Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

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3.6 Oscillating reactions* 69 (2) Rate equations: =+ 1 [A] 0 ([X] + ) − 2 ([X] + )([Y] + ) (3.184) =+ 1 [A] 0 [X] + 1 [A] 0 − 2 [X] [Y] | {z } = 1 [A] 0 [X] − 2 [X] − 2 [Y] − 2 | {z } = 1 [A] 0 = − 2 [X] − 2 (3.185) =+ 2 ([X] + )([Y] + ) − 3 ([Y] + ) (3.186) = 2 [X] [Y] + 2 [X] + 2 [Y] + 2 − |{z} 3 [Y] − 3 |{z} = 2 [X] = 2 [X] =+ 2 [Y] + 2 (3.187) (3) Simplified DE’s for small displacements: We may neglect the terms containing . Thus, we obtain two coupled first-order DE’s = − 2 [X] (3.188) =+ 2 [Y] (3.189) which can be converted to a single second-order DE by the ansatz · = − (3.190) · =+ (3.191) Second-order DE: ·· = − · = − (3.192) Formal solution of this second-order DE: () =[X()] − [X] = 1 + 2 − (3.193) (4) Eigenvalues: To determine the eigenvalue belonging to the linear DE’s, we have to solve the determinant ¯ ¯0 − − + 0 − ¯ − 2 [X] ¯ 2 [Y] ¯ =0 (3.194) y 2 + 2 [X] [Y] =0 (3.195) y The solutions for are purely imaginary: 12 = ± ¡ ¢ 2 2 12 [X] [Y] (3.196) = ± ( 1 3 [A] 0 ) 12 (3.197)
3.6 Oscillating reactions* 70 (5) Solution for (): () = 1 + 2 − = 0 cos (3.198) with =( 1 3 [A] 0 ) 12 (3.199) This means that if the system is displaced from its steady-state, the species concentrations will start to oscillate and the displacements may even grow in magnitude until the amplitude reaches the limiting value 0 (see limit cycle diagram in Fig. 3.6). I Figure 3.6: Limit cycle diagram for chemical oscillations according to the Lotka mechanism. b) The Brusselator 32 Reaction scheme: A → 1 X (1) B+X 2 → R+Y (2) Y+2X (3.200) 3 → 3X (3) X → 4 S (4) 32 The name ‘Brusselator’ stems from the workplace of its inventor Prigogine (Brussels). The model does not correspond to a real chemical system.
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Physical Chemistry 3: — Chemical
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iii 2.7 Temperature dependence of r
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v 7.1.2 Formal kinetic model 149 7.
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vii 12 Photochemical reactions 241
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ix Preface This scriptum contains l
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xi Disclaimer Use of this text is e
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xiii I Figure 2: Organisational mat
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xv I Figure 6: Recommended textbook
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1.2 Time scales for chemical reacti
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1.2 Time scales for chemical reacti
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1.3 Historical events 6 I Empirical
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1.4 References 8 2. Formal kinetics
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2.1 Definitions and conventions 10
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2.2 Kinetics of irreversible first-
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Page 51 and 52: 2.7 Temperature dependence of rate
Page 57 and 58: 3.1 Determination of the order of a
Page 59 and 60: 3.2 Application of the steady-state
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Page 63 and 64: 3.4 Generalized first-order kinetic
Page 73 and 74: 3.5 Numerical integration 58 I Surv
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Page 81 and 82: 3.6 Oscillating reactions* 66 3.6 O
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Page 91 and 92: 3.7 References 76 4. Experimental m
Page 93 and 94: 4.3 The discharge flow (DF) techniq
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Page 97 and 98: 4.4 Flash photolysis (FP) 82 4.4 Fl
Page 99 and 100: 4.4 Flash photolysis (FP) 84 I Figu
Page 101 and 102: 4.6 Stopped flow studies 86 4.6 Sto
Page 103 and 104: 4.8 Relaxation methods 88 4.8 Relax
Page 105 and 106: 4.9 Femtosecond spectroscopy 90 I F
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Page 109 and 110: 4.10 References 94 5. Collision the
Page 111 and 112: 5.1 Hardspherecollisiontheory 96
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Page 115 and 116: 5.1 Hard sphere collision theory 10
Page 117 and 118: 5.2 Kinetic gas theory 102 I The 1D
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Page 121 and 122: 5.2 Kinetic gas theory 106 5.2.2 Th
Page 123 and 124: 5.2 Kinetic gas theory 108 • Resu
Page 125 and 126: 5.3 Transport processes in gases 11
Page 131 and 132: 5.4 Advanced collision theory 116 W
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5.4 Advanced collision theory 120 5
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5.4 Advanced collision theory 122 I
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5.4 Advanced collision theory 126 a
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5.4 Advanced collision theory 130 (
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5.4 Advanced collision theory 136
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5.4 Advanced collision theory 138 6
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6.2 Polyatomic molecules 140 I Figu
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6.2 Polyatomic molecules 142 I Mode
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6.3 Trajectory Calculations 144 •
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6.3 Trajectory Calculations 146 7.
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7.1 Foundations of transition state
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7.2 Applications of transition stat
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7.3 Thermodynamic interpretation of
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7.4 Transition state spectroscopy 1
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8.1 Experimental observations 176 (
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8.2 Lindemann mechanism 178 8.2 Lin
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8.2 Lindemann mechanism 180 I Half-
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8.3 Generalized Lindemann-Hinshelwo
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8.4 The specific unimolecular react
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8.5 Collisional energy transfer* 20
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8.6 Recombination reactions 202 8.7
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9.1 Chain reactions and chain explo
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9.1 Chain reactions and chain explo
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9.2 Heat explosions 208 • positiv
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9.2 Heat explosions 210 (3) We can
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9.2 Heat explosions 212 9.2.3 Explo
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9.2 Heat explosions 214 10. Catalys
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10.1 Kinetics of enzyme catalyzed r
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10.2 Kinetics of heterogeneous reac
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11.1 Qualitative model of liquid ph
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11.2 Diffusion-controlled reactions
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11.2 Diffusion-controlled reactions
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11.3 Activation controlled reaction
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11.4 Electron transfer reactions (M
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11.5 Reactions of ions in solutions
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11.5 Reactions of ions in solutions
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12.2 Fluorescence quenching (Stern-
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12.4 Radiationless processes in pho
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14 Combustion chemistry* 258 15. As
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16 Energy transfer processes* 260 1
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Appendix B 262 Appendix A: Useful p
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Appendix B 264 The Marquardt-Levenb
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Appendix C 266 • Convolution of t
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Appendix C 268 C.2 Application to t
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Appendix C 270 C.3 Application to p
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Appendix D 272 Final solutions for
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Appendix D 274 D.2 Special matrices
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Appendix D 276 I Multiplication by
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Appendix D 278 D.4 Coordinate trans
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Appendix D 280 D.5 Systems of linea
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Appendix D 282 D.6 Eigenvalue equat
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Appendix E 284 Secular equation: de
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Appendix E 286 Appendix E: Laplace
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Appendix F 288 Appendix F: The Gamm
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Appendix G 290 Appendix G: Ergebnis
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Appendix G 292 I Anschauliche Bedeu
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Appendix G 294 Ergebnis: = 1 X
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Appendix G 296 (2) Grenzfall für
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Appendix G 298 G.4 Mikrokanonische
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Appendix G 300 G.5 Statistische Int
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Appendix G 302 y = (G.7
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Appendix G 304 G.8 Zustandssumme f
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Appendix G 306 G.9 Zustandssumme f
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Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

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