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452 Chapter 20 Radioactivity and Nuclear Chemistry Solution: = 0.693 k Rearrange to solve for k.k = 0.693 = 3.8 0.693 days = 0.18237 day⁻¹ Because In = kt + In N₀ = (0.18237 day) + In (1.55 g) = -0.56478 Nₜ = -0.56478 = 0.57 g. Check: The units (g) are correct. The amount is consistent with a time between one and two half-lives. 20.49 Given: F-18 initial decay rate = 1.5 for F-18 = 1.83 h Find: t to decay rate of 2.5 X 10³/s Conceptual Plan: Radioactive decay implies first-order kinetics, k then Rate₀, k t. Solution: = 0.693 Rearrange to solve for k.k = 0.693 = 1.83 0.693 = 0.378689 Because In Rate₀ Rate, = kt rearrange to solve for t. t h Check: The units (h) are correct. The time is between five and six half-lives, and the rate is just over 1/2⁶ of the origi- nal amount. 20.51 Given: boat analysis, C-14/C-12 = 72.5% of living organism Find: t Other: for decay of C-14 = 5715 years Conceptual Plan: Radioactive decay implies first-order kinetics, k then 72.5% of k Solution: Rearrange to solve for k. = 0.693 = 5715 0.693 yr = 1.212598 X 10⁻⁴ yr⁻¹ then [C-14], = Because In = kt + In rearrange to solve for t. t 1.212598 X 1 0.725 = 2.65 X 10³ yr Check: The units (yr) are correct. The time to 72.5% decay is consistent with a time less than one half-life. 20.53 Given: skull analysis, C-14 decay rate = 15.3 dis/min gC in living organisms and 0.85 dis/min gC in skull Find: t Other: for decay of C-14 = 5715 years Conceptual Plan: Radioactive decay implies first-order kinetics, k then Rate₀, k t. Solution: 0.693 Rearrange to solve for k.k = 0.693 = 5715 0.693 = 1.212598 X 10⁻⁴ Because = -kt, rearrange to solve for t. t = 1 = 1.212598 X 1 10⁻⁴ yr⁻¹ 0.85 15.3 dis/min.gC = 2.38 X 10⁴ yr Check: The units (yr) are correct. The rate is 6% of initial value, and the time is consistent with a time just more than four half-lives. 20.55 Given: rock analysis, 0.438 g Pb-206 to every 1.00 g U-238, no Pb-206 initially Find: age of rock Other: for decay of U-238 to Pb-206 = 4.5 X years Conceptual Plan: Radioactive decay implies first-order kinetics, k then g Pb-206 mol Pb-206 mol U-238 g U-238 then k t. 1 mol Pb-206 1 mol U-238 238 g U-238 In = -kt + In N₀ 206 g Pb-206 1 mol Pb-206 1 mol U-238 Copyright © 2017 Pearson Education, Inc.