem 2.64 O. A. Pringle Show that the fusion energy that could be liberated in H2+H2 from the deuterium in.0 kg of seawater is about 600 times greater than the 47 MJ/kg heat of combustion of gasoline. About 0.05 percent by mass of the hydrogen content of seawater is deuterium. kg of seawater contains = kg of deuterium 8 The factor 2/8 is needed because 2/8 of the mass of water is hydrogen. Beiser already calculated the energy released in the fusion of deuterium in equations 2.28 and 2.29 on page 462. This calculation is just like problem I won't repeat it here. Because Beiser asks for the amount that "could" be liberated, let's take the more energetic reaction, equation The mass of a deuterium is m.. H kg It takes 2 of these to liberate 4 MeV of energy. A kg of water contains number pairs of deuteriums number = m H2 Energy_released number (converted to Joules) Energy_released = Joules Compared with dynamite: ratio Energy_released ratio = Beiser meant 60 instead of

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