Canonical rate constant

The quasi-equilibrium assumption in the above canonical fonn of the transition state theory usually gives an upper bound to the real rate constant. This is sometimes corrected for by multiplying (A3.4.98) and (A3.4.99) with a transmission coefifiwient 0 < k < 1. 

The rate of hydrogen transfer can be calculated using the direct dynamics approach of Truhlar and co-workers which combines canonical variational transition state theory (CVT) [82, 83] with semi-classical multidimensional tunnelling corrections [84], The rate constant is calculated using [83] ... 

For non-linear chemical reactions that are fast compared with the local micromixing time, the species concentrations in fluid elements located in the same zone cannot be assumed to be identical (Toor 1962 Toor 1969 Toor and Singh 1973 Amerja etal. 1976). The canonical example is a non-premixed acid-base reaction for which the reaction rate constant is essentially infinite. As a result of the infinitely fast reaction, a fluid element can contain either acid or base, but not both. Due to the chemical reaction, the local fluid-element concentrations will therefore be different depending on their stoichiometric excess of acid or base. Micromixing will then determine the rate at which acid and base are transferred between fluid elements, and thus will determine the mean rate of the chemical reaction. 

In equation (4.68), dk is a canonical rate constant, relating to energized molecule having energy between E and E + dE. The expression can be integrated between the limits from minimum energy ( Eq ) to infinity to get the ordinary rate constant k, i.e. 

The result looks familiar and well it should. Equation 14.28 is just the expression for the rate constant in conventional canonical transition state theory with the... 

Rate constants and Arrhenius parameters for the reaction of Et3Si radicals with various carbonyl compounds are available. Some data are collected in Table 5.2 [49]. The ease of addition of EtsSi radicals was found to decrease in the order 1,4-benzoquinone > cyclic diaryl ketones, benzaldehyde, benzil, perfluoro propionic anhydride > benzophenone alkyl aryl ketone, alkyl aldehyde > oxalate > benzoate, trifluoroacetate, anhydride > cyclic dialkyl ketone > acyclic dialkyl ketone > formate > acetate [49,50]. This order of reactivity was rationalized in terms of bond energy differences, stabilization of the radical formed, polar effects, and steric factors. Thus, a phenyl or acyl group adjacent to the carbonyl will stabilize the radical adduct whereas a perfluoroalkyl or acyloxy group next to the carbonyl moiety will enhance the contribution given by the canonical structure with a charge separation to the transition state (Equation 5.24). 

At high temperatures and low pressures, the unimolecular reactions of interest may not be at their high-pressure limits, and observed rates may become influenced by rates of energy transfer. Under these conditions, the rate constant for unimolecular decomposition becomes pressure- (density)-dependent, and the canonical transition state theory would no longer be applicable. We shall discuss energy transfer limitations in detail later. 

Variational transition-state theory has been formulated on various levels [5, 23-27]. At first, there is the group of canonical VTST (CVTST) treatments, which correspond to the search for a maximum of the free energy AG(r) along the reaction path r [23, 24]. It was noticed early that for barri-erless potentials this approach leads to an overestimate of the rate constant because, in the language of SACM, channels are included that are closed. Therefore, an improved version (ICVTST) was proposed [25] that truncates Q at the position r of the minimum of (t(r) by including only states... 

Rel ation of Exact TST to the Harmonic Approximation. In the canonical ensemble, the most familiar TST expression for the rate constant is probably... 

Such a method has recently been developed by Miller. et. al. (28). It uses short lengths of classical trajectory, calculated on an upside-down potential energy surface, to obtain a nonlocal correction to the classical (canonical) equilibrium probability density Peq(p, ) at each point then uses this corrected density to evaluate the rate constant via eq. 4. The method appears to handle the anharmonic tunneling in the reactions H+HH and D+HH fairly well (28), and can... 

Definition of Critical and Rate-Limiting Bottlenecks" The hypothesis of local equilibrium within the reservoirs means that the set of transitions from reservoir to reservoir can be described as a Markov process without memory, with the transition probabilities given by eq. 4. Assuming the canonical ensemble and microscopic reversibility, the rate constant Wji, for transitions from reservoir i to reservoir j can be written... 

We see that the rate constant may be determined as the time integral of the canonical averaged flux autocorrelation function for the flux across the dividing surface between reactants and products. It is also clear that we only need to calculate the flux correlation function for trajectories starting on the dividing surface, for otherwise F(p(0), q(0)) = 0 and there will be no contributions to the product formation. 

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. 

Thus, according to RRKM theory for an apparent unimolecular reaction, Eq. (7.58) gives the (canonical) rate constant for such an elementary reaction ... 

Chemical kinetic rate methods including conventional transition state theory (TST), canonical variational transition state theory (CVTST) and Rice-Ramsper-ger-Kassel-Marcus in conjunction with master equation (RRKM/ME) and separate statistical ensemble (SSE) have been successfully applied to the hydrocarbon oxidation. Transition state theory has been developed and employed in many disciplines of chemistry [41 4]. In the atmospheric chemistry field, conventional transition state theory is employed to calculate the high-pressure-limit unimole-cular or bimolecular rate constants if a well-defined transition state (i.e., a tight... 

In the high-pressure limit the energy in each mode is Boltzmann distributed. The high-pressure canonical rate constant can then be found by averaging the rate of Eq. (50) over the Boltzmann distribution. The integrations are not elementary, but are detailed by Slater [56, Chapter 5] and yield the result... 

Fig. 2.9. Plot of the canonical VRC-RRKM predictions for the high pressure CH3 + H addition rate constant for different fixed pivot point locations.

Figure 1 (a) Illustration of the unimolecular dissociation of a reactant molecule ABC into products A and BC. p( ), Rts and Nts are the density of reactant states, the intermolecular distance of the transition state (TS) and the number of open channels at the transition state, respectively. The vertical axis on the left-hand side shows a spectrum dominated by sharp resonances, (b) Schematic representation of the energy dependence of the micro-canonical rate constant k E) (solid line). The dots represent the state-specific quantum mechanical rates kn-... 

The unimolecular rate constant k E), for a micro-canonical ensemble of reactant states, is identical with the RRKM rate constant. If N 0) is the number of reactant molecules excited at t = 0 in accord with a micro-... 

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.. 

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