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282 Chapter 14 Chemical Kinetics 14.55 Given: decomposition of SO₂Cl₂, first order; k = 1.42 X 10⁻⁴ Find: (a) (b) t to decrease to 25% of 0, (c) t to 0.78 M when [SO₂Cl₂]₀ = and (d) after 2.00 X 10² S and 5.00 X 10² S when = 0.150 M Conceptual Plan: (a) k (b) 25% of (c) (d) Solution: (a) = 0.693 k = 1.42 0.693 = 4.88 X 10³ (b) = 0.25 Because In = -kt + In rearrange to solve for t. t = 1 = 1.42 X 1 = 9.8 X 10³ (c) = 0.78 = M Because In = kt + rearrange to solve for t.t = 1 = 1.42 X 1 10⁻⁴ 1.00 = 1.7 X 10³ (d) = 0.150 M and 2.00 X 10² S in In = (1.42 X 10⁻⁴ (2.00 X 10² + In 0.150 M = 1.92552 = -1.92552 = 0.146 M [SO₂Cl₂]₀ = 0.150 M and 5.00 X 10² in In = (1.42 X 10⁻⁴ (5.00 X + In 0.150 M = -1.96812 = e -1.96812 = 0.140 M Check: The units (s, s, and M) are correct. The rate law is a common form. The half-life is consistent with a small value of k. The time to 25% is consistent with two half-lives. The time to 0.78 M is consistent with being less than one half-life. The final concentrations are consistent with the time being less than one half-life. 14.57 Given: for radioactive decay of U-238 = 4.5 billion years and independent of Find: t to decrease by 10%; number U-238 atoms today, when 1.5 X 10⁸ atoms formed 13.8 billion years ago Conceptual Plan: independent of concentration implies first-order kinetics, k then 90% of [U-238]₀, k t and [U-238]₀,t,k [U-238], Solution: Rearrange to solve for k.k = 0.693 = 4.5 0.693 X = 1.54 X then [U-238], = 0.10 Because In [U-238], = kt + U-238 rearrange to solve for t. t = k 1 [U-238]₀ [U-238], = 1.54 1 = 6.8 X yr and = 1.15 10¹⁸ atoms; t = 13.8 X yr = kt + In U-238 = (1.54 X X X 10¹⁸ atoms) = 39.726797 [U-238], = = 1.8 X 10¹⁷ atoms Check: The units (yr and atoms) are correct. The time to 10% decay is consistent with less than one half-life. The final concentration is consistent with the time being about three half-lives. Copyright © 2017 Pearson Education, Inc.