Effectiveness factor plot second-order

Curve B of Figure 12.3 [adopted from Wheeler (38)] represents the dependence of the effectiveness factor on Thiele modulus for second-order kinetics. Values of r for first-order and zero-order kinetics in straight cylindrical pores are shown as curves A and C, respectively. Each curve is plotted versus its appropriate modulus. 

Plots of effectiveness factors versus corresponding Thiele moduli for zero-, first-, and second-order kinetics based on straight cylindrical pore model. For large hr, values of r are as follows ... 

Presumably less nucleophilically assisted solvolyses could show higher a-deuterium isotope effects, and there is a linear relationship between the magnitude of nucleophilic solvent assistance (Table 2) and the a-deuterium isotope effect for solvolyses of 2-propyl sulpho-nates (Fig. 7). Another measure of nucleophilic assistance is the ratio k2 (OH )/, where k2 is the second-order rate constant for nucleophilic attack by OH and kx is the first-order rate constant for reaction with the solvent water, and a linear correlation was obtained by plotting the ratio versus the experimentally observed isotope effects for methyl and trideuteriomethyl sulphonates, chlorides, bromides and iodides (Hartman and Robertson, 1960). Using fractionation factors the latter correlation may also be explained by a leaving group effect on initial state vibrational frequencies (Hartshorn and Shiner, 1972), but there seems to be no sound evidence to support the view that Sn2 reactions must give a-deuterium isotope effects of 1-06 or less. 

This is illustrated in Figure 7.4 where the effectiveness factor is plotted versus the low ij Aris number An0 for a bimolecular reaction with (1,1) kinetics, and for several values of/ . P lies between 0 and 1, calculations were made with a numerical method. Again all curves coincide in the low tj region, because rj is plotted versus An0. For p = 0, the excess of component B is very large and the reaction becomes first order in component A. For p = 1, A and B match stoichiometrically and the reaction becomes pseudosecond order in component A (and B for that matter). Hence the rj-An0 graphs for simple first- and second-order reactions are the boundaries when varying p. 

A plot of the effectiveness factor as a function of the Thiele modulus is shown in Figure 12-5. Figure l2-3(a) shows t) as a function of the Thiele modulus < )j for a spherical catalyst pellet for reactions of zero, first, and second order. Figure 12-5(b) corresponds to a first-order reaction occurring in three differently shaped pellets of volume Vp and external surface area Ap, and the Thiele modulus for a first-order reaction, < >], is defined difierently for each shape. When volume change accompanies a reaction (i.e., 0) the corrections shown in Figure 12-6 apply to the effectiveness factor for a first-order reaction. 

Plotted on the ppm scale, spectral features arising from second order quadrupole interaction effects would increase by a factor of 1.9, for the 96 MHz spectrum and 3.4, for the 76 MHz spectrum, when compared with the 132 MHz spectrum. Inspection of the spectra in Figure lb indicates that this is not occurring, establishing that the spectral features are associated with separate sodium-23 MASNMR lines. This result can be more clearly observed for the partially exchanged, hydrated Y zeolites discussed below. 

The initial velocity of quinuclidine substitution is significantly faster than that of triethylamine at the same temperature (Table III, runs 21-23), even though the former was investigated in a mixed solvent. Similar results were found in the quaternization of the model compound. If a steric effect were considered to be the sole factor producing the decrease in k with respect to kQ, one would expect that (1) hQ/k2 for quinuclidine substitution should be smaller than hQ/k2 for TEA substitution and/or (2) the initial linearity in the second order plot would extend beyond 52% conversion where deviation occurs in the triethylamine system. Experimental results refute these expectations rate retardation is enhanced in quinuclidine reactions, furtherfore, the break point is almost the same for both cases. 

From Eq. (7.174), we see that the ideal terms obtained with infinite GB have been corrupted by additional terms caused by finite GB. To see the effects graphically, we select several values of Q and then factor the polynomial in Eq. (7.175) for many values of GB in order to draw loci. The polynomial in Eq. (7.175) is fifth order, but three of the roots are in the far left-half normalized s plane. The dominant poles are a pair of complex poles that correspond to the ideal poles in Eq. (7.172) but are shifted because of finite GB. Figure 7.117 shows the family of loci generated, one locus for each value of Q selected. Only the lod of the dominant second quadrant pole are plotted. The other dominant pole is the conjugate. 

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