Reaction mechanisms RRKM theory

The reactions of the bare sodium ion with all neutrals were determined to proceed via a three-body association mechanism and the rate constants measured cover a large range from a slow association reaction with NH3 to a near-collision rate with CH3OC2H4OCH3 (DMOE). The lifetimes of the intermediate complexes obtained using parameterized trajectory results and the experimental rates compare fairly well with predictions based on RRKM theory. The calculations also accounted for the large isotope effect observed for the more rapid clustering of ND3 than NH3 to Na+. 

RRKM theory represents the state of the art in understanding unimolecular reaction kinetics. However, because of the rigorous treatment of molecular energetics and quantum mechanics, it requires rather sophisticated numerical software to evaluate the rate constant. Computer programs to evaluate RRKM rate expressions are widely available examples are UNIMOL by Gilbert and Smith [143], and a program by Hase and Bunker [166]. 

In addition to experiments, a range of theoretical techniques are available to calculate thermochemical information and reaction rates for homogeneous gas-phase reactions. These techniques include ab initio electronic structure calculations and semi-empirical approximations, transition state theory, RRKM theory, quantum mechanical reactive scattering, and the classical trajectory approach. Although still computationally intensive, such techniques have proved themselves useful in calculating gas-phase reaction energies, pathways, and rates. Some of the same approaches have been applied to surface kinetics and thermochemistry but with necessarily much less rigor. 

In more detail, our approach can be briefly summarized as follows gas-phase reactions, surface structures, and gas-surface reactions are treated at an ab initio level, using either cluster or periodic (plane-wave) calculations for surface structures, when appropriate. The results of these calculations are used to calculate reaction rate constants within the transition state (TS) or Rice-Ramsperger-Kassel-Marcus (RRKM) theory for bimolecular gas-phase reactions or unimolecular and surface reactions, respectively. The structure and energy characteristics of various surface groups can also be extracted from the results of ab initio calculations. Based on these results, a chemical mechanism can be constructed for both gas-phase reactions and surface growth. The film growth process is modeled within the kinetic Monte Carlo (KMC) approach, which provides an effective separation of fast and slow processes on an atomistic scale. The results of Monte Carlo (MC) simulations can be used in kinetic modeling based on formal chemical kinetics. 

Nowadays, the basic framework of our understanding of elementary processes is the transition state or activated complex theory. Formulations of this theory may be found in refs. 1—13. Recent achievements have been the Rice—Ramsperger—Kassel—Marcus (RRKM) theory of unimol-ecular reactions (see, for example, ref. 14 and Chap. 4 of this volume) and the so-called thermochemical kinetics developed by Benson and co-workers [15] for estimating thermodynamic and kinetic parameters of gas phase reactions. Computers are used in the theory of elementary processes for quantum mechanical and statistical mechanical computations. However, this theme will not be discussed further here. 

This standard mechanistic analysis has a long successful history. Organic chemistry textbooks are filled with PESs and discussions of the implication of single-step versus multiple-step mechanisms, concerted TSs, and so on. - Transition state theory (TST) and Rice-Ramsperger-Kassel-Marcus (RRKM) theory provide tools for predicting rates based upon simple assumptions built upon the notion of reaction on the PES following the reaction coordinate. " ... 

TST22.23 also makes the statistical approximation and invokes an equihbrium between reactant and TS. TST invokes constant temperature instead of a micro-canonical ensemble as in RRKM theory. Using statistical mechanics, the reaction rate is given by the familiar equation... 

With these theories in mind, we now turn to a number of examples of organic reactions studied using direct dynamics. In all of these cases, some aspect of the application of the statistical approximation is found to be in error. At minimum, the collected weight of these trajectory studies demonstrates the caution that need be used when applying TST or RRKM. More compelling though is that these studies question the very nature of the meaning of reaction mechanism. 

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

Modern unimolecular theory has its origins in the work of Rice, Ramsberger and Kassel [44] who investigated the rate of dissociation of a molecule as a function of energy. Marcus and Rice [44] subsequently extended the theory to take account of quantum mechanical features. This extended theory, referred to as RRKM theory, is currently the most widely used approach and is usually the point of departure for more sophisticated treatments of unimolecular reactions. The key result of RRKM theory is that the microcanonical rate coefficient can be expressed as... 

A later modification of the RRK theory presented by R. A. Marcus (J. Chem. Phys., 1952, 20, 359) (Nobel Prize in 1992) resulted in the so-called RRKM theory. In this case, the mechanism of the reaction consists of the following steps... 

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 experimental results were interpreted in line with the reaction mechanism above mentioned, and discussed in the light of RRKM theory which quantitatively deals with the behavior of chemical activation system such as the decomposition rates of hot alkyl radicals. 

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