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

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8.4 The specific unimolecular reaction rate constants () 191 8.4 The specific unimolecular reaction rate constants k (E) We may intuitively expect that the specific uinimolecular reaction rate constants () depend on the energy content (i.e., the excitation energy ) of the energetically activated molecules molecules A ∗ . More precisely, we will also have to take into account the dependence of the specific rate constants on other conserved degrees of freedom, in particular total angular momentum (⇒ ()), and perhaps other quantities (symmetry, components of if -mixing is weak, (⇒ (; ; ))). 8.4.1 Rice-Ramsperger-Kassel (RRK) theory I RRK model: In order to derive an expression for the specific rateconstants (), werealizethatitisnotenoughthatthemoleculesareenergizedtoabove 0 . Indeed, for the reaction to take place, the energy also has to be in the right oscillator, namely in the RC. In particular, for the reaction to take place, of the total excitation energy , the amount † has to be concentrated in the RC such that † ≥ 0 . Thus, we can write the reaction scheme as ∗ † → † → (8.74) where † denotes those excited molecules that have enough energy in the RC to overcome 0 . We assume that once this critical configuration ( † ) is reached, the molecules will immediately cross the TS and go to the product side. I Probabilities of A † and A ∗ : The probability of † compared to ∗ can be derived using statistical arguments by considering the ratio of the density of states. We consider amoleculewith • oscillators, • = quanta which have to be distributed over the oscillators, • a threshold energy of 0 such that a minimum of = 0 quanta have to be concentrated in the RC and only − quanta can be distributed freely, • À and − À . The probability of † relative to ∗ is given by ¡ †¢ ( ∗ ) = ¡ †¢ ( ∗ ) (8.75) where ( ∗ ) ∝ ∗ () = 1 ( + − 1)! (8.76) !( − 1)! and, since in the critical configuration only − quanta can be distributed freely, ¡ †¢ ∝ † () = 1 ( − + − 1)! ( − )! ( − 1)! (8.77)
8.4 The specific unimolecular reaction rate constants () 192 y ¡ †¢ ( ∗ ) ( − + − 1)! !( − 1)! = × ( − )! ( − 1)! ( + − 1)! ( − + − 1)! ! = × ( − )! ( + − 1)! (8.78) (8.79) I Specific rate constant: The critical configuration † is reached with the rate coefficient (≡ frequency factor) † (see the reaction scheme above). Since the energy is now in place (in the RC), † will cross the TS to products. The specific rateconstant () is therefore simply y () = † × () = † × ¡ †¢ ( ∗ ) ( − + − 1)! ( − )! × ! ( + − 1)! (8.80) (8.81) I Kassel formula: With the approximations according to Eq. 8.49), ( + − 1)! ! ≈ −1 for À (8.82) and we obtain ( − + − 1)! ( − )! ¡ †¢ ( ∗ ) ≈ ( − ) −1 for − À , (8.83) = ( − )−1 −1 = µ −1 − (8.84) y ¡ †¢ µ −1 − ( ∗ ) = 0 (8.85) so that the specific rateconstantbecomes This expression is known as the Kassel formula for (). µ −1 − () = † 0 (8.86) I Notes on the Kassel formula: • Owing to the approximations À and − À , the Kassel formula is valid only at À 0 .
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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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2.2 Kinetics of irreversible first-
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2.3 Kinetics of reversible first-or
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2.3 Kinetics of reversible first-or
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2.4 Kinetics of second-order reacti
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2.4 Kinetics of second-order reacti
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2.5 Kinetics of third-order reactio
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2.6 Kinetics of simple composite re
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2.7 Temperature dependence of rate
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3.1 Determination of the order of a
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3.2 Application of the steady-state
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3.2 Application of the steady-state
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3.4 Generalized first-order kinetic
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3.5 Numerical integration 58 I Surv
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3.5 Numerical integration 60 2 ord
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3.5 Numerical integration 62 3.5.5
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3.5 Numerical integration 64 , Func
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3.6 Oscillating reactions* 66 3.6 O
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3.6 Oscillating reactions* 68 • B
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3.6 Oscillating reactions* 70 (5) S
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3.6 Oscillating reactions* 72 I Fig
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3.6 Oscillating reactions* 74 [X]
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3.7 References 76 4. Experimental m
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4.3 The discharge flow (DF) techniq
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4.3 The discharge flow (DF) techniq
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4.4 Flash photolysis (FP) 82 4.4 Fl
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4.4 Flash photolysis (FP) 84 I Figu
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4.6 Stopped flow studies 86 4.6 Sto
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4.8 Relaxation methods 88 4.8 Relax
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4.9 Femtosecond spectroscopy 90 I F
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4.9 Femtosecond spectroscopy 92 I F
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4.10 References 94 5. Collision the
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5.1 Hardspherecollisiontheory 96
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5.1 Hardspherecollisiontheory 98 I
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5.1 Hard sphere collision theory 10
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5.2 Kinetic gas theory 102 I The 1D
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5.2 Kinetic gas theory 104 I Figure
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5.2 Kinetic gas theory 106 5.2.2 Th
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5.2 Kinetic gas theory 108 • Resu
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5.3 Transport processes in gases 11
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5.4 Advanced collision theory 116 W
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5.4 Advanced collision theory 118 I
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5.4 Advanced collision theory 120 5
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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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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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