Rate-independent process

As for a rate-independent process, the hysteresis results from an irreversibility of the domain wall motion, with the two limiting cases for the hardening-softening process being defined with respect to the number of trapping sites for the domain wall in its potential energy landscape ... 

In spite of its wide application, the mechanisms of this reaction remain obscure. Many diverse arguments have been published since the reaction was first investigated in 1897 (Bl, C5, C9, F7, J6, M5, P9, R2, S5, W2, W4, Yl, Y4). Cooper et al. (C9) introduced this method as a yardstick for the measurement of volumetric mass-transfer coefficients in gas-liquid contacting. Karow et al. (Kl) later concluded that the sulfite oxidation is suitable for fermentation process scale-up studies. Cooper et al. established that the reaction proceeds at a rate independent of sulfite ion concentration over wide concentration ranges. In their work they considered the sulfite oxidation to be of zero order with respect to both sulfite and sulfate concentration. 

For example, for alkyl (8-16) glycoside (Plantacare 818 UP) non-ionic surfactant solution of molecular weight 390 g/mol, an increase in surfactant concentration up to 300 ppm (CMC concentration) leads to a significant decrease in surface tension. In the range 300 < C < 1,200 ppm the surface tension was almost independent of concentration. In all cases an increase in liquid temperature leads to a decrease in surface tension. This surface tension relaxation is a diffusion rate-dependent process, which typically depends on the type of surfactant, its diffusion/absorption kinetics, micellar dynamics, and bulk concentration levels. As the CMC is approached the absorption becomes independent of the bulk concentration, and the surfactant... 

The intermediates 74 and 76 can now lose OR to give the acid (not shown in the equations given), or they can lose OH to regenerate the carboxylic ester. If 74 goes back to ester, the ester will still be labeled, but if 76 reverts to ester, the 0 will be lost. A test of the two possible mechanisms is to stop the reaction before completion and to analyze the recovered ester for 0. This is just what was done by Bender, who found that in alkaline hydrolysis of methyl, ethyl, and isopropyl benzoates, the esters had lost 0. A similar experiment carried out for acid-Catalyzed hydrolysis of ethyl benzoate showed that here too the ester lost However, alkaline hydrolysis of substimted benzyl benzoates showed no loss. This result does not necessarily mean that no tetrahedral intermediate is involved in this case. If 74 and 76 do not revert to ester, but go entirely to acid, no loss will be found even with a tetrahedral intermediate. In the case of benzyl benzoates this may very well be happening, because formation of the acid relieves steric strain. Another possibility is that 74 loses OR before it can become protonated to 75. Even the experiments that do show loss do not prove the existence of the tetrahedral intermediate, since it is possible that is lost by some independent process not leading to ester hydrolysis. To deal with this possibility. Bender and Heck measured the rate of loss in the hydrolysis of ethyl trifluorothioloacetate- 0 ... 

The chromic acid oxidation of dithionic acid is independent of oxidant concentration and its rate is equal to that of the acid-catalysed hydrolysis to sulphite and sulphate, which must therefore constitute the rate-determining process . ... 

In general the rate equation for a heterogeneous reaction accounts for more than one process. The present consideration is directed to the general problem of combining the rates for processes of different kinds. Let r1( r2,..., rn be the rates of changes for the individual processes that are to be accounted for by an overall rate. If the changes occur by parallel paths, then the overall rate will be greater than the rate for any individual path. In fact, if the different parallel paths are independent of each other, the overall rate will be simply the sum of all the individual rates, or... 

If the single-electron mechanism has not been demonstrated to be the rate-controlling process by an independent method, then, in the publication of the experimental results, it is preferable to replace the assumed quantity ax by the conventional value cm, provided that the charge number of the overall reaction is known (e.g. in an overall two-electron reaction it is preferable to replace = 0.5 by or = 0.25). If the independence of the charge transfer coefficient on the potential has not been demonstrated for the given potential range, then it is useful to determine it for the given potential from the relation for a cathodic electrode reaction (cf. Eq. 5.2.37) ... 

Transfer of aluminum ions from the oxide into the solution was considered as a statistically independent process, whose kinetics are governed by a rate equation similar to Eq. (24), i.e. (neglecting the return of the ions into the oxide),... 

It appears that all these possibilities can be excluded. If reactions (a) or (gf) were rate-limiting the reaction velocity would be independent of the concentration of the substrate, while reaction (e) (identical with (Z)) would predict no catalysis by acids or bases. If reactions (b), (d) or (h) determined the rate the reaction would show specific catalysis by hydrogen or hydroxide ions, in place of the general acid-base catalysis actually observed. Reactions (c), (f) and (m) are unacceptable as rate-limiting processes, since they involve simple proton transfers to and from oxygen. Reactions (j) and (k) might well be slow, but their rates would depend upon the nucleophilic reactivity of the catalyst towards carbon rather than on its basic strength towards a proton as shown in Section IV,D it is the latter quantity which correlates closely with the observed rates. 

If the boundary motion is controlled by an independent process, then the boundary motion velocity is independent of diffusion. This can happen if the magma is gradually cooling and crystal growth rate is controlled both by temperature change and mass diffusion. This problem does not have a name. In this case, u depends on time or may be constant. If the dependence of u on time is known, the problem can also be solved. The Stefan problem and the constant-w problem are covered below. 

Data obtained with m-trifluoromethylnitrobenzene are given in Table V. Again, at low nitroaromatic concentrations, the reaction is first-order in the nitroaromatic. However, at high nitroaromatic concentrations the oxidation rate becomes independent of nitroaromatic concentration, as expected for an ionization rate-limited process. 

The effect of methyl substituents on the (3-carbon on the rate of (3-elimination is complicated, different effects are observed for the acid dependent and acid independent process and for different central cations (Table V) (136). Methyl substituents on the (3-carbon enhance the specific rate of the acid independent path for both chromium(III) and copper(II) complexes, however no such effect was observed for the analogues (tspc)Co(III) complexes (136). The effect on the acid catalyzed reaction is even opposite for the chromium(III) and copper(II) complexes (136). 

An attempt is often made to relate T] and T2 to the molecular dynamics of a system. For this purpose a relationship is sought between T1 or T2 and the correlation time tc of the nuclei under investigation. The correlation time is the time constant for exponential decay of the fluctuations in the medium that are responsible for relaxation of the magnetism of the nuclei. In general, l/xc can be thought of as a rate constant made up of the sum of all the rate constants for various independent processes that lead to relaxation. One of the most important of these (1 /t2) is for molecular tumbling. 

More correct ideas about the polymerization came out after the observation was made, that the polymerization induced by sodium caprolactam starts with a characteristic induction period (67, 68). Beginning with a practically zero rate, the polymerization goes on with increasing velocity up to a maximal value (curve A on fig. I). The induction period indicates that certain active groups are formed by an independent process. The previous assumption of a disproportionation reaction adopted by Hanford and Joyce (42) can be supported by model reactions like the disproportionation of acetamide, benzamide (24, 57) or N-butylacetamide (68) in the presence of the respective sodium salts. 

We assume a fixed total population N = 217 to which a GCKS provides key management, i.e., the number of potential members is 217. Each member independently decides to join or leave the group. We approximate the members arrival by a fixed-rate Poisson process and assume that the lifetimes are independently and identically distributed random variables. In the simulation, we evaluate different lifetime distribution... 

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