Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...
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<strong>Physical</strong> <strong>Chemistry</strong> 3:<br />
<strong>—</strong> <strong>Chemical</strong> <strong>Kinetics</strong> <strong>—</strong><br />
F. Temps<br />
Institute of <strong>Physical</strong> <strong>Chemistry</strong><br />
<strong>Christian</strong>-<strong>Albrechts</strong>-University Kiel<br />
Olshausenstr. 40<br />
D-24098 Kiel, Germany<br />
Email: temps@phc.uni.kiel.de<br />
URL: http://www.uni-kiel.de/phc/temps<br />
Summer Term 2013<br />
This scriptum contains notes for the lecture “<strong>Physical</strong> <strong>Chemistry</strong> 3:<br />
<strong>Chemical</strong> <strong>Kinetics</strong>” (MNF-chem0405) in the summer term 2013 at the<br />
Institute of <strong>Physical</strong> <strong>Chemistry</strong> of <strong>Christian</strong>-<strong>Albrechts</strong>-University (CAU)<br />
Kiel. This module is the last of the three semester lecture course<br />
in <strong>Physical</strong> <strong>Chemistry</strong> for B.Sc. students of <strong>Chemistry</strong> and Business<br />
<strong>Chemistry</strong> at CAU Kiel. It includes 3 hours weekly (3 SWS) for lectures<br />
and 1 hour weekly (1 SWS) for an exercise class.<br />
c° F. Temps (2004 <strong>—</strong> 2013) Last update: June 22, 2013
ii<br />
Contents<br />
Preface<br />
ix<br />
Disclaimer<br />
xi<br />
Organisational matters<br />
xii<br />
1 Introduction 1<br />
1.1 Types of chemical reactions 1<br />
1.2 Time scales for chemical reactions 2<br />
1.3 Historical events 5<br />
1.4 References 7<br />
2 Formal kinetics 8<br />
2.1 Definitions and conventions 8<br />
2.1.1 The rate of a chemical reaction 8<br />
2.1.2 Rate laws and rate coefficients 9<br />
2.1.3 The order of a reaction 10<br />
2.1.4 The molecularity of a reaction 11<br />
2.1.5 Mechanisms of complex reactions 12<br />
2.2 <strong>Kinetics</strong> of irreversible first-order reactions 16<br />
2.3 <strong>Kinetics</strong> of reversible first-order reactions (relaxation processes) 20<br />
2.4 <strong>Kinetics</strong> of second-order reactions 24<br />
2.4.1 A + A → products 24<br />
2.4.2 A + B → products 25<br />
2.4.3 Pseudo first-order reactions 26<br />
2.4.4 Comparison of first-order and second-order reactions 27<br />
2.5 <strong>Kinetics</strong> of third-order reactions 28<br />
2.6 <strong>Kinetics</strong> of simple composite reactions 29<br />
2.6.1 Consecutive first-order reactions 29<br />
2.6.2 Reactions with pre-equilibrium 32<br />
2.6.3 Parallel reactions 33<br />
2.6.4 Simultaneous first- and second-order reactions* 33<br />
2.6.5 Competing second-order reactions* 34
iii<br />
2.7 Temperature dependence of rate coefficients 35<br />
2.7.1 Arrhenius equation 35<br />
2.7.2 Deviations from Arrhenius behavior 37<br />
2.7.3 Examples for non-Arrhenius behavior 38<br />
2.8 References 40<br />
3 <strong>Kinetics</strong> of complex reaction systems 41<br />
3.1 Determination of the order of a reaction 41<br />
3.2 Application of the steady-state assumption 43<br />
3.2.1 The reaction H 2 + Br 2 → 2 HBr 43<br />
3.2.2 Further examples 45<br />
3.3 Microscopic reversibility and detailed balancing 47<br />
3.4 Generalized first-order kinetics* 48<br />
3.4.1 Matrix method* 48<br />
3.4.2 Laplace transforms* 53<br />
3.5 Numerical integration 57<br />
3.5.1 Taylor series expansions 58<br />
3.5.2 Euler method 58<br />
3.5.3 Runge-Kutta methods 59<br />
3.5.4 Implicit methods (Gear): 61<br />
3.5.5 Extrapolation methods (Bulirsch-Stoer) 62<br />
3.5.6 Example: Consecutive first-order reactions 62<br />
3.6 Oscillating reactions* 66<br />
3.6.1 Autocatalysis 66<br />
3.6.2 <strong>Chemical</strong> oscillations 67<br />
3.7 References 75<br />
4 Experimental methods 76<br />
4.1 End product analysis 76<br />
4.2 Modulation techniques 77<br />
4.3 The discharge flow (DF) technique 78<br />
4.4 Flash photolysis (FP) 82<br />
4.5 Shock tube studies of high temperature kinetics 85<br />
4.6 Stopped flow studies 86
iv<br />
4.7 NMR spectroscopy 87<br />
4.8 Relaxation methods 88<br />
4.9 Femtosecond spectroscopy 89<br />
4.10 References 93<br />
5 Collision theory 94<br />
5.1 Hard sphere collision theory 95<br />
5.1.1 The gas kinetic collision frequency 95<br />
5.1.2 Allowance for a threshold energy for reaction 99<br />
5.1.3 Allowance for steric hindrance 100<br />
5.2 Kinetic gas theory 101<br />
5.2.1 The Maxwell-Boltzmann speed distributions of gas molecules in 1D and 3D 101<br />
5.2.2 The rate of wall collisions in 3D 106<br />
5.3 Transport processes in gases 110<br />
5.3.1 The general transport equation 110<br />
5.3.2 Diffusion 112<br />
5.3.3 Heat conduction 113<br />
5.3.4 Viscosity 114<br />
5.4 Advanced collision theory 115<br />
5.4.1 Fundamental equation for reactive scattering in crossed molecular beams 116<br />
5.4.2 Crossed molecular beam experiments 117<br />
5.4.3 From differential cross sections () to reaction rate constants ( ) 120<br />
5.4.4 Laplace transforms in chemistry 121<br />
5.4.5 Bimolecular rate constants from collision theory 123<br />
5.4.6 Long-range intermolecular interactions 132<br />
6 Potential energy surfaces for chemical reactions 138<br />
6.1 Diatomic molecules 138<br />
6.2 Polyatomic molecules 139<br />
6.3 Trajectory Calculations 143<br />
7 Transition state theory 146<br />
7.1 Foundations of transition state theory 146<br />
7.1.1 Basic assumptions of TST 147
v<br />
7.1.2 Formal kinetic model 149<br />
7.1.3 The fundamental equation for ( ) 149<br />
7.1.4 Tolman’s interpretation of the Arrhenius activation energy 153<br />
7.2 Applications of transition state theory 154<br />
7.2.1 The molecular partition functions 154<br />
7.2.2 Example 1: Reactions of two atoms A + B → A···B ‡ → AB 156<br />
7.2.3 Example 2: The reaction F + H 2 → F···H···H ‡ → FH + H 157<br />
7.2.4 Example 3: Deviations from Arrhenius behavior (T dependence of ( ))* 159<br />
7.3 Thermodynamic interpretation of transition state theory 160<br />
7.3.1 Fundamental equation of thermodynamic transition state theory 160<br />
7.3.2 Applications of thermodynamic transition state theory 164<br />
7.3.3 Kinetic isotope effects 166<br />
7.3.4 Gibbs free enthalpy correlations 168<br />
7.3.5 Pressure dependence of 168<br />
7.4 Transition state spectroscopy 170<br />
7.5 References 174<br />
8 Unimolecular reactions 175<br />
8.1 Experimental observations 175<br />
8.2 Lindemann mechanism 178<br />
8.3 Generalized Lindemann-Hinshelwood mechanism 182<br />
8.3.1 Master equation 182<br />
8.3.2 Equilibrium state populations 184<br />
8.3.3 The density of vibrational states () 185<br />
8.3.4 Unimolecular reaction rate constant in the low pressure regime 188<br />
8.3.5 Unimolecular reaction rate constant in the high pressure regime 190<br />
8.4 The specific unimolecular reaction rate constants () 191<br />
8.4.1 Rice-Ramsperger-Kassel (RRK) theory 191<br />
8.4.2 Rice-Ramsperger-Kassel-Marcus (RRKM) theory 193<br />
8.4.3 The SACM and VRRKM model 198<br />
8.4.4 Experimental results 199<br />
8.5 Collisional energy transfer* 200<br />
8.6 Recombination reactions 201
vi<br />
8.7 References 202<br />
9 Explosions 203<br />
9.1 Chain reactions and chain explosions 203<br />
9.1.1 Chain reactions without branching 203<br />
9.1.2 Chain reactions with branching 204<br />
9.2 Heat explosions 207<br />
9.2.1 Semenov theory for “stirred reactors” ( =const) 207<br />
9.2.2 Frank<strong>—</strong>Kamenetskii theory for “unstirred reactors” 211<br />
9.2.3 Explosion limits 212<br />
9.2.4 Growth curves and catastrophy theory 212<br />
10 Catalysis 214<br />
10.1 <strong>Kinetics</strong> of enzyme catalyzed reactions (homogeneous catalysis) 214<br />
10.1.1 The Michaelis-Menten mechanism 215<br />
10.1.2 Inhibition of enzymes: 217<br />
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 218<br />
10.2.1 Reaction steps in heterogeneous catalysis 218<br />
10.2.2 Physisorption and chemisorption 218<br />
10.2.3 The Langmuir adsorption isotherm 219<br />
10.2.4 Surface Diffusion 222<br />
10.2.5 Types of heterogeneous reactions 223<br />
11 Reactions in solution 227<br />
11.1 Qualitative model of liquid phase reactions 227<br />
11.2 Diffusion-controlled reactions 230<br />
11.2.1 Experimental observations 230<br />
11.2.2 Derivation of 230<br />
11.3 Activation controlled reactions 234<br />
11.4 Electron transfer reactions (Marcus theory) 235<br />
11.5 Reactions of ions in solutions 238<br />
11.5.1 Effect of the ionic strength of the solution 238<br />
11.5.2 Kinetic salt effect 238<br />
11.6 References 240
vii<br />
12 Photochemical reactions 241<br />
12.1 Basic radiative processes in a two-level system; Einstein coefficients 241<br />
12.2 Fluorescence quenching (Stern-Volmer equation) 242<br />
12.3 The radiative lifetime 0 and the fluorescence quantum yield Φ 243<br />
12.4 Radiationless processes in photoexcited molecules 244<br />
12.4.1 The Born-Oppenheimer approximation and its breakdown 244<br />
12.4.2 Avoided crossings of two potential curves of a diatomic molecule 248<br />
12.4.3 Quantum mechanical treatment of a coupled 2×2 system 250<br />
12.4.4 Radiationless processes in polyatomic molecules: Conical intersections 252<br />
13 Atmospheric chemistry* 256<br />
14 Combustion chemistry* 257<br />
15 Astrochemistry* 258<br />
16 Energy transfer processes* 259<br />
17 Reaction dynamics* 260<br />
18 Electrochemical kinetics* 261<br />
A Useful physicochemical constants 262<br />
B The Marquardt-Levenberg non-linear least-squares fitting algorithm 263<br />
C Solution of inhomogeneous differential equations 267<br />
C.1 General method 267<br />
C.2 Application to two consecutive first-order reactions 268<br />
C.3 Application to parallel and consecutive first-order reactions 270<br />
D Matrix methods 273<br />
D.1 Definition 273<br />
D.2 Special matrices and matrix operations 274<br />
D.3 Determinants 277<br />
D.4 Coordinate transformations 278<br />
D.5 Systems of linear equations 280<br />
D.6 Eigenvalue equations 282
viii<br />
E Laplace transforms 286<br />
F The Gamma function Γ () 288<br />
G Ergebnisse der statistischen Thermodynamik 290<br />
G.1 Boltzmann-Verteilung 291<br />
G.2 Mittelwerte 293<br />
G.3 Anwendung auf ein Zweiniveau-System 295<br />
G.4 Mikrokanonische und makrokanonische Ensembles 298<br />
G.5 Statistische Interpretation der thermodynamischen Zustandsgrößen 300<br />
G.6 Chemisches Gleichgewicht 301<br />
G.7 Zustandssumme für elektronische Zustände 303<br />
G.8 Zustandssumme für die Schwingungsbewegung 304<br />
G.9 Zustandssumme für die Rotationsbewegung 306<br />
G.10 Zustandssumme für die Translationsbewegung 307
ix<br />
Preface<br />
This scriptum contains lecture notes for the module “<strong>Physical</strong> <strong>Chemistry</strong> 3: <strong>Chemical</strong><br />
<strong>Kinetics</strong>” (chem0405), the last part of the 3 semester course in <strong>Physical</strong> <strong>Chemistry</strong> for<br />
B.Sc. students of <strong>Chemistry</strong> and Business <strong>Chemistry</strong> at CAU Kiel. It is assumed (but<br />
not formally required) that students have previously taken courses on chemical thermodynamics<br />
(“<strong>Physical</strong> <strong>Chemistry</strong> 1: <strong>Chemical</strong> Equilibrium” or equivalent) and elementary<br />
quantum mechanics (“<strong>Physical</strong> <strong>Chemistry</strong> 2: Structure of Matter” or equivalent) as<br />
well as “Mathematics for <strong>Chemistry</strong>” (parts 1 and 2, or equivalent). Module chem0405<br />
includes 3 hours weekly (3 SWS) for lectures and 1 SWS of exercises (56 contact hours<br />
combined). Students who started their studies at CAU Kiel should normally be in their<br />
4th semester.<br />
The scriptum gives a summary of the material covered in the scheduled lectures to<br />
allow students to repeat the material more economically. It covers basic material that<br />
all chemistry students should learn irrespective of their possible inclination towards inorganic,<br />
organic or physical chemistry, but goes well beyond the standard <strong>Physical</strong> <strong>Chemistry</strong><br />
textbooks used in the PC-1 and PC-2 courses. This is done in recognition of the<br />
established research focus at the Institute of <strong>Physical</strong> <strong>Chemistry</strong> at CAU to enable students<br />
to pursue their B.Sc. thesis project in <strong>Physical</strong> <strong>Chemistry</strong>. Some more specialized<br />
sections have been marked by asterisks and may be omitted on first reading towards the<br />
B.Sc. degree. Useful additional reference material is given in the Appendix.<br />
A brief lecture scriptum can never replace a textbook. For additional review, students<br />
are strongly encouraged (and at places required) to consult the recommended books<br />
on <strong>Chemical</strong> <strong>Kinetics</strong> beyond the standard <strong>Physical</strong> <strong>Chemistry</strong> textbooks. The book<br />
by Logan 1 is a comprehensive overview of the field of chemical kinetics and a nice<br />
introduction; it also has the advantage that it is written in German. The book by Pilling<br />
and Seakins 2 gives a good introduction especially to gas phase kinetics. The book by<br />
Houston 3 includes more information on the dynamics of chemical reactions. The book<br />
by Barrante 4 is highly recommended for everyone who wants to brush-up some math<br />
skills. A highly recommended web site for looking up mathematical definitions and<br />
recipies is the MathWorld online encyclopedia (http://mathworld.wolfram.com).<br />
The computer has become indispensable in modern research. This applies especially<br />
to <strong>Chemical</strong> <strong>Kinetics</strong>, where only computers can solve (by numerical integration) the<br />
coupled nonlinear partial differential equation systems that describe important complex<br />
reaction systems (e.g., related to combustion, the atmosphere or climate change). The<br />
integration of differential equations, a major task in chemical kinetics, can be performed<br />
using Wolfram Alpha (http://www.wolframalpha.com). More complicated problems can<br />
be solved with versatile computer algebra systems, e.g. (in order of increasing power)<br />
Derive, 5 MuPad, MathCad, 6 Maple, Mathematica 7 . A graphics program (e.g., Origin, 8<br />
1 S. R. Logan, Grundlagen der Chemischen Kinetik, Wiley-VCH, Weinheim, 1996.<br />
2 M. J. Pilling, P. W. Seakins, Reaction <strong>Kinetics</strong>, Oxford University Press, Oxford, 1995.<br />
3 P. L. Houston, <strong>Chemical</strong> <strong>Kinetics</strong> and Reaction Dynamics, McGraw-Hill, 2001.<br />
4 J. R. Barrante, Applied Mathematics for <strong>Physical</strong> <strong>Chemistry</strong>, Prentice Hall, 2003 ($ 47.11).<br />
5 Derive is available as shareware without charge.<br />
6 MathCad is installed in the PC lab.<br />
7 Mathematica has been used by the author for some of the lecture material.<br />
8 Student versions of Origin are available for ≈ 35 − 120 (depending on run time).
x<br />
or QtiPlot) may be needed for data evaluation and presentation. 9<br />
As practically all original scientific literature is published in English (British and American;<br />
in fact many chemists at some time in their professional lifes have to master the<br />
subtle differences between both), this scriptum is written in English to introduce students<br />
to the predominating language of science at an early stage. Students should also<br />
note that the working language of some of the research groups (including that of the<br />
author) in the <strong>Chemistry</strong> Department at CAU is English.<br />
Finally, I would like to encourage all students to ask questions when they appear, whether<br />
during the lecture or in the exercise classes!<br />
Kiel, June 22, 2013,<br />
F. Temps<br />
9 Excel also allows you to plot simple functions and data, but is not sufficient for serious scientific<br />
applications.
xi<br />
Disclaimer<br />
Use of this text is entirely at the reader’s risk. Some sections are incomplete, some<br />
equations still have to be checked, some figures need to be redrawn, references are still<br />
missing, etc. No warranties in any form whatever are given by the author.
xii<br />
Organisational matters<br />
• List of participants<br />
• Lecture schedule<br />
• Exams and grades<br />
<strong>—</strong> Homework assignments<br />
<strong>—</strong> Short questions<br />
<strong>—</strong> Written final exam (“Klausur”)<br />
I<br />
Table 1: List of participants.<br />
<strong>Chemistry</strong> (B.Sc.) 82<br />
Business <strong>Chemistry</strong> (B.Sc.) 25<br />
Biochemistry (B.Sc.)<br />
<strong>Chemistry</strong> (2x-B.Sc.)<br />
Physics (M.Sc.) 1<br />
Other ) 1<br />
Total 109<br />
) Not registered for exercise classes and final written exam (“Klausur”).<br />
I<br />
Figure 1: <strong>Physical</strong> chemistry courses at CAU Kiel.
xiii<br />
I<br />
Figure 2: Organisational matters.<br />
I<br />
Figure 3: Requirements and grading.
xiv<br />
I<br />
Figure 4: Lecture contents.<br />
I<br />
Figure 5: Table of contents (continued). a<br />
a Topics marked with asterisks will be omitted in this course for lack of time. They are the subject of<br />
specialized lectures for advanced students.
xv<br />
I<br />
Figure 6: Recommended textbooks on chemical kinetics.<br />
References to original publications and other recommended literature for further reading<br />
willbegivenattheendofeachchapter.
1<br />
1. Introduction<br />
1.1 Types of chemical reactions<br />
• Homogeneous reactions:<br />
<strong>—</strong> Reactions in the gas phase, e.g.,<br />
H 2 +Cl 2 → 2HCl<br />
<strong>—</strong> Reactions in the liquid phase, e.g., H + -transfer, reduction/oxidation reactions,<br />
hydrolysis, . . . (most of organic and inorganic chemistry)<br />
<strong>—</strong> Reactions in the solid phase (usually very slow, except for special systems<br />
and special conditions; not covered in this lecture).<br />
• Heterogeneous reactions:<br />
<strong>—</strong> Reactions at the gas-solid interface, e.g.,<br />
C()+12O 2<br />
() → CO()<br />
<strong>—</strong> Reactions at the gas-liquid interface, e.g., heterogeneous catalysis, atmospheric<br />
reactions on aerosols (e.g., Cl release into the gas phase from<br />
polar stratospheric clouds in antarctic spring)<br />
ClONO 2 ()+HCl() → Cl 2 ()+HNO 3 ()<br />
<strong>—</strong> Reactions at the liquid-solid boundary, e.g., solvation, heterogeneous catalysis,<br />
electrochemical reactions (electrode kinetics), . . .<br />
• Irreversible reactions, e.g.,<br />
H 2 +12O 2 → H 2 O<br />
• Reversible reactions, e.g.,<br />
H 2 +I 2 À 2HI<br />
• Composite reactions, e.g.,<br />
CH 4 +2O 2 → CO 2 +2H 2 O<br />
• Elementary reactions, e.g.,<br />
OH + H 2 → H 2 O+H<br />
CH 3 NC → CH 3 CN
1.2 Time scales for chemical reactions 2<br />
1.2 Time scales for chemical reactions<br />
A chemical reaction requires some “contact” between the respective reaction partners.<br />
The “time between contacts” thus determines the upper limit for the rate of a reaction.<br />
I<br />
Reactions in the gas phase:<br />
• Free mean pathlength À molecular dimension (diameter) .<br />
• Reaction rate is determined by time between collisions.<br />
• Time between two collisions @ :<br />
≈ 500 m s (1.1)<br />
y<br />
y<br />
≈ 10 −7 m (1.2)<br />
∆ ≈ ≈ 2 × 10 −10 s (1.3)<br />
• Gas phase reactions take place on the nanosecond time scale:<br />
I @ ≤ 1 ∆ =5× 109 s −1 (1.4)<br />
I<br />
Reactions in the liquid phase:<br />
• Molecules are permanently in contact with each other.<br />
• Typical distance between neighboring molecules (upper limit allowing for solvation<br />
shell):<br />
≈ 2 × 10 −8 10 −7 cm (1.5)<br />
• Maximal reaction rate is determined by the rate of diffusion of the reacting molecules<br />
in and out of the solvent cage.<br />
I<br />
Reactions in the solid phase:<br />
• Atoms and molecules localized on fixed lattice positions.<br />
• Reaction rate is determined by rate of diffusion (“hopping”) of the atoms and<br />
molecules via vacancies (unoccupied lattice positions, “Fehlstellen”) or interstitial<br />
sites (“Zwischengitterplätze”).<br />
• Hopping from one lattice position to another usually requires a corresponding high<br />
activation energy.<br />
• Solid state reactions are usually very slow.
1.2 Time scales for chemical reactions 3<br />
I<br />
Time scale of molecular vibrations:<br />
• The frequencies of the vibrational motions of the atoms in a molecule with respect<br />
to each other determine an upper limit for the rate of a unimolecular reaction.<br />
• Uncertainty principle (Fourier relation between frequency and time domain):<br />
y<br />
With = = ˜:<br />
∆ ∆ ≥ ~ (1.6)<br />
∆ ∆ ≥ 1<br />
2<br />
1<br />
∆ ≈<br />
2 ∆˜<br />
• Application to a C-C stretching vibration:<br />
y<br />
y<br />
(1.7)<br />
(1.8)<br />
˜(C-C) ≈ 1000 cm −1 (1.9)<br />
= = ˜ =3× 10 13 s −1 (1.10)<br />
= −1 =333 × 10 −14 s ≈ 33 fs (1.11)<br />
• Application to slow torsional motions of large molecular groups in solution:<br />
˜ ≈ 100 cm −1 (1.12)<br />
y<br />
∆ ≈ 330 fs (1.13)<br />
I Femtochemistry: We can observe ultrafast chemical reactions directly nowadays by<br />
following the motions of atoms or groups of atoms in the course of a reaction in real<br />
time by femtosecond spectroscopy and femtosecond diffraction techniques. This new<br />
area of “femtochemistry” has grown enormously in importance in the last decade, since<br />
the award of the Nobel prize in chemistry to A.H. Zewail in 1999.<br />
1 femtosecond =1fs=10 −15 s (1.14)<br />
I Timescale of e − -jump processes: Only electron jumps are faster than vibrational<br />
motions (by factor of roughly ).<br />
• Estimated time for − -jump to another orbital due to absorption/emission of a<br />
photon:<br />
∆ ≈ 10 −18 s (1.15)<br />
There are indications that electron correlation may slow this time to some<br />
10 −17 s. 10<br />
10 The group of F. Krausz (MPI for Quantum Optics, Munich) recently used attosecond metrology<br />
based on an interferometric method to measure a delay of 21 ± 5as in the photo-induced emission<br />
of electrons from the 2 orbitals of neon atoms with respect to emission from the 2 orbital by a<br />
100 eV light pulse (Schultze et al, Science 328, 1658 (2010)).
1.2 Time scales for chemical reactions 4<br />
• Other fast electron transfer processes:<br />
<strong>—</strong> Autoionization of a doubly excited state, by one electron falling to a lower<br />
orbital, while the other is excited into the ionization continuum.<br />
• But “chemical” intra- and intermolecular − -transfer reactions are again slower:<br />
<strong>—</strong> The electron jump itself is very fast (≈ 10 −18 s,asabove).<br />
<strong>—</strong> However, in the Franck-Condon limit, the reactions usually require substantial<br />
molecular rearrangements due to the structural differences between the<br />
reactants and products (⇒ Marcus theory, Section 11.4). The net reaction<br />
rate is thus again limited by the time scale of vibrational motion.<br />
I<br />
Conclusions:<br />
• The time scales of chemical reactions span the enormous range of some 30 orders<br />
of magnitude, from 1fs=10 −15 s up to 10 9 y=10 16 s!<br />
• In view of this huge span, a factor-to-two experimental uncertainty in an experimental<br />
measurement or a theoretical prediction of a rate constant really means<br />
rather little.<br />
• The enormous range of reaction rates, their complex dependencies on temperature,<br />
pressure, and species concentrations, and the question of the underlying<br />
reaction mechanisms that determine the reaction rates, make the field of chemical<br />
reaction kinetics and dynamics very attractive and rewarding!
1.3 Historical events 5<br />
1.3 Historical events<br />
I Sucrose inversion reaction (Wilhelmy, 1850): Kinetic investigation of the H + catalyzed<br />
hydrolysis of sucrose by monitoring the change as function of time of the optical<br />
activity of the solution from reactant (sucrose) to products (glucose and fructose). 11<br />
I<br />
Table 1.1: Optical activity parameters for the sucrose inversion reaction.<br />
C 12 H 22 O 11 +H 2 O −→ C 6 H 12 O 6 + C 6 H 12 O 6<br />
sucrose glucose fructose<br />
(1 M) ) +665 ◦ <strong>—</strong> +520 ◦ −920 ◦<br />
(1 M) ) Ø =+665 ◦ Ø = −200 ◦<br />
) (1 M) =specific angle of rotation of a 1 M solution at (Na-D).<br />
I<br />
Figure 1.1: Observed concentration-time profile of sucrose.<br />
11 This experiment is carried out in the <strong>Physical</strong> <strong>Chemistry</strong> Lab Course 1 at CAU (“Grundpraktikum”).
1.3 Historical events 6<br />
I<br />
Empirical rate law:<br />
ln <br />
0<br />
= − (1.16)<br />
Differentiation gives<br />
or<br />
y Empirical rate law:<br />
ln <br />
<br />
− ln 0<br />
= − ()=− (1.17)<br />
| {z } <br />
=0<br />
1 <br />
<br />
= − (1.18)<br />
<br />
= − (1.19)<br />
Reaction rate :<br />
≡− <br />
<br />
(1.20)<br />
Notice that even though there is one molecule of H 2 O taking part in the reaction, H2 O<br />
does not appear in the rate law, because it is practically constant in the dilute solution.<br />
I<br />
Figure 1.2: Milestones in chemical kinetics.
1.3 Historical events 7<br />
1.4 References<br />
Arrhenius 1889 S. A. Arrhenius, Über die Reaktionsgeschwindigkeit bei der Inversion<br />
von Rohrzucker durch Säuren, Z.Physik.Chem.4, 226 (1889).<br />
Bunker 1966 D. L. Bunker, Theory of Elementary Gas Reaction Rates, Pergamon,<br />
Oxford, 1966.<br />
Glasstone 1941 S. Glasstone, K. Laidler, H. Eyring, TheTheoryofRateProcesses,<br />
Mc Graw-Hill, New York, 1941.<br />
Homann 1975 K. H. Homann, Reaktionskinetik, Grundzüge der Physikalischen<br />
Chemie Bd. IV, R. Haase (Ed.), Steinkopf, Darmstadt, 1975.<br />
Houston 1996 P. L. Houston, <strong>Chemical</strong> <strong>Kinetics</strong> and Reaction Dynamics, McGraw-<br />
Hill, Boston, 2001.<br />
Johnston 1966 H. S. Johnston, Gas Phase Reaction Rate Theory, Ronald,NewYork,<br />
1966.<br />
Kassel 1932 L. S. Kassel, The <strong>Kinetics</strong> of Homogeneous Gas Reactions, Chem.Catalog,<br />
New York, 1932.<br />
Logan 1996 S. R. Logan, Grundlagen der Chemischen Kinetik, Wiley-VCH, Weinheim,<br />
1996.<br />
Pilling 1995 M. J. Pilling, P. W. Seakins, Reaction <strong>Kinetics</strong>, Oxford University Press,<br />
Oxford, 1995.<br />
Smith 1980 I. W. M. Smith, <strong>Kinetics</strong> and Dynamics of Elementary Gas Reactions,<br />
Butterworths, London, 1980.<br />
Steinfeld 1989 J. I. Steinfeld, J. S. Francisco, W. L. Haase, <strong>Chemical</strong> <strong>Kinetics</strong> and<br />
Dynamics, Prentice Hall, Englewood Cliffs, 1989.
1.4 References 8<br />
2. Formal kinetics<br />
2.1 Definitions and conventions<br />
2.1.1 The rate of a chemical reaction<br />
I<br />
Rate of a simple reaction of type A + B → C+D:<br />
≡− A<br />
= − B<br />
=+ C<br />
=+ D<br />
<br />
= − [A]<br />
<br />
= − [B]<br />
<br />
=+ [C]<br />
<br />
=+ [D]<br />
<br />
(2.1)<br />
(2.2)<br />
We will usually write the concentrations of a species A by using square brackets, i.e.,<br />
[A] and understand that [A] is time dependent, i.e., [A ()].<br />
I Definition of the rate of reaction: 12 Using the standard convention for the signs<br />
of the stoichiometric coefficients of the reactants (−1) and products (+1), we write a<br />
generic chemical reaction as<br />
1 B 1 + 2 B 2 + → B + +1 B +1 + (2.3)<br />
I<br />
The progress of the reaction from reactants to products can be described using the<br />
reaction progress coordinate :<br />
= <br />
(2.4)<br />
<br />
The time dependence leads to the definition of the reaction rate.<br />
Definition 2.1: The reaction rate is defined as<br />
≡ 1 <br />
<br />
<br />
(2.5)<br />
For the above reaction, therefore,<br />
= − 1 1<br />
| 1 | = − 1 2<br />
| 2 | = =+ 1 <br />
| | =+ 1 +1<br />
| +1 | <br />
= (2.6)<br />
I Units: 13<br />
• SI unit for the reaction rate:<br />
[] = mol m 3 s (2.7)<br />
12 Reaction rate = Reaktionsgeschwindigkeit. Von manchen Schulen wird auch der Begriff “Reaktionsrate”<br />
verwendet. Die bevorzugte “einzig wahre korrekte Übersetzung” wird von der jeweils anderen<br />
Seite sehr ernst genommen, allerdings gilt in Kiel in dieser Hinsicht Waffenstillstand.<br />
13 For general information on the SI system see (Mills 1988) or (Homann 1995).
2.1 Definitions and conventions 9<br />
• Convenient units for gas phase reactions:<br />
<strong>—</strong> molar units:<br />
[] = mol cm 3 s (2.8)<br />
<strong>—</strong> molecular units: 14 [] =molecules cm 3 s ⇒ cm −3 s −1 (2.9)<br />
<strong>—</strong> We prefer to use molar units for rate constants, because rate constants are<br />
inherently quantities that are “averages” of some other quantity, e.g., a molecular<br />
cross section weighted by the relevant statistical distribution function<br />
(usually the Boltzmann distribution). Rate constants are thus properties of<br />
an ensemble of molecules, they are not original molecular quantities.<br />
• Convenient units for liquid phase reactions:<br />
[] = mol ls=M s (2.10)<br />
2.1.2 Rate laws and rate coefficients<br />
I Therateequation(ratelaw): Experiments show that the reaction rate generally<br />
depends on the concentrations of the reactants. The functional dependence is expressed<br />
by the rate equation (also called rate law), 15 e.g.<br />
for reaction 2.3.<br />
= − 1 [B 1 ]<br />
= <br />
| 1 | <br />
(2.11)<br />
= ([B 1 ] [B 2 ] [B ] [B +1 ] ) (2.12)<br />
Theratelawsofcomplexreactions(i.e.,the ([B 1 ] , [B 2 ] , , [B ] , [B +1 ] , ))<br />
are often rather complicated; only rarely (as in the case of elementary reactions) are<br />
they simple and intuitive.<br />
Often (but not always), ([B 1 ] , [B 2 ] , , [B ] , [B +1 ] , ) can be written as a<br />
product of a rate coefficient and some power series of the concentrations, for<br />
example<br />
= − 1 [B 1 ]<br />
= [B 1 ] [B 2 ] ... (2.13)<br />
| 1 | <br />
Note that the exponents . . . are not usually equal to 1 2 ...<br />
I Therateconstant(ratecoefficient): The rate constant (also called rate coefficient)<br />
16 usually depends on temperature ( = ( )) and sometimes also on pressure<br />
( = ()), but not on the concentrations ( 6= ()).<br />
14 IUPAC has decided that “molecules” is not a unit; see (Mills 1989) or (Homann 1995).<br />
15 Rate law = Geschwindigkeits- oder Zeitgesetz.<br />
16 Rate constant = Geschwindigkeitskonstante. Es wird manchmal auch die Übersetzung “Ratenkonstante”<br />
verwendet (aber bitte nicht “Ratekonstante”).
2.1 Definitions and conventions 10<br />
I Rate laws for first-order and second-order reactions: Simple rate laws describe<br />
first-order and second-order reactions:<br />
• Example of a first-order reaction: CH 3 NC → CH 3 CN<br />
= − [CH 3NC]<br />
<br />
=+ [CH 3CN]<br />
<br />
= [CH 3 NC] (2.14)<br />
• Example of a second-order reaction: C 4 H 9 +C 4 H 9 → C 8 H 18<br />
= − 1 2<br />
[C 4 H 9 ]<br />
<br />
=+ [C 8H 18 ]<br />
<br />
= [C 4 H 9 ] 2 (2.15)<br />
I Question: How can we determine reaction rates and rate constants ?<br />
I<br />
Figure 2.1: The easiest <strong>—</strong> and always applicable <strong>—</strong> method for determining the reaction<br />
rate and rate coefficient is the tangent method. To determine , yousimplytake<br />
the slope of a plot of () at time ; then follows via Eq. 2.13.<br />
We’ll learn about better methods for determining for elementary reactions (first-order<br />
and second-order reactions) further below.<br />
2.1.3 The order of a reaction<br />
I<br />
Definition 2.2: The order of a reaction is defined as the sum of the exponents in the<br />
rate law (Eq. 2.13).
2.1 Definitions and conventions 11<br />
I Example: Considering a generic rate law, like<br />
− [A] = | | [A] [B] (2.16)<br />
<br />
the total order of the reaction is = + . We also say, the reaction is -th order in<br />
[A] and -th order in B, ...<br />
The order of a reaction may be determined by the tangent method (Fig. 2.1): Its effects<br />
on the reaction rate are observable by increasing (e.g., doubling) the concentrations of<br />
the reaction partners one by one. Better methods will be explored later (⇒ homework<br />
assignments).<br />
I<br />
Caveats:<br />
• The order of a reaction is, in general, an empirical quantity; it has to be determined<br />
by an experimental measurement.<br />
• The order of a complex reaction cannot be determined by simply looking at the<br />
overall reaction.<br />
• There are reactions with integer order, like the first-order or second-order reactions<br />
above (Eqs. 2.14 and 2.15). However, the order can also be fractional, as for the<br />
following reactions:<br />
(1) CH 3 CHO → CH 4 +CO:<br />
[CH 4 ]<br />
<br />
= [CH 3 CHO] 32 (2.17)<br />
(2) H 2 +Br 2 → 2HBr:<br />
[HBr]<br />
<br />
= 0 [H 2 ][Br 2 ] 12<br />
1+ 00 [HBr] [Br 2 ]<br />
(2.18)<br />
• A negative order in a concentration means that the respective species acts as an<br />
inhibitor.<br />
• A positive order in a product concentration means that the reaction rate is enhanced<br />
by autocatalysis.<br />
2.1.4 The molecularity of a reaction<br />
I<br />
Definition 2.3: The molecularity of a reaction is the number of molecules in an elementary<br />
reaction step at the microscopic (molecular) level.
2.1 Definitions and conventions 12<br />
I<br />
There are only three possible cases for the molecularity (one example for each<br />
case):<br />
• monomolecular (= unimolecular) reactions:<br />
CH 3 NC → CH 3 CN (2.19)<br />
• bimolecular reactions:<br />
O+H 2 → OH + H (2.20)<br />
• termolecular reactions:<br />
H+O 2 +M→ HO 2 +M (2.21)<br />
• Events where more than three species come together in the right configuration<br />
and react are extremely unlikely and do not play a role.<br />
I Caveat: The order of a reaction can be integer or non-integer. However, the molecularity<br />
can be only 1 or 2, or at the most 3.<br />
2.1.5 Mechanisms of complex reactions<br />
A stoichiometric formula rarely describes the reactive events happening at the microscopic,<br />
molecular level. It is, for example, completely impossible that the combustion<br />
of 1 -heptane molecule with 11 molecules of O 2<br />
-C 7 H 16 +11O 2 → 7CO 2 +8H 2 O (2.22)<br />
in an internal combustion engine occurs in a single collision and gives 15 product molecules.<br />
Instead, the net reaction is the result of a complex sequence of a very large<br />
number (thousands, in this case) of unimolecular, bimolecular, and termolecular elementary<br />
reactions.<br />
The identification of the individual elementary reactions, the determination of their rates<br />
as a function of and , andtheverification of the ensuing reaction mechanism is one<br />
of the main tasks in the field of chemical kinetics.<br />
I<br />
Examples for complex reaction systems:<br />
• H 2 /O 2 system (Fig. 2.2),<br />
• Combustion of CH 4 (Fig. 2.3),<br />
• NO formation and removal in flames (Fig. 2.4).
2.1 Definitions and conventions 13<br />
I<br />
Figure 2.2: Reactions in the H 2 /O 2 system.<br />
I Rate equations for the H 2 /O 2 system: Assuming that we know the reaction mechanism<br />
(i.e., all elementary reactions contributing), we can write down the rate equations<br />
for all species. Doing so, we obtain a set of coupled non-linear differential equations<br />
(DE’s) which can be solved usually only numerically, if we want to describe the<br />
net reaction:<br />
[H]<br />
<br />
[OH]<br />
<br />
[O]<br />
<br />
= − 1 [H] [O 2 ]+ 2 [O] [H 2 ]+ 3 [OH] [H 2 ] − 4 [H] [O 2 ][M] (2.23)<br />
− 5 [H] [HO 2 ] − 6 [H] [HO 2 ]+<br />
= 1 [H] [O 2 ]+ 2 [O] [H 2 ] − 3 [OH] [H 2 ]+2 6 [H] [HO 2 ] (2.24)<br />
− 7 [OH] [HO 2 ] − 2 8 [OH] 2 [M]<br />
+ (2.25)<br />
= (2.26)
2.1 Definitions and conventions 14<br />
I Figure 2.3: Mechanism of CH 4 combustion (Warnatz 1996).<br />
I Figure 2.4: NO in combustion processes (Warnatz 1996).
2.1 Definitions and conventions 15<br />
I<br />
How we will proceed:<br />
• In the remainder of this Chapter, we shall learn how to analytically solve the rate<br />
equations for elementary reactions and for simple composite reactions.<br />
• In Chapter 3, we shall then consider different methods for solving coupled nonlinear<br />
differential equations for complex reaction systems.<br />
<strong>—</strong> These methods are well established for stationary reaction systems.<br />
<strong>—</strong> However, the modeling of instationary reaction systems (such as ignition<br />
processes, explosions and detonations, 3D or 4D models of atmospheric<br />
chemistry and climate evolution, . . . ) continues to be a tremendously challenging<br />
task.
2.2 <strong>Kinetics</strong> of irreversible first-order reactions 16<br />
2.2 <strong>Kinetics</strong> of irreversible first-order reactions<br />
A → B (2.27)<br />
I Solution of the rate equations for A and B:<br />
− [A]<br />
<br />
=+ [B]<br />
<br />
= [A] (2.28)<br />
Mass balance:<br />
[A]<br />
<br />
+ [B]<br />
<br />
=0 (2.29)<br />
Integration of the rate equation (Eq. 2.28):<br />
Z<br />
− [A] = (2.30)<br />
[A]<br />
Z<br />
Z<br />
ln [A] = −= − (2.31)<br />
y<br />
ln [A] = −+ (2.32)<br />
Initial value condition at =0:<br />
[A ( =0)]=[A] 0<br />
(2.33)<br />
y<br />
Solution for [A ()]:<br />
y<br />
Solution for [B ()]:<br />
y<br />
=ln[A] 0<br />
(2.34)<br />
ln [A] = −+ln[A] 0<br />
(2.35)<br />
[A ()] = [A] 0<br />
− (2.36)<br />
[B ()] = [A] 0<br />
− [A ()] (2.37)<br />
¡<br />
[B ()] = [A] ¢ 0 1 − <br />
−<br />
(2.38)
2.2 <strong>Kinetics</strong> of irreversible first-order reactions 17<br />
I<br />
Figure 2.5: <strong>Kinetics</strong> of first-order reactions.<br />
I Time constant: The rate coefficient has the dimension of an inverse time. Thus,<br />
= 1 <br />
(2.39)<br />
hasdimensionoftimeandisthetimeafterwhich[A] has dropped to 1 <br />
its initial value:<br />
[A] =<br />
=[A] 0<br />
− = [A] 0<br />
<br />
≈ [A] 0<br />
2718<br />
(=368 %) of<br />
(2.40)<br />
I Half lifetime: The half lifetime 12 is the time after which [A] has dropped to 1 2 of<br />
its initial value (Fig. 2.5):<br />
y<br />
£<br />
A<br />
¡<br />
= 12<br />
¢¤<br />
=[A]0 − 12<br />
= [A] 0<br />
2<br />
− 12<br />
= 1 2<br />
(2.41)<br />
(2.42)<br />
y<br />
12 =ln2 (2.43)<br />
12 = ln 2<br />
<br />
(2.44)
2.2 <strong>Kinetics</strong> of irreversible first-order reactions 18<br />
I<br />
Conclusion:<br />
Thehalflifetimeforafirst-order reactions is independent of the initial concentration!<br />
I Experimental determination of k:<br />
(1) Plot of ln [A] vs. gives a straight line with slope = − (see Fig. 2.5):<br />
ln ([A] [A] 0<br />
)=− (2.45)<br />
Advantage: We do not need to know the absolute concentration of [A].<br />
(2) Numerical fit of to the exponential decay curve using the Marquardt-Levenberg<br />
non-linear least-squares fitting algorithm (Fig. 2.6; see Appendix B for details).<br />
This method is useful in particular for analyzing experimental decay curves with<br />
a constant background term. Again, we do not need to know the absolute concentration<br />
of [A].<br />
Advantages: Can be applied in presence of a constant background signal (noise,<br />
weak DC signal). Can also be applied for analyzing data from pump-probe experiments,<br />
when the duration of the pump is not short to the decay (deconvolution).<br />
I Figure 2.6: Exponential decay curve of a particular vibration-rotation state of<br />
the CH 3 O radical resulting from its unimolecular dissociation reaction according to<br />
CH 3 O → H+H 2 CO (Dertinger 1995). The small box is the output box from a fit<br />
using the software package ORIGIN.
2.2 <strong>Kinetics</strong> of irreversible first-order reactions 19<br />
I<br />
Figure 2.7: Excited-state relaxation dynamics of the adenine DNA dinucleotide after<br />
UV excitation.<br />
I Note (and warning): Anybody who wants to do least-squares fitting is urged to<br />
consult appropriate literature before doing the fitting! The book by Bevington 17 is<br />
considered required reading for any physical chemistry graduate student.<br />
As the saying goes: With three parameters, you can draw an elephant. With a fourth<br />
parameter, you can make him walk.<br />
17 P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the <strong>Physical</strong> Sciences,<br />
McGraw-Hill, Boston, 1992.
2.3 <strong>Kinetics</strong> of reversible first-order reactions (relaxation processes) 20<br />
2.3 <strong>Kinetics</strong> of reversible first-order reactions (relaxation processes)<br />
A<br />
1 À<br />
−1<br />
B (2.46)<br />
I<br />
Rate equation:<br />
[A]<br />
<br />
= − 1 [A] + −1 [B] (2.47)<br />
I<br />
Equilibrium at t →∞:<br />
[A]<br />
<br />
y<br />
= − [B]<br />
<br />
= − 1 [A] + −1 [B] = 0 (2.48)<br />
[B] ∞<br />
[A] ∞<br />
= 1<br />
−1<br />
= (2.49)<br />
• Therateconstantforthereversereaction −1 can be calculated from 1 and .<br />
• Or the equilibrium constant can be determined from measurements of the<br />
forward and reverse rate constants. From ,wecanthendetermineimportant<br />
thermochemical quantities, for example for reactive free radicals, via<br />
= ( B ª )( C ª )<br />
=exp<br />
∙− ∆ ¸<br />
ª<br />
(2.50)<br />
( A ª )<br />
<br />
• Important note:<br />
<strong>—</strong> In the frequently encountered case that the number of species on the reactant<br />
and product side of a reaction differ, one has to be very careful with the<br />
units of and .<br />
Consider, for example, a gas phase reaction of the type<br />
In this case, has the dimension of mol cm 3 :<br />
B C<br />
<br />
= 1 [B] [C]<br />
= =<br />
−1 [A]<br />
0 has the dimension of bar:<br />
A → B+C (2.51)<br />
A<br />
<br />
= 1 B C<br />
A<br />
= 0 <br />
<br />
(2.52)<br />
0 = B C<br />
(2.53)<br />
A<br />
But the thermodynamic equilibrium constant is dimensionless:<br />
= ( ∙<br />
B ª )( C ª )<br />
= 0 <br />
( A ª ) =exp − ∆ ¸<br />
ª<br />
(2.54)<br />
ª <br />
<strong>—</strong> General relation between 0 ,and :<br />
= 0 × ¡ ª¢ −Σ ( ) = × ¡ ª¢ Σ ( )<br />
(2.55)
2.3 <strong>Kinetics</strong> of reversible first-order reactions (relaxation processes) 21<br />
I<br />
Solution of the rate equations for [A] and [B]:<br />
Returning to the time dependence of the reaction<br />
A<br />
1 À<br />
−1<br />
B (2.56)<br />
we now solve the rate equation<br />
[A]<br />
<br />
= − 1 [A] + −1 [B] (2.57)<br />
• Mass balance:<br />
[B] = [A] 0<br />
− [A] (2.58)<br />
• Integration:<br />
[A]<br />
<br />
= − 1 [A] + −1 [B] (2.59)<br />
= − 1 [A] + −1 ([A] 0<br />
− [A]) (2.60)<br />
= − ( 1 + −1 )[A]+ −1 [A] 0<br />
(2.61)<br />
y<br />
[A]<br />
( 1 + −1 )[A]− −1 [A] 0<br />
= − (2.62)<br />
• Solution by substitution:<br />
=( 1 + −1 )[A]− −1 [A] 0<br />
(2.63)<br />
y<br />
y<br />
<br />
[A] =( 1 + −1 ) (2.64)<br />
[A] =<br />
<br />
( 1 + −1 )<br />
(2.65)<br />
<br />
= −<br />
( 1 + −1 ) <br />
(2.66)<br />
<br />
= − ( 1 + −1 ) (2.67)<br />
ln = − ( 1 + −1 ) + (2.68)<br />
• Initial value condition at =0:<br />
y<br />
y<br />
= 0 (2.69)<br />
=ln 0 (2.70)<br />
<br />
0<br />
=exp[− ( 1 + −1 ) ] (2.71)
2.3 <strong>Kinetics</strong> of reversible first-order reactions (relaxation processes) 22<br />
I Solution for [A(t)] (Fig. 2.8):<br />
or<br />
( 1 + −1 )[A]− −1 [A] 0<br />
( 1 + −1 )[A] 0<br />
− −1 [A] 0<br />
=exp[− ( 1 + −1 ) ] (2.72)<br />
−1<br />
[A] − [A]<br />
1 + 0<br />
−1<br />
[A] 0<br />
−<br />
=exp[− ( 1 + −1 ) ] (2.73)<br />
−1<br />
[A]<br />
1 + 0<br />
−1<br />
This expression is not so easy to memorize, but we may recast it in a simple way:<br />
Using<br />
1<br />
= [B] ∞<br />
= [A] 0 − [A] ∞<br />
(2.74)<br />
−1 [A] ∞<br />
[A] ∞<br />
which gives<br />
[A] ∞<br />
=<br />
−1<br />
[A]<br />
1 + 0<br />
(2.75)<br />
−1<br />
we obtain<br />
[A ()] − [A] ∞<br />
=exp[− ( 1 + −1 ) ] (2.76)<br />
[A] 0<br />
− [A] ∞<br />
With ∆ [A] <br />
=[A()]−[A] ∞<br />
and ∆ [A] 0<br />
=[A] 0<br />
−[A] ∞<br />
, we write this result in compact<br />
form as<br />
∆ [A] <br />
∆ [A] 0<br />
=exp[− ( 1 + −1 ) ] (2.77)<br />
I<br />
Figure 2.8: <strong>Kinetics</strong> of reversible first-order reactions.
2.3 <strong>Kinetics</strong> of reversible first-order reactions (relaxation processes) 23<br />
I<br />
Relaxation processes:<br />
• The transition from a deviation from equilibrium into the equilibrium is called<br />
relaxation.<br />
• ( 1 + −1 ) −1 has the dimension of time and is called the relaxation time,<br />
=( 1 + −1 ) −1 (2.78)<br />
• The temporal evolution of relaxation processes from a nonequilibrium to the equilibtium<br />
state is determined by the sum of the respective forward and reverse rate<br />
constants!
2.4 <strong>Kinetics</strong> of second-order reactions 24<br />
2.4 <strong>Kinetics</strong> of second-order reactions<br />
2.4.1 A + A → products<br />
A+A→ P (2.79)<br />
I<br />
Solution of the rate equation:<br />
Integration:<br />
y<br />
Initial value condition at =0:<br />
− 1 [A]<br />
= [A] 2 (2.80)<br />
2 <br />
[A]<br />
2<br />
= −2 (2.81)<br />
[A]<br />
− 1 = −2 + (2.82)<br />
[A]<br />
[A ( =0)]=[A] 0<br />
(2.83)<br />
y<br />
Solution for [A ()] (Fig. 2.15):<br />
= − 1<br />
[A] 0<br />
(2.84)<br />
1<br />
[A] = 1 +2 (2.85)<br />
[A] 0<br />
I Experimental determination of k:<br />
(1) Graphical method:<br />
1<br />
[A] = 1<br />
[A] 0<br />
+2 (2.86)<br />
⇒ Plot of [A] −1 vs. gives a straight line with slope =2 (Fig. 2.9).<br />
⇒ We do need to know the absolute concentration of [A]!<br />
(2) We can again use a nonlinear least squares fitting method (Appendix B).
2.4 <strong>Kinetics</strong> of second-order reactions 25<br />
I<br />
Figure 2.9: <strong>Kinetics</strong> of second-order reactions.<br />
I<br />
Half lifetime:<br />
y<br />
y<br />
y<br />
£<br />
A<br />
¡<br />
12<br />
¢¤<br />
=<br />
[A] 0<br />
2<br />
(2.87)<br />
2<br />
[A] 0<br />
= 1<br />
[A] 0<br />
+2 12 (2.88)<br />
12 =<br />
1<br />
2 [A] 0<br />
(2.89)<br />
The half lifetime for a second-order reactions depends on the initial concentration!<br />
2.4.2 A + B → products<br />
A+B→ P (2.90)
2.4 <strong>Kinetics</strong> of second-order reactions 26<br />
I<br />
Solution of the rate equation:<br />
− [A]<br />
<br />
= − [B]<br />
<br />
= [A] [B] (2.91)<br />
(1) Substitution:<br />
=([A] 0<br />
− [A] <br />
)=([B] 0<br />
− [B] <br />
) (2.92)<br />
[A] = [A] 0<br />
− (2.93)<br />
y<br />
<br />
= ([A] 0 − )([B] 0<br />
− ) (2.94)<br />
(2) Integration using partial fractions (skipped here):<br />
(3) General solution:<br />
<br />
[B] 0<br />
[A] <br />
[A] 0<br />
[B] <br />
=exp([A] 0<br />
− [B] 0<br />
) (2.95)<br />
2.4.3 Pseudo first-order reactions<br />
In experimental studies of second-order reactions of the type A+B→ C, onelikesto<br />
use a high excess of one of the reactions partners (e.g., [B] À [A]) sothat[B] ≈ const<br />
Under this condition, the reaction is said to be pseudo first-order, because<br />
[B] = 0 (2.96)<br />
y Solution of [A ()]:<br />
[A] = [A] 0<br />
−0 =[A] 0<br />
−[B] (2.97)<br />
I<br />
Experimental investigation of pseudo first-order reactions:<br />
(1) Measurements of the pseudo first-order decay of [A] vs. at different concentrations<br />
of [B]: Plotofln [A] vs. give pseudo first-order rate constant 0 = [B]<br />
for each value of [B].<br />
(2) Plot of 0 = [B] vs. [B] gives straight line with slope .<br />
⇒ We do not need the absolute concentration of [A], only the concentration of [B] is<br />
needed.
2.4 <strong>Kinetics</strong> of second-order reactions 27<br />
I<br />
Figure 2.10: <strong>Kinetics</strong> of pseudo first-order reactions.<br />
2.4.4 Comparison of first-order and second-order reactions<br />
Consider a first-order and a second-order reaction with the same initial concentrations<br />
and the same initial rate at = 0. As shown in Fig. 2.11 below, the rate of the<br />
second-order reaction rapidly slows down as time increases.<br />
I<br />
Figure 2.11: Comparison of first-order and second-order reactions.
2.5 <strong>Kinetics</strong> of third-order reactions 28<br />
2.5 <strong>Kinetics</strong> of third-order reactions<br />
y<br />
A+B+C→ D (2.98)<br />
− [A]<br />
<br />
= [A] [B] [C] (2.99)<br />
I Pseudo second-order combination reactions: In combination reactions of two<br />
species A + A → A 2 or A + B → AB (for example, the recombination of two free<br />
radicals), one of the species (bath gas M) is usually present in large excess ([M] À [A],<br />
[B]), so that the kinetics becomes pseudo second-order, giving the two cases:<br />
•<br />
•<br />
A+A+M→ A 2 +M (2.100)<br />
A+B+M→ AB + M (2.101)<br />
M plays the role of an inert collision partner which removes the excess energy from the<br />
newly formed A 2 or AB molecules.
2.6 <strong>Kinetics</strong> of simple composite reactions 29<br />
2.6 <strong>Kinetics</strong> of simple composite reactions<br />
2.6.1 Consecutive first-order reactions<br />
A −→ 1<br />
B (2.102)<br />
B −→ 2<br />
C (2.103)<br />
I<br />
Coupled differential equations:<br />
[A]<br />
<br />
= − 1 [A] (2.104)<br />
[B]<br />
<br />
=+ 1 [A] − 2 [B] (2.105)<br />
[C]<br />
<br />
=+ 2 [B] (2.106)<br />
In this case, since the DE’s are linear, there is an exact solution which we will examine<br />
first. We will also look at an approximate solution which can be obtained with the<br />
quasi steady-state approximation and at methods to obtain numerical solutions, which<br />
we have to use for complex non-linear inhomogeneous DE systems.<br />
I<br />
Mass balance:<br />
[A] + [B] + [C] = [A] 0<br />
(2.107)<br />
I<br />
Initial conditions:<br />
[A ( =0)] = [A] 0<br />
(2.108)<br />
[B ( =0)] = 0 (2.109)<br />
[C ( =0)] = 0 (2.110)<br />
I<br />
Exact solution:<br />
• First-order decay of [A]:<br />
y<br />
[A]<br />
<br />
= − 1 [A] (2.111)<br />
[A] = [A] 0<br />
− 1<br />
(2.112)<br />
• Inhomogeneous DE for [B]:<br />
[B]<br />
<br />
=+ 1 [A] − 2 [B] (2.113)
2.6 <strong>Kinetics</strong> of simple composite reactions 30<br />
• Solution for [B ()] 18 :<br />
⎧<br />
1 [A] ⎪⎨ 0<br />
¡<br />
<br />
− 1 − ¢ − 2 <br />
if 1 6= 2<br />
[B ()] =<br />
2 − 1<br />
(2.114)<br />
⎪⎩<br />
1 [A] 0<br />
− 1 <br />
if 1 = 2<br />
• Reaction products by mass balance:<br />
[C ()] = [A] 0<br />
− [A ()] − [B ()] = (2.115)<br />
I<br />
Figure 2.12: <strong>Kinetics</strong> of consecutive first-order reactions (case I).<br />
18 ThesolutionisdetailedinAppendixC.
2.6 <strong>Kinetics</strong> of simple composite reactions 31<br />
I<br />
Figure 2.13: <strong>Kinetics</strong> of consecutive first-order reactions (case II).<br />
I Caveat: As seen, the general solution for [B] is the difference of two exponentials,<br />
[B] = 1 [A] 0<br />
2 − 1<br />
¡<br />
<br />
− 1 − − 2 ¢ , (2.116)<br />
One exponential describes the rise, the other the fall of [B]. It is sometimes implicitly,<br />
but wrongly assumed that the rise time corresponds to 1 and the decay time to 2 .<br />
The truth is that we cannot tell just from the shape of the concentration-time profile<br />
whether 1 or 2 correspond to the rise or to the fall of [B].<br />
I Quasi steady-state approximation: Under the condition that<br />
2 À 1 (2.117)<br />
we can find a simple solution for the DE’s. Under this condition, after a short initial<br />
induction time (see Fig. 2.13), the change of [B] is very small compared to that of<br />
[A]. Therefore,wehaveapproximately<br />
[B]<br />
<br />
≈ 0 (2.118)<br />
This important approximation is known as the (quasi)steady-state approximation. 19<br />
19 Deutsch: Quasistationaritätsannahme.
2.6 <strong>Kinetics</strong> of simple composite reactions 32<br />
I Quasi steady-state solution for [B (t)]: For we have<br />
y<br />
y<br />
[B]<br />
<br />
= 1 [A] − 2 [B] ≈ 0 (2.119)<br />
1 [A] ≈ 2 [B] (2.120)<br />
[B] <br />
= 1<br />
2<br />
[A] = 1<br />
2<br />
[A] 0<br />
− 1<br />
(2.121)<br />
For , [C ()] is therefore given by<br />
y<br />
[C]<br />
<br />
= 2 [B] ≈ 1 [A] (2.122)<br />
[C] = [A] 0<br />
¡<br />
1 − <br />
− 1 ¢ (2.123)<br />
2.6.2 Reactions with pre-equilibrium<br />
A<br />
1 À<br />
−1<br />
B (2.124)<br />
B 2 → C (2.125)<br />
I<br />
Equilibrium between [A] and [B]:<br />
1 −1 À 2 (2.126)<br />
y<br />
[B]<br />
[A] = 1<br />
= <br />
−1<br />
(2.127)<br />
[B] = [A] (2.128)<br />
I<br />
Time dependence of [C]:<br />
[C]<br />
<br />
= 2 [B] = 2 [A] (2.129)
2.6 <strong>Kinetics</strong> of simple composite reactions 33<br />
2.6.3 Parallel reactions<br />
A −→ 1<br />
B (2.130)<br />
A −→ 2<br />
C (2.131)<br />
A −→ 3<br />
D (2.132)<br />
y<br />
y<br />
and<br />
[A]<br />
<br />
= − ( 1 + 2 + 3 )[A] (2.133)<br />
[A] = [A] 0<br />
−( 1+ 2 + 3 )<br />
(2.134)<br />
[B] =<br />
[C] =<br />
[D] =<br />
1 [A] 0<br />
¡ ¢ 1 − <br />
−( 1 + 2 + 3 )<br />
( 1 + 2 + 3 )<br />
(2.135)<br />
2 [A] 0<br />
¡ ¢ 1 − <br />
−( 1 + 2 + 3 )<br />
( 1 + 2 + 3 )<br />
(2.136)<br />
3 [A] 0<br />
¡ ¢ 1 − <br />
−( 1 + 2 + 3 )<br />
( 1 + 2 + 3 )<br />
(2.137)<br />
Note that the decay of [A] is determined by the total removal rate constant =<br />
1 + 2 + 3<br />
2.6.4 Simultaneous first- and second-order reactions*<br />
A −→ 1<br />
C (2.138)<br />
A+A−→ 2<br />
D (2.139)<br />
y<br />
y<br />
[A]<br />
= − 1 [A] − 2 2 [A] 2<br />
<br />
(2.140)<br />
µ<br />
1<br />
[A] = −2 2 22<br />
+ + 1 <br />
exp (− 1 )<br />
1 1 [A] 0<br />
(2.141)<br />
I Approximation for k 1 t ¿ 1: Power series expansion of the exponential gives<br />
1<br />
[A] ≈ 1 µ <br />
1<br />
+ +2 2 × (2.142)<br />
[A] 0<br />
[A] 0
2.6 <strong>Kinetics</strong> of simple composite reactions 34<br />
2.6.5 Competing second-order reactions*<br />
A+A−→ 1<br />
A 2 (2.143)<br />
A+B−→ 2<br />
P 1 (2.144)<br />
B −→ 3<br />
P 2 (2.145)<br />
I<br />
Solution for the case [A] À [B]:<br />
(1)<br />
[A]<br />
<br />
= −2 1 [A] 2 − 2 [A] [B] (2.146)<br />
≈−2 1 [A] 2 (2.147)<br />
y<br />
or<br />
1<br />
[A] − 1<br />
[A] 0<br />
=2 1 (2.148)<br />
[A] = ¡ [A] 0 −1 +2 1 ¢ −1<br />
(2.149)<br />
=<br />
[A] 0<br />
1+2 1 [A] 0<br />
<br />
(2.150)<br />
(2)<br />
[B]<br />
= − 2 [A] [B] − 3 [B] (2.151)<br />
<br />
This is another inhomogeneous first-order DE, for which the solution can be found<br />
as outlined in Appendix C, giving<br />
[B] = [B] 0<br />
× (1 + 2 1 [A] 0<br />
) − 22 1<br />
× exp (− 3 ) (2.152)<br />
I<br />
Example:<br />
1<br />
CH 3 +CH 3 −→ C 2 H 6 (2.153)<br />
CH 3 +OH−→ 2<br />
CH 2 +H 2 O (2.154)<br />
OH −→ 3<br />
P 2 (2.155)<br />
In the experimental study of reaction 2.154 (Deters 1998), the rate coefficient for the<br />
reaction ( 2 according to Eq. 2.152) was fitted to measured OH concentration-time<br />
profiles in the presence of high concentrations of CH 3 using the Marquardt-Levenberg<br />
nonlinear least squares fitting algorithm (Press 1992). The rate coefficients 1 and<br />
3 were determined by independent measurements. The absolute CH 3 concentrations,<br />
which are needed for the evaluation of 2 according to Eq. 2.152, were obtained using<br />
a quantitative titration reaction.
2.7 Temperature dependence of rate coefficients 35<br />
2.7 Temperature dependence of rate coefficients<br />
2.7.1 Arrhenius equation<br />
I Arrhenius: 20 Svante Arrhenius (Arrhenius 1889) found the following empirical relation<br />
describing the temperature dependence of the reaction rate constant:<br />
ln <br />
(1 ) = − <br />
<br />
(2.156)<br />
I<br />
Definition 2.4: is called the Arrhenius activation energy of the reaction:<br />
= − ln <br />
(1 )<br />
(2.157)<br />
I<br />
It is very important to keep in mind the following points:<br />
• Eq. 2.156 is nothing else but a simple and very convenient way to describe the<br />
empirical temperature dependence of reaction rate constants!<br />
• Eq. 2.156 says that, within experimental error, a plot of ln vs. 1 should give<br />
a straight line. We will see later, however, that this is rarely exactly true. The<br />
study and the understanding of non-Arrhenius behavior is in fact an important<br />
topicinmodernkinetics.<br />
• as determined by Eqs. 2.156 and 2.157 is therefore a purely empirical quantity!<br />
• We will use Eq. 2.157 simply as the definition and recipy for determining from<br />
experimental data and to relate theoretical models to the experimental data!<br />
I The rationale behind the Arrhenius equation (2.156): The rationale for the<br />
Arrhenius equation is that molecules need additional energy so that chemical bonds can<br />
be broken, atoms can rearrange, and new bonds can be formed (Fig. 2.14).<br />
Van’t Hoff’s equation:<br />
∆ = 2 ln <br />
<br />
= 2 ln −→ <br />
<br />
= 2 ln −→ ←− <br />
<br />
− 2 ln ←− <br />
<br />
(2.158)<br />
(2.159)<br />
= −→ − ←− (2.160)<br />
Thus we can write<br />
ln <br />
<br />
= <br />
2 (2.161)<br />
20 Arrhenius is generally regarded as one of the founders of physical chemistry (together with Faraday,<br />
van’t Hoff, andOstwald).
2.7 Temperature dependence of rate coefficients 36<br />
or 21<br />
ln <br />
(1 ) = − <br />
<br />
(2.164)<br />
I<br />
Figure 2.14: Derivation of the Arrhenius equation.<br />
I Experimental determination of E (see Fig. 2.14): Aplotofln vs. 1 gives<br />
a straight line with slope − <br />
I Integration of Eq. 2.156: Assuming that is independent of ,Eq.2.156canbe<br />
integrated:<br />
ln = <br />
(2.165)<br />
<br />
2<br />
y<br />
ln = − <br />
+ (2.166)<br />
<br />
y<br />
( )= − <br />
(2.167)<br />
Equation 2.167 is used to represent ( ) over a limited -range using the two parameters<br />
and :<br />
• = Arrhenius activation energy (measure for the energy barrier of the reaction)<br />
• = pre-exponential factor (frequency factor, collision frequency).<br />
21<br />
y<br />
(1 )<br />
<br />
= − 1<br />
2 (2.162)<br />
= − 2 (1 ) (2.163)
2.7 Temperature dependence of rate coefficients 37<br />
I<br />
I<br />
It is important to realize, however, that the value of E is not equal to the<br />
true height of the potential energy barrier for the reaction!<br />
As said above, is an empirical quantity! Its relation to the true threshold energy 0 ,<br />
which is the minimum energy above which reaction may occur, will be a main subject<br />
in the theory of bimolecular and unimolecular reactions (Sections 5 <strong>—</strong> 8). There is also<br />
a statistical interpretation of (see Section 8).<br />
Figure 2.15: The reaction HCO → H + CO (G. Friedrichs).<br />
2.7.2 Deviations from Arrhenius behavior<br />
In general, is dependent on temperature, i.e., = ( ). For gas phase reactions,<br />
for example, there is at least the √ dependence of the hard sphere collision frequency.<br />
The -dependence of is often small compared to so that within experimental<br />
error the dependence of ( ) is determined by .<br />
If the -dependence of is not small compared to ,wewillhavetouseamore<br />
general expression for ( ), as explained in the following.<br />
I Generalized three-parameter expression for k (T ): To account for deviations<br />
from Arrhenius because of ( ), one conveniently employs the three parameter expression<br />
( )= − 0<br />
(2.168)<br />
with and 0 now being -independent quantities.<br />
Range of values for specific bimolecular reactions (justified by theory; see later):<br />
− 15 ≤ ≤ +3 (2.169)
2.7 Temperature dependence of rate coefficients 38<br />
I<br />
I<br />
Question: What is the relation of Eq. 2.168 with the Arrhenius equation?<br />
Answer: We have to evaluate the Arrhenius parameters from Eq. 2.168 according to<br />
the definitions of and :<br />
(1) :<br />
=+ 2 ln <br />
(2.170)<br />
<br />
=+ 2 µ<br />
ln + ln − <br />
0<br />
(2.171)<br />
<br />
<br />
y<br />
= 0 + (2.172)<br />
Onlyinthecasethat ¿ 0 can we neglect the term compared to 0 !<br />
(2) : From the expression for ,wehave<br />
y<br />
0 = − (2.173)<br />
( )= −( − )<br />
= − <br />
= − <br />
(2.174)<br />
(2.175)<br />
(2.176)<br />
Comparing coefficients, we find<br />
= (2.177)<br />
2.7.3 Examples for non-Arrhenius behavior<br />
Complex forming bimolecular reactions may show extreme non-Arrhenius behavior. An<br />
extreme case is the reaction OH + CO → H+CO 2 , which incidentally is the single<br />
most important reaction in hydrocarbon combustion next to H+O 2 → OH + O.
2.7 Temperature dependence of rate coefficients 39<br />
I Figure 2.16: The reaction OH + CO → H+CO 2 (Fulle 1996, Kudla 1992).<br />
Another case is encountered if tunneling plays a role (for example, H + transfer reactions<br />
at low temperatures). In this case, ( ) tends to a plateau value as → 0 (Eisenberger<br />
1991).
2.7 Temperature dependence of rate coefficients 40<br />
2.8 References<br />
Arrhenius 1889 S. A. Arrhenius, Über die Reaktionsgeschwindigkeit bei der Inversion<br />
von Rohrzucker durch Säuren, Z.Physik.Chem.4, 226 (1889).<br />
Bevington 1992 P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis<br />
for the <strong>Physical</strong> Sciences, McGraw-Hill, Boston, 1992.<br />
Dertinger 1995 S. Dertinger, A. Geers, J. Kappert, F. Temps, J. W. Wiebrecht,<br />
Rotation-Vibration State Resolved Unimolecular Dynamics of Highly Vibrationally<br />
Excited CH 3 O( 2 ): III. State Specific Dissociation Rates from Spectroscopic Line<br />
Profiles and Time Resolved Measurements, Faraday Discuss. Roy. Soc. 102, 31<br />
(1995).<br />
Deters 1998 R. Deters, M. Otting, H. Gg. Wagner, F. Temps, B. László, S. Dóbé,<br />
T. Bérces, A Direct Investigation of the Reaction CH 3 + OH at = 298K:<br />
Overall Rate Coefficient and CH 2 Formation, Ber. Bunsenges. Phys. Chem. 102,<br />
58 (1998).<br />
Eisenberger 1991 H. Eisenberger, B. Nickel, A. A. Ruth, W. Al-Soufi, K.H.Grellmann,<br />
M. Novo, Keto-Enol Tautomerization of 2-(2 0 -Hydroxyphenyl)benzoxazole<br />
and 2-(2 0 -Hydroxy-4 0 -methylphenyl)benzoxazole in the Triplet State: Hydrogen<br />
Tunneling and Isotope Effects. 2. Dual Phosphorescence Studies, J. Phys. Chem.<br />
95, 10509 (1991).<br />
Fulle 1996 D. Fulle, H. F. Hamann, H. Hippler, J. Troe, High Pressure Range of Addition<br />
Reactions of HO. II. Temperature and Pressure Dependence of the Reaction<br />
HO + CO À HOCO → H+CO 2 ,J.Chem.Phys.105, 983 (1996).<br />
Homann 1995 K. H. Homann, SI-Einheiten, VCH,Weinheim,1995.<br />
Kudla 1992 K. Kudla, A. G. Koures, L. B. Harding, G. C. Schatz, A quasiclassical<br />
trajectory study of OH rotational excitation in OH + CO collisions using ab initio<br />
potential surfaces, J. Chem. Phys. 96, 7465 (1992).<br />
Mills 1988 I. M. Mills et al., Units and Symbols in <strong>Physical</strong> <strong>Chemistry</strong>, Blackwell,<br />
Oxford, 1988.<br />
Press 1992 W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical<br />
Recipes in Fortran, Cambridge University Press, Cambridge, 1992. Versions are<br />
also available for C and Pascal.<br />
Warnatz 1996 J. Warnatz, U. Maas, R. W. Dibble, Combustion. <strong>Physical</strong> and <strong>Chemical</strong><br />
Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation,<br />
Springer, Heidelberg, 1996.
2.8 References 41<br />
3. <strong>Kinetics</strong> of complex reaction systems<br />
Real reaction systems are usually much more complicated:<br />
• Net reaction mechanisms may consist of ≈ 100 - 10 000 elementary reactions.<br />
• No analytical solutions for the rate laws.<br />
• In favorable cases, one may be able to use certain approximations to simplify the<br />
reaction systems, like<br />
<strong>—</strong> apply steady-state approximation where possible,<br />
<strong>—</strong> take into account reactions in equilibrium,<br />
<strong>—</strong> take into account microscopic reversibility (detailed balancing).<br />
• For more acurate descriptions, however, one usually needs numerical solutions<br />
for the rate equations. Using modern computers, this is not a problem even for<br />
10 000 reactions, as long as the transport processes are not too complicated<br />
(as, for example, 1D reaction systems such premixed flames or 1D models of atmospheric<br />
chemistry). However, numerical solutions are still extremely challenging<br />
for instationary 3D reaction systems, such as<br />
<strong>—</strong> 3D models of atmospheric chemistry including daily/annual variations and<br />
couplings to the ocean,<br />
<strong>—</strong> ignition processes (internal combustion engines).<br />
3.1 Determination of the order of a reaction<br />
Any reaction mechanism which we may postulate has to explain the observed reaction<br />
order. Thus, the determination of the reaction order is a central topic.<br />
We explore some general methods for determining the reaction order by considering an<br />
example:<br />
= − [A] = [A] (3.1)<br />
<br />
I First-order and second-order reactions: Test whether plots of ln vs. or 1 vs.<br />
, give the straight lines expected for first-order or second-order reactions, respectively.<br />
I Log-log plot of reaction rate vs. concentration: We see from Eq. 3.1 that a<br />
log-log plot of vs. [A] gives a straight line with slope :<br />
lg ∝ lg [A] (3.2)
3.1 Determination of the order of a reaction 42<br />
I Half lifetime method (for n 6= 1): 22<br />
y<br />
Z<br />
− [A] = [A] (3.3)<br />
<br />
Z<br />
[A] − [A] = − (3.4)<br />
y<br />
−<br />
1<br />
( − 1) [A]−(−1) = −+ (3.5)<br />
Initial value condition at =0:<br />
[A ( =0)]=[A] 0<br />
(3.6)<br />
y<br />
y<br />
Half lifetime:<br />
y<br />
y<br />
y<br />
= − 1<br />
( − 1) [A]−(−1) 0<br />
(3.7)<br />
1<br />
[A] −1 − 1<br />
[A] −1<br />
0<br />
2 −1<br />
[A] −1<br />
0<br />
=( − 1) (3.8)<br />
£<br />
A<br />
¡<br />
12<br />
¢¤<br />
=<br />
1<br />
2 [A] 0<br />
(3.9)<br />
− 1<br />
[A] −1<br />
0<br />
12 = 2−1 − 1<br />
( − 1)<br />
=( − 1) 12 (3.10)<br />
1<br />
[A] −1<br />
0<br />
(3.11)<br />
lg 12 ∝−( − 1) lg [A] 0<br />
(3.12)<br />
y A log-log plot of 12 vs. [A] 0<br />
gives a straight line with slope −( − 1).<br />
22 We leave aside a usually occuring stoichiometric factor ν here, since we can always define νk = k 0<br />
as rate constant k 0 .
3.2 Application of the steady-state assumption 43<br />
3.2 Application of the steady-state assumption<br />
The steady state assumption allows us to reduce a large reaction mechanism to a smaller<br />
one by eliminating the respective intermediate species.<br />
3.2.1 The reaction H 2 + Br 2 → 2 HBr<br />
The reaction between H 2 and Br 2 (Bodenstein et al.; <strong>Christian</strong>sen, Polanyi & Herzfeld)<br />
runs as a thermal reaction or as a photochemically induced reaction (initiated by light).<br />
In the latter case, it is a chain reaction with a quantum yield of<br />
Φ ≈ 10 6 (3.13)<br />
I Definition 3.1: Quantum yield Φ:<br />
Φ =<br />
number of product molecules<br />
number of absorbed photons<br />
(3.14)<br />
I<br />
Reaction mechanism (elementary reactions) describing the H 2 -Br 2 system:<br />
Br 2 → Br + Br chain initiation by thermal dissociation (1)<br />
Br 2 + → Br + Br chain initiation by photolysis () (1 0 )<br />
Br + H 2 → HBr + H chain propagation (2)<br />
H+Br 2 → HBr + Br chain propagation (3)<br />
H+HBr → H 2 +Br inhibition (4)<br />
Br + HBr 6→ Br 2 +H endothermic (∆ ª =+170kJ mol) (6)<br />
y reaction (6) can be neglected<br />
Br + Br<br />
+M → Br 2 chain termination (5)<br />
(3.15)
3.2 Application of the steady-state assumption 44<br />
I Rates equations: With the steady state approximations (i.e., [X] ≈ 0) for[Br]<br />
and [H], weobtain 23<br />
[H]<br />
<br />
[Br]<br />
<br />
[HBr]<br />
<br />
= + 2 [Br] [H 2 ] − 3 [H] [Br 2 ] − 4 [H] [HBr] ≈ 0 (3.16)<br />
y 2 [Br] [H 2 ] − 4 [H] [HBr] = + 3 [H] [Br 2 ] (3.17)<br />
2 [Br] [H 2 ]<br />
y [H] <br />
=<br />
3 [Br 2 ]+ 4 [HBr]<br />
(3.18)<br />
(3.19)<br />
= +2 1 [Br 2 ] − 2 [Br] [H 2 ]+ 3 [H] [Br 2 ]+ 4 [H] [HBr]<br />
| {z }<br />
=−<br />
[H]<br />
≈0<br />
−2 5 [Br] 2 (3.20)<br />
= +2 1 [Br 2 ] − 2 5 [Br] 2 ≈ 0 (3.21)<br />
µ 12 1<br />
y [Br] <br />
= [Br 2 ]<br />
(3.22)<br />
5<br />
µ 12 1<br />
2 [H 2 ]<br />
y [H] <br />
= [Br 2 ] ×<br />
(3.23)<br />
5 3 [Br 2 ]+ 4 [HBr]<br />
= 2 [Br] [H 2 ]+ 3 [H] [Br 2 ] − 4 [H] [HBr] (3.24)<br />
Using 2 [Br] [H 2 ] − 4 [H] [HBr] = + 3 [H] [Br 2 ] (Eq. 3.17) and substituting [H] <br />
(Eq.<br />
3.18), the last line simplifies to<br />
[HBr]<br />
<br />
=2 3 [H] [Br 2 ] (3.25)<br />
µ 12 1<br />
2 [H 2 ]<br />
=2 3 [Br 2 ] × [Br 2 ] ×<br />
5 3 [Br 2 ]+ 4 [HBr]<br />
(3.26)<br />
= 2 2 ( 1 5 ) 12 [H 2 ][Br 2 ] 12<br />
1+( 4 3 )[HBr] [Br 2 ]<br />
(3.27)<br />
=<br />
0 [H 2 ][Br 2 ] 12<br />
1+ 00 [HBr] [Br 2 ]<br />
(3.28)<br />
I<br />
Result:<br />
with<br />
[HBr]<br />
<br />
= 0 [H 2 ][Br 2 ] 12<br />
1+ 00 [HBr] [Br 2 ]<br />
(3.29)<br />
0 =2 2 ( 1 5 ) 12 and 00 = 4 3 (3.30)<br />
I Conclusion: We see that HBr acts as an inhibitor of the total reaction.<br />
23 For the photolysis induced reaction, 1 has to be replaced by 0 1 ,where is the light intensity and<br />
0 1<br />
the corresponding photolysis rate constants (related to the absorption coefficient).
3.2 Application of the steady-state assumption 45<br />
3.2.2 Further examples<br />
I<br />
I<br />
Exercise 3.1: The thermal decomposition of acetaldehyde according to the overall<br />
reaction<br />
CH 3 CHO → CH 4 +CO (3.31)<br />
is known to proceed via the following elementary steps (Rice-Herzfeld mechanism):<br />
CH 3 CHO → CH 3 +HCO (1)<br />
CH 3 +CH 3 CHO → CH 4 +CH 3 CO (2)<br />
(3.32)<br />
CH 3 CO → CH 3 +CO (3)<br />
CH 3 +CH 3 → C 2 H 6 (4)<br />
Derive an expression for the production rate of [CH 4 ]. (Hint: Use the steady-state<br />
approximations for [CH 3 CO] and [CH 3 ] and neglect the HCO for the derivation of the<br />
expression.) ¤<br />
A more complete mechanism takes into account the thermal decomposition of the HCO<br />
radicals according to HCO → H + CO (5) followed by the reaction H + CH 3 CHO →<br />
H 2 +CH 3 CO (6).<br />
Solution 3.1: With the steady state approximations for [CH 3 CO] and [CH 3 ],weobtain<br />
[CH 3 CO]<br />
<br />
= 2 [CH 3 ][CH 3 CHO] − 3 [CH 3 CO] ≈ 0 (3.33)<br />
y [CH 3 CO] <br />
= 2 [CH 3 ][CH 3 CHO]<br />
3<br />
(3.34)<br />
and<br />
y<br />
[CH 3 ]<br />
<br />
= 1 [CH 3 CHO] − 2 [CH 3 ][CH 3 CHO] (3.35)<br />
+ 3 [CH 3 CO] − 2 4 [CH 3 ] 2 (3.36)<br />
= 1 [CH 3 CHO] − 2 [CH 3 ][CH 3 CHO] (3.37)<br />
+ 3 2 [CH 3 ][CH 3 CHO]<br />
− 2 4 [CH 3 ] 2<br />
3<br />
(3.38)<br />
= 1 [CH 3 CHO] − 2 4 [CH 3 ] 2 ≈ 0 (3.39)<br />
µ 12 1 [CH 3 CHO]<br />
y [CH 3 ] <br />
=<br />
(3.40)<br />
2 4<br />
[CH 4 ]<br />
<br />
[CH 4 ]<br />
<br />
= 2 [CH 3 ][CH 3 CHO]<br />
= 2<br />
µ<br />
1<br />
2 4<br />
<br />
[CH 3 CHO] 32 (3.41)<br />
¥<br />
I Exercise 3.2: The thermal decomposition of ethane gives mainly H 2 +C 2 H 4 . In<br />
addition, small amounts of CH 4 are formed. The decay of the C 2 H 6 concentration can<br />
be described by the reaction steps<br />
C 2 H 6 → CH 3 +CH 3 (1)<br />
CH 3 +C 2 H 6 → CH 4 +C 2 H 5 (2)<br />
C 2 H 5 → C 2 H 4 +H (3)<br />
(3.42)<br />
H+C 2 H 6 → C 2 H 5 +H 2 (4)<br />
H+C 2 H 5 → C 2 H 6 (5)
3.2 Application of the steady-state assumption 46<br />
Using the steady-state approximations for the radicals [CH 3 ], [C 2 H 5 ],and[H], derivea<br />
rate law describing the overall decay of [C 2 H 6 ]. Note: From the C-H bond dissociation<br />
energies in C 2 H 6 and C 2 H 5 ,itisknownthat 3 À 1 . ¤<br />
I Solution 3.2:<br />
− [C 2H 6 ]<br />
<br />
=<br />
" µ<br />
3 <br />
2<br />
2 1 + 1<br />
4 + 12<br />
#<br />
1 3 4<br />
[C 2 H 6 ] (3.43)<br />
5<br />
¥
3.3 Microscopic reversibility and detailed balancing 47<br />
3.3 Microscopic reversibility and detailed balancing<br />
The principle of microscopic reversibility is a consequence of the time reversal<br />
symmetry of classical and quantum mechanics: If all atomic momenta of a molecule<br />
that has reached some product state from an initial state are reversed, the molecule will<br />
return to its initial state via thesamepath.<br />
A macroscopic manifestation of the principle of microscopic reversibility is detailed<br />
balancing which states that<br />
(1) in equilibrium, the forward and reverse rates of a process are equal, i.e., for a<br />
1<br />
reaction A 1 À A 2 ,<br />
−1<br />
and<br />
1 [A 1 ]= −1 [A 2 ] (3.44)<br />
(2) if molecules A 1 and A 2 are in equilibrium and A 2 and A 3 are in equilibrium, both<br />
with respect to each other, there is also equilibrium between A 1 and A 3 :<br />
A 1<br />
1<br />
À<br />
−1<br />
A 2<br />
2<br />
À<br />
−2<br />
A 3 (3.45)<br />
y<br />
A 1<br />
3<br />
À<br />
−3<br />
A 3 (3.46)<br />
Thus, we can write<br />
[A 2 ] <br />
= 1<br />
= 12<br />
[A 1 ] <br />
−1<br />
(3.47)<br />
[A 3 ] <br />
= 2<br />
= 23<br />
[A 2 ] <br />
−2<br />
(3.48)<br />
[A 3 ] <br />
= 3<br />
= 13<br />
[A 1 ] <br />
−3<br />
(3.49)<br />
= 12 23 = 1 2<br />
−1 −2<br />
(3.50)<br />
Detailed balancing allows us to simplify complex reaction systems by eliminating − 1<br />
out of a sequence of reversible reactions that are in equilibrium.
3.4 Generalized first-order kinetics* 48<br />
3.4 Generalized first-order kinetics*<br />
3.4.1 Matrix method*<br />
For coupled complex reaction systems with only first-order reactions, the solution of the<br />
rate equations can be reduced to an eigenvalue problem (Jost 1974). The solution of<br />
the rate equations, i.e., the calculation of the time-dependent concentration profiles,<br />
requires finding the eigenvalues and eigenvectors of the rate constant matrix of the<br />
system. 24<br />
Eigenvalue problems are readily solved in practice using numerical methods (Press 1992).<br />
We consider an example which we solved using conventional methods in Section 2.3.<br />
I<br />
Example:<br />
The rate equations<br />
A 1<br />
1<br />
À<br />
2<br />
A 2 (3.51)<br />
[A 1 ]<br />
<br />
= − 1 [A 1 ]+ 2 [A 2 ] (3.52)<br />
[A 2 ]<br />
<br />
=+ 1 [A 1 ] − 2 [A 2 ] (3.53)<br />
canbewritteninamorecompactformasamatrix equation:<br />
⎛ ⎞<br />
[A 1 ] µ µ <br />
⎜<br />
⎝ ⎟ −1 +<br />
[A 2 ]<br />
⎠ =<br />
2 [A1 ]<br />
(3.54)<br />
+ 1 − 2 [A 2 ]<br />
<br />
With the concentration vector<br />
and the rate constant matrix<br />
K =<br />
A =<br />
µ <br />
[A1 ]<br />
[A 2 ]<br />
µ <br />
−1 + 2<br />
+ 1 − 2<br />
(3.55)<br />
(3.56)<br />
this becomes<br />
·<br />
A = K · A (3.57)<br />
I<br />
Generalized matrix rate equation for coupled first-order reaction systems:<br />
·<br />
A = K · A (3.58)<br />
24 A brief overview on matrices and determinants is given in Appendix D.
3.4 Generalized first-order kinetics* 49<br />
I Ansatz: The ansatz for solving Eq. 3.58 assumes that we can express A in terms of<br />
another vector B, 25 A = P · B (3.59)<br />
where P · B means the dot product, using a rotation matrix P which is defined so that<br />
it diagonalizes K via the eigenvalue equation K·P = P · Λ , which by multiplication<br />
from the left with P −1 yields<br />
P −1 · K · P = Λ (3.60)<br />
The eigenvalue matrix Λ is a diagonal matrix of the form<br />
⎛ ⎞<br />
1 0 0<br />
Λ = ⎝ 0 2 0<br />
0 0 . ⎠ (3.61)<br />
..<br />
As defined by Eq. 3.60, its elements on the diagonal { 1 2 } are the eigenvalues of<br />
the rate constant matrix. ThematrixP is called the eigenvector matrix associated<br />
with K. The columns of P are the respective eigenvectors <br />
associated with the<br />
respective eigenvalues .<br />
I Solution: Inserting the ansatz<br />
A = P · B (3.62)<br />
into our matrix equation 26 ·<br />
A = K · A (3.63)<br />
we have<br />
or<br />
(P · B)<br />
<br />
Muliplication of this equation from the left by P −1 gives<br />
= K · P · B (3.64)<br />
P · ·<br />
B = K · P · B (3.65)<br />
P −1 · P · ·<br />
B = P −1 · K · P · B (3.66)<br />
or<br />
·<br />
B = Λ · B (3.67)<br />
This DE can be immediately integrated since Λ is diagonal. The solution is<br />
B = Λ B 0 (3.68)<br />
where Λ is a diagonal matrix with elements © 1 2 ª , and Λ B 0 means<br />
element-wise multiplication. The result gives the time dependence of B, i.e., B()<br />
starting from the initial values B 0 .<br />
25 The components of B will be seen to be linear combinations of the components of the concentration<br />
vector A. B is immediately obtained once we have the matrix of eigenvectors P (see below).<br />
26 The dot above a concentration vector indicates differentiation by .
3.4 Generalized first-order kinetics* 50<br />
Having B(), we can transform back into the original concentration basis by using<br />
B = P −1 · A (3.69)<br />
and<br />
B 0 = P −1 · A 0 (3.70)<br />
InsertingintooursolutionforB (Eq. 3.68), we obtain<br />
P −1 · A = Λ P −1 · A 0 (3.71)<br />
The final step is now a multiplication from the left with P which yields the solution for<br />
Eq. 3.58<br />
A = P · Λ P −1 · A 0 (3.72)<br />
I Note: This may look complicated to someone who is not used to it. However, once<br />
we have set up K, which is straightforward, everything is done by the computer: The<br />
computer computes the<br />
• eigenvalues of the rate constant matrix K,<br />
• the eigenvector matrix P oftherateconstantmatrixK,<br />
• all necessary inverse matrices, etc.,<br />
• and does all the matrix multiplications.<br />
All you usually have to do is to program the K matrix.<br />
I<br />
Application to our example (solution “by hand”):<br />
• Evaluation of the eigenvalue matrix Λ by solution of the eigenvalue equation<br />
det (K − Λ · I) =0 (3.73)<br />
with I as unity matrix:<br />
y<br />
¯ − 1 − + 2<br />
+ 1 − 2 − ¯ =0 (3.74)<br />
(− 1 − ) · (− 2 − ) − ( 1 · 2 )=0 (3.75)<br />
1 2 + 1 + 2 + 2 − 1 2 =0 (3.76)<br />
( +( 1 + 2 )) = 0 (3.77)<br />
y<br />
1 =0 (3.78)<br />
2 = − ( 1 + 2 ) (3.79)<br />
y<br />
Λ =<br />
µ <br />
0 0<br />
0 − ( 1 + 2 )<br />
(3.80)
3.4 Generalized first-order kinetics* 51<br />
• Conclusions:<br />
(1) The largest (negative) eigenvalue ( 2 in this case) determines the shortest<br />
time scale and thus the short-time evolution of the reactant concentrations!<br />
(2) The smallest (negative) eigenvalue ( 1 in this case) determines the longest<br />
time scale of the reaction, i.e., the evolution of the reactant concentrations<br />
at long times. The smallest (negative) eigenvalue thus determines the net<br />
reaction rate!<br />
(3) Note that in the example the concentrations at long times are constant<br />
(equilibrium!) and therefore 1 =0.<br />
• Evaluation of the eigenvectors p 1 und p 2 by inserting the eigenvalues { 1 2 }<br />
one by one into<br />
K · P = Λ · P (3.81)<br />
yields, for each and the respective p , a linear equation of the type<br />
K · p = · p (3.82)<br />
y<br />
y<br />
(K − ) · p =0 (3.83)<br />
(− 1 − ) 1 + 2 2 =0 (3.84)<br />
1 1 +(− 2 − ) 2 =0 (3.85)<br />
• Inserting 1 =0:<br />
− 1 1 + 2 2 =0 (3.86)<br />
1 1 + − 2 2 =0 (3.87)<br />
y<br />
y<br />
1 1 = 2 2 (3.88)<br />
1 = 2 · (3.89)<br />
2 = 1 · (3.90)<br />
y<br />
• Inserting 2 = − ( 1 + 2 ):<br />
p 1 =<br />
µ <br />
2 · <br />
1 · <br />
(3.91)<br />
(− 1 + 1 + 2 ) 1 + 2 2 =0 (3.92)<br />
1 1 +(− 2 + 1 + 2 ) 2 =0 (3.93)<br />
y<br />
2 1 + 2 2 =0 (3.94)<br />
1 1 + 1 2 =0 (3.95)
3.4 Generalized first-order kinetics* 52<br />
y<br />
y<br />
1 = − 2 (3.96)<br />
1 = (3.97)<br />
2 = − (3.98)<br />
y<br />
µ <br />
<br />
p 2 =<br />
−<br />
(3.99)<br />
• Eigenvector matrix (= rotation matrix P):<br />
µ <br />
2 <br />
P =<br />
1 −<br />
(3.100)<br />
• The arbitrary factor is determined by the initial conditions to be =1,sothat<br />
µ <br />
2 1<br />
P =<br />
(3.101)<br />
−1<br />
1<br />
• Solution for A ():<br />
A = P · Λ P −1 · A 0 (3.102)<br />
I<br />
Solution using symbolic algebra programs:<br />
• Rate constant matrix K:<br />
K =<br />
µ <br />
−1 + 2<br />
+ 1 − 2<br />
(3.103)<br />
• eigenvalues(K) ={− 1 − 2 0} :<br />
µ <br />
− (1 + <br />
Λ =<br />
2 )0<br />
0 0<br />
(3.104)<br />
• eigenvectors(K) =<br />
½µ<br />
−1<br />
1<br />
¾<br />
(Ã 1<br />
!)<br />
<br />
↔− 1 − 2 <br />
2<br />
1 ↔ 0:<br />
1<br />
⎛<br />
P = ⎝ −1 ⎞<br />
2<br />
1<br />
⎠ (3.105)<br />
+1 1<br />
• inverse(P) :<br />
⎛<br />
− ⎞<br />
1 2<br />
P −1 ⎜<br />
= ⎝<br />
1 + 2 1 + 2<br />
⎟<br />
1 1<br />
⎠ (3.106)<br />
1 + 2 1 + 2
3.4 Generalized first-order kinetics* 53<br />
• B 0 = P −1 A 0 :<br />
• Solution for B ():<br />
• Solution for A ():<br />
B 0 = P −1 A 0 = P −1 ·<br />
µ<br />
10<br />
0<br />
⎛<br />
<br />
⎜<br />
= ⎝<br />
− 10<br />
1<br />
1 + 2<br />
10<br />
1<br />
1 + 2<br />
⎞<br />
⎟<br />
⎠ (3.107)<br />
B = Λ B 0 (3.108)<br />
µ ⎛ ⎞<br />
1<br />
<br />
−( 1 + 2<br />
=<br />
) −<br />
0 ⎜ 10<br />
0 1 ⎝<br />
1 + 2<br />
⎟<br />
1<br />
⎠ (3.109)<br />
10<br />
1 + 2<br />
=<br />
⎛<br />
⎜<br />
⎝<br />
− 10<br />
1<br />
1 + 2<br />
× −( 1+ 2 )<br />
10<br />
1<br />
1 + 2<br />
× 1<br />
A = P · Λ P −1 · A 0 = P · Λ P −1 ·<br />
µ µ<br />
<br />
−( 1 + 2<br />
= P ·<br />
) 0<br />
P<br />
0 1<br />
−1 10<br />
·<br />
0<br />
= P ·<br />
= P ·<br />
=<br />
=<br />
⎜<br />
⎝<br />
⎞<br />
⎟<br />
⎠ (3.110)<br />
µ<br />
10<br />
<br />
µ ⎛ 1<br />
<br />
−( 1 + 2 ) −<br />
0 ⎜ 10<br />
0 1 ⎝<br />
1 + 2<br />
1<br />
10<br />
1 + <br />
⎛ µ<br />
⎞ 2<br />
−( 1+ 2 )<br />
1<br />
− 10<br />
1 + 2<br />
1<br />
µ<br />
10<br />
1<br />
1 + 2<br />
<br />
⎛<br />
2<br />
−( 1+ 2 )<br />
10 + 10 1<br />
⎜ 1 + 2 1 + 2<br />
⎝ 1<br />
−( 1+ 2 )<br />
10 − 10 1<br />
1 + 2<br />
⎛<br />
2 + 1 −( 1+ 2 )<br />
10<br />
⎜ 1 + 2<br />
⎝ 1 − −( 1+ 2 )<br />
10 1<br />
1 + 2<br />
1 + <br />
⎞ 2<br />
0<br />
⎞<br />
<br />
(3.111)<br />
(3.112)<br />
⎟<br />
⎠ (3.113)<br />
⎟<br />
⎠ (3.114)<br />
⎞<br />
⎟<br />
⎠ (3.115)<br />
⎟<br />
⎠ (3.116)<br />
This result is identical to the result for [A ()] obtained in Section 2.3.<br />
3.4.2 Laplace transforms*<br />
The concept of Laplace and inverse Laplace transforms is extremely useful in chemical<br />
kinetics because
3.4 Generalized first-order kinetics* 54<br />
(1) Laplace transforms provide a convenient method for solving differential equations,<br />
(2) Laplace transforms form the connection between microscopic molecular properties<br />
and statistically (Boltzmann) averaged quantities.<br />
I<br />
Definition 3.2: The Laplace transform L [ ()] of a function () is defined as the<br />
integral<br />
Z ∞<br />
() =L [ ()] = () − (3.117)<br />
where<br />
0<br />
• is a real variable,<br />
• () is a real function of the variable with the property () =0for 0,<br />
• is a complex variable,<br />
• () =L [ ()] is a function of the variable .<br />
I<br />
Definition 3.3: The inverse Laplace transform L −1 [ ()] of the function () is<br />
defined as the integral<br />
() =L −1 [ ()] = 1 Z<br />
2<br />
+∞<br />
−∞<br />
() (3.118)<br />
where<br />
• is an arbitrary real constant.<br />
I Relation between f (t) and F (p): Since L −1 [ ()] recovers the original function<br />
(), the pair of functions () and () is said to form a Laplace pair:<br />
L −1 [ ()] = L −1 [L [ ()]] = () (3.119)<br />
and<br />
L [ ()] = L £ L −1 [ ()] ¤ = () (3.120)<br />
A list of Laplace and inverse Laplace transforms of some functions can be found in<br />
Appendix E (Table E.1).
3.4 Generalized first-order kinetics* 55<br />
I Solution of differential equations using Laplace transforms: Consider the<br />
Laplace transform of the derivative () of a function (): 27<br />
L [() ] =<br />
Z ∞<br />
We can solve this equation for () via<br />
0<br />
()<br />
<br />
= () −¯¯∞<br />
− (3.121)<br />
Z<br />
0 − ∞<br />
= − ( =0)+<br />
0<br />
Z ∞<br />
0<br />
() ¡ −¢ (3.122)<br />
() − (3.123)<br />
= − 0 + () (3.124)<br />
() = L [() ]+ 0<br />
<br />
and recover the solution () by inverse Laplace transformation,<br />
() =L −1 ∙ L [() ]+0<br />
<br />
¸<br />
(3.125)<br />
(3.126)<br />
I Example: Application to two consecutive first-order reactions. As an example,<br />
we consider the DE’s for two consecutive first-order reactions<br />
A 1<br />
→ B 2<br />
→ C (3.127)<br />
We already solved this problem previously (in Section 2.6.1) using conventional methods,<br />
and repeat the solution now using the Laplace transform method.<br />
Towards these ends, in this example, we shall denote the concentrations [A ()] and<br />
[B ()] as () and () and their Laplace transforms as () and (). Therespective<br />
initial concentrations shall be [A] 0<br />
= 0 and [B] 0<br />
= 0 =0. Thus we have:<br />
• DE for the concentration ():<br />
= 1 0 − 1 − 2 (3.128)<br />
• Laplace transform: () can be found using Eq. 3.124:<br />
L [] =− 0 + () (3.129)<br />
Taking the Laplace transform of the r.h.s. of Eq. 3.128: 28<br />
L [] =L £ 1 0 − 1 ¤ − L [ 2 ()] (3.130)<br />
= 1 0 L £ − 1 ¤ − 2 L [ ()] = 1 0<br />
+ 1<br />
− 2 () (3.131)<br />
= − 0 + () (Eq. 3.129) (3.132)<br />
27 We used in line 2: integration by parts; in line 3: the differivative ( − ) = − − ;inline4:<br />
the definition () =L [ ()] and the abbreviation ( =0)= 0 .<br />
28 We used L [ ]= 1 from Table E.1 and L [ ()] = ().<br />
−
3.4 Generalized first-order kinetics* 56<br />
Since we have 0 =0from the initial condition, this becomes<br />
()+ 2 () = 1 0<br />
+ 1<br />
(3.133)<br />
• Solution for ():<br />
() = 1 0<br />
1<br />
+ 1<br />
1<br />
+ 2<br />
(3.134)<br />
• Inverse Laplace transform: 29<br />
() =L [ ()] = 1 0<br />
2 − 1<br />
¡<br />
<br />
− 1 − − 2 ¢ (3.135)<br />
which is the same as obtained in Section 2.6.1.<br />
∙<br />
¸<br />
29 FromTableE.1,weseethatL −1 1<br />
1 ¡<br />
= − ¢ .<br />
( − )( − ) ( − )
3.5 Numerical integration 57<br />
3.5 Numerical integration<br />
I Initial value problems: For complex reaction systems, we have to integrate the coupled<br />
nonlinear DE systems numerically using the computer. We start with the known<br />
initial values of the concentrations at time 0 and the associated DE’s ( · ) and want<br />
to compute the values of at time 0 . In numerics, these problems are known as<br />
initial value problems. With additional conditions, for example for the spatial distributions<br />
of the concentration (fixed values at ( 0 0 0 1 )), they become boundary<br />
value problems.<br />
In solving these problems, we generally have to deal with systems of stiff nonlinear<br />
coupled (ordinary) differential equations (DEsystemsaresaidtobestiff, ifthe<br />
rate constants (more precisely: the eigenvalues) vary by many orders of magnitude;<br />
under these conditons, the changes of the dependable variables differ by many orders of<br />
magnitude).<br />
Thus we have to solve<br />
• equations (kinetic equations + transport processes), with<br />
• variables (concentrations), and the given<br />
• initial values and boundary values.<br />
I<br />
Applications of numerical integration methods:<br />
• Combustion chemistry:<br />
<strong>—</strong> Stationary combustion (for example diffusion flames),<br />
<strong>—</strong> Instationary combustion (ignition processes, turbulent combustion) - first<br />
successful attempts have been made,<br />
<strong>—</strong> Problem: For all but the simplest fuels, we have to deal with hundreds of<br />
species and thousands of elementary reactions.<br />
• Atmospheric chemistry:<br />
<strong>—</strong> 2D “box” models ( ( )) are straightforward.<br />
<strong>—</strong> 4D models are required (time, latitude, longitude, height)!<br />
<strong>—</strong> Huge problems, when it comes to<br />
∗ feedback with oceans, etc.?<br />
∗ do we know all reactions?<br />
∗ heterogeneous reactions?<br />
• <strong>Physical</strong> organic chemistry:<br />
<strong>—</strong> Elucidation of reaction mechanisms.<br />
• Macromolecular chemistry (including industrial macromolecular synthesis):<br />
<strong>—</strong> Prediction of chain length and other propertiesofpolymersandco-polymers.
3.5 Numerical integration 58<br />
I<br />
Survey of methods of numerical integration methods:<br />
• explicit methods (e.g., Runge-Kutta),<br />
• implicit methods (e.g., Gear),<br />
• extrapolation methods (e.g., Richardson, Bulirsch-Stoer, Deuflhard).<br />
In the following, we consider the basics of the most convenient methods.<br />
3.5.1 Taylor series expansions<br />
The Taylor expansion is the basis for all numerical integration methods. We start from<br />
the initial value of ( 0 ) at some time 0<br />
( 0 )= 0 (3.136)<br />
and want to determine the value ( 0 + ) at time 0 + . UsingtheDEfor <br />
we expand in a Taylor series around 0 :<br />
Recursion formula:<br />
·<br />
() =( ) (3.137)<br />
( 0 + ) = ( 0 )+ · · ( 0 0 )+ 2<br />
2! · ··<br />
( 0 0 )+ (3.138)<br />
= ( 0 )+ · ( 0 0 )+ 2<br />
2! · ·<br />
( 0 0 )+ (3.139)<br />
+1 = + · ( )+ 2<br />
2! · ·<br />
( )+ (3.140)<br />
The first-order term ( ) is given by the rate equation at , the second-order<br />
term ( ·<br />
) (which is taken into account for example by the 2 order Runge-Kutta<br />
method) and higher order terms (⇒ higher order Runge-Kutta methods) have to be<br />
evaluated numerically by finite differences.<br />
3.5.2 Euler method<br />
The Euler method is based on a 1 order Taylor expansion. If the step size is small<br />
enough, the higher order terms can be neglected:<br />
( 0 + ) = ( 0 )+ · ·<br />
( 0 )+O( 2 ) (3.141)<br />
Initial value problem:<br />
·<br />
= ( ) (3.142)
3.5 Numerical integration 59<br />
with<br />
( 0 )= 0 (3.143)<br />
Result by substituting ·<br />
( 0 )= ( 0 ):<br />
( 0 + ) = ( 0 )+( 0 )+O ¡ 2¢ (3.144)<br />
Recursion formula (see Fig. 3.1):<br />
+1 = + (3.145)<br />
+1 = + ( ) (3.146)<br />
Problem: Large errors for practical step sizes.<br />
I<br />
Figure 3.1: Euler method.<br />
3.5.3 Runge-Kutta methods<br />
I<br />
2 order Runge-Kutta method:<br />
Initial value problem:<br />
with<br />
·<br />
= ( ) (3.147)<br />
( 0 )= 0 (3.148)
3.5 Numerical integration 60<br />
2 order Runge-Kutta ansatz = Taylor expansion to 2 order:<br />
( 0 + ) = ( 0 )+ · ( 0 )+ 2 ··<br />
<br />
2! ( 0 )+O( 3 ) (3.149)<br />
Ã<br />
≈ ( 0 )+ ·<br />
·<br />
( 0 )+ 2 ( 0 + ) − ·<br />
!<br />
( 0 )<br />
+ O( 3 ) (3.150)<br />
2 <br />
= ( 0 )+ ³ ·(0 + )+ ·( 0 )´<br />
2 | {z }<br />
=2× slope of () at midpoint<br />
+ O( 3 ) (3.151)<br />
Recursion formula:<br />
+1 = + (3.152)<br />
+1 = + 2 + O( 3 ) (3.153)<br />
1 = ( ) (3.154)<br />
2<br />
µ<br />
= <br />
+ 2 + 1<br />
2<br />
<br />
(3.155)<br />
I<br />
Figure 3.2: Illustration of the 2 order Runge-Kutta method.
3.5 Numerical integration 61<br />
I 4 order Runge-Kutta method: In practice, one frequently uses the 4 order<br />
Runge-Kutta method.<br />
I<br />
Figure 3.3: Illustration of the 4 order Runge-Kutta method.<br />
3.5.4 Implicit methods (Gear):<br />
The Runge-Kutta method becomes unstable for stiff DE systems. In this case, it is<br />
preferable to employ implicit (also called iterative or backward) integration methods.<br />
The starting point is the implicit form of the Euler method,<br />
( 0 + ) = ( 0 )+ · ·<br />
( 0 + )+O( 2 ) (3.156)<br />
For the initial value problem ·<br />
= ( ) with ( 0 )= 0 this leads to the recursion<br />
formula:<br />
+1 = + (3.157)<br />
+1 = + ( +1 +1 ) (3.158)<br />
Using the so-called predictor-corrector methods, +1 is predicted in a first step by<br />
starting from using one of the explicit methods described above. The obtained trial<br />
value (0)<br />
+1 is then corrected by using a backward integration scheme (Gear 1971a, Gear<br />
1971a, Press 1992) to obtain a better estimate +1. (1) Then, and (1)<br />
+1 are used for<br />
a new prediction-corrrection cycle, giving +1. (2) The procedure is repeated until the<br />
difference between successive iterations falls below some tolerance limit .<br />
The predictor-corrector method of Gear has been the method of choice for a long time<br />
(Gear 1971a, Gear 1971a).
3.5 Numerical integration 62<br />
3.5.5 Extrapolation methods (Bulirsch-Stoer)<br />
The idea behind extrapolation methods is that the final answer for +1 is approximated<br />
by an analytical function in the step size (Richardson extrapolation). The best strategy<br />
is to use a rational extrapolating function as the ansatz, and not a polynomial function<br />
(Bulirsch-Stoer). This function is determined and fitted to the problem of interest by<br />
performing calculations with various values of . The extrapolating function is then<br />
used for extrapolating from to +1 .<br />
The optimal strategy for stiff DE systems is to follow the Bulirsch-Stoer method (Stoer<br />
1980) in the form implemented by Deuflhard (Bader 1983, Deuflhard 1983, Deuflhard<br />
1985). For Fortran subroutines, see (Press 1992).<br />
I<br />
Figure 3.4: The principle of extrapolation methods.<br />
3.5.6 Example: Consecutive first-order reactions<br />
As an example, we solve the DE’s for two consecutive first-order reactions<br />
A 1<br />
−→ B 2<br />
−→ C (3.159)<br />
(a)exactly,usingsymbolicalgebrasoftwarelikeMathCad,Maple,Mathematica,or<br />
MuPad (the math engine built-in into SWP, the program used to write this script) and<br />
(b) using numeric integration.
3.5 Numerical integration 63<br />
I<br />
Exact solution using symbolic algebra software:<br />
(1) To set-up the initial value problem for the program, 30 we enter the following a<br />
1 × 6 matrix:<br />
⎡<br />
0 ⎤<br />
= − 1 <br />
0 =+ 1 − 2 <br />
0 =+ 2 <br />
(3.160)<br />
⎢ (0) = 0 ⎥<br />
⎣<br />
(0) = 0<br />
⎦<br />
(0) = 0<br />
(2) To obtain the solution, we select the menu item "SolveODE, Exact". The program<br />
gives the output:<br />
Exact solution is:<br />
⎡<br />
⎢ () = 1 µ<br />
⎤<br />
− 1<br />
0 1 − 0 1 2 + 0 2 − 2<br />
1<br />
1 2 − 1 2 − ⎥ 1<br />
⎢<br />
⎣<br />
− 1<br />
− 2<br />
() = 0 1 − 0 1<br />
2 − 1 2 − 1<br />
() = 1 µ<br />
<br />
− 1<br />
0 1 2 − 0 2 − 1<br />
1<br />
1 2 − 1 2 − 1<br />
⎥<br />
⎦<br />
(3.161)<br />
(3) After simplifying, factorizing, and rewriting the output, we obtain<br />
⎡<br />
() = 0<br />
¡ ¢ ⎤<br />
2 − 1 + 1 − 2<br />
− 2 − 1<br />
2 − 1 ⎢<br />
1<br />
¡ ¢<br />
⎣ () = 0 <br />
− 2<br />
− − 1 ⎥<br />
1 − ⎦<br />
2<br />
() = 0 − 1<br />
(3.162)<br />
We immediately recognize these results from section 2.<br />
I Numerical solutions: Numerical integration is performed by the different symbolic<br />
algebra programs using one of the methods outlined in the preceding subsection. We<br />
have to specify the DE’s, the initial values, and the values for the rate constants. 31<br />
(1) We adopt the rate constant values 1 =100 and 2 =100.<br />
(2) We define the problem (now using capital letters):<br />
⎡<br />
0 ⎤<br />
= −<br />
0 =+ − 10<br />
0 =+10<br />
⎢ (0) = 1 ⎥<br />
⎣<br />
(0) = 0<br />
⎦<br />
(0) = 0<br />
(3.163)<br />
30 The different software packages differinthewaysinwhichtheproblemshavetobesetup.<br />
31 SWP can handle the problem only if the rate constant values are explicitly entered in the DE’s.<br />
Other software does better.
3.5 Numerical integration 64<br />
, Functions defined: <br />
y 1.0<br />
0.5<br />
0.0<br />
0 2 4<br />
(3) Application of "SolveODE, Numeric" gives the output:<br />
, Functions defined: <br />
This means that the program now “knows” these functions.<br />
(4) We can plot , , and using the Plot2D menu point:<br />
1<br />
c<br />
0.75<br />
0.5<br />
0.25<br />
0<br />
0<br />
1.25<br />
2.5<br />
3.75<br />
5<br />
t<br />
(Weaddtheplotsfor and by dragging both symbols to the plot window)<br />
(5) With 0 =1, 1 =1,and 2 =10, we also plot the exact solutions , , and<br />
= 0<br />
2 − 1<br />
¡<br />
2 − 1 + 1 − 2<br />
− 2 − 1 ¢ (3.164)<br />
1<br />
¡<br />
= 0 <br />
− 2<br />
− ¢ − 1<br />
1 − 2<br />
(3.165)<br />
= 0 − 1<br />
(3.166)
3.5 Numerical integration 65<br />
1<br />
c<br />
0.75<br />
0.5<br />
0.25<br />
0<br />
0<br />
1.25<br />
2.5<br />
3.75<br />
5<br />
t<br />
(6) The rest is just some fixing up: We combine the two plots, with an increased<br />
number of points plotted, add axis labels and colors:<br />
1<br />
c(t)<br />
0.75<br />
0.5<br />
0.25<br />
0<br />
0<br />
1.25<br />
Numerical solution of the DE system defined by Eq. 3.163. Full lines: numeric<br />
solutions. Dotted lines: exact solutions.<br />
2.5<br />
t<br />
3.75<br />
5<br />
The exact and the numerical solutions are identical within numeric precision.
3.6 Oscillating reactions* 66<br />
3.6 Oscillating reactions*<br />
The reactant and product concentrations normally approach equilibrium in a smooth<br />
(often exponenential) fashion. Under special circumstances, however, the concentrations<br />
may also show oscillations and/or bistability as they approach equilibrium. Such<br />
behavior can occur only if there is a chemical feedback by autocatalysis.<br />
3.6.1 Autocatalysis<br />
Example for autocatalysis:<br />
A+B → B+B (3.167)<br />
Rate law:<br />
− [A]<br />
<br />
= [A] [B] (3.168)<br />
To solve this DE, we substitute for [B ()] using the mass balance relation [A] 0<br />
−[A ()] =<br />
[B ()] − [B] 0<br />
and introduce =[A()] as reaction progress variable:<br />
[B ()] = [A] 0<br />
+[B] 0<br />
− [A ()] (3.169)<br />
=[A] 0<br />
+[B] 0<br />
− (3.170)<br />
This gives the DE<br />
y<br />
Z<br />
− <br />
= ([A] 0 +[B] 0<br />
− ) (3.171)<br />
Z<br />
<br />
([A] 0<br />
+[B] 0<br />
) − = − (3.172)<br />
2<br />
Solution:<br />
or<br />
[B ()] =<br />
µ <br />
1 [A]0 [B ()]<br />
ln<br />
= − (3.173)<br />
[A] 0<br />
+[B] 0<br />
[B] 0<br />
[A ()]<br />
[A] 0<br />
+[B] 0<br />
1+([A] 0<br />
[B] 0<br />
)exp[− ([A] 0<br />
+[B] 0<br />
) ]<br />
(3.174)
3.6 Oscillating reactions* 67<br />
I<br />
Figure 3.5: Autocatalysis.<br />
• Ascanbeseenfromthesigmoid(-shaped) concentration-time profile of [B] in<br />
Fig. 3.5, the rate of increase of [B] increases up to the inflection point ∗ .<br />
• This curve is typical for a population increase from a low plateau to a new (higher)<br />
plateau after a favorable change of the environment due to, e.g., the availability<br />
of more food, a technical revolution, a reduction of mortality by a medical<br />
breakthrough, ...<br />
Many interesting games which can be played based on these ideas are described in (Eigen<br />
1975).<br />
3.6.2 <strong>Chemical</strong> oscillations<br />
The presence of one or more autocatalytic steps in a reaction mechanism can lead to<br />
chemical oscillations.<br />
a) The Lotka mechanism<br />
The origin of chemical oscillations can be understood by considering the reaction mechanism<br />
proposed by Lotka in 1910 (see Lotka 1920):<br />
(3.175)<br />
A+X 1<br />
→ X+X (1)<br />
X+Y 2<br />
→ Y+Y (2)<br />
Y<br />
<br />
→<br />
3<br />
Z (3)
3.6 Oscillating reactions* 68<br />
• Both reactions (1) and (2) are autocatalytic.<br />
• The Lotka mechanism illustrates the principle, but it does not correspond to an<br />
existing chemical reaction system.<br />
• A real oscillating reaction is the Belousov-Zhabotinsky reaction; this reaction may<br />
be described using a more complicated mechanism (see below).<br />
In order to model the resulting chemical oscillations, we assume that the reaction takes<br />
place in a flow reactor, which is constantly supplied with new A such that the concentration<br />
of A stays constant ([A] = [A] 0<br />
), while the product Z is constantly removed.<br />
I<br />
Rate equations and steady-state solutions:<br />
[X]<br />
<br />
[Y]<br />
<br />
=+ 1 [A] 0<br />
[X] − 2 [X] [Y] (3.176)<br />
=+ 2 [X] [Y] − 3 [Y] (3.177)<br />
Steady-state solutions:<br />
[X]<br />
<br />
[Y]<br />
<br />
=+ 1 [A] 0<br />
[X] − 2 [X] [Y] = 0 (3.178)<br />
=+ 2 [X] [Y] − 3 [Y] = 0 (3.179)<br />
Dividing these equations by [X] and [Y], respectively, we find<br />
2 [Y] <br />
= 1 [A] 0<br />
(3.180)<br />
2 [X] <br />
= 3 (3.181)<br />
Since [A] = [A] 0<br />
, the steady state solutions for [X] and [Y] are independent of time!<br />
I Displacements from steady-state solutions: What happens if the concentrations<br />
are displaced from the steady-state values by small amounts and ?<br />
(1) Ansatz:<br />
[X] = [X] <br />
+ (3.182)<br />
[Y] = [Y] <br />
+ (3.183)
3.6 Oscillating reactions* 69<br />
(2) Rate equations:<br />
<br />
=+ 1 [A] 0<br />
([X] <br />
+ ) − 2 ([X] <br />
+ )([Y] <br />
+ ) (3.184)<br />
=+ 1 [A] 0<br />
[X] <br />
+ 1 [A] 0<br />
− 2 [X] <br />
[Y] | {z }<br />
= 1 [A] 0<br />
[X] <br />
− 2 [X] <br />
− 2 [Y] <br />
− 2 <br />
| {z }<br />
= 1 [A] 0<br />
<br />
= − 2 [X] <br />
− 2 (3.185)<br />
<br />
=+ 2 ([X] <br />
+ )([Y] <br />
+ ) − 3 ([Y] <br />
+ ) (3.186)<br />
= 2 [X] <br />
[Y] <br />
+ 2 [X] <br />
+ 2 [Y] <br />
<br />
+ 2 − |{z} 3 [Y] <br />
− 3 |{z}<br />
= 2 [X] <br />
= 2 [X] <br />
=+ 2 [Y] <br />
+ 2 (3.187)<br />
(3) Simplified DE’s for small displacements: We may neglect the terms containing<br />
. Thus, we obtain two coupled first-order DE’s<br />
<br />
= − 2 [X] <br />
(3.188)<br />
<br />
=+ 2 [Y] <br />
(3.189)<br />
which can be converted to a single second-order DE by the ansatz<br />
·<br />
= − (3.190)<br />
·<br />
=+ (3.191)<br />
Second-order DE:<br />
··<br />
= − · = − (3.192)<br />
Formal solution of this second-order DE:<br />
() =[X()] − [X] <br />
= 1 + 2 − (3.193)<br />
(4) Eigenvalues: To determine the eigenvalue belonging to the linear DE’s, we have<br />
to solve the determinant<br />
¯<br />
¯0 − −<br />
+ 0 − ¯ − 2 [X] <br />
¯ 2 [Y] <br />
¯ =0 (3.194)<br />
y<br />
2 + 2 [X] <br />
[Y] <br />
=0 (3.195)<br />
y The solutions for are purely imaginary:<br />
12 = ± ¡ ¢<br />
2 2 12<br />
[X] <br />
[Y] <br />
(3.196)<br />
= ± ( 1 3 [A] 0<br />
) 12 (3.197)
3.6 Oscillating reactions* 70<br />
(5) Solution for ():<br />
() = 1 + 2 − = 0 cos (3.198)<br />
with<br />
=( 1 3 [A] 0<br />
) 12 (3.199)<br />
This means that if the system is displaced from its steady-state, the species concentrations<br />
will start to oscillate and the displacements may even grow in magnitude<br />
until the amplitude reaches the limiting value 0 (see limit cycle diagram in<br />
Fig. 3.6).<br />
I<br />
Figure 3.6: Limit cycle diagram for chemical oscillations according to the Lotka mechanism.<br />
b) The Brusselator 32<br />
Reaction scheme:<br />
<br />
A →<br />
1<br />
X (1)<br />
B+X 2<br />
→ R+Y (2)<br />
Y+2X (3.200)<br />
3<br />
→ 3X (3)<br />
<br />
X →<br />
4<br />
S (4)<br />
32 The name ‘Brusselator’ stems from the workplace of its inventor Prigogine (Brussels). The model<br />
does not correspond to a real chemical system.
3.6 Oscillating reactions* 71<br />
Here, A and B are reactants, R and S are final products, and X and Y are intermediates.<br />
The autocatalytic reaction is reaction (3).<br />
Rate equations (using “dimensionless time” = 4 ) and steady-state concentrations:<br />
[X]<br />
<br />
[X]<br />
<br />
Steady-state concentrations:<br />
= ·<br />
[X] = 1 [A] − 2 [B] [X] + 3 [X] 2 [Y] − 4 [X]<br />
4<br />
(3.201)<br />
= ·<br />
[Y] = 2 [B] [X] − 3 [X] 2 [Y]<br />
4<br />
(3.202)<br />
[X] <br />
[Y] <br />
= 1 [A]<br />
4<br />
(3.203)<br />
= 2 4 [B]<br />
1 3 [A]<br />
(3.204)<br />
We now investigate the effect of small displacements from steady-state by [X] and<br />
[Y]. The reaction rates respond to these displacements according to<br />
⎛ ⎞<br />
⎛ ⎞<br />
⎝ [X]<br />
·<br />
⎠ [X] and ⎝ [Y]<br />
·<br />
⎠ [Y] (3.205)<br />
[X]<br />
[Y]<br />
y<br />
• The steady-state is stable if<br />
⎛ ⎞<br />
⎝ [X]<br />
·<br />
⎠<br />
[X]<br />
• The steady-state is unstable if<br />
⎛ ⎞<br />
⎝ [X]<br />
·<br />
⎠<br />
[X]<br />
• Stability criterion:<br />
⎛ ⎞<br />
⎝ [X]<br />
·<br />
⎠<br />
[X]<br />
<br />
<br />
⎛ ⎞<br />
+ ⎝ [Y]<br />
·<br />
⎠<br />
[Y]<br />
<br />
<br />
<br />
⎛ ⎞<br />
+ ⎝ [Y]<br />
·<br />
⎠<br />
[Y]<br />
⎛ ⎞<br />
+ ⎝ [Y]<br />
·<br />
⎠<br />
[Y]<br />
<br />
<br />
<br />
0 (3.206)<br />
≥ 0 (3.207)<br />
= 2 []<br />
− 2 1 3 [] 2<br />
− 1 (3.208)<br />
4 4<br />
2<br />
c) Belousov-Zhabotinsky reaction<br />
Oxidation of malonic acid to CO 2 by bromate, catalyzed by cerium ions in acidic solution:<br />
2BrO − 3 +3CH 2 (COOH) 2<br />
+2H + −→ 2BrCH(COOH) 2<br />
+3CO 2 +4H 2 O (3.209)<br />
The reaction is probed by addition of the redox indicator ferroin/ferriin: The color varies<br />
between red and blue depending on the concentrations of Ce 4+ and Ce 3+
3.6 Oscillating reactions* 72<br />
I<br />
Figure 3.7: Mechanism of the Belousov-Zhabotinsky reaction.<br />
I<br />
Figure 3.8: Phases I - III of the Belousov-Zhabotinsky reaction.
3.6 Oscillating reactions* 73<br />
I<br />
Figure 3.9: Key steps of the Belousov-Zhabotinsky reaction.<br />
d) The Oregonator 33<br />
TheOregonatormodelcanbeusedtodescribetheBelousov-Zhabotinsky reaction.<br />
I<br />
Schematic model and kinetic equations:<br />
A+Y 1<br />
→ X+R (1)<br />
X+Y 2<br />
→ 2R (2)<br />
A+X 3<br />
→ 2X+2Z (3)<br />
(3.210)<br />
B+Z 5<br />
→ 1 2 Y (5)<br />
2X<br />
<br />
→<br />
4<br />
A +R (4)<br />
with A=BrO − 3 B=CH 2 (COOH) 2 + BrCH(COOH) 2 R=HOBr X=HBrO 2 <br />
Y=Br − Z=Ce 4+ .<br />
Using dimensionless time ( = 5 [B] ), we obtain<br />
33 The name ‘Oregonator’ stems from the workplace of its inventor Noyes (University of Oregon).
3.6 Oscillating reactions* 74<br />
[X]<br />
<br />
[Y]<br />
<br />
[Z]<br />
<br />
= 1 [A] [Y] − 2 [X] [Y] + 3 [A] [X] − 2 4 [X 2 ]<br />
5 [B]<br />
− 1 [A] [Y] − 2 [X] [Y] + 1<br />
=<br />
2 5 [B] [Z]<br />
5 [B]<br />
= 2 3 [A] [X] − 5 [B] [Z]<br />
5 [B]<br />
(3.211)<br />
(3.212)<br />
(3.213)<br />
I Figure 3.10: Numerical integration of the Oregonator model (rate constants in<br />
lmol −1 s −1 : 1 =0005, 2 =10, 3 =10, 4 =10, 5 = 0004; initial concentrations<br />
in mol l −1 : [A] = 001, [B] = 10, [X] 0<br />
=00, [Y] 0<br />
=00, [Z] 0<br />
=000025.)
3.6 Oscillating reactions* 75<br />
3.7 References<br />
Bader 1983 G. Bader, P. Deuflhard, Numerische Mathematik 41, 373 (1985)<br />
Deuflhard 1983 P. Deuflhard, Numerische Mathematik 41, 399 (1985).<br />
Deuflhard 1985 P. Deuflhard, SIAM Review 27, 505 (1985).<br />
Eigen 1975 M. Eigen, R. Winkler, Das Spiel, Piper, München, 1975.<br />
Gear 1971a C.W.Gear,Commun.Am.Comput.Mach.14, 1976 (1971).<br />
Gear 1971b C. W. Gear, Numerical Initial Value Problems in Ordinary Differential<br />
Equations, Prentice-Hall, Englewood Cliffs, 1971.<br />
Houston 1996 P. L. Houston, <strong>Chemical</strong> <strong>Kinetics</strong> and Reaction Dynamics, McGraw-<br />
Hill, Boston, 2001.<br />
Jost 1974 W. Jost, in <strong>Physical</strong> <strong>Chemistry</strong>: An Advanced Treatise, Ed.:W.Jost,Vol.<br />
VIa, Academic Press, New York, 1974.<br />
Lotka 1920 A. J. Lotka, Undamped Oscillations Derived from the Law of Mass Action,<br />
J. Am. Chem. Soc. 44, 1595 (1920).<br />
Press 1992 W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical<br />
Recipes in Fortran, Cambridge University Press, Cambridge, 1992.<br />
Scott 1994 S. K. Scott, Oscillations, Waves, and Chaos in <strong>Chemical</strong> <strong>Kinetics</strong>, Oxford<br />
Science Publications, Oxford, 1994.<br />
Stoer 1980 J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer,New<br />
York, 1980.
3.7 References 76<br />
4. Experimental methods<br />
In this section we consider experimental techniques for determining rate coefficients.<br />
We’ll get to faster reactions as we go down the list.<br />
Special techniques used in quantum state-resolved reaction dynamics, photodissociation,<br />
or transition state spectroscopy (molecular beams, state specific detection, photofragment<br />
translational spectroscopy, photofragment imaging, other pump-probe techniques,<br />
fluorescence up-conversion, fs-CARS, fs-DFWM) will be considered later (in advanced<br />
courses), but we’ll briefly touch photochemistry and “femtochemistry”.<br />
4.1 End product analysis<br />
• firststepforsettingupreactionmechanisms,<br />
• all convenient analytical techniques (HPLC, GC, GC-MS, MS, UV, FTIR, . . . ),<br />
• Problem: No “direct” kinetic information because of lack of time resolution.<br />
I<br />
Figure 4.1: FTIR spectrometer for kinetic studies of photolysis-induced reactions.
4.2 Modulation techniques 77<br />
4.2 Modulation techniques<br />
• especially for photo-induced reactions.<br />
• photolysis light has to be modulated at a frequency that is comparable to that of<br />
the reaction y timescaleof10’sofmsandlonger.<br />
• kinetic information is derived from the “in-phase” and the “in-quadrature” components<br />
of the reactant and product concentrations as function of modulation<br />
frequency.<br />
I Figure 4.2:
4.3 The discharge flow (DF) technique 78<br />
4.3 The discharge flow (DF) technique<br />
DF technique is used for gas phase reactions on ms timescale:<br />
• laminar flow along tubular reactor:<br />
∆ = ∆<br />
<br />
with =meanflow speed, ∆ = distance along the flow tube.<br />
(4.1)<br />
• y first-order reaction gives exponential decay of concentration along ∆.<br />
• in reality ⇒ Hagen-Poisseuille flow (parabolic flow profile),<br />
• but fast radial diffusion of reactants (at pressures of 1 to 10 mbar) gives “plug<br />
shaped” radial concentration profiles despite parabolic flow profile.<br />
• more complicated, when “back diffusion” and radial diffusionhavetobetaken<br />
into account (at higher pressures).<br />
• problem: heterogeneous reactions lead to first-order decay of reactive radicals<br />
even in the absence of a reactant.<br />
Combinations with numerous highly sensitive detection techniques, for instance:<br />
• mass spectrometry (MS): universal, but often low sensitivity; problems with fragmentationofmoleculesintheionsource,<br />
• laser induced fluorescence (LIF): excitation to electronically excited state followed<br />
by relaxation by spontaneous emission. very sensitive (e.g., for OH ( 2 Σ ← 2 Π)<br />
detection limit below 10 6 molecules cm 3 ),<br />
• electron spin resonance (ESR): for paramagnetic atoms by exploiting the Zeeman<br />
effect depending on magnetic field <br />
= 0 + (4.2)<br />
∆ =1y ∆ = (4.3)<br />
ESR transitions are in the microwave region (X band: 9GHz).<br />
• laser magnetic resonance (LMR): very sensitive; based on observation of rotational<br />
transitions between Zeeman-shifted rotational levels of paramagnetic radicals<br />
( rot = rotational constant).<br />
00 = rot ( +1)+ 00<br />
00 (4.4)<br />
0 = rot ( +1)+ 0 0 (4.5)<br />
∆ = ±1 ∆ =0 ±1 : (4.6)<br />
y 0 = rot ( 00 +1)( 00 +2)+ 0 0 (4.7)<br />
y ∆ = =2 rot ( 00 +1)+ ( 0 0 − 00<br />
00 ) (4.8)<br />
LMR transitions are in the FIR region (100 m ≤ ≤ 2000 m).
4.3 The discharge flow (DF) technique 79<br />
I<br />
Figure 4.3: Schematic diagram of a DF/MS combination.<br />
I<br />
Figure 4.4: Schematic diagram of a DF/LIF combination.
4.3 The discharge flow (DF) technique 80<br />
I<br />
Figure 4.5: Principles of Electron Spin Resonance (ESR) and Laser Magnetic Resonance<br />
(LMR).<br />
I<br />
Figure 4.6: Laser Magnetic Resonance (LMR).
4.3 The discharge flow (DF) technique 81<br />
I<br />
Figure 4.7: Schematic diagram of a DF/LMR combination<br />
I<br />
Figure 4.8: Flow reactor with laser flash photolysis and time-resolved mass spectrometric<br />
detection of transient reactants and products.
4.4 Flash photolysis (FP) 82<br />
4.4 Flash photolysis (FP)<br />
• “pump-probe” technique:<br />
<strong>—</strong> generation of transient species by flash photolysis using fast flashlampor<br />
pulsed lasers (“pump”),<br />
<strong>—</strong> fast detection of transient species by UV absorption spectrometry, laser induced<br />
fluorescence (LIF), cavity ringdown spectroscopy (CRDS) (”probe”).<br />
• for gas and liquid phase reactions on nanosecond timescale (Norrish & Porter).<br />
I<br />
Figure 4.9: Typical “Pump-Probe” set-up for kinetics studies by pulsed laser photolysis<br />
and detection by laser-induced fluorescence.<br />
I<br />
Cavity Ringdown Spectroscopy (CRDS):<br />
• CRDS is an ideal method for absorption spectroscopy with pulsed lasers despite<br />
of their large (±5 %) amplitude fluctuations.<br />
• The absorption measurement is transformed into the time domain:<br />
<strong>—</strong> time dependence of signal transmitted through a resonantor:<br />
= 0 exp (−) (4.9)
4.4 Flash photolysis (FP) 83<br />
<strong>—</strong> decay of intensity for pass through cavity (length , mirrorreflectivity ,<br />
absorption path length ):<br />
µ <br />
<br />
1 − <br />
= −<br />
<br />
−<br />
<br />
(4.10)<br />
| {z } | {z }<br />
due to reflection losses at mirrors due to absorption<br />
<strong>—</strong> with<br />
y<br />
<strong>—</strong> empty cavity ( =0):<br />
<strong>—</strong> with absorber ( 6= 0):<br />
= (4.11)<br />
<br />
= − [(1 − )+] (4.12)<br />
<br />
−1 (1 − )<br />
0 =<br />
<br />
(4.13)<br />
−1 =<br />
[(1 − )+]<br />
<br />
(4.14)<br />
<strong>—</strong> The difference of the ringdown times measures the absorption coefficient<br />
and is related to absorber concentration [A] by<br />
• Performance of a CRD spectrometer:<br />
<strong>—</strong> mirrors:<br />
<strong>—</strong> measuredemptycavitydecaytime:<br />
<strong>—</strong> effective absorption pathlength (1 m cell):<br />
• applications to chemical kinetics:<br />
−1 − −1<br />
0 = = [A] (4.15)<br />
=99999% (4.16)<br />
0 ≈ 300 s (4.17)<br />
eff =90km (4.18)<br />
<strong>—</strong> slow reactions: pump-probe scheme with variable time delay.<br />
first-order reaction with rate constant 1 :<br />
[A] = [A] 0<br />
− 1<br />
(4.19)<br />
y<br />
ln ¡ ¢<br />
−1 − −1<br />
0 = − 1 (4.20)<br />
<strong>—</strong> fast reactions: kinetics and ringdown occur on the same time scale. y<br />
(extended) <strong>Kinetics</strong> and Ringdown model ((e)SKaR); see Brown2000 and<br />
Guo2003 for details.
4.4 Flash photolysis (FP) 84<br />
I Figure 4.10:<br />
I Figure 4.11:
4.5 Shock tube studies of high temperature kinetics 85<br />
4.5 Shock tube studies of high temperature kinetics<br />
• Detection of atoms by atomic resonance absorption spectrometry (ARAS),<br />
• Detection of small radicals by frequency modulation absorption spectrometriy<br />
(FM-AS)<br />
I Figure 4.12:<br />
I Figure 4.13:
4.6 Stopped flow studies 86<br />
4.6 Stopped flow studies<br />
• the analogue of the flow tube technique for liquid phase reactions.<br />
• applications mainly kinetics of enzyme reactions, studies of protein folding dynamics<br />
I Figure 4.14:
4.7 NMR spectroscopy 87<br />
4.7 NMR spectroscopy<br />
• line coalescence<br />
• good for interconversion of two conformers (“isomerization”)<br />
I Figure 4.15:
4.8 Relaxation methods 88<br />
4.8 Relaxation methods<br />
• application to fast reactions in the liquid phase (Eigen)<br />
• -jump, -jump, -jump, . . . :<br />
(1) ⇒ displacement from equilibrium<br />
(2) ⇒ measurements of relaxation into new equilibrium state by transient absorption<br />
spectroscopy, conductivity, . . .
4.9 Femtosecond spectroscopy 89<br />
4.9 Femtosecond spectroscopy<br />
I<br />
Figure 4.16: Femtosecond pump-probe spectroscopy for the investigation of ultrafast<br />
reactions (time resolution down to the order of 10 fs).<br />
I<br />
Figure 4.17: Signals in broadband femtosecond transient absorption spectroscopy.
4.9 Femtosecond spectroscopy 90<br />
I Figure 4.18: Broadband femtosecond transient absorption spectroscopy of photochromic<br />
molecular switches in SFB 677.<br />
I<br />
Figure 4.19: Broadband femtosecond transient absorption spectroscopy of Disperse<br />
Red 1.
4.9 Femtosecond spectroscopy 91<br />
I Figure 4.20: Broadband femtosecond transient absorption spectroscopy of Furyl<br />
Fulgides.<br />
I<br />
Figure 4.21: Ultrafast Electronic Deactivation in DNA Building Blocks.
4.9 Femtosecond spectroscopy 92<br />
I<br />
Figure 4.22: Femtosecond fluorescence up-conversion data for DNA strands.<br />
I<br />
Figure 4.23: Ultrafast Electronic Deactivation in DNA Building Blocks: d(pApA) as<br />
Model System.
4.9 Femtosecond spectroscopy 93<br />
4.10 References<br />
Busch 1999 K. W. Busch, M. A. Busch, Cavity-Ringdown Spectroscopy, ACSSymposium<br />
Series, Vol 720, American <strong>Chemical</strong> Society, Washington, 1999.<br />
Brown 2000 S.S.Brown,A.R.Ravishankara,H.Stark,J.Phys.Chem.A104, 7044<br />
(2000).<br />
Friebolin 1999 H.Friebolin, Ein- und zweidimensionale NMR-Spektroskopie, Wiley-<br />
VCH, Weinheim, 1999.<br />
Guo 2003 Y. Guo, M. Fikri, G. Friedrichs, F. Temps, An Extended Simultaneous <strong>Kinetics</strong><br />
and Ringdown Model: Determination of the Rate Constant for the Reaction<br />
SiH 2 +O 2 .Phys.Chem.Chem.Phys.5, 4622 (2003).<br />
OKeefe 1988 A.O’Keefe,D.A.G.Deacon,Rev.Sci.Instrum.59, 2544 (1988).
4.10 References 94<br />
5. Collision theory<br />
So far, we have considered more or less complex chemical reactions following more or<br />
less complicated rate laws, but we have only considered the rate coefficients appearing<br />
in the rate laws as empirical quantities, which had to be determined experimentally.<br />
We shall now ask the question, whether we can rationalize and predict those rate coefficients<br />
using theoretical models. Towards these ends, we have to look at the reactions<br />
at a microscopic, molecular level. In particular, we have to<br />
• consider the relation between the microscopic (individual molecule) properties and<br />
the macroscopic (thermally averaged) rate quantities (rate constants, product<br />
branching ratios)<br />
• develop models for the microscopic (molecular level) progress of individual elementary<br />
reactions, including the different molecular degrees of freedom (quantum<br />
states).<br />
• develop sufficiently simplified models averaging over the molecular degrees of<br />
freedom which still allow us to make practically useful, relevant, and sufficiently<br />
accurate predictions.<br />
We shall do so in the next four chapters.<br />
I Collision theory: To begin with, this chapter considers bimolecular reactions in the<br />
gas phase. These obviously require a collision between two molecules as condition for a<br />
reaction to take place.<br />
• We start with the hard sphere collision model of structureless particles as the<br />
simplest model for bimolecular reactions. Although this model has two major<br />
flaws, it leads to the right result because of a compensation of errors.<br />
• We then consider results of kinetic gas theory, which allow us to properly take<br />
into account the molecular speed distributions.<br />
• We apply the hard sphere collision model to understand transport processes in the<br />
gas phase (diffusion, heat conductivity, viscosity).<br />
• We then proceed with a correct derivation of advanced collision theory models<br />
starting from differential cross sections (which can be measured in crossed molecular<br />
beam experiments). Advanced collision theory is the method of choice for fast<br />
gas phase reactions which can be studied in detail by molecular beam experiments<br />
and handled by quantum theory.
5.1 Hardspherecollisiontheory 95<br />
5.1 Hard sphere collision theory<br />
5.1.1 The gas kinetic collision frequency<br />
I<br />
The gas kinetic collision frequency as upper limit for rate constants of bimolecular<br />
reactions: The gas kinetic collision frequency constitues an upper limit for the<br />
rate constant of bimolecular gas phase reactions. We assume that the molecules A and<br />
B are structureless hard spheres (radii , ) moving around with fixedmeanspeeds<br />
and and look at the number of collisions between A and B per unit time.<br />
I<br />
Figure 5.1: Derivation of the hard sphere gas kinetic collision frequency.<br />
I<br />
Discussion of the molecular quantities affecting the rate constant:<br />
• collision parameter (= projected distance between centers of gravity):<br />
(5.1)<br />
• maximal collision parameter = max value leading to a collision max :<br />
≤ max = + = (5.2)<br />
• collision cross section = 2 (= reactive cross section) 34 :<br />
= 2 = ( + ) 2 (5.3)<br />
34 Deutsch: reaktiver Wirkungsquerschnitt.
5.1 Hardspherecollisiontheory 96<br />
• mean speed of A and B:<br />
=<br />
=<br />
r r<br />
8 8<br />
=<br />
(5.4)<br />
<br />
r r <br />
8 8<br />
=<br />
(5.5)<br />
<br />
• mean relative speed (2 different molecules, i.e., A 6= B): 35<br />
=<br />
r 8 <br />
<br />
(5.9)<br />
with the reduced mass<br />
=<br />
<br />
+ <br />
(5.10)<br />
• mean relative speed (single type of molecules, i.e., A = B):<br />
y<br />
=<br />
<br />
+ <br />
= 1 2 (5.11)<br />
= √ 2 ×<br />
r<br />
8 <br />
<br />
(5.12)<br />
I<br />
Hard sphere gas kinetic collision frequency:<br />
• We want to determine the number of collisions of a single molecule A with<br />
molecules B in time . This is given by the product of the (volume that A has<br />
swept in ) × (number density of B):<br />
<strong>—</strong> volume of cylinder that A has swept in time :<br />
volume = cross section × (5.13)<br />
y<br />
2 × (5.14)<br />
35 The mean relative speed of two molecules with velocities u and u is found by looking at the<br />
square of the difference speed<br />
2 = |u − u | 2 = | | 2 + | | 2 − 2 | || | cos (5.6)<br />
The third term averaged over cos is 0, therefore<br />
Since and<br />
2 = | | 2 + | | 2 = 3 <br />
<br />
+ 3 <br />
= 3 ( + )<br />
= 3 <br />
<br />
³<br />
2´12 differ only by a constant factor, we also find that<br />
=<br />
s<br />
8 <br />
<br />
(5.7)<br />
(5.8)
5.1 Hardspherecollisiontheory 97<br />
<strong>—</strong> number density of B<br />
(5.15)<br />
<strong>—</strong> Number of collisions of a single A in time : ⇒ number of molecules B in<br />
that volume<br />
= 2 × × (5.16)<br />
• Number of all collisions between A and B:<br />
= 2 × × (5.17)<br />
• Modification for collisions between identical molecules (collisions must not be<br />
counted twice):<br />
= 1 2 × 2 × × (5.18)<br />
=<br />
√<br />
2<br />
2 × 2 × × (5.19)<br />
• Comparison with rate expression:<br />
y<br />
− <br />
<br />
= − <br />
<br />
= × = (5.20)<br />
• Expression for rate constant for collisions A + B:<br />
<br />
max = 2 × = × (5.21)<br />
y<br />
µ 12<br />
max = 2 8 <br />
×<br />
(5.22)<br />
<br />
• Modification for collisions between identical molecules (A + A):<br />
max = 1 2 2 × = √ 1 µ 12<br />
2 8 <br />
×<br />
(5.23)<br />
2 <br />
Theseexpressionsgivetheupper limits for the rate constants of bimolecular<br />
reactions. The upper limits are reached only if<br />
• there are no energetic restrictions (no energy barriers),<br />
• all collisions lead to reaction (no steric constraints),<br />
• molecules can be approximated as hard spheres (unrealistic, usually one has <br />
),<br />
• all molecules collide with the same relative speed (in reality there is a distribution<br />
of speeds).
5.1 Hardspherecollisiontheory 98<br />
I<br />
Mean free pathlength:<br />
• The mean free pathlength is the distance that a molecule flies between two successive<br />
collisions:<br />
y<br />
=<br />
distance<br />
# of collisions = distance/time<br />
# of collisions/time = <br />
(5.24)<br />
<br />
=<br />
1<br />
2 × <br />
∝ 1 <br />
• Modification for collisions between identical molecules (A + A):<br />
(5.25)<br />
= 1 √<br />
2<br />
1<br />
2 × <br />
(5.26)<br />
I<br />
Time between two collisions:<br />
1<br />
<br />
=<br />
1<br />
2 × <br />
(5.27)<br />
I<br />
Example 5.1: Collision frequency for O 2 -N 2 @ =298K= 1000 mbar:<br />
Molecular parameters:<br />
2 = 2 2 =3467 Å (5.28)<br />
2 = 2 2 =3798 Å (5.29)<br />
y =36 Å =36 × 10 −10 m (5.30)<br />
= 2 =41Å 2 (5.31)<br />
28 × 32<br />
=<br />
28 + 32 × 10−3 kg mol × 1<br />
<br />
s<br />
<br />
(5.32)<br />
=<br />
8<br />
<br />
=650m s (5.33)<br />
(but (O 2 )=444m s) (5.34)<br />
Rate constant:<br />
µ 12<br />
max = 2 8 <br />
×<br />
<br />
(5.35)<br />
=26 × 10 −10 cm 3 s (5.36)<br />
=16 × 10 14 cm 3 mol s (5.37)<br />
Mean free pathlength:<br />
=1000mbar (5.38)<br />
=40 × 10 −5 mol cm 3 (5.39)<br />
=25 × 10 25 m 3 (5.40)<br />
=36 × 10 −10 m (5.41)<br />
=1× 10 −7 m (5.42)
5.1 Hardspherecollisiontheory 99<br />
Rule-of-thumb:<br />
Time between collisions:<br />
=03mbary =03mm (5.43)<br />
=1000mbar (5.44)<br />
1 1<br />
=<br />
2 × <br />
(5.45)<br />
=15 × 10 −10 s (5.46)<br />
≈ 01ns (5.47)<br />
¤<br />
5.1.2 Allowance for a threshold energy for reaction<br />
I The threshold energy E 0 : The majority of reactions exhibit a threshold energy, below<br />
which the reaction does not occur (quantum mechanical tunneling is excluded here).<br />
The molecules must have a minimum energy min to overcome this reaction threshold energy<br />
0 . For our structureless spherical molecules, we consider only translational energy<br />
: 36 0 : no reaction (5.48)<br />
0 : reaction (5.49)<br />
I Boltzmann distribution: Fraction of molecules with translational energy 0 :<br />
µ<br />
( 0 )<br />
=exp − µ<br />
0<br />
=exp − <br />
0<br />
(5.50)<br />
<br />
<br />
<br />
I<br />
Resulting rate constant:<br />
or<br />
µ 12<br />
max = 2 8 <br />
×<br />
× − 0 <br />
(5.51)<br />
<br />
µ 12<br />
max = 2 8 <br />
×<br />
× − 0<br />
<br />
(5.52)<br />
I Arrhenius activation energy: Applying the definition of the apparent (Arrhenius)<br />
activation energy ,wefind that<br />
= 2 ln <br />
<br />
= 0 + 1 (5.53)<br />
2<br />
As we see, is not constant but (slightly) -dependent: The slope of a plot of ln <br />
max<br />
vs. −1 increases as →∞(or −1 → 0). The Arrhenius preexponential factor <br />
would not be constant as well, it is also slightly -dependent.<br />
36 We use and 0 to indicate the energies in molecular units, and and 0 for the respective<br />
values in molar units.
5.1 Hard sphere collision theory 100<br />
5.1.3 Allowance for steric hindrance<br />
I The sterical factor p: In addition, even if 0 , not every collision will lead to<br />
a reaction, because real molecules are not structureless: A reaction requires a specific<br />
orientation of the molecules. This is taken into account by introducing a steric factor<br />
:<br />
µ 12<br />
max = × 2 8 <br />
×<br />
× − 0<br />
(5.54)
5.2 Kinetic gas theory 101<br />
5.2 Kinetic gas theory<br />
5.2.1 The Maxwell-Boltzmann speed distributions of gas molecules in 1D and 3D<br />
I<br />
Figure 5.2: Experimental measurement of molecular speed distribution using a rotating<br />
disk velocity selector and an effusive molecular beam (orrifice diameter ¿ ).<br />
I<br />
Rotating disk velocity selector:<br />
• Flight time of molecules with velocity between two rotating disks (distance ):<br />
1 = <br />
(5.55)<br />
• Time delay of second slit with respect to first slit ( =angle, = angular velocity):<br />
2 = <br />
(5.56)<br />
• Transmission only if<br />
y<br />
1 = 2 (5.57)<br />
= × <br />
(5.58)<br />
In practice, one uses a number of velocity selectors in series. A measurement of the 1D<br />
speed distribution ( ) = ( )<br />
<br />
using this method gives
5.2 Kinetic gas theory 102<br />
I The 1D Maxwell-Boltzmann speed distribution (Fig. 5.3):<br />
r<br />
µ<br />
<br />
( ) =<br />
2 × exp − 2 <br />
2 <br />
<br />
(5.59)<br />
Note that this distribution is already normalized for<br />
Z ∞<br />
( ) =1 We check this<br />
by substituting<br />
r<br />
r<br />
r<br />
<br />
<br />
=<br />
2 y =<br />
2 2 <br />
y = (5.60)<br />
<br />
y<br />
Z ∞<br />
r Z ∞ r<br />
2 <br />
( ) =<br />
= 1 Z∞<br />
√ <br />
2 −2 −2 = 1 √<br />
√ =1 (5.61)<br />
<br />
−∞<br />
−∞<br />
−∞<br />
−∞<br />
I<br />
Figure 5.3: 1D Maxwell-Boltzmann speed distribution.
5.2 Kinetic gas theory 103<br />
I Velocity distribution in 3 dimensions: Since the velocities in direction are<br />
independent, the probability that a molecule has a velocity between and + ,<br />
and + ,and and + is given be the product of ( ) , ( ) ,<br />
and ( ) :<br />
( )<br />
<br />
= ( ) ( ) ( ) <br />
µ Ã<br />
32 <br />
=<br />
× exp − ¡ ¢ !<br />
2 + 2 + 2 <br />
(5.62)<br />
2 <br />
2 <br />
We are, however, not interested in the distribution of velocities pointing into a specific<br />
direction ( ), but in the probability that a molecule has a certain speed || in<br />
any direction, i.e., that the velocity vector ends somewhere on a sphere with radius<br />
q<br />
|| = 2 + 2 + 2 (5.63)<br />
To obtain this distribution, we must integrate over all polar angles and . This<br />
corresponds to replacing the volume element with the volume 4 2 of<br />
a spherical shell with radius and thickness y<br />
I The 3D Maxwell-Boltzmann speed distribution (Fig. 5.4):<br />
() =<br />
µ 32 <br />
<br />
× 4 2 × exp<br />
µ− 2 (5.64)<br />
2 <br />
2 <br />
Note that this distribution is also normalized for<br />
Z ∞<br />
() =1 We check this by<br />
substituting<br />
y<br />
Z ∞<br />
0<br />
r<br />
r<br />
µ 12 <br />
<br />
=<br />
2 y = 2 y = 2 <br />
<br />
<br />
(5.65)<br />
= 2 <br />
2 (5.66)<br />
2<br />
() =<br />
µ 32 <br />
× 4<br />
2 <br />
0<br />
Z ∞<br />
0<br />
0<br />
µ<br />
2 2 <br />
2 −2 <br />
12<br />
(5.67)<br />
µ 32 Z∞<br />
µ 32 1 1<br />
=4 × 2 −2 =4 × × 1 √ =1 (5.68)<br />
4
5.2 Kinetic gas theory 104<br />
I<br />
Figure 5.4: 3D Maxwell-Boltzmann speed distribution.<br />
I<br />
Notice the differences between the following quantities:<br />
• Value at maximum of ():<br />
• Mean speed:<br />
• Root mean square value:<br />
max =<br />
=<br />
r<br />
2 <br />
<br />
r<br />
8 <br />
<br />
³<br />
2´12 =<br />
r<br />
3 <br />
<br />
(5.69)<br />
(5.70)<br />
(5.71)<br />
I<br />
Table 5.1: Mean molecular speed values for some gases @ =298K<br />
gas [m s]<br />
H 2511<br />
H 2 1776<br />
He 1255<br />
O 2 444<br />
CO 2 379<br />
Hg 178<br />
238 U 163
5.2 Kinetic gas theory 105<br />
I Kinetic energy distribution function: We may substitute for in Eq. 5.64:<br />
= 1 2 2 (5.72)<br />
µ 12 2<br />
y =<br />
(5.73)<br />
<br />
y<br />
µ 12 2<br />
= × 1 µ 12 1<br />
2 × ( ) −12 = ( ) −12 (5.74)<br />
2<br />
µ 12 1<br />
y = ( ) −12 (5.75)<br />
2<br />
Insertion into Eq. 5.64 gives (see Fig. 5.5)<br />
( ) =<br />
y<br />
µ 32 µ <br />
µ 12 µ 12 <br />
2<br />
× 4 × − 1 1<br />
×<br />
(5.76)<br />
2 <br />
<br />
2 <br />
µ 32 1<br />
( ) =2<br />
( ) 12 − (5.77)<br />
<br />
I<br />
Figure 5.5: Maxwell-Boltzmann distribution function for the kinetic energy.<br />
The fraction of molecules with 0 is temperature dependent.
5.2 Kinetic gas theory 106<br />
5.2.2 The rate of wall collisions in 3D<br />
Earlier, we calculated the rate of wall collisions of the molecules of a gas with mass <br />
and number density in a container of volume and the resulting gas pressure using<br />
asimplified model by assuming that 1 r<br />
6 of all molecules have a fixed speed = 8 <br />
<br />
in + direction (cf. Fig. 5.6). In this simple picture, we neglected the 3D molecular<br />
velocity distribution. We obtained the correct result only due to a cancelation of two<br />
errors (assumption of fixed 1 and 3D distribution). In the following,<br />
6<br />
(1) we briefly review the simplified picture,<br />
(2) we then proceed to derive the correct results by appropriate averaging over the<br />
velocity/speed distribution.<br />
I<br />
Figure 5.6: Simplified model for the calculation of the pressure of an ideal gas.<br />
I<br />
Simplified model for the calculation of the pressure of an ideal gas:<br />
• In the time , all gas molecules (mass ) inthevolume + hit the wall<br />
area .<br />
• Number of wall collisions in time (with number density ):<br />
= × 1 × × (5.78)<br />
6
5.2 Kinetic gas theory 107<br />
• Momentum change due to wall collision, force on wall, and pressure:<br />
=2 × 1 (5.79)<br />
6<br />
• Gas pressure:<br />
y<br />
= = 1 <br />
= 1 × 2 × 1 (5.80)<br />
6<br />
= 1 3 2 (5.81)<br />
• Result for 1mol ( = <br />
<br />
with =6022 1 × 10 23 mol −1 and kin = 2<br />
2 ):<br />
= 1 3 2 = 2 3 kin = 2 3 × 3 2 = (5.82)<br />
I<br />
Correct derivation of the gas kinetic wall collision frequency:<br />
• Number of molecules with velocities in a volume of with =<br />
p <br />
2 + 2 + 2 (cf. Fig. 5.7) that will hit the area in the time :<br />
( )=| | ( )<br />
<br />
<br />
( )<br />
<br />
( )<br />
<br />
(5.83)<br />
• To find the number of all molecules hitting in , we have to integrate over all<br />
positive and over all (positive and negative) and , using the 1D-Maxwell-<br />
Boltzmann distributions ( ) = ( )<br />
<br />
,etc.:<br />
µ 32 <br />
= <br />
×<br />
2 <br />
Z ∞ Z ∞ Z ∞ µ<br />
| | exp − 2 <br />
2 <br />
0 −∞ −∞<br />
µ µ <br />
exp − 2 <br />
exp − 2 <br />
<br />
2 2 <br />
(5.84)<br />
We know that the distributions over and are normalized to 1. Hence, we<br />
only have to solve the integral over , which we do by the substitution<br />
µ 12 µ 12 µ 12 <br />
<br />
2 <br />
=<br />
y =<br />
y =<br />
(5.85)<br />
2 <br />
2 <br />
<br />
y<br />
Z ∞<br />
0<br />
µ <br />
| | exp − 2 <br />
=<br />
2 <br />
Z ∞<br />
0<br />
= 2 <br />
<br />
µ 2 <br />
<br />
Z ∞<br />
0<br />
12<br />
|| −2 µ 2 <br />
<br />
−2 = 2 <br />
<br />
1<br />
2 = <br />
<br />
12<br />
(5.86)<br />
(5.87)
5.2 Kinetic gas theory 108<br />
• Result: 37<br />
= <br />
µ 12 <br />
× <br />
2 <br />
µ 12 = × <br />
(5.88)<br />
2<br />
y<br />
wall = µ 12<br />
= × <br />
(5.89)<br />
2<br />
This can be rewritten using the mean molecular speed , because<br />
With<br />
we thus obtain the<br />
wall = × 1 4<br />
µ 16 <br />
2<br />
=<br />
12<br />
= × 1 4<br />
µ 8 <br />
<br />
• Final result for the gas kinetic collision frequency:<br />
µ 8 <br />
<br />
12<br />
(5.90)<br />
12<br />
(5.91)<br />
wall = 1 4 (5.92)<br />
I<br />
Figure 5.7: On the derivation of the gas kinetic wall collision frequency.<br />
µ 12 <br />
37 Notice that two of the three<br />
factors in Eq. 5.84 above are compensated by the<br />
2 <br />
µ 12 <br />
integrations over and ,onlyone<br />
factor remains.<br />
2
5.2 Kinetic gas theory 109<br />
I<br />
Correct derivation of the gas pressure and the ideal gas equation:<br />
• Likewise, we take the differential number of wall collisions for specific speeds(Eq.<br />
5.83) to compute the momentum change ( ) in time <br />
( )=2 | |×| | × ( ) ( ) ( ) <br />
(5.93)<br />
• Inserting again the 1D-Maxwell-Boltzmann distributions and integrating over all<br />
positive and all ,wefind the total momentum change in time :<br />
µ 32 <br />
= <br />
× 2×<br />
2 <br />
Z ∞ Z ∞ Z ∞ µ<br />
2 exp − 2 <br />
2 <br />
0 −∞ −∞<br />
µ µ <br />
exp − 2 <br />
exp − 2 <br />
<br />
2 2 <br />
• Pressure for = µ 12<br />
<br />
<br />
again using the substitution = <br />
2 <br />
y<br />
y<br />
= <br />
= <br />
<br />
= <br />
<br />
µ 12 <br />
× 2<br />
2 <br />
µ 1<br />
<br />
12<br />
2 2 <br />
<br />
Z ∞<br />
0<br />
Z ∞<br />
0<br />
(5.94)<br />
µ <br />
2 exp − 2 <br />
(5.95)<br />
2 <br />
2 −2 = <br />
(5.96)<br />
= (5.97)<br />
Thus, we have fixed our earlier simple derivations by appropriately integrating over the<br />
Maxwell-Boltzmann velocity distribution of the molecules.
5.3 Transport processes in gases 110<br />
5.3 Transport processes in gases<br />
5.3.1 The general transport equation<br />
I Definition of the flux −→ J of a quantity: We consider a generic transport property<br />
Γ (5.98)<br />
which differs in magnitude along the direction, i.e., has the gradient<br />
grad Γ (5.99)<br />
The flux of Γ throughanarea in time is defined as<br />
andcanbewrittenas<br />
−→ Γ = Γ<br />
<br />
(5.100)<br />
−→ Γ = − grad Γ (5.101)<br />
Kinetic gas theory allows us to derive a general equation for the proportionality constant<br />
.<br />
I<br />
Figure 5.8: Derivation of the general transport equation.
5.3 Transport processes in gases 111<br />
I The general transport equation: We consider the net flux of Γ through an area <br />
at = 0 that is transported by gas molecules coming from an area above at = 0 +<br />
and from below from an area above at = 0 − (where isthemeanfreepathlength).<br />
Denoting the respective transport quantity per molecule as Γ, these partial fluxes are<br />
given by the number of gas kinetic collisions hitting on ( = 0 ) times the Γ carried<br />
by each gas molecule (number density ):<br />
• Using the gas kinetic wall collision frequency wall = 1 (Eq. 5.92), we obtain<br />
4<br />
Γ + = 1 µ Γ<br />
µΓ<br />
4 × × 0 + (5.102)<br />
<br />
and<br />
• Net effect:<br />
Γ − = 1 4 × × µΓ 0 −<br />
µ Γ<br />
(5.103)<br />
<br />
Γ = Γ + − Γ − = 1 µ Γ<br />
2 × × (5.104)<br />
<br />
• Net flux (with − sign to account for the fact that the flux is directed along the<br />
downhill gradient of Γ):<br />
−→ Γ = Γ<br />
µ Γ<br />
= −1 2 <br />
with given by (see the derivation of wall above)<br />
=<br />
µ 8 <br />
<br />
(5.105)<br />
12<br />
(5.106)<br />
and, for only one type of gas molecules A (i.e., A - A collisons),<br />
= 1 √<br />
2<br />
1<br />
2 × <br />
(5.107)<br />
• If also varies along (diffusion), Eq. 5.105 may be recast by defining the overall<br />
transport quantity Γ 0 from the transport quantity per molecule Γ as<br />
Γ 0 = Γ (5.108)<br />
into the form<br />
−→ Γ = − 1 2 Ã<br />
!<br />
Γ 0<br />
<br />
(5.109)<br />
In the following, ³ we shall apply Eqs. 5.105 or 5.109 to determine the self-diffusion coefficient<br />
Γ =1 Γ 0 = <br />
´, the heat conductivity ¡ Γ = ¢ ,andtheviscosity ¡ ¢<br />
Γ = <br />
of gases.
5.3 Transport processes in gases 112<br />
5.3.2 Diffusion<br />
I<br />
Figure 5.9: The diffusion in a mixture A + B of one sort of molecules A against a<br />
time-independent concentration gradient <br />
<br />
is described by Fick’s first law.<br />
• Fick’s first law:<br />
−→ = µ <br />
<br />
= − <br />
<br />
(5.110)<br />
• In order to derive an expression for the self-diffusion coefficient (diffusion of an<br />
isotope of A in normal A), we make the ansatz<br />
Γ =1 Γ 0 = (5.111)<br />
y<br />
• Result:<br />
−→ = − 1 µ <br />
2 <br />
<br />
(5.112)<br />
= 1 (5.113)<br />
2<br />
• Fick’s second law (time-dependent concentration gradient):<br />
• Einstein’s diffusion law in 1D:<br />
• Einstein’s diffusion law in 3D:<br />
<br />
<br />
= 2 <br />
2 (5.114)<br />
∆ 2 =2 (5.115)<br />
∆ 2 =6 (5.116)
5.3 Transport processes in gases 113<br />
5.3.3 Heat conduction<br />
I<br />
Figure 5.10: Heat conduction is described by Fourier’s law.<br />
• Fourier’s law (for constant temperature gradient):<br />
−→ =<br />
µ <br />
= − <br />
(5.117)<br />
• Ansatz:<br />
y<br />
• Result:<br />
Γ = = internal energy per molecule (5.118)<br />
−→ <br />
= − 1 µ <br />
2 <br />
= − 1 µ µ <br />
<br />
2 <br />
= − 1 µ µ<br />
2 <br />
<br />
= − 1 µ <br />
2 <br />
<br />
<br />
(5.119)<br />
(5.120)<br />
(5.121)<br />
(5.122)<br />
= 1 2 (5.123)<br />
with = number density and = specific heat per molecule.
5.3 Transport processes in gases 114<br />
5.3.4 Viscosity<br />
I<br />
Figure 5.11: Newton’s law describes the inner friction in laminar flow.<br />
• Newton’s law:<br />
• Ansatz:<br />
µ <br />
<br />
= − <br />
<br />
(5.124)<br />
Γ = = momentum per molecule in direction (5.125)<br />
y<br />
• Result:<br />
−→ = − 1 µ <br />
2 ( )<br />
<br />
= 1 ( )<br />
= 1 <br />
= <br />
= 1 (5.126)<br />
2<br />
with = number density and = mass of molecule.
5.4 Advanced collision theory 115<br />
5.4 Advanced collision theory<br />
The hard sphere model is a very primitive model. In reality, we have to account for the<br />
following:<br />
• The reaction probability depends on exact details of the collision, i.e.,<br />
hard sphere model → ( ) (5.127)<br />
• Molecules are not hard spheres with = 2 but “soft”, i.e., the reactive cross<br />
section depends on the relative speed (or kinetic energy)<br />
→ () → ( ) (5.128)<br />
• The speed distribution follows the Maxwell-Boltzmann distribution, i.e.,<br />
→ () → ( ) (5.129)<br />
• Molecules in different quantum states behave differently, i.e.,<br />
→ ( ; ) (5.130)<br />
I Examples: The complexity of the problem is illustrated in Fig. 5.12 by two examples:<br />
D( )+H 2 ( 2 2 2 ) −→ HD( )+H( ) (5.131)<br />
OH( Ω )+H 2 ( 2 2 2 )<br />
→ H 2 O( 2 1 2 3 )+H( ) (5.132)<br />
As can be seen, the reaction rate is expected to depend on (speed, translational<br />
energy), (vibrational quantum numbers ( 1 = symmetric stretch, 2 = bend, 3 =<br />
antisymmetric stretch)), (rotational quantum numbers), Ω (fine structure<br />
state, electronic state). In addition, the products are scattered into the spatial angles<br />
.<br />
I<br />
Figure 5.12: Quantum state-specific elementary reactions.
5.4 Advanced collision theory 116<br />
We shall now derive the fundamental equation of collision theory that allows us to<br />
account for those effects.<br />
5.4.1 Fundamental equation for reactive scatteringincrossedmolecularbeams<br />
• Consider the following reactive scattering process (Fig. 5.13):<br />
A+B→ C (5.133)<br />
• IntensityofbeamofA:<br />
= × (5.134)<br />
Is this reasonable? ⇒ Look at the physical dimensions of × :<br />
[cm s] × £ molecules/ cm 3¤ = £ molecules cm 2 s ¤ =[intensity] (5.135)<br />
• Depletion of intensity of A due to scattering by B:<br />
= − () × × × <br />
↑<br />
↑<br />
↑<br />
<br />
<br />
(5.136)<br />
y rate equation:<br />
<br />
= −() (5.137)<br />
<br />
• Rate coefficient for a given value of :<br />
() =() (5.138)<br />
• Averaging over the speed distribution () leads to the fundamental equation<br />
of all improved collision theories:<br />
( )=h()i (5.139)<br />
i.e.,<br />
( )= ∞ R<br />
0<br />
() ( ) (5.140)<br />
• Frequently used approximation: Since the explicit form of () is usually not<br />
known, we write<br />
( )=hi h()i (5.141)<br />
with<br />
and<br />
hi =<br />
µ 8 <br />
<br />
12<br />
(5.142)<br />
h()i = -averaged (reactive) cross section (5.143)
5.4 Advanced collision theory 117<br />
I<br />
Figure 5.13: Derivation of the fundamental equation.<br />
5.4.2 Crossed molecular beam experiments<br />
I Differential and integral reactive cross sections: Experiments with crossed molecular<br />
beams allow us to measure differential and integral reactive cross sections<br />
(Herschbach 1987, Lee 1987).<br />
We consider the reaction<br />
( )+ ( ) −→ ( )+ ( ) (5.144)<br />
where denotes the speeds of the molecules and stands for the quantum numbers<br />
of their energy states (electronic, vibrational, rotational, nuclear spin). In an ideal<br />
experiment (which has yet to be performed), all of the reactants are defined by, e.g.,<br />
a specific excitation using laser, and the of the products are determined by some<br />
probe laser. The can be determined in a molecular beam experiment using a rotating<br />
disk velocity selector (see Fig. 5.14).
5.4 Advanced collision theory 118<br />
I<br />
Figure 5.14: Measurement of differential cross sections in crossed molecular beam<br />
experiments.<br />
I<br />
Differential and integral cross sections from crossed molecular beam experiments:<br />
• We define the differential cross section ( ) for scattering<br />
of products C into the solid angle element Ω at such that<br />
( ) Ω<br />
= ( )<br />
<br />
× × × ( ) × × ( ) × Ω (5.145)<br />
• Corresponding integral cross sections are obtain by integration over and .<br />
• Depending on the scattering process of interest, we distinguish between<br />
<strong>—</strong> elastic cross sections ( ): measure of the size of the molecules,<br />
<strong>—</strong> inelastic cross sections ( ):measureoftheefficiency of energy transfer<br />
processes,<br />
<strong>—</strong> reactive cross sections ( ): measure of reactivity.
5.4 Advanced collision theory 119<br />
I<br />
Figure 5.15: Measurement of differential cross sections in crossed molecular beam<br />
experiments.
5.4 Advanced collision theory 120<br />
5.4.3 From differential cross sections σ (...) to reaction rate constants k (T )<br />
I<br />
Figure 5.16: The strategy.<br />
I<br />
1 → 2:Integration over angles:<br />
0<br />
( )=<br />
Z 2 Z <br />
0<br />
0<br />
( ) 0<br />
( )sin (5.146)<br />
I 2 → 3:Summation over all product quantum states α 0 :<br />
( )= X 0<br />
0<br />
( ) (5.147)<br />
I 3 → 4:Thermal averaging over initial quantum states α:<br />
( )= X <br />
× ( ) (5.148)<br />
with the Boltzmann factor and the partition function <br />
=<br />
− <br />
P<br />
− <br />
= − <br />
<br />
(5.149)<br />
= X <br />
− <br />
(5.150)<br />
y<br />
( )= X <br />
( ) × − <br />
<br />
(5.151)
5.4 Advanced collision theory 121<br />
I 5 → 6:Thermal averaging over all ² :<br />
( )= X <br />
× ( )<br />
= X − <br />
× ( )<br />
<br />
<br />
Z<br />
− <br />
=<br />
× ( ) <br />
<br />
<br />
I<br />
3 → 5 and 4 → 6:Laplace Transformation:<br />
Z ∞<br />
( ) ∝ × ( ) × − (5.152)<br />
0<br />
This expression follows from the fundamental equation (Eq. 5.140) by substituting <br />
for . The exact transformation (giving the correct factors before the integral) will be<br />
carried out below (see Eq. 5.162).<br />
5.4.4 Laplace transforms in chemistry<br />
Formally, the transformation (Eq. 5.152) is a Laplace transformation (see Appendix<br />
E). A Laplace transform generally described the transformation from a microscopic<br />
(“microcanonical”) quantity such as ( ) to a macroscopic (“macrocanonical”, i.e.,<br />
dependent) quantity such as ( ).<br />
I Definition of the Laplace Transformation: The Laplace transform L [ ()] of a<br />
function () is defined as the integral<br />
() =L [ ()] =<br />
Z ∞<br />
0<br />
() − (5.153)<br />
I<br />
Inverse Laplace Transformation:<br />
with an arbitrary real number .<br />
() =L −1 [ ()] = 1<br />
2<br />
Z<br />
+∞<br />
−∞<br />
() (5.154)
5.4 Advanced collision theory 122<br />
I<br />
Notes:<br />
(1) The so far best known Laplace transform to us is the partition function<br />
( )= X <br />
− =<br />
Z ∞<br />
0<br />
() − (5.155)<br />
(2) While the pathway from ( ) to ( ) by forward Laplace transformation is<br />
straightforward, it is not generally straightforward to derive ( ) by inverse<br />
Laplace transformation from ( ), because ( ) would be needed from =0<br />
to = ∞.
5.4 Advanced collision theory 123<br />
5.4.5 Bimolecular rate constants from collision theory<br />
I Transformation from velocity to energy space: Taking our results from kinetic<br />
gas theory, we are now in the position to apply the fundamental equation of collision<br />
theory (Eq. 5.140)<br />
( )=<br />
Z ∞<br />
0<br />
() ( ) (5.156)<br />
using various functional forms for . Towards these ends, we first transform from to<br />
by inserting () from Eq. 5.64 (with the reduced mass instead of ). We thus<br />
obtain<br />
µ <br />
( )=4<br />
2 <br />
32<br />
∞<br />
As done above, we substitute for according to<br />
and obtain<br />
( )=<br />
Z<br />
0<br />
3 ()exp<br />
µ− 2 (5.157)<br />
2 <br />
= 1 2 2 (5.158)<br />
µ 12 2<br />
y =<br />
(5.159)<br />
<br />
y<br />
µ 12 2 1<br />
=<br />
2<br />
µ 1<br />
2<br />
y =<br />
µ 12 µ 1 2<br />
<br />
µ 12 µ 12 µ 12 1 1 1<br />
=<br />
(5.160)<br />
2 <br />
12 µ 1<br />
<br />
12<br />
(5.161)<br />
32<br />
∞<br />
Z<br />
0<br />
( )exp<br />
µ<br />
− <br />
<br />
<br />
<br />
(5.162)<br />
The integration from 0 to ∞ takes into account that a reaction can occur only if<br />
0 .<br />
This expression is interpreted as follows:<br />
• The function − describes the fraction of molecules with energy .<br />
• The function ( ) in the integrand is called the excitation function 38 . This<br />
can, in general, be a rather complicated (and not necessarily smooth) function<br />
(see Fig. 5.17).<br />
• The complete integrand ( ) − is called the reaction function 39 ,<br />
because it describes the reactive fraction of the molecules.<br />
• Therateconstant ( ) corresponds to the area under the reaction function.<br />
38 “Anregungsfunktion”<br />
39 “Reaktionsfunktion”
5.4 Advanced collision theory 124<br />
I<br />
Figure 5.17: Advanced collision theory: The excitation function.<br />
I<br />
Figure 5.18: Advanced collision theory: The reaction function.
5.4 Advanced collision theory 125<br />
I Non-equilibrium energy distribution:* The above expression for ( ) is valid if<br />
the reacting molecules are in internal equilibrium, i.e., the energy states are populated<br />
according to the Boltzmann distribution.<br />
Deviations from the Boltzmann distributions will occur in the case of very fast reactions,<br />
if the reaction is faster than the relaxation between the energy states. In this case, ( )<br />
decreases because the mean energy of the reacting molecule decreases.<br />
Mean energy of the reacting molecules (see Fig. 5.19):<br />
= X <br />
=<br />
Z ∞<br />
0<br />
() (5.163)<br />
(1) For thermal equilibrium:<br />
() =() (5.164)<br />
with () being the Boltzmann distribution.<br />
y<br />
(2) Non-equilibrium:<br />
() () (5.165)<br />
non-eq. eq. (5.166)<br />
I<br />
Figure 5.19: Mean energy of the reacting molecules in thermal equilibrium and for a<br />
non-eqilibrium situation.<br />
I Conclusion: The non-equilibrium distribution leads to a reduced rate coefficient.
5.4 Advanced collision theory 126<br />
a) Advanced collision theory for hard spheres (Fig. 5.20)<br />
I<br />
Figure 5.20: Thehardspheremodel.<br />
(1) Ansatz for the reactive cross section:<br />
( )=<br />
(2) Inserting this ansatz into Eq. 5.162:<br />
( )=<br />
(3) Result for ( ): With<br />
we obtain<br />
y<br />
µ 12 µ 1 2<br />
<br />
Z<br />
½<br />
2 0<br />
0 0<br />
(5.167)<br />
32<br />
∞<br />
Z<br />
0<br />
2 exp<br />
µ<br />
− <br />
<br />
(5.168)<br />
<br />
= <br />
( − 1) (5.169)<br />
2 µ 12 µ 32 1 2<br />
( )=<br />
2<br />
<br />
µ<br />
× ( ) 2 × exp − µ<br />
<br />
− ∞<br />
<br />
<br />
¯¯¯¯ − 1 (5.170)<br />
0<br />
( )= 2 µ 8 <br />
<br />
12 µ<br />
1+ µ<br />
0<br />
exp − <br />
0<br />
<br />
(5.171)
5.4 Advanced collision theory 127<br />
(4) Arrhenius activation energy:<br />
y<br />
= 2 ln <br />
<br />
ln ∝ 1 µ<br />
2 ln +ln 1+ <br />
0<br />
− 0<br />
<br />
ln <br />
= 1 µ<br />
1<br />
2 + 1<br />
− µ <br />
0 0<br />
+<br />
1+ 0 2 2<br />
= 0 + 1 2 − 0<br />
1+ 0 <br />
(5.172)<br />
(5.173)<br />
(5.174)<br />
(5.175)<br />
b) Line-of-Centers model (Figs. 5.21 - 5.22)<br />
Even for hard spheres, because of angular momentum conservation (see below), not the<br />
entire collision energy, but only the fraction “directed” along the line of centers ()<br />
is effective.<br />
In particular, we have<br />
• for a collision with =0:<br />
= (5.176)<br />
y central collisions are most effective,<br />
• for a collision with = :<br />
=0 (5.177)<br />
y glancing collisions are ineffective because =0,<br />
• general expression (see below):<br />
= <br />
µ<br />
1 − 2<br />
2 <br />
(5.178)<br />
y decreases with increasing .
5.4 Advanced collision theory 128<br />
I<br />
Figure 5.21: Collision energy along line-of-centers<br />
Functional expression for ( ):<br />
(1) We assume that a reaction can occur only if ≥ 0 . Thus, we first determine<br />
(see Fig. 5.21):<br />
y<br />
= cos (5.179)<br />
2 = 2 cos 2 (5.180)<br />
= ¡ 2 1 − sin 2 ¢ (5.181)<br />
µ <br />
= 2 1 − 2<br />
(5.182)<br />
2<br />
= <br />
µ<br />
1 − 2<br />
2 <br />
(5.183)<br />
(2) We see from this expression that decreases with increasing . Thus, for a<br />
given ,wehaveamaximalvalueof (i.e., a value max ), for which ≥ 0 .<br />
We determine max from the condition that<br />
µ <br />
= 1 − 2<br />
≥ <br />
2 0 (5.184)<br />
y<br />
y<br />
2 ≤ 2 µ<br />
1 − 0<br />
<br />
<br />
= 2 max (5.185)<br />
( )= 2 max = 2 µ<br />
1 − 0<br />
<br />
<br />
(5.186)
5.4 Advanced collision theory 129<br />
y<br />
⎧<br />
⎨<br />
( )=<br />
⎩<br />
2 µ<br />
1 − 0<br />
<br />
<br />
0<br />
(5.187)<br />
0 0<br />
( ) is shown as function of in Fig. 5.22. Crossed molecular beam experiments<br />
have indeed shown similar dependencies for a number of reactions.<br />
I<br />
Figure 5.22: Reactive cross section for the line-of-centers model.<br />
(3) Result for ( ) from Eq. 5.162:<br />
y<br />
µ 1<br />
( )=<br />
<br />
Z ∞ µ<br />
× 2<br />
0<br />
=<br />
µ 1<br />
<br />
×<br />
Z ∞<br />
0<br />
( )=<br />
12 µ 2<br />
<br />
32<br />
1 − <br />
0<br />
<br />
12 µ 2<br />
32<br />
<br />
2 ( − 0 )exp<br />
µ 8 <br />
<br />
µ<br />
exp − <br />
<br />
(5.188)<br />
<br />
µ<br />
− <br />
<br />
(5.189)<br />
<br />
12 µ<br />
2 exp − <br />
0<br />
<br />
(5.190)<br />
This result is identical to that of our primitive “hard sphere-fixedmeanrelative<br />
speed” model from Section 5.1!
5.4 Advanced collision theory 130<br />
(4) Arrhenius activation energy:<br />
y<br />
= 2 ln <br />
<br />
(5.191)<br />
= 0 + 1 (5.192)<br />
2<br />
c) Allowance for the “steric effect”<br />
In order to account for the “structures” of the molecules which can undergo a reaction<br />
only if the reaction partners have a specificrelativeorientation,weintroducea(generally<br />
-dependent) steric factor ( ):<br />
µ 12 µ<br />
8 <br />
( )=( )<br />
2 exp − <br />
0<br />
(5.193)<br />
<br />
<br />
In ( ), we include all sorts of factors leading to deviations from the predicitions by simple<br />
collision theory. In favorable cases, ( ) can be predicted by theory. Towards these<br />
ends, we define a reaction probability ( ) (also called the opacity function,<br />
andcalculatedbytheory)andintegrateaccordingto<br />
( )=<br />
Z<br />
0 max<br />
( )2 (5.194)<br />
0<br />
In other cases, it may be convenient to define an angle dependent reaction threshold<br />
energy ∗ 0:<br />
∗ 0 = 0 + 0 (1 − cos ) (5.195)<br />
y<br />
∗ 0 = 0 for =0<br />
∗ (5.196)<br />
0 0 for 6= 0<br />
Such models are helpful for explaining steric factors in the range of 1 ≤ ≤ 0001.
5.4 Advanced collision theory 131<br />
I<br />
Figure 5.23: Collision theory: Allowance for steric effects.<br />
I<br />
Table 5.2: Comparison of predictions by collision theory with experimental data.
5.4 Advanced collision theory 132<br />
I Harpooning: The value of ≥ 1 for the steric factor of the K + Br 2 reaction can<br />
be understood by the “harpooning mechanism”: An electron jumps from the K to the<br />
Br 2 at a long intermolecular distance. The subsequent collision and reaction at closer<br />
distance then occurs between a K + and Br − 2 . The collision between these two is strongly<br />
affectedbytheCoulombattraction.<br />
5.4.6 Long-range intermolecular interactions<br />
To this point, we have neglected any interactions between the reacting molecules other<br />
than the the collision itself. Thus, only straight line trajectories were allowed. In reality,<br />
however, there will be attractive or repulsive interactions between the colliding molecules<br />
(see Fig. 5.24). These will inevitably affect the collision dynamics. An extreme example<br />
is the K + Br 2 reaction mentioned above.<br />
We will investigate the potential energy hypersurfaces governing the reactions in Section<br />
6. Here, we will consider only long-range intermolecular interactions which can be easily<br />
implementedincollisiontheory.<br />
I<br />
Figure 5.24: Examples of attractive and repulsive trajectories for the collision between<br />
two molecules.<br />
a) Types of long-range interactions<br />
The long-range interactions between molecules can be described using multipole expansions<br />
of the potentials (charge: 1; dipole: 2; induced dipole: 3; quadrupole: 3 . . . In<br />
general, the interaction potential between two multipoles of ranks and 0 is proportional<br />
to − with = + 0 − 1:<br />
µ <br />
() =−<br />
(5.197)
5.4 Advanced collision theory 133<br />
I<br />
Table 5.3: Types of long-range intermolecular interactions.<br />
system<br />
()<br />
charge-charge (Coulomb) () = 1 2 2<br />
4 0 <br />
charge-dipole<br />
( ) = −→ · −→ = − cos <br />
4 0 2<br />
dipole-dipole ( 1 2 )=<br />
charge-quadrupole Θ ∝ −3<br />
dipole-quadrupole<br />
Θ ∝ −4<br />
charge-induced dipole ∝ −4<br />
quadrupole-quadrupole ΘΘ ∝ −5<br />
induced dipole-induced dipole <br />
∝ −6<br />
μ 1 · μ 2 − 3(μ 1 · r)(μ 2 · r)<br />
2<br />
4 0 3<br />
I Table 5.4: Long-range intermolecular interactions ( =15 D, =3Å 3 , =1,<br />
=0;energies in kJ mol; forcomparison: =25kJ mol @ =298K).<br />
³ ´ ³ ´<br />
system =5Å =10Å<br />
ion-ion 278 139<br />
ion-dipole 17.4 4.35<br />
ion-induced dipole 3.34 0.21<br />
dipole-dipole 2.17 0.27<br />
b) Lennard-Jones (6-12) potential for nonpolar molecules<br />
A realistic intermolecular potential for non-polar molecules is the Lennard-Jones (6-12)<br />
potential<br />
∙ ³LJ ´12 ³ LJ<br />
´6¸<br />
LJ () =4 LJ − (5.198)<br />
<br />
LJ is usually given in reduced form in units of K as<br />
∗ LJ = LJ<br />
<br />
(5.199)
5.4 Advanced collision theory 134<br />
I<br />
Figure 5.25: Schematic plot of the Lennard-Jones potential.<br />
I Estimation of LJ parameters: Values for LJ and LJ are typically derived from<br />
transport properties (e.g., viscosity), thermodynamic data (vdW parameters), or <strong>—</strong> if<br />
there is no better way <strong>—</strong> from empirical correlations (see, e.g., Svehla1963) with critical<br />
data, boiling temperatures, . . .<br />
I<br />
Table 5.5: Lennard-Jones for some gases (Mourits1977).<br />
LJ (pm) ( LJ ) (K) LJ (pm) ( LJ ) (K)<br />
He 2560 102 N 2 3738 820<br />
Ne 2822 320 O 2 3480 1026<br />
Ar 3465 1135 CH 4 3790 1421<br />
Kr 3662 1780 CF 4 4486 1673<br />
Xe 4050 2302 CCl 4 5611 4155<br />
H 2 2930 370 CO 2 3943 2009<br />
SF 6 5199 2120
5.4 Advanced collision theory 135<br />
I<br />
Figure 5.26: Lennard-Jones interaction potentials for He, Ne, Ar, Kr, and Xe.<br />
c) Orbital angular momentum and centrifugal barrier*<br />
In the case of a non-central collision, part of the translational energy is converted<br />
to “rotational energy” (orbital angular momentum; see Fig. 5.27). This is seen by<br />
partitioning the velocity vector between the direction ( )andtheradial<br />
direction ( ). The component corresponds to a rotational motion.<br />
I<br />
Figure 5.27: Orbital angular momentum in binary collisions.
5.4 Advanced collision theory 136<br />
• Radial velocity component:<br />
y<br />
• Kinetic energy:<br />
• Angular velocity:<br />
=<br />
cos = = <br />
<br />
= <br />
1<br />
2 2 +<br />
| {z }<br />
translation via LC<br />
1<br />
2 2 <br />
| {z }<br />
= 1 2 2 2 (rotation)<br />
(5.200)<br />
(5.201)<br />
(5.202)<br />
= <br />
= <br />
= <br />
= <br />
<br />
= <br />
2 (5.203)<br />
• Kinetic energy:<br />
= 1 2 ( ) 2 + 1 2 ( ) 2 (5.204)<br />
= 1 2 · 2 + 1 2<br />
2 2 2 (5.205)<br />
= 1 2 · 2 + 1 2 2<br />
<br />
2 (5.206)<br />
= 1 2 · 2 + 1 2 2 2<br />
2 2 (5.207)<br />
• Orbital angular momentum: is the orbital angular momentum <br />
= 2 <br />
= = (5.208)<br />
y<br />
= 1 2 · 2 +<br />
2<br />
2 2 (5.209)<br />
From quantum mechanics, 2 = ~ 2 ( +1)with as orbital angular momentum<br />
quantum number. y<br />
= 1 2 · 2 + ~2 ( +1)<br />
2 2 (5.210)<br />
= 1 2 · 2 + 2 ( +1) (5.211)<br />
where 2 = ~2<br />
is formally an -dependent rotational constant.<br />
22 • Centrifugal barrier: The centrifugal kinetic energy term ~2 ( +1)<br />
acts as a<br />
2 2<br />
centrifugal barrier<br />
centrifugal = 1 2 2<br />
<br />
= ~2 ( +1)<br />
(5.212)<br />
2 2 2
5.4 Advanced collision theory 137<br />
• Withalong-rangeattractivetermoftheform− ¡ <br />
¢ <br />
<br />
, the intermolecular potential<br />
thus has the form<br />
µ 2<br />
() =− + (5.213)<br />
<br />
µ 2 <br />
= − + ~2 ( +1)<br />
(5.214)<br />
2 2<br />
i.e.,<br />
() ∝− − + −2 (5.215)<br />
<strong>—</strong> The −2 term is important in practice if 3. In effect, it leads to an<br />
effective energy barrier (“centrifigual” barrier).<br />
<strong>—</strong> The larger for a given , the larger the orbital angular momentum .<br />
I<br />
Figure 5.28: Potential energy curves with centrifugal barriers for a diatomic system.<br />
d) Langevin “capture” rate constant for ion-molecule reactions*<br />
An nice illustration of the above results is the rate constant for ion-molecule reactions.<br />
The intermolecular interaction is governed by the point charge-induced dipole interaction.<br />
40 In addition, the centrifugal barrier has to be taken into account, so that<br />
() =−<br />
µ 2 ()2 <br />
(4 0 ) 2 2 + 4 (5.216)<br />
<br />
The resulting expression for the rate constant is<br />
à !<br />
4 () 2 12 µ 1<br />
( )=<br />
(4 0 ) 2 × Γ<br />
(5.217)<br />
2<br />
Typical values of are 100 × <br />
40 () needs to be checked again!!
5.4 Advanced collision theory 138<br />
6. Potential energy surfaces for chemical reactions<br />
6.1 Diatomic molecules<br />
I<br />
Morse potential (anharmonic oscillator):<br />
() = [1 − exp (− ( − ))] 2 (6.1)<br />
I<br />
Vibrational states of a harmonic oscillator:<br />
µ<br />
= ~ + 1 µ<br />
= + 1 <br />
2<br />
2<br />
(6.2)<br />
I<br />
Vibrational states of an anharmonic oscillator:<br />
µ<br />
= ~ + 1 µ<br />
− ~ + 1 2<br />
+ (6.3)<br />
2<br />
2<br />
I Dissociation energies D and D 0 :<br />
= 0 + 1 2 ~ − 1 4 ~ + (6.4)<br />
I<br />
Figure 6.1: Potential energy curve of diatomic molecules.
6.2 Polyatomic molecules 139<br />
6.2 Polyatomic molecules<br />
I<br />
I<br />
The potential energy becomes dependent of the different internal coordinates.<br />
Question: How many internal coordinates (dimensions) do we need?<br />
Answer:<br />
• Total number of degrees of freedom of an -atomic molecule:<br />
• for translational motion of center of gravity:<br />
• for rotational motion around center of gravity<br />
• resulting number of vibrational coordinates:<br />
3 (6.5)<br />
3 (6.6)<br />
2 for linear molecules (6.7)<br />
3 for nonlinear molecules (6.8)<br />
<strong>—</strong> linear molecules:<br />
<strong>—</strong> nonlinear molecules:<br />
3 − 5 (6.9)<br />
3 − 6 (6.10)<br />
I<br />
3N − 6 dimensional potential energy hypersurface:<br />
• The internal coordinates are described using the normal coordinates (vibrational<br />
displacements).<br />
• The potential energy function<br />
( 1 2 3−6 ) (6.11)<br />
is a hypersurface in a 3 − 6 (3 − 5) dimensional hyperspace. But we can plot<br />
only in 2-D or 3-D space.
6.2 Polyatomic molecules 140<br />
I<br />
Figure 6.2: Potential energy hypersurface for a simple atom transfer reaction (plot for<br />
fixed angle, e.g., collinear collision and other fixed coordinates depending on dimensionality<br />
of the system).<br />
I<br />
Figure 6.3: Schematic potential energy hypersurface of a triatomic molecule.
6.2 Polyatomic molecules 141<br />
I<br />
Figure 6.4: Potential energy hypersurface of HCO.<br />
I<br />
Figure 6.5: Potential energy hypersurface of HCO.
6.2 Polyatomic molecules 142<br />
I<br />
Models of potential energy surfaces:<br />
(1) Empirical or semiempirical models (e.g., LEPS),<br />
(2) Single points ( ) by ab initio calculations using quantum chemistry. Problem:<br />
Fitting the surface to the computed points.<br />
I<br />
<strong>Kinetics</strong> and Dynamics:<br />
(1) TST = Transition State Theory (statistical theory),<br />
(2) Classical trajectory calculations (by solving Newton’s equations),<br />
(3) Fully quantum mechanical wave packet calculations (by solving the timedependent<br />
Schrödinger equation).
6.3 Trajectory Calculations 143<br />
6.3 Trajectory Calculations<br />
The “exact” dynamics of the molecules on a potential energy hypersurface can be<br />
followed by classical trajectory calculations.<br />
I Phase Space: 6-dimensional space spanned by the spatial coordinates ( )and<br />
the conjugated momenta ( ):<br />
↔ (6.12)<br />
where<br />
= <br />
<br />
<br />
= <br />
·<br />
(6.13)<br />
I<br />
I<br />
Hamilton function:<br />
= + = 2<br />
+ () (6.14)<br />
2<br />
Note that depends only on and depends only on .<br />
Equations of motion:<br />
• for a 1 particle system:<br />
(1)<br />
= = <br />
<br />
= −<br />
()<br />
<br />
y<br />
·<br />
= − <br />
= − <br />
since contains only not .<br />
(2)<br />
y<br />
y<br />
• for an atomic system:<br />
= + =<br />
(6.15)<br />
(6.16)<br />
= 2<br />
2 + (6.17)<br />
<br />
<br />
= 1 × = 1 µ<br />
× <br />
·<br />
3X<br />
=1<br />
·<br />
= <br />
<br />
= ·<br />
(6.18)<br />
(6.19)<br />
<br />
2 2 + ( 1 2 3−6 ) (6.20)<br />
·<br />
= − <br />
(6.21)<br />
<br />
·<br />
= <br />
(6.22)<br />
<br />
Thus we obtain a set of coupled differential equations which can be solved using<br />
the Runge-Kutta or Gear methods.
6.3 Trajectory Calculations 144<br />
• Thermal rate constants ( ) follow by suitable averaging over the initial conditions<br />
( vibrational states, rotational states, angles, ...).<br />
• Major problems: Zero-point energy? Tunneling? Surface crossings?<br />
I<br />
Figure 6.6: Trajectory calculations.<br />
I<br />
Figure 6.7: Trajectory calculations.
6.3 Trajectory Calculations 145<br />
I<br />
Figure 6.8: The reaction H + Cl 2 isareactionwithanearlyTS.Thereactionisknown<br />
to yield HCl with an inverted vibrational state distribution (⇒ HCl-Laser).<br />
I<br />
Figure 6.9: Bimolecular reaction with a late TS.
6.3 Trajectory Calculations 146<br />
7. Transition state theory<br />
7.1 Foundations of transition state theory<br />
The concept of transition state theory (TST) or activated complex theory (ACT) goes<br />
back to H. Eyring and M. Polanyi (1935). TST is a statistical theory; it does not give<br />
information on the exact dynamics of a reaction. It does, however, give (statistical)<br />
information on product state distributions.<br />
There are two forms of TST:<br />
• Macrocanonical TST ⇒ ( ) for a macrocanonical ensemble of molecules with<br />
aconstant (an ensemble of molecules with energy states populated according<br />
to the Boltzmann distribution - each state being equally likely.)<br />
• Microcanonical TST (TST) ⇒ () for a microcanonical ensemble of molecules<br />
with specific energy.<br />
We will consider only the first version.<br />
The detailed application of TST also depends on whether or not the reaction has a<br />
distinctive potential energy barrier (⇒ conventional TST or variational TST).<br />
I<br />
Figure 7.1: Transition State Theory (TST): Fundamental equations and applications.
7.1 Foundations of transition state theory 147<br />
7.1.1 Basic assumptions of TST<br />
(1) The Born-Oppenheimer approximation (separation of electronic and nuclear motion)<br />
is a valid approximation. Thus we can describe the nuclear motion during<br />
the reaction on a single electronic potential energy surface (PES; see Fig. 7.2). 41<br />
(2) The reactant molecules are in internal equilibrium (the quantum states are populated<br />
according to the Boltzmann distribution).<br />
(3) “Transition state” and “dividing surface“:<br />
a) Conventional TST (valid for so-called “type I” PES’s = PES’s with a distinctive<br />
energy barrier along the reaction coordinate (RC)): We can localize<br />
a distinctive “transition state” (TS) for the reaction at the saddle point on<br />
the PES that separates the reactant and product “valleys”. The TS is located<br />
at the maximum of the PES along the minimum energy path (MEP;<br />
see Fig. 7.3).<br />
b) Variational TST (version that can be used for so-called “type II” PES’s =<br />
PES’s without a clear energy barrier along the RC (we can’t localize a simple<br />
TS)): We can localize a 3 − 7-dimensional “dividing surface” (DS) on the<br />
3 − 6-dimensional PES that separates the reactant and product regions<br />
on the PES in such a way that the “reactive flux” through the DS becomes<br />
minimal (Fig. 7.2). The TS is then placed at the point where the MEP<br />
intersects the DS. In this case, the position of the TS depends strongly on<br />
the finer details such as angular momentum (because of centrifugal barriers<br />
which move inwards with increasing or increasing ).<br />
(4) The motion along the minimum energy path across the TS (through the DS) can<br />
be separated from other degrees of freedom and treated as translational motion.<br />
AttheTS,wethereforehave1translationaldegreeoffreedom ‡ (the RC) and<br />
3 − 7 (vibrational) degrees of freedom orthogonal to the RC.<br />
(5) Reactant molecules passing through the DS never come back (no recrossing).<br />
(6) The states of the TS are in quasi-equilibrium with the states of the reactant<br />
molecule even if we have no equilibrium between reactants and products (the<br />
individual molecules don’t know that this equilibrium is not present - this quasiequilibrium<br />
assumption can be dropped in dynamical derivations of TST; see<br />
below).<br />
I<br />
Conditions for TS at the saddle point:<br />
(1) has maximum along ‡ :<br />
<br />
=0and 2 <br />
0 (7.1)<br />
‡ <br />
‡2<br />
y 1 imaginary frequency (criterion for TS in quantum chemical calculations)!<br />
41 TST cannot be used at least in the simple form considered in this lecture when two potential surfaces<br />
are involved that are energetically close to each other (or even become degenerate) and the reaction<br />
involves a “non-adiabatic transition” between these two potentials.
7.1 Foundations of transition state theory 148<br />
(2) has minimum along all 3 − 7 other coordinates:<br />
<br />
<br />
=0and 2 <br />
2 <br />
0 (7.2)<br />
y 3 − 7 real other frequencies.<br />
I<br />
Figure 7.2: Transition state theory: The dividing surface with minimal reactive flux.<br />
I<br />
Figure 7.3: Minimum energy path (MEP) and transition state (TS) of a simple atom<br />
transfer reaction.
7.1 Foundations of transition state theory 149<br />
7.1.2 Formal kinetic model<br />
In order to derive the TST expression for ( ), wefirst look at a simple formal kinetic<br />
model. This formal kinetic model has its roots in the original idea that the TS can be<br />
considered as some sort of an “activated complex” with a local energy minimum on<br />
the PES. It is not a good derivation of the TST expression for ( ), but leads to the<br />
correct expression.<br />
I<br />
Activated complex model of TST:<br />
y<br />
A+BC 1 À<br />
2<br />
ABC ‡ 3 → P (7.3)<br />
£<br />
ABC<br />
‡ ¤ = 1 [A] [BC]<br />
2 + 3<br />
(7.4)<br />
I Assumption: 3 ¿ 2 y<br />
£ ¤ ABC<br />
‡ = 1<br />
[A] [BC] = <br />
12 [A] [BC] (7.5)<br />
2<br />
y Equilibrium between the reactants (A+BC)andtheTS(ABC ‡ ):<br />
y<br />
[P]<br />
<br />
= 3<br />
£<br />
ABC<br />
‡ ¤ <br />
(7.6)<br />
= 3 | {z 12 [A] [BC] (7.7)<br />
}<br />
=<br />
I<br />
ACT expression for rate constant:<br />
= 3 12 (7.8)<br />
Below, we will find that<br />
and<br />
3 = ‡ = <br />
<br />
<br />
12 ∝ exp<br />
µ− ∆‡<br />
<br />
(7.9)<br />
(7.10)<br />
7.1.3 The fundamental equation for k (T )<br />
The above derivation of the TST rate constant was based on an oversimplified picture,<br />
since the TS is not at a local minimum but at a saddle point on the PES. In this section,<br />
we derive the TST fundamental equation more carefully by bringing in some dynamical<br />
aspects. In (1) - (4), we explore the implications of the quasi-equilibrium assumption,<br />
and in (5) - (9) we develop a dynamical semiclassical model: 42<br />
42 Semiclassical because the translation is handled classically while all other degrees of freedom are<br />
described quantum mechanically.
7.1 Foundations of transition state theory 150<br />
(1) We start by assuming equilibrium between the reactants and products:<br />
A+BC 1 À<br />
2<br />
ABC ‡ 3<br />
À<br />
4<br />
P (7.11)<br />
(2) We consider two dividing surfaces at the TS separated by a distance of ≤ .<br />
is the “length” of a “phase space cell” in semiclassical theory which is determined<br />
by Heisenberg’s uncertainty principle:<br />
∆ ‡ × ∆ ‡ = × ∆ ‡ = (7.12)<br />
(3) We now consider the number densities of molecules passing through these surfaces<br />
from left to right ( −→ ‡ )andfromrighttoleft( ←−<br />
‡ ). The total number density ‡<br />
at the TS in equilibrium is their sum, i.e.,<br />
‡ = −→ ‡ + ←−<br />
‡ = ‡ A BC (7.13)<br />
In equilibrium, we must have<br />
Thus,<br />
−→<br />
‡ = ←−<br />
‡ (7.14)<br />
−→<br />
‡ = ←−<br />
‡ = 1 2 ‡ = ‡ <br />
2 A BC (7.15)<br />
(4) We now suddenly remove all the products. Then ←−<br />
‡ =0 However, there is no<br />
reason for −→ ‡ to change, because the molecules on the reactant side remain in<br />
internal equilibrium (this is the quasi-equilibrium assumption). 43 y<br />
−→<br />
‡ = ‡<br />
2 = ‡ <br />
2 A BC (7.16)<br />
Thus, we can still express −→ ‡ using the equilibrium constant ‡ even if there is<br />
no equilibrium between reactants and products.<br />
(5) We now turn to the reaction rate, which is given by the number density of molecules<br />
−→ ‡ in the time interval that are crossing the DS in the forward direction.<br />
We can write this as<br />
−→ ‡<br />
= −→ −→<br />
<br />
‡ ‡ ‡<br />
=<br />
<br />
‡ (7.17)<br />
where<br />
• −→ ‡ is the decay rate of the TS in the forward direction (or the frequency<br />
factor for the molecules crossing the DS in the forward direction),<br />
43 The quasi-equilibrium assumption for −→ ‡ is made in canonical TST. In a microcanonical derivation<br />
· ·<br />
−→ ←−<br />
of the TST expression for (), we consider the fluxes ‡ and ‡ through the dividing surface<br />
at the TS. Then, the individual molecules “dont’t know” that the products have been removed and<br />
·<br />
·<br />
←−<br />
−→<br />
‡ =0. Since they don’t know about this, the flux ‡ has to remain unchanged.
7.1 Foundations of transition state theory 151<br />
• −→ ‡ = −→ ‡ is the mean speed of the molecules crossing the DS in the forward<br />
direction.<br />
(6) We now use the equilibrium constant to determine ‡ via<br />
‡ = ‡ ABC<br />
A BC<br />
(7.18)<br />
y ‡ = ‡ ABC = ‡ A BC (7.19)<br />
y<br />
−→ −→<br />
‡ <br />
‡<br />
=<br />
‡ A BC = A BC (7.20)<br />
with<br />
−→<br />
<br />
‡<br />
=<br />
‡ (7.21)<br />
(7) From statistical thermodynamics (see Appendix G), we have the following expression<br />
for the equilibrium constant ‡ ( ) in terms of the molecular partition<br />
functions:<br />
‡ ( )=<br />
∗ <br />
=<br />
µ<br />
<br />
exp − ∆ <br />
0<br />
(7.22)<br />
<br />
• The ’s are the respective molecular partition functions (per unit volume).<br />
• ∗ is written with respect to the zero-point level of the reactants.<br />
• The exp (−∆ 0 ) factor arises subsequently, because we now express<br />
the value of partition function for the TS relative to the zero-point level of<br />
the TS ( ∗ = exp (−∆ 0 )).<br />
(8) Separating the motion along the reaction coordinate ‡ from the other degrees of<br />
freedom, we write<br />
= ‡ ‡ <br />
(7.23)<br />
where ‡ is the 1-D (translational) partition function describing the translational<br />
motion of the molecules along ‡ across the TS and ‡ <br />
is the partition function<br />
for all 3 − 7 other (rotational-vibrational-electronic) degrees of freedom.<br />
Here,<br />
• the 1-D translational partition function is<br />
‡ =<br />
µ 2 <br />
2 12<br />
× (7.24)<br />
• the mean speed along ‡ for crossing the DS in the forward direction is given<br />
by<br />
R<br />
−→ ∞<br />
‡ −2 2 µ <br />
<br />
12<br />
0<br />
= R ∞<br />
−∞ −2 2 = <br />
(7.25)<br />
2
7.1 Foundations of transition state theory 152<br />
(9) Collecting and inserting the results into<br />
−→<br />
<br />
‡<br />
( )=<br />
( ) (7.26)<br />
we obtain<br />
( )= 1 µ 12 µ 12 µ<br />
2 <br />
×<br />
× ‡ <br />
exp − ∆ <br />
0<br />
(7.27)<br />
2<br />
2<br />
<br />
y<br />
( )= ‡ µ<br />
<br />
exp − ∆ <br />
0<br />
(7.28)<br />
<br />
I Result: In order to account () forsomerecrossingoftheTSand() for quantum<br />
mechanical tunneling, we add an additional factor called the transmission coefficient.<br />
For well-behaved reactions, ≈ 1. Thus, we end up with the fundamental equation<br />
of TST in the form:<br />
( )= <br />
<br />
‡ µ<br />
<br />
exp − ∆ <br />
0<br />
<br />
Note that ‡ <br />
is the partition function of the TS excluding the RC.<br />
Equation 7.29 allows us to calculate ( ) using the following data:<br />
(7.29)<br />
• The threshold energy for the reaction ∆ 0 (or, per mol, ∆ 0 ). ∆ 0 has to be<br />
obtained from ab initio quantum chemistry calculations, and<br />
• structural information on the reactants and the TS (i.e., bond lengths and angles,<br />
vibrational frequencies).<br />
I Figure 7.4: Illustration of ∆ ‡ , ∆ 0 ,and .
7.1 Foundations of transition state theory 153<br />
∆ ‡ : Born-Oppenheimer potential energy barrier (7.30)<br />
∆ 0 : zero-point corrected threshold energy for the reaction (7.31)<br />
: Arrhenius activation energy (7.32)<br />
∆ 0 is given with respect to the zero-point energies of the reactants and the TS,<br />
= 1 2<br />
X<br />
(7.33)<br />
7.1.4 Tolman’s interpretation of the Arrhenius activation energy<br />
I We now understand the difference between E and ∆E 0 :<br />
2 ln ( )<br />
= <br />
<br />
Ã<br />
= 2 <br />
ln <br />
<br />
<br />
‡ µ<br />
<br />
exp<br />
<br />
= ∆ 0 + + 2 ln ‡ <br />
−<br />
| {z }<br />
=h reacting moleculesi<br />
− ∆ 0<br />
<br />
(7.34)<br />
! (7.35)<br />
µ<br />
2 ln <br />
+ 2 ln <br />
<br />
(7.36)<br />
<br />
<br />
| {z }<br />
=h all molecules i<br />
• ∆ 0 is difference of the zero-point levels of the reactants and the TS,<br />
• the term is the mean translational energy in the RC (from the factor),<br />
• the 2 ln terms are the internal (vibration-rotation) energies of the TS and<br />
<br />
the reactants, respectively. y<br />
I Tolman’s interpretation of E (see Fig. 7.4):<br />
= h reacting molecules i − h all molecules i (7.37)
7.2 Applications of transition state theory 154<br />
7.2 Applications of transition state theory<br />
( )= <br />
<br />
‡ µ<br />
<br />
exp − ∆ <br />
0<br />
<br />
(7.38)<br />
TST has gained enormous importance in all areas of physical chemistry because it<br />
provides the basis for understanding – and/or calculating –<br />
• quantitative values of preexponential factors,<br />
• ∆ 0 from measured values,<br />
• -dependence of and deviations from Arrhenius behavior,<br />
• gas phase and liquid phase reactions (proton transfer, electron transfer, organic<br />
reactions, reactions of ions in solutions),<br />
• pressure dependence of reaction rate constants (activation volumes),<br />
• kinetic isotope effects,<br />
• structure-reactivity relations (e.g., Hammett relations in organic chemistry),<br />
• features of biological reactions (enzyme reactions),<br />
• heterogeneous reactions (heterogeneous catalysis, electrode kinetics), . . .<br />
7.2.1 The molecular partition functions<br />
In order to evaluate Eq. 7.29 for a given reaction, we need, besides ∆ 0 , the molecular<br />
partition functions for the reactants and the TS. We summarize only the main points<br />
here; further information is given in Appendix G.<br />
I<br />
Definition:<br />
= P − <br />
(7.39)<br />
I<br />
<strong>Physical</strong> interpretation:<br />
• The molecular partition function is a number which describes how many states<br />
are available to the molecule at a given temperature .<br />
• The ratio<br />
‡ <br />
(7.40)<br />
<br />
in Eq. 7.29 is thus the ratio of the number of states available to the TS and<br />
‡ <br />
the reactant molecules. Since each state is equally likely, is the relative<br />
<br />
probability of the TS vs. the probability of the reactants.<br />
Note again that ‡ <br />
motion along the RC.<br />
is the partition function of the TS excluding the translational
7.2 Applications of transition state theory 155<br />
I Separation of degrees of freedom: For separable degrees of freedom, we can write<br />
the molecular partition function as<br />
= × × × (7.41)<br />
Nuclear spin ( ) may have to be taken into account in some cases as well.<br />
I<br />
Table 7.1: Molecular partition functions.<br />
type of motion energy partition function magnitude<br />
µ <br />
translation (1-D) 2 2<br />
12 2 <br />
10 8 × <br />
8 2<br />
<br />
µ µ 2 <br />
translation (3-D) 2 <br />
2 32<br />
<br />
+ 2 <br />
+ 2 2 <br />
10 24 × <br />
8 2 2 2 <br />
µ 2 <br />
rotation (1-D) a ~ 2<br />
12<br />
2 2<br />
2 <br />
10<br />
µ ~ 2 <br />
rotation (2-D) b ~ 2<br />
2 <br />
( +1)<br />
10 2<br />
2 ~ 2 <br />
µ <br />
rotation (3-D) c 12<br />
32 2 <br />
or <br />
(<br />
~ 2 ) 12 10 3 10 4<br />
vibration d [1 − exp (− )] −1 1 10<br />
electronic <br />
P exp (− ) 1<br />
a Free internal rotor. For a hindered rotor (torsional vibrational mode), the partition function has to<br />
be determined by an explicit calculation over the hindered rotor quantum states.<br />
b Linear molecule.<br />
c Nonlinear molecule.<br />
d For each harmonic oscillator degree of freedom measured from zero-point level.
7.2 Applications of transition state theory 156<br />
7.2.2 Example 1: Reactions of two atoms A + B → A···B ‡ → AB<br />
This example is somewhat artifical unless there is some mechanism to remove the bond<br />
energy (for example, in associative ionization, A + B → A···B ‡ → AB + + − ).<br />
‡ µ<br />
<br />
exp − ∆ <br />
0<br />
<br />
Ã<br />
! Ã !<br />
= ‡ <br />
‡ <br />
( )( ) 1<br />
<br />
= µ<br />
2 ( + ) 2<br />
<br />
2 2 <br />
µ<br />
× exp − ∆ <br />
0<br />
<br />
= µ<br />
<br />
2<br />
+ <br />
2 <br />
µ<br />
× exp − ∆ <br />
0<br />
<br />
( )= <br />
<br />
<br />
µ<br />
exp − ∆ <br />
0<br />
<br />
2<br />
2 <br />
32 µ 8 2 ‡ <br />
2<br />
32<br />
Ã<br />
!<br />
8 2 ( + ) 2<br />
2 + <br />
y<br />
µ 12 µ<br />
8 <br />
( )=<br />
( + ) 2 exp − ∆ <br />
0<br />
<br />
<br />
This result is identical to that obtained by the LC model in collision theory!<br />
(7.42)<br />
(7.43)<br />
(7.44)<br />
(7.45)<br />
(7.46)
7.2 Applications of transition state theory 157<br />
7.2.3 Example 2: The reaction F + H 2 → F···H···H ‡ → FH + H<br />
I<br />
Exercise 7.1: Evaluatetherateconstantforthereaction<br />
F+H 2 → F ···H ···H ‡ → FH + H (7.47)<br />
using the data in Table 7.2 at =200, 300, 500, 1000 and 2000 K, determinethe<br />
Arrhenius parameters, and compare the results with the experimental value of<br />
=2× 10 14 exp (−800 )cm 3 mol s (7.48)<br />
¤<br />
I Table 7.2: Properties of the reactants and transition state of the reaction F + H 2 .<br />
parameter F ···H ···H ‡ a F H 2<br />
2 (F ···H) ( Å) 1602<br />
1 (H ···H) ( Å) 0756<br />
07417 b<br />
1 (cm −1 ) 40076 43952<br />
2 (cm −1 ) 3979<br />
3 (cm −1 ) 3979<br />
4 (cm −1 ) 3108 c<br />
³ (u) 21014 189984 2016<br />
u Å 7433<br />
0277<br />
1 2<br />
<br />
d<br />
4 4 1<br />
0 ¡ kJ mol −1¢ 657 000<br />
a The transition state FHH ‡ is assumed to be linear.<br />
b = (H-H).<br />
c One imaginary frequency describes the TS along the RC.<br />
d The electronic ground state of F is 2 32 . We neglect the 2 12 spin-orbit component at ( 2 12 )=<br />
404 cm −1 because it does not correlate with the products H + HF. The electronic state of linear<br />
FHH ‡ is 2 Π. The ground state of H 2 is 2 Σ + .<br />
I Solution 7.1:<br />
( )= ‡ µ<br />
<br />
exp − ∆ <br />
0<br />
2 <br />
Ã<br />
!<br />
= ‡ <br />
( )( 2 )<br />
µ<br />
× exp − ∆ <br />
0<br />
<br />
<br />
à !<br />
‡ <br />
2<br />
<br />
à !<br />
‡ <br />
2<br />
<br />
(7.49)<br />
à !<br />
‡ <br />
2<br />
<br />
(7.50)
7.2 Applications of transition state theory 158<br />
The different factors can be evaluated separately:<br />
Ã<br />
!<br />
‡ <br />
µ 32 µ <br />
+ 2 <br />
2 32<br />
=<br />
(7.51)<br />
( )( 2 )<br />
2 2 <br />
<br />
à !<br />
‡ µ <br />
<br />
8 2 ‡ 2<br />
‡<br />
=<br />
= <br />
2 8 2 2 2 2 2 (7.52)<br />
2<br />
<br />
à !<br />
‡ <br />
1 − exp (− 2 )<br />
=<br />
(7.53)<br />
2<br />
3Y<br />
´i<br />
<br />
à !<br />
‡ <br />
2<br />
<br />
=1<br />
= 4<br />
4 × 1<br />
h<br />
1 − exp<br />
³<br />
− ‡ <br />
(7.54)<br />
Note: The difference between the TST result and experiment can be ascribed (a)<br />
to tunneling and (b) deficiencies of the ab initio PES. Indeed, according to newer ab<br />
initio calculations, the TS is actually slightly bent and the TS parameters differ slightly,<br />
yielding better agreement with experiment. ¥
7.2 Applications of transition state theory 159<br />
7.2.4 Example 3: Deviations from Arrhenius behavior (T dependence of k (T ))*<br />
I<br />
Exercise 7.2: The temperature dependence of bimolecular gas phase reactions of the<br />
type A + B → productscanusuallybeexpressedintheform<br />
Using the TST expression for ( )<br />
( )= × × exp (−∆ 0 ) (7.55)<br />
( )= <br />
<br />
‡ µ<br />
<br />
exp<br />
<br />
− ∆ 0<br />
<br />
<br />
, (7.56)<br />
determine the values of for the types of reactions in Table 7.3, assuming that the<br />
vibrational degrees of freedom are not excited. ¤<br />
I Table 7.3:<br />
A + B → TS ‡ <br />
transl. rot. <br />
<br />
atom atom linear TS 1 32 1<br />
+05<br />
32 32 1<br />
atom linear molecule linear TS 1 −32 1<br />
−05<br />
1<br />
atom linear molecule nonlin. TS 1 −32 32<br />
00<br />
1<br />
atom nonlin. molecule nonlin. TS 1 −32 32<br />
−05<br />
32<br />
linear molecule linear molecule linear TS 1 −32 1<br />
−15<br />
1 1<br />
linear molecule linear molecule nonlin. TS 1 −32 32<br />
−10<br />
1 1<br />
linear molecule nonlin. molecule nonlin. TS 1 −32 32<br />
−15<br />
1 32<br />
nonlin. molecule nonlin. molecule nonlin. TS 1 −32 32<br />
−20<br />
32 <br />
32<br />
I Solution 7.2: See Table 7.3. ¥<br />
Using the above results, we can recast ( ) and determine the following expression for<br />
the Arrhenius activation energy, with as above:<br />
= ∆ 0 + + 2 ln ‡ <br />
<br />
− 2<br />
X<br />
Reactants<br />
ln reactants<br />
<br />
<br />
(7.57)<br />
We see that the value for depends on the vibrational frequencies of the TS and the<br />
reactants. Often, the frequencies of the reactants are very high so that reactant<br />
=1.<br />
However, the TS may have soft vibrations so that ‡ <br />
may approach the classical value<br />
=[1− exp (− )] −1 → <br />
for →∞ (7.58)<br />
<br />
Thus, each “soft” vibrational mode in the TS increases by 1, each “soft” vibrational<br />
mode in the reactants decreases by 1.
7.3 Thermodynamic interpretation of transition state theory 160<br />
7.3 Thermodynamic interpretation of transition state theory<br />
7.3.1 Fundamental equation of thermodynamic transition state theory<br />
As shown above,<br />
( )= <br />
‡ (7.59)<br />
<br />
‡ can often be interpreted using thermodynamic arguments, i.e., ∆ ‡ , ∆ ‡ ,and<br />
∆ ‡ . For liquid phase reactions, this is straightforward. For gas phase reactions, we<br />
have to take into account the difference between ‡ and .<br />
‡<br />
a) Thermodynamic transition state theory for unimolecular gas phase reactions<br />
For unimolecular isomerization and dissociation reactions of the type<br />
or<br />
we have<br />
ABC → ABC ‡ → ACB ‡ (7.60)<br />
ABC → ABC ‡ → A+BC ‡ , (7.61)<br />
£ ¤<br />
ABC<br />
‡<br />
‡ ( )=<br />
[ABC]<br />
= ABC ‡<br />
ABC<br />
= ‡ ( ) (7.62)<br />
‡ ( )= ‡ ( )= ‡ ( ) is dimensionless (∆ ‡ =0) and we do not have to worry<br />
about a difference between ‡ ( )= ‡ ( ) and about units or different standard<br />
states.<br />
I Results: Eq. 7.59 thus gives the following results:<br />
• Rate constant:<br />
y<br />
( )= <br />
‡ ( )= µ <br />
<br />
exp − ∆‡<br />
<br />
( )= µ <br />
<br />
∆<br />
‡<br />
exp exp<br />
µ− ∆‡<br />
<br />
<br />
(7.63)<br />
(7.64)<br />
• Arrhenius activation energy:<br />
y<br />
= 2 ln <br />
<br />
= 2 ∙ µ<br />
ln ∆<br />
‡<br />
exp <br />
<br />
exp<br />
¸<br />
µ− ∆‡<br />
<br />
(7.65)<br />
<br />
= ∆ ‡ + + 2 ∆‡<br />
<br />
= ∆ ‡ + + ∆‡ <br />
<br />
(7.66)<br />
(7.67)<br />
If ∆ ‡ 6= ( ) in the temperature range of interest, we obtain<br />
= ∆ ‡ + (7.68)
7.3 Thermodynamic interpretation of transition state theory 161<br />
• Arrhenius preexponential factor:<br />
∆ ‡ = − (7.69)<br />
y<br />
y<br />
( )= µ µ<br />
∆<br />
‡<br />
exp exp − <br />
<br />
<br />
= µ <br />
∆<br />
‡<br />
exp <br />
(7.70)<br />
(7.71)<br />
b) Thermodynamic transition state theory for bimolecular gas phase reactions<br />
I Conversion from K<br />
‡ (T ) to K‡ <br />
(T ): For bimolecular reactions<br />
A+BC→ ABC ‡ , (7.72)<br />
we have to account for the different numbers of species ∆ ‡ = ‡ − reactants :<br />
We therefore have to convert from ‡ ( ) to ‡ ( ) via<br />
∆ ‡ = −1 . (7.73)<br />
£ ¤ <br />
ABC<br />
‡ ABC<br />
‡ ABC ‡ ª<br />
‡ ( )=<br />
[A] [BC] = <br />
A BC<br />
= <br />
A ª BC ª × 1<br />
µ −∆<br />
‡<br />
<br />
= ª ‡ ( ) ×<br />
.<br />
ª<br />
<br />
(7.74)<br />
Here, ‡ ( ) is conveniently given in units of lmol −1 . 44 However, , ‡ ∆ ‡ , ∆ ‡ and<br />
∆ ‡ arerelatedtothestandardstateat ª =1bar. 45 To keep track of units in the<br />
conversion , it is useful to apply in<br />
lbar by writing<br />
mol K<br />
=83145<br />
J<br />
mol K =83145 m3 Pa<br />
lbar<br />
=0083145<br />
mol K mol K = 0 (7.76)<br />
44 With ‡ ( ) in units of ¡ mol l −1¢ ∆<br />
, we get as close as possible to the standard state ∗ =<br />
1molkg −1 for reactions in aqueous solution.<br />
45 We recall that the thermodynamic equilibrium constant ( )=exp(−∆ ª ) referred to<br />
the standarrd state at ª =1baris dimensionless. Instead of the dimensionless ‡ ( ), onemay<br />
also decide to use the equilibrium constant in pressure units ‡ 0 :<br />
hABC ‡i ABC ‡<br />
‡ ( )=<br />
[A] [BC] = <br />
A BC<br />
= ‡ 0 ( ) × 0 (7.75)
7.3 Thermodynamic interpretation of transition state theory 162<br />
so that, with ∆ ‡ = −1, weobtain<br />
Thus,viaEq.7.59:<br />
‡ ( )=<br />
( )= <br />
<br />
ABC ‡ ª<br />
A ª<br />
<br />
<br />
BC ª<br />
<br />
× 1<br />
µ 0<br />
= <br />
ª ‡ ( ) × . (7.77)<br />
ª<br />
0 <br />
ª ‡ ( )= µ<br />
0 <br />
exp ª<br />
− ∆ª‡<br />
<br />
<br />
, (7.78)<br />
in units of<br />
µ µ <br />
0 <br />
dim ( ( )) = dim × dim = l<br />
(7.79)<br />
<br />
ª mol s<br />
and with the standard state for the thermodynamics quantities explicitly indicated by<br />
the ª symbol.<br />
I<br />
Results:<br />
• Rate constant:<br />
( )= <br />
<br />
µ <br />
0 ∆<br />
ª‡<br />
exp exp<br />
ª <br />
µ− ∆ª‡<br />
<br />
<br />
(7.80)<br />
• Arrhenius activation energy:<br />
= 2 ln <br />
<br />
(7.81)<br />
If ∆ ª‡ 6= ( ) in the temperature range of interest:<br />
= ∆ ª‡ +2 (7.82)<br />
• Arrhenius preexponential factor:<br />
∆ ª‡ = − 2 (7.83)<br />
y<br />
( )= <br />
<br />
= 2 <br />
<br />
µ <br />
0 ∆<br />
ª‡<br />
exp ª <br />
µ<br />
0 ∆<br />
ª‡<br />
exp ª <br />
µ<br />
exp<br />
<br />
exp<br />
− <br />
− 2<br />
<br />
µ<br />
− <br />
<br />
<br />
(7.84)<br />
(7.85)<br />
Comparison with the Arrhenius expression = exp (− ) thus gives<br />
= 2 <br />
<br />
µ <br />
0 ∆<br />
ª‡<br />
exp ª <br />
(7.86)
7.3 Thermodynamic interpretation of transition state theory 163<br />
• Note: and are given above in units of lmol −1 s −1 (we may denote this by<br />
writing them as and ), while ∆ ª‡ and ∆ ª‡ used above are referred to<br />
the standard state for gases of ª =1bar.<br />
<strong>—</strong> For ideal gases, ∆ ‡ and ∆ ‡ are related via<br />
∆ ª‡ = ∆ ª‡ + ∆ ‡ ( )=∆ ª‡ − = ∆ +‡ − (7.87)<br />
<strong>—</strong> Likewise, ∆ ª‡ and ∆ +‡ for are related via<br />
∆ +‡ − ∆ ª‡ =<br />
y<br />
Z <br />
+<br />
ª<br />
Z<br />
<br />
+<br />
=<br />
ª<br />
<br />
= Z <br />
+<br />
ª<br />
<br />
= Z <br />
+<br />
ª<br />
∆ ‡ <br />
(7.88)<br />
<br />
= ∆ ‡ +<br />
ln<br />
+ 0 = +<br />
ª ∆‡ ln<br />
+ 0 = ª<br />
ª ∆‡ ln (7.89)<br />
+ 0 <br />
= ∆ ‡ ln + 0 <br />
(7.90)<br />
ª<br />
with ª =1barand + = +<br />
+ =1moll−1 .<br />
∆ ª‡ = ∆ +‡ − ln + 0 <br />
ª (7.91)<br />
c) Thermodynamic transition state theory for reactions in the liquid phase<br />
For reactions in condensed phases,<br />
∆ ≈ ∆ (7.92)<br />
and<br />
y<br />
£<br />
ABC<br />
‡ ¤<br />
‡ ( ) =<br />
[A] [BC] = ¡ +¢ ∆ ‡ exp<br />
≈ ¡ +¢ <br />
∆ ‡ exp<br />
µ− ∆+‡<br />
exp<br />
<br />
• for unimolecular reactions:<br />
( )= µ<br />
<br />
exp<br />
• for bimolecular reactions:<br />
( )= µ<br />
<br />
+ exp<br />
− ∆+‡<br />
µ− ∆+‡<br />
<br />
− ∆+‡<br />
<br />
<br />
<br />
µ− ∆+‡<br />
<br />
exp<br />
<br />
exp<br />
<br />
<br />
+‡<br />
∆<br />
exp<br />
µ−<br />
<br />
<br />
µ− ∆+‡<br />
<br />
µ− ∆+‡<br />
<br />
<br />
<br />
(7.93)<br />
(7.94)<br />
(7.95)<br />
(7.96)<br />
Note: The conversion between values for ∆ +‡ and ∆ ª‡ and, likewise, between<br />
values for ∆ +‡ and ∆ ª‡ for a given reaction in the liquid and gas phase requires<br />
some caution. 46<br />
46 See,e.g.,D.M.Golden,J.Chem.Ed.48, 235 (1971).
7.3 Thermodynamic interpretation of transition state theory 164<br />
7.3.2 Applications of thermodynamic transition state theory<br />
a) Comparison of ∆S ‡ -values for different types of transition states<br />
I<br />
Bimolecular reactions:<br />
A+BC→ ABC ‡ (7.97)<br />
y<br />
∆ ‡ 0 (7.98)<br />
or even<br />
∆ ‡ ¿ 0 (7.99)<br />
Furthermore, for chemically otherwise similar molecules<br />
(nonlinear molecule) (linear molecule) À (cyclic molecule) (7.100)<br />
Thus,<br />
¯<br />
¯∆ ‡ (nonlinear molecule)¯¯ ¯¯∆ ‡ (linear molecule)¯¯ ¿ ¯¯∆ ‡ (cyclic molecule)¯¯<br />
(7.101)<br />
For example:<br />
⎛<br />
A ··· B<br />
∆ ‡ ⎜<br />
⎝<br />
¯<br />
y<br />
...<br />
⎞ ‡ ⎛<br />
‡ ⎞ ⎛<br />
⎟ A ··· B<br />
<br />
¯<br />
C<br />
⎠¯¯¯¯¯¯¯ ¯∆‡ ⎝ A ··· B ··· C ⎠¯<br />
⎞ ‡ ¯ ¿<br />
∆ ‡ ⎜ ⎟<br />
⎝<br />
. .<br />
¯ C ··· D<br />
⎠¯¯¯¯¯¯¯<br />
(7.102)<br />
cyclic ¿ linear nonlinear (7.103)<br />
I<br />
Unimolecular reactions:<br />
ABC → ABC ‡ (7.104)<br />
• For isomerization reactions with a tight transition state:<br />
∆ ‡ ⎛<br />
⎝<br />
• For bond fission reactions with a loose transition state:<br />
‡<br />
B<br />
⎞<br />
Á Â ⎠ . 0 (7.105)<br />
A – C<br />
∆ ‡ ¡ AB ···C ‡¢ 0 (7.106)<br />
y<br />
loose tight (7.107)
7.3 Thermodynamic interpretation of transition state theory 165<br />
b) Thermochemical estimation of A-factors*<br />
Equations 7.71 and 7.86 allow us to estimate the -factors of chemical reaction rate<br />
constants by the following procedure (Benson1976):<br />
(1) We setup a model for the TS, with a certain structure and vibrational freuqencies.<br />
(2) We take a reference molecule that is similar to the TS and use the ref of the<br />
reference molecule for a zero-order estimation of ∆ ‡ according to<br />
∆ ‡ = ref −<br />
X<br />
( reactants ) (7.108)<br />
reactants<br />
(3) We apply corrections for the differences between ref and ‡ basedonthestatistical<br />
thermodynamics result of<br />
= ln + ln <br />
(7.109)<br />
<br />
y<br />
µ <br />
∆ cor = ‡ − ref = ln ‡ <br />
‡<br />
+ <br />
ref ln (7.110)<br />
ref<br />
Usually, the correction term accounts for differences of the entropies for translation,<br />
rotation, internal rotations, vibrations, symmetry, electron spin, plus sometimes<br />
other terms (e.g., optical isomers).<br />
I Example 7.1: Estimation of the -factor for the reaction O+CH 4 →<br />
(O ···H ···CH 3 ) ‡ → OH + CH 3 .<br />
∆ ‡ = (O ···H ···CH 3 ) ‡ − (O) − (CH 4 ) (7.111)<br />
ref = (CH 3 F) (7.112)<br />
∆ cor thus depends on a comparison of CH 3 F and OHCH ‡ 3 (see Table ). ¤<br />
I Table 7.4: Estimation of the -factor for the reaction O+CH 4 →<br />
(O ···H ···CH 3 ) ‡ → OH + CH 3 .<br />
Contribution ∆ ‡ (Jmol −1 K −1 )<br />
ª (CH 3F) − ª (O) − ª (CH 4)<br />
−1243<br />
Spin: + ln 3 + 91<br />
external rotation: 1 3 ln ( 1 2 3 ) ‡<br />
( 1 2 3 ) ref + 79<br />
translation − 04<br />
CF =1000cm −1 → − 04<br />
OH =2000cm −1 + 00<br />
2 OHC ()<br />
³<br />
=600cm −1 ´<br />
+ 42<br />
∆ ‡ = ª <br />
O ···H ···CH ‡ 3 − ª (O) − ª (CH 4) −1039<br />
Correction ∆ ‡ → ∆ ‡ : + ln ( 0 ) y ∆ ‡ = −198 Jmol −1 K −1 .
7.3 Thermodynamic interpretation of transition state theory 166<br />
Estimated -value:<br />
= 2 µ <br />
∆<br />
‡<br />
exp (7.113)<br />
<br />
µ −197Jmol −1<br />
=739 × 625 × 10 12 K −1 <br />
× exp<br />
831 J mol −1 K −1 cm 3 mol −1 s −1 (7.114)<br />
=43 × 10 12 cm 3 mol −1 s −1 (7.115)<br />
Experimental -value:<br />
expt =19 × 10 12 cm 3 mol −1 s −1 (7.116)<br />
The difference is probably due to a poor estimate of the TS structure <strong>—</strong> one can do<br />
much better, I just didn’t try hard enough here. 47<br />
7.3.3 Kinetic isotope effects<br />
I<br />
Figure 7.5: The kinetic isotope effect due to the difference in the zero-point energies<br />
of the reactants and the respective TS structures.<br />
47 For H abstraction reactions from a series of hydrocarbons by 3 CH 2 , the agreement that was achieved<br />
was better than ±30 %.
7.3 Thermodynamic interpretation of transition state theory 167<br />
Kinetic isotope effects occur in particular in the case of H/D atom transfer reactions.<br />
The reason is that the threshold energy ∆ ‡ 0 depends on the vibrational zero-point<br />
energies.<br />
We consider the reactions<br />
where<br />
A+H<strong>—</strong>R → A<strong>—</strong>H<strong>—</strong>R ‡ → products (7.117)<br />
A+D<strong>—</strong>R → A<strong>—</strong>D<strong>—</strong>R ‡ → products (7.118)<br />
∆ ‡ 0 (H<strong>—</strong>R) = ∆ ‡ + ‡ (A<strong>—</strong>H<strong>—</strong>R) − (H<strong>—</strong>R) (7.119)<br />
∆ 0(D<strong>—</strong>R) ‡ = ∆ ‡ + ‡ (A<strong>—</strong>D<strong>—</strong>R) − (D<strong>—</strong>R) (7.120)<br />
with<br />
y<br />
= <br />
1<br />
2<br />
X<br />
(7.121)<br />
<br />
∆ ‡ 0(DH) = ∆‡ 0(D<strong>—</strong>R) − ∆ 0(H<strong>—</strong>R) ‡ (7.122)<br />
=+ ‡ (A<strong>—</strong>D<strong>—</strong>R) − ‡ (A<strong>—</strong>H<strong>—</strong>R) − ( (D<strong>—</strong>R) − (H<strong>—</strong>R))<br />
(7.123)<br />
y<br />
à !<br />
(D<strong>—</strong>R)<br />
(H<strong>—</strong>R) =exp − ∆‡ 0(D/H)<br />
<br />
(7.124)<br />
I<br />
Example 7.2: R-H/R-D substitution. The vibrational frequencies depend on the force<br />
constants and reduced masses :<br />
= 1 µ 12 <br />
(7.125)<br />
2 <br />
For H/D substitution:<br />
(HR)<br />
(DR) ≈ 1 (7.126)<br />
2<br />
y<br />
µ 12<br />
(HR) (HR)<br />
(DR) = ≈ √ 2 (7.127)<br />
(DR)<br />
With (HR) = 3000 cm −1 and (DR) = 2100 cm −1 , we find ∆ =<br />
1<br />
2 (3000 − 2100) cm−1 =450cm −1 , by which the DR reactant is stabilized compared<br />
to the HR reactant. For =300K:<br />
µ<br />
(DR)<br />
(HR) =exp − 450 <br />
≈ −225 ≈ 01 (7.128)<br />
200<br />
For more realistic estimates, one has to take into account the change of all vibrational<br />
frequencies in the reacting molecules and the TS.<br />
¤
7.3 Thermodynamic interpretation of transition state theory 168<br />
7.3.4 Gibbs free enthalpy correlations<br />
I<br />
Figure 7.6: Linear free enthalpy correlations.<br />
Application of the TST expression<br />
= µ <br />
<br />
exp − ∆‡<br />
(7.129)<br />
<br />
to a series of related reactions (1) and (2) often gives a relation of the type 48<br />
∆ ‡ 1 − ∆ ‡ 2 = ¡ ¢<br />
∆ ª 1 − ∆ ª 2<br />
(7.130)<br />
y<br />
ln 1<br />
= ln 1<br />
(7.131)<br />
2 2<br />
7.3.5 Pressure dependence of k<br />
Liquid phase reactions often show a pressure dependence of the rate constant. Using<br />
the relation<br />
µ <br />
= (7.132)<br />
<br />
<br />
we can predict this pressure dependence. Starting with<br />
= µ <br />
<br />
exp − ∆‡<br />
(7.133)<br />
<br />
we find<br />
µ ln <br />
= − ∆ ‡<br />
(7.134)<br />
<br />
<br />
48 A well-known example is the Hammett correlation in organic chemistry.
7.3 Thermodynamic interpretation of transition state theory 169<br />
I<br />
Activation volume:<br />
∆ ‡ (7.135)<br />
I Example 7.3: Comparison of the pressure dependencies of the rate constants for 1<br />
and 2 reactions:<br />
(CH 3<br />
) 3<br />
CCl + H 2 O → (CH 3<br />
) 3<br />
C + +Cl − +H 2 O → ∆ ‡ 0<br />
y ↑ y ↓ 1<br />
HO − +CH 3 I<br />
⎡<br />
⎤<br />
|<br />
→ ⎣ HO ···C ···I ⎦<br />
ÁÂ<br />
−<br />
→ ∆ ‡ 0<br />
y ↑ y ↑ 2<br />
¤
7.4 Transition state spectroscopy 170<br />
7.4 Transition state spectroscopy<br />
Until a few years ago, the “transition state” was a rather “elusive and strange” species.<br />
This situation has completely changed (a) because of the rapidly improving quantum<br />
chemical methods for computing TS properties and (b) because of transition state<br />
spectroscopy, in the frequency domain (J. Polanyi, Kinsey), and in the time domain<br />
(Zewail).<br />
I<br />
Figure 7.7: Transition state spectroscopy: The reaction K + NaCl.<br />
I<br />
Figure 7.8: Transition state spectroscopy: Raman emission of dissociating molecules.
7.4 Transition state spectroscopy 171<br />
I Figure 7.9: Transition state spectroscopy: The reaction F + H 2 .<br />
I<br />
Figure 7.10: Transition state spectroscopy: NaI Photodissociation.
7.4 Transition state spectroscopy 172<br />
I<br />
Figure 7.11: Transition state spectroscopy: NaI Photodissociation.<br />
I<br />
Figure 7.12: Transition state spectroscopy: Reaction a in van-der-Waals Complex.
7.4 Transition state spectroscopy 173<br />
I<br />
Figure 7.13: Transition state spectroscopy: Stimulated Emission Pumping.
7.4 Transition state spectroscopy 174<br />
7.5 References<br />
Benson 1976 S. W. Benson, Thermochemical <strong>Kinetics</strong>, Methods for the Estimation<br />
of Thermochemical Data and Rate Parameters, 2nd Ed., Wiley, New York, 1976.<br />
Evans 1935 M. G. Evans, M. Polanyi, Trans. Faraday Soc. 31, 875 (1935).<br />
Eyring 1935 H. Eyring, J. Chem. Phys. 3, 107 (1935).<br />
Herschbach 1987 D. R. Herschbach, Molekulardynamik einfacher chemischer Reaktionen,Angew.Chem.99,<br />
1251 (1987).<br />
Lee 1987 Y. T. Lee, Molekularstrahluntersuchungen chemischer Elementarprozesse,<br />
Angew. Chem. 99, 967 (1987).<br />
Polanyi 1987 J. C. Polanyi, Konzepte der Reaktionsdynamik, Angew. Chem. 99, 981<br />
(1987).<br />
Wigner 1938 E. Wigner, Trans. Faraday Soc. 34, 29 (1938).
7.5 References 175<br />
8. Unimolecular reactions<br />
Unimolecular reactions are reactions in which only one species experiences chemical<br />
change.<br />
I<br />
Figure 8.1: Examples of unimolecular reactions.<br />
8.1 Experimental observations<br />
(1) There are many reactions governed by a first-order rate law,<br />
[A]<br />
= − ∞ [A] , (8.1)<br />
<br />
for example isomerization and dissociation reactions like<br />
Cyclopropane → Propene (8.2)<br />
-Butene → -Butene (8.3)<br />
C 2 H 6 → 2CH 3 (8.4)<br />
Cycloheptatriene → Toluene (8.5)<br />
(2) In these reactions, no other species than the reactants experience any chemical<br />
change.<br />
(3) These reactions have the same rates in the gas phase and in solution,<br />
gas phase ≈ solution phase , (8.6)<br />
i.e., the environment has no effect on the reaction rates!
8.1 Experimental observations 176<br />
(4) There are many similar other reactions, however, for which the rate law is secondorder,<br />
i.e.,<br />
[A]<br />
= − 0 [A] [M] (8.7)<br />
<br />
Examples are<br />
Br 2 → Br + Br (8.8)<br />
H 2 O 2 → OH + OH (8.9)<br />
(5) More detailed measurements show that at a given temperature the order of the<br />
reaction may depend on pressure (see Fig. 8.2), for instance for the reaction<br />
one has found that<br />
CH 3 NC → CH 3 CN (8.10)<br />
a) for → 0:<br />
[CH 3 NC]<br />
= − 0 [CH 3 NC] [M] (8.11)<br />
<br />
This is called the “low pressure range” for the reaction.<br />
b) for →∞:<br />
[CH 3 NC]<br />
= − ∞ [CH 3 NC] (8.12)<br />
<br />
This is called the “high pressure range” for the reaction”.<br />
(6) The transition from the low to the high pressure range depends<br />
a) onthesizeofthemolecule(seeFig.8.3),<br />
b) for a given molecule, it depends on temperature.<br />
(7) The activation energy in the low pressure regime is much smaller than in the high<br />
pressure regime:<br />
= 2 ln <br />
(8.13)<br />
<br />
where<br />
0 ∞ (8.14)<br />
and<br />
∞ ≈ 0 (8.15)<br />
(8) The preexponential factors in the low pressure regime are much higher than the<br />
collision frequencies, i.e.,<br />
0 (8.16)<br />
(9) For all these reactions, the molecules must somehow be “energized”. There are<br />
other reactions, in which the molecules are energized chemically (“chemically<br />
activated reactions”) or by photons. At the microscopic, i.e., molecular level,<br />
these reactions should have related mechanisms.
8.1 Experimental observations 177<br />
I<br />
Questions:<br />
(1) What are the differences of the reactions under (1), (4) and (5)?<br />
(2) How can we rationalize their reaction mechanisms?<br />
I<br />
Figure 8.2: Schematic fall-off curve of the unimolecular rate constant.<br />
I<br />
Figure 8.3: Experimentally observed fall-off curves of unimolecular reactions.
8.2 Lindemann mechanism 178<br />
8.2 Lindemann mechanism<br />
First attempts to explain the pressure dependence of unimolecular reaction rate constants<br />
due to radiation and the dissociation dynamics due to centrifugal forces (rotational<br />
excitation) proved to be wrong. In addition, the historic examples for unimolecular reactions,<br />
for instance the N 2 O 5 decomposition, were later proven to have more complex<br />
reaction mechanisms. The first “realistic” model that explained the pressure dependence<br />
of was that proposed by Lindemann (1922).<br />
I<br />
Figure 8.4: Lindemann mechanism.<br />
I<br />
Figure 8.5: Lindemann mechanism: High pressure regime.
8.2 Lindemann mechanism 179<br />
I<br />
Figure 8.6: Lindemann mechanism: Low pressure regime.<br />
I<br />
Figure 8.7: Fall-off curve: The proper way to plot log is vs. log [M]. ab<br />
a<br />
Since 0 ∝ [M], wecanalsoplotlog vs. log 0 . This gives the so-called doubly reduced fall-off<br />
curves.<br />
b In reality, the fall-off regime extends over several orders of magnitude in [M].
8.2 Lindemann mechanism 180<br />
I Half-pressure: We have found the following results:<br />
=<br />
2 1 [M]<br />
−1 [M] + 2<br />
(8.17)<br />
Thus, if −1 [M] = 2 ,wehave<br />
1<br />
∞ = 2<br />
−1<br />
(8.18)<br />
0 = 1 [M] (8.19)<br />
=<br />
2 1 [M]<br />
= 2 1 [M]<br />
−1 [M] + 2 2 −1 [M] = 1 2 1<br />
= 1 2 −1 2 ∞ (8.20)<br />
Half “pressure”:<br />
[M] 12<br />
= 2<br />
−1<br />
(8.21)<br />
I Lifetimes of energized molecules: It is instructive to look at the reciprocal values:<br />
1<br />
−1 [M] = 1 2<br />
(8.22)<br />
1<br />
• is the time between two collisions. This is, in fact, the lifetime of the<br />
−1 [M]<br />
energized molecules with respect to collisional deactivation.<br />
• 1 is the lifetime of the excited molecules with respect to reaction.<br />
2<br />
Thus, at the half pressure, we have<br />
Collisional Deactivation = Unimolecular Decay (8.23)<br />
I Determination of k ∞ :<br />
y<br />
=<br />
2 1 [M]<br />
−1 [M] + 2<br />
(8.24)<br />
1<br />
= −1<br />
+ 1 1<br />
1 2 1 [M]<br />
(8.25)<br />
= 1 + 1 1<br />
∞ 1 [M]<br />
(8.26)<br />
= 1 + 1 ∞ 0<br />
(8.27)<br />
y Plot of −1 vs. [M] −1 should give a straight line with slope 1 −1 and intercept<br />
∞ −1 .
8.2 Lindemann mechanism 181<br />
I<br />
Deficiencies of the Lindemann model:<br />
I Figure 8.8:<br />
• Activation energies:<br />
In particular:<br />
0 ∞ (8.28)<br />
∞ ≈ 0 (8.29)<br />
0 ≈ 0 − ( − 15) (8.30)
8.3 Generalized Lindemann-Hinshelwood mechanism 182<br />
8.3 Generalized Lindemann-Hinshelwood mechanism<br />
8.3.1 Master equation<br />
I<br />
Figure 8.9: Strong collision model.<br />
I<br />
Figure 8.10: Generalized Lindemann-Hinshelwood weak collision model.
8.3 Generalized Lindemann-Hinshelwood mechanism 183<br />
I<br />
Figure 8.11: Generalized Lindemann-Hinshelwood weak collision model.<br />
I<br />
Figure 8.12: Generalized Lindemann-Hinshelwood model: Rate constants in the high<br />
and low pressure (strong and weak collision) limits.
8.3 Generalized Lindemann-Hinshelwood mechanism 184<br />
8.3.2 Equilibrium state populations<br />
As seen from the master equation analysis, the unimolecular reaction rate constant<br />
in the high and in the low pressure regimes depends on the equilibrium (Boltzmann)<br />
state distributions (for discrete states) or () (continuous state distributions). In<br />
a polyatomic molecule, due to the large number of vibrational degrees of freedom, the<br />
number of vibrational states increases very rapidly with increasing vibrational excitation<br />
energy. The quanta of vibrational excitation can be distributed over the oscillators. In<br />
order to calculate the number of combinations, we start with a single oscillator and<br />
expand our scope first to two oscillators and then to =3 − 6 oscillators.<br />
a) Equilibrium state population for a single oscillator:<br />
I<br />
Boltzmann distribution:<br />
with the energy quanta<br />
the degeneracy factor<br />
and the partition function<br />
= − <br />
<br />
(8.31)<br />
= × ; =0 1 2 (8.32)<br />
=<br />
(8.33)<br />
1<br />
1 − − <br />
(8.34)<br />
I Limiting case for high T (transition to classical mechanics): ¿ y<br />
classical<br />
<br />
= <br />
<br />
(8.35)<br />
since<br />
− ≈ 1 − for → 0 (8.36)<br />
Note that this classical description of vibrations is not so bad for unimolecular reactions,<br />
where we are interested in the kinetic behavior at high temperatures.<br />
b) Equilibrium state population for s oscillators:<br />
y<br />
() → () (8.37)<br />
() = () − <br />
(8.38)<br />
<br />
with () being the density of vibrational states at the energy and<br />
=<br />
Y<br />
=1<br />
1<br />
1 − − <br />
(8.39)<br />
and<br />
=3 − 6 (resp. =3 − 5) (8.40)
8.3 Generalized Lindemann-Hinshelwood mechanism 185<br />
8.3.3 The density of vibrational states ρ (E)<br />
Thedensityofvibrationalstatesisdefined as<br />
() =<br />
number of states in energy interval<br />
energy interval<br />
(8.41)<br />
i.e.,<br />
() =<br />
()<br />
<br />
(8.42)<br />
In a polyatomic molecule, as we shall see in the following, () increases with very<br />
rapidly owing to the large number of overtone and combination states, and reaches truly<br />
astronomical values:<br />
a) s =1:<br />
() = 1<br />
<br />
(8.43)<br />
b) s =2:<br />
We consider the ways, in which we can distribute two identical energy quanta between<br />
the two oscillators:<br />
Osc. 1 Osc. 2 Osc. 1 Osc. 2 Osc. 1 Osc. 2 Osc. 1 Osc. 2 <br />
0 - - 1<br />
| - - | 2<br />
2 || - | | - || 3<br />
3 ||| - || | | || - ||| 4<br />
4 5<br />
5 6<br />
.<br />
.<br />
.<br />
.<br />
(8.44)<br />
c) s oscillators:<br />
• For simplicity, we consider a molecule with identical harmonic oscillators.<br />
• This molecule shall be excited with quanta, eachofsize, i.e., the molecule<br />
has the excitation energy = <br />
I<br />
Question: How can we distribute these quanta over the oscillators?
8.3 Generalized Lindemann-Hinshelwood mechanism 186<br />
I<br />
Answer: We are asking for the number of distinguishable possibilities to distribute <br />
quanta between oscillators, (). This is identical to the problem of distributing <br />
balls between boxes, for example for =11and =8:<br />
••• • • • •• • •• (8.45)<br />
That number is identical to the number of distinguishable permutations of balls ( • )<br />
and − 1 walls ( | ),<br />
( + − 1)!<br />
() = (8.46)<br />
!( − 1)!<br />
where ( + − 1)! is the total number of permutations, which is corrected by dividing<br />
through !( − 1)! to account for the number of indistinguishable permutations among<br />
the balls (!) andwalls(( − 1)!) among each other (which give indistinguishable results).<br />
I Answer: We make the following approximation for À : 49<br />
( + − 1)!<br />
!( − 1)!<br />
≈<br />
−1<br />
( − 1)!<br />
(8.49)<br />
I<br />
Question: How can we compute ()?<br />
I<br />
Answer:<br />
() =<br />
()<br />
<br />
(8.50)<br />
(1) We consider the energy interval<br />
∆ = (8.51)<br />
at the excitation energy<br />
= × (8.52)<br />
In this energy interval, we have the number of states<br />
∆() = () (8.53)<br />
y<br />
() =<br />
()<br />
<br />
∆ ()<br />
≈<br />
∆ = 1 ( + − 1)!<br />
!( − 1)!<br />
(8.54)<br />
49 The approximation<br />
( + − 1)!<br />
≈ −1 (8.47)<br />
!<br />
can be derived using Stirling’s formula (ln ! = ln − ) and the power series expansion of<br />
for || ¿1 (see homework assignment).<br />
ln (1 − ) ≈ (8.48)
8.3 Generalized Lindemann-Hinshelwood mechanism 187<br />
(2) Using the approximation<br />
( + − 1)!<br />
!( − 1)!<br />
≈<br />
−1<br />
( − 1)!<br />
(Eq. 8.49) for À gives<br />
() = 1 −1<br />
( − 1)!<br />
(8.55)<br />
(3) Inserting = into (), weobtain<br />
() = 1 () −1<br />
( − 1)!<br />
(8.56)<br />
(4) Result:<br />
() =<br />
−1<br />
( − 1)! () (8.57)<br />
d) Notes:*<br />
(1) The above derivation may seem artificial because in a real molecule not all oscillators<br />
are identical. However, the derivation can be easily generalized.<br />
(2) For instance, we can start from the number of states of the first oscillator 1 ( 1 )<br />
and convolute this with the density of states 2 () of the second oscillator<br />
2 ( 2 )= 2 ( − 1 )= 1<br />
(8.58)<br />
<br />
and compute the total (combined) density of states for two oscillators by convolution<br />
according to<br />
() =<br />
Z <br />
etc. up to oscillators. The result is<br />
0<br />
1 ( 1 ) 2 ( − 1 ) 1 (8.59)<br />
() =<br />
−1<br />
( − 1)! Q <br />
=1 <br />
(8.60)<br />
(3) We can also use inverse Laplace transforms: Since is the Laplace transform<br />
of () according to<br />
=<br />
Z ∞<br />
0<br />
() − = L [ ()] (8.61)<br />
we can determine () from (which we know) by inverse Laplace transformation.<br />
(4) With corrections for zero-point energy:<br />
() =<br />
( + ) −1<br />
( − 1)! Q <br />
=1 <br />
(8.62)
8.3 Generalized Lindemann-Hinshelwood mechanism 188<br />
(5) With (empirical) Whitten-Rabinovitch corrections:<br />
() = ( + () ) −1<br />
( − 1)! Q <br />
=1 <br />
(8.63)<br />
where () is a correction factor that is of the order of 1 except at very low<br />
energies.<br />
(6) Exact values of () for harmonic oscillators can be obtained by direct state<br />
counting algorithms.<br />
(7) Corrections for anharmonicity can be applied using different means.<br />
8.3.4 Unimolecular reaction rate constant in the low pressure regime<br />
From the master equation analysis above, we obtained the thermal unimolecular reaction<br />
rate constant in the low pressure regime as<br />
0 = X X<br />
−1() [M] <br />
(8.64)<br />
<br />
<br />
<br />
The reduction of the excited state populations compared to the equilibrium (Boltzmann)<br />
distribution is shown in Fig. 8.13. Two cases can be distinguished:<br />
(1) With the strong collision assumption (h∆ iÀ), we can apply the so-called<br />
equilibrium theories which assume that = for ≤ 0 . Thus, the state<br />
population below 0 remains as in the high pressure regime, whereas it is reduced<br />
above 0 .Then,<br />
y<br />
0 = X <br />
X<br />
<br />
−1() [M] <br />
<br />
= X <br />
Z ∞<br />
0 = [M]<br />
X<br />
−1() [M] = [M] X<br />
<br />
<br />
(8.65)<br />
0<br />
() (8.66)<br />
Using the expressions for () () and in the classical limit ( ¿ )<br />
derived above and doing the integration by parts, we obain the Hinshelwood<br />
equation for 0 :<br />
µ −1 0 1<br />
0 = [M]<br />
( − 1)! − 0 <br />
(8.67)<br />
This equation is usually used to describe experimental data with as a fit parameter<br />
(usually about half the real ).<br />
Since ∝ 05 and = 2 ln , the low pressure activation energy<br />
becomes<br />
0 = 0 − ( − 15) (8.68)<br />
Thus, may be significantly smaller than 0 . This can be understood as<br />
illustrated in Fig. 8.14.
8.3 Generalized Lindemann-Hinshelwood mechanism 189<br />
(2) With the weak collision assumption (h∆ i ≤ ) made by the so-called nonequilibrium<br />
theories, the bottleneck in the state population falls below the equilibrium<br />
distribution already below 0 (see Fig. 8.14). Using the weak collision<br />
assumption, Troe derived the expression<br />
0 = [M] ( 0 )<br />
<br />
− 0 <br />
(8.69)<br />
where is the so-called collision efficiency ( ≤ 1) whichcanbeexpressedin<br />
terms of the average energy transferred per collision h∆ i. becomes even<br />
smaller than the Hinshelwood above. A simple estimate of the weak collision<br />
effect based on Fig. 8.14 may lead to<br />
0 ≈ 0 − ( − 05) (8.70)<br />
I<br />
Figure 8.13: State populations.
8.3 Generalized Lindemann-Hinshelwood mechanism 190<br />
I<br />
Figure 8.14: Activation energies for unimolecular reactions in the high pressure, the<br />
low pressure strong collision, and the low pressure weak collision limits.<br />
8.3.5 Unimolecular reaction rate constant in the high pressure regime<br />
From the master equation analysis above, the thermal unimolecular reaction rate constant<br />
in the high pressure regime is (see Fig. 8.12)<br />
with the Boltzmann distribution<br />
asshowninFig.8.13.<br />
∞ =<br />
Z ∞<br />
0<br />
() () (8.71)<br />
() = () − <br />
<br />
(8.72)<br />
Averaging over the Boltzmann distribution gives the TST expression<br />
( )= <br />
<br />
‡<br />
− 0 <br />
(8.73)
8.4 The specific unimolecular reaction rate constants () 191<br />
8.4 The specific unimolecular reaction rate constants k (E)<br />
We may intuitively expect that the specific uinimolecular reaction rate constants ()<br />
depend on the energy content (i.e., the excitation energy ) of the energetically activated<br />
molecules molecules A ∗ . More precisely, we will also have to take into account<br />
the dependence of the specific rate constants on other conserved degrees of freedom,<br />
in particular total angular momentum (⇒ ()), and perhaps other quantities<br />
(symmetry, components of if -mixing is weak, (⇒ (; ; ))).<br />
8.4.1 Rice-Ramsperger-Kassel (RRK) theory<br />
I RRK model: In order to derive an expression for the specific rateconstants (),<br />
werealizethatitisnotenoughthatthemoleculesareenergizedtoabove 0 . Indeed,<br />
for the reaction to take place, the energy also has to be in the right oscillator, namely<br />
in the RC. In particular, for the reaction to take place, of the total excitation energy ,<br />
the amount † has to be concentrated in the RC such that † ≥ 0 . Thus, we can<br />
write the reaction scheme as<br />
∗ †<br />
→ † → (8.74)<br />
where † denotes those excited molecules that have enough energy in the RC to overcome<br />
0 . We assume that once this critical configuration ( † ) is reached, the molecules<br />
will immediately cross the TS and go to the product side.<br />
I Probabilities of A † and A ∗ : The probability of † compared to ∗ can be derived<br />
using statistical arguments by considering the ratio of the density of states. We consider<br />
amoleculewith<br />
• oscillators,<br />
• = quanta which have to be distributed over the oscillators,<br />
• a threshold energy of 0 such that a minimum of = 0 quanta have to be<br />
concentrated in the RC and only − quanta can be distributed freely,<br />
• À and − À .<br />
The probability of † relative to ∗ is given by<br />
¡ †¢<br />
( ∗ ) = ¡ †¢<br />
( ∗ )<br />
(8.75)<br />
where<br />
( ∗ ) ∝ ∗ () = 1 ( + − 1)!<br />
(8.76)<br />
!( − 1)!<br />
and, since in the critical configuration only − quanta can be distributed freely,<br />
¡ †¢ ∝ † () = 1 ( − + − 1)!<br />
( − )! ( − 1)!<br />
(8.77)
8.4 The specific unimolecular reaction rate constants () 192<br />
y<br />
¡ †¢<br />
( ∗ )<br />
( − + − 1)! !( − 1)!<br />
= ×<br />
( − )! ( − 1)! ( + − 1)!<br />
( − + − 1)! !<br />
= ×<br />
( − )! ( + − 1)!<br />
(8.78)<br />
(8.79)<br />
I Specific rate constant: The critical configuration † is reached with the rate coefficient<br />
(≡ frequency factor) † (see the reaction scheme above). Since the energy is now<br />
in place (in the RC), † will cross the TS to products.<br />
The specific rateconstant () is therefore simply<br />
y<br />
() = † ×<br />
() = † × ¡ †¢<br />
( ∗ )<br />
( − + − 1)!<br />
( − )!<br />
×<br />
!<br />
( + − 1)!<br />
(8.80)<br />
(8.81)<br />
I Kassel formula: With the approximations according to Eq. 8.49),<br />
( + − 1)!<br />
!<br />
≈ −1 for À (8.82)<br />
and<br />
we obtain<br />
( − + − 1)!<br />
( − )!<br />
¡ †¢<br />
( ∗ )<br />
≈ ( − ) −1 for − À , (8.83)<br />
=<br />
( − )−1<br />
−1 =<br />
µ −1 − <br />
(8.84)<br />
y<br />
¡ †¢ µ −1 −<br />
( ∗ ) = 0<br />
(8.85)<br />
<br />
so that the specific rateconstantbecomes<br />
<br />
This expression is known as the Kassel formula for ().<br />
µ −1 −<br />
() = † 0<br />
(8.86)<br />
<br />
I<br />
Notes on the Kassel formula:<br />
• Owing to the approximations À and − À , the Kassel formula is valid<br />
only at À 0 .
8.4 The specific unimolecular reaction rate constants () 193<br />
• The Kassel formula describes experimental data qualitatively well. However, instead<br />
of the full value of =3 − 6, one usually has to use an effective number<br />
eff . As suggested by Troe, eff can be estimated from the specific heatofthe<br />
reactant molecules via<br />
=<br />
µ <br />
<br />
<br />
<br />
= <br />
<br />
<br />
µ<br />
2 ln <br />
<br />
using the known expression for ,<br />
Y 1<br />
=<br />
1 − exp (− )<br />
=1<br />
<br />
≈ eff (8.87)<br />
(8.88)<br />
• Often, eff is equal to about one half of the number of oscillators,<br />
eff ≈<br />
(3 − 6)<br />
2<br />
(8.89)<br />
I<br />
Figure 8.15: Specific rate constants: RRK formula.<br />
8.4.2 Rice-Ramsperger-Kassel-Marcus (RRKM) theory<br />
RRK theory fails in the following points:<br />
• Itcanbeusedonlywith as an effective parameter ( eff ).<br />
• It leads to the wrong results close to threshold, where → 0 (see the conditions<br />
above).<br />
• It is difficult to derive for oscillators with different frequencies.<br />
• RRK theory is essentially a classical theory. It does not know about zero-point<br />
energy.
8.4 The specific unimolecular reaction rate constants () 194<br />
I RRKM expression: These problems are essentially solved by RRKM theory which<br />
gives the expression<br />
() = ‡ ( − 0 )<br />
()<br />
(8.90)<br />
where<br />
• ‡ ( − 0 ) is the number of open channels (essentially the number of states at<br />
the transition state configuration exluding the RC), counted from the zero-point<br />
level of the TS, at the energy − 0 ,<br />
• () is the density of states of the reacting molecules at the energy .<br />
These quantities are the two critical quantities in microcanonical transition state theory.<br />
I<br />
Figure 8.16: RRKM expression for ().
8.4 The specific unimolecular reaction rate constants () 195<br />
I<br />
Figure 8.17: RRKM expression for () with angular momentum conservation.<br />
I<br />
Figure 8.18: The sum and the density of states.
8.4 The specific unimolecular reaction rate constants () 196<br />
I RRKM expression with angular momentum conservation: Eq. 8.90 can be generalized<br />
to include other conserved quantities, e.g., total angular momentum,<br />
() = ‡ ( − 0 )<br />
()<br />
(8.91)<br />
I Relation between RRKM and RRK theory: To elucidate the relation between Eqs.<br />
8.86 and 8.90, we look at () and ( − 0 ) more closely.<br />
(1) Density of vibrational states ():<br />
() =<br />
number of states in energy interval<br />
energy interval<br />
=<br />
()<br />
<br />
(8.92)<br />
a) RRK result for the number of indistinguishable permutations of quanta<br />
over oscillators; = À :<br />
y<br />
() = 1 ( + − 1)!<br />
!( − 1)!<br />
() =<br />
≈ 1 −1<br />
( − 1)! = 1 () −1<br />
( − 1)!<br />
(8.93)<br />
−1<br />
( − 1)! () (8.94)<br />
b) It is relatively easy to show that for oscillators with different frequencies, the<br />
expression for () becomes<br />
() =<br />
−1<br />
( − 1)! Q <br />
=1 <br />
(8.95)<br />
c) With corrections for zero-point energy:<br />
() =<br />
( + ) −1<br />
( − 1)! Q <br />
=1 <br />
(8.96)<br />
d) With (empirical) Whitten-Rabinovitch correction:<br />
() = ( + () ) −1<br />
( − 1)! Q <br />
=1 <br />
(8.97)<br />
where () is a correction factor that is of the order of 1 except at very<br />
low energies.<br />
e) Exact values of () for harmonic oscillators can be obtained by direct<br />
state counting algorithms. Corrections for anharmonicity can be applied<br />
using different means.
8.4 The specific unimolecular reaction rate constants () 197<br />
(2) Number of states () and ‡ ( − 0 ): Since () = () , the<br />
number of states from =0to can be obtained by integration,<br />
a) RRK expression:<br />
y<br />
() =<br />
Z <br />
0<br />
() =<br />
() =<br />
Z <br />
0<br />
Z <br />
0<br />
() (8.98)<br />
−1<br />
( − 1)! () =<br />
() =<br />
Z<br />
1 1<br />
( − 1)! () −1 <br />
0<br />
(8.99)<br />
<br />
!() (8.100)<br />
b) Allowing for different frequencies, we modify this expression to<br />
() =<br />
<br />
! Q <br />
=1 <br />
(8.101)<br />
c) As above, a correction for zero-point energy gives<br />
() = ( + ) <br />
! Q <br />
=1 <br />
(8.102)<br />
d) With the Whitten-Rabinovitch correction, this becomes<br />
() = ( + () ) <br />
! Q <br />
=1 <br />
(8.103)<br />
e) Correction for → 0: At =0,wemusthaveatleastonestate,i.e.,<br />
‡ () =1+ ( + () ) <br />
! Q <br />
=1 <br />
− ( (0) ) <br />
! Q <br />
=1 <br />
(8.104)<br />
(3) Application to unimolecular reactions:<br />
a) At the TS, we have only − 1 oscillators since the RC has to be excluded.<br />
Furthermore, the states range from the zero-point energy 0 of the TS to<br />
, while we count from the zero-point of the reactants. y<br />
‡ ( − 0 )=1+<br />
³<br />
− 0 + ‡ ( − 0 ) ‡ ´−1<br />
<br />
−<br />
³<br />
‡ (0) ‡ ´−1<br />
<br />
( − 1)! Q −1<br />
=1 ‡ ( − 1)! Q −1<br />
=1 ‡ <br />
(8.105)<br />
This expression gives the correct value of ‡ ( − 0 )=1at = 0 .
8.4 The specific unimolecular reaction rate constants () 198<br />
b) Density of states:<br />
() = ( + () ) −1<br />
( − 1)! Q <br />
=1 <br />
(8.106)<br />
c) Specific rate constant:<br />
³<br />
() = 1 − 0 + ‡ ( − 0 ) ‡ ´−1<br />
() + <br />
()( − 1)! Q −<br />
−1<br />
=1 ‡ <br />
(4) For energies sufficiently above 0 , we can use the expressions<br />
and<br />
to obtain<br />
y<br />
‡ ( − 0 )=<br />
() = ‡ ( − 0 )<br />
()<br />
() =<br />
() =<br />
=<br />
³<br />
− 0 + ‡ ´−1<br />
<br />
( − 1)! Q −1<br />
=1 ‡ <br />
( + ) −1<br />
( − 1)! Q <br />
=1 <br />
³<br />
‡ (0) ‡ ´−1<br />
<br />
()( − 1)! Q −1<br />
=1 ‡ <br />
(8.107)<br />
(8.108)<br />
(8.109)<br />
³<br />
Q <br />
=1 − 0 + ‡ ´−1<br />
<br />
<br />
Q −1<br />
=1 ‡ <br />
( + ) −1 (8.110)<br />
Q <br />
=1 <br />
Q −1<br />
=1 ‡ <br />
The resemblence to the Kassel formula is obvious.<br />
Ã<br />
− 0 + ‡ <br />
+ <br />
! −1<br />
(8.111)<br />
8.4.3 The SACM and VRRKM model<br />
More advanced models allow for the tightening of the TS with increasing (variational<br />
RRKM theory = VRRKM theory) or even for individual adiabatic channel potential<br />
curves (statistical adiabatic channel model = SACM). The latter model has to be used<br />
for reactions with loose transition states.
8.4 The specific unimolecular reaction rate constants () 199<br />
8.4.4 Experimental results<br />
I<br />
Figure 8.19: Experimental methods for the preparation of highly vibrationally excited<br />
molecules with definedexcitationenergy.<br />
I<br />
Figure 8.20: Comparison of theoretically predicted specificrateconstants () for unimolecular<br />
isomerization of cycloheptatrienes with directly measured experimental data.
8.5 Collisional energy transfer* 200<br />
8.5 Collisional energy transfer*<br />
Skipped for lack of time.
8.6 Recombination reactions 201<br />
8.6 Recombination reactions<br />
See homework assignment.
8.6 Recombination reactions 202<br />
8.7 References<br />
Lindemann 1922 F. A. Lindemann, Trans. Faraday Soc. 17, 598 (1922).<br />
Hinshelwood 1927 C. N. Hinshelwood, Proc. Roy. Soc. A113, 230 (1927).<br />
Kassel 1928a J. Phys. Chem. 32, 225 (1928).<br />
Kassel1928b J. Phys. Chem. 32, 1065 (1928).<br />
Kassel 1932 L. S. Kassel, The <strong>Kinetics</strong> of Homogeneous Gas Reactions, Chem.Catalog,<br />
New York, 1932.<br />
Rice 1927 O. K. Rice, H. C. Ramsperger, J. Am. Chem. Soc. 49, 1617 (1927).<br />
Rice 1928 O. K. Rice, H. C. Ramsperger, J. Am. Chem. Soc. 50, 617 (1928).<br />
Troe1979 J.Troe,J.Phys.Chem.83, 114 (1979).<br />
Troe1994 J. Chem. Soc. Faraday Trans. 90, 2303 (1994).
8.7 References 203<br />
9. Explosions<br />
9.1 Chain reactions and chain explosions<br />
9.1.1 Chain reactions without branching<br />
I<br />
Reaction scheme:<br />
(1) A → X chain initiation<br />
(2) X+B→ P+X chain propagation<br />
(4) X → A chain termination<br />
I<br />
Rate laws:<br />
[X]<br />
<br />
= 1 [A] − 4 [X] (9.1)<br />
[P]<br />
<br />
= 2 [B] [X] (9.2)<br />
As we see, reaction (2) does not appear in the rate law because → !<br />
I Solutions of Eq. 9.1 for two limiting cases (cf. Fig. 9.1):<br />
(1) Solution for → 0 and [A] ≈ const by substitution, using<br />
= 1 [A] − 4 [X]<br />
y<br />
y<br />
y<br />
y<br />
=0:<br />
y<br />
<br />
[X] = − 5 y [X] = − <br />
4<br />
− 1 4<br />
<br />
= y <br />
= − 4<br />
= − 4 = 1 [A] − 4 [X] (9.3)<br />
[X] = 1<br />
4<br />
[A] − − 4<br />
(9.4)<br />
[X] = 0 y = 1<br />
[]<br />
4<br />
(9.5)<br />
[X] = 1 [A]<br />
³ ´<br />
1 − − 4<br />
4<br />
(9.6)
9.1 Chain reactions and chain explosions 204<br />
(2) Solution with steady state assumption for [X] at À (i.e, after the induction<br />
time ):<br />
[X]<br />
≈ 0 (9.7)<br />
<br />
y<br />
[X] <br />
= 1<br />
[A] (9.8)<br />
4<br />
y Considering the free radical concentration profile [X ()], no desasters are happening.<br />
I<br />
Figure 9.1: Evolution of the free radical concentration in a normal chain reaction.<br />
9.1.2 Chain reactions with branching<br />
I<br />
Reaction scheme:<br />
(1) A → X chain initiation<br />
(3) X+B→ P+ X chain branching<br />
= branching factor<br />
(4) X → A chain termination<br />
I<br />
Most important example for a chain branching reaction:<br />
H+O 2 → OH + O (9.9)<br />
This reaction is responsible for flame propagation in all hydrocarbon flames: Since H is<br />
small, it has the highest diffusion coefficient and diffuses rapidly into the unburnt gases.
9.1 Chain reactions and chain explosions 205<br />
I<br />
Rate laws:<br />
[X]<br />
<br />
[P]<br />
<br />
= 1 [A] + ( − 1) 3 [X] [B] − 4 [X] (9.10)<br />
= 3 [B] [X] (9.11)<br />
I<br />
Solutions for three limiting cases:<br />
(1) =1or 4 3 ( − 1) [B]:<br />
[X]<br />
≈ 0 (9.12)<br />
<br />
y<br />
1 [A]<br />
[X] =<br />
(9.13)<br />
4 − ( − 1) 3 [B]<br />
y<br />
[P]<br />
= 3 [B] [X] = (9.14)<br />
<br />
The overall reaction is stable, because the radical concentration is stable. The<br />
product formation rate is final (constant, well behaved).<br />
(2) 4 = 3 ( − 1) [B]: Atshorttimes,[A] [B] ≈ const y<br />
[X]<br />
= 1 [A] = const (9.15)<br />
<br />
y<br />
[X] →∞ (9.16)<br />
The overall reaction turns unstable, because the radical concentration goes to<br />
infinity and thus the product formation rate too, leading to chain explosion!<br />
(3) 4 3 ( − 1) [B]: Atshorttimes,[A] [B] ≈ const y<br />
[X]<br />
+ 4 [X] − 3 ( − 1) [X] [B] = 1 [A]<br />
<br />
(9.17)<br />
[X]<br />
+[X]( 4 − 3 ( − 1) [B]) = 1 [A]<br />
<br />
(9.18)<br />
y<br />
1 [A]<br />
³<br />
´<br />
[X] =<br />
× exp [( − 1) 3 [B] ] − 1 (9.19)<br />
( − 1) 3 [B] − 4<br />
The overall reaction turns unstable, because the radical concentration increases<br />
exponentially, leading to an exponentially growing product formation rate, and<br />
therefore chain explosion!
9.1 Chain reactions and chain explosions 206<br />
I<br />
Figure 9.2: Evolutionofthefreeradicalconcentrationsinachainreactionwithbranching<br />
(three limiting cases).
9.2 Heat explosions 207<br />
9.2 Heat explosions<br />
We consider an exothermic bimolecular reaction in a closed reactor<br />
A+B→ C+D ∆ ª 0 (9.20)<br />
and look at the radial temperature profile in the reactor.<br />
I<br />
Figure 9.3: Temperature profilesinastirredandinanunstirredreactor.<br />
9.2.1 Semenov theory for “stirred reactors” (T =const)<br />
I Heat production in an exothermic reaction (∆ H ª < 0):<br />
• Energy released per volume:<br />
∆ ∆ ª £ ¤ kJ cm<br />
3<br />
(9.21)<br />
• Reaction rate:<br />
• heat release:<br />
<br />
<br />
= − [][] (9.22)<br />
<br />
<br />
= ∆ ª <br />
<br />
= ∆ ª 0 − <br />
| {z }<br />
| {z }<br />
Arrhenius rate constant in pressure units<br />
(9.23)<br />
(9.24)
9.2 Heat explosions 208<br />
• positive feedback effect:<br />
reaction rate increases<br />
y temperature ↑<br />
y reaction rate ↑<br />
y heat explosion !!<br />
I<br />
Figure 9.4: Heat explosions.<br />
I Heat removal: At constant temperature in the stirred reactor, we use Newton’s law<br />
<br />
= ( − 0 ) (9.25)<br />
where<br />
: heat conduction coefficient (depends on material)<br />
: reactor surface area<br />
: temperature<br />
0 : wall temperature<br />
(9.26)<br />
As we see, the heat removal increases (becomes steeper), when we increas the reactor<br />
surface area (at constant reactor volume).<br />
I<br />
Stability limits:<br />
(1) 0 = 1 : Heat removal wins over heat production y negative feedback y stable.<br />
(2) 0 = 2 : Heat removal wins over heat production y negative feedback y stable.<br />
(3) 0 = 3 : Heat production wins over heat removal y positive feedback y heat<br />
explosion, but the system should never reach point [3].<br />
(4) 0 = 4 : tangent y stability limit, the system runs away upon the smallest<br />
perturbation!<br />
(5) 0 = 5 : always unstable!
9.2 Heat explosions 209<br />
I<br />
Conclusions:<br />
• The quantity that determines whether the system is stable or not is the wall<br />
temperature 0 ,because 0 determines the initial point of the heat removal line.<br />
• Smaller surface-to-volume ratio is dangerous, because smaller surface at a given<br />
volume leads to a smaller slope of the heat removal line y instability!<br />
• Security measure: internal cooling system (cold water pipes, heat exchanger).<br />
• Emergency measure: when cooling system fails, the reaction mixture should be<br />
strongly diluted by addition of solvent immediately!<br />
I Quantitative estimation of the stability limits:* At point (4) in Fig. 9.4, we have:<br />
<br />
= <br />
(9.27)<br />
<br />
<br />
<br />
y ( − 0 )= ∆ ª 0 − (9.28)<br />
= <br />
<br />
<br />
(9.29)<br />
y = ∆ ª 0 − <br />
<br />
2 (9.30)<br />
Eq<br />
Eq y 2<br />
<br />
=( − 0 ) (9.31)<br />
y 2 − <br />
+ <br />
y = <br />
2 ± s µ<br />
2<br />
0 =0 (9.32)<br />
2<br />
− <br />
0 (9.33)<br />
Solution is: ⎧<br />
⎪⎨<br />
= 1 ³<br />
+ p ´ ⎫<br />
( 2 2<br />
− 4 0 )<br />
⎪⎬<br />
⎪ ⎩ = 1 ³<br />
− p ´<br />
( 2 2<br />
− 4 0 )<br />
⎪⎭<br />
This can be simplified:<br />
(9.34)<br />
(1) For practically important reactions, we have:<br />
À 0 y 0 ¿ <br />
<br />
(9.35)<br />
(2) We also can exclude unreasonably hight temperatures y only the −-sign before<br />
∗ s<br />
= µ 2<br />
<br />
2 − − <br />
2 0 (9.36)<br />
= <br />
2 − r<br />
<br />
1 − 4 0<br />
(9.37)<br />
2 <br />
√ counts: y<br />
= <br />
2<br />
à r<br />
1 −<br />
!<br />
1 − 4 0<br />
<br />
(9.38)
9.2 Heat explosions 210<br />
(3) We can expand the √ :<br />
= 4 0<br />
; 1; (9.39)<br />
<br />
y √ 1 − =1− 1 2 − 1<br />
2 · 4 2 − (9.40)<br />
y ∗ = µ<br />
<br />
1 − 1+ 2 0<br />
+ 16 · <br />
2 0<br />
2<br />
2 2 · 4 · ( ) 2 + (9.41)<br />
y ∗ = 0 + 2 0<br />
<br />
(9.42)<br />
y ∗ − 0 = 2 0<br />
<br />
(9.43)<br />
(4) Insert into in (I) and power expansion:<br />
· ·<br />
(5) Explosion limit:<br />
2<br />
0<br />
2<br />
· exp<br />
µ <br />
2<br />
0<br />
<br />
∙<br />
− ¸<br />
<br />
=<br />
0<br />
⎡<br />
<br />
= · ∆ ª · 0 · exp ⎣−<br />
´ ⎦ · · (9.44)<br />
0<br />
³1+ 0<br />
<br />
∙<br />
= · ∆ ª · 0 · · exp − ¸<br />
<br />
· 2 · · (9.45)<br />
0<br />
· · <br />
· ∆ ª · · · 0 · · <br />
= (9.46)<br />
⎤<br />
y<br />
∝ · <br />
(9.47)<br />
Power expansion of the exponent in step (4):<br />
⎡<br />
⎤<br />
∙<br />
<br />
exp ⎣−<br />
´ ⎦ =exp −<br />
0<br />
³1+ 1 ¸<br />
1<br />
0<br />
1+<br />
<br />
∙<br />
=exp − 1 <br />
≈ exp<br />
∙− 1 (1 − )¸<br />
<br />
∙<br />
=exp 1 − 1 ¸<br />
<br />
∙<br />
= 1 · exp − 1 ¸<br />
<br />
∙<br />
= · exp − ¸<br />
<br />
0<br />
¡<br />
1 − + 2 ¢¸<br />
= 0<br />
<br />
¿ 1(9.48)<br />
(9.49)<br />
(9.50)<br />
(9.51)<br />
(9.52)<br />
(9.53)
9.2 Heat explosions 211<br />
I Figure 9.5:<br />
9.2.2 Frank<strong>—</strong>Kamenetskii theory for “unstirred reactors”<br />
I<br />
Figure 9.6: Temperature profilesinanunstirredreactor.<br />
Boundary effects are neglected in the Figure. In reality, there is a smooth transition in<br />
the boundary layer.
9.2 Heat explosions 212<br />
9.2.3 Explosion limits<br />
I<br />
Figure 9.7: Explosion limits in the H 2 /O 2 system.<br />
9.2.4 Growth curves and catastrophy theory<br />
I<br />
Exponential growth:<br />
∝ · y <br />
<br />
= (9.54)<br />
I<br />
Hyperbolic growth:<br />
∝<br />
1<br />
( − ) y <br />
= 1+ mit 1+1 (9.55)<br />
I<br />
Heat explosion:<br />
<br />
∝ quadratic, i.e. we get hyperbolic growth ! (9.56)
9.2 Heat explosions 213<br />
I<br />
Figure 9.8: Growth curves.
9.2 Heat explosions 214<br />
10. Catalysis<br />
The net rate of a chemical reaction may be affected by the presence of a third species<br />
even though that species is neither consumed nor produced by the reaction. If the rate<br />
increases, we speak of catalysis. If the rate decreases, we speak of inhibition.<br />
In general, we have to distinguish<br />
• homogeneous catalysis,<br />
• heterogeoues catalysis.<br />
10.1 <strong>Kinetics</strong> of enzyme catalyzed reactions (homogeneous catalysis)<br />
The mechanism describing the homogeneous catalytical activity of enzymes has first<br />
been elucidated by Michaelis and Menten (1914).<br />
I<br />
Experimental observations on enzyme catalysis:<br />
(1) For constant substrate concentration [S] 0<br />
, the reaction rate is proportional to<br />
the concentration of the enzyme [E] 0<br />
∝ [E] 0<br />
for [S] 0<br />
= const (10.1)<br />
(2) At an applied enzyme concentration [E] 0<br />
,thereactionrate first increases with<br />
increasing substrate concentrations [S] 0<br />
andthenreachessaturation,asisshown<br />
in Fig. 10.1.<br />
I<br />
Figure 10.1: Enzyme catalysis: Michelis-Menten kinetics.
10.1 <strong>Kinetics</strong> of enzyme catalyzed reactions (homogeneous catalysis) 215<br />
10.1.1 The Michaelis-Menten mechanism<br />
Basic reaction scheme:<br />
E+S 1<br />
À ES (10.2)<br />
−1<br />
ES 2<br />
→ E+P (10.3)<br />
Typical values of 2 :<br />
2 ≈ 10 2 10 3 s −1 (in some cases up to 10 6 s −1 ) (10.4)<br />
Reaction rate:<br />
[P]<br />
<br />
= − [S]<br />
<br />
= 2 [ES] (10.5)<br />
[ES]<br />
<br />
= 1 [E] [S] − −1 [ES] − 2 [ES] (10.6)<br />
≈ 0 (10.7)<br />
Mass balance:<br />
Steady state ES concentration:<br />
[E] 0<br />
= [E]+[ES] (10.8)<br />
y [E] = [E] 0<br />
− [ES] (10.9)<br />
[S] 0<br />
= [S]+[ES]+[P] (10.10)<br />
y [S] = [S] 0<br />
− [ES] − [P] (10.11)<br />
[ES]<br />
<br />
= 1 ([E] 0<br />
− [ES]) ([S] 0<br />
− [ES] − [P]) − ( −1 + 2 )[ES] (10.12)<br />
≈ 0 (10.13)<br />
Approximations:<br />
y<br />
[ES]<br />
<br />
[E] 0<br />
¿ [S] 0<br />
(10.14)<br />
[ES] ¿ [S] 0<br />
(10.15)<br />
[P] ≈ 0 for early times (10.16)<br />
= 1 [E] 0<br />
[S] 0<br />
− 1 [S] 0<br />
[ES] − ( −1 + 2 )[ES] (10.17)<br />
≈ 0 (10.18)<br />
Steady state ES concentration:<br />
[ES] <br />
=<br />
1 [E] 0<br />
[S] 0<br />
1 [S] 0<br />
+ −1 + 2<br />
(10.19)
10.1 <strong>Kinetics</strong> of enzyme catalyzed reactions (homogeneous catalysis) 216<br />
Resulting rate expression:<br />
y<br />
= [P] = 2 [ES] (10.20)<br />
<br />
1 2 [E]<br />
=<br />
0<br />
[S] 0<br />
(10.21)<br />
1 [S] 0<br />
+ −1 + 2<br />
2 [E]<br />
=<br />
0<br />
1+ (10.22)<br />
−1 + 2<br />
1 [S] 0<br />
=<br />
with the Michaelis-Menten constant<br />
2 [E] 0<br />
1+ [S] 0<br />
(10.23)<br />
= −1 + 2<br />
1<br />
(10.24)<br />
Limiting cases:<br />
(1) [S] 0<br />
→∞y <br />
[S] 0<br />
¿ 1: y<br />
= [P]<br />
<br />
= 2 [E] 0<br />
(10.25)<br />
(2) [S] 0<br />
→ 0 y <br />
[S] 0<br />
À 1: y<br />
= [P]<br />
<br />
= 2 [E] 0<br />
[S] 0<br />
<br />
(10.26)<br />
Determination of 2 and from a Lineweaver-Burk plot (Fig. 10.2):<br />
1<br />
= 1 +<br />
1<br />
(10.27)<br />
2 [E] 0<br />
2 [E] 0<br />
[S] 0<br />
⇒ Aplotof −1 vs. [S] −1 0<br />
should give a straight line with intercept ( 2 [E] 0<br />
) −1 and<br />
slope 2 [E] 0<br />
. y<br />
slope<br />
intercept = = −1 + 2<br />
(10.28)<br />
1
10.1 <strong>Kinetics</strong> of enzyme catalyzed reactions (homogeneous catalysis) 217<br />
I<br />
Figure 10.2: Lineweaver-Burk plot.<br />
10.1.2 Inhibition of enzymes:<br />
Extended reaction mechanism:<br />
E+S 1<br />
À ES (10.29)<br />
−1<br />
ES 2<br />
→ E+P (10.30)<br />
E+I 3<br />
À EI (10.31)<br />
−3<br />
The EI complex is catalytically inactive, since I blockstheactivesiteofE.<br />
Inhibition constant:<br />
= [EI]<br />
[E] [I] = 3<br />
−3<br />
(10.32)<br />
I<br />
Reaction rate (Fig. 10.2):<br />
y<br />
[ES]<br />
<br />
= ≈ 0 (10.33)<br />
1<br />
= 1 +(1+ [I]) ×<br />
1<br />
(10.34)<br />
2 [E] 0<br />
2 [E] 0<br />
[S] 0<br />
y The slope of the plot of −1 vs. [S] −1 0<br />
goes up (see Fig. 10.2).<br />
Exercise 10.1: Derive Eq. 10.34 with a reasonable assumption for the concentrations<br />
of I. ¤
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 218<br />
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions)<br />
I Examples for heterogeneous reactions (gas-solid interface): Many reactions,<br />
which are very slow in the gas phase, take place as heterogeneous reactions in the<br />
presence of a heterogeneous catalyst:<br />
• catalytic hydrogenation<br />
• NH 3 synthesis (Haber-Bosch)<br />
• catalytic exhaust gas treatment (internal combustion engines)<br />
• catalytic DeNO processes<br />
• CO oxidation<br />
• oxidative coupling reactions of CH 4 (oxide catalysts)<br />
• atmospheric reactions (at gas-solid and gas-liquid interface)<br />
10.2.1 Reaction steps in heterogeneous catalysis<br />
Heterogeneous reactions take place via a series of steps:<br />
• Diffusion in gas (or liquid) phase to catalyst surface.<br />
• Adsorption:<br />
<strong>—</strong> “physical” adsorption (“physisorption”),<br />
<strong>—</strong> “chemical” adsorption (“chemisorption”).<br />
• Diffusiononthesurface.<br />
• <strong>Chemical</strong> reaction on the surface.<br />
• Desorption.<br />
• Diffusion away from surface.<br />
10.2.2 Physisorption and chemisorption<br />
We distinguish between “physical” adsorption (“physisorption”) and “chemical” adsorption<br />
(“chemisorption”) on a surface based upon the adsorption energies (surface binding<br />
energies). ⇒ Fig. 10.3.<br />
• Physisorption:<br />
<strong>—</strong> van-der-Waals type interaction,<br />
<strong>—</strong> typical bond energy: ∆ ≤−40 kJ mol −1 ,
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 219<br />
<strong>—</strong> example: Ar on a metal or oxide surface.<br />
• Chemisorption:<br />
<strong>—</strong> formation of a chemical bond between adsorbate and surface.<br />
<strong>—</strong> typical bond energy: ∆ ≤−100 − 300 kJ mol −1 ,<br />
<strong>—</strong> example: CO on Pd:<br />
O<br />
||<br />
C<br />
|<br />
− Pd − Pd −<br />
(10.35)<br />
∗ chemical bond by overlap of orbital of the C atom with free bands of<br />
Pd,<br />
∗ back donation from occupied bands into ∗ orbital of the CO substrate.<br />
∗ Experiment shows that (Pd − C=O)≈ 236 cm −1 .<br />
I<br />
Figure 10.3: Potential curves for physisorption and chemisorption.<br />
10.2.3 The Langmuir adsorption isotherm<br />
I<br />
Reaction scheme:<br />
A() +<br />
−<br />
|<br />
S − 1<br />
À<br />
−1<br />
−<br />
A<br />
|<br />
S − (10.36)<br />
− | S− = “active surface site” (10.37)
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 220<br />
I Fraction of occupied active surface sites Θ:<br />
• Adsorption rate:<br />
• Desorption rate:<br />
• Steady state:<br />
y<br />
y<br />
1 × (1 − Θ) × (10.38)<br />
−1 × Θ (10.39)<br />
1 (1 − Θ) = −1 Θ (10.40)<br />
1 =( −1 + 1 ) Θ (10.41)<br />
I Langmuir isotherm (Fig. 10.4):<br />
Θ =<br />
1 <br />
−1 + 1 =<br />
<br />
1+ <br />
(10.42)<br />
with<br />
= 1<br />
−1<br />
(10.43)<br />
I<br />
Figure 10.4: Langmuir adsorption isotherm.
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 221<br />
I<br />
Determination of V , V ∞ , and the adsorption enthalpy ∆H : At a series<br />
of temperatures , we need to measure Θ ( )= ( )<br />
∞ ( ) as function of by<br />
pressure/volume measurements in an ultrahigh vacuum apparatus.<br />
• From Eq. 10.42:<br />
• Multiplication of l.h.s. and r.h.s. with<br />
• Aplotof<br />
<br />
<br />
=<br />
1<br />
Θ = <br />
∞<br />
=1+ 1<br />
<br />
1<br />
∞<br />
<br />
:<br />
<br />
∞<br />
+ <br />
∞<br />
<br />
(10.44)<br />
= + (10.45)<br />
<br />
<br />
vs. gives a straight line (see the inset in Fig. 10.4) with slope<br />
=( ∞<br />
) −1 (10.46)<br />
and intercept<br />
=( ∞<br />
) −1 (10.47)<br />
•<br />
• van’t Hoff plot:<br />
=<br />
slope<br />
intercept = <br />
ln <br />
(1 ) = −∆ <br />
<br />
(10.48)<br />
(10.49)<br />
I Desorption rate: If ∆ 0 0 we can describe the desorption rate by TST (see Figs.<br />
10.3 und 10.6).<br />
−1 = µ<br />
‡<br />
exp − ∆ <br />
0<br />
(10.50)<br />
<br />
The ’s are partition functions per unit area.
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 222<br />
10.2.4 Surface Diffusion<br />
I<br />
Figure 10.5: Surface Diffusion.<br />
I<br />
Diffusion is an activated process:<br />
µ<br />
= 0 exp − ∆ <br />
<br />
<br />
∆ = energy barrier of jump of adsorbate from one site to another.<br />
(10.51)<br />
I<br />
Fick’s 2nd law:<br />
( )<br />
<br />
= × 2 ( )<br />
2 (10.52)<br />
I<br />
Einstein’s diffusion law:<br />
• 1D:<br />
∆ 2 =2 (10.53)<br />
• 2D:<br />
∆ 2 =4 (10.54)
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 223<br />
10.2.5 Types of heterogeneous reactions<br />
a) Dissociative Adsorption<br />
I<br />
Scheme:<br />
A 2 () +<br />
−<br />
|<br />
S −<br />
|<br />
S − 2<br />
À<br />
−2<br />
−<br />
A<br />
|<br />
S −<br />
A<br />
|<br />
S − (10.55)<br />
I<br />
Adsorption isotherm:<br />
• Adsorption rate:<br />
• Desorption rate:<br />
y<br />
with<br />
Θ =<br />
2 2 (1 − Θ) 2 (10.56)<br />
−2 Θ 2 (10.57)<br />
( 2<br />
× ) 12<br />
1+( 2 × ) 12 (10.58)<br />
= 2<br />
−2<br />
(10.59)<br />
I<br />
Figure 10.6: Potential energy curves for dissociative adsorption.
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 224<br />
b) Surface catalyzed unimolecular dissociation<br />
I<br />
Scheme:<br />
A() +<br />
−<br />
|<br />
S − 3<br />
À<br />
−3<br />
−<br />
A<br />
|<br />
S − 4<br />
→ Products (10.60)<br />
I Example: Decomposition of PH 3 on a W metal surface.<br />
I<br />
Decomposition rate and two limiting cases:<br />
[P]<br />
<br />
= 4 Θ = 4 × A<br />
1+ A<br />
(10.61)<br />
(1) A ¿ 1: The reaction becomes first-order in A .<br />
[P]<br />
<br />
= 4 A (10.62)<br />
(2) A À 1: The reaction becomes zeroth order, i.e., independent of A !<br />
[P]<br />
<br />
= 4 (10.63)<br />
c) Bimolecular reactions<br />
I<br />
Scheme:<br />
A() +<br />
−<br />
|<br />
S − <br />
À<br />
−<br />
−<br />
A<br />
|<br />
S − (10.64)<br />
−<br />
A<br />
|<br />
S − + −<br />
B() +<br />
B<br />
−<br />
|<br />
S − 5<br />
→<br />
|<br />
S −<br />
−<br />
<br />
À<br />
−<br />
A − B<br />
−<br />
B<br />
|<br />
S − (10.65)<br />
|<br />
S − S − 6<br />
→ Products (10.66)<br />
I Reaction rate: Assuming that A and B compete for the same active sites, we have<br />
(1 − Θ − Θ )= − Θ (10.67)<br />
y<br />
[P]<br />
<br />
(1 − Θ − Θ )= − Θ (10.68)<br />
= 5 Θ Θ =<br />
5 <br />
(1 + + ) 2 (10.69)
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 225<br />
I Limiting cases: It is often the case that one species is adsorbed much more strongly<br />
than the other.<br />
• Assume, for instance, that ¿ . Then,<br />
[P]<br />
<br />
= 5 <br />
(1 + ) 2 (10.70)<br />
(1) À 1:<br />
[P]<br />
<br />
= 5 <br />
<br />
(10.71)<br />
The reaction is inhibited by B, which is strongly adsorbed: At high ,all<br />
active sites are blocked by B.<br />
(2) ¿ 1:<br />
[P]<br />
= 5 (10.72)<br />
<br />
The reaction is truly second order.<br />
⇒ We can see from these two cases that, as increases, the rate [P]<br />
<br />
increases, reaches a maximum, and then decreases again.<br />
first<br />
d) Example 1: Heterogeneous oxidation of CO<br />
I<br />
Figure 10.7: Heterogeneous oxidation of CO on Pt. (a) Langmuir-Hinshelwood and<br />
Eley-Rideal mechanisms, (b) potential energy diagram (values in kJ mol −1 ).
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 226<br />
e) Example 2: Haber-Bosch NH 3 synthesis<br />
I Figure 10.8: Potential energy diagram for the catalytic reaction N 2 + 3H 2 → 2NH 3<br />
on a Fe 3 O 4 catalyst according to Ertl and co-workers (1982). Energies in kJ mol −1 .<br />
f) How can we control and accelerate heterogeneous reactions?<br />
The answer depends on which step is rate determining:<br />
• If diffusion to/from the surface is the rate determining step, we can accelerate<br />
the reaction by stirring<br />
• If the adsorption step is rate determining (because of an activation energy), we<br />
can accelerate the reaction by increasing and and, in particular, by increasing<br />
surface area.
10.2 <strong>Kinetics</strong> of heterogeneous reactions (surface reactions) 227<br />
11. Reactions in solution<br />
“Everyone has problems - chemists have solutions.”<br />
I<br />
Comparison of gas and liquid phase reactions (@ T = 298 K and p =1bar):<br />
• Molecules in the gas phase occupy ≈ 02 % of the total volume, while molecules<br />
in the liquid phase occupy ≈ 50 % of the total volume.<br />
• Molecules in liquid are in permanent contact with each other, and undergo permanent<br />
collisions with each other.<br />
• The crossing of a potential barrier in a reaction is not continuous as in gas phase<br />
but affected by multiple collisions during the crossing.<br />
• As a result of the permanent collisions, we can expect the reactant molecules to<br />
be in thermodynamic equilibrium even above 0 . Thus TST should be applicable.<br />
• However, it is very difficult to evaluate the partition functions in liquids. Hence,<br />
one usually employs the thermodynamic version TST.<br />
• Solvent molecules act as an efficient heat bath.<br />
I Observations on liquid phase reactions: Experience has shown that reactions in<br />
liquids have similar rates as in the gas phase, except if<br />
• the reaction occurs with the solvent,<br />
• there are strong interactions of the reactant molecules with the solvent (e.g., ionic<br />
reactions; ⇒ solvent shells),<br />
• the reactions are diffusion controlled.<br />
11.1 Qualitative model of liquid phase reactions<br />
In contrast to gas phase reactions for which we have advanced theories (⇒ transition<br />
state theory, unimolecular rate theory), the theory for liquid phase reactions is much<br />
less well developed. However, we can understand the required basic reaction steps as<br />
sketched in Fig. 11.1 and a number of limiting cases resulting from these steps.
11.1 Qualitative model of liquid phase reactions 228<br />
I<br />
Figure 11.1: Structure and dynamics of solutions.<br />
I Formal kinetics: We distinguish between the formation of the encounter complex in<br />
the solvent cage (by diffusion) and the actual reaction:<br />
+ <br />
<br />
À<br />
−<br />
{}<br />
<br />
→ (11.1)<br />
y<br />
y<br />
with<br />
[{}]<br />
<br />
[ ]<br />
<br />
= [][] − − [{}] − [{}] ≈ 0 (11.2)<br />
[{}] =<br />
<br />
− + <br />
[][] (11.3)<br />
= [{}] = <br />
− + <br />
[][] = [][] (11.4)<br />
=<br />
<br />
− + <br />
(11.5)<br />
I<br />
Rate constant for liquid phase reactions:<br />
=<br />
<br />
− + <br />
(11.6)
11.1 Qualitative model of liquid phase reactions 229<br />
I<br />
Two limiting cases:<br />
(1) Diffusion control: À − y<br />
= (11.7)<br />
This case is observed when =0and the diffusion coefficient is small.<br />
We shall find below that<br />
=4 (11.8)<br />
(2) Reaction (or activation) control: ¿ − y<br />
= <br />
−<br />
= {} (11.9)<br />
In this case, the reaction is said to be activation controlled because usually it has<br />
a sizable .<br />
can be described by TST:<br />
= <br />
∆+ −∆+ <br />
(11.10)
11.2 Diffusion-controlled reactions 230<br />
11.2 Diffusion-controlled reactions<br />
11.2.1 Experimental observations<br />
The transition to diffusion control can be observed in the gas phase by studying the<br />
kinetics in highly compressed gases (up to =1000bar) or, even better, in supercritical<br />
media. At ≈ 1000 bar, the gas density reaches the density of the liquid phase.<br />
I Smoluchowsky-Kramers limit: At high densities, the rate constant becomes<br />
∝ ∝ 1 <br />
(11.11)<br />
I<br />
Figure 11.2: Diffusion control of a unimolecular reaction: At very high pressures, the<br />
fragments recombine before they become separated by diffusion y the reaction becomes<br />
diffusion controlled.<br />
11.2.2 Derivation of k <br />
In order to derive a theoretical expression for , we consider the distribution of the<br />
molecules around a molecule inthecenterofasphereasshowninFig.11.3.
11.2 Diffusion-controlled reactions 231<br />
I<br />
Figure 11.3: Radial distribution of reactants in a diffusion-controlled reaction.<br />
(1) We start by considering the boundary conditions for the concentration [] (cf.<br />
Fig. 11.3):<br />
• At a distance →∞, the concentration of is identical to the mean<br />
concentration in the solution, i.e.,<br />
→∞y [] <br />
=[] =∞<br />
=[] (11.12)<br />
• At the distance = for a diffusion controlled reaction, the concentration<br />
of vanishes because of its fast reaction with , i.e.,<br />
→ 0 y [] =0<br />
=0 (11.13)<br />
(2) The result is a concentration gradient [] <br />
around , which gives rise to a net<br />
<br />
flux ←− by diffusion as described by Fick’s first law:<br />
= × × [] <br />
(11.14)<br />
<br />
We define this flux, measured in units of molecules/s, in the direction from = ∞<br />
(high concentration) to = (low concentration); thus there is no − sign in<br />
the usually written form of Fick’s law.<br />
is the mean diffusion coefficient, measured in m 2 s,<br />
= + (11.15)
11.2 Diffusion-controlled reactions 232<br />
(3) With =4 2 as the surface of the sphere with radius around , weobtain<br />
= × 4 2 × [] <br />
<br />
(11.16)<br />
(4) To determine the concentration profile [] <br />
, we integrate over [] <br />
:<br />
Z []∞ Z ∞<br />
<br />
[] <br />
=<br />
<br />
[] <br />
4 2 <br />
(11.17)<br />
y<br />
∞<br />
[]| ∞ <br />
<br />
= −<br />
4 <br />
¯¯¯¯<br />
(11.18)<br />
y<br />
<br />
[] − [] <br />
= − 0 (11.19)<br />
4 <br />
(5) is found from the boundary conditions that [] <br />
=0at = :<br />
<br />
=4 × [] (11.20)<br />
(6) The concentration gradient of [] is obtained by inserting the above result for <br />
into Eq. 11.19:<br />
[] − [] <br />
= []<br />
(11.21)<br />
<br />
y<br />
³<br />
[] <br />
=[] 1 − ´<br />
<br />
(11.22)<br />
<br />
This expression is plotted in Fig. 11.3.<br />
(7) To determine , we write the rate of the diffusion controlled reaction as<br />
[ ]<br />
= [] (11.23)<br />
<br />
Inserting the expression for from Eq. 11.20, we obtain<br />
[ ]<br />
=4 × [][] (11.24)<br />
<br />
The diffusion-limited rate constant is thus (in molecular units)<br />
=4 (11.25)<br />
(8) Nernst-Einstein relation between diffusion constant and viscosity for spherical particles<br />
with radius :<br />
= <br />
(11.26)<br />
6 <br />
y<br />
∝ <br />
(11.27)<br />
<br />
(9) Keepinmindthatdiffusion is an activated process:<br />
= 0 − <br />
Typical value for H 2 O: ≈ 15 kJ mol −1 .<br />
(11.28)
11.2 Diffusion-controlled reactions 233<br />
I<br />
Typical values:<br />
y Typical magnitude of :<br />
≈ 2 × 10 −5 cm 2 s −1 (11.29)<br />
≈ 5 × 10 −8 cm (11.30)<br />
≈ 8 × 10 9 lmol −1 s −1 (11.31)<br />
I<br />
Table 11.1: Diffusion constants of ions in aqueous solution at =298K.<br />
ion (cm 2 s −1 ) ion (cm 2 s −1 )<br />
H + 91 × 10 −5 OH − 52 × 10 −5<br />
Li + 10 × 10 −5 Cl − 20 × 10 −5<br />
Na + 13 × 10 −5 Br − 21 × 10 −5<br />
I<br />
Examples of diffusion controlled reaction in solution:<br />
• radical recombination reactions, e.g.<br />
+ → 2 : =13 × 10 10 lmol −1 s −1 (11.32)<br />
• electronic deactivation (quenching) in solution<br />
• H + transfer reactions<br />
+ + − → 2 : =14 × 10 11 lmol −1 s −1 (11.33)<br />
This reaction is ten times faster than a typical diffusion controlled reaction, because<br />
H + transfer is a concerted process (Grotthus mechanism, see Fig. 11.4)!<br />
I<br />
Figure 11.4: Structures involved in fast H + transfer processes in liquid water.
11.3 Activation controlled reactions 234<br />
11.3 Activation controlled reactions<br />
The rate constant for an activation controlled reaction is<br />
= {} (11.34)<br />
I Estimation of K {} : We can estimate the equilibrium constant {} using simple<br />
arguments based on the coordination number. Accordingly, the concentration of the<br />
encounter complex is<br />
[{}] =[] × (probability of finding next to ) (11.35)<br />
=[] × []<br />
[]<br />
(11.36)<br />
where [] is the solvent concentration and is the coordination number. y<br />
{} = [{}]<br />
[][] = <br />
[]<br />
(11.37)<br />
For H 2 Oat =298K, [] =555moll −1 . Thus, taking for example =8,wefind<br />
that<br />
{} =014 l mol −1 (11.38)<br />
I Application of thermodynamic TST: TSTcanbeappliedintwoways:<br />
(1) Considering the overall reaction, we can write<br />
= <br />
exp ¡ −∆ ‡ ¢ (11.39)<br />
where ∆ ‡ is the overall Gibbs free enthalphy for the reaction.<br />
(2) We can also separate the effects from and {} by writing<br />
´<br />
{} =exp<br />
³−∆ ª {} <br />
(11.40)<br />
and<br />
= <br />
exp ¡ −∆ + ¢ (11.41)<br />
∆ ª {}<br />
is the Gibbs free enthalpy change for the formation of the encounter<br />
complex, and ∆ + is the Gibbs free enthalphy of activation from the encounter<br />
complex. Thus,<br />
= ³<br />
<br />
exp −<br />
³∆ ∆+´ ´<br />
ª<br />
{} + (11.42)<br />
I Pressure dependence of the rate constant: ⇒ See section 7.3.5.
11.4 Electron transfer reactions (Marcus theory) 235<br />
11.4 Electron transfer reactions (Marcus theory)<br />
Electron transfer is what happens in oxidation-reduction reactions. Hence, electron<br />
transfer reactions are extremely important. They are strongly affected by the solvent<br />
because the change of the charge distribution results in a large reorganization energy.<br />
I Figure 11.5:<br />
I Marcus theory: For a simplified derivation of ( ) we refer to Fig. 11.5.<br />
(1) The energy of the donor and acceptor molecules will generally depend on all 3−6<br />
internal coordinates. We consider the dependence on an effective coordinate ,<br />
which describes the minimum energy path from reactant to product (Fig. 11.5).<br />
(2) The exergonicity of the ET reaction ∆ ª is negative for an exergonic reaction<br />
(convention; cf. Fig. 11.5).<br />
(3) For the donor (D) and the acceptor (A), the Gibbs energy profiles () and<br />
() shall be of parabolic form, which we may write as<br />
() = 2 (11.43)<br />
and<br />
() =( − ) 2 + ∆ ª . (11.44)<br />
The origin =0has been placed at the minimum of the donor; the acceptor has<br />
its minimum at = . 50<br />
50 We assume that ∝ 2 , taking any proportionality constant into account by proper rescaling of the<br />
abscissa.
11.4 Electron transfer reactions (Marcus theory) 236<br />
(4) The ET rate constant is described by TST as<br />
= <br />
exp ¡ −∆ + ¢ (11.45)<br />
The Gibbs free energy of activation<br />
∆ + = 2 (11.46)<br />
is the difference from the donor minimum to the TS at the donor/acceptor curve<br />
crossing at point .<br />
(5) In order to relate ∆ + to the potential curves, we define the “reorganization<br />
energy” as the energy that the acceptor would have at the equilibrium<br />
geometry of the donor. This is the energy of the acceptor parabola () at<br />
=0.<br />
If we count from the minimum of the acceptor parabola (), wehave<br />
=(− ) 2 = 2 (11.47)<br />
(6) We now determine the energy of the intersection point of the two parabolas<br />
from the condition<br />
( )= ( ) . (11.48)<br />
Inserting into Eqs. 11.43 and 11.44 yields, and using Eq. 11.47, yields<br />
2 =( − ) 2 + ∆ ª (11.49)<br />
= 2 − 2 + 2 + ∆ ª (11.50)<br />
= 2 − 2 + + ∆ ª (11.51)<br />
y<br />
y<br />
y<br />
y<br />
2 = + ∆ ª (11.52)<br />
∆ + = 2 = ( + ∆ ª ) 2<br />
( )= <br />
<br />
= + ∆ ª<br />
2 <br />
(11.53)<br />
= ( + ∆ ª ) 2<br />
(11.54)<br />
4 <br />
2 4 <br />
Ã<br />
!<br />
exp − ( + ∆ ª ) 2<br />
(11.55)<br />
4 <br />
(7) Limiting cases: We can distinguish between<br />
• −∆ ª : Regular region – the reaction becomes faster, the more<br />
exergonic the reaction becomes.<br />
• −∆ ª = : turning point<br />
• −∆ ª : Inverted region – the reaction becomes slower, the more<br />
exergonic the reaction becomes.
11.4 Electron transfer reactions (Marcus theory) 237<br />
I Conclusion: Equation 11.55 predicts the ET rate constant to increase with increasing<br />
exoergicity (−∆ ª ) of the reaction. However, once −∆ ª , the rate constant<br />
is predicted to decrease again (see Figs. 11.6D and 11.7). The prediction of this<br />
unexpected inverted Marcus regime before any experimental data were available is<br />
the hallmark of Marcus theory.<br />
I Figure 11.6:<br />
I Figure 11.7:
11.5 Reactions of ions in solutions 238<br />
11.5 Reactions of ions in solutions<br />
Reactions of ionic species in liquids are one exception to the rule that reactions in the<br />
liquid phase usually have similar rates as in the gas phase (the other exception being<br />
electron transfer reactions). The reason is that the charged species are very sensitive<br />
to their environment, in particular solvation. Changes of the charge distribution are<br />
accompanied by correspondingly large solvation shell rearrangements.<br />
11.5.1 Effect of the ionic strength of the solution<br />
The main effect arises from the stabilization of shielding every ion in solution by the oppositely<br />
charged ”ionic atmosphere” or “ion cloud” (⇒ Debye-Hückel theory, Appendix<br />
??).<br />
I<br />
Equilibrium constant for<br />
hAB ‡ i<br />
:<br />
‡ = ‡ <br />
<br />
= ‡ <br />
<br />
£<br />
<br />
‡ ¤<br />
[][]<br />
(11.56)<br />
I<br />
Activity coefficient from Debye-Hückel theory:<br />
log = − 2 12<br />
(11.57)<br />
with (for water at =298K, according to Debye-Hückel theory)<br />
=0509 l 12 mol −12 (11.58)<br />
I<br />
Ionic strength:<br />
= 1 X<br />
2 (11.59)<br />
2<br />
<br />
11.5.2 Kinetic salt effect<br />
I<br />
TST rate constant:<br />
[ ]<br />
<br />
= £ ‡ ‡¤ = ‡ <br />
‡ [][] (11.60)<br />
‡ <br />
Defining 0 as the rate constant for the ideal solution, where all =1, this becomes<br />
= 0<br />
<br />
‡ <br />
(11.61)
11.5 Reactions of ions in solutions 239<br />
I<br />
Dependence of the rate constant on the ionic strength:<br />
³<br />
´<br />
log =log 0 − 2 + 2 − ‡ 2<br />
<br />
12<br />
(11.62)<br />
with ‡ = + .<br />
Inserting ‡ = + ,weobtain<br />
log =log 0 − ¡ 2 + 2 − ( + ) 2¢ 12<br />
(11.63)<br />
y<br />
log 0<br />
=+2 12<br />
(11.64)<br />
I<br />
Conclusions:<br />
• If and havethesamesign,then will increase with increasing , because<br />
the repulsion is reduced by the shielding effect of the ion cloud.<br />
• If and have opposite sign, then will decrease with increasing , because<br />
the attraction is reduced by the shielding effect of the ion cloud.<br />
I<br />
Figure 11.8: Kinetic salt effect for reactions of ions in solution.
11.5 Reactions of ions in solutions 240<br />
11.6 References<br />
Eigen 1954a M. Eigen, Disc. Faraday Soc. 17, 194 (1954).<br />
Eigen 1954b M.EigenZ.Physik.Chem.NF1, 176 (1954).
11.6 References 241<br />
12. Photochemical reactions<br />
I<br />
Primary photochemical processes:<br />
• Absorption of a photon (depends on transition dipole moment ¯¯ ¯¯2),<br />
• Intramolecular processes:<br />
<strong>—</strong> fluorescence (≡ spontaneous emission of a photon, FL),<br />
<strong>—</strong> stimulated emission (SE), (⇒ Einstein coefficients)<br />
<strong>—</strong> chemical reaction in the excited state (R),<br />
<strong>—</strong> intramolecular vibrational (energy) redistribution (IVR),<br />
<strong>—</strong> internal conversion (IC),<br />
<strong>—</strong> intersystem crossing (ISC),<br />
<strong>—</strong> phosphorescence (PHOS).<br />
• Intermolecular (i.e., collisional) processes:<br />
<strong>—</strong> electronic deactivation (quenching, Q),<br />
∗ may affect fluorescence as well as phosphorescence,<br />
<strong>—</strong> vibrational energy transfer (VET),<br />
<strong>—</strong> rotational energy transfer (RET),<br />
<strong>—</strong> collision-induced intersystem crossing (CIISC).<br />
12.1 Basic radiative processes in a two-level system; Einstein coefficients<br />
I<br />
Figure 12.1: Absorption, stimulated emission, and spontaneous emission.<br />
2<br />
<br />
= − 1<br />
<br />
= 12 () 1 − 21 () 2 − 21 2 (12.1)
12.2 Fluorescence quenching (Stern-Volmer equation) 242<br />
12.2 Fluorescence quenching (Stern-Volmer equation)<br />
I<br />
Kinetic model:<br />
A + → A ∗ 1<br />
A ∗ → A + <br />
A ∗ +Q → A + Q <br />
(12.2)<br />
If any other radiationless processes can be neglected, we find<br />
[A ∗ ] <br />
= 1 [A]<br />
+ [Q]<br />
(12.3)<br />
I Fluorescence intensity I 0 in the absence of the quencher Q:<br />
0 ∝ [A ∗ ] <br />
= 1 [A] (12.4)<br />
I Fluorescence intensity I in the presence of the quencher Q:<br />
∝ [A ∗ ] <br />
= <br />
1 [A]<br />
+ [Q]<br />
(12.5)<br />
I<br />
Stern-Volmer equation:<br />
y<br />
0<br />
<br />
= 1 [A]<br />
1 [A]<br />
+ [Q]<br />
= + [Q]<br />
<br />
=1+ [Q] (12.6)<br />
0<br />
<br />
=1+ 0 [Q] (12.7)<br />
0 =( ) −1 is the radiative lifetime.<br />
I Conclusion: Aplotof 0 vs. [Q] should give a straight line with intercept 1 and<br />
slope 0 [Q].
12.3 The radiative lifetime 0 and the fluorescence quantum yield Φ 243<br />
12.3 The radiative lifetime τ 0 and the fluorescence quantum yield Φ <br />
I Determination of the radiative lifetime τ 0 : Assuming that there are no competing<br />
nonradiative processes, we can determine the radiative lifetime 0 by<br />
(1) time-resolved measurements of the (exponential) fluorescence decay after pulsed<br />
excitation,<br />
(2) high-resolution measurement of the natural (Lorentzian) linewidth according to<br />
∆ ∆ = ~ (12.8)<br />
y<br />
0 =<br />
1<br />
2 ∆ 0<br />
(12.9)<br />
(3) evaluation from the molar absorption coefficient (Lambert-Beer) using the relationship<br />
between the Einstein coefficients 21 and 21 , 51<br />
21 = 83<br />
3 21 (12.10)<br />
(4) calculate the value of ¯¯ ¯¯2 (⇒ 12 21 21 )fromfirst principles.<br />
I Fluorescence quantum yield: Apart from quenching, the fluorescence is reduced by<br />
other radiationless processes (IC, ISC, R, . . . ). The experimentally measured fluorescence<br />
lifetime is therefore smaller (often much smaller) than the radiative lifetime 0 .<br />
The ratio is the fluorescence quantum yield:<br />
Φ =<br />
<br />
+ + + + [Q] + = 0<br />
(12.11)<br />
51 The actual procedure, which involves integration over the absorption band, corrects for the influence<br />
of the refraction index of the solvent, etc., is called the Strickler-Berg analysis.
12.4 Radiationless processes in photoexcited molecules 244<br />
12.4 Radiationless processes in photoexcited molecules<br />
I<br />
Figure 12.2: Jablonski Diagram.<br />
Elementary Photochemical Processes: Jablonski Diagram<br />
Energy<br />
RXN<br />
(10 -14 .. 10 -6 s)<br />
S 2<br />
T 2<br />
IC<br />
S 2 -S 1 Abs.<br />
T 1<br />
T 2 -T 1 Abs.<br />
IVR<br />
IC<br />
VET* S 1<br />
ISC<br />
(10 -9 ..10 -3 s)<br />
S 0 VET* (10 -11 s)<br />
IVR (10 -13 ..10 -9 s)<br />
Absorption<br />
(10 -18 s)<br />
Fluorescence<br />
(10 -9 ..10 -8 s)<br />
Phosphorescence<br />
(10 -3 ..10 -6 s)<br />
*also known as VR<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 253<br />
12.4.1 The Born-Oppenheimer approximation and its breakdown<br />
a) The full Hamiltonian<br />
We consider a non-moving, non-rotating molecule in the laboratory framework. The<br />
molecule is described by the Schrödinger equation (SE)<br />
ˆ(r R) = (r R) (12.12)<br />
or ³<br />
ˆ − ´<br />
(r R) =0 (12.13)<br />
with being the energy eigenvalue associated with the wavefunction (r R) as function<br />
of the electronic (e) coordinates r and the nuclear (N) coordinates R.<br />
The Hamilton operator ˆ appearingintheSEcanbewrittenas<br />
ˆ = ˆ + ˆ = ˆ (r)+ ˆ (R)+ (r R) (12.14)<br />
where<br />
ˆ = − ~2<br />
2 <br />
X <br />
=1<br />
∇ 2 (12.15)<br />
ˆ = − ~2 X <br />
∇ 2 (12.16)<br />
2 <br />
=1
12.4 Radiationless processes in photoexcited molecules 245<br />
and<br />
with<br />
(r R) = (R)+ (r R)+ (r) (12.17)<br />
<br />
<br />
<br />
=+ 2<br />
4 0<br />
X<br />
<br />
X<br />
0 0 =1<br />
= − 2<br />
4 0<br />
<br />
X<br />
=1<br />
X <br />
=1<br />
=+ 2 X X <br />
4 0<br />
0<br />
0<br />
<br />
0 0 0<br />
=1<br />
(12.18)<br />
<br />
<br />
(12.19)<br />
1<br />
(12.20)<br />
Electronic spin, spin-orbit interaction, and nuclear spins are neglected here. Further,<br />
we have left aside the so-called electronic mass and electronic nuclear cross polarization<br />
terms, which appear in the case of a moving molecule as result of the separation of the<br />
center-of-mass.<br />
b) Adiabatic separation of nuclear and electronic motion<br />
We now want to separate the slow motion of the nuclei from the the fast motion of<br />
the electrons. This separation can be made because the fast electrons can be assumed<br />
to very rapidly follow the slow nuclei. The kinetic energy of the nuclei is assumed to be<br />
small compared to the electronic energy.<br />
Thus, for every nuclear configuration R, there is an electronic wavefunction el<br />
(r R)<br />
belonging to the electronic state |i specified by quantum number . This el<br />
(r R)<br />
depends on the nuclear coordinates R, but only very little on the nuclear velocities.<br />
We say that the electrons adiabatically follow the periodic vibrational motion of the<br />
nuclei. This approximation is therefore called adiabatic approximation.<br />
I Ansatz for Ĥ:<br />
with<br />
ˆ 0 = ˆ + (r R) =− ~2<br />
2 <br />
ˆ = ˆ 0 + ˆ 0 (12.21)<br />
X <br />
=1<br />
∇ 2 + (r R) (12.22)<br />
ˆ 0 = ˆ = − ~2 X <br />
∇ 2 (12.23)<br />
2 <br />
=1<br />
I The zero-order Hamiltonian Ĥ 0 :<br />
• The zero-order Hamiltonian ˆ 0 depends only on the electron kinetic energy operator<br />
and the potential energy and describes the rigid molecule (all R are<br />
fixed!) via the unperturbed SE<br />
ˆ 0 el (r; R) = (0) (R) el (r; R) (12.24)
12.4 Radiationless processes in photoexcited molecules 246<br />
• The zero-order electronic wave functions el<br />
(r; R) describe the electrons and<br />
el<br />
their distribution<br />
¯¯ (r; R)¯¯2 for the case of fixed nuclei in the electronic state<br />
|i withtheelectronicenergy el<br />
.<br />
• The el<br />
(r; R) depend on R only as a parameter, not as a variable (we do not<br />
differentiate or integrate with respect to the in the zero-order SE).<br />
• The el<br />
(r; R) form a complete orthonormal system.<br />
• The eigenvalues (0) (R) of the electronic SE are what we have come to know as<br />
the molecular potential energy functions (or hypersurfaces) (R) of the molecule<br />
with all nuclei at rest.<br />
I The perturbation Ĥ 0 :<br />
• The perturbation ˆ 0 is the nuclear kinetic energy operator ˆ .<br />
c) Solution of the complete SE<br />
• To solve the complete SE<br />
³<br />
ˆ 0 + ˆ 0 − ´<br />
(r R) =0 (12.25)<br />
we use the ansatz<br />
(r R) = X (R) · el (r; R) (12.26)<br />
with the expansion coefficients = (R) depending only on R but not on r.<br />
• Inserting the ansatz 12.26 into the complete SE,<br />
³<br />
ˆ 0 + ˆ 0 − ´ X<br />
(R) el<br />
(r; R) =0 (12.27)<br />
and doing some algebraic manipulation, we arrive at a<br />
• system of coupled equations for the electronic wavefunctions el (r; R) for the<br />
electronic state el<br />
(r; R) and the nuclear wavefunctions (R):<br />
and<br />
ˆ 0 el<br />
(r; R) = <br />
(0)<br />
(R) el<br />
(r; R) (12.28)<br />
ˆ 0 (R)+ X (R) (R) = ¡ − (0)<br />
(R) ¢ (R) (12.29)<br />
with the coupling coefficients
12.4 Radiationless processes in photoexcited molecules 247<br />
(R) =<br />
Z<br />
∗ (r; R) ˆ 0 (r; R) r<br />
r<br />
−~ 2 Z<br />
r<br />
∗ (r; R) X <br />
1<br />
<br />
<br />
<br />
(r; R) r<br />
<br />
<br />
(12.30)<br />
• We interprete these Eqs. as follows:<br />
(1) <br />
(0) (R) can be interpreted as the zero-order potential functions of the th<br />
zero-order electronic state that is given by the solution of the electronic SE<br />
(Eq. 12.24) for fixed nuclei R.<br />
(2) Without the (R), Eq. 12.29 describes the kinetic energy of the nuclei<br />
in the potential (0) (R). 52 Eq. 12.29 is thus simply the vibrational SE for<br />
the electronic potential <br />
(0) (R).<br />
(3) The coefficients (R) defined by Eq. 12.30 are coupling terms that connect<br />
(i.e., mix) the electronic states and . They describe how the<br />
different electronic states and are coupled by the nuclear motion (the<br />
two terms on the RHS resulting from the nuclear kinetic energy operator<br />
ˆ ;seeEq.12.30).<br />
d) Born-Oppenheimer approximation<br />
The Born-Oppenheimer (BO) approximation completely neglects the coupling coefficients<br />
. The complete SE is therefore reduced to two uncoupled equations, which<br />
we denote as electronic SE<br />
ˆ 0 el<br />
(r) = <br />
(0)<br />
(R) el<br />
(r) (12.31)<br />
and nuclear (vibrational) SE<br />
h i<br />
ˆ + <br />
(0) (R) = (R) (12.32)<br />
which describe, respectively, the electronic wavefunction el<br />
for fixed nuclear coordinates<br />
R in the electronic state el<br />
and the set of nuclear wavefunctions (R) for the energy<br />
state of the nuclei in the electronic state el<br />
.<br />
e) Breakdown of the Born-Oppenheimer approximnation<br />
When two (or more) electronic states el<br />
(R), el<br />
(R) become nearly isoenergetic or,<br />
at a crossing, even degenerate, the coupling matrix element (R) may no longer be<br />
neglected, because through the (R), thenuclear motion leads to a mixing between<br />
the different electronic BO-states.<br />
52 We used to denote these potential energy functions (≡ potential energy hypersurfaces) as (R).
12.4 Radiationless processes in photoexcited molecules 248<br />
I<br />
This breakdown of the BO approximnation leads to non-adiabatic coupling of<br />
electronic states:<br />
• For a diatomic molecule, the coupling leads to an avoided crossing of the two<br />
electronic potential curves.<br />
• For polyatomic molecules, the coupling can lead to a conical intersection (CI)<br />
between the two electronic potential energy hypersurfaces.<br />
As will be outlined in the following, the existence of conical intersections between different<br />
excited electronic potential energy hypersurfaces and between an excited and<br />
the ground electronic potential energy hypersurface has dramatic consequences for the<br />
dynamics of photochemical reactions.<br />
The recognition since about the years 2000 - 2005 that conical intersections in polyatomic<br />
molecules are the rule rather than the exception has indeed revolutionized our<br />
understanding of photochemical reactions.<br />
12.4.2 Avoided crossings of two potential curves of a diatomic molecule<br />
I<br />
I<br />
Illustration of an avoided crossing using two diabatic model potential functions<br />
+ coupling term:<br />
1 () = 1 [1 − exp [− 1 ( − 1 )]] 2 (12.33)<br />
2 () = 2 + exp [− 2 ( − 2 )] (12.34)<br />
12 () = 1 [1 + tanh [− 2 ( − 12 )]] (12.35)<br />
12 () is just a convenient model function with the physically correct behavior<br />
12 () → 0 for →∞.<br />
Figure 12.3: Diabatic model potential curves for a diatomic molecule.<br />
Avoided Crossing of Two Potentials of a Diatomic Molecule<br />
Diabatic model potential curves: V1<br />
( R) 0 <br />
H <br />
<br />
0 V2<br />
( R)<br />
<br />
1exp ( ) <br />
2<br />
V R D <br />
R R<br />
1 1 1 1<br />
<br />
exp ( ) <br />
V R D R R<br />
2 2 2 2<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 254
12.4 Radiationless processes in photoexcited molecules 249<br />
I<br />
Figure 12.4: Avoided crossing of two potential curves of a diatomic molecule.<br />
Avoided Crossing of Two Potentials of a Diatomic Molecule<br />
Adiabatic potential curves: V1<br />
( R) W( R)<br />
<br />
H <br />
<br />
W( R) V2( R)<br />
<br />
V<br />
<br />
<br />
V1<br />
( R) W( R)<br />
<br />
R<br />
Eigenvalues <br />
WR ( ) V2( R)<br />
<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 255<br />
I<br />
Figure 12.5: Avoided crossing of two potential curves of a diatomic molecule. At the<br />
avoided crossing the characters of the adiabatic potential curves switch.<br />
Avoided Crossing of Two Potentials of a Diatomic Molecule<br />
V E W V V V V<br />
1 2 1 2 1 2<br />
0 E<br />
W<br />
with: ; <br />
W V2<br />
E<br />
2 2<br />
<br />
2<br />
V ( R) ( R) ( R) W( R)<br />
"Avoided Crossing" with<br />
Splitting of 2 |W( R x ) |<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 256
12.4 Radiationless processes in photoexcited molecules 250<br />
I<br />
Figure 12.6: Conical intersection of potential energy surfaces of a polyatomic molecule.<br />
Conical Intersections of Potential Surfaces in Polyatomic Molecules<br />
V E W V V V V<br />
1 2 1 2 1 2<br />
0 E<br />
W<br />
with: ; <br />
W V2<br />
E<br />
2 2<br />
V<br />
S 2<br />
Conical Intersections<br />
Diatomic Molecules: s = 1<br />
<br />
2<br />
V ( R) ( R) ( R) W( R)<br />
S 1<br />
S 0<br />
Q 1<br />
Q 2<br />
Polyatomic Molecules: s = 3N-6<br />
V ( Q , Q ) ( Q , Q )<br />
<br />
1 2 1 2<br />
<br />
( Q , Q ) W( Q , Q ) <br />
<br />
1 2 1 2<br />
2<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 257<br />
12.4.3 Quantum mechanical treatment of a coupled 2×2 system<br />
We can easily verify the “avoided crossing” of two coupled potential energy curves as<br />
follows:<br />
I<br />
Schrödinger equation:<br />
( − ) |i =0 (12.36)<br />
I<br />
Hamilton-Operator:<br />
= (0) + (1) (12.37)<br />
I<br />
Ansatz:<br />
|i = 1 | 1 i + 2 | 2 i (12.38)<br />
with orthonormal basis vectors that are eigenvectors of (0) , i.e.,<br />
½<br />
® 1<br />
| = =<br />
0<br />
for = <br />
for 6= <br />
(12.39)<br />
and the zero-order energies<br />
(0) | 1 i = (0)<br />
1 | 1 i (12.40)<br />
(0) | 2 i = (0)<br />
2 | 2 i (12.41)
12.4 Radiationless processes in photoexcited molecules 251<br />
I Solution of the Schrödinger equation: Insertion of the ansatz (Eq. 12.38) into the<br />
SE and multiplication from the left by either h 1 | or by h 2 | gives the two equations:<br />
1 h 1 | ( − ) | 1 i + 2 h 1 | ( − ) | 2 i =0 (12.42)<br />
1 h 2 | ( − ) | 1 i + 2 h 2 | ( − ) | 2 i =0 (12.43)<br />
This is a system of coupled linear equations (“secular equations”):<br />
with the “matrix elements” (integrals over all r resp. R):<br />
Specifically, we have<br />
1 ( 11 − )+ 2 12 =0 (12.44)<br />
1 21 + 2 ( 22 − ) =0 (12.45)<br />
= h | | i (12.46)<br />
= h | (0) + (1) | i (12.47)<br />
= h | (0) | i + h | (1) | i (12.48)<br />
11 = h 1 | (0) | 1 i = (0)<br />
1 (12.49)<br />
22 = h 2 | (0) | 2 i = (0)<br />
2 (12.50)<br />
and<br />
12 = h 1 | (1) | 2 i = h 2 | (1) | 1 i = 21 (12.51)<br />
Therefore, we obtain the secular equations as follows:<br />
I<br />
Secular equations:<br />
³ ´<br />
1 (0)<br />
1 − + 2 12 =0 (12.52)<br />
³ ´<br />
1 21 + 2 (0)<br />
2 − =0 (12.53)<br />
I Secular determinant: A non-trivial solution for the secular equations requires that<br />
(0)<br />
1 − 12<br />
¯ 21 (0)<br />
=0 (12.54)<br />
2 − ¯<br />
I<br />
Eigenvalues:<br />
0=<br />
³ ´³<br />
(0)<br />
1 − <br />
= (0)<br />
1 (0)<br />
2 − <br />
´<br />
(0)<br />
2 − − | 12 | 2 (12.55)<br />
³ ´<br />
(0)<br />
1 + (0)<br />
2 + 2 − | 12 | 2 (12.56)<br />
with<br />
because (1) is Hermitian.<br />
| 12 | 2 = 12 21 (12.57)
12.4 Radiationless processes in photoexcited molecules 252<br />
With some rewriting 53 ,weobtain<br />
± = 1 ³ ´<br />
(0)<br />
1 + (0)<br />
2 ± 1 r ³<br />
(0)<br />
1 + (0)<br />
´2<br />
(0)<br />
2 − 4 1 (0)<br />
2 +4| 12 | 2 (12.58)<br />
2<br />
2<br />
y<br />
± = 1 2<br />
³ ´<br />
(0)<br />
1 + (0)<br />
2 ± 1 r ³<br />
(0)<br />
1 − (0)<br />
´2<br />
2 +4|12 | 2 (12.59)<br />
2<br />
I Shorthand form: Using the abbreviations<br />
± (R) = ± (12.60)<br />
1 (R) = (0)<br />
1 (12.61)<br />
2 (R) = (0)<br />
2 (12.62)<br />
2 (R) =| 12 | 2 (12.63)<br />
Σ (R) = 1 2 ( 1 (R)+ 2 (R)) (12.64)<br />
∆ (R) = 1 2 ( 1 (R) − 2 (R)) (12.65)<br />
we can rewrite the solution for the eigenvalues + and − in a compact form as<br />
± (R) =Σ (R) ±<br />
q<br />
(∆ (R)) 2 +( (R)) 2 (12.66)<br />
12.4.4 Radiationless processes in polyatomic molecules: Conical intersections<br />
I “Traditional” view on the cis-trans isomerization of C 2 H 4 :<br />
• Reaction coordinate can be approximated by torsion angle (0 ≤ ≤ 180 ◦ ).<br />
• Reaction proceeds via a perpendicular transition state ( =90 ◦ ), in which the C<br />
atoms are connected by a single bond.<br />
• Correlation diagram shows that the orbital of the trans-isomer transforms into<br />
the ∗ orbital of the cis-isomer and vice versa.<br />
• Crossing of the two diabatic states corresponds to an avoided crossing of the<br />
adiabatic states.<br />
53<br />
³<br />
(0)<br />
1 + (0)<br />
´2 ³<br />
(0)<br />
2 − 4 1 (0) 2 =<br />
(0)<br />
1<br />
´2<br />
+<br />
³<br />
³<br />
=<br />
(0)<br />
2<br />
(0)<br />
1<br />
´2<br />
+2<br />
(0)<br />
´2<br />
+<br />
³<br />
1 (0) 2 − 4 (0)<br />
1 (0) 2<br />
(0)<br />
2<br />
´2<br />
− 2<br />
(0)<br />
1 (0) 2 =<br />
³<br />
(0)<br />
1 − (0)<br />
´2<br />
2
12.4 Radiationless processes in photoexcited molecules 253<br />
• The reaction is thus thermally forbidden because of the high energy barrier at the<br />
avoided crossing between the trans and cis isomers on the left and right.<br />
• The reaction is photochemically allowed because of the barrierless adiabatic correlation<br />
between the ∗ states on the left and right.<br />
I<br />
Figure 12.7: Cis-trans isomerization of C 2 H 4 : Transformation of the and ∗ orbitals.<br />
Photochemical cis-trans isomerization of C 2 H 4<br />
Traditional picture: Diabatic potentials<br />
<br />
<br />
diabatic crossing<br />
<br />
Topics in <strong>Chemical</strong> <strong>Kinetics</strong> and Dynamics • © F. Temps, IPC Kiel 35<br />
I<br />
Figure 12.8: Cis-trans isomerization of C 2 H 4 : HMO picture with an avoided crossing.<br />
Photochemical cis-trans isomerization of C 2 H 4<br />
Traditional picture: Avoided crossing<br />
<br />
<br />
avoided crossing<br />
<br />
Topics in <strong>Chemical</strong> <strong>Kinetics</strong> and Dynamics • © F. Temps, IPC Kiel 36
12.4 Radiationless processes in photoexcited molecules 254<br />
I<br />
Figure 12.9: Internal coordinates relevant for the cis-trans isomerization of C 2 H 4 .The<br />
out-of-plane bending angle plays a role as coupling mode leading to a CI; the angles<br />
and play a role in the isomerization to H 3 CCH (also by a CI).<br />
Photochemical cis-trans isomerization of C 2 H 4<br />
Important internal coordinates:<br />
Torsion<br />
Out-of-plane bend<br />
Barbatti et al., 2004<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 258<br />
I Figure 12.10: Diabatic model PES’s for C 2 H 4 .<br />
Photochemical cis-trans isomerization of C 2 H 4<br />
Diabatic PES:<br />
<br />
(s1)-dimensional<br />
crossing seem<br />
<br />
<br />
<br />
<br />
<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 259
12.4 Radiationless processes in photoexcited molecules 255<br />
I Figure 12.11: Adiabatic model PES’s for C 2 H 4 with the conical intersection at =90 ◦ .<br />
Photochemical cis-trans isomerization of C 2 H 4<br />
Adiabatic PES:<br />
<br />
<br />
conical intersection<br />
(s-2) dimensional<br />
<br />
<br />
<br />
<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 260<br />
I<br />
Figure 12.12: Wave packet motion in a complex photo-induced reaction.<br />
Dynamics of Photo-Induced Reactions: Photochemical Funnels<br />
Conical Intersection =<br />
"Photochemical Funnel"<br />
Energy<br />
Q b<br />
S 0<br />
S 1<br />
S 0<br />
Reaction Coordinate<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 262
12.4 Radiationless processes in photoexcited molecules 256<br />
13. Atmospheric chemistry*<br />
(skipped for lack of time in the semester)
13 Atmospheric chemistry* 257<br />
14. Combustion chemistry*<br />
(skipped for lack of time in the semester)
14 Combustion chemistry* 258<br />
15. Astrochemistry*<br />
(skipped for lack of time in the semester)
15 Astrochemistry* 259<br />
16. Energy transfer processes*<br />
(See lecture in M.Sc. course.)
16 Energy transfer processes* 260<br />
17. Reaction dynamics*<br />
(See lecture in M.Sc. course.)
17 Reaction dynamics* 261<br />
18. Electrochemical kinetics*<br />
(skipped for lack of time in the semester)
Appendix B 262<br />
Appendix A: Useful physicochemical constants<br />
I Table A.1: List of useful physical constants (Cohen 1986).<br />
physical constant<br />
symbol value<br />
Loschmidt (or Avogadro) number =60221367 × 10 23 mol −1<br />
gas constant =8314511 J mol −1 K −1<br />
Boltzmann constant =1380658 × 10 −23 JK −1<br />
electron charge =160217733 × 10 −19 C<br />
Faraday constant = 96485309 C mol −1<br />
speed of light =299792458 × 10 8 ms −1<br />
Planck’s constant =66260755 × 10 −34 Js<br />
~ ~ =10547266 × 10 −34 Js<br />
electron mass =91093897 × 10 −31 kg<br />
proton mass =16726231 × 10 −27 kg<br />
neutron mass =16749286 × 10 −27 kg<br />
atomic mass unit =16605402 × 10 −27 kg<br />
Rydberg constant e e =10973731534 × 10 7 m −1<br />
Bohr radius 0 0 =529177249 × 10 −11 m<br />
electric field constant 0 0 =1( 0 2 )<br />
0 =8854187817 × 10 −12 Fm −1<br />
magnetic field constant 0 0 =4 × 10 −7 NA −2<br />
0 =125664 × 10 −6 NA −2<br />
I<br />
References:<br />
Cohen 1986 E. R. Cohen and B. N. Taylor, The 1986 Adjustment of the Fundamental<br />
Constants, CODATA Bull. 63, 1 (1986). An updated list is contained in every<br />
August issue of Physics Today.
Appendix B 263<br />
Appendix B: The Marquardt-Levenberg non-linear least-squares<br />
fitting algorithm<br />
With the omnipresence of the computer, the Marquardt-Levenberg algorithm (Bevington<br />
1992, Press 1992) has become an extremely important data analysis tool. It is generally<br />
themethodofchoiceincaseswhereitisnotpossibletomakesimplelinearplotsto<br />
determine rate constants.<br />
I Problem: Experimentally, we measure the values of some quantity at the points ,<br />
, . Supposewetake measurements.Nowsupposewewouldliketodescribeour<br />
data points by some model. Towards these ends, we take a physically reasonable<br />
model function<br />
= ( ;( 1 ))<br />
(B.1)<br />
which we would like to fit to our data in such a way that () describes the data<br />
points in “the best possible way”. The model function depends directly on the<br />
variables , and it also depends parametrically on nonlinear parameters .In<br />
kinetics, for example, is usually some concentration, e.g. , which depends on time<br />
i.e., = (). The parameters may be the rate constants (or decay times )<br />
and the initial concentration 0 . Our job is to adjust and optimize these parameters<br />
by somehow “fitting” them. The method of choice is, of course, “least-squares fitting”:<br />
We start by defining the so-called figure-of-merit or goodness-of-fit function 54<br />
2 =<br />
"<br />
#<br />
X [ − ( )] 2<br />
=1<br />
<br />
2<br />
(B.2)<br />
Here, the and the independent variables , are the measured quantities, the<br />
are the experimental uncertainties of the ,and ( ) is the calculated value<br />
of the model fit function at the point { } (the parameters 1 are<br />
omitted here in ). What we have to do is to adjust the parameters in the model<br />
function such that 2 reaches a minimum, i.e. we have to search the -dimensional<br />
parameter space for the (global) minimum of 2 .<br />
I Solution: The Marquardt-Levenberg algorithm is a method to determine this minimum<br />
of 2 as function of the fit parameters in the model fit function.<br />
The minimum of 2 as function of the fit parameters a is defined by the conditions<br />
"<br />
#<br />
2<br />
= X [ − ( )] 2<br />
=0 (B.3)<br />
<br />
<br />
2<br />
=1<br />
<br />
X<br />
∙ ¸<br />
[ − ( )] ( )<br />
= −2<br />
(B.4)<br />
<br />
2 <br />
=1<br />
Thus, taking the partial derivatives of 2 with respect to each of the parameters<br />
,weobtainasetof coupled equations in the unknown parameters . These<br />
equations are, in general, nonlinear in the .<br />
54 Deutsch: Fehlerquadratsumme.
Appendix B 264<br />
The Marquardt-Levenberg algorithm gives a recipy for finding the minimum of 2 using<br />
a combination of the method of steepest descent (following the gradient of the hypersurface<br />
defined by 2 as a function of the parameters 1 and, near the minimum<br />
of 2 , a parabolic Taylor series expansion of the fitting function around the minimum).<br />
At the end of a fit, we shall usually report the (± 2 standard deviations), the<br />
standard deviation of the ’s, and the so-called reduced- 2 ,whichis 2 divided by the<br />
number of degrees of freedom, = − , i.e.,<br />
"<br />
#<br />
2 = 1 X [ − ( )] 2<br />
(B.5)<br />
− <br />
<br />
2 <br />
=1<br />
I Example 1: Exponential decay. Suppose we measure the time dependent concentration<br />
() of a molecule in some quantum state, which is populated at some time 0<br />
by a laser pulse and then decays monoexponentially. As a model fit function, we use<br />
the expression<br />
() = 1 exp [− 2 ( − 3 )] + 4 + 5 <br />
(B.6)<br />
The parameters we are interested in are 2 (the decay rate coefficient) and 1 (the<br />
initial amplitude or initial concentration). The parameter 3 just describes the trigger<br />
delay time (= 0 ). We have also added the parameters 4 and 5 to describe a constant<br />
background term (due to some offset of our experimental signal) and a linear drift in<br />
this background term (perhaps due to some experimental instability, such as a gradual<br />
temperature increase in the lab). Our figure of merit is then<br />
" #<br />
X<br />
2 [ − ()] 2<br />
=<br />
(B.7)<br />
=1<br />
The (in general nonlinear) conditions for the minimum of 2 , from which we determine<br />
the parameters ,are<br />
" #<br />
2<br />
= X [ − ()] 2<br />
X<br />
∙ ¸<br />
[ − ()] ()<br />
= −2<br />
=0<br />
1 1 <br />
2<br />
=1 <br />
2<br />
=1 1<br />
(B.8)<br />
2<br />
2<br />
= (B.9)<br />
<br />
(B.10)<br />
Figure B.1 below shows a simple two-parameter fit (parameters 1 and 1) tosucha<br />
signal.<br />
<br />
2
Appendix B 265<br />
I<br />
Figure B.1: Exponential decay curve of a particular vibration-rotation state of the<br />
CH 3 O radical resulting from the unimolecular dissociation reaction of the radical according<br />
to CH 3 O → H+H 2 CO (Dertinger 1995). The small box is the output box<br />
from a fit usingtheORIGINprogram.<br />
I Example 2: Multiexponential decay. In femtosecond spectroscopy, we often observe<br />
ultrafast multiexponential decays of laser-excited molecules. The laser prepares<br />
an excited wavepacket, which usually does not decay single-exponentially. Further, we<br />
need to take into account the final duration of the pump laser pulse (by deconvolution,<br />
or by forward convolution).<br />
• Model function to be fitted to measured data:<br />
() = X exp (− )<br />
(B.11)<br />
with adjustable parameters and .<br />
• Instrument response function (IRF): Often represented by a Gaussian centered at<br />
time 0 Ã !<br />
1<br />
() = √ exp − ( − 0) 2<br />
(B.12)<br />
IRF 2 2 2 IRF<br />
with width parameter IRF related to the full width at half maximum (FWHM)<br />
of the IRF by<br />
FWHM = √ 8ln2≈ 2355 IRF<br />
(B.13)
Appendix C 266<br />
• Convolution of the molecular intensity () and () gives the signal function<br />
() =<br />
Z<br />
+∞<br />
−∞<br />
( 0 ) ( − 0 ) 0 =<br />
where ⊗ denotes the convolution.<br />
Z<br />
+∞<br />
−∞<br />
( − 0 ) ( 0 ) 0 = () ⊗ ()<br />
(B.14)<br />
• Resulting model function to be fitted to the data:<br />
() = 1 X<br />
∙ 1 2 IRF<br />
exp − ( − ¸ ∙ µ ¸<br />
0)<br />
( − 0 ) − 2 IRF<br />
1+erf √ + <br />
2<br />
2 2 <br />
<br />
2IRF <br />
(B.15)<br />
where erf () is the error function and is a simple constant background term (can<br />
be replaced by background + drift + ).<br />
I<br />
Figure B.2: Excited-state relaxation dynamics of the adenine dinucleotide after UV<br />
photoexcitation.<br />
I<br />
References:<br />
Bevington 1992 P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis<br />
for the <strong>Physical</strong> Sciences, McGraw-Hill, Boston, 1992.<br />
Dertinger 1995 S. Dertinger, A. Geers, J. Kappert, F. Temps, J. W. Wiebrecht,<br />
Rotation-Vibration State Resolved Unimolecular Dynamics of Highly Vibrationally<br />
Excited CH 3 O( 2 ): III. State Specific Dissociation Rates from Spectroscopic Line<br />
Profiles and Time Resolved Measurements, Faraday Discuss. Roy. Soc. 102, 31<br />
(1995).<br />
Press 1992 W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical<br />
Recipes in Fortran, Cambridge University Press, Cambridge, 1992. Versions are<br />
also available for C and Pascal.
Appendix C 267<br />
Appendix C: Solution of inhomogeneous differential equations<br />
C.1 General method<br />
An inhomogeneous DE is a DE of the type<br />
0 + () × = ()<br />
(C.1)<br />
In order to solve Eq. C.1, we first find a solution of the homogeneous DE<br />
0 + () × =0<br />
(C.2)<br />
and then determine a particular solution of the inhomogeneous DE:<br />
• Solution of the homogeneous DE by separation of variables:<br />
0 + () × =0<br />
0 = − () × <br />
(C.3)<br />
(C.4)<br />
y<br />
ln <br />
<br />
= − () (C.5)<br />
= × − ()<br />
(C.6)<br />
• Determination of a particular solution of the inhomogeneoues DE by variation of<br />
constant:<br />
→ ()<br />
(C.7)<br />
0 () =<br />
(C.8)<br />
Insertion of () and 0 () into Eq. C.1 and integration gives<br />
= <br />
(C.9)<br />
The general solution of Eq. C.1 is then given by<br />
= + <br />
(C.10)
Appendix C 268<br />
C.2 Application to two consecutive first-order reactions<br />
Consider the reaction system<br />
A 1<br />
−→ B<br />
B 2<br />
−→ C<br />
(C.11)<br />
(C.12)<br />
which is described by the following rate equations<br />
[A]<br />
<br />
[B]<br />
<br />
[C]<br />
<br />
= − 1 [A] (C.13)<br />
=+ 1 [A] − 2 [B]<br />
(C.14)<br />
=+ 2 [B]<br />
(C.15)<br />
The solution for Eq. C.13 is<br />
[A] = [A] 0<br />
− 1 <br />
(C.16)<br />
Thus we have to find the solution of the inhomogeneous DE for [B] (Eq. C.14)<br />
[B]<br />
<br />
+ 2 [B] = 1 [A] 0<br />
− 1 <br />
(C.17)<br />
• Solution of the homogeneous DE by separation of variables:<br />
y<br />
[B]<br />
<br />
= − 2 [B] (C.18)<br />
[B] <br />
= × − 2 <br />
(C.19)<br />
• Determination of a particular solution by variation of constant:<br />
y<br />
[B] <br />
<br />
= ()<br />
[B] <br />
= () × − 2 <br />
= ()<br />
<br />
× − 2 − () × 2 − 2 <br />
(C.20)<br />
(C.21)<br />
(C.22)<br />
Insertion of [B] <br />
and [B] <br />
<br />
into Eq. C.17 gives<br />
()<br />
<br />
which simplifies to<br />
× − 2 − () × 2 − 2 + () × 2 − 2 = 1 [A] 0<br />
− 1 <br />
()<br />
<br />
= 1 [A] 0<br />
( 2− 1 )<br />
This DE is easily integrated to obtain ():<br />
(C.23)<br />
(C.24)
Appendix C 269<br />
<strong>—</strong> If 1 6= 2 we obtain<br />
and thus<br />
() = 1 [A] 0<br />
2 − 1<br />
( 2− 1 )<br />
(C.25)<br />
[B] <br />
= () − 2 <br />
= 1 [A] 0<br />
2 − 1<br />
( 2− 1 ) − 2 <br />
= 1 [A] 0<br />
2 − 1<br />
− 1 <br />
(C.26)<br />
(C.27)<br />
(C.28)<br />
<strong>—</strong> If 1 = 2 we obtain<br />
()<br />
<br />
= 1 [A] 0<br />
( 2− 1 )<br />
(C.29)<br />
= 1 [A] 0<br />
(C.30)<br />
y<br />
y<br />
() = 1 [A] 0<br />
<br />
(C.31)<br />
[B] <br />
= () − 2 <br />
= 1 [A] 0<br />
− 2 <br />
(C.32)<br />
(C.33)<br />
• General solution:<br />
y<br />
Initial value condition at =0:<br />
[B] = [B] <br />
+[B] <br />
(C.34)<br />
⎧<br />
⎪⎨ × − 2 + 1 [A] 0<br />
− 1 <br />
if 1 6= 2<br />
[B] =<br />
2 − 1<br />
(C.35)<br />
⎪⎩<br />
× − 2 + 1 [A] 0<br />
− 2 <br />
if 1 = 2<br />
[B ( =0)]=0 (C.36)<br />
y<br />
⎧<br />
⎪⎨ − 1 [A] 0<br />
if 1 6= 2<br />
=<br />
2 − 1<br />
(C.37)<br />
⎪⎩<br />
0 if 1 = 2<br />
Final solutions for [B ()]:<br />
⎧<br />
1 [A] ⎪⎨ 0<br />
¡<br />
<br />
− 1 − ¢ − 2 <br />
if 1 6= 2<br />
[B ()] =<br />
2 − 1<br />
(C.38)<br />
⎪⎩<br />
1 [A] 0<br />
− 2 <br />
if 1 = 2
Appendix C 270<br />
C.3 Application to parallel and consecutive first-order reactions<br />
Consider the reaction system<br />
A 1<br />
−→ P<br />
A 1<br />
−→ B 2<br />
−→ P<br />
(C.39)<br />
(C.40)<br />
which is described by the following rate equations<br />
[A]<br />
<br />
[B]<br />
<br />
[P]<br />
<br />
= − ( 1 + 1 )[A]=− 1 [A] (C.41)<br />
=+ 1 [A] − 2 [B]<br />
(C.42)<br />
=+ 1 [A] + 2 [B]<br />
(C.43)<br />
with 1 =( 1 + 1 ). 55<br />
The solution for Eq. C.41 is<br />
[A] = [A] 0<br />
−( 1+ 1 ) =[A] 0<br />
− 1 <br />
(C.44)<br />
Thus we have to find the solution of the inhomogeneous DE for [B] (Eq. C.42)<br />
[B]<br />
+ 2 [B] = + 1 [A]<br />
<br />
0<br />
− 1 <br />
using the above standard procedure:<br />
(C.45)<br />
• Solution of the homogeneous DE by separation of variables:<br />
y<br />
[B]<br />
<br />
= − 2 [B] (C.46)<br />
[B] <br />
= × − 2 <br />
(C.47)<br />
• Determination of a particular solution by variation of constant:<br />
y<br />
[B] <br />
<br />
= ()<br />
[B] <br />
= () × − 2 <br />
= ()<br />
<br />
× − 2 − () × 2 − 2 <br />
(C.48)<br />
(C.49)<br />
(C.50)<br />
55 An example is the radiationless deactivation of a ∗ electronically excited molecule A directly to<br />
the ground state P or via an intermediatate optically dark ∗ state B, which decays to the ground<br />
statemoreslowly.
Appendix C 271<br />
Insertion of [B] <br />
and [B] <br />
<br />
into Eq. C.45 gives<br />
()<br />
<br />
× − 2 − () 2 − 2 + 2 () − 2 = 1 [A] 0<br />
− 1 <br />
(C.51)<br />
The terms with ± () 2 − 2 cancel, so that<br />
()<br />
× − 2 = 1 [A]<br />
<br />
0<br />
− 1 <br />
which we rewrite in order to solve () as<br />
()<br />
<br />
= 1 [A] 0<br />
− 1 × + 2 <br />
or<br />
()<br />
= 1 [A]<br />
<br />
0<br />
( 2− 1 ) <br />
This DE is easily integrated to obtain ():<br />
(C.52)<br />
(C.53)<br />
(C.54)<br />
<strong>—</strong> If 2 6= 1 =( 1 + 1 ) we obtain<br />
() = 1 [A] 0<br />
2 − 1<br />
( 2− 1 )<br />
(C.55)<br />
and thus<br />
[B] <br />
= () − 2 <br />
= 1 [A] 0<br />
2 − 1<br />
( 2− 1 ) − 2 <br />
= 1 [A] 0<br />
2 − 1<br />
− 1 <br />
(C.56)<br />
(C.57)<br />
(C.58)<br />
<strong>—</strong> The case 2 = 1 does not interest us here, as the ∗ state B should be a<br />
longer-lived one.<br />
• General solution for 2 6= 1 :<br />
y<br />
Initial value condition at =0:<br />
[B] = [B] <br />
+[B] <br />
[B] = × − 2 + 1 [A] 0<br />
2 − 1<br />
− 1 <br />
(C.59)<br />
(C.60)<br />
[B ( =0)]=0 (C.61)<br />
y<br />
= − 1 [A] 0<br />
2 − 1<br />
(C.62)
Appendix D 272<br />
Final solutions for [B ()]:<br />
i.e.<br />
[B ()] = − 1 [A] 0<br />
2 − 1<br />
× − 2 + 1 [A] 0<br />
2 − 1<br />
− 1 <br />
= 1 [A] 0<br />
2 − 1<br />
− 1 − 1 [A] 0<br />
2 − 1<br />
× − 2 <br />
[B ()] = 1 [A] 0<br />
1 − 2<br />
× ¡ − 2 − − 1 ¢<br />
(C.63)<br />
(C.64)<br />
(C.65)<br />
• In the time-resolved fluorescence measurements, we observe<br />
() ∝ [A ()] + [B ()]<br />
(C.66)<br />
with some unknown factor accounting for the (lower) transition probability of<br />
state B, i.e., the fluorescence-time profile () is<br />
() =[A] 0<br />
− 1 + 1 [A] 0<br />
1 − 2<br />
× ¡ − 2 − − 1 ¢<br />
=[A] 0<br />
− 1 + 1 [A] 0<br />
1 − 2<br />
× − 2 − 1 [A] 0<br />
1 − 2<br />
× − 1 <br />
= µ<br />
1 [A] 0<br />
× − 2 + [A]<br />
1 − 0<br />
− <br />
1 [A] 0<br />
× − 1 <br />
2 1 − 2<br />
= 2 × − 2 + 1 × − 1 <br />
(C.67)<br />
(C.68)<br />
(C.69)<br />
(C.70)<br />
This is the same result as would be obtained by a fit toasumoftwoexponentials<br />
with the rate constants 1 = 1 + 1 and 2 , i.e., time constants<br />
1 =( 1 + 1 ) −1 and 2 = 2 .
Appendix D 273<br />
Appendix D: Matrix methods<br />
This appendix has been taken from the “Introduction to Molecular Spectroscopy” script,<br />
not all subsections apply to chemical kinetics.<br />
D.1 Definition<br />
I Matrices: A matrix is a rectangular, × dimensional array of numbers :<br />
A =<br />
⎛<br />
⎜<br />
⎝<br />
⎞<br />
11 12 13 1<br />
21 22 23 2<br />
⎟<br />
. . . . ⎠<br />
1 2 3 <br />
(D.1)<br />
The are called the matrix elements.<br />
We will need to consider only × dimensional square matrices, which we can think<br />
of as a collection of column vectors a :<br />
⎛<br />
⎞<br />
11 12 1<br />
<br />
A = ⎜ 21 22 2<br />
⎟<br />
⎝ . . . ⎠<br />
(D.2)<br />
1 2 <br />
I Notation: Matrices will be denoted by bold symbols:<br />
A<br />
(D.3)<br />
I Matrix manipulation by computer programs: Matrix manipulations are carried<br />
out efficiently using computer programs (see, e.g., Press1992) or with symbolic algebra<br />
programs. 56<br />
56 The most commmon symbolic math programs are MathCad, MuPad, Maple, andMathematica.<br />
MathCad and Mathematica are available in the PC lab.
Appendix D 274<br />
D.2 Special matrices and matrix operations<br />
I Identity matrix: I = 1<br />
I = 1 =<br />
⎛<br />
⎜<br />
⎝<br />
10 0<br />
01 0<br />
. . .<br />
00 1<br />
⎞<br />
⎟<br />
⎠<br />
(D.4)<br />
We frequently use the Kronecker :<br />
= <br />
(D.5)<br />
where<br />
½<br />
1 for = <br />
=<br />
0 for 6= <br />
(D.6)<br />
I<br />
Diagonal matrices:<br />
⎛<br />
⎜<br />
⎝<br />
⎞<br />
11 0<br />
0 22 0<br />
⎟<br />
. . . . ⎠<br />
0 0 <br />
(D.7)<br />
I<br />
Block diagonal matrices:<br />
⎛<br />
⎞<br />
11 12 0<br />
<br />
⎜ 21 22 0<br />
⎟<br />
⎝ . . . . ⎠<br />
0 0 <br />
(D.8)<br />
I Complex conjugate (c.c.) of a matrix: A ∗<br />
To obtain the complex conjugate A ∗ of the matrix A, change to −:<br />
( ∗ ) <br />
=( ) ∗ (D.9)<br />
I Transpose of a matrix: <br />
To obtain the transpose A of the matrix A, interchange rows and colums:<br />
¡<br />
<br />
¢ =( ) (D.10)
Appendix D 275<br />
I Hermitian conjugate (“transpose conjugate”) of a matrix: A †<br />
To obtain the Hermitian conjugate (or “transpose conjugate” or “adjoint”) A † of the<br />
matrix A, take the complex conjugate and transpose:<br />
† = ¡ ∗¢ = ¡ ¢ ∗<br />
(D.11)<br />
where ¡<br />
<br />
† ¢ =( ) ∗ (D.12)<br />
I Hermitian matrices: AmatrixA is Hermitian, if<br />
A † =(A ∗ ) = A<br />
(D.13)<br />
i.e., ¡<br />
<br />
† ¢ =( ) ∗ = (D.14)<br />
I Inverse of a matrix: A −1<br />
AmatrixA −1 is the inverse of the original matrix A, if<br />
AA −1 = A −1 A = I<br />
(D.15)<br />
I Orthogonal matrices: AmatrixA is orthogonal, iftheinverseofA equals its transpose:<br />
A −1 = A <br />
(D.16)<br />
I Unitary matrices: AmatrixU is unitary, iftheinverseofU equals its transpose<br />
conjugate:<br />
U −1 = U † =(U ∗ ) <br />
(D.17)<br />
y<br />
U † U = I<br />
(D.18)<br />
I<br />
Trace of a matrix:<br />
tr (A) =<br />
X<br />
=1<br />
<br />
(D.19)<br />
I<br />
Matrix addition:<br />
C = A + B<br />
(D.20)<br />
with<br />
= + <br />
(D.21)
Appendix D 276<br />
I<br />
Multiplication by a scalar:<br />
C = A<br />
(D.22)<br />
with<br />
= <br />
(D.23)<br />
I<br />
Matrix multiplication:<br />
C = A · B = AB<br />
(D.24)<br />
with<br />
=<br />
X<br />
<br />
(D.25)<br />
=1<br />
Example:<br />
µ µ µ µ <br />
12 56 1 × 5+2× 71× 6+2× 8 19 22<br />
=<br />
34 78 3 × 5+4× 73× 6+4× 8 43 50<br />
(D.26)<br />
Notes:<br />
(1) Matrix multiplication is defined only if the number of columns of the first matrix<br />
is identical to the number of rows of the second matrix.<br />
(2)<br />
AB6= BA<br />
(D.27)<br />
I Singular matrices: AmatrixA is called singular, if its determinant vanishes:<br />
det (A) =|A| =0<br />
(D.28)
Appendix D 277<br />
D.3 Determinants<br />
I Determinant of a matrix: A determinant of a matrix is, in general, a number which<br />
is evaluated according to the rule<br />
det (A)=|A| =<br />
X<br />
<br />
=1<br />
(D.29)<br />
The are called the co-factors which are obtained by omitting the ’th row and ’th<br />
column (marked below in red) and multiplying by (−1) + :<br />
11 1 1<br />
. . .<br />
=(−1)<br />
¯¯¯¯¯¯¯¯¯¯¯<br />
+ 1 <br />
. . .<br />
1 <br />
¯¯¯¯¯¯¯¯¯¯¯<br />
(D.30)<br />
I<br />
Evaluation of simple determinants:<br />
¯ <br />
¯ = − <br />
(D.31)<br />
11 12 13<br />
21 22 23 = (D.32)<br />
¯ 31 32 33<br />
¯¯¯¯¯¯<br />
= 11 22 33 + 12 23 31 + 13 21 32<br />
− ( 13 22 31 + 11 23 32 + 12 21 33 )<br />
where we “extended” the original determinant according to the scheme<br />
11 12 13 11 12<br />
21 22 23 21 22<br />
(D.33)<br />
¯ 31 32 33<br />
¯¯¯¯¯¯ 31 32
Appendix D 278<br />
D.4 Coordinate transformations<br />
I Multiplication of a vector by a matrix: x 0 = Ax<br />
The multiplication of an -dimensional column vector x by an × matrix A gives a<br />
new vector x 0 with components corresponding to those of x in a transformed coordinate<br />
system.<br />
I Example: Multiplication of a 2-dim vector<br />
x =<br />
µ<br />
1<br />
2<br />
<br />
(D.34)<br />
by a matrix<br />
gives<br />
µ <br />
cos − sin <br />
A =<br />
sin cos <br />
(D.35)<br />
µ µ <br />
cos − sin 1<br />
Ax =<br />
(D.36)<br />
sin cos 2<br />
µ <br />
1 cos − <br />
=<br />
2 sin <br />
(D.37)<br />
1 sin + 2 cos <br />
= x 0 (D.38)<br />
I<br />
Figure D.1: Rotation of a coordinate system.
Appendix D 279<br />
I<br />
The result is a rotation:<br />
• Intheoriginalcoordinatesystem,thenewvectorx 0 corresponds to the result of<br />
an anti-clockwise rotation of x around .<br />
• Alternatively, we can say that x 0 isgiveninanewcoordinatesystemthatis<br />
obtained by a clockwise rotation of the old coordinate system around .<br />
I Rotation matrices and rotation operators: A is called a rotation matrix. Weshall<br />
also call A a rotation operator.<br />
I<br />
Exercise D.1: Show that Eqs. D.36 - D.38 does describe an anti-clockwise rotation<br />
of the vector x to the new vector x 0 and determine its coordinates using trigonometric<br />
arguments. ¤
Appendix D 280<br />
D.5 Systems of linear equations<br />
I Solutions of linear equations (I): Cramer’s rule. Matrices and determinants can<br />
be generally used for solving systems of linear equations of the type<br />
11 1 + 12 2 + 13 3 + 1 = 1<br />
21 1 + 22 2 + 23 3 + 2 = 2<br />
= <br />
(D.39)<br />
1 1 + 2 2 + 3 3 + <br />
= <br />
These equations can be written in matrix form:<br />
⎛<br />
⎞ ⎛ ⎞ ⎛ ⎞<br />
11 12 1 1 1<br />
<br />
⎜ 21 22 2<br />
<br />
⎟ ⎜ 2<br />
⎟<br />
⎝ . . . ⎠ ⎝ . ⎠ = <br />
⎜ 2<br />
⎟<br />
⎝ . ⎠<br />
1 2 <br />
or<br />
Ax= c<br />
(D.40)<br />
(D.41)<br />
Simple algebraic manipulations (see any math textbook) gives the expressions<br />
|A| = |A |<br />
(D.42)<br />
where |A| is the determinant of the coefficient matrix A and |A | is the respective<br />
determinant of |A| in which the th column is replaced by the 1 2 3 ,e.g.<br />
|A 2 | =<br />
⎛<br />
⎞<br />
11 1 1<br />
<br />
⎜ 21 2 2<br />
⎟<br />
⎝ . . . ⎠<br />
1 <br />
(D.43)<br />
Solutions for the : From Eq. D.42 we obtain the non-trivial solutions for the <br />
according to<br />
= |A |<br />
(D.44)<br />
|A|<br />
under the condition that the A matrix is not singular, i.e., the determinant of coefficients<br />
does not vanish<br />
|A| 6= 0<br />
(D.45)<br />
(The trivial and uninteresting solutions are 1 2 3 =0).<br />
I Solutions of linear equations (II): In the following, we shall only consider the special<br />
linear equations of the type<br />
11 1 + 12 2 + 13 3 + 1 =0<br />
21 1 + 22 2 + 23 3 + 2 =0<br />
= .<br />
1 1 + 2 2 + 3 3 + =0<br />
(D.46)
Appendix D 281<br />
or<br />
Ax=0<br />
(D.47)<br />
The trivial and uninteresting solutions of this equation are 1 2 3 =0.<br />
The interesting solutions require that the matrix of coefficients A is singular, i.e., the<br />
determinant of coefficients has to vanish<br />
|A| =0<br />
(D.48)<br />
This type of equations is encountered in eigenvalue problems.
Appendix D 282<br />
D.6 Eigenvalue equations<br />
Consider a set of linear equations as considered in the previous section which can be<br />
writtenintheform<br />
Ax= x<br />
(D.49)<br />
where<br />
(1) A is an × -dimensional matrix,<br />
(2) is a scalar constant, called an eigenvalue of A ( is any one of the set of <br />
eigenvalues of A), and<br />
(3) x is an -dimensional column vector (which can in general be complex), called<br />
the eigenvector of A belonging to the particular eigenvalue.<br />
Eq. D.49 is called an eigenvalue equation. It has the following property: The multiplication<br />
of x by (and hence that of x by the matrix A) changes the length of x (by<br />
the factor ), but not the direction.<br />
I Eigenvalues and eigenvectors of the molecular Hamiltonian: The determination<br />
of eigenvalues and eigenvectors of the molecular Hamiltonian H, i.e., the “energy<br />
matrix”, or the matrix of the Hamilton operator b , is the most important problem in<br />
spectroscopy (in general, it is the key problem!!).<br />
I Secular equation and eigenvalues λ : Eq. D.49 can be rewritten as<br />
(A − I) x = 0 (D.50)<br />
In order not to obtain the trivial solutions 1 = 2 = 3 = =0, det (A − I) has<br />
to vanish, i.e., det (A − I) =0. This is the so-called characteristic equation or<br />
secular equation:<br />
|A − I| =<br />
¯<br />
The secular equation has roots<br />
which are called the eigenvalues.<br />
11 − 12 1<br />
21 22 − 2<br />
. . .<br />
=0 (D.51)<br />
1 2 − ¯<br />
{ 1 2 } (D.52)
Appendix D 283<br />
I Eigenvectors x : For each eigenvalue , we can write an eigenvalue equation<br />
In matrix notation, this becomes<br />
Ax 1 = 1 x 1 (D.53)<br />
Ax 2 = 2 x 2 (D.54)<br />
. (D.55)<br />
Ax = x (D.56)<br />
with the diagonal eigenvalue matrix<br />
Λ =<br />
AX= X Λ<br />
⎛<br />
⎜<br />
⎝<br />
⎞<br />
1 0 0<br />
0 2 0<br />
⎟<br />
. . . ⎠<br />
0 0 <br />
(D.57)<br />
(D.58)<br />
The different eigenvectors x are usually normalized to unity, i.e.,<br />
|x| = ¡ x † x ¢ 12<br />
=1<br />
(D.59)<br />
I Matrix diagonalization: The matrix of the eigenvectors X can be determined by<br />
multiplying the equation<br />
AX= X Λ<br />
(D.60)<br />
from the left with X −1 ,whichgives<br />
X −1 AX= X −1 X Λ<br />
(D.61)<br />
i.e.,<br />
y<br />
X −1 AX= Λ<br />
(D.62)<br />
The determination of the eigenvector matrix is equivalent to finding a matrix X that<br />
transforms the matrix A into diagonal form (Λ).<br />
I<br />
Example:<br />
A =<br />
µ <br />
2 3<br />
310<br />
(D.63)<br />
Eigenvalue equations:<br />
Ax = x <br />
(D.64)<br />
or in matrix form:<br />
AX= X Λ<br />
(D.65)
Appendix E 284<br />
Secular equation:<br />
det (A − I) =<br />
¯ 2 − 3<br />
3 10− ¯<br />
=(2− )(10− ) − 9=0<br />
(D.66)<br />
(D.67)<br />
y<br />
0=20− 2 − 10 + 2 − 9= 2 − 12 +11 (D.68)<br />
12 =+ 12 2 ± s µ12<br />
2<br />
2<br />
− 11 = 6 ± √ 36 − 11 (D.69)<br />
=6± 5 (D.70)<br />
y<br />
Eigenvalues:<br />
1 =1 (D.71)<br />
2 =11 (D.72)<br />
Eigenvectors: Inserting the eigenvalues one by one into the eigenvalue equation yields:<br />
(1) µ µ µ µ <br />
2 3 11 11 11<br />
= <br />
310 1 =11<br />
21 21 21<br />
y<br />
µ<br />
−3<br />
1<br />
(2) µ µ µ µ <br />
2 3 12 12 12<br />
= <br />
310 2 =1<br />
22 22 22<br />
y µ <br />
1<br />
3<br />
<br />
(D.73)<br />
(D.74)<br />
(D.75)<br />
(D.76)<br />
Unnormalized eigenvalue matrix:<br />
µ <br />
−31<br />
1 3<br />
(D.77)<br />
Normalized eigenvalue matrix:<br />
X =<br />
⎛<br />
⎜<br />
⎝<br />
−3 1<br />
√ √<br />
10 10<br />
1<br />
√<br />
10<br />
3<br />
⎞<br />
⎟<br />
⎠<br />
√<br />
10<br />
(D.78)
Appendix E 285<br />
I Final note: In practice, matrix manipulations such as matrix inversion, the evaluation<br />
of determinants, the determination of eigenvalues and eigenvectors, etc., arecarriedout<br />
numerically by using computers.<br />
I<br />
References<br />
Press 1992 W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical<br />
Recipes in Fortran, Cambridge University Press, Cambridge, 1992. Versions are<br />
also available for C and Pascal.
Appendix E 286<br />
Appendix E: Laplace transforms<br />
The concept of Laplace and inverse Laplace transforms is extremely useful in chemical<br />
kinetics for two reasons:<br />
(1) They connect microscopic molecular properties and statistically averaged quantities:<br />
a) () ↔ ( ): Collision cross section vs. thermal rate constant for bimolecular<br />
reactions (section ??),<br />
b) () ↔ ( ): Specific rateconstantvs. thermal rate constant for unimolecular<br />
reactions (section 8),<br />
c) () ↔ ( ): Density of states vs. partition function in statistical rate<br />
theories 8).<br />
(2) They provide a convenient method for solving of differential equations (section<br />
3.4.2).<br />
I<br />
Definition E.1: The Laplace transform L [ ()] of a function () is defined as the<br />
integral<br />
Z ∞<br />
() =L [ ()] = () − <br />
(E.1)<br />
where<br />
0<br />
• is a real variable,<br />
• () is a real function of the variable with the property () =0for 0,<br />
• is a complex variable,<br />
• () =L [ ()] is a function of the variable .<br />
I<br />
Definition E.2: The inverse Laplace transform L −1 [ ()] of the function () is<br />
defined as the integral<br />
() =L −1 [ ()] = 1 Z<br />
2<br />
+∞<br />
−∞<br />
() <br />
(E.2)<br />
where<br />
• is an arbitrary real constant.
Appendix F 287<br />
I<br />
Notes:<br />
• Since L −1 [ ()] recovers the original function (), the pair of functions ()<br />
and () is said to form a Laplace pair.<br />
• ThepropertiesoftheLaplaceandinverseLaplacetransformscanbederivedfrom<br />
those of Fourier transforms (Zachmann 1972).<br />
• Laplace transforms can be computed using symbolic algebra computer programs<br />
(Maple, Mathematica, etc.). In favorable cases, it is also possible to obtain inverse<br />
Laplace transforms by computer programs. However, it is often more convenient<br />
to take the Laplace and inverse Laplace transforms from corresponding Tables. 57<br />
I<br />
Table E.1: Table of Laplace transforms.<br />
() () =L [ ()] () () =L [ ()]<br />
()<br />
()<br />
<br />
() = R ∞<br />
0<br />
() − <br />
() − ( =0) 1<br />
− <br />
1<br />
1<br />
<br />
1<br />
( − ) 2<br />
<br />
1<br />
2<br />
sin <br />
<br />
2 + 2<br />
2 2<br />
1<br />
3<br />
cos <br />
<br />
2 + 2<br />
−1<br />
Γ ()<br />
1<br />
1 ¡<br />
− ¢ 1<br />
( − )<br />
( − )( − )<br />
Γ () is the Gamma function. For integer , Γ () =( − 1)! (see Appendix F).<br />
I<br />
References:<br />
Houston 1996 P. L. Houston, <strong>Chemical</strong> <strong>Kinetics</strong> and Reaction Dynamics, McGraw-<br />
Hill, Boston, 2001.<br />
Steinfeld 1989 J. I. Steinfeld, J. S. Francisco, W. L. Haase, <strong>Chemical</strong> <strong>Kinetics</strong> and<br />
Dynamics, Prentice Hall, Englewood Cliffs, 1989.<br />
Zachmann 1972 H. G. Zachmann, Mathematik für Chemiker, Verlag Chemie, Weinheim,<br />
1972.<br />
57 Extensive Tables of Laplace transforms are given by (Steinfeld 1989).
Appendix F 288<br />
Appendix F: The Gamma function Γ (x)<br />
• If is a positive integer, the Gamma function is<br />
Γ () =( − 1)!<br />
(F.1)<br />
• The Gamma function smoothly interpolates between all points ( ) given by<br />
=( − 1)! for positive non-integer (cf. Fig. F.1).<br />
• The Gamma function is defined for all complex numbers with a positive real<br />
part as<br />
Z ∞<br />
Γ () = −1 − <br />
(F.2)<br />
0<br />
This integral converges for complex numbers with a positive real part. It has poles<br />
for complex numbers with a negative real part (cf. Fig. F.2).<br />
I<br />
Figure F.1: The Gamma function.
Appendix G 289<br />
I<br />
Figure F.2: For negative arguments the Gamma function has poles (at all negative<br />
integers).
Appendix G 290<br />
Appendix G: Ergebnisse der statistischen Thermodynamik<br />
In diesem Kapitel sollen wichtige Ergebnisse der statistischen Thermodynamik zusammengefasst<br />
werden. Im Vordergrund steht die Berechnung der thermodynamischen Zustandsfunktionen<br />
und die Berechnung von chemischen Gleichgewichten aus molekularen<br />
Eigenschaften. Wir benutzen die Ergebnisse der Quantenmechanik:<br />
• Ein Molekül kann nur bestimmte Energiezustände einnehmen.<br />
• Diese Energiezustände sind im Prinzip nicht kontinuierlich sondern diskret verteilt.<br />
• Jeder Energiezustand hat ein bestimmtes statistisches Gewicht . Das statistische<br />
Gewicht gibt an, wie oft ein Zustand bei einer bestimmten Energie vorkommt.<br />
Beispiele:<br />
=1 eindimensionaler harmonischer Oszillator<br />
=2 isotroper zweidimensionaler harmonischer Oszillator<br />
(z.B. Biegeschwingung von CO 2 )<br />
=2 +1Rotation eines linearen Moleküls<br />
(G.1)<br />
• Die Ergebnisse der statischen Thermodynamik unterscheiden sich, je nachdem<br />
ob das betrachtete System aus unterscheidbaren oder aus ununterscheidbaren<br />
Teilchen besteht:<br />
Beispiele für unterscheidbare Teilchen: Atome im Kristallgitter<br />
Beispiele für ununterscheidbare Teilchen: 1.) Gasmoleküle<br />
2.) Elektronen (Fermi-Statistik)<br />
(G.2)<br />
• Die anzuwendende Statistik hängt vom Spin der Teilchen ab:<br />
Spin gerade: Bose-Einstein-Statistik<br />
Spin ungerade: Fermi-Dirac-Statistik<br />
(G.3)<br />
Auf die Unterschiede zwischen der Bose-Einstein- und der Fermi-Dirac-Statistik<br />
wird an anderer Stelle eingegangen (⇒ Vorlesung “Statistische Thermodynamik”).<br />
• Im Grenzfall À gilt die Boltzmann-Statistik.<br />
• Im Folgenden werden behandelt: Elektronische Anregung, Schwingungsanregung,<br />
Rotationsanregung, Translationsanregung.<br />
Die Translation erfordert abhängig vom Problem manchmal eine etwas andere<br />
Behandlung (klassische statistische Thermodynamik bzw. Semiklassik). Quantenmechanik<br />
und klassische Mechanik stimmen im Ergebnis für À überein.
Appendix G 291<br />
G.1 Boltzmann-Verteilung<br />
Die Besetzungswahrscheinlichkeit eines Zustands wird durch die Boltzmann-Verteilung<br />
angegeben. Diese ergibt sich als die wahrscheinlichste Verteilung aus dem Boltzmann-<br />
Ansatz für die Entropie<br />
= ln <br />
(G.4)<br />
wird als die thermodynamische Wahrscheinlichkeit bezeichnet.<br />
Die Boltzmann-Verteilung kann hergeleitet werden, wenn man entsprechend dem 2.<br />
Hauptsatz den Maximalwert für sucht (Methode der Lagrange’schen Multiplikatoren).<br />
Hier setzen wir die Boltzmann-Verteilung als gegeben voraus (z.B. als empirisches, experimentelles<br />
Ergebnis):<br />
I Boltzmann-Verteilung: Für diskrete Energiezustände ist die Besetzungswahrscheinlichkeit<br />
eines Zustands proportional zu − :<br />
= <br />
∝ − <br />
(G.5)<br />
Bei Berücksichtigung einer möglichen Entartung von Zuständen (Entartungsfaktor bzw.<br />
statistisches Gewicht ) und mit einer Proportionalitätskonstante 0 zur Normierung<br />
gilt<br />
= <br />
= 0 × × − <br />
(G.6)<br />
Dabei ist die Zahl der Teilchen im Energiezustand (mit dem statistischen Gewicht<br />
) und die Gesamtzahl der Teilchen.<br />
Die Funktion gibt die Wahrscheinlichkeit der Besetzung des Zustands an. Die<br />
Normierungskonstante 0 ergibt sich aus der Normierungsbedingung, dass die Gesamtwahrscheinlichkeit<br />
zu 1 herauskommt:<br />
P<br />
<br />
= X = 0 × X − =1<br />
(G.7)<br />
<br />
<br />
<br />
y 0 1<br />
= P . (G.8)<br />
− <br />
y<br />
I<br />
Normierte Boltzmann-Verteilungsfunktion:<br />
= <br />
=<br />
− <br />
P − <br />
(G.9)<br />
I Molekulare Zustandssumme: Die Summe P − im Nenner der o.a. Gleichung<br />
wird als molekulare Zustandssumme bezeichnet:<br />
= P − <br />
(G.10)<br />
Die Zustandssumme ist die zentrale Grösse der statistischen Thermodynamik. ist<br />
von der Temperatur abhängig.
Appendix G 292<br />
I<br />
Anschauliche Bedeutung der molekularen Zustandssumme:<br />
= Zahl der bei der Temperatur im Mittel besetzten Energiezustände eines Molekül.<br />
I<br />
Boltzmann-Verteilungsfunktion für kontinuierlich verteilte Energiezustände:<br />
Für kontinuierlich verteilte Energiezustände ersetzt man durch () und erhält<br />
analog eine kontinuierliche Besetzungswahrscheinlichkeit:<br />
() = ()<br />
<br />
= () × − <br />
<br />
(G.11)<br />
I Zustandsdichte: () = Zustandsdichte (=Anzahl der Zustände pro Energie-<br />
Intervall):<br />
() = ()<br />
<br />
(G.12)
Appendix G 293<br />
G.2 Mittelwerte<br />
I Mittlere Energie ²: In der Thermodynamik interessieren Mittelwerte, z.B. die mittlere<br />
Energie . Für Größen, die durch eine Verteilungsfunktion beschrieben werden,<br />
erhält man den Mittelwert nach<br />
=<br />
P<br />
P <br />
<br />
=<br />
P<br />
<br />
P<br />
<br />
=<br />
P<br />
<br />
= X P<br />
<br />
= P<br />
<br />
mit = <br />
und P =1.<br />
Übergang zu einer kontinuierlichen Verteilungsfunktion liefert<br />
Z ∞<br />
()<br />
(G.13)<br />
=<br />
0<br />
Z ∞<br />
0<br />
()<br />
(G.14)<br />
Diese Ausdrücke vereinfachen sich, wenn () auf 1 normiert ist.<br />
I<br />
Allgemeine Berechnung der Mittelwerts y einer Größe y(²):<br />
X<br />
<br />
=<br />
<br />
X<br />
<br />
<br />
(G.15)<br />
bzw.<br />
Z ∞<br />
()()<br />
=<br />
0<br />
Z ∞<br />
()<br />
(G.16)<br />
0<br />
I<br />
Example G.1: Berechnung der mittleren Energie für ein 3-Niveausystem:<br />
0 =0 1 = 1 2 =2 1<br />
0 =3 2 =2 2 =1<br />
= X =6<br />
(G.17)<br />
(G.18)
Appendix G 294<br />
Ergebnis:<br />
=<br />
1 X<br />
= X <br />
<br />
= X <br />
(G.19)<br />
<br />
<br />
<br />
= 1 6 (3 0 +2 1 +1 2 ) (G.20)<br />
= 1 6 (3 0 +2 1 +2 1 ) (G.21)<br />
= 1 6 (0 + 4 1) (G.22)<br />
= 4 6 1 (G.23)<br />
= 2 3 1<br />
(G.24)<br />
¤<br />
I Mittlere Energie eines Moleküls: Bei Gültigkeit der Boltzmann-Verteilung kann die<br />
mittlere Energie eines Moleküle aus der molekularen Zustandssumme berechnet werden:<br />
=<br />
=<br />
1 X<br />
= X <br />
<br />
<br />
P<br />
− <br />
P<br />
− <br />
<br />
= X <br />
<br />
(G.25)<br />
(G.26)<br />
y<br />
=<br />
P<br />
− <br />
= 2<br />
<br />
= 2<br />
<br />
= 2<br />
<br />
= 2<br />
<br />
<br />
X<br />
<br />
X<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
2 − <br />
|× 2<br />
2<br />
(G.27)<br />
(G.28)<br />
¡<br />
<br />
− <br />
<br />
|(durch differenzieren zeigen) (G.29)<br />
X<br />
− <br />
(G.30)<br />
<br />
= 2 ln <br />
<br />
(G.31)<br />
(G.32)
Appendix G 295<br />
G.3 Anwendung auf ein Zweiniveau-System<br />
I<br />
Chemisch relevante Beispiele für Zweiniveau-Systeme:<br />
• Paramagnetische Atome u. Moleküle<br />
• NMR-Spektroskopie: Atomkerne mit Kernspin. Typischer Frequenzbereich:<br />
100 MHz.<br />
• Molekül mit cis-trans-Isomeren oder 2 Konformeren<br />
I<br />
Figure G.1: Zwei-Niveau-System: Energiezustände<br />
The Two Level System: Energy Levels<br />
<br />
<br />
<br />
<br />
g <br />
<br />
g <br />
<br />
Zustandssumme:<br />
Q <br />
i<br />
0 / kT 1<br />
/ kT<br />
1e<br />
1e<br />
1e<br />
i<br />
/ kT<br />
i<br />
g e<br />
1<br />
/ kT<br />
Besetzungsverhältnis:<br />
0/<br />
kT<br />
0 g0e<br />
N<br />
<br />
N<br />
N<br />
N<br />
Q<br />
1<br />
/ kT<br />
1 g1e<br />
N<br />
<br />
N<br />
<br />
Q<br />
1<br />
/ kT<br />
1 g1e<br />
<br />
0<br />
/ kT<br />
0 ge 0<br />
e<br />
1<br />
/ kT<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 173<br />
I<br />
Besetzungsverhältnis:<br />
mit<br />
1<br />
= 1 − 1<br />
<br />
(G.33)<br />
= 0 − 0 + 1 − 1<br />
=1+ − 1<br />
(G.34)<br />
(G.35)<br />
(1) Grenzfall für → 0:<br />
1<br />
→∞y − 1 → −∞ =0<br />
y Bei → 0 ist nur der tiefste Energiezustand besetzt.<br />
(G.36)
Appendix G 296<br />
(2) Grenzfall für →∞:<br />
1<br />
→ 0 y − 1 → −0 =1<br />
(G.37)<br />
y Bei →∞ergibt sich eine Gleichbesetzung der Energiezustände. Dies ist die<br />
maximale Besetzung im oberen Zustand!<br />
• Eine Besetzungsinversion ( 1 0 ) würde formal eine negative Temperatur<br />
erfordern (→ Laser).<br />
I<br />
Figure G.2: Zwei-Niveau-System: Boltzmann-Besetzung der Zustände (rot: unterer<br />
Zustand, blau: oberer Zustand).<br />
The Two Level System: Boltzmann Distribution<br />
N i<br />
N<br />
1<br />
= 1000 J/mol<br />
0.75<br />
0.5<br />
0.25<br />
0<br />
0<br />
250<br />
500<br />
750<br />
1000<br />
T /(K)<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 174<br />
I<br />
Mittlere thermische Energie:<br />
(1) 1 =125kJ mol (entsprechend =1000cm −1 ; z.B. C-C-Schwingung im IR<br />
oder Energieabstand zwischen cis-trans-Isomeren):<br />
½<br />
1<br />
= − 1 8 × 10 =<br />
−3 @ = 300K<br />
(G.38)<br />
0 023 @ =1000K<br />
(2) 1 =004 kJ mol (entsprechend Energieabstand zwischen Kernspinzuständen bei<br />
100 MHz NMR):<br />
1<br />
0<br />
= − 1 =0999984 @ =300K<br />
(G.39)<br />
Reihenentwicklung:<br />
−1 =1− 1<br />
+ ≈ 1 − 1<br />
(G.40)<br />
<br />
y sehr kleiner Besetzungsunterschied y Wunsch der NMR-Spektroskopiker zu<br />
immer größeren Frequenzen (Rekord heute 950 MHz).
Appendix G 297<br />
I<br />
Figure G.3: Zwei-Niveau-System: Mittlere thermische Energie.<br />
The Two Level System: Energy<br />
E<br />
1000<br />
= 1000 J/mol<br />
750<br />
500<br />
250<br />
0<br />
0<br />
250<br />
500<br />
750<br />
1000<br />
T /(K)<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 175<br />
I<br />
Spezifische Wärme:<br />
= <br />
<br />
(G.41)<br />
I<br />
Figure G.4: Zwei-Niveau-System: Spezifische Wärme.<br />
The Two Level System: Specific Heat<br />
c v<br />
5<br />
= 1000 J/mol<br />
4<br />
3<br />
2<br />
1<br />
0<br />
0<br />
250<br />
500<br />
750<br />
1000<br />
T /(K)<br />
<strong>Physical</strong> <strong>Chemistry</strong> III: <strong>Chemical</strong> <strong>Kinetics</strong> • © F. Temps, IPC Kiel 176
Appendix G 298<br />
G.4 Mikrokanonische und makrokanonische Ensembles<br />
Die statistische Thermodynamik macht Aussagen über statistische Ensembles<br />
(Ansammlungen von Systemen, Molekülen). Wir betrachten verschiedene Ensembles<br />
von miteinander wechselwirkenden Molekülen.<br />
I Der Begriff des Ensembles: Wir betrachten ein abgeschlossenes System bei konstantem<br />
Volumen, konstanter Zusammensetzung und konstanter Temperatur (bzw. bestimmter<br />
Energie (s.u.)). Ein Ensemble erhalten wir, wenn wir uns N Replikationen<br />
dieser Systeme denken, die wir unter Einhaltung bestimmter Regeln erhalten (Regel =<br />
Canon). Alle Systeme in dem Ensemble sollen im thermischen Gleichgewicht miteinander<br />
(sie dürfen zwischen einander Energie austauschen).<br />
Es werden verschiedene Sorten von Ensembles unterschieden:<br />
• Mikrokanonisches Ensemble: Ensemble von Systemen (z.B. Atomen,<br />
Molekülen) mit konstanter Teilchenzahl, konstantem Volumen, konstanter Energie.<br />
• Makrokanonisches Ensemble (kurz kanonisches Ensemble genannt): Ensemble<br />
von Systemen (z.B. Atomen, Molekülen) mit konstanter Teilchenzahl, konstantem<br />
Volumen, konstanter Temperatur.<br />
• Großkanonisches Ensemble: Ensemble von offenen Systemen (z.B. Teilchen,<br />
Molekülen) mit konstantem Volumen, konstanter Temperatur, die zwischeneinander<br />
Teilchen austauschen können.<br />
I Systemzustandssumme (= kanonische Zustandssumme): Molekulare Zustandssumme:<br />
= P alle Molekülzustände − <br />
(G.42)<br />
Systemzustandssumme (= kanonische Zustandssumme):<br />
= P alle Systemzustände − <br />
(G.43)<br />
Innere Energie des Systems:<br />
= 2 ln <br />
<br />
(G.44)<br />
Bei mehreren Teilchensorten (A, B, C, . . . ): Zustandsummen sind multiplikativ,<br />
da die Energien additiv sind, y<br />
= × × × <br />
(G.45)
Appendix G 299<br />
• Systemzustandssumme für unterscheidbare Teilchen der Sorte (z.B.<br />
Kristall):<br />
Y <br />
= = <br />
<br />
(G.46)<br />
y für 1mol:<br />
= 2 ln <br />
<br />
<br />
2<br />
ln <br />
= <br />
<br />
(G.47)<br />
• Systemzustandssumme für nicht unterscheidbare Teilchen der Sorte (z.B.<br />
Gas):<br />
y für 1mol ( = ):<br />
= <br />
<br />
!<br />
(G.48)<br />
= 2 ln <br />
<br />
= 2 <br />
ln <br />
!<br />
(G.49)
Appendix G 300<br />
G.5 Statistische Interpretation der thermodynamischen Zustandsgrößen<br />
I<br />
Innere Energie:<br />
= 2 ln <br />
<br />
(G.50)<br />
I<br />
Helmholtz-Energie:<br />
<br />
= − <br />
(G.51)<br />
y<br />
y<br />
<br />
<br />
µ <br />
<br />
<br />
<br />
=<br />
<br />
µ <br />
<br />
<br />
2<br />
= − <br />
2<br />
<br />
<br />
− <br />
−<br />
µ <br />
<br />
<br />
− <br />
= − ln <br />
= −<br />
<br />
2<br />
<br />
= − ln <br />
(G.54)<br />
= − ln <br />
(G.52)<br />
(G.53)<br />
(G.55)<br />
I<br />
Entropie:<br />
= −<br />
µ <br />
<br />
<br />
= − <br />
<br />
= (G.56)<br />
I<br />
Druck:<br />
= −<br />
µ <br />
<br />
<br />
µ ln <br />
= <br />
<br />
<br />
= (G.57)
Appendix G 301<br />
G.6 Chemisches Gleichgewicht<br />
y<br />
bzw.<br />
+ À <br />
=exp<br />
µ− ∆ <br />
ª<br />
<br />
∆ ª = − ln <br />
(G.58)<br />
(G.59)<br />
(G.60)<br />
I Gleichgewichtskonstante: Wir berechnen zunächst<br />
= − ln ,<br />
(G.61)<br />
= <br />
<br />
indem wir <br />
!<br />
ln − anwenden:<br />
etc.. . . einsetzen und die Stirling’sche Näherung ln ! =<br />
= − [ln +ln +ln ]<br />
"<br />
#<br />
= − ln <br />
<br />
<br />
<br />
! +ln <br />
! +ln <br />
!<br />
= − [( ln − ln !)<br />
+( ln − ln !)<br />
+( ln − ln !)]<br />
= − [( ln − ln + )<br />
+( ln − ln + )<br />
+( ln − ln + )]<br />
(G.62)<br />
(G.63)<br />
(G.64)<br />
(G.65)<br />
I Gleichgewichtsbedingung: Im chemischen Gleichgewicht muss gelten<br />
µ <br />
=0 (G.66)<br />
<br />
y<br />
0= [( ln − ln + )<br />
<br />
+( ln − ln + )<br />
+( ln − ln + )]<br />
(G.67)<br />
µ<br />
<br />
1<br />
= ln − − ln +1<br />
<br />
µ<br />
<br />
<br />
1<br />
+ ln − − ln +1<br />
<br />
µ<br />
<br />
<br />
1<br />
− ln − − ln +1<br />
(G.68)<br />
<br />
=(ln − ln )+(ln − ln ) − (ln − ln ) (G.69)<br />
=ln <br />
<br />
− ln <br />
<br />
(G.70)
Appendix G 302<br />
y<br />
<br />
<br />
=<br />
<br />
<br />
(G.71)<br />
Übergang auf Teilchendichten und volumenbezogenen Zustandssummen:<br />
y<br />
( )<br />
( )( ) = ( )<br />
( )( )<br />
=<br />
( )<br />
( )( ) = ∗ <br />
∗ ∗ <br />
(G.72)<br />
(G.73)<br />
Bezug auf die getrennten molekularen Nullpunktsenergien:<br />
=<br />
0 <br />
0 0 <br />
× −∆ 0 <br />
(G.74)<br />
mit 0 = die auf das Volumen bezogene Molekülzustandssumme:<br />
0 = <br />
<br />
× × × (G.75)<br />
I Endergebnis: Die Indices ∗ und 0 wurden hier zur Klarheit benutzt. Der Einfachheit<br />
halber werden diese üblicherweise weggelassen.<br />
y<br />
=<br />
<br />
<br />
× −∆ 0 <br />
(G.76)
Appendix G 303<br />
G.7 Zustandssumme für elektronische Zustände<br />
Die elektronische Zustandssumme muss unter Einschluss aller elektronischen Entartungen<br />
explizit berechnet werden:<br />
= X <br />
− <br />
(G.77)<br />
Speziell sind zu berücksichtigen:<br />
• Spinentartung (Spinmultiplizität ):<br />
=2 +1<br />
(G.78)<br />
• elektronischer Bahndrehimpuls ( (Atome) bzw. Λ (lin. Moleküle) bzw. Symmetrierasse<br />
oder für nichtlin. Moleküle):<br />
• elektronische Feinstrukturkomponenten (Index im Atom-Termsymbol):<br />
=2 +1<br />
(G.79)<br />
Nur niedrig liegende Elektronenzustände liefern einen Beitrag. Die Zustandsumme ist<br />
explizit auszurechnen.
Appendix G 304<br />
G.8 Zustandssumme für die Schwingungsbewegung<br />
I<br />
Harmonischer Oszillator:<br />
= =0 1 2 (G.80)<br />
I<br />
Schwingungszustandssumme:<br />
∞X<br />
= − =<br />
y<br />
=0<br />
∞X<br />
=0<br />
−·<br />
mit = <br />
<br />
(G.81)<br />
= © 1+ − + −2 + −3 + ª mit = − 1 (G.82)<br />
=1+ + 2 + 3 + <br />
(G.83)<br />
= 1 für 1<br />
1 − <br />
(G.84)<br />
=<br />
∙ µ<br />
1<br />
1 − = 1 − exp − <br />
¸−1<br />
− <br />
(G.85)<br />
I<br />
s Oszillatoren:<br />
=<br />
Y<br />
=1<br />
∙ µ<br />
1 − exp − ¸−1<br />
<br />
<br />
(G.86)<br />
I<br />
typische Werte bei Zimmertemperatur:<br />
≈ 1 10<br />
(G.87)<br />
I Grenzwert µ von Q für T →∞bzw. für ν → 0: Für →∞bzw. für → 0<br />
kann exp − <br />
entwickelt werden:<br />
<br />
µ<br />
exp − <br />
≈ 1 − <br />
+ <br />
(G.88)<br />
y<br />
1<br />
1<br />
= µ<br />
1 − exp − = µ<br />
1 − 1 − (G.89)<br />
<br />
+ <br />
y<br />
≈ <br />
(G.90)<br />
<br />
Für klassische Oszillatoren:<br />
≈<br />
Y<br />
=1<br />
<br />
<br />
(G.91)
Appendix G 305<br />
I<br />
Abweichungen vom harmonischen Oszillator:<br />
• Für anharmonische Oszillatoren ist die Zustandssumme am besten durch explizite<br />
numerische Berechnung nach P ∞<br />
=0 − zu ermitteln.<br />
• Torsionsschwingungen können bei tiefen Temperaturen als harmonische Oszillatoren<br />
beschrieben werden, bei hohen Temperaturen als freie 1D-Rotatoren. Im<br />
Zwischenbereich muss die Zustandssumme ebenfalls explizit berechnet werden<br />
(wobei die Energiezustände für den gehinderten 1D-Rotator zugrundegelegt werden<br />
müssen).
Appendix G 306<br />
G.9 Zustandssumme für die Rotationsbewegung<br />
I<br />
lineare Moleküle:<br />
= ~2<br />
( +1) =0 1 2 (G.92)<br />
2<br />
~ 2<br />
2 = <br />
(G.93)<br />
I<br />
Rotationszustandssumme:<br />
(2) = 1 <br />
= 1 <br />
= 1 <br />
∞X<br />
− <br />
(G.94)<br />
=0<br />
∞X<br />
(2 +1) −~2 (+1)2<br />
=0<br />
Z ∞<br />
(G.95)<br />
(2 +1) −~2 (+1)2 (G.96)<br />
=0<br />
y<br />
(2) = 2<br />
~ 2 = <br />
<br />
(G.97)<br />
I<br />
Symmetriezahl:<br />
<br />
(G.98)<br />
I Rotationszustandssumme für nichtlineare Moleküle: ist nur für symmetrische<br />
Kreisel analytisch darstellbar, nicht für asymmetrische Kreisel. Für nicht zu tiefe Temperaturen<br />
ist<br />
(3) = 12<br />
<br />
µ 2<br />
~ 2 32<br />
( ) 12 = 12<br />
()32 () −12 (G.99)<br />
I<br />
typische Werte bei Zimmertemperatur:<br />
≈ 100 1000<br />
(G.100)
Appendix G 307<br />
G.10 Zustandssumme für die Translationsbewegung<br />
I<br />
Teilchen im 1D-Kasten:<br />
³<br />
= 2 <br />
´2<br />
8 × =1 2 3 (G.101)<br />
<br />
I<br />
1D-Translationszustandssumme:<br />
y<br />
=<br />
=<br />
∞X<br />
− =<br />
<br />
Z ∞<br />
1<br />
−2 2 8 2 <br />
(G.102)<br />
µ 2<br />
2 12<br />
× ¡ in cm −1¢ (G.103)<br />
<br />
<br />
= µ 2<br />
2 12<br />
(G.104)<br />
I<br />
Teilchen im 3D-Kasten:<br />
" µ 2 µ 2 µ # 2<br />
= 2<br />
8 × <br />
+ +<br />
<br />
(G.105)<br />
I<br />
3DTranslationszustandssumme:<br />
y<br />
µ 32 2<br />
=<br />
× <br />
<br />
<br />
2<br />
=<br />
¡<br />
in cm<br />
−3 ¢ (G.106)<br />
µ 2<br />
2 32<br />
(G.107)<br />
I<br />
typische Werte:<br />
≈ 10 24 × <br />
(G.108)<br />
y<br />
<br />
<br />
≈ 10 24 cm −3<br />
(G.109)