Bell-like dependence

Such a bell-like dependence of Wsp on the surface density of transition metal ions has also been observed in other catalytic reactions (hydroformylation, oxidation, polymerization, etc.), and is probably one of the specific features of catalysis by immobilized metal complexes. While there is no well-founded explanation of the rising branch of the plot, the diminishing trend may be coimected with the formation and fiirther growth of low- and/or zero-valent transition metal ion associations, diminishing the catalytic efficiency. Active centers of immobilized catalysts are localized on the boimdaries of cluster-like substances with stabilization by their electron systems. 

A bell-like dependence of the gelation efficiency on the process temperature is inherent in cryotropic gel formation. 

The simplest case of comb polymer is the H-shaped structure in which two side arms of equal length are grafted onto each end of a linear cross-bar [6]. In this case the backbones may reptate, but the reptation time is proportional to the square of Mj, rather than the cube, because the drag is dominated by the dumb-bell-like frictional branch points at the chain ends [45,46]. In this case the dependence on is not a signature of Rouse motion - the relaxation spectrum itself exhibits a characteristic reptation form. The dynamic structure factor would also point to entangled rather than free motion. 

The shape of Bragg peaks is usually represented by a bell-like function -the so-called peak shape function. The latter is weakly dependent on the crystal structure and is the convolution of various individual functions, established by the instrumental parameters and to some extent by the properties of the specimen, see Table 2.7. The shape of each peak can be modeled using instrumental and specimen characteristics, although in reality ab initio modeling is difficult and most often it is performed using various empirically selected peak shape functions and parameters. If the radiation is not strictly monochromatic, i.e. when both Kai and Kaa components are present in the diffracted beam, the resultant peak should include contributions from both components as shown in Figure 2.39. 

Summing up all the above considerations it should be expected that the dependence of the propagation rate constant kp (not the rate itself) on bond polarity in the active center should be a bell-like curve schematically shown in Fig. 3 (curve 1). 

The authors [91] proposed description of organic phase influence on limiting characteristics of polyurethanearylates (PUAr) interfacial polycondensation. As it is known [55], one from the methods of polymer solubility parameter 5 experimental determination is plotting of the dependence of intrinsic viscosity [t ], measured in several solvents, on this solvents solubility parameter 5 value. The smaller difference 6p-5J or the better solvent thermodynamical quality in respect of polymer is, the larger [q] is. The dependences [q](5 ) have usually belllike shape and such dependence maximum corresponds to 5 [55]. In Fig. 23 the dependence of on 5 of solvents, used as organic phase at PUAr interfacial polycondensation is adduced. The dependence q /S ) bell-like shape is obtained again and its maximum corresponds to 5 10 (cal/cm ), that is a reasonable estimation for PUAr [36, 55]. Let us note that all q values were determined in one solvent, which was not used at synthesis, namely, in mixture phenol-simm-tetrachloroethane. The dependence qj 4(5 ), adduced in Fig. 23, allows to make two conclusions. Firstly, the value q, reached in PUAr interfacial polycondensation process, is controlled by solvent thermodynamical qnality and the greatest... 

All these favorable and unfavorable factors and mechanisms are effective to a different extent during cryotropic gelation in both aqueous and organic media. For instance. Fig. 15 shows the bell-like temperature dependence of the gel-fraction yield when poly(styrene) was crosslinked with 4,4 -xylylene dichloride in nitrobenzene (see Fig.Sa for the reaction scheme). At temperatures 30-40 °C lower than room temperature, the efficiency of polymer crossUnking is higher than in unfrozen solutions (curve with open circles in Fig. 15) [27]. Thus, close similarity in the character of such temperature dependences for the processes in frozen aqueous and... 

Figure 3.3 demonstrates how the conversion factor depends on parameter a for different values of parameter e, which is determined by the intrii sic properties of the device (fe and b) and the load (mass m), s = mg/kb. These dependencies have a bell-like shape with the maximum values of determined by the parameter e. Figure 3.4 demonstrates that the extreme values of... 

From the various types of envelopes for windowing available. Phase Vocoder normally employs either the Hamming function or the Kaiser function. Both functions have a bell-like shape. One may provide better analysis results than the other, depending upon the nature of the sound analysed and the kind of result expected. 

It has been shown that the rate of formation of oximes is at a maximum at a pH that depends on the substrate but is usually 4, and that the rate decreases as the pH is either raised or lowered from this point. We have previously seen (p. 425) that bell-shaped curves like this are often caused by changes in the rate-determining step. In this case, at low pH values step 2 is rapid (because it is acid catalyzed), and step 1... 

As the understanding of the ionic intermediates has progressed, advantage has been taken of the fact that bromination, like SN1 heterolysis, is a carbocation-forming reaction. Kinetic data on this addition have therefore been used to examine in detail how the basic concepts of physical organic chemistry work as regards transition-state shifts with reactivity (Ruasse et al, 1984). Bromination lends itself particularly well to the quantitative application of the BEMA HAPOTHLE (acronym for Bell, Marcus, Hammond, Polanyi, Thornton and Leffler Jencks, 1985). In particular, it has been possible to evaluate the transition-state dependence on the solvent and substituents. The major disadvantage that bromination shares with many... 

This expression was derived by Bell (1978), who used Kramers theory to show that bond lifetime ean be shortened by an applied force in processes such as cell adhesion. Although Eq. (3.2) is quite useful, it is in practice limited, most notably by the fact that it assumes that xp is constant. Typically, measurements of force dependency are made under conditions in which force changes with time, and it is likely that the position of the transition state will move as the shape of the potential surface is perturbed by an applied force (Evans and Ritchie 1997 Hummer and Szabo 2003). Theoretical and empirical treatments of various cases have been put forth in the hterature, but they are outside the scope of this chapter and will not be reviewed here. 

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