Model parabolic

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

For a certain choice of g cj)) and f 4>) the above equation can be solved analytically for simple spatially modulated phases, such as lamellar or hexagonal [22]. This is possible for the piecewise parabolic model of /(0) ... 

Frequently, the relationship between biological activity and log P is curved and shows a maximum [ 18]. In that case, quadratic and non-linear Hansch models have been proposed [19]. The parabolic model is defined as ... 

In this case, the bilinear model of eq. (37.10) fits the data as well as the parabolic one. A possible reason for the lack of improvement is that the part of the model which accounts for the higher values of log P is not well covered by the data (Fig. 37.1). The parabolic model yields an optimal value for log P of 5.10 while the optimum of the bilinear model is found at 5.18. 

PARABOLIC MODEL OF BIMOLECULAR HOMOLYTIC REACTION 4.4.1 Main Equations of IPM... 

The parabolic model is, in essence, empirical because the parameter a is calculated from spectroscopic fa and v ) and atomic (/q and /q) data, while the parameter bre (or Ee0) is found from the experimental activation energies E(E= RT a(A/k)), where A is the pre-exponential factor typical of the chosen group of reactions, and k is the rate constant. The enthalpy of reaction is calculated by Equation (4.6). The calculations showed that = const, for structurally similar reactions. The values of a and bre for reactions of different types are given in Table 4.16. 

The following parameters are used to characterize the elementary step in the parabolic model (see Figure 6.1). 

FIGURE 6.1 Parabolic model of the reaction R02 + RH in coordinates potential energy versus amplitude of vibration of the reacting bonds (for symbols, see text). 

The activation energy for any individual reaction within the limits of the given group of reactions may be calculated correctly from the parameter bre. Within the framework of the parabolic model, the quantity Ee was calculated by one of the following equations [4,11] ... 

A clear-cut dependence of the activation energy on the heat (enthalpy) of the reaction, which is equal, in turn, to the difference between the dissociation energies of the ruptured (Z> ) and the formed (D j bonds, was established for a great variety of radical abstraction reactions [1,2,16]. In parabolic model, the values of Dei and Def, incorporating the zero-point energy of the bond vibrations, are examined. The enthalpy of reaction AHe, therefore, also includes the difference between these energies (see Equation [6.7]). 

In terms of the parabolic model, it is possible to obtain simple and physically clear equations for the estimation of A/ n as a function of a, hr, and AHe. The following simple... 

A comparative analysis of the kinetics of the reactions of atoms and radicals with paraffinic (R1 ), olefinic (R2H), and aromatic alkyl-substituted (R3H) hydrocarbons within the framework of the parabolic model permitted a new important conclusion. It was found that the tt-C—C bond occupying the a-position relative to the attacked C—H bond increases the activation energy for thermally neutral reaction [11]. The corresponding results are presented in Table 6.9. 

Within the framework of the parabolic model, the radical addition... 

Parameters of Various Classes of the Addition of Atoms and Radicals to Multiple Bonds Used in the Parabolic Model [40-45]... 

The polar effect involved in radical addition has been repeatedly discussed in the scientific literature. The parabolic model opens up new prospects for the correct estimation of the polar effect (see Section 6.2.7). It permits one to determine the contribution of this effect to the activation energy using experimental data. This contribution (AE ) is estimated by choosing a reference reaction that involves the same reaction center but in which one or both reactants... 

It is apparent that these reactions are close in their enthalpies, but greatly differ in the rate constants. The peroxyl radical reacts with p-cresol by four orders of magnitude more rapidly than with ethylbenzene. Such a great difference in the reactivities of RH and ArOH is due to the different activation energies of these reactions, while their pre-exponential factors are close. This situation was analyzed within the scope of the parabolic model of transition states (see Chapter 6 and Refs. [33-38]). 

One can see that the rate constants for the disproportionation of the aminyl and peroxyl radicals involving the C—H bond are substantially lower than the rate constants for the addition of R02 to the aromatic ring of the aminyl radical (k = 3 x 108 L mol-1 s ). Conversely, the reaction of Am with H02 occurs very rapidly and can compete successfully with the addition reaction. Thus, the result obtained within the scope of the parabolic model is in good agreement with the experiment. 

The estimation based on the equations of the parabolic model indicates that a reaction of the type (ArO + H02 —> ArOH + 02) involving phenoxyl radicals also requires no activation energy (in this case, AH> A emin = 57kJ mol-1). However, the addition of the peroxyl radical to the aromatic ring of the phenoxyl radical occurs very rapidly. Hence, the rate constant for this reaction is determined by diffusion processes. The data on the Ee0 values are also consistent with this. For the ArO + HOOR reactions with the O H O reaction center and for Am + HOOR reactions with the N H O reaction center, these values are 45.3 and 39.8 kJ mol-1, respectively [23]. At the same time, the calculation of the preexponential factor in terms of the parabolic model indicates that the rate constant k 7 for the reaction of ROOH with the participation of the aminyl radical is several times higher than that for the reaction involving the phenoxyl radical, where the enthalpies of these reactions... 

In the present example, the total degrees of freedom is ten. The parabolic model that is to be fit. 

Fitting the expanded (parabolic) model to the data in Equation 11.16 gives the parameter estimates... 

The experimental design represented by Equation 11.6 requires only half as many experiments as the design represented by Equation 11.12. Each design can be used to fit the same parabolic model. What are the advantages and disadvantages of each design ... 

---