Diffusional constant approximation

All the ko values listed in Table VII are much larger than the contact diffusional constants ko = 4nuD, so that the reactions are strongly in the diffusional limit, as was expected [248]. The same is true for the diffusional k(c), which is larger everywhere than the Stern-Volmer constant obtained in the contact approximation. The family of such curves compared in Figure 3.90... 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

If Rm and Rq are of the same order, the diffusional rate constant is approximately equal to 8RT/3tj. 

Because formation ofexcimer E is a diffusion-controlled process, Eqs (4.11)-(4.13) apply to the diffusional rate constant ki for excimer formation. Under the approximation that ki is time-independent, the d-pulse responses, under the initial conditions (at t = 0), [M ] = [M ]o and [E ]o = 0, are... 

This result implicitly requires (a) that no strongly attractive/repulsive electrostatic interaction occurs as the particles approach each other, and (b) that a stationary diffusional concentration field surrounds the particle. [Note Einstein s result for spherical particle diffusivity i.e.,D = k Tlfm-qr, where is the Boltzmann constant, T is temperature in kelvin, 17 is the viscosity of the medium, and r is the radius) indicates that RuD will be approximately Ak TKyni].]... 

The diffusional rate constant kD is calculated on the basis of the Debye-Hiickel theory (Equation 6.107), where the distance tr is the sum of A and B radii in the hard-sphere approximation. 

Figure 3.62. The light dependence of the Stem—Volmer constant K, i (/, i for diffusional quenching with given exponential rate (Wq — 103 exp[—2(r — cr)/L] ns-1), but at different diffusion in pairs containing the excited molecules (a) D =0.1 Dd, (6) D = D= 10 5cm2/s (dashed line—the same, but in contact approximation), (c) D = WD. Other parameters a = 5A, L = 1.0 A, i = 10ns, ko = f Wq(r)d3r = 1.9 x 105 A3/ns. (From Ref. 200.)...

In the lowest approximation with respect to c, the Stern-Volmer constant in DET, Eq. (3.27), is the same as in IET, Eq. (3.273). Under diffusional control the difference between k and Ko is expected to be the largest, but not in this approximation when... 

Figure 3.87. The Stern-Volmer constants as functions of the dimensionless concentration % = 4tiac3/3 obtained in the contact approximation and under diffusional control at t /td = 0.01. The thick line represents DET, which is exact for immobile donors and independently moving acceptors. The rest of the curves are obtained with the approximate methods, SA, MET, and IET. (From Ref. 133.)...

According to the Smoluchowski theory of diffusion-controlled bimolecular reactions in solutions, this constant decreases with time from its kinetic value, k0 to a stationary (Markovian) value, which is kD under diffusional control. In the contact approximation it is given by Eq. (3.21), but for remote annihilation with the rate Wrr(r) its behavior is qualitatively the same as far as k(t) is defined by Eq. (3.34)... 

Conceptual models of percutaneous absorption which are rigidly adherent to general solutions of Pick s equation are not always applicable to in vivo conditions, primarily because such models may not always be physiologically relevant. Linear kinetic models describing percutaneous absorption in terms of mathematical compartments that have approximate physical or anatomical correlates have been proposed. In these models, the various relevant events, including cutaneous metabolism, considered to be important in the overall process of skin absorption are characterized by first-order rate constants. The rate constants associated with diffusional events in the skin are assumed to be proportional to mass transfer parameters. Constants associated with the systemic distribution and elimination processes are estimated from pharmacokinetic parameters derived from plasma concentration-time profiles obtained following intravenous administration of the penetrant. 

Equal increments in i correspond to approximately equal decrements in In 4tAbuik-As illustrated in Table 23-8, ai, a2, and aj depend on the aspect ratio for rectangular channels. The asymptotic Nusselt number for mass transfer is given by cKi for constant transverse diffusional flux at the catalytic wall. The exponent m in equation (23-82) is either for plug flow or for viscous flow. 

The rate of hexobarbital detoxification using a Bio-Fiber 80/5 cell culture tube was then compared to the rate of detoxification using an Amlcon Vitaflber and to that of a pS suspension in a test tube. The steady-state rates of all three were the same (Figure 5). It is Important to note that in these systems containing approximately 2.8 mg pS protein the steady-state rates were constant for the 3 hour duration of the experiments. These experiments also indicate that at this low concentration of pS no diffusional limitations exist for either hexobarbital... 

Another simple case is a homogeneously growing single-phase layer, which allows only a slow diffusional transport of corrosive medium to the substrate. Here the steady state means for Eq. (5) that a layer, and hence the diffusion path length /, is growing with time while the absolute concentration difference Ac stays constant. At the substrate-oxide interface, the equilibrium with the material is achieved or the concentration of the agent can be approximated to zero, while the concentration at the surface of the scale is constant at the solubility limit of the scale material. 

Quantitative formulations have been worked out for the rate constants for both full and partial diffusion control. Diffusion rates do not vary greatly from one system to another and for uncharged reactants in aqueous solution at 25°C diffusional rate constants are approximately 7 x 10 dm mol s If the chemical rate constant is substantially greater than this there is therefore appreciable diffusion control. Most reactions, however, have much smaller rate constants because of an energy barrier to reaction, and are therefore not affected by diffusion. For reactions between ions the diffusion rates and chemical rates are increased if the ions are of opposite signs, and they are decreased if they are of the same sign. 

Particle Size Effect For the moment, let it be assumed that the alkylation reaction is approximately first order in butene concentration. It then follows that the apparent order should also be unity in the presence of diffusional limitations. If it is assumed that our alkylation reactor can be modeled as a plug-flow reactor, then the apparent rate constant for each particle size can be readily estimated using equation 1. 

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