RRKM theory rate constant

The RRKM theory was designed to account for the pressure dependence of the unimolecular rate constant. The theory has been successfully applied to many reactions, e.g., cyclopropane propene, CgHg C2H4 + and CH3NC Of these the latter is a particularly... 

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 / =... 

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

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. 

As a result of possible recrossings of the transition state, the classical RRKM lc(E) is an upper bound to the correct classical microcanonical rate constant. The transition state should serve as a bottleneck between reactants and products, and in variational RRKM theory [22] the position of the transition state along q is varied to minimize k E). This minimum k E) is expected to be the closest to the truth. The quantity actually minimized is N (E - E ) in equation (A3.12.15). so the operational equation in variational RRKM theory is... 

For some systems qiiasiperiodic (or nearly qiiasiperiodic) motion exists above the unimoleciilar tlireshold, and intrinsic non-RRKM lifetime distributions result. This type of behaviour has been found for Hamiltonians with low uninioleciilar tliresholds, widely separated frequencies and/or disparate masses [12,, ]. Thus, classical trajectory simulations perfomied for realistic Hamiltonians predict that, for some molecules, the uninioleciilar rate constant may be strongly sensitive to the modes excited in the molecule, in agreement with the Slater theory. This property is called mode specificity and is discussed in the next section. 

RRKM theory, since steps are expected in M( ) even if all the states of the reactant do not participate in p( ). However, if the measured tln-eshold rate constant k(Eo) equals the inverse of the accurate anliannonic density of states for the reactant (difficult to detemiine), RRKM theory is verified. 

Figure B2.5.7 shows the absorption traces of the methyl radical absorption as a fiinction of tune. At the time resolution considered, the appearance of CFt is practically instantaneous. Subsequently, CFl disappears by recombination (equation B2.5.28). At temperatures below 1500 K, the equilibrium concentration of CFt is negligible compared witli (left-hand trace) the recombination is complete. At temperatures above 1500 K (right-hand trace) the equilibrium concentration of CFt is appreciable, and thus the teclmique allows the detennination of botli the equilibrium constant and the recombination rate [54, M]. This experiment resolved a famous controversy on the temperature dependence of the recombination rate of methyl radicals. Wliile standard RRKM theories [, ] predicted an increase of the high-pressure recombination rate coefficient /r (7) by a factor of 10-30 between 300 K and 1400 K, the statistical-adiabatic-chaunel model predicts a...

If crossing the central barrier is not rate-controlling in TST, then trapping in the ion-dipole complex must be incorporated into the statistical model and it is more difficult to represent the effect of central barrier recrossings correcting TST with the K factor is not sufficient. The recrossings and presence of both intermolecular and intramolecular complexes are expected to affect the k, kisom, and k rate constants in equation 6. The value for k should be smaller than that of a capture model, and kisom and k 8 should disagree with the predictions of 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+. 

Vibrational frequencies for various normal modes must be estimated and active as well as inactive energies should be decided. Numerical methods may be used to calculate rate constant k at various concentrations obtained by RRKM theory. The rate constant has been found to be same as given by conventional transition state theory, i.e. 

The RRKM theory is the most widely used of the microcanonical, statistical kinetic models It seeks to predict the rate constant with which a microcanonical ensemble of molecules, of energy E (which is greater than Eq, the energy of the barrier to reaction) will be converted to products. The theory explicitly invokes both the transition state hypothesis and the statistical approximation described above. Its result is summarized in Eq. 2... 

Of course, in a thermal reaction, molecules of the reactant do not all have the same energy, and so application of RRKM theory to the evaluation of the overall unimolecular rate constant, k m, requires that one specify the distribution of energies. This distribution is usually derived from the Lindemann-Hinshelwood model, in which molecules A become activated to vibrationally and rotationally excited states A by collision with some other molecules in the system, M. In this picture, collisions between M and A are assumed to transfer energy in the other direction, that is, returning A to A ... 

In RRKM theory, the activation rate constant kact of Eq. 10.146 in QRRK theory is replaced by the more rigorous... 

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]. 

Activation parameters for the high pressure gas-phase approach of 1,2-d2-cyclopropanes to cis, trans equilibrium (equation 1) have been reported as log A, a(kcal mol"1) of 16.0,64.2 and 16.4,65.176,77. From pressure-dependent measurements of rate constants and calculations based on RRKM theory, the threshold energy E for the cis, trans isomerization has been estimated to be 61.1 kcal mol"1 and 61.3 kcal mol"11 16 1, s. 

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). 

Fig. 7.3.4 Schematic illustration of the rate constant k(E) according to RRKM theory.

The dimension of the factor IIf=1z/j/IIlz v is that of a frequency. If the frequencies of the reactant and the activated complex are not too different, this frequency is roughly a typical vibrational frequency vr (typically in the range 1013 to 1014 s 4). Since the energy-dependent factor is less than one, we have that the microcanonical rate constant k(E) < i/r, i.e., it is less than a typical vibrational frequency. The energy dependence as a function of the number of vibrational degrees of freedom was illustrated in Fig. 7.3.2, and as shown previously in Eq. (7.38) it can be interpreted as the probability that the energy in one out of s vibrational modes exceeds the energy threshold Eq for the reaction. Note that if we make the identification vr n =1z/i/n 11i/ , we have recovered RRK theory, Eq. (7.39), from RRKM theory. 

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