Diffusion-limited coefficient

There is no doubt that the coefficient of the third term on the right hand side of this equation is much larger than the coefficient of the second term at the bulk diffusion limit, and this justifies our original form (5,29). However, on constructing from and using equation (5,32) and its... 

Effective diffusion coefficient, in porous medium at bulk diffusion limit, 14... 

Pore diffusion limitation was studied on a very porous catalyst at the operating conditions of a commercial reactor. The aim of the experiments was to measure the effective diffiisivity in the porous catalyst and the mass transfer coefficient at operating conditions. Few experimental results were published before 1970, but some important mathematical analyses had already been presented. Publications of Clements and Schnelle (1963) and Turner (1967) gave some advice. 

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. 

Torkelson and coworkers [274,275] have developed kinetic models to describe the formation of gels in free-radical pol5nnerization. They have incorporated diffusion limitations into the kinetic coefficient for radical termination and have compared their simulations to experimental results on methyl methacrylate polymerization. A basic kinetic model with initiation, propagation, and termination steps, including the diffusion hmitations, was found to describe the gelation effect, or time for gel formation, of several samples sets of experimental data. 

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

We have also measured the rate constant for the association reaction of two Mn(C0)5 radicals generated on photolysis of Mn2(CO)io- With appropriate assumptions regarding the absorption coefficient for Mn(C0)5, the rate constant for this reaction was determined to be (2.7 0.6) x 10 1 mole l s [6,10]. This is compatible with the diffusion limited rate constant for this reaction that has been measured in solution and is within an order of magnitude of a gas kinetic rate constant as would be expected for an essentially unactivated radical-radical association reaction [33a]. 

Mass transfer considerations are critical in any bioprocess. In typical, aerobic, suspended cell fermentations, the major concern is the oxygen transfer rate, determined by the overall mass transfer coefficient, kft, and the driving force. In three-phase biofluidization, in which the cells are immobilized as a biofilm or within carrier particles, the situation is further complicated by possible intraparticle diffusion limitations. Numerous recent studies have addressed these issues. 

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

As mentioned earlier, ascorbate and ubihydroquinone regenerate a-tocopherol contained in a LDL particle and by this may enhance its antioxidant activity. Stocker and his coworkers [123] suggest that this role of ubihydroquinone is especially important. However, it is questionable because ubihydroquinone content in LDL is very small and only 50% to 60% of LDL particles contain a molecule of ubihydroquinone. Moreover, there is another apparently much more effective co-antioxidant of a-tocopherol in LDL particles, namely, nitric oxide [125], It has been already mentioned that nitric oxide exhibits both antioxidant and prooxidant effects depending on the 02 /NO ratio [42]. It is important that NO concentrates up to 25-fold in lipid membranes and LDL compartments due to the high lipid partition coefficient, charge neutrality, and small molecular radius [126,127]. Because of this, the value of 02 /N0 ratio should be very small, and the antioxidant effect of NO must exceed the prooxidant effect of peroxynitrite. As the rate constants for the recombination reaction of NO with peroxyl radicals are close to diffusion limit (about 109 1 mol 1 s 1 [125]), NO will inhibit both Reactions (7) and (8) and by that spare a-tocopherol in LDL oxidation. 

All these reactions are thermodynamically favourable in the direction of proton transfer to hydroxide ion but the rate coefficients are somewhat below the diffusion-limited values. In broad terms, the typical effect of an intramolecular hydrogen bond on the rate coefficient for proton removal is to reduce the rate coefficient by a factor of up to ca 105 below the diffusion limit. Correspondingly the value of the dissociation constant of the acid is usually decreased by a somewhat smaller factor from that of a non-hydrogen-bonded acid. There are exceptions, however. 

In this region, the equilibrium constant for the proton-transfer step in Scheme 7 has a value K2> 1 and the proton transfer step is strongly favourable thermodynamically in the forward direction. This reaction step is a normal proton transfer between an oxygen acid which does not possess an intramolecular hydrogen bond and a base (B) and will therefore be diffusion-limited with a rate coefficient k2 in the range 1 x 109 to 1 x 1010dm3mol-1 s 1. It follows from (65) that kB will have a value which is... 

The reactions, with rate coefficients well below the diffusion-limited values, are thought to occur by direct proton transfer from the donor acid into the molecular cavity. The kinetic isotope effect for proton transfer was observed to vary as a function of the pX-value of HA and to pass through a maximum value kHA/kDA 4.0, the maximum occurring for a reaction with ApA" = pA (HA) — pA ([2.1.1]H22+) = ca + 1. A similar large kinetic isotope effect kHA/kDA = 3.9 was observed for protonation of the cryptand by H20 and D20 in the isotopically different solvents (Kjaer et al., 1979). 

In any case, exceptions to the FIAM have been pointed out [2,11,38,44,74,76,78]. For example, the uptake has been shown to depend on the Cj M or rMI (e.g. in the case of siderophores [11] or hydrophobic complexes [43,50]), rather than on the free c M. Several authors [11,12,15] showed that a scheme taking into account the kinetics of parallel transfer of M from several solution complexes to the internalisation transporter ( ligand exchange ) can lead to exceptions to the FIAM, even if there is no diffusion limitation. Adsorption equilibrium has been assumed in all the models discussed so far in this chapter, and the consideration of adsorption kinetics is kept for Section 4. Within the framework of the usual hypotheses in this Section 3, we would expect that the FIAM is less likely to apply for larger radii and smaller diffusion coefficients (perhaps arising from D due to the labile complexation of M with a large macromolecule or a colloid particle, see Section 3.3). 

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

It should be emphasized that, because small molecules in usual solvents have diffusion coefficients <10 cm2 s-1, the rapid diffusion limit can be attained only for donors with lifetimes of 1 ms. This is the case for lanthanide ions for instance, the lifetime of Tb3+ chelated to dipicolinate is 2.2 ms. Stryer and coworkers (1978) showed that using Tb3+ as a donor and rhodamine B as an acceptor, the concentration of rhodamine B resulting in 50% transfer was 6.7 x 10 6 M, which is three orders of magnitude less than the concentration corresponding to 50% transfer in the static limit. 

Since the rate constants of bimolecular diffusion-limited reactions in isotropic solution are proportional to T/ these data testify to the fact that the kt values are linearly dependent on the diffusion coefficient D in water, irrespective of whether the fluorophores are present on the surface of the macromolecule (human serum albumin, cobra neurotoxins, proteins A and B of the neurotoxic complex of venom) or are localized within the protein matrix (ribonuclease C2, azurin, L-asparaginase).1 36 1 The linear dependence of the functions l/Q and l/xF on x/t] indicates that the mobility of protein structures is correlated with the motions of solvent molecules, and this correlation results in similar mechanisms of quenching for both surface and interior sites of the macromolecule. 

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