Kinetics of Complex Reactions: The Steady-State and Equilibrium Approximations

5.60 Spring 2007 Lecture #32 page 1 Complex Reactions and Mechanisms (continued) IV) Steady State and Equilibrium Approximations a) Steady State Approximation A k1 k-1 B k2 C Assume that [B] is small and slowly varying d[B] ≈ 0 and (k2 + k-1) >> k1 dt e.g. [B] reaches a steady state concentration [B]SS and remains there d[B] = k1 [A] − k−1 [B]SS − k2 [B]SS ≈ 0 dt Steady State approximation Solving… So − [B]SS = k1 [A] k−1 + k2 d[A] = k1 [A] − k−1 [B]SS dt − d[A] k1k2 [A] = dt k−1 + k2 5.60 Spring 2007 Lecture #32 page 2 k k [A] d[C] d[A] = k2 [B]SS = 1 2 =− dt k −1 + k2 dt Looks like k' A (first order) with k'= C k1k2 k −1 + k2 **Necessary Condition for use of Steady State Approximation** Data must be taken after B has built up to a steady i) state value. (k2 + k-1) >> k1 ⇒ [B]SS is small ii) ______________________________________ b) Equilibrium Approximation A k1 k-1 Assume That is… B B k2 C k2 << k-1 and k1 k2 C is the rate limiting step. Then… A and B quickly come into equilibrium, while C slowly builds up. Keq = k1 [B] ≈ k −1 [A] [B] = k1 [A] = Keq [A] k −1 Equilibrium approximation 5.60 Spring 2007 So… Lecture #32 kk d[C] = k2 [B] = k2Keq [A] = 1 2 [A] dt k−1 k k d[C ] = 1 2 [A ] dt k −1 Or, Looks like A k' C (first order) with k' = In general, for a mechanism with multiple pre-equilibria… e.g. page 3 A A A A ↔ ↔ ↔ ↔ I1 I2 I3 I4 In ↔ B K1 K2 K3 K4 (Rate Determining) ⎡n ⎤ d[B] =k n [In ] = kn ⎢∏ Ki ⎥[A] dt ⎣ i=1 ⎦ k1k2 k−1 5.60 Spring 2007 Lecture #32 page 4 Examples: A) Apparent Termolecular Reactions (Reaction Chaperones) k I+I+M I2 + M M is a rare gas molecule or the wall of the reaction vessel Mechanism: k1 I+I I2* k-1 I2* + M (M* k2 I2 + M* M) where (k2 + k-1) >> k1 , So that is the Steady State approximation! d[I2* ] = k1 [I] 2 − k −1 [I2* ]SS − k2 [I2* ]SS [M] ≈ 0 dt Steady State approximation Solving… And … [I2* ]SS k1 [I] 2 = k −1 + k2 [M] d[I2 ] k1 [I] 2 * = k2 [I2 ]SS [M] = k2 [M] dt k−1 + k2 [M] 5.60 Spring 2007 Lecture #32 page 5 Limiting Cases i) k2[M] >> k-1 d[I2 ] =k 1 [I] 2 dt then (high pressure) ii) k2[M] << k-1 second order d[I2 ] k 1 k2 = [M][I] 2 dt k −1 then (low pressure) B) third order Gas decomposition (Lindemann Mechanism) A(g) → products Mechanism: k1 A+M A* k-1 k2 A* + M products (B + …) M is a rare gas molecule and/or A, k1 is fast, k-1 So… (k2 + k-1) >> k1 , is very fast, k2 is slow Steady State approximation again. 5.60 Spring 2007 Lecture #32 page 6 d[A * ] = k1 [A][M] − k −1 [A * ]SS [M] − k2 [A * ]SS ≈ 0 dt Steady State approximation [A * ]SS = − k1 [A][M] k −1 [M] + k2 k k [A][M] d[A] d[B] = = k2 [A * ]SS = 1 2 dt dt k−1 [M] + k2 Limiting Cases i) High pressure (1 bar) − ii) k-1[M] >> k2 d[A] k1k2 = [A] = k∞ [A] dt k −1 Low pressure (~10-4 bar) − d[A] = k1 [A][M] dt (1st order) k-1[M] << k2 (if M≡A, then 2nd order in A)