32.1 5.111 Lecture 32 Kinetics Topics: Radioactive Decay, Second Order Integrated Rate Laws, Kinetics and Chemical Equilibrium, Introduction to Reaction Mechanisms Chapter 13 (p 498-501, 507-508) and Chapter 17 (p 660-664) Radioactive Decay The decay of a nucleus is ____________ of the number of surrounding nuclei that have decayed. We can apply first order integrated rate laws: [A] = [A]0 e-k t and t1/2 = 0.6931 k However, instead of concentration, the first order integrated rate law is expressed in terms of N (number of nuclei) N = Noe-k t k ≡ decay constant t ≡time N0 ≡ number of nuclei originally present Chemical kinetics – monitor changes in concentration over time Nuclear kinetics – monitor rate of occurrence of decay events with a Geiger counter (radiation detector) Decay rate is also called Activity (A) Activity = A = -dN = k N dt because activity is proportional to the number of nuclei (N): N = Noe-k t can be expressed as A = Aoe-k t A ≡ Activity A0 ≡ original activity S.I. unit for Activity is the becquerel (Bq) 1 Bq ≡ 1 radioactive disintegration per second Older unit is the curie (Ci) 1 Ci = 3.7 x 1010 disintegrations per sec Table of types of radiation p. 701 alpha decay – mass change of helium-4 nucleus (2 protons, 2 neutrons) beta decay – no mass change (particle = electron) 32.2 238 Table of half-lives p. 713 U 9 4.5 x 10 y Uranium238 decay series p. 706 (A = atomic mass, Z= atomic number) 234 Th α Decay 230 1600 y A 3.8 d 218 20 min 210 Tl Pb Ra 92 8 x 104 y Ra Po 3 min 214 91 Th β Decay 222 U 2.5 x 10 y 226 Days of Our Half-Lives by Chemistry Poet: Mala Radhakrishnan Pa 5 Bi Po 200 µs Pb Bi Po 138 d Pb 206 81 Radioactive Decay Example 82 83 84 85 86 Z 87 88 Find N0 Find k Find A0 Find A 90 Figure by MIT OpenCourseWare. Find the original activity and the activity after 17 years (5.4 x 108 sec) of 0.50 g of 239 Pu (t = 24,000 years (7.6 x 1011 sec)) 1/2 94 89 32.3 Medical uses of Radioactive Decay. Example: Technetium-99 is the most widely used radioactive nuclide in medicine. It is used for diagnostic organ imaging and bone scans, with over 7 million uses annually in the US alone. One of the patent holders for technetium, "cardioliteTM", is our own, Professor of Chemistry, Alan Davison. Second Order Integrated Rate Laws (Chapter 13.6) 2A → B rate of disappearance of A = - d[A] dt = k [A]2 Separate concentration and time terms d[A] 1 [A] [A]t ![A] 1 d[A] [A]2 = 0 1 - = -k dt 2 -k ! t dt 0 1 - =-kt [A]0 [A]t slope = k 1 1 = 1 kt + [A]0 [A]t [A]t intercept 1 [A]0 y = mx + b time Second order half-life 1 = 1 kt + [A]0 [A]t 1 ([A]0/2) = 1 k t 1/2 + [A]0 32.4 2 [A]0 1 1 - = k t 1/2 [A]0 = k t 1/2 [A]0 t 1/2 = 1 Second order half-life depends on ________________________________ k[A]0 In real life, need to experimentally determine if reaction is first or second order. 1 ln[A]t [A]t time first-order plot ln[A]t = -kt + ln [A]o time second-order plot 1 [A]t = 1 kt + [A]0 Kinetics and Chemical Equilibrium At equilibrium, the rates of the forward and reverse reactions are equal. The equilibrium constant for a chemical reaction that has form A + B ⇔ C + D is K= 32.5 Suppose experiments show both the forward reaction and reverse reaction are second order, with the following rate laws: k1 A+B ⇔C+D k-1 forward reaction reverse reaction ratef = rater = At equilibrium, these rates are equal: k1 [A][B] = k-1 [C][D] and [C][D] [A][B] Therefore K = k1 k-1 = k1 k-1 The equilibrium constant for a reaction is equal to the ratio of the rate constants for the forward and reverse elementary reactions that contribute to the overall reaction. Equilibrium constants in kinetics terms: K>1 k1 ? k-1 K<1 k1 ? k-1 Reactions do not typically occur in 1 step, but proceed through a series of steps. Each step is called an elementary reaction. For an overall reaction, the order and the rate law cannot be derived from the stoichiometry of the balanced reaction. For an elementary reaction, the order and rate law can be predicted. Elementary reactions occur exactly as written. 32.6 Example: decomposition of ozone 2O3 (g) → 3O2 (g) proposed mechanism has two elementary reaction steps 1st step O3 →O + O2 unimolecular rate = k [O3] 2nd step O + O3 →O2 + O2 bimolecular rate = molecularity ≡ number of reactant molecules that come together to form product. Unimolecular – 1 reactant ex. Decomposition, radioactive decay Bimolecular – 2 reactants ex. Two reactants collide to form product Termolecular – 3 reactants ex. Three reactants collide to form product (rare) Individual steps (elementary reactions) can be added together to get the overall equation for the reaction. 1st step 2nd step O3 →O + O2 O + O3 →O2 + O2 “O” is a reaction intermediate Overall O3 + O3 →O2 + O2 + O2 “O” does not appear in the overall reaction equation Reaction mechanisms (which are a series of elementary reaction steps) must be tested experimentally. Reaction mechanisms cannot be proven to be correct. At best, data are consistent with a reaction mechanism.