Third-order kinetics, effectiveness factors

Therefore, the problems which faced the would-be designers of chain reactors early in 1941 were (1) the choice of the proper moderator to uranium ratio, and (2) the size and shape of the uranium lumps which would most likely lead to a self-sustaining chain reaction, i.e., give the highest multiplication factor. In order to solve these problems, one had to understand the behavior of the fast, of the resonance, and of the thermal neutrons. We were concerned with the second problem which itself consisted of two parts. The first was the measurement of the characteristics of the resonance lines of isolated uranium atoms, the second, the composite effect of this absorption on the neutron spectrum and total resulting absorption. One can liken the first task to the measurement of atomic constants, such as molecular diameter, the second one, to the task of kinetic gas theory which obtains the viscosity and other properties of the gas from the properties of the molecules. The first task was largely accomplished by Anderson and was fully available to us when we did our work. Anderson s and Fermi s work on the absorption of uranium, and on neutron absorption in general, also acquainted us with a number of technics which will be mentioned in the third and fourth of the reports of this series. Finally, Fermi, Anderson, and Zinn carried out, in collaboration with us in Princeton, one measurement of the resonance absorption. This will be discussed in the third article of this series. 

Consider the estimation of the center concentration, Q. As indicated earlier, the center concentration can be set to zero for 4>c > 3. On the other hand, the reaction can be considered to be free from diffusional effects for all practical purposes when G is less than 0.3. Therefore, estimation is necessary only for the intermediate range of o, i.e., 0.3 < G < 3. As the results in Figure 4.9 show, the maximum error in using the effectiveness factor of a first-order reaction for a third-order reaction is less than 10% when the generalized modulus is used. This suggests that the center concentration corresponding to a first-order reaction may be used for arbitrary kinetics ... 

With 77 % aqueous acetic acid, the rates were found to be more affected by added perchloric acid than by sodium perchlorate (but only at higher concentrations than those used by Stanley and Shorter207, which accounts for the failure of these workers to observe acid catalysis, but their observation of kinetic orders in hypochlorous acid of less than one remains unaccounted for). The difference in the effect of the added electrolyte increased with concentration, and the rates of the acid-catalysed reaction reached a maximum in ca. 50 % aqueous acetic acid, passed through a minimum at ca. 90 % aqueous acetic acid and rose very rapidly thereafter. The faster chlorination in 50% acid than in water was, therefore, considered consistent with chlorination by AcOHCl+, which is subject to an increasing solvent effect in the direction of less aqueous media (hence the minimum in 90 % acid), and a third factor operates, viz. that in pure acetic acid the bulk source of chlorine ischlorineacetate rather than HOC1 and causes the rapid rise in rate towards the anhydrous medium. The relative rates of the acid-catalysed (acidity > 0.49 M) chlorination of some aromatics in 76 % aqueous acetic acid at 25 °C were found to be toluene, 69 benzene, 1 chlorobenzene, 0.097 benzoic acid, 0.004. Some of these kinetic observations were confirmed in a study of the chlorination of diphenylmethane in the presence of 0.030 M perchloric acid, second-order rate coefficients were obtained at 25 °C as follows209 0.161 (98 vol. % aqueous acetic acid) ca. 0.078 (75 vol. % acid), and, in the latter solvent in the presence of 0.50 M perchloric acid, diphenylmethane was approximately 30 times more reactive than benzene. 

The previous sections have documented that the hydrolysis reaction of tetraalkoxy- and organoalkoxysilanes is influenced by sterlc and inductive effects and appears to be specific-acid-(H30 ) and specific-base-(OH ) catalyzed. The reaction order with respect to water and silicate is observed to be two and one, resulting in third- and second-order overall kinetics, respectively. Based on these factors, it is generally argued that hydrolysis proceeds by bimolecular nucleophilic displacement reactions (S y2-Si reactions) involving pentacoordinate intermediates or transition states... 

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