Canonical transition-state theory

A3.4.7 STATISTICAL THEORIES BEYOND CANONICAL TRANSITION STATE THEORY... 

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... 

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. 

This set of approximations is essentially similar to that for the more familiar canonical transition state theory, apart from the final assumption, that the state is described by a microcanonical distribution (i.e. fixed energy) rather than the Boltzmann distribution (thermal equilibrium) of canonical transition state theory. 

Variational transition state theory was suggested by Keck [36] and developed by Truhlar and others [37,38]. Although this method was originally applied to canonical transition state theory, for which there is a unique optimal transition state, it can be applied in a much more detailed way to RRKM theory, in which the transition state can be separately optimized for each energy and angular momentum [37,39,40]. This form of variational microcanonical transition state theory is discussed at length in Chapter 2, where there is also a discussion of the variational optimization of the reaction coordinate. 

It is interesting to consider the relationship between the reaction degeneracy and the molecular symmetries in canonical transition state theories. In the latter, the rate constants are expressed in terms of the partition functions, including the rotational partition functions, so that the molecular symmetries are automatically included. On the other hand, in the microcanonical TST, the rotational density of states is often not part of the rate constant expression (see discussion of rotational effects in the following chapter). Thus, the reaction degeneracy must be included separately. 

Chapter 6 presents estimations of thermochemical properties of intermediates, transition states and products important to destruction of the aromatic ring in the phenyl radical + O2 reaction system. We have employed both DFT and high-level ab initio methods to analyze the substituent effects on a number of chemical reactions and processes involving alkyl and peroxyl radicals. Partially based on the results obtained in the vinyl system, high-pressure-limit kinetic parameters are obtained using canonical Transition State Theory. An elementary reaction mechanism is constructed to model experimental data obtained in a combustor at 1 atm, and in high-pressure turbine systems (5-20 atm), as well as in supercritical water [31]. 

ThermKin (Thermodynamic Estimation of Radical and Molecular Kinetics) evolved (see Sheng s thesis [74]) from a previously developed computer code, i.e. THERMRXN (included in THERM) [82] which calculates equilibrium thermodynamic properties for any given reaction. Additionally, ThermKin determines the forward rate constants, k(T), based on the canonical transition state theory (CTST). 

High Pressure limit kinetic parameters are obtained from canonical Transition State Theory calculations. Multifrequency Quantum Rice-Ramsperger-Kassel (QRRK) analysis is used to calculate k(E) data and master equation analysis is applied to evaluate fall-off in this chemically activated reaction system. 

The reactions for which thermochemical properties of transition states are calculated by ab initio or Density Functional Theory methods, oo s are fit by three parameters n, and over the temperature range of 298-2000 K, kao = A T)T exp( a/ 7). Entropy differences between the reactants and transition state structures are used to determine the pre-exponential factor. A, via canonical Transition State Theory [197]. 

Table 6.6 lists the high pressure limit kinetic parameters for the elementary reaction steps in this complex phenyl + O2 reaction system. These parameters are derived from the canonical transition state theory, the statistical mechanics from the DFT and ab initio data and from evaluation of literature data. Rate constants to all channels illustrated are calculated as function of temperature at different pressure. A reduced mechanism is proposed in Appendix F for the Phenyl + O2 system, for a temperature range of 600K different pressures 0.01 atm, 0.1 atm, latm, and 10 atm. 

The kinetic parameters of each path are determined as a function of temperature and pressure using the bimolecular chemical activation analysis. High Pressure limit kinetic parameters from the calculation results are obtained with the canonical Transition State Theory. The multifrequency Quantum Rice-Ramsperger-Kassel analysis is utilized to obtain k(E) and Master Equation analysis is used for the evaluation of pressure fall-off in this complex bimolecular chemical activation reaction. Results are applicable to elementary experiments at low pressures, ambient combustion studies at one atmosphere, as well as higher-pressure turbine systems. 

Thermal Rate Constants Maximum Fkee Energy GrUcrion.—In canonical transition-state theory as applied to the calculation of c T) lor unimolecular reactions, it has long been proposed that the ccnrect location q of the ar ivated complex be at a maximum of the free energy. " - This may or may not coincide with the... 

At the high-pressure limit, equation (23) can be integrated analytically and it can be shown that k-am(X) obtained from RRKM theory as described here is similar but not equal to the high-pressure rate obtained via canonical transition state theory ... 

Thermodynamic calculations were performed at 0 K, room temperature and 600 K, in all cases assuming atmospheric pressure. Rate constants were calculated using die standard canonical transition state theory with Eckart s correction for tunneling [43] and both with and without hindered rotor treatment of the internal rotation. Rate coefficients are calculated based on conventional transition state theory in the high-pressure Umit. For the bimolecular radical addition, the rate coefficient is calculated according to Eq. 1 ... 

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