RRKM calculations oscillators

Only in the high-energy limit does classical statistical mechanics give accurate values for the sum and density of states tenns in equation (A3.12.15) [3,14]. Thus, to detennine an accurate RRKM lc(E) for the general case, quantum statistical mechanics must be used. Since it is difficult to make anliannonic corrections, both the molecule and transition state are often assumed to be a collection of hannonic oscillators for calculating the... 

Fig. 16. Product relative translational energy distribution72 P(E T) points are experimental solid line RRKM calculations using an effective S( 16-9,151 and 13-9, respectively) dashed line RRKM calculations using full number of oscillators (S = 27).

If another X group is added onto the aniline molecule, the An—X2 dissociation rate is more likely to be determined by the statistical dissociation step because with each additional nonlinear monomer the number of van der Waals modes increases by six. Because the van der Waals modes are extremely anharmonic and coupled to each other, a proper RRKM calculation should use anharmonic densities and sums. However, these are not yet generally available for the systems of interest. In all cases it is best to use the quantum density of states (i.e., RRKM) and not the classical approximation of it (RRK). With a binding energy of say 480 cm and six oscillators, the average energy per van der Waals mode is 60 cm. Since these frequencies typically vary between 20 and about 400 cm, it is evident that the average number of quanta excited per mode is only about 1 or 2, which does not correspond to the classical limit. 

RRKM theory has been used widely to interpret measurements of unimolecular rate constants. However, harmonic state counting procedures are usually used in the RRKM calculations. This is not because enharmonic effects are thought to be unimportant, but because they are difficult to account for. The only comprehensive attempt to include the effect of anharmonicity has involved treating the vibrational degrees of freedom as separable Morse oscillators. However, since this correction is an obvious oversimplification it has not been widely used. The importance of anharmonicity is illustrated by comparing the trajectory unimolecular rate constant for C2H5 H + C2Hi dissociation at 100 kcal/mol (Fig. 4b), which is about 4.7 X 10 with that predicted by harmonic classical RRKM... 

The RRK theory assumes that the Arrhenius high-pressure thermal A-factor is given by the frequency for the critical oscillator, which is in the range of 10 to 10 " sec V However, for many reactions Arrhenius high-pressure thermal A-factors are in fact larger than lO " sec . This inability is overcome by the RRKM (Marcus) theory, or by use (in this study) of a preexponential factor determined by quantum calculation of the thermochemical properties of the transition state structure. 

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