RRKM theory rate constants from

For reactions that are unimolecular in one or both directions, the reaction rate is expected to be pressure dependent, as discussed in detail in an earlier chapter of this text. In the high-pressure limit, conventional transition state theory as described in the previous section can be applied to estimate the rate constant. The only change in equation (20) is that only a single reactant partition function appears in the denominator. The pressure dependence can then be described at various levels of sophistication, from QRRK theory to RRKM theory, to full master equation treatments using microcanonical rate constants from RRKM theory, as described in the chapter by Carstensen and Dean. Because these approaches have been described in detail there, they are not treated in the present chapter. 

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. 

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+. 

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 ... 

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. 

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). 

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. 

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. 

Reported gas-phase NMR studies have compared experimental pressure-dependent rate constants obtained from lineshape analyses with those calculated using RRKM theory which assumes stochastic IVR. This method is sensitive to significant departures from RRKM theory but cannot distinguish smaller departures due... 

Figure 6. Logarithmic plot values of LUm/fc versus pressure for the Cope rearrangement of bullvalene (torr at the experimental temperature of 356 K). Experimental values are signified by solid circles. Pressures are the total sample pressure at 356 K. Errors in frUni/fc are reported to 2o. The solid (upper) line represents the values calculated from RRKM theory using the biradicaloid transition state model. The lower line represents calculated rate constants using the aromatic transition-state model. The collision diameter was 3.6 A in both cases.

For most of the molecules discussed above, the experimentally determined pressure-dependent gas-phase rate constants can be modeled adequately with RRKM theory. SF4 and possibly aziridine are the exceptions. Due to uncertainties in the model parameters used in RRKM calculations, and the sensitivity of the calculations to these parameters, only qualitative conclusions can be drawn from the observed agreement. Since major departures are not observed, it can be... 

With this simplification, Gray, Rice, and Davis obtained reasonably accurate values for the predissociation rate constant as a function of initial vibrational excitation. The rate constant thus obtained is larger than that from exact trajectory calculations by about a factor of two. By contrast, the RRKM theory would give a rate constant that is about three orders of magnitude larger than is observed. 

When the effect of intramolecular energy transfer is taken into account, more accurate rate constants can be obtained. We first compare the rate constants associated with the intramolecular bottleneck from the MRRKM theory with those from the Davis-Gray turnstile approach. As seen in Table III, they are in reasonable agreement. Hence, the Davis-Gray theory and the MRRKM theory predict similar overall reaction rates. This is demonstrated in Table IV. Table IV also shows that the predissociation rate constants would have been overestimated by a factor more than 100 if the RRKM theory were to be directly applied. 

MRRKM theory, from RIT, from direct trajectory simulations, and, for reference purposes, from the RRKM theory. In particular, a test of the effect of the RRKM choice of transition state on the predicted rate of isomerization is made by neglecting the contribution of intramolecular energy transfer (Model No. 1). It is seen that the RRKM choice of transition state leads to considerable error the isomerization rate constant predicted is greater than those from the MRRKM theory and RIT by as much as a factor of 4. With intramolecular bottlenecks taken into account, both RIT and the MRRKM theory agree well with trajectory calculations. 

Table XXIII displays the rate constants obtained from the MRRKM theory and the reaction path analysis. It is seen that the former are about a factor of two smaller, and the latter about a factor of two larger, than those derived from direct trajectory calculations. We infer that, since both the RRKM and the MRRKM calculated rate constants are smaller than that calculated from trajectory calculations, there is a nonstatistical contribution to the isomerization rate that is not captured by the MRRKM theory.

The Classical Isomerization Rate Constants of Cyclobutanone from the RRKM Theory, Gray-Rice theory, MRRKM Theory, and Trajectory Calculations (in 10 a.u.)... 

RRKM theory, developed from RRK theory by Marcus and others [20-23], is the most commonly applied theory for microcanonical rate coefficients, and is essentially the formulation of transition state theory for isolated molecules. An isolated molecule has two important conserved quantities, constants of the motion , namely its energy and its angular momentum. The RRKM rate coefficient for a unimolecular reaction may depend on both of these. For the sake... 

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