Constant of diffusion

All reactions collected in Table 19.6 are slow. They occur with rate constants that are sufficiently lower than the rate constants of diffusion in polymer, as well as the frequency of reactant orientation in the cage (vor =vrx P). Hence, physical processes are not limited by the rates of these reactions. However, polymer media influences the kinetics of these reactions. 

It can be proposed that superimposed upon the intrinsic random walk of molecules in the NFI channel network are processes of non-diffusional molecular re-orientation leading to an optimal sorbate arrangement. These processes are slow for the relatively "stiff" 2-butyne molecule (due to its triple bond) but fast for the "flexible" n-butane, i.e. the additional regime of sorption kinetics becomes observable if the time constant of diffusion (k /D) is small compared to the time constant of re-orientation. Since the latter process should be Independent of crystal size, size variation will give further evidence for appropriate systems. 

Figure 23 shows the dependence of CTL intensity at 400 and 500 °C on flow velocity. The CTL intensity I at 400 °C is almost independent of the mean flow velocity v of the sample gas, i.e., the catalytic oxidation is under reaction-controlled conditions. On the other hand, at 500 °C the curve plotted as log I vs. log v has a slope of 1/2. Equation 16 shows that the rate of catalytic oxidation ry is proportional to the square root of the flow velocity uo, so that the catalytic oxidation at 500 °C is under diffusion-controlled conditions. As the rate constant of diffusion k ) increases with increasing m0> the value of rx... 

Most important macroscopic transport properties (i.e., permeabilities, solubilities, constants of diffusion) of polymer-based membranes have their foundation in microscopic features (e.g., free-volume distribution, segmental dynamics, distribution of polar groups, etc.) which are not sufficiently accessible to experimental characterization. Here, the simulation of reasonably equilibrated and validated atomistic models provides great opportunities to gain a deeper insight into these microscopic features that in turn will help to develop more knowledge-based approaches in membrane development. 

The permeation of small molecules in amorphous polymers is typically following the solution diffusion model, that is, the permeability P of a feed component i can be envisioned as the product of the respective solubility S and constant of diffusion Dj. Both parameters can be obtained experimentally and in principle also by atomistic simulations. 

It is obvious that if V is large at certain places I will get contributions practically only from these places. If the solution is ideal and no field of force is applied to the column, we may say that

different point of view. The dissolved molecules diffuse between the solvent molecules, which cause them to move in a force field of very variable intensity. If this were not so we should expect the diffusion constant to be the same everywhere. Denoting by D this constant of diffusion, we get... 

This scheme mimics, e.g., CO or H2 oxidation on the noble metal catalysts Pt, Pd, or Rh, where symbol A stands for CO or H, and B2 for 02. The reaction was simulated on a 2D lattice of adsorption sites. To compare the rates of diffusion and reaction, it is useful to employ the Arrhenius form to represent the rate constants of diffusion jumps of A and B particles to nearest-neighbor vacant sites and for the reaction between two nearest-neighbor reactants, respectively. The diffusion of A is usually rapid when compared to the LH step, while the rate constant for the LH step might be higher, close to, or lower than that for the diffusion of B2. The MC algorithm used to simulate the A + B2 reaction is as follows ... 

Scheme 24 summarizes the possible processes that can occur during a free-radical polymerization induced via an intermolecular electron transfer process (PET) in the presence of aromatic amines kdii is the rate constant of diffusive encounters be-... 

Time resolved studies on dye molecules can help to elucidate the solvation dynamics and can give information on the time constants of diffusion of the ionic components of an RTIL [69-75], Time resolved fluorescence studies show the diffusional motion of the dissolved solutes [76], Luminescence quenching of fluorescent transition metal dyes by oxygen has been used in case of so-called core-shell soft-sphere ionic liquids [77] to monitor the oxygen permeability of these ILs [78],... 

Reactions between molecules A and B via an encounter complex A -B may be treated by the kinetic Scheme 2.3, where kd is equal to the bimolecular rate constant of diffusion, k d is the first-order rate constant for escape from the encounter complex and kP is the first-order rate constant of product formation in the encounter complex. 

The partition ratio kBl(k d + kB) defines the efficiency of product formation from the encounter complex (see also Section 3.7.4). For the limiting case k dobserved rate constant of reaction approaches the rate constant of diffusion, kx kd. In 1917, von Smoluchowski derived Equation 2.27 from Fick s first law of diffusion for the ideal case of large spherical solutes. 

Equation 2.29 Rate constant of diffusion of large spherical solutes... 

Note that r and the diffusion coefficient D have cancelled from Equation 2.29, because D is inversely proportional to the molecular radii r /2. Hence the rate constant kd depends only on temperature and solvent viscosity in this approximation. A selection of viscosities of common solvents and rate constants of diffusion as calculated by Equation 2.29 is given in Table 8.3. The effect of diffusion on bimolecular reaction rates is often studied by changing either the temperature or the solvent composition at a given temperature. For many solvents,54-56 although not for alcohols,57 the dependence of viscosity on temperature obeys an Arrhenius equation, that is, plots of log rj versus 1 IT are linear over a considerable range of temperatures and so are plots of log(kdr]/T) versus 1/T.56... 

For example, the multiplicity of radicals with one unpaired electron, S = V2, is 2S + 1=2. Each of four spin states is then expected to form with equal probability upon encounter of two radicals 2A and 2B, a = 1/4. Three of these are sublevels of the encounter complex with triplet multiplicity, S = SA + SB = 1, 2S + 1 = 3, and the fourth is the singlet encounter pair, S = SA + SB — 1 = 0, 2S + 1 = 1. Only the latter can undergo radical recombination to form a singlet product P=A B without undergoing ISC. The above considerations therefore suggest that the rate constant for radical recombination will not exceed one-quarter of the rate constant of diffusion, because only every fourth encounter will lead to recombination. 

As long as r > 3R0, the fluorescence decay is close to exponential, the lifetime of the donor fluorescence decreases linearly with increasing concentration of A and fluorescence quenching obeys Stern Volmer kinetics (Section 3.9.8, Equation 3.36). However, the bimolecular rate constants ket of energy transfer derived from the observed quenching of donor fluorescence often exceed the rate constants of diffusion kd calculated by Equation 2.26, because resonance energy transfer does not require close contact between D and A. Finally, when r < 3R0, at high concentrations and low solvent viscosity, the kinetics of donor fluorescence become complicated, but an analysis is possible,109,110 if required. 

Calculate the concentration required to quench 99% of singlet excited naphthalene, given that the lifetime of the naphthalene singlet state is 100 ns in the absence of dioxygen and A. /V-di methyl aniline and that the rate constant of diffusion /cdill is 5 x 109m 1 s [0.2m]... 

The reader will be relieved to know that Krogh s constant of diffusion is not defined in such an unhelpful way but, rather, in the more obvious form of nmols-1 cm-1 torr-1. This formulation... 

The rate constant of diffusion-controlled quenching of photo-excited )S-naphthylamine by CCI4 (in isooctane) is 1.3 x lO mol dm s , and in cyclohexane 7.2 x 10 mol dm s at 298 K [52]. Another example is a reaction between ions with opposite charge. At 298 K, the rate constant of hydroxyl and ammonium ion association in water is 3 x 10 moP dm s and that of oxonium and chloroacetate ion association... 

The rate constants of diffusion-controlled reactions are proportional to the diffusion rates of both reaction components. They are therefore a function of solvent viscosity inversely proportional to it according to Einstein [54]. Except in extreme cases, propagation is not diffusion-controlled. Monomer addition to the radical requires the crossing of an energy barrier of about 10 kJ mol and larger the steric factor of this reaction is 10 and less. Therefore the rate constant of propagation does not decrease even at medium conversion at considerably increased viscosity (see Chap. 4, Tables 3 and 4). 

It is important to note that the constant a describes the rate constant of diffusion relative to the surface reaction rate constant. A large value of the constant a... 

P (Q Q) and Q(Q -> n) are the probabilities that a molecule will rotate in the redistribution processes of trans cis optical transition and cis =i trans thermal recovery respectively. The orientational hole burning is represented by a probability proportional to cos 9. The last terms on the right-hand side of Eq. 11 describe the rotational diffusion due to Brownian motion. This is a Smoluchowski equation for the rotational diffusion characterized by a constant of diffusion for the cis (trans) configur-... 

The derivation of a theoretical expression for the rate constant of diffusion along a concentration gradient has been described in a straightforward manner by Collins (238) and Noyes (3), for the reaction... 

Tdifr Time constant of diffusion-limited uptake... 

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