This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Step 1: Understand the concept of Binding Energy. The binding energy (BE) of a nucleus is the energy required to separate it into its constituent protons and neutrons. It is calculated from the mass defect ( m), which is the difference between the total mass of the individual nucleons and the actual mass of the nucleus. The relationship is given by Einstein's mass-energy equivalence: BE = m c^2. In nuclear physics, it's common to use the conversion factor 1 amu = 931.5 MeV. The binding energy per nucleon is the total binding energy divided by the number of nucleons (protons + neutrons) in the nucleus. Given masses: Mass of deuteron M(^2_1H) = 2.013554 amu Mass of alpha particle M(^4_2He) = 4.002604 amu Mass of proton M_p = 1.007825 amu Mass of neutron M_n = 1.008665 amu i) Calculate the binding energy per nucleon when two deuterons combine to form an -particle. The nuclear reaction is: ^2_1H + ^2_1H ^4_2He Step 1.1: Calculate the total mass of the reactants. M_reactants = 2 × M(^2_1H) M_reactants = 2 × 2.013554 amu M_reactants = 4.027108 amu Step 1.2: Identify the mass of the product. M_product = M(^4_2He) = 4.002604 amu Step 1.3: Calculate the mass defect ( m). m = M_reactants - M_product m = 4.027108 amu - 4.002604 amu m = 0.024504 amu Step 1.4: Calculate the total binding energy (BE). BE = m × 931.5 MeVamu BE = 0.024504 amu × 931.5 MeVamu BE = 22.828276 MeV Step 1.5: Calculate the binding energy per nucleon. The -particle (^4_2He) has 4 nucleons (2 protons and 2 neutrons). BE_per nucleon = (BE)/(Number of nucleons) BE_per nucleon = 22.828276 MeV4 5.7071 MeV/nucleon ii) Calculate the binding energy per nucleon when two neutrons and two protons combine to form an -particle. The nuclear reaction is: 2n + 2p ^4_2He Step 2.1: Calculate the total mass of the reactants. M_reactants = (2 × M_n) + (2 × M_p) M_reactants = (2 × 1.008665 amu) + (2 × 1.007825 amu) M_reactants = 2.017330 amu + 2.015650 amu M_reactants = 4.032980 amu Step 2.2: Identify the mass of the product. M_product = M(^4_2He) = 4.002604 amu Step 2.3: Calculate the mass defect ( m). m = M_reactants - M_product m = 4.032980 amu - 4.002604 amu m = 0.030376 amu Step 2.4: Calculate the total binding energy (BE). BE = m × 931.5 MeVamu BE = 0.030376 amu × 931.5 MeVamu BE = 28.298964 MeV Step 2.5: Calculate the binding energy per nucleon. The -particle (^4_2He) has 4 nucleons. BE_per nucleon = (BE)/(Number of nucleons) BE_per nucleon = 28.298964 MeV4 7.0747 MeV/nucleon That's 2 down. 3 left today — send the next one.