Long bulk diffusion

Chapcer 4. GAS MOTION IN A LONG TlfBE AT THE LIMIT OF BULK DIFFUSION AND VISCOUS FLOW... 

While electrical conductivity, diffusion coefficients, and shear viscosity are determined by weak perturbations of the fundamental diffu-sional motions, thermal conductivity is dominated by the vibrational motions of ions. Heat can be transmitted through material substances without any bulk flow or long-range diffusion occurring, simply by the exchange of momentum via collisions of particles. It is for this reason that in liquids in which the rate constants for viscous flow and electrical conductivity are highly temperature dependent, the thermal conductivity remains essentially the same at lower as at much higher temperatures and more fluid conditions. 

The value of[Tl ] in the solution bulk remains essentially constant since only a tiny proportion of the overall amount of Tl is oxidized, but at the surface of the electrode we can say, to a good approximation, that [Tt ] = 0. Very soon after the potential is stepped, Tl from the bulk diffuses toward the electrode, thereby attempting to even out the concentration gradient, i.e. to replenish the Tt" that was consumed at the commencement of the step. W need to recognize, however, that these thallium ions will not remain as Tl for long as they will be oxidized immediately to form Tl, i.e. as soon as they impinge on the electrode. The end result is that a concentration gradient will soon form after the potential has been stepped. 

The reduction of the long-range diffusivity, Di by a factor of four with respect to bulk water can be attributed to the random morphology of the nanoporous network (i.e., effects of connectivity and tortuosity of nanopores). For comparison, the water self-diffusion coefficient in Nafion measured by PFG-NMR is = 0.58 x 10 cm s at T = 15. Notice that PFG-NMR probes mobilities over length scales > 0.1 /rm. Comparison of QENS and PFG-NMR studies thus reveals that the local mobility of water in Nafion is almost bulk-like within the confined domains at the nanometer scale and that the effective water diffusivity decreases due to the channeling of water molecules through the network of randomly interconnected and tortuous water-filled domains. ... 

Treatments of this problem are greatly simplified by uncoupling the long-term diffusion due to Edc from the rapid diffusional fluctuations due to Eac- We do that by recognizing that Edc sets up mean surface concentrations that look like bulk values to the ac perturbation because of the difference in time scale. In Section 10.3, we defined the faradaic impedance in terms of bulk concentrations thus the current response in ac voltammetry as a function of is readily obtained by substituting the surface concentrations imposed by directly into these impedance relations. Since this strategy is simple and intuitive, we will pursue it. More rigorous treatments are available in the literature for the interested reader (2, 3, 5). The results are the same by either approach. 

Figure 3.15. Adsorbates on the surface which exist as a two-dimensional phase protected by large potential barriers from desorption and from bulk diffusion. Because of their long residence time on the surface, they come to equilibrium among the various surface sites.

This law describes the surfactant flux from the bulk to the interface and along the interface during the adsorption or desorption process. As long as the surface concentration T is lower than the respective equilibrium value Tg the adsorption flux dominates while in the opposite case a desorption flux results. If any flow and fluxes other than bulk diffusion is neglected the boundary condition has the following very simple form... 

The relaxation time t, especially Tad, was determined to be shorter than Is, irrespective of the initial electrode potential. All of the capacitors were fully charged before the initial time of 10 s, which indicates that the time of 10 s is long enough for one to analyze the CTs in terms of bulk diffusion of lithium. 

Fig. 5.5 Illustration of the tube model for a long chain diffusing in the bulk phase...

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