Effective kinetic rate constant

The nature of the channel data can be determined by evaluating how the effective kinetic rate constant fceff varies as a function of the effective time scale eff at which it is measured. We developed several different methods to determine this function from the experimental data (10, II), and subsequently our methods have been improved by others (12, 13). 

The effective kinetic rate constant fceff is the probability for changing states when we observe the data at temporal resolution feff. Note that because 1 < d < 2, keff increases when we observe the channel at finer temporal resolution teff. That is, the faster we can look, the faster we see the channel open and close. If log fceff is plotted versus log eff, then eq 3 is a straight line. When the data are not fractal, this plot has other forms. For example, when there are only a few discrete states, such as those predicted by the Markov model, then there are a few well separated plateaus on this plot (10). Thus, without making any a priori assumptions about the data, we determine the function fceff(leff) and thus plot log keS versus log Ieff. The form of this plot can thus tell us the characteristics of the channel kinetics. 

As shown in Figure 4, for the channel in the corneal endothelium, we found (10) that the logarithm of the effective kinetic rate constant fceff as a function of the logarithm of the effective time scale feff is a straight line, which is consistent with eq 3. Thus, this channel has fractal kinetics. We also found a similar form for the currents recorded through channels in cultured hippocampal neurons (II). 

Figure 4. The effective kinetic rate constant is the probability that the channel changes state when the current through the channel is measured at an effective time resolution tejr. The data shown were measured from a potassium channel in the corneal endothelium and have the power law form of eq 3, which is indicative of self-similar fractal behavior. (Reproduced with permission from reference 38. Copyright 1990 New York Academy of Sciences.)...

The continuous nature of the effective kinetic rate constant function keff(feff) suggests that there is actually a broad continuum of many channel states. That is, the energy structure of the channel must have a very large number of shallow local minima, rather than the few deep minima suggested by the Markov model. 

The entire previous discussion assumed that the switching of the ion channel from one conformational state to another is an inherently stochastic or random process. The kinetic rate constant, or the effective kinetic rate constant, tells us the probability per second that the channel will switch from one state to another, but it does not tell us exactly when this switch will occur. Are these transitions from one state to another really stochastic ... 

Dispersion of the two phases. Rates always depend on mixing. Effective kinetic rate constants can be formulated as a function of energy dissipation or interfacial area. 

Findings with Bench-Scale Unit. We performed this type of process variable scan for several sets of catalyst-liquid pairs (e.g., Figure 2). In all cases, the data supported the proposed mechanism. Examination of the effect of temperature on the kinetic rate constant produced a typical Arrhenius plot (Figure 3). The activation energy calculated for all of the systems run in the bench-scale unit was 18,000-24,000 cal/g mole. 

Figure 7. Effect of temperature on kinetic rate constant with...

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

Picosecond kinetics, 266 Pre-equilibria, 133-135 Pre-steady-state region, 116 Pressure, effect on rate constants, 166-167... 

A kinetic study for the polymerization of styrene, initiated with n BuLi, was designed to explore the Trommsdorff effect on rate constants of initiation and propagation and polystyryl anion association. Initiator association, initiation rate and propagation rates are essentially independent of solution viscosity, Polystyryl anion association is dependent on media viscosity. Temperature dependency correlates as an Arrhenius relationship. Observations were restricted to viscosities less than 200 centipoise. Population density distribution analysis indicates that rate constants are also independent of degree of polymerization, which is consistent with Flory s principle of equal reactivity. 

Methods similar to those discussed in this chapter have been applied to determine free energies of activation in enzyme kinetics and quantum effects on proton transport. They hold promise to be coupled with QM/MM and ab initio simulations to compute accurate estimates of nulcear quantum effects on rate constants in TST and proton transport rates through membranes. 

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

At 20°C, an induction period was observed in hexane. At 5°C an induction period of about 2 hours was seen in hexane, toluene, and styrene. This suggests that the Pt(II) complex [C6H5CH=CH2PtCl2]2 became catalytic after conversion to C6H5CH=CH2PtH(—Si=). This conversion was not instantaneous thus an induction period was noted, influenced by temperature and by solvents. The kinetic rate constant for formation of the phenylethylsilane was temperature dependent but nearly free of solvent effects. 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

No carrier is completely specific for a given trace metal metals of similar ionic radii and coordination geometry are also susceptible to being adsorbed at the same site. The binding of a competing metal to an uptake site will inhibit adsorption as a function of the respective concentrations and equilibrium constants (or kinetic rate constants, see below) of the metals. Indeed, this is one of the possible mechanisms by which toxic trace metals may enter cells using transport systems meant for nutrient metals. The reduced flux of a nutrient metal or the displacement of a nutrient metal from a metabolic site can often explain biological effects [92]. 

The secondary /3-deuterium KIEs observed for the reaction of the same substrate with hydroxide ion and with tris(hydroxymethyl)methylamine in aqueous solution at 25°C were small, i.e. kH/kD = 1.09 0.01 and 1.10 0.01, respectively. While Kresge argued that the EIE was primarily due to hyperconjugation, the secondary /3-deuterium KIEs were attributed partly to hyperconjugation and partly to a polar (inductive) effect. The rate constants for the evaluation of both the EIE and the KIEs were determined in separate kinetic runs by following the increase in the absorbance due to the nitronate ion by UV spectroscopy. 

Choosing a method to determine isotope effects on rate constants, and selecting a particular set of techniques and instrumentation, will very much depend on the rate and kind of reaction to be studied, (i.e. does the reaction occur in the gas, liquid, or solid phase , is it 1st or 2nd order , fast or slow , very fast or very slow , etc.), as well as on the kind and position of the isotopic label, the level of enrichment (which may vary from trace amounts, through natural abundance, to full isotopic substitution). Also, does the isotopic substitution employ stable isotopes or radioactive ones, etc. With such a variety of possibilities it is useless to attempt to generate methods that apply to all reactions. Instead we will resort to discussing a few examples of commonly encountered strategies used to study kinetic isotope effects. 

The model predicted that the for chloroform metabolism without correcting for core temperature effects was 14.2 mg/hour/kg (2/3 of that reported in the Corley model) and the Iv was 0.25 mg/L. Without body temperature corrections, the model underpredicted the rate of metabolism at the 5,500 ppm vapor concentration. Addition of a first-order kinetic rate constant (kl=l. 86 hour ) to account for liver... 

Structures have been determined for [Fe(gmi)3](BF4)2 (gmi = MeN=CHCF[=NMe), the iron(II) tris-diazabutadiene-cage complex of (79) generated from cyclohexanedione rather than from biacetyl, and [Fe(apmi)3][Fe(CN)5(N0)] 4F[20, where apmi is the Schiff base from 2-acetylpyridine and methylamine. Rate constants for mer fac isomerization of [Fe(apmi)3] " were estimated indirectly from base hydrolysis kinetics, studied for this and other Schiff base complexes in methanol-water mixtures. The attenuation by the —CH2— spacer of substituent effects on rate constants for base hydrolysis of complexes [Fe(sb)3] has been assessed for pairs of Schiff base complexes derived from substituted benzylamines and their aniline analogues. It is generally believed that iron(II) Schiff base complexes are formed by a template mechanism on the Fe " ", but isolation of a precursor in which two molecules of Schiff base and one molecule of 2-acetylpyridine are coordinated to Fe + suggests that Schiff base formation in the presence of this ion probably occurs by attack of the amine at coordinated, and thereby activated, ketone rather than by a true template reaction. ... 

In this case of uncharged, nonpolar reactions, there is little interaction between the reactants and the solvent. As a result, the solvent does not play an important role in the kinetics per se, except through its role in determining the solubility of reactive species and cage effects. The rate constants for such reactions therefore tend to be similar to those for the same reactions occurring in the gas phase. Thus, as we saw earlier, diffusion-controlled reactions in the gas phase have rate constants of 10-ll) cm3 molecule-1 s-1, which in units of L mol-1 s-1 corresponds to 6 X 1010 L mol-1 s-1, about equal to (usually slightly greater than) that for diffusion-controlled reactions in solution. 

The reaction with 4-hydroxy-TEMPO leading to the ether is particularly interesting as the reaction could be spin allowed (i.e., doublet - - triplet —> doublet) if sufficient interaction between the O—H and nitroxide centers takes place. However, the EPR parameters for 4-hydroxy-TEMPO suggest that the interaction between the two sites is small.The magnitude of the relaxation of spin conservation rules seems unclear, but the kinetic results show virtually no effect. The rate constant for insertion at the O—H bond is 2 x 10 s, which is essentially the... 

Chemical reactions at supercritical conditions are good examples of solvation effects on rate constants. While the most compelling reason to carry out reactions at (near) supercritical conditions is the abihty to tune the solvation conditions of the medium (chemical potentials) and attenuate transport limitations by adjustment of the system pressure and/or temperature, there has been considerable speculation on explanations for the unusual behavior (occasionally referred to as anomalies) in reaction kinetics at near and supercritical conditions. True near-critical anomalies in reaction equilibrium, if any, will only appear within an extremely small neighborhood of the system s critical point, which is unattainable for all practical purposes. This is because the near-critical anomaly in the equilibrium extent of the reaction has the same near-critical behavior as the internal energy. However, it is not as clear that the kinetics of reactions should be free of anomalies in the near-critical region. Therefore, a more accurate description of solvent effect on the kinetic rate constant of reactions conducted in or near supercritical media is desirable (Chialvo et al., 1998). 

It has been suggested that the experimental isothermal kinetic rate constants of some reactions at near and supercritical conditions could not be explained solely by the thermodynamic pressure effect, but from the combination of local composition enhancement and density augmentation around reactants. 

Diffusional effects were combined into apparent kinetic rate constants by using commercial-sized catalysts in kinetic experiments. The experiments were designed so that no significant external transport and axial dispersion effects occurred. 

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