# Chapter 13 Rates of Reaction

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1 Chapter 13 Rates of Reaction Chemical Kinetics: Rates of Chemical Reactions & Factors that Affect Rates of Reactions How fast does a chemical reaction occur? rates of reactants consumed rates of products formed define and calculate rates relationship between rate & concentration rate law & experimental determination of rate law relationship between concentration & time integrated rate law & half-life of reaction dependence of rate on temperature & catalyst Arrhenius equation reaction mechanisms Reaction Rates concentration rate of chemical reaction = time [X] [X]2 [X]1 rate = = t t2 t1 Reaction Rates typically a plot of [X] vs. time will not yield a straight line, but instead a curve; concentration vs. time curve consider the difference in shape of concentration vs. time curve for reactants & products [X] = molar concentration of reactant or product if reactant: [X]2 < [X]1 rate of consumption of reactant [react] [prod] if product: [X]2 > [X]1 rate of formation of product time time

2 Consider the decomposition of hydrogen peroxide: 2 H2O2 (aq) 2 H2O (l) + O2 (g) If the initial concentration of H2O2 is M, and the concentration of H2O2 decreases to M after 10 seconds of reaction, calculate the rate of decomposition of H2O2 over this time interval. [H2O2]0 = M at t = 0 s [H2O2] = M at t = 10 s Consider the reaction: 4 NH3 (g) + 5 O2 (g)! 4 NO (g) + 6 H2O (g) The initial concentration of NH3 is M; after 25 seconds, [NH3] has decreased to M. Determine the following: a. the rate of consumption of NH3 b. the rate of consumption of O2 c. the rate of formation of H2O. a closer look at some experimental data: 2 H2O2 (aq)! 2 H2O (l) + O2 (g) Concentration vs. Time Curve for Decomposition of H 2 O 2 Time, s [H2O2], M Change in [H2O2], M Rate, M/s [H 2 O 2 ], M things to note: [H2O2] decreases over time b/c it is a reactant consumed rate of reaction is not constant over time as [H2O2] decreases, so does rate Time, s

3 Average vs. Instantaneous Reaction Rates average rate of reaction - rate of reaction over some time interval ex. What is the rate of decomposition of H2O2 between 1200 and 1600 seconds? to calculate an instantaneous rate of reaction, draw a line tangent to the concentration vs. time curve at the point of interest slope of tangent line = rate at time, t; units M s 1 instantaneous rate of reaction - rate of reaction at a specific time point ex. What is the rate of decomposition of H2O2 at 1400 seconds? initial rate of reaction - instantaneous rate of reaction at t = 0 s to calculate an initial rate of reaction, draw a line tangent to the concentration vs. time curve at t = 0 slope of tangent line = initial rate of reaction; units M s 1 Rate Laws rate law equation that shows the dependence of a reaction s rate on concentration for a reaction: A + B! C + D Rate = k[a] m [B] n *** for a reaction, the rate law must be determined experimentally ***

4 Rate = k[a] m [B] n m and n order of reaction with respect to A & B typically small, positive, whole numbers negative numbers and fractions are possible k rate constant dependent on specific reaction dependent on temperature independent of concentration units are variable; depend on orders of reaction Rate = k[a] m [B] n some rate law terminology: if m = 1, the reaction is 1 st order in A if n = 2, the reaction is 2 nd order in B overall order of the reaction = m + n reactions of the same overall order will have similar characteristics specifically we will discuss 0, 1 st, and 2 nd order overall reactions Reaction Order & Rate Change with Concentration consider the reaction: if 1 st order overall: Rate = k[a] 1 A! B + C if [A] increases by factor of 2, rate will increase by factor of 2 1 OR rate will double if 2 nd order overall: Rate = k[a] 2 if [A] increases by factor of 3, rate will increase by factor of 3 2 OR rate will increase 9x Reaction Order & Rate Change with Concentration consider the reaction: A! B + C if 1 order overall: Rate = k[a] 1 if [A] increases by factor of 2, rate will change by factor of 2 1 OR rate will decrease by half if 0 order overall: Rate = k[a] 0 rate is constant; changes in [A] have no effect on the rate as long as some A is present

5 Reaction Order & Rate Change with Concentration in general, for the reaction: A! B + C Rate = k[a] m if [A] changes by a factor of x, the rate will change by a factor of x m when [A] is doubled: m = rate multiplied by =! = = = = 8 when [A] is tripled: m = rate multiplied by = ⅓ = = = = 27 Reaction Order & Units of k order rate law k = k units 1 st Rate = k[a] k = Rate/[A] s 1 2 nd Rate = k[a] 2 k = Rate/[A] 2 M 1 s 1 3 rd Rate = k[a] 3 k = Rate/[A] 3 M 2 s 1 0 Rate = k k = Rate M s 1 Determination of Rate Law from Experimental Data: Method of Initial Rates reaction run multiple times at the same temperature ( same k) with different initial concentrations initial rates of reaction are determined analyze how rate changes as concentration changes determine order with respect to each reactant write rate law do calculations a simple 2 N2O5 (g)! 4 NO2 (g) + O2 (g) Rate = k[n2o5] m experiment [N2O5]0 initial rate M 4.8 x 10 6 M s M 9.6 x 10 6 M s 1 [N2O5] increase by factor of 2 rate increases by factor of 2 reaction is 1 st order in N2O5 m = 1

6 2 HgCl2 (aq) + C2O4 2 (aq)! 2 Cl (aq) + 2 CO2 (g) + Hg2Cl2 (s) Rate = k[hgcl2] m [C2O4 2 ] n experiment [HgCl2]0 [C2O4 2 ]0 initial rate M 0.15 M 1.8 x 10 5 M min M 0.30 M 7.1 x 10 5 M min M 0.30 M 3.5 x 10 5 M min 1 compare experiments 1 & 2 to determine order with respect to C2O4 2 ; i.e. determine n compare experiments 2 & 3 to determine order with respect to HgCl2; i.e. determine m example (continued): comparing experiments 1 & 2: determination of n Rate2 k[hgcl2]2 = m [C2O4 2 ]2 n Rate1 k[hgcl2]1 m [C2O4 2 ]1 n 7.1 x 10 5 k(0.105) m (0.30) n = 1.8 x 10 5 k(0.105) m (0.15) n 4 = 2 n so: n = 2 the reaction is 2 nd order in C2O4 2 example (continued): comparing experiments 2 & 3: determination of m Rate2 k[hgcl2]2 = m [C2O4 2 ]2 2 Rate3 k[hgcl2]3 m [C2O4 2 ] x 10 5 k(0.105) m (0.30) 2 = 3.5 x 10 5 k(0.052) m (0.30) 2 2 = 2 m so: m = 1 the reaction is 1 st order in HgCl2 example (continued): 2 HgCl2 (aq) + C2O4 2 (aq)! 2 Cl (aq) + 2 CO2 (g) + Hg2Cl2 (s) Rate = k[hgcl2][c2o4 2 ] 2 experiment [HgCl2]0 [C2O4 2 ]0 initial rate M 0.15 M 1.8 x 10 5 M min M 0.30 M 7.1 x 10 5 M min M 0.30 M 3.5 x 10 5 M min 1 Determine the value of k. Determine the rate of reaction when [HgCl2] = M and [C2O4 2 ] = M.

7 2 NO (g) + O2 (g)! 2 NO2 (g) Rate = k[no] m [O2] n example (continued): comparing experiments 1 & 2: determination of n experiment [NO]0 [O2]0 initial rate M M 7.10 M s M M 28.4 M s M M M s 1 compare experiments 1 & 2 to determine order with respect to O2; i.e. determine n compare experiments 2 & 3 to determine order with respect to NO; i.e. determine m Rate2 k[no]2 = m [O2]2 n Rate1 k[no]1 m [O2]1 n 28.4 k(0.0010) m (0.0040) n = 7.10 k(0.0010) m (0.0010) n 4 = 4 n so: n = 1 the reaction is 1 st order in O2 example (continued): comparing experiments 2 & 3: determination of m Rate3 k[no]3 = m [O2]3 Rate2 k[no]2 m [O2] k(0.0030) m (0.0040) = 28.4 k(0.0010) m (0.0040) 9 = 3 m so: m = 2 the reaction is 2 nd order in NO 1 st Order Overall Reactions for a reaction: A! products rate law: Rate = k[a] rate law provides information about how rate changes relative to changes in [A] Other questions we d like to answer? What is [A] at some time, t? What % of A has been consumed after t? How much time elapses before the reaction is 50% complete? these questions all probe the relationship between [A] and time use the Integrated Rate Law to answer these questions

8 1 st Order Overall Reactions: Integrated Rate Law [A]t ln = kt [A]0 ln [A]t ln[a]0 = kt ln [A]t = kt + ln[a]0 2 H2O2 (aq)! 2 H2O (l) + O2 (g) The decomposition of hydrogen peroxide is a 1 st order reaction with k = 7.30 x 10 4 s 1. If [H2O2]0 = 2.32 M, determine [H2O2] after 1200 seconds of reaction. look at this as an equation for a straight line: plot of ln[a] vs. t will yield a straight line slope = k y intercept = ln[a]0 2 H2O2 (aq)! 2 H2O (l) + O2 (g) The decomposition of hydrogen peroxide is a 1 st order reaction with k = 7.30 x 10 4 s 1. What percent of H2O2 initially present has decomposed after 500 s of reaction? At what time after the start of the reaction the sample of H2O2 ⅔ decomposed? Half-Life of a Reaction, t! The half-life (t!) of a reaction is the time required for the [reactant] to decrease to half of its initial value. for a 1 st order reaction: t = t! [A]t =![A]0 t! = ln 2 = k k note: for a 1 st order reaction, t! is constant t! is dependent on k but not on [A]0

9 Half-Life of a 1 st Order Reaction Half-Life of a 1 st Order Reaction 2 nd Order Overall Reactions for a reaction: A! products Zero (0) Order Overall Reactions for a reaction: A! products rate law: Rate = k[a] 2 rate law provides information about how rate changes relative to changes in [A] rate law: Rate = k[a] 0 ; Rate = k the rate of a 0 order reaction is constant and does not depend on [A] integrated rate law: 1 = kt + 1 [A]t [A]0 plot of 1/[A] vs. t will give a straight line slope = k half-life equation: t! = 1 k[a]0 y intercept = 1/[A]0 integrated rate law: [A]t = kt + [A]0 plot of [A] vs. t will give a straight line slope = k half-life equation: [A]0 t! = 2k y intercept = [A]0

10 Consider a reaction with k = M s 1. After 180 seconds of reaction, [A] = M. What was the initial concentration of A? Consider the following 2 nd order reaction: 2 A! B For this reaction, k = M 1 s 1. If [A]0 = 1.10 M, what will be the [B] after 45.0 seconds? Collision Theory of Reactions reactions occur when particles collide not all collisions result in the formation of product 2 factors influence whether or not product forms: energy of collision orientation of particle at collision Energy Considerations molecular collisions are successful (i.e. result in the formation of product) when they occur with enough energy to break bonds minimum amount of E required for reaction to occur is the activation energy, Ea, for the reaction Ecollision > Ea required for product formation in general, the larger the Ea for a reaction, the slower the rate a smaller fraction of molecular collisions will occur with sufficient energy to form products

11 Energy Considerations Energy Considerations energy of molecular collision is affected by temperature at higher T s: kinetic energy of particles is greater speed is greater force of collisions will be greater frequency of collisions will be higher fraction of collisions with E > Ea is greater Orientation of Collisions consider the reaction: A2 + B2! 2 AB favorable orientation: Orientation of Collisions NO (g) + Cl2 (g)! NOCl (g) + Cl (g) A A B A +!! + B A B A B A B B favorable orientation: unfavorable orientation: A A + B B! no reaction unfavorable orientation:

12 consider the species in the middle of this reaction: A A B A +!! + B B A B A B A B Reaction Profile track how energy changes as the reaction progresses from reactant! transition state! products reactants transition state products a short-lived, transient species formed in between reactants and products can be called: transition state activated complex reaction intermediate Temperature Dependence of Reaction Rates reaction rates almost always increase with increasing temperature if rate increases while [A] is held constant, k must also increase temperature dependence of k is given by the Arrhenius equation: k = Ae Ea/RT OR Ea ln k = ln A RT Ea = activation energy, kj/mol A = frequency factor, s 1 R = J/K mol

13 Arrhenius Equation look at the logarithmic form of the Arrhenius equation as a straight line equation: Ea ln k = + ln A RT plot of ln k vs. 1/T yields a straight line with: slope = Ea/R this is called an Arrhenius plot you can derive a 2-point equation: y intercept = ln A k2 Ea 1 1 ln( ) = ( ) R k1 T1 T2 Consider the following 1 st order reaction: 2 N2O5 (g)! 4 NO2 (g) + O2 (g) For this reaction at 50 C, A = 4.0 x s 1 and Ea = 88 kj/mol. Determine the rate of this reaction at 50 C if [N2O5]0 = 2.68 M. Determine the value of k at 30 C. Consider a reaction with Ea = 35.7 kj/mol. Determine the factor by which the rate of a reaction increases as temperature is increased from 50 C to 75 C. Catalysts A catalyst is a substance that increases the rate of a chemical reaction without being consumed by it. a catalyst provides an alternate path from reactant to product path with lower Ea frequently multiple step path a catalyst participates in one step of the mechanism, and is regenerated in a later step

14 Reaction Profile for a Catalyzed Reaction: Catalysts heterogeneous catalyst catalyst is in a different phase than the reacting species usually a solid catalyst in contact with liquid or gas phase reaction mixture reactants adsorb onto catalyst surface physical adsorption chemisorption homogeneous catalyst catalyst in the same phase as reacting species Catalytic Hydrogenation - Heterogeneous Catalysis Catalytic Converters - Heterogeneous Catalysis conversion of CO (g), NO (g), and O2 (g) to CO2 (g) and N2 (g) released into atmosphere

15 Enzymes - Biochemical Catalysts Arrhenius Equation and Catalyst Considerations for a reaction, the addition of a catalyst causes: rate increase ratecat > rateuncat k increase kcat > kuncat Ea decrease Ea-cat < Ea-uncat consider writing the Arrhenius equation in terms of both a catalyzed and uncatalyzed reaction: kcat = Ae Eacat/RT and kuncat = Ae Eauncat/RT kcat = kuncat Ae Eacat/RT Ae Eauncat/RT Reaction Mechanisms a collection of elementary steps that describe the path from reactants to products in a chemical reaction for a mechanism to be plausible, it must meet 2 criteria: match the overall or net chemical reaction after steps are added together produce a rate law that is consistent with the experimentally determined rate law Elementary Processes individual steps in reaction mechanism rate law can be written by inspection; orders equivalent to stoichiometric coefficients identified by their molecularity may be unimolecular, bimolecular, termolecular may be fast, reversible (equilibrium) steps Rateforward = Ratereverse may be slow rate determining step (RDS) largest Ea ** determines the overall rate of reaction Raterxn = RateRDS

16 Reaction Intermediates & Catalysts in Mechanisms reaction intermediates and catalysts are both species that may be involved in a mechanism but not appear in the overall chemical equation or its rate law reaction intermediate generated in one step, then consumed in a later step catalyst consumed in one step, then regenerated in a later step proposed mechanism for the depletion of ozone (O3) by Cl: step 1: Cl (g) + O3 (g)! ClO (g) + O2 (g) step 2: ClO (g) + O (g)! Cl (g) + O2 (g) overall rxn: O3 (g) + O (g)! 2 O2 (g) ClO is a reaction intermediate b/c it is formed in step 1 and consumed in step 2 Cl is a catalyst (homogeneous) b/c it is consumed in step 1 and regenerated in step 2 Consider the following reaction: 2 ICl (g) + H2 (g)! I2 (g) + 2 HCl (g) The experimentally determined rate law is: Rate = k[h2][icl] Is this a plausible mechanism? step 1: H2 (g) + ICl (g)! HI (g) + HCl (g) (slow) step 2: HI (g) + ICl (g)! I2 (g) + HCl (g) (fast) Consider the following reaction: 2 ICl (g) + H2 (g)! I2 (g) + 2 HCl (g) The experimentally determined rate law is: Rate = k[h2][icl] Is this a plausible mechanism? step 1: H2 (g) + ICl (g)! HI (g) + HCl (g) (RDS) step 2: HI (g) + ICl (g)! I2 (g) + HCl (g) (fast) overall rxn: H2 (g) + 2 ICl (g)! I2 (g) + 2 HCl (g) rate law from mechanism: Raterxn = RateRDS Rate = k[h2][icl]

17 Consider the following reaction: 2 ICl (g) + H2 (g)! I2 (g) + 2 HCl (g) The experimentally determined rate law is: Rate = k[h2][icl] Is this a plausible mechanism? step 1: H2 (g) + ICl (g)! HI (g) + HCl (g) (RDS) step 2: HI (g) + ICl (g)! I2 (g) + HCl (g) (fast) overall rxn: H2 (g) + 2 ICl (g)! I2 (g) + 2 HCl (g) rate law from mechanism: Raterxn = RateRDS Rate = k[h2][icl] Yes this is a plausible mechanism. Ozone reacts with nitrogen dioxide to produce oxygen and dinitrogen pentoxide by the following proposed mechanism: step 1: O3 (g) + NO2 (g)! NO3 (g) + O2 (g) (slow) step 2: NO2 (g) + NO3 (g)! N2O5 (g) (fast) Determine the overall reaction & rate law consistent with this mechanism. Ozone reacts with nitrogen dioxide to produce oxygen and dinitrogen pentoxide by the following proposed mechanism: step 1: O3 (g) + NO2 (g)! NO3 (g) + O2 (g) (RDS) step 2: NO2 (g) + NO3 (g)! N2O5 (g) (fast) overall rxn: O3 (g) + 2 NO2 (g)! O2 (g) + N2O5 (g) rate law from mechanism: Raterxn = RateRDS Rate = k[o3][no2] Consider the following proposed mechanism for the reaction between F2 and N2O4: Ea, kj/mol step 1: N2O4 (g) 2 NO2 (g) 59 step 2: NO2 (g) + F2 (g)! F (g) + FNO2 (g) 83 step 3: F (g) + NO2 (g)! FNO2 (g) 9 What is the overall reaction predicted by this mechanism? Identify any reaction intermediates or catalysts.

18 Consider the following proposed mechanism for the reaction between F2 and N2O4: Ea, kj/mol step 1: N2O4 (g) 2 NO2 (g) 59 step 2: NO2 (g) + F2 (g)! F (g) + FNO2 (g) 83 step 3: F (g) + NO2 (g)! FNO2 (g) 9 overall rxn: N2O4 (g) + F2 (g)! 2 FNO2 (g) NO2 and F are reaction intermediates Consider the following proposed mechanism for the reaction between F2 and N2O4: Ea, kj/mol step 1: N2O4 (g) 2 NO2 (g) 59 step 2: NO2 (g) + F2 (g)! F (g) + FNO2 (g) 83 step 3: F (g) + NO2 (g)! FNO2 (g) 9 overall rxn: N2O4 (g) + F2 (g)! 2 FNO2 (g) Which step {1, 2, or 3} is the fastest step? Which step {1, 2, or 3} is the rate-determining step? Consider the following proposed mechanism for the reaction between F2 and N2O4: Ea, kj/mol step 1: N2O4 (g) 2 NO2 (g) 59 step 2: NO2 (g) + F2 (g)! F (g) + FNO2 (g) 83 step 3: F (g) + NO2 (g)! FNO2 (g) 9 overall rxn: N2O4 (g) + F2 (g)! 2 FNO2 (g) Which step {1, 2, or 3} is the fastest step? step 3 because smallest Ea Which step {1, 2, or 3} is the rate-determining step? step 2 because largest Ea For the proposed mechanism for the reaction between F2 and N2O4, consider both the Ea values and H values for steps 1 3, and the overall reaction to sketch the reaction profile. Ea, kj/mol H, kj step 1: step 2: step 3: overall rxn:

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