Driving force steady-state diffusion

The driving force for the transfer process was the enhanced solubility of Br2 in DCE, ca 40 times greater than that in aqueous solution. To probe the transfer processes, Br2 was recollected in the reverse step at the tip UME, by diffusion-limited reduction to Br . The transfer process was found to be controlled exclusively by diffusion in the aqueous phase, but by employing short switching times, tswitch down to 10 ms, it was possible to put a lower limit on the effective interfacial transfer rate constant of 0.5 cm s . Figure 25 shows typical forward and reverse transients from this set of experiments, presented as current (normalized with respect to the steady-state diffusion-limited current, i(oo), for the oxidation of Br ) versus the inverse square-root of time. 

When the driving force is small, it may be taken to be linearly related to the flux. On this basis, an equation can be derived for the rate of steady-state diffusion, which is identical in form to the empirical first law of Pick,... 

Fig. 38. Steady state solutions for boundary conditions typical for diffusion controlled transport of photo-generated electrons through the substrate. The film thickness is 3 microns, and illumination is from the electrolyte side. The absorption depth (1/a) is 1/10 of the thickness d. a) the generation profile ttl(x) = dj x)/dx, b) the electron flux J x), c) concentration gradient dn x) dx, d) the profile of excess free carrier concentration n x), e) the driving force for electron diffusion in eV cm". ...

The driving force for steady-state diffusion is the concentration gradient (dC/dx). 

The steady-state diffusive flux function for organic colloidal particles in bed pore-waters is similar to Equation 12.20. The concentration difference in units of mass organic carbon per volume of water is the driving force, AC, described using the particle concentration at depth (z = h), Cp, g-OC m , subtracted from that at the interface (z = 0), Cpi. The chemical flux from the bed is the product of the particle flux and the loading of chemical onto the particles, ITpi (mg g ), at the interface... 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

Chemical reaction steps Even if the overall electrochemical reaction involves a molecular species (O2). it must first be converted to some electroactive intermediate form via one or more processes. Although these processes are ultimately driven by depletion or surplus of intermediates relative to equilibrium, the rate at which these processes occur is independent of the current except in the limit of steady state. We therefore label these processes as chemical processes in the sense that they are driven by chemical potential driving forces. In the case of Pt, these steps include dissociative adsorption of O2 onto the gas-exposed Pt surface and surface diffusion of the resulting adsorbates to the Pt/YSZ interface (where formal reduction occurs via electrochemical-kinetic processes occurring at a rate proportional to the current). 

One limit of behavior considered in the models cited above is an entirely bulk path consisting of steps a—c—e in Figure 4. This asymptote corresponds to a situation where bulk oxygen absorption and solid-state diffusion is so facile that the bulk path dominates the overall electrode performance even when the surface path (b—d—f) is available due to existence of a TPB. Most of these models focus on steady-state behavior at moderate to high driving forces however, one exception is a model by Adler et al. which examines the consequences of the bulk-path assumption for the impedance and chemical capacitance of mixed-conducting electrodes. Because capacitance is such a strong measure of bulk involvement (see above), the results of this model are of particular interest to the present discussion. 

In field-flow fractionation, a component undergoing flow transport through a thin channel is forced sideways against a wall by an applied field or gradient. The component is confined to a narrow region adjacent to the wall, by a combination of the wall s surface, which it cannot pass, and the driving force, which prevents its escape toward the center of the channel. The component molecules or particles soon establish a thin steady-state distribution in which outward diffusion balances the steady inward drift due to the field. The structure and dimensions of this layer determine its behavior in the separation process. 

Under steady-state conditions we can also show that NJ)i + N1D2 = NtD. Note that the driving force for the convection must be a pressure gradient in the gas opposing the diffusion of the more rapid species. 

Suppose that when diffusion is occurring, the driving force Fq and the flux/reach values that do not change with time. The system can be said to have attained a steady state. Then the as-yet-unknown relation between the diffusion flux J and the diffu-sional force can be represented quite generally by a power series... 

As in diffusion, the relationship between the steady-state flux J of ions and the driving force of the electric field will be represented by the expression... 

This model, developed by Whitman [16] in 1923, is still used. The assumption is that adjacent to the interface there exists a thin film that is laminar in character and through which transfer is by molecular diffusion only. In this film, the entire concentration driving force exists, i.e., outside the film, in the bulk fluid, the concentration is constant at some steady state. Flow outside the film may be laminar or turbulent, but inside the film, flow conditions are completely laminar. Figure... 

CONCENTRATION POLARIZATION. The nearly complete rejection of solute by the membrane leads to a higher concentration at the membrane surface than in the bulk solution, and this effect is called concentration polarization. At steady state, the solute carried to the membrane by the water flux almost equals the amount of solute diffusing back to the solution. The gradient may be relatively small, as shown in Fig. 26.17, or the solute concentration at the membrane surface may be several times the bulk concentration. Concentration polarization reduces the flux of water because the increase in osmotic pressure reduces the driving force for water transport. The solute rejection decreases both because of the lower water flux and because the greater salt concentration at the surface increases the flux of solute. 

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