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.
ScanSolve AI Answer
Step 1: Determine the initial number of protons and neutrons. For the initial radioactive nucleus with mass number A and atomic number Z: Number of protons (initial) = Z Number of neutrons (initial) = A - Z Step 2: Account for the emission of 3 -particles. An -particle is a helium nucleus (^4_2He). Each -particle emission reduces the mass number by 4 and the atomic number by 2. For 3 -particles: Change in mass number = 3 × 4 = 12 Change in atomic number = 3 × 2 = 6 After 3 -emissions, the intermediate nucleus has: New mass number (A') = A - 12 New atomic number (Z') = Z - 6 Step 3: Account for the emission of 2 positrons. A positron (^+ particle) is represented as ^0_1e. Each positron emission converts a proton into a neutron, so the mass number remains unchanged, but the atomic number decreases by 1. For 2 positrons: Change in mass number = 2 × 0 = 0 Change in atomic number = 2 × 1 = 2 (atomic number decreases by 2) After 2 positron emissions, the final nucleus has: Final mass number (A'') = A' - 0 = A - 12 Final atomic number (Z'') = Z' - 2 = (Z - 6) - 2 = Z - 8 Step 4: Calculate the number of protons and neutrons in the final nucleus. Number of protons in the final nucleus = Z'' = Z - 8 Number of neutrons in the final nucleus = A'' - Z'' = (A - 12) - (Z - 8) = A - 12 - Z + 8 = A - Z - 4 Step 5: Determine the ratio of the number of neutrons to the number of protons in the final nucleus. Ratio = Number of neutronsNumber of protons Ratio = (A - Z - 4)/(Z - 8) Comparing this result with the given options, it matches option (C). The final answer is (C). That's 2 down. 3 left today — send the next one.