Diffusion dependencies

This situation seems highly probable for step-growth polymerization because of the high activation energy of many condensation reactions. The constants for the diffusion-dependent steps, which might be functions of molecular size or the extent of the reaction, cancel out. 

Back-diffusion is the transport of co-ions, and an equivalent number of counterions, under the influence of the concentration gradients developed between enriched and depleted compartments during ED. Such back-diffusion counteracts the electrical transport of ions and hence causes a decrease in process efficiency. Back-diffusion depends on the concentration difference across the membrane and the selectivity of the membrane the greater the concentration difference and the lower the selectivity, the greater the back-diffusion. Designers of ED apparatus, therefore, try to minimize concentration differences across membranes and utilize highly selective membranes. Back-diffusion between sodium chloride solutions of zero and one normal is generally [Pg.173]

The diffusion coefficient in these phases D,j is usually considerably smaller than that in fluid-filled pores however, the adsorbate concentration is often much larger. Thus, the diffusion rate can be smaller or larger than can be expected for pore diffusion, depending on the magnitude of the flmd/solid partition coefficient. 

The switch from feed to displacer changes M roots, generating M boundaries. All of these boundaries are self-sharpening, except the boundary associated with the transition from to Oti that can be self-sharpening or diffuse depending on the relative value of these... 

The overall growth kineties is thus seeond order and exhibits diffusion depend-enee, with intrinsie surfaee integration kineties (not shown) being greater than seeond order implying a polynuelear growth meehanism. 

Adiabatic Head Developed per Single-Stage Wheel. The head developed hy a single stage of compression, consisting of an impeller and diffuser, depends upon the design, efficiency, and capacity and is related to its speed. 

According to the transition state theory, the pre-exponential factor A is related to the frequency at which the reactants arrange into an adequate configuration for reaction to occur. For an homolytic bond scission, A is the vibrational frequency of the reacting bond along the reaction coordinates, which is of the order of 1013 to 1014 s 1. In reaction theory, this frequency is diffusion dependent, and therefore, should be inversely proportional to the medium viscosity. Also, since the applied stress deforms the valence geometry and changes the force constants, it is expected... 

The catalytic reaction can be conveniently divided into a number of sequential steps, all of which impact on the overall efficiency of the reaction. First the reactants must diffuse to the catalyst surface the rate of diffusion depends on several factors including fluid density, viscosity and fluid flow rate. Whilst some reaction will take place at the external surface, the majority of reactants will need to diffuse into the internal pores. For a... 

The second term on the right-hand side is the numerical diffusivity, depending on the flow velocity in the x-direction, u, and the length of the control volume in the %-direction, h. Similar expressions hold for the other coordinate directions. Apparently, the numerical diffusion increases with increasing flow velocity and... 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t —> oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

The method preferred in our laboratory for determining the UWL permeability is based on the pH dependence of effective permeabilities of ionizable molecules [Eq. (7.52)]. Nonionizable molecules cannot be directly analyzed this way. However, an approximate method may be devised, based on the assumption that the UWL depends on the aqueous diffusivity of the molecule, and furthermore, that the diffusivity depends on the molecular weight of the molecule. The thickness of the unstirred water layer can be determined from ionizable molecules, and applied to nonionizable substances, using the (symmetric) relationship Pu = Daq/ 2/iaq. Fortunately, empirical methods for estimating values of Daq exist. From the Stokes-Einstein equation, applied to spherical molecules, diffusivity is expected to depend on the inverse square root of the molecular weight. A plot of log Daq versus log MW should be linear, with a slope of —0.5. Figure 7.37 shows such a log-log plot for 55 molecules, with measured diffusivities taken from several... 

Movement through the body of a solid is called volume, lattice, or bulk diffusion. In a gas or liquid, bulk diffusion is usually the same in all directions and the material is described as isotropic. This is also true in amorphous or glassy solids and in cubic crystals. In all other crystals, the rate of bulk diffusion depends upon the direction taken and is anisotropic. Bulk diffusion through a perfect single crystal is dominated by point defects, with both impurity and intrinsic defect populations playing a part. 

For simple flows where the mean velocity and/or turbulent diffusivity depend only weakly on the spatial location, the Eulerian PDF algorithm described above will perform adequately. However, in many flows of practical interest, there will be strong spatial gradients in turbulence statistics. In order to resolve such gradients, it will be necessary to use local grid refinement. This will result in widely varying values for the cell time scales found from (7.13). The simulation time step found from (7.15) will then be much smaller than the characteristic cell time scales for many of the cells. When the simulation time step is applied in (7.16), one will find that Ni must be made unrealistically large in order to satisfy the constraint that Nf > 1 for all k. 

The latter, in contrast to nuclear Overhauser enhancement and exchange spectroscopy (NOESY), always feature positive NOEs (negative cross-peaks with respect to diagonal), eliminating known problems of NOEs vanishing or spin diffusion, depending on correlation time, when high field spectrometers are used for measurements of medium-size compounds. 

The chemical stability of an amorphous formulation is usually also a function of its storage temperatme relative to Tg. The enhanced molecular mobility achieved near the glass transition translates into an increase in translational diffusion-dependent degradation pathways, such as aggregation in proteins. It should be noted that the reaction kinetics near the Tg do not obey Arrhenius kinetics, and that extrapolation of the accelerated stability data generated near the Tg to stability at the storage temperature should be viewed with extreme caution. Amorphous materials must be stored well below the glass transition (at least 10°C, and typically 40 to 50°C below Tg) to maintain their physical and chemical stability. 

A quasireversible electrode reaction is controlled by the film thickness parameter A, and additionally by the electrode kinetic parameter k. The definition and physical meaning of the latter parameter is the same as for quasireversible reaction under semi-infinite diffusion conditions (Sect. 2.1.2). Like for a reversible reaction, the dimensionless net peak current depends sigmoidally on the logarithm of the thickness parameter. The typical region of restricted diffusion depends slightly on K. For instance, for log( If) = -0.6, the reaction is under restricted diffusion condition within the interval log(A) < 0.2, whereas for log(if) = 0.6, the corresponding interval is log(A) <0.4. 

The highest level, at structural scales >10 nm, is that over which long-range transport takes place and diffusion depends on the degree of connectivity of the water pockets, which involves the concept of percolation. The observed decrease in water permeation with decreasing water volume fraction is more pronounced in sulfonated poly(ether ketone) than in Nafion, owing to differences in the state of percolation. Proton conductivity decreases in the same order, as well. 

The concentration of phosphocreatine is usually greater than that of ATP and that of creatine greater than that of ADP so that these metabolites diffuse more rapidly. This is because diffusion depends upon the concentrations of participants and the concentration gradient the larger the gradient, the greater is the rate of diffusion. Consequently, the shuttle is important in cells where the distance between the sites of ATP utilisation and the mitochondria is large... 

In a centrifugal field, dissolved molecules or suspended particles either sediment (if their density exceeds that of the pure solvent), or flotate for the opposite case (negative or inverse sedimentation). Under otherwise identical experimental conditions, the velocity of the molecules or particles depends on the viscosity of the solution or suspension and - very importantly - on the mass and shape of the dissolved species. Sedimentation and flotation are antagonized by the diffusion. Depending on the rotor speed and the molar mass of the dissolved/dis-... 

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