HW07solutionsF11

67 combination of equations 2 8 and 9 gives dd k 3k1

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Unformatted text preview: + k ) t k1 A0 e 1 2 −k t + C1e 3 k3 − k1 − k2 (6) (7) (8) (9) At t= 0 k1 A0 + C1 k 3 − k1 − k2 Combination of equations 6 and 7 gives k1 A0 Ⱥ e− ( k1 + k2 ) t − e −k3t Ⱥ B= Ⱥ k3 − k2 − k1 Ⱥ B=0= From the rate expression for species D dD = k2 A + k3 B dt Fall 2011 p. 6/7 Combination of equations 2, 8 and 9 gives dD k 3k1 A0 − (k + k )t − (k + k )t = k2 A0 e 1 2 + e 1 2 − e −k 3t dt k3 − k2 − k1 Separation of variables and integration give t t k3 k1 A0 −( k1 + k 2 )t − (k + k )t −k t D = k2 A0 ∫ e dt + ∫ e 1 2 − e 3 dt k3 − k2 − k1 [ 0 = [ 0 − (k1 +k 2 )t k2 A0 1 − e k1 + k2 + [ ( ) k3k1 A0 Ⱥ 1 − e − k1 + k2 t 1 − e −k 3t Ⱥ − k3 − k 2 − k1 Ⱥ k1 + k2 k 3 Ⱥ Ⱥ Ⱥ (10) From the rate expression for species C dC = (k1 + k2 ) A dt Combination of equations 2 and 11 gives (11) ` (12) (13) (14) (15) (16) Separation of variables and integration give −( k + k ) t C = A0 Ⱥ1 − e 1 2 Ⱥ Ⱥ Ⱥ From the differential equation...
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