Physics Solution Manual for 1100 and 2101

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Unformatted text preview: energy of 1.712 513 u a binding energy per nucleon of 1593 F5 J g931.1 uMeV I = 1595 MeV and G H K 1595 MeV = 7.90 MeV/nucleon . 202 nucleons 14. REASONING a. The gravitational potential energy of the earth-boulder system is given by PE = mgh (Equation 6.5), where m is the boulder’s mass, g = 9.8 m/s2 is the acceleration due to gravity, and h is the height of the shaft. Equation 6.5 gives the amount of gravitational potential energy the system loses as the boulder falls to the bottom, which is the binding energy when the boulder is at the bottom of the shaft. b. The earth-boulder system loses energy as it falls, so it also loses mass. The binding energy is proportional to the change ∆m in the system’s mass, according to Equation 31.3: Binding energy = ( ∆m) c2 . Here, c = 3.00 108 m/s is the speed of light in a vacuum. SOLUTION a. The binding energy is equal to the potential energy the earth-boulder system loses when the boulder falls down the mineshaft. Therefore, we have that ( )( ) Binding energy = PE = mgh = ( 245 kg ) 9.8 m/s 2 3.0 × 103 m = 7.2 × 106 J b. Solving the expression Binding energy = ( ∆m) c2 (Equation 31.3) for ∆m, we obtain ∆m = Binding energy 7.2 ×106 J = = 8.0 ×10−11 kg 2 8 c2 ( 3.00 ×10 m/s ) 15. REASONING The mass defect ∆m is related to the total binding energy and the square of the speed c of light. Thus, to determine ∆m we will use Figure 31.5 to obtain the binding energy per nucleon for the oxygen 16 O nucleus and multiply this value by the total number 8 of nucleons (16) to find the total binding energy. Note that the data in Figure 31.5 is given in MeV per nucleon and that an MeV is not an SI unit compatible with kilograms. Thus, we will need to convert MeV to joules (J). 1594 NUCLEAR PHYSICS AND RADIOACTIVITY SOLUTION The mass defect ∆m is given by ∆m = Binding energy c2 (31.3) In Figure 31.5 we can see that the binding energy per nucleon for oxygen 16 O is 8.00 MeV 8 per nucleon. Therefore, the total binding energy for the 16 nucleons in the nucleus is Binding energy = ( 8.00 MeV/nucleon ) (16 nucleons ) = 128 MeV Using this value in Equation 31.3 and converting the energy units of MeV into joules (J), we find that ∆m = Binding energy = c2 1×106 eV 1.60 ×10−19 1 eV 1 MeV (128 MeV ) (3.00 ×108 m/s ) 2 J = 2.28 × 10−28 kg 16. REASONING The binding energy of a nucleus is the energy required to separate the nucleus into its constituent protons and neutrons. The binding energy of the nucleus is equivalent to a certain amount ∆m of mass, called the mass defect of the nucleus. According to Equation 31.3 the relation between the two is Binding energy = (∆m)c2 where c is the speed of light in a vacuum. The mass defect of a nucleus is the total mass of the separated protons and neutrons minus the mass of the intact nucleus. As discussed in the text this expression is equivalent to 1 u = 931.5 MeV. The energy required to break all the nuclei into their constituent protons and neutrons is equal to the number of atoms (and, hence, nuclei) in the metal times the binding energy of a single nucleus. As discussed in Section 14.1, the number N of nuclei (or atoms) in a material is equal to the number n of moles times Avogadro’s number NA (the number of nuclei per mole). Thus, N = nNA. However, the number of moles is equal to the mass m of the metal divided by its mass per mole. Recall that the mass per mole (in grams per mole) has the same numerical value as the atomic mass of the substance. 63 SOLUTION The binding energy of a nucleus is the mass defect for the 29 Cu nucleus, expressed in energy units of MeV (1 u = 931.5 MeV). Since the copper nucleus has 29 protons and 34 neutrons, its mass defect is Chapter 31 Problems ∆ m = 29 (1.007 825 u ) + 34 (1.008 665 u ) − 1442443 1442443 29 hydrogen atoms (protons plus electrons) 34 neutrons 62.939 598 u 14 244 4 3 1595 = 0.581 937 u Intact copper atom (including 29 electrons) The binding energy (in MeV) is 931.5 MeV Binding energy = ( 0.581 937 u ) = 542 MeV 1u The number N of nuclei in the penny is equal to the number n of moles times Avogadro’s number NA, so N = nNA. The number of moles is equal to the mass m (3.00 g) of the penny divided by its mass per mole. Since the mass (in grams) per mole has the same numerical value as the atomic mass of the substance, the mass per mole is 62.939 598 g/mol. Thus, the number of nuclei is m N = n NA = NA Mass per mole 3.00 g ( 23 22 ) = 6.02 ×10 nuclei/mol = 2.87 × 10 nuclei 62.939 598 g/mol The energy required to break all the copper nuclei into their constituent protons and neutrons is ( 542 MeV ) ( 2.87 × 10 22 nuclei ) = 1.56 × 10 25 MeV 17. SSM REASONING Since we know the difference in binding energies for the two isotopes, we can determine the corresponding mass defect. Also knowing that the isotope with the larger binding energy contains one more neutron than the other isotope gives us enough information to calculate the atomic mass difference between the two isotopes. SOLUTION The mas...
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This note was uploaded on 04/30/2013 for the course PHYS 1100 and 2 taught by Professor Chastain during the Spring '13 term at LSU.

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