Frequency factor estimation

Frequency factors arc often determined from data obtained within a narrow temperature window. For this reason, it has been recommended4 that when extrapolating rate constants less error might be introduced by adopting the standard values for frequency factors (above) than by using experimentally measured values. The standard values may also be used to estimate activation energies from rate constants measured at a single temperature. 

By using the method of Levenbeig-Marquardt [4] the activation energies and frequency factors for individual rate constants are determined as given in Table 2 and the reaction orders with respect to CPD and ethylene are estimated to be 2i = 22 = 0.94, ... 

For fitting such a set of existing data, a much more reasonable approach has been used (P2). For the naphthalene oxidation system, major reactants and products are symbolized in Table III. In this table, letters in bold type represent species for which data were used in estimating the frequency factors and activation energies contained in the body of the table. Note that the rate equations have been reparameterized (Section III,B) to allow a better estimation of the two parameters. For the first entry of the table, then, a model involving only the first-order decomposition of naphthalene to phthalic anhydride and naphthoquinone was assumed. The parameter estimates obtained by a nonlinear-least-squares fit of these data, are seen to be relatively precise when compared to the standard errors of these estimates, s0. The residual mean square, using these best parameter estimates, is contained in the last column of the table. This quantity should estimate the variance of the experimental error if the model adequately fits the data (Section IV). The remainder of Table III, then, presents similar results for increasingly complex models, each of which entails several first-order decompositions. 

Analysis of the microwave spectrum of piperidine and of A-deuteropiper-idine suggests that the strongest Q-branch series together with the associated R-branch lines arise from the N-H-axial conformer.144 From this absorption and from a weaker series of Q branches, /eq//aj (the relative intensities of the type-A lines of /V-Hcq and N-Hax conformers corrected for the frequency factor) was estimated as 1/6 at — 34°C. This ratio is related to AE = En Hax - jv-Hcq by the expression... 

It is also interesting to estimate the maximum value of the frequency factor in the case of purely quantum nuclear motion. This can be done with the help of the formula W 2nV2Sp, where V2 exp(—2yR) is the exchange matrix element, S is the Franck-Condon factor, p 1 jco is the density of the vibrational levels, and co 1000 cm-1 is the characteristic vibrational frequency of the nuclei. In the atomic unit system, the multiplier 2np has the order 103 and the atomic unit of frequency is 4.13 x 1016s-1 consequently, in the usual unit system, the frequency factor is of the order 4 x 1019Ss-1. The frequency factor reaches its maximum value when S 1. Thus, in the case of purely quantum nuclear motion, the maximum value of the frequency factor is also 1019-102°s-1. 

An estimate of the specific frequency factor can be made according to the principles laid down in IV, 2. Assuming that the kinetics of the benzene hydrogenation is about zero order in the benzene pressure and first order in the hydrogen pressure, one would expect. 

The discussion of the previous section suggests that the linear combination of the shifted and scaled Fourier transforms of the analysis window in Equation (9.72) must be explicitly accounted for in achieving separation. The (complex) scale factor applied to each such transform corresponds to the desired sine-wave amplitude and phase, and the location of each transform is the desired sine-wave frequency. Parameter estimation is difficult, however, due to the nonlinear dependence of the sine-wave representation on phase and frequency. 

In order to estimate the kinetic parameters for the addition and condensation reactions, the procedure proposed in [11, 14] has been used, where the rate constant kc of each reaction at a fixed temperature of 80°C is computed by referring it to the rate constant k° at 80°C of a reference reaction, experimentally obtained. The ratio kc/k°, assumed to be temperature independent, can be computed by applying suitable correction coefficients, which take into account the different reactivity of the -ortho and -para positions of the phenol ring, the different reactivity due to the presence or absence of methylol groups and a frequency factor. In detail, the values in [11] for the resin RT84, obtained in the presence of an alkaline catalyst and with an initial molar ratio phenol/formaldehyde of 1 1.8, have been adopted. Once the rate constants at 80°C and the activation energies are known, it is possible to compute the preexponential factors ko of each reaction using the Arrhenius law (2.2). 

Coombes et al (27) have studied the nitration of toluene under homogeneous conditions by using very small amounts of toluene totally dissolved in the acid phase. Their data were used to estimate the frequency factor. 

P/P )]2- As pointed out by LaMer (LI), this term dominates the rate expression, making possible the prediction of critical supersaturation ratios within 10%, despite a hundredfold error in estimating the frequency factor. Close agreement between theory and experiment in the condensation of various vapors is demonstrated by the data of Volmer and Flood (V8) and discussed by Pound (P3). 

Calculated from the data of Melville and Robb on the reverse reaction (assuming that 6 = 2 Kcal) and A s 6 from Table XIII.8. [H. W. Melville and J. C. Robb, Proc. Roy, Soc, London)j A196 494 (1949)]. The frequency factor is 10 fold less than that quoted by Steacie, Atomic and Free Radical Reactions, 2d ed., ACS monograph No. 125, Reinhold Publishing Co., New York, 1954, which arises in part from the estimate of Ks.e. 

Despite these difficulties of (piantitative interpretation, there are sufficient data on the most important steps in the proposed mechanism [Eq. (XIII. 13.2)] to indicate that at least (lualitatively it is a correct scheme. In Table XIII. 13 arc collected some estimates of thermodynamic data pertinent to the reaction scheme. The data on reactions 3 and 4 are included in Tables XI1.6 and XII.8, respectively. Volman and Graven loc, cit.) have estimated the activation energy of reaction 7 as 13.5 db 2 Kcal, which is very close to the heat of reaction estimated at 10 Kcal (Table XIII. 13). There are no quantitative data on the value of the frequency factor of 7, but from the low activation energy one might expect that it should show pressure dependence. ... 

This could result in an apparently low frequency factor. Thus if fc is in the region of being a bimolecular reaction and has the form At(M), we can compute an upper limit for A from the estimated entropy change in the reaction (Table XIII, 13) and the... 

These values indicate a collision eflSciency for the recombination of NO2 + NO3 of about 1 in 300 collisions, which is very close to the value observed for NO2 + NO2. The frequency factor of reaction 1 is very much higher than the usual factor for unimolecular fissions but lower than the estimated value for the fission of N2O4. The frequency factor for fcs is about 10 lower than collision frequencies and is in the expected range of values for atom transfers between large molecules. It is the activation energy for this reaction, Ez = 3.65 Kcal, which seems surprising. From Table XIII. 16, one finds that AH3 = 5.1 Kcal ... 

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