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

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Appendix D 283 I Eigenvectors x : For each eigenvalue , we can write an eigenvalue equation In matrix notation, this becomes Ax 1 = 1 x 1 (D.53) Ax 2 = 2 x 2 (D.54) . (D.55) Ax = x (D.56) with the diagonal eigenvalue matrix Λ = AX= X Λ ⎛ ⎜ ⎝ ⎞ 1 0 0 0 2 0 ⎟ . . . ⎠ 0 0 (D.57) (D.58) The different eigenvectors x are usually normalized to unity, i.e., |x| = ¡ x † x ¢ 12 =1 (D.59) I Matrix diagonalization: The matrix of the eigenvectors X can be determined by multiplying the equation AX= X Λ (D.60) from the left with X −1 ,whichgives X −1 AX= X −1 X Λ (D.61) i.e., y X −1 AX= Λ (D.62) The determination of the eigenvector matrix is equivalent to finding a matrix X that transforms the matrix A into diagonal form (Λ). I Example: A = µ 2 3 310 (D.63) Eigenvalue equations: Ax = x (D.64) or in matrix form: AX= X Λ (D.65)
Appendix E 284 Secular equation: det (A − I) = ¯ 2 − 3 3 10− ¯ =(2− )(10− ) − 9=0 (D.66) (D.67) y 0=20− 2 − 10 + 2 − 9= 2 − 12 +11 (D.68) 12 =+ 12 2 ± s µ12 2 2 − 11 = 6 ± √ 36 − 11 (D.69) =6± 5 (D.70) y Eigenvalues: 1 =1 (D.71) 2 =11 (D.72) Eigenvectors: Inserting the eigenvalues one by one into the eigenvalue equation yields: (1) µ µ µ µ 2 3 11 11 11 = 310 1 =11 21 21 21 y µ −3 1 (2) µ µ µ µ 2 3 12 12 12 = 310 2 =1 22 22 22 y µ 1 3 (D.73) (D.74) (D.75) (D.76) Unnormalized eigenvalue matrix: µ −31 1 3 (D.77) Normalized eigenvalue matrix: X = ⎛ ⎜ ⎝ −3 1 √ √ 10 10 1 √ 10 3 ⎞ ⎟ ⎠ √ 10 (D.78)
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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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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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Page 277 and 278: Appendix B 262 Appendix A: Useful p
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Page 281 and 282: Appendix C 266 • Convolution of t
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Page 285 and 286: Appendix C 270 C.3 Application to p
Page 287 and 288: Appendix D 272 Final solutions for
Page 289 and 290: Appendix D 274 D.2 Special matrices
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Page 303 and 304: Appendix F 288 Appendix F: The Gamm
Page 305 and 306: Appendix G 290 Appendix G: Ergebnis
Page 307 and 308: Appendix G 292 I Anschauliche Bedeu
Page 309 and 310: Appendix G 294 Ergebnis: = 1 X
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Page 317 and 318: Appendix G 302 y = (G.7
Page 319 and 320: Appendix G 304 G.8 Zustandssumme f
Page 321 and 322: Appendix G 306 G.9 Zustandssumme f
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Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

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