RRKM theory assumptions

In the statistical description of ununolecular kinetics, known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory [4,7,8], it is assumed that complete IVR occurs on a timescale much shorter than that for the unimolecular reaction [9]. Furdiemiore, to identify states of the system as those for the reactant, a dividing surface [10], called a transition state, is placed at the potential energy barrier region of the potential energy surface. The assumption implicit m RRKM theory is described in the next section. 

RRKM theory assumes a microcanonical ensemble of A vibrational/rotational states within the energy interval E E + dE, so that each of these states is populated statistically with an equal probability [4]. This assumption of a microcanonical distribution means that the unimolecular rate constant for A only depends on energy, and not on the maimer in which A is energized. If N(0) is the number of A molecules excited at / =... 

The rapid IVR assumption of RRKM theory means that a microcanonical ensemble is maintained as the A molecnles decompose so that, at any time t, k(E) is given by... 

Figure A3,12.2(a) illnstrates the lifetime distribution of RRKM theory and shows random transitions among all states at some energy high enongh for eventual reaction (toward the right). In reality, transitions between quantum states (though coupled) are not equally probable some are more likely than others. Therefore, transitions between states mnst be snfficiently rapid and disorderly for the RRKM assumption to be mimicked, as qualitatively depicted in figure A3.12.2(b). The situation depicted in these figures, where a microcanonical ensemble exists at t = 0 and rapid IVR maintains its existence during the decomposition, is called intrinsic RRKM behaviour [9].

The above describes the fundamental assumption of RRKM theory regarding the intramolecular dynamics of A. The RRKM expression for k E) is now derived. 

In deriving the RRKM rate constant in section A3.12.3.1. it is assumed that the rate at which reactant molecules cross the transition state, in the direction of products, is the same rate at which the reactants fonn products. Thus, if any of the trajectories which cross the transition state in the product direction return to the reactant phase space, i.e. recross the transition state, the actual unimolecular rate constant will be smaller than that predicted by RRKM theory. This one-way crossing of the transition state, witii no recrossmg, is a fiindamental assumption of transition state theory [21]. Because it is incorporated in RRKM theory, this theory is also known as microcanonical transition state theory. 

In the above discussion it was assumed that the barriers are low for transitions between the different confonnations of the fluxional molecule, as depicted in figure A3.12.5 and therefore the transitions occur on a timescale much shorter than the RRKM lifetime. This is the rapid IVR assumption of RRKM theory discussed in section A3.12.2. Accordingly, an initial microcanonical ensemble over all the confonnations decays exponentially. However, for some fluxional molecules, transitions between the different confonnations may be slower than the RRKM rate, giving rise to bottlenecks in the unimolecular dissociation [4, ]. The ensuing lifetime distribution, equation (A3.12.7), will be non-exponential, as is the case for intrinsic non-RRKM dynamics, for an mitial microcanonical ensemble of molecular states. 

Because QRRK theory was developed long before computing became readily available, it had to employ significant physical approximations to obtain a tractable result. The most significant assumption was that the molecule is composed of s vibrational modes with identical frequency i and that other molecular degrees of freedom are completely ignored. RRKM theory relies on neither approximation and thus has a much sounder physical basis. In the limit of infinite pressure, RRKM theory matches the transition state theory discussed in Section 10.3. 

RRKM theory, an approach to the calculation of the rate constant of indirect reactions that, essentially, is equivalent to transition-state theory. The reaction coordinate is identified as being the coordinate associated with the decay of an activated complex. It is a statistical theory based on the assumption that every state, within a narrow energy range of the activated complex, is populated with the same probability prior to the unimolecular reaction. The microcanonical rate constant k(E) is given by an expression that contains the ratio of the sum of states for the activated complex (with the reaction coordinate omitted) and the total density of states of the reactant. The canonical k(T) unimolecular rate constant is given by an expression that is similar to the transition-state theory expression of bimolecular reactions. 

First, we want to derive an expression for the microcanonical rate constant k(E) when the total internal energy of the reactant is in the range E to E + dE. From Eq. (7.43), the rate of reaction is given by the rate of disappearance of A or, equivalently, by the rate at which activated complexes A pass over the barrier, i.e., the flow through the saddle-point region. The essential assumptions of RRKM theory are equivalent to the assumptions underlying transition-state theory. 

The first assumption, that phase space is populated statistically prior to reaction, implies that the ratio of activated complexes to reactants is obtained by the evaluation of the ratio between the respective volumes in phase space. If this assumption is not fulfilled, then the rate constant k(E, t) may depend on time and it will be different from rrkm(E). If, for example, the initial excitation is localized in the reaction coordinate, k(E,t) will be larger than A rrkm(A). However, when the initially prepared state has relaxed via IVR, the rate constant will coincide with the predictions of RRKM theory (provided the other assumptions of the theory are fulfilled). 

Perhaps the point to emphasise in discussing theories of translational energy release is that the quasiequilibrium theory (QET) neither predicts nor seeks to describe energy release [576, 720], Neither does the Rice— Ramspergei Kassel—Marcus (RRKM) theory, which for the purposes of this discussion is equivalent to QET. Additional assumptions are necessary before QET can provide a basis for prediction of energy release (see Sect. 8.1.1) and the nature of these assumptions is as fundamental as the assumption of energy randomisation (ergodic hypothesis) or that of separability of the transition state reaction coordinate (Sect. 2.1). The only exception arises, in a sense by definition, with the case of the loose transition state [Sect. 8.1.1(a)]. 

The RRKM theory is a transition state theory with the reaction coordinate treated classically. It inherits any defects of the parent, separability of coordinates, non-equilibrium effects, and the assumption of unit transmission coefficient (trajectories do not turn back to regenerate X ). It is expected to give an upper bound to the reaction rate in cases where tunnelling through the potential energy barrier is... 

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. " ... 

One of the explicit assumptions of RRKM theory, and of transition state theory, is that of a unique dividing surface on the reaction path, which acts as a point of no return. This assumption leads to the conclusion that the rate of reaction can be equated to the rate at which trajectories cross this dividing surface in the direction of products. This assumption is not correct - it has been known for some time that some trajectories return after passing through the transition... 

The basic assumptions of SACM appear to be entirely different from the RRKM theory. In SACM the adiabatic channels do not mix, whereas in RRKM theory the rate coefficient is calculated on the assumption of rapid redistribution of vibrational energy into the reaction coordinate. However, as stated above, the adiabatic property of SACM does not hold strictly, it is only necessary for the counting of channels to be correct. In this sense SACM can be thought of as a fully quantized version of RRKM theory, as stated in Chapter 2. 

With this replacement of the strong collider assumption now commonplace, the term RRKM theory has become largely synonymous with quantum TST for unimolecular reactions, and we use this terminology here. The foundations of RRKM theory have been tested in depth with a wide variety of inventive theoretical and experimental studies [9]. While these tests have occasionally indicated certain limitations in its applicability, for example to timescales of a picosecond or longer, the primary conclusion remains that RRKM theory is quantitatively valid for the vast majority of conditions of importance to chemical kinetics. The H + O2 HO2 OH + O reaction is an example of an important reaction where deviations from RRKM predictions are significant [10, 11]. The foundations of RRKM theory and TST have been aptly reviewed in various places [7, 9, 12-15]. Thus, the present chapter begins with only a brief... 

The key further assumption of RRKM theory is that there is no recrossing of the dividing surface. In other words, it is assumed that all trajectories passing through the transition state in the direction of products continue directly on to products. With this assumption, the reactivity function is approximated by a step function in the velocity through the dividing surface, yielding... 

It is important to note that this assumption yields an RRKM rate coefficient, RRKM, that is an upper bound to the ergodic rate coefficient, ergodic, since every reactive trajectory (with xr J) necessarily has a positive velocity through the dividing surface. Thus, RRKM theory may be implemented in a variational manner, with the best approximation to ergodic obtained from the dividing surface S that provides the smallest rrkm-... 

The fundamental assumption of RRKM theory is that the classical motion of the reactant is sufficiently chaotic so that a micro-canonical ensemble of states is maintained as the reactant decomposes [6,324]. This assumption is often referred to as one of a rapid intramolecular vibrational energy redistribution (IVR) [12]. By making this assumption, at any time k E) is given by Eq. (62). As a result of the fixed time-independent rate constant k(E), N(t) decays exponentially, i.e.. 

The variation of the first-order rate coefficient with pressure can be interpreted in terms of rrk theory using s = 6, and by rrkm theory on the assumption that all of the vibrations are active . ... 

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