Diffusion-limited rate constant

Keys et al. (2000) explored five approaches to modeling the pharmacokinetics of di- -butyl phthalate and mono- -butyl phthalate. In a flow-limited version of the model, transfers between blood and tissues are simulated as functions of blood flow, tissue concentrations of di- -butyl phthalate or mono-n-butyl phthalate, and tissue blood partition coefficients, assuming instantaneous partitioning of the compounds between tissue and blood (Ramsey and Anderson 1984). In an enterohepatic circulation version of the model, the transfer of mono-n-butyl phthalate from the liver to the small intestine is represented with a first order rate constant (diffusion-limited) and a time delay constant for the subsequent reabsorption of mono- -butyl phthalate from the small intestine. In a diffusion-limited version of the model, the tissue transfers include a first order rate term (referred to as the permeation constant) that relates the intracellular-to-extracellular concentration gradient to the rates of transfer. This model requires estimates of extracellular tissue volume (ECV) and intracellular volume (ICV) ECV is assumed to be equal to tissue blood volume and ICV is assumed to be equal to the difference between tissue blood volume and... 

K. At very low values of kf the normalized peak currents approach zero whereas for very high kf they approach the diffusion limited values. In reduction processes, the positive peak currents, Ip, are always higher than Ip (DPP curves with negative pulses), cf. Fig. 34. Both dependences coincide for very low and very high rate constants (diffusion limited zones). 

It appears that the rate constant for most of the boards has a smaller temperature dependence than the initial maximum rate, the corresponding "activation energy" E3 being around or less than 5 kcal/mol. An important conclusion is that the rate is diffusion limited. This has to be compared to a mean activation energy around 20 kcal/mole for the initial maximum rate of heat release for the commercial boards. As a consequence thereof the total heat release extrapolated over infinite time does increase to a significant extent with temperature from 150 to 230°C. 

The value of = 1 X 10 s for the first-order rate constant for collapse of an ion pair between Me-4 and pentaflourobenzoate ion is larger than the second-order rate constant rcoo = 5x10 M s reported for the bimolecular addition of alkane carboxylates to Me-4. This second-order rate constant is limited by the rate constant for formation of an ion pair between Me-4 and a carboxylate ion. The larger barrier to encounter-limited reactions of carboxylate ions compared with the diffusion-limited reactions of anions such as azide ion, = 5 X 10 represents the barrier to desolvation of nucleophile that must precede formation of an ion pair between Me-4 and a carboxylate ion (Scheme 13). ... 

Because of the appearance of equation 65, B is called the parabolic rate constant. This limiting case is the diffusion-controlled oxidation regime that occurs when oxidant availability at the Si-Si02 interface is limited by transport through the oxide (thick-oxide case). 

AG, Free Energy of Activation Rate Constant Upper Limit on Concentration Diffusion-Controlled Limit Dropping the AG by 1.36 kcal/mol (5.73 kJ/mol) Increases the Rate of Reaction Tenfold at Room Temperature Reasonable Rate at 25°C Half-Life Lifetime of an intermediate Rate-Determining Step Transition State Position Reactivity vs. Selectivity Thermodynamic vs. Kinetic AG = AH -TAS, Enthalpy of Transition Entropy of Transition Stabilization of Intermediates Stabilization of Reactants... 

Currently available thermodynamic and kinetic data bases are incomplete to support quantitative modeling of many corrosion systems, particularly those where predictions of behavior under extreme conditions or over extended periods of time are desired. Because the unavailability of data limits the use of models, a critical need exists to upgrade and expand the sources of information on the thermodynamic properties of chemical species, exchange current densities, activity coefficients, rate constants, diffusion coefficients, and transport numbers, particularly where concentrated electrolytes under extreme conditions are involved. Many of these data are obtained in disciplines that traditionally have been on the periphery of corrosion science, so it will be necessary to encourage interdisciplinary collaboration to meet the need. 

A two-step binding mechanism is often invoked to explain an apparent second-order rate constant slower than diffusion. Estimates of the rate of diffusion-limited binding of a substrate to an enzyme active site vary considerably, but fall in the range of approximately 10 sec" (5). The apparent second-order rate... 

For sufficiently small particles 0 0 and 1, so the measured rate constant approaches the intrinsic rate constant (k). By making replicate measurements under similar conditions, with different particle size fractions it is possible to determine both the intrinsic rate constant and the effective interparticle diffusivity. Haag [67] suggested that this approach could be used to determine intracrystalline diffusivities in zeolite crystals. A more complete experimental study in which the diffusivity of 2,2-dimethyl butane in HZSM-5 was determined both chromatographically and from measurements of the cracking rate under diffusion-limited conditions was reported by Post et al. [68] - see Fig. 11. This approach has the advantage that it makes steady-state rather than transient measurements, but it is limited to sorbates for which a suitable catalytic reaction occurs. 

With reasonable values for molecular parameters this yields < 10 -10 M sec" when T 500 K. A similar bound can be found for k if, for the reaction to occur, it is assumed that two molecules must be within a collision diameter when struck by a third. The result is k < 10 -10 M sec (Problem 5.23). In solution the rate constant is limited by the rate of diffusion (Sections 5.7 and 9.13). 

Although scan rates of few ten thousand volts per second (ie. 0/v 1 xs) seem to be a limit for quantitative analytical purposes (without using deconvolution procedures) for electrodes of few micrometers radii it is possible to use cyclic voltammetry in the low megavolt per second domain ( 0/v 25 ns) provided that the target is only identification of transient intermediates and determination of thermodynamic figures. Besides its intrinsic attractive value, such an order of magnitude of scan rates constitutes a milestone in electrochemical kinetics. Indeed bimolecular reactions in liquids cannot proceed faster than molecules encounter. Thus bimolecular rate constants are limited by diffusion limit rate... 

Hydroxyl radical (OH ) is one of the most reactive species present in both the atmosphere and surface waters. For many compounds, an encounter with a molecule of OH- almost always results in a reaction (i.e., the rate is diffusion-limited). In air, such a diffusion-limited rate constant is of the order of 10 cm /(molecule sec) in water, it is of the order of 10 /(M sec). Typical concentrations of OH- in sunlit air and water are 10 molecules/cm and 10 M, respectively. 

Straub J E and Berne B J 1986 Energy diffusion in many dimensional Markovian systems the consequences of the competition between inter- and intra-molecular vibrational energy transfer J. Chem. Phys. 85 2999 Straub J E, Borkovec M and Berne B J 1987 Numerical simulation of rate constants for a two degree of freedom system in the weak collision limit J. Chem. Phys. 86 4296... 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

The diffusion of H and D atoms in the molecular crystals of hydrogen isotopes was explored with the EPR method. The atoms were generated by y-irradiation of crystals or by photolysis of a dopant. In the H2 crystals the initial concentration of the hydrogen atoms 4x 10 mol/cm is halved during 10 s at 4.2 K as well as at 1.9 K [Miyazaki et al. 1984 Itskovskii et al. 1986]. The bimolecular recombination (with rate constant /ch = 82cm mol s ) is limited by diffusion, where, because of the low concentration of H atoms, each encounter of the recombinating partners is preceded by 10 -10 hops between adjacent sites. 

The quantity kcat/Km is a rate constant that refers to the overall conversion of substrate into product. The ultimate limit to the value of k at/Km is therefore set by the rate constant for the initial formation of the ES complex. This rate cannot be faster than the diffusion-controlled encounter of an enzyme and its substrate, which is between 10 to 10 per mole per second. The quantity kcat/Km is sometimes called the specificity constant because it describes the specificity of an enzyme for competing substrates. As we shall see, it is a useful quantity for kinetic comparison of mutant proteins. 

In these circumstances a decision must be made which of two (or more) kinet-ically equivalent rate terms should be included in the rate equation and the kinetic scheme (It will seldom be justified to include both terms, certainly not on kinetic grounds.) A useful procedure is to evaluate the rate constant using both of the kinetically equivalent forms. Now if one of these constants (for a second-order reaction) is greater than about 10 ° M s-, the corresponding rate term can be rejected. This criterion is based on the theoretical estimate of a diffusion-controlled reaction rate (this is described in Chapter 4). It is not physically reasonable that a chemical rate constant can be larger than the diffusion rate limit. 

This treatment obviously is oversimplified. At the next level of development, it is necessary to incorporate the intermolecular forces between A and B. ° If A and B are ions of opposite charge, it is found that the diffusion-limited rate constant is about 10 M s . ... 

From these considerations we conclude that diffusion-limited bimolecular rate constants are of the order 10 -10 M s . If an experimentally measured rate constant is of this magnitude, the usual conclusion is, therefore, that it is diffusion limited. For example, this extremely important reaction (in water)... 

Equations (4-5) and (4-7) are alternative expressions for the estimation of the diffusion-limited rate constant, but these equations are not equivalent, because Eq. (4-7) includes the assumption that the Stokes-Einstein equation is applicable. Olea and Thomas" measured the kinetics of quenching of pyrene fluorescence in several solvents and also measured diffusion coefficients. The diffusion coefficients did not vary as t) [as predicted by Eq. (4-6)], but roughly as Tf. Thus Eq. (4-7) is not valid, in this system, whereas Eq. (4-5), used with the experimentally measured diffusion coefficients, gave reasonable agreement with measured rate constants. 

In Section 3-3 we discussed the problem of kinetically equivalent rate terms. Suppose one of the rate constants evaluated for such a rate equation were larger than the diffusion-limited value this is a reasonable basis upon which to reject the formulation of the rate equation leading to this result. Jencks has given examples of this argument. 

Table 4-1 lists some rate constants for acid-base reactions. A very simple yet powerful generalization can be made For normal acids, proton transfer in the thermodynamically favored direction is diffusion controlled. Normal acids are predominantly oxygen and nitrogen acids carbon acids do not fit this pattern. The thermodynamicEilly favored direction is that in which the conventionally written equilibrium constant is greater than unity this is readily established from the pK of the conjugate acid. Approximate values of rate constants in both directions can thus be estimated by assuming a typical diffusion-limited value in the favored direction (most reasonably by inspection of experimental results for closely related... 

But k must always be greater than or equal to k h / (A i + kf). That is, the reaction can go no faster than the rate at which E and S come together. Thus, k sets the upper limit for A ,. In other words, the catalytic effieiency of an enzyme cannot exceed the diffusion-eontroUed rate of combination of E and S to form ES. In HgO, the rate constant for such diffusion is approximately (P/M - sec. Those enzymes that are most efficient in their catalysis have A , ratios approaching this value. Their catalytic velocity is limited only by the rate at which they encounter S enzymes this efficient have achieved so-called catalytic perfection. All E and S encounters lead to reaction because such catalytically perfect enzymes can channel S to the active site, regardless of where S hits E. Table 14.5 lists the kinetic parameters of several enzymes in this category. Note that and A , both show a substantial range of variation in this table, even though their ratio falls around 10 /M sec. 

D-Methylmalonyl-CoA, the product of this reaction, is converted to the L-isomer by methylmalonyl-CoA epunerase (Figure 24.19). (This enzyme has often and incorrectly been called methylmalonyl-CoA racemase. It is not a racemase because the CoA moiety contains five other asymmetric centers.) The epimerase reaction also appears to involve a carbanion at the a-position (Figure 24.20). The reaction is readily reversible and involves a reversible dissociation of the acidic a-proton. The L-isomer is the substrate for methylmalonyl-CoA mutase. Methylmalonyl-CoA epimerase is an impressive catalyst. The for the proton that must dissociate to initiate this reaction is approximately 21 If binding of a proton to the a-anion is diffusion-limited, with = 10 M sec then the initial proton dissociation must be rate-limiting, and the rate constant must be... 

The rate of hydrolysis of sarin on Dowex-50 cation exchange resin is insensitive to the stirring rate. However, with a more active catalyst (Amberlite-IRA 400), the rate constant at 20°C was 5.3, 7.5, and 8.5 h at 60,800 and 1000 revolutions/min , respectively, suggesting that film diffusion was the rate-limiting. step. Thus, the mechanism of the rate-limiting step depends on the nature of the catalyst [34]. 

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