Diffusion effects independence

A paper by McCabe and Stevens (M4) illustrates an empirical approach to the correlation of growth rate coefficients. These authors reasoned that mass transfer to the interface consists of two parallel processes, a diffusion effect independent of velocity and a flow effect linear in velocity. They expressed this by a rate term (r + /Jit). This was related to rg, the over-all growth rate coefficient, by the conventional expression for rate processes in series,... 

For the light molecules He and H2 at low temperatures (below about 50°C.) the classical theory of transport phenomena cannot be applied because of the importance of quantum effects. The Chapman-Enskog theory has been extended to take into account quantum effects independently by Uehling and Uhlenbeck (Ul, U2) and by Massey and Mohr (M7). The theory for mixtures was developed by Hellund and Uehling (H3). It is possible to distinguish between two kinds of quantum effects— diffraction effects and statistics effects the latter are not important until one reaches temperatures below about 1°K. Recently Cohen, Offerhaus, and de Boer (C4) made calculations of the self-diffusion, binary-diffusion, and thermal-diffusion coefficients of the isotopes of helium. As yet no experimental measurements of these properties are available. 

For irradiation times of J short with respect to the relaxation time of / the NOE extent is independent of the relaxation time of the nucleus and provides a direct measurement of time required to saturate signal J is not negligible compared with t, the response of the system is not linear [18]. The truncated NOE is independent of paramagnetism as it does not depend on p/, which contains the electron spin vector S in the R[m term, and only depends on gkj), which does not contain S. If then the steady state NOE is reached, the value of p/ can also be obtained. This is the correct way to measure p/ of a nucleus, provided saturation of J can be considered instantaneous. In general, measurements at short t values minimize spin diffusion effects. In fact, in the presence of short saturation times, the transfer of saturation affects mainly the nuclei directly coupled to the one whose signal is saturated. Secondary NOEs have no time to build substantially. As already said, this is more true in paramagnetic systems, the larger the R[m contribution to p/. 

It is convenient to express the effect of diffusion by means of a diffusion factor rj, which is defined as the ratio of the observed reaction rate to the reaction rate when no diffusion effects are present. The diffusion factor r/ is a function of a parameter which contains all the independent variables of the system. For example, for a first-order reaction... 

Figure 24 illustrates the dependence of Type III selectivity on intraparticle and interphase diffusion effects by plotting the apparent overall selectivity from eqs 159, 167 and 168 for Bim/fa = 1, against the conversion of reactant Ai. From this figure, it appears that the influence of intraparticle diffusion may reduce the overall selectivity in Type III reactions by a factor of about two. Wheeler [113] reported that this degree of reduction is independent of the intrinsic selectivity factor AA . It may therefore serve as a general rule of thumb. 

At steady state the flux JA is independent of time. It also does not depend on the distance z in view of the mass conservation. Provided the effective diffusivity is independent of the concentration and position inside the pellet, integration of Equation 5.1 with respect to z gives... 

In fast flames and shock tube flows such as are considered here, the concentration gradients in the recombination region are such that diffusion effects can be neglected. The recombination can also be considered as taking place in the presence of effectively constant concentrations of the bulk species Hj, HjO and N2 or Ar. As was first pointed out by Sugden and co-workers [133] the radical concentrations do not behave independently during the approach to full equilibrium. The observed relationships... 

The elimination of diffusion effects or their treatment as an independent correction factor will be described in Section IV, and it will therefore be sufficient to use the expression (4) as a starting point. For those reactions involving pressure change, this expression must be translatable in terms of a variation of gas pressure with time. For this purpose the reaction to be expected needs to be known, for then dn/dt will be expressible... 

Hence, one may conclude that in the limit k —> 0 the dynamics of the charge fluctuations is completely determined by relaxation processes with the finite (nonzero) relaxation time. In this sense we can speak about the fast kinetic-like behavior of the charge fluctuations in the model considered. This results in the effective independence of the other hydrodynamic Eqs. (44), (46), and (47), from the time evolution of fast charge subsystem, so that the hydrodynamics of a binary mixture of charge particles becomes rather similar to the case of simple liquids. However, we have to remember that in the hydrodynamic limit the additional (comparing with simple liquids) well-defined transport coefficients, namely the mutual D and />r thermal diffusion coefficients, exist in the system that play a crucial role in the electric and the thermoelectric properties, respectively. 

An experimental test to verify the absence of significant concentration gradients inside the catalyst particle is based on the inverse proportional relation between the effectiveness factor and the particle diameter for strong internal diffusion limitations. Hence, a measured rate that is independent of the particle size indicates that internal diffusion limitations can be neglected. Care should be taken to avoid artefacts. External heat transfer effects also depend on particle size and for exothermic reactions might balance the internal diffusion effects. Furthermore, if the catalyst particle consists of a support with a non-uniformly distributed active phase, crushing and sieving to obtain smaller particles is hazardous. 

The effect of flow rate has been examined as a factor that influences bandwidth, and therefore resolution. Figure 14.6 shows a typical plot of band width as a function of flow rate. At low flow rates, longitudinal diffusion is significant, and leads to larger bandwidths than at the optimum. At high flow rates, the kinetics of partitioning becomes the dominant factor that leads to increased bandwidths and decreased resolution. The contribution of eddy diffusion is independent of flow rate. 

If we take the limit as Ar 0 and assume that the effective diffusivity is independent of the concentration of reactant (see Sec. 11-2), this difference equation becomes... 

Does the rate depend on linear velocity With external diffusion, the rate at constant space velocity increases with linear velocity, at about 1/2 order. Velocity independence is a good indication of the absence of external diffusion effects. 

All commitments greater than zero tend to reduce the observed isotope effect, and as most effects of mechanistic interest in glycosyl transfer are small, they compound the experimental problem. Rather than trying to deconvolute a system with non-zero commitments, it is often easier to use a poor substrate and/or a non-optimal pH. Obviously, external commitments arising from diffusion are independent of pH, and in practice protein conformational changes or product release steps giving rise to internal commitments are less sensitive than bond-breaking steps to pH. 

Thus, the most suitable route for obtaining information on the adsorption/desorption kinetics is from the short-time transient behavior. Under these conditions, surface diffusion effects are negligible and the short-time current response depends only on Ka, Kd, and A for a given tip/substrate separation. Provided that an independent measurement of A can be made, an absolute assignment of the interfacial kinetics is possible. Furthermore, analysis of the long-time current allows the importance, and magnitude, of surface diffusion to be determined. 

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