Statistical theories comparison

The crucial test of all of the theories based on solvation would be the absence of the isokinetic relationship in the gas phase, but the experimental evidence is ambiguous. Rudakov found no relationship for atomization of simple molecules (6), whereas Riietschi claimed it for thermal decomposition of alky] chlorides (96) and Denisov for several radical reactions (107) however, the first series may be too inhomogeneous and the latter ones should be tested with use of better statistics. A comparison of the same reaction series in the gas phase on the one hand and in solution on the other hand would be most desirable, but such data seem not to be available. 

The preceding equations will, of course, be somewhat in error owing to the neglect of intramolecular condensations. Very large species will be suppressed relatively more on this account. All conceivable errors can do no more, however, than to effect a distortion of the quantitative features of the predictions, which will be small in comparison with the vast difference between the branched polymer distribution and that usually prevailing in linear polymers. From this point of view, the statistical theory given offers a useful description of the state of affairs. 

The greatest need in model performance testing and validation is clearly the use of quantitative measures to describe comparisons of observed and predicted values. As noted above, although a rigorous statistical theory for model performance assessments has yet to be developed, a variety of statistical measures has been used in various combinations and the frequency of use has been increasing in recent years. The FAT workshop (3.) identified three general types of comparisons that are often made in model performance testing ... 

III. Comparisons between Statistical Theory Calculations and Experiment... 

III. COMPARISONS BETWEEN STATISTICAL THEORY CALCULATIONS AND EXPERIMENT FOR Cl- + CH3Br CICH3 + Br ... 

Finally, accurate theoretical kinetic and dynamical models are needed for calculating Sn2 rate constants and product energy distributions. The comparisons described here, between experimental measurements and statistical theory predictions for Cl"+CHjBr, show that statistical theories may be incomplete theoretical models for Sn2 nucleophilic substitution. Accurate kinetic and dynamical models for SN2 nucleophilic substitution might be formulated by introducing dynamical attributes into the statistical models or developing models based on only dynamical assumptions. 

Oros, F.J., Davis, J.M. (1992). Comparison of statistical theories of spot overlap in two-dimensional separations and verification of means for estimating the number of zones. J. Chromatogr. 591, 1-18. 

Comparison with Statistical Theory, Small-strain Behaviour. Calculations of theoretical moduli, using the phantom theory and the affine theory as limiting cases, were carried out in order to compare the theoretical predictions with values found experimentally (Table IV). 

Comparison with Statistical Theory at Moderate Strains. So far we have shown, that a transition between the two limiting classical theories, i.e. affine theory and phantom theory, is possible by a suitable choice of the network microstructure. This argument goes beyond the revised theory by Ronca and Allegra and by Flory, which predicts such a transition as a result of increasing strain, thus explaining the experimentally observed strain dependence of the reduced stress. 

An approximate analysis of polymer adsorption as a set of sequential reactions leads to a simple equation for the adsorption isotherm expressed in terms of three parameters. Comparison of the model with recently published statistical theories reveals remarkable agreement in both the general shape of the isotherms and the predicted effects of molecular weight. The problems of applying such models to experimental data are discussed. 

It is of considerable interest to compare the present model with the predictions of the statistical theories. In particular, the results will be compared with the theories of Scheutjens and Fleer (9,10,13). These authors have pointed out that their results are in general agreement with other published models. Comparison of the adsorption isotherms shown in Figure 1 with those given by Scheutjens and Fleer (see, for example, Figure 5 of Reference 13) shows excellent qualitative agreement. In each case, the isotherms consist of three regions as described above. 

Figure 2. Comparison of adsorption isotherms based on the present model (solid lines) with the statistical theory of Scheutjens and Fleer (symbols). The following values of the parameters were used...

Fig. 10.18. Rotational state distributions of CN(X2E+) (upper part) and NO(X2IIi/2) respectively NO(X2n3/2) (lower part) following the photodissociation of NCNO at three wavelengths. The middle part depicts the absorption spectrum. Comparison of experimental data (open circles) with the predictions of the statistical theory (filled circles). Reproduced from Qian et al. (1985).

The concentration dependence of the Stern-Volmer constant has been used already as a test for the validity of the theories of intermolecular (Section XII.B) and intramolecular electron transfer (Section IV.E). However, the statistical theory of Felderhof and Jones [256], which addressed the latter case but did not withstand the test [257], was excluded from the abovementioned comparison. The same should be done with a few theories of dissociation/association. These are the theory of Vogelsang and Hauser [243] and the so-called phenomenological rate equations proposed by Lee and Karplus [139] and used in a number of other works [145,174,176]. 

A comparison of the predictions of the statistical theory with experiments on diepoxy-diamine systems can be found in papers by Dusek et al. Topol-... 

In cases like D2CO or NO2 comparison with experimental data on a state-specific level are ruled out entirely and one has to retreat to more averaged quantities like the average dissociation rate, (fc), and the distribution of rates, Q(k). If the dynamics is ergodic — the basic assumption of all statistical theories — one can derive a simple expression for Q k), which had been established in nuclear physics in order to describe the neutron emission rates of heavy nuclei [280]. These concepts have since developed into the field of random matrix theory (RMT) and statistical spectroscopy [281-283] and have also found applications in the dissociation of energized molecules [121,284-286]. 

Statistical theories of thermodynamics yield many correct and practical results. For example, they yield the canonical and grand canonical distributions for petit and grand systems, respectively these distributions, which were proposed by Gibbs, have been shown by innumerable comparisons with experiments to describe accurately the properties of quasistable states. Again, they predict the equality of temperatures of systems in mutual stable equilibrium, the Maxwell relations, and the Gibbs equation. 

In comparing measured and reference data the statistical theory of hypothesis testing applies. The null hypothesis Hq for every comparison is that the unknown compound is identical to the reference compound. The test parameter is the difference quantity Aq, representing in our example, the difference between two retention values ... 

First of all, the formula (51.HI) provides an insight into the relations between collision and statistical theories in chemical kinetics, which will be discussed later. This goal can be achieved by a comparison with an alternative formulation of the rate constant, more closely related to the statistical approach, which is suitable in the usual situation where the potential energy surface has a col. This formulation will be given in the next section. 

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