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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Here are the solutions to the questions: a) i) The half-life of 4.5 × 10^9 years means that it takes 4.5 × 10^9 years for half of the initial amount of a radioactive sample of Uranium-238 to decay. a) ii) The radioactive decay equation for Uranium-238 by alpha emission is: ^238_92U ^234_90Th + ^4_2He a) iii) Step 1: Identify the given values and the decay formula. Initial mass (N_0) = 5 g Final mass (N) = 10\% of 5 g = 0.10 × 5 g = 0.5 g Half-life (t_1/2) = 4.5 × 10^9 years The radioactive decay formula is N = N_0 ((1)/(2))^t/t_1/2. Step 2: Substitute the values into the formula. 0.5 = 5 ((1)/(2))^t/(4.5 × 10^9) Step 3: Isolate the exponential term. (0.5)/(5) = ((1)/(2))^t/(4.5 × 10^9) 0.1 = ((1)/(2))^t/(4.5 × 10^9) Step 4: Take the natural logarithm of both sides. (0.1) = (((1)/(2))^t/(4.5 × 10^9)) (0.1) = (t)/(4.5 × 10^9) (0.5) Step 5: Solve for t. t = ((0.1))/((0.5)) × 4.5 × 10^9 years t = (-2.302585)/(-0.693147) × 4.5 × 10^9 years t ≈ 3.3219 × 4.5 × 10^9 years t ≈ 1.495 × 10^10 years a) iv) A. The same element can have different atomic masses because of the existence of isotopes. Isotopes are atoms of the same element that have the same number of protons (and thus the same atomic number) but different numbers of neutrons, leading to different mass numbers. a) iv) B. Step 1: Use the formula for relative atomic mass (RAM). RAM = (mass of isotope × relative abundance)100 Step 2: Substitute the given data for Barium. RAM = ((134 × 2.2) + (135 × 5.5) + (136 × 7.7) + (137 × 12.1) + (138 × 72.5))/(100) Step 3: Calculate the sum of the products. RAM = (294.8 + 742.5 + 1047.2 + 1657.7 + 10005)/(100) RAM = (13747.2)/(100) Step 4: Calculate the relative atomic mass. RAM = 137.472 b) i) A rate law is an equation that expresses the relationship between the rate of a chemical reaction and the concentrations of the reactants, raised to appropriate powers (reaction orders), at a specific temperature. b) ii) Step 1: Determine the order with respect to each reactant based on the given information. • Doubling the concentration of A_2 doubles the reaction rate, so the reaction is first order with respect to A_2. (Rate [A_2]^1) • Tripling the concentration of B triples the reaction rate, so the reaction is first order with respect to B. (Rate [B]^1) • Doubling the concentration of C has no effect on the rate, so the reaction is zero order with respect to C. (Rate [C]^0) Step 2: Combine the orders to write the rate law. Rate = k[A_2]^1[B]^1[C]^0 Rate = k[A_2][B] b) iii) A mechanism consistent with the rate law Rate = k[A_2][B] would have the slowest (rate-determining) step involving one molecule of A_2 and one molecule of B. Reactant C is not involved in the rate-determining step. Proposed Mechanism: Step 1 (Slow): A_2 + B AB + A (Rate-determining step) Step 2 (Fast): A + C AC Overall Reaction: A_2 + B + C AB + AC 3 done, 2 left today. You're making progress.