Zumdahl s Chapter 12 Chemical Kinetics Chapter Contents Introduction Rates of Reactions Differential Reaction Rate Laws Experimental Determinations Initial Rates Saturation Methods Integrated Rate Laws 0th Order 1st Order Life 2nd Order Multiple Reactants Reaction Mechanisms Models for Kinetics Catalysis It s d j vu all over again Kinetics of processes have appeared before Kinetic Theory has been invoked several times In the origin of pressure As van der Waal s pressure correction P a n V 2 n V 2 is a concentration dependence on collision rates As a justification for Raoult s Law In the development of the Mass Action Law kf A B kr C D K kf kr C D A B A two pronged approach The speed with which chemical reactions proceed is governed by two things The rate at which reactants come into one another s proximity collide and The probability that any given collision will prove effective in turning reactants to products We look first at the macroscopic measurement of reaction rates Reactants vanish in time so reactant is a falling function of t Likewise product is a rising function of t The shape of these functions tells us about concentration dependence Concentration Change of Concentration in Time 0 A 0 A B t d d B B A 0 t Time Concentration A B Reaction Rate 0 A 0 A B t d d B d A dt 0 t Time B A Stoichiometry requires d A dt d B dt But d A dt can itself be a function of time It falls rapidly initially Then it approaches its equilibrium value as A on the graph asymptotically K B A aA bB aA bB Rxn Rate Rate 1 a d A dt Concentration d A dt a b d B dt is the new stoichiometric condition dt d B Because that equals 0 Now neither differential is the reaction rate But we can fix this by Rate 1 b d B dt B A 0 d A dt 0 t Time A aA bB cC dD If z is the stoichiometric coefficient of the general compound Z and z takes on positive signs for products and negative signs for reactants Rate 1 z d Z dt is rate of reaction M s d Z dt is easy if Z f t is known but it isn t All we can measure is Z t and use the Fundamental Theorem of Calculus to approximate d Z dt as Z t as t 0 Estimating Experimental Rates For reasons soon apparent we will often want the t 0 value of d A dt That requires an extrapolation of A t to t 0 where it is varying rapidly t 0 A 0286 0014 2714 1 5 2 2 A 3000 0300 5 1 A 0028 0258 2456 Why d A dt at t 0 Ask the question the other way around At t 0 are there additional complications Sure At the very least the reverse reaction of products to produce reactants changes the rate of loss of A An added headache Also A is changing most rapidly at t 0 minimizing the small difference of large numbers error Simplified Rate Laws Not laws like Laws of Thermodynamics but rather rate rules for simple reactions Two versions of the Rate Laws Differential like d A dt k A n Integral like A 1 n A 01 n n 1 kt But they must be consistent for the same reaction As these happen to be iff n 2 of course Rate exponents are often not stoichiometric Simplified INITIAL Rate Laws Since products are absent at t 0 such laws include only rate dependence on reactants Simple reactions often give power rate laws E g Rate 1 a d A dt k A n B m The n and m are often integers A s dependence is studied in excess B since B 0 will be fixed So k B 0m A n Reaction Rate Orders Rate k A n B m The n and m are called the order of the reaction with regard to A and B respectively The reaction is said to have an overall order O that is the sum of the species orders e g n m The significance of overall order is simply that increasing all species by a factor f increases the reaction rate by a factor f O We find a species order by changing only species Determining Reaction Order If we use only initial rates all species remain at species 0 Then by fixing all species except one we find its order by knowing at least two initial rates where its concentrations differ A B k A n B m 0 1 0 1 0 5 M s 0 2 0 1 2 0 M s 0 2 0 2 4 0 M s This data is consistent with n 2 and m 1 and we find k 500 M 2 s 1 as a bonus expts must match unknowns In k A n B m we had k n m unknown So we needed at least 3 experiments More if we want self consistency checks This is just like linear equations in fact ln k A n B m ln k n ln A m ln B So we ll need at least 3 ln Rate experiments in order to find n m and ln k unambiguously The Big Three 0th Order or d A dt k0 A 0 k0 A A 0 k0 t 1st Order d A dt k1 A 1 or A 1 d A d ln A k1 dt hence ln A ln A 0 k1 t 2nd Order 2 n 1 r 1 o F lide s e e s d A dt k2 A 2 or A 2 dt d A 1 k2 dt hence A 1 A 0 1 k2 t Integrated Law Curve Shapes A 0 same values of k and A 0 1st order trick Curve falls by equal factors in equal times 2nd order 0th order 0 A linear with t confirms 0th order 0 t Slope k 1st order 2t t Confirming 1st Order ln A 0 A straight line in ln A vs t ln A 0 2nd order ln A 0th order 0 Slope k 1st order t ln2 k t Confirming 2nd Order 1 A 1st order 0th order 2nd order Slope k A straight line in 1 A vs t 1 A 0 0 t Caveat The 0th Law plot showed A 0 which presumes there is no reverse reaction The reaction is quantitative Indeed all these plots ignore all reactants products and intermediates except A In reality these shapes can be trusted only under conditions of initial rate and where A is overwhelming the limiting reactant Multiple Reactants What about A B P Rate k2 A B where P is any combination of products What s an integrated law for d P dt k2 A B By stoichiometry d A dt d B dt d P dt Via those substitutions we can produce kt 1 B 0 A 0 P …