Statistical theories phase-space theory

The tetrafluoromethane ion has also been found to decay before electronic randomisation has occurred [129, 769] (see Sect. 5.3 and 5.9 for other perfluorinated molecules). The breakdown diagrams for CF3X molecules (X = a halogen atom other than F) have been reported [690]. Translational energy release distributions have also been measured for these molecules and shown to be in agreement with the predictions of statistical theory (phase space theory) [691]. Carbonyl chloride and fluoride have been studied [451] (see Sect. 8). 

The determination of the microcanonical rate coefficient k E) is the subject of active research. A number of techniques have been proposed, and include RRKM theory (discussed in more detail in Section 2.4.4) and the derivatives of this such as Flexible Transition State theory. Phase Space Theory and the Statistical Adiabatic Channel Model. All of these techniques require a detailed knowledge of the potential energy surface (PES) on which the reaction takes place, which for most reactions is not known. As a consequence much effort has been devoted to more approximate techniques which depend only on specific PES features such as reaction threshold energies. These techniques often have a number of parameters whose values are determined by calibration with experimental data. Thus the analysis of the experimental data then becomes an exercise in the optimization of these parameters so as to reproduce the experimental data as closely as possible. One such technique is based on Inverse Laplace Transforms (ILT). 

Phase space theory (PST) has been widely used for estimation of rates and energy partitioning for ion dissociations. It can be considered within the framework of transition-state theory as the limiting case of a loose transition state, in which all product degrees of freedom are statistically fully accessible at the transition state. As such, it is expected to give an upper limit for dissociation rates and to be best suited to barrierless dissociations involving reaction coordinates with simple bond cleavage character. 

Figure 3. Thermal rate constants for capture of HC1 by H3 (PST locked-dipole capture corresponding to phase-space theory, Eq. (16) SACM statistical adiabatic channel model, Eqs. (26)-(34) [15] SACMci classical SACM, Eqs. (28H31) [15] CT classical trajectories, Eqs. (26) and (27) [1]).

The development of quantum theory, particularly of quantum mechanics, forced certain changes in statistical mechanics. In the development of the resulting quantum statistics, the phase space is divided into cells of volume hf. where h is the Planck constant and / is the number of degrees of freedom. In considering the permutations of the molecules, it is recognized that the interchange of two identical particles does not lead to a new state. With these two new ideas, one arrives at the Bose-Einstein statistics. These statistics must be further modified for particles, such as electrons, to which the Pauli exclusion principle applies, and the Fermi-Dirac statistics follow. 

Statistical phase-space theory Statistical phase-space theory Statistical phase-space theory... 

In more recent work, Bass et al. (1983) applied the statistical phase space theory to clustering reactions of CH3OH2, (CH3)2OH +, and (CH3OH)2H+ with CH jOH. Generally good agreement was found between the experimental and the... 

It must be emphasized that such phenomena are to be expected for a statistical system only in the regime of low level densities. Theories like RRKM and phase space theory (PST) (Pechukas and Light 1965) are applicable when such quantum fluctuations are absent for example, due to a large density of states and/or averaging over experimental parameter such as parent rotational levels in the case of incomplete expansion-cooling and/or the laser linewidth in ultrafast experiments. However, in the present case, it is unlikely that such phenomena can be invoked to explain why different rates are obtained when using ultrafast pump-probe methods that differ only in experimental detail. 

Statistical methods represent a background for, e.g., the theory of the activated complex (239), the RRKM theory of unimolecular decay (240), the quasi-equilibrium theory of mass spectra (241), and the phase space theory of reaction kinetics (242). These theories yield results in terms of the total reaction cross-sections or detailed macroscopic rate constants. The RRKM and the phase space theory can be obtained as special cases of the single adiabatic channel model (SACM) developed by Quack and Troe (243). The SACM of unimolecular decay provides information on the distribution of the relative kinetic energy of the products released as well as on their angular distributions. 

Phase space theory, flexible RRKM theory, and the statistical adiabatic channel model... 

The average lifetime is defined by Eqs. (13), (15), (20) and (22). According to a general prescription in statistical reaction theory or the phase-space theory due to Light [38], the lifetime of an isomer in a basin a should be given by... 

Figure 5. Comparison of the observed product rotational state distributions (solid bars) and the results of two statistical models the statistical adiabatic channel model (SACM, open bars) and phase space theory (PST, hatched bars). The distributions are an average of those observed, or those calculated, at several excitation wavelengths in the region of (a) the 5tbu main band, (b) the 5voh combination band, (c) the 6vOH main band, and (d) the 6v0H combination band of HOOH. (Reproduced with permission from Ref. 41.)...

In this instance, the solid curve is phase space theory, which turns out to be consistent with the product distributions associated with the 103 criteria applied to Eq. (2.17). Each of the product distributions as well as the total reactive cross section is, in fact, given by the statistical theory based on an SCR characterized by trajectories that diverge by > 103 from nearby neighbors in phase space. 

Among statistical models, the phase-space theory has been significantly extended to include closed channels and to calculate angular distributions. Studies have made clear limitations of the theory in predicting product energy and angular distributions, and have shown that fission and transition-state models are derivable from each other. Many neutral- and ion-reactions have been treated by means of the phase-space theory, including, more... 

There are several ways to derive the RRKM equation (Forst, 1973). The one adopted here is based on classical transition state theory and was first proposed by Wigner (Wigner, 1937 Hirschfelder and Wigner, 1939). Although there are several other statistical formulations of the unimolecular rate [phase space theory (Pechukas and Light, 1965), statistical adiabatic channel model (SACM) (Quack and Troe, 1974),... 

The statistical dissociation rate constant can be calculated from the point of view of the reverse reaction, namely the recombination of the products to form a complex. This approach, commonly referred to as phase space theory (PST) (Pechukas and Light, 1965 Pechukas et al., 1966 Nikitin, 1965 Klots, 1971, 1972) is limited to reactions with no reverse activation energy, that is, reactions with very loose transition states. PST assumes the decomposition of a molecule or collision complex is governed by the phase space available to each product under strict conservation of energy and angular momentum. The loose transition state limit assumes that the reaction potential energy surface is of no importance in determining the unimolecular rate constant. 

Detailed Analysis of Kinetic Energy Release Distributions for Type I Surfaces using Phase Space Theory. The model for the statistical phase space theory calculations(S) begins with Equation 1, where is the flux through the... 

In accordance with the considerations outlined in the previous section, a statistical kinetic energy release distribution should be observed for this system. As shown in Figure 7, the experimental distribution for this process can be fit very closely using statistical phase space theory, which yields a bond dissociation energy Do°(Fe -C5H5) = 55 + 5 kcal/mol. A reaction coordinate diagram for... 

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