Reaction diffusion-limited

Rice S A 1985 Diffusion-limited reactions Oomprehensive Ohemical Kinetics vol 25, ed C H Bamford, C F H Tipper and R G Compton (Amsterdam Elsevier)... 

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 effectiveness of a porous catalyst T] is defined as the actual diffusion-limited reaction rate divided by the reaction rate that could have been achieved if all the internal surface had been at bulk concentration conditions. 

KINETIC EXPRESSIONS DERIVED FOR DIFFUSION-LIMITED REACTIONS... 

A diffusion-limited reaction proceeding in spherical particles (radius r) obeys a rate expression obtained by combining eqn. (10) with the contracting volume relation [eqn. (7), n = 3], viz. 

In a diffusion-limited reaction, movement of species occurs by a "hopping motion. 

However, there is another operative timescale in solution. This is that timescale for reaction with other photolytically generated species or with added reactants. This reaction cannot take place faster than the diffusion-limited reaction rate which is concentration dependent (59). Typical diffusion-controlled reaction rate constants are 109-1010 dm3 mol"1 second-1. By comparison, a typical gas-kinetic rate con-... 

Absolute values of the rate constants ks (s ) and kp (s ). In most cases these rate constants were determined from the values of kaz/ks (M 1) or kaz/kp (M-1) for partitioning of the carbocation between reaction with azide ion and solvent, by using the diffusion-limited reaction of azide ion, kaz = 5 x 109m 1s, as a clock for the slower reactions of solvent.7 8 13 32 82... 

The intrinsic barriers for the reaction of [12+] correspond to intrinsic rate constants of ( mcoh)o = 1 x 108m-1 s-1 and (kp)0 = 450 s-1 (equation 4). This analysis shows that the thermoneutral addition of methanol to [12+] is an intrinsically fast reaction, with a rate constant that is only 50-fold smaller than that for a diffusion-limited reaction.16... 

On the other hand, the low temperature dependance of the rate constants with activation energies around 5 kcal/mole indicates a diffusion limited reaction rate which could refer to diffusion of oxygene into the fibers of the board, i.e. into the fiberwalls. The corresponding negative activation energy for the groundwood based hardboard and the effect of fire retardants there upon are difficult to understand. 

Traytak, S. D. (1996). Competition effects in steady-state diffusion-limited reactions renormalization group approach, J. Chem. Phys., 105, 10 860-10 867. 

Qualitatively, one can now deduct the apparent activation energy As for the case of a diffusion-limited reaction. If we plot the logarithm of the product of the efficiency factor and the constant for the speed of reaction ln[kij] against 1/T, a typical curve with three regimes can be seen (see Figure 11.15). 

A similar prediction can be made for the concentration distribution of reagents for a diffusion limited reaction occurring at the phase boundary. The concentration of the reactants decreases around the phase boundary, as this is the site where they are consumed. In Figure 2.14, it is assumed that the reactant A has about one tenth of the solubility in phase 2 compared to phase 1, thus in most cases some of reactant A will diffuse across the phase boundary into this phase. As in phase 1, the concentration distribution will not be equal throughout the phase, but it will be lower in proximity to the phase boundary. If the reaction is very fast, reactant A will be consumed at the phase boundary and will therefore not enter phase 2. 

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. 

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). ... 

The diffusion-related molecular processes occurring within a Cig stationary-phase have also been investigated using pyrene as a fluorescent probe [169]. Particular spectral bands were attributed to pyrene excimers formed in a diffusion-limited reaction. Rate constants for this formation were then used to estimate the microviscosity of the stationary-phase. A similar application of total internal reflection fluorescence... 

Rice, S. A. Diffusion-limited reactions Elsevier Amsterdam 1985. 

X 10 M s . This illustrates the degree to which electrostatic interactions can change a reaction from a diffusion-limited process to one that involves other types of protein-ligand interactions. See Chemical Kinetics Diffusion-Limited Reaction... 

ENCOUNTER-CONTROLLED RATE DIFFUSION-LIMITED REACTION CHEMICAL KINETICS DIFFUSION OF LIGAND TO RECEPTOR DIFFUSION OF MOLECULES INTO A PORE... 

Figure 7-20 Sketch of tube wall reactor with a porous catalyst film of thickness t on walls. Expected reactant concentration profiles with reaction-limited, mass-transfer-limited, and pore-diffusion-limited reaction.

In combustion of solids such as charcoal, a reaction occurs between carbon and O2 from the air in diffusion-limited reactions. More than 1000 years ago the Chinese found that when... 

See, for example Rice, S.A. Diffusion-Limited Reactions. Bamford, C.H. Tipper, C.F.H., Compton, G. eds. Comprehensive Chemical Kinetics Elsevier Amsterdam, 1985 Vol. 25. 

Also in case of diffusion-limited reactions where the overall effectiveness factor is used to describe the effect of diffusion on the rate of biocatalysis, the mathematics are the same as in the case of the batch reactor. Substitution of Equation (11.56) in Equation (11.23) thus yields ... 

An unusual "critical density phenomenon is encountered with AN/fuel expls. This has been attributed by Gordon to diffusion-limited reaction. This case is reviewed, and the implications concerning the question of detonability in solid composite proplnts are examined... 

This study employed conventional diffusion-reaction theory, showing that with diffusion-limited reactions the internal effectiveness factor of a heterogeneous catalyst is inversely related to the Thiele modulus. Using a standard definition of the Thiele modulus [100], the observed reaction rate of an immobilized-enzyme reaction will vary with the square root of the immobilized-enzyme concentration in a diffusion-limited system. In this case, a plot of the reaction rate versus the enzyme loading in the catalyst formulation will be nonlinear. 

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