Transition state theory and statistical mechanics

Natural Atomic Orbital and Natural Bond Orbital Analysis 230 9.7 Computational Considerations 232 9.8 Examples 232 References 234 10 Molecular Properties 235 104 Examples 236 References 294 12 Transition State Theory and Statistical Mechanics 296 12.1 Transition State Theory 296 12.2 Statistical Mechanics 298 12.2.1 ans 299 12.2.2 300... 

Table 6.6 lists the high pressure limit kinetic parameters for the elementary reaction steps in this complex phenyl + O2 reaction system. These parameters are derived from the canonical transition state theory, the statistical mechanics from the DFT and ab initio data and from evaluation of literature data. Rate constants to all channels illustrated are calculated as function of temperature at different pressure. A reduced mechanism is proposed in Appendix F for the Phenyl + O2 system, for a temperature range of 600K different pressures 0.01 atm, 0.1 atm, latm, and 10 atm. 

RRKM theory is a microcanonical transition state theory and as such, it gives the connection between statistical unimolecular rate theory and the transition state theory of thermal chemical reaction rates. Isomerization or dissociation of an energized molecule A is assumed in RRKM theory to occur via the mechanism... 

The early to mid 1930s was a time of intensive activity in the formulation of transition-state theory (TST). Laidler and King [1] have provided an excellent review of the early history of TST, tracing the development of rate theories using treatments based upon thermodynamics, kinetic theory, and statistical mechanics, and focusing on Eyring s 1935 contribution to the formulation of TST [2]. A snapshot of the state of the development of TST and some of the controversy surrounding it in 1937 is captured in volume 34 of the Transactions of the Faraday... 

An important area of application for QM methods has been determining and describing reaction pathways, energetics, and transition states for reaction processes between small species. QM-derived first and second derivatives of energy calculated at stable and saddle points on PES can be used under statistical mechanics formulations [33, 34] to yield enthalpies and free energies of structures in order to determine their reactivity. Transition state theory and idealized thermodynamic relationships (e.g., AG[Po—>P] = kTln[P/Po]) allow temperature and pressure regimes to be spanned when addressing simple gas phase and gas-surface interactions. 

In this book, we demonstrate the use of transition-state theory to describe catalytic reactions on surfaces. In order to do this we start by treating the kinetics of catalytic reactions (Chapter 2) and provide some background information on important catalytic processes (Chapter 3). In Chapter 4 we introduce the statistical mechanical basis of transition-state theory and apply it to elementary surface reactions. Chapter 5 deals with the physical justification of the transition-state theory. We also discuss the consequences of media effects and of lateral interactions between adsorbates on surfaces for the kinetics. In the final chapter we present the principles of catalytic kinetics, based on the application of material given in earlier chapters. 

Chemical dynamics simulations of the gas phase 5 2 reactions of methyl halides have been carried out at many different levels of theory and compared with experimental measurements and predictions based on transition state theory and RRKM (Rice-Ramsperger-Kassel-Marcus) theory. Although many 5 2 reactions occur by the traditional pre-reaction complex, transition state, post-reaction complex mechanism, three additional non-statistical mechanisms were detected when the F -CH3-I reaction was analysed at an atomic level (i) a direct rebound mechanism where F attacks the backside of the carbon and CH3-F separates (bounces off) from the iodine ion, (ii) a direct stripping mechanism where F approaches CH3-I from the side and strips away the CH3 group, and (iii) an indirect reaction where the pre-reaction complex activates the C-I bond causing a CH3-I rotation and then the 5 2 reaction. The presence of these processes demonstrate that three non-statistical effects, (i) recrossing of the transition state is important, (ii) the transfer of the translational energy from the reactants into the rotational and vibrational modes of the substrate is inefficient, and (iii) there is... 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

The derivation of the transition state theory expression for the rate constant requires some ideas from statistical mechanics, so we will develop these in a digression. Consider an assembly of molecules of a given substance at constant temperature T and volume V. The total number N of molecules is distributed among the allowed quantum states of the system, which are determined by T, V, and the molecular structure. Let , be the number of molecules in state i having energy e,- per molecule. Then , is related to e, by Eq. (5-17), which is known as theBoltzmann distribution. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

A well defined theory of chemical reactions is required before analyzing solvent effects on this special type of solute. The transition state theory has had an enormous influence in the development of modern chemistry [32-37]. Quantum mechanical theories that go beyond the classical statistical mechanics theory of absolute rate have been developed by several authors [36,38,39], However, there are still compelling motivations to formulate an alternate approach to the quantum theory that goes beyond a theory of reaction rates. In this paper, a particular theory of chemical reactions is elaborated. In this theoretical scheme, solvent effects at the thermodynamic and quantum mechanical level can be treated with a fair degree of generality. The theory can be related to modern versions of the Marcus theory of electron transfer [19,40,41] but there is no... 

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