Gradients of concentration

A molecule colliding with the pore wall is reflected in a specular manner so that the direction of the molecule leaving the surface has no correlation with that of the incident molecule. This leads to a Fickian mechanism, known as Knudsen diffusion, in which the flux is proportional to the gradient of concentration of partial pressure. The Knudsen diffusivity is independent of pressure and varies only weaMy with temperature ... 

The tme driving force for any diffusive transport process is the gradient of chemical potential rather than the gradient of concentration. This distinction is not important in dilute systems where thermodynamically ideal behavior is approached. However, it becomes important at higher concentration levels and in micropore and surface diffusion. To a first approximation the expression for the diffusive flux may be written... 

Dispersion The movement of aggregates of molecules under the influence of a gradient of concentration, temperature, and so on. The effect is represented hy Tick s law with a dispersion coefficient substituted for molecular diffusivity. Thus, rate of transfer = —Dj3C/3p). 

The mean turbulent flux of concentration in the vertical direction z is w x. Assuming that this turbulent flux is proportional to the gradient of concentration, and in the direction from higher to lower concentrations, an overall diffusivity K can be defined ... 

Where F is the flux of diffusing species and dc/dx is its gradient of concentration c in the x direction. From the equation of continuity,... 

The current is proportional to the gradient of concentration of product at the electrode surface ... 

If cross-diffusion due to the chemical reaction is neglected, the diffusive currents are proportional to the gradients of concentrations... 

The logical conclusion reached while considering these data is as follows. In liquid phase (THF), under the conditions of a regular volume continuum without gradients of concentration and potential, all anion-radicals of azoxybenzene can be stabilized just after formation due to their bonding with potassium cations. This yields the coordinative complex. The complex is diamagnetic and, therefore, azoxybenzene anion-radicals cannot be revealed by ESR spectroscopy (Scheme 2.15). 

In most applications, the axial dispersion in both phases can be considered to be negligible (Smith, 1981). Moreover, no radial gradients of concentration and velocity exist for the gas or liquid. 

When transport is not able to do its job adequately and there is a change in the interfacial concentrations of electron acceptors and donors from the bulk values, there is a variation of concentration with distance from the interface toward the bulk of the solution. What matters, however, as far as the charge-transfer reaction is concerned, is the gradient of concentration at the interface because it is this gradient that drives the diffusion flux Jjy Even when there is convection with a laminar flow of electrolyte, the transport in the (assumed) stagnant layer adjacent to the electrode is by diffusion... 

The diffusion-layer concept is an artifice for handling the flux arising from what would be, if treated in a proper hydrodynamic way, a complicated space variation of concentration at the interface. There is always some gradient of concentration at the interface there is an initial region in which the concentration changes linearly with distance, but there is, in the real case, no sharply defined layer of definite thickness, even when convection (natural or forced) produces a steady-state concentration... 

Figure 6.3b shows the idealized concentration profile of an absorbed component A, obtained by the Hatta theory, for the case of a relatively slow reaction that is either first-order or pseudo first-order with respect to A. As A is consumed gradually while diffusing across the film, the gradient of concentration of A that is required for its diffusion gradually decreases with increasing distance from the interface. The enhancement factor for such cases is given by the Hatta theory as r... 

As a byproduct, we can learn from Eq. 18-50 that, in fact, it is not the gradient of concentration, C but the chemical activity, a that drives diffusion. Since at constant C activity changes with temperature, ionic strength, and other parameters, a diffusive flux may actually occur even if the concentration gradient is zero. 

Unless carried out very carefully, data from flow reactors may be influenced by experimental uncertainties. Potential problems with the flow reactor technique include imperfect mixing of reactants, radial gradients of concentration and temperature, and catalytic effects on reactor walls. Uncertainties in induction times, introduced by finite rate mixing of reactants, presence of impurities, or catalytic effects, may require interpretation of the data in terms of concentration gradients, rather than just exhaust composition [442]. 

In the frog Xenopus the ovum accumulates an RNA (Vgl mRNA) that encodes a growth factor of the TGF-P family (Chapter 30, Section A,6).280 Initially present throughout the ovum, it moves and forms a gradient of concentration that is highest at the vegetal pole. 

This result is called the continuity equation for mass [4, 52]. The cause of the flux is called a driving force , which is not used in the Newtonian sense, but instead names any source of perturbation. In the case of mass transport, this cause is typically a gradient (of concentration, electrical potential, or density). 

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