Diffusion limiting equivalent resistance

Differential capacitance measurements by Niki et for cytochrome C3 from D. vulgaris, strain Miyazaki, were consistent with irreversible, diffusion-limited adsorption for 4-s drop times above a concentration of 10 fiM. The surface excess of cytochrome C3 was calculated to be 0.92 x 10 " mole/cm. Niki etal also investigated the a.c. polarographic behavior of cytochrome C3 at the reversible half-wave potential. The capacitive peak height was frequency independent while the resistive peak height decreased with increasing frequency to a value of zero above 2000 Hz. These results were fit to a Laitinen-Randles equivalent circuit yielding an n value of... 

Figure 4.5.1 compares the impedance spectrum of reflective finite-length diffusion with the spectrum of its limiting equivalent circuit, a limiting resistance in series with a limiting capacitance. It can be seen that these responses approach each other as the frequency deaeases. 

In some limiting cases, one can obtain an analytical solution of Eq. (41). In particular, if the number M of the barriers is large, M 1, then the resistance of each barrier is small compared to the net resistance, Rk approximate solution of Eq. (41) is g = , and the CGF coincides with that for diffusive wire, Eq. (44). In the tunnel limit, when the resistance of each barrier much exceeds the net resistance of diffusive segments, Rk Rn, the first term in Eq. (41) can be neglected. Then an analytical expression for the parameter g and the CGF at arbitrary M can be obtained in the case of equivalent barriers, rk = 1/M,... 

The previous definitions can be interpreted in terms of ionic-species diffusivities and conductivities. The latter are easily measured and depend on temperature and composition. For example, the equivalent conductance A is commonly tabulated in chemistry handbooks as the limiting (infinite dilution) conductance A and at standard concentrations, typically at 25°C. A = 1000 K/C = X+ + X = A + flC), (cmVohm gequiv) K = a/R = specific conductance, (ohm cm) C = solution concentration, (gequiv/ ) a = conductance cell constant (measured), (cm ) R = solution electrical resistance, which is measured (ohm) and/(C) = a complicated function of concentration. The resulting equation of the electrolyte diffusivity is... 

The electrolyte resistance Re is added in series with the previous impedance. If the electrochemical reaction is mass-trai3sport limited, the previous equivalent circuit is still valid, but the Faradaic impedance includes a diffusion impedance as described in Chapter 11. 

When the polymer flhn is oxidized, its electronic conductivity can exceed the ionic conductivity due to mobile counterions. Then, the film behaves as a porous metal with pores of limited diameter and depth. This can be represented by an equivalent circuit via modified Randles circuits such as those shown in Figure 8.4. One Warburg element, representative of linear finite restricted diffusion of dopants across the film, is also included. The model circuit includes a charge transfer resistance, associated with the electrode/fllm interface, and a constant phase element representing the charge accumulation that forms the interfacial double... 

Coulometric titration techniques were used to measure chemical diffusion at between 700 and lOOOC. The transient current response to a potentiostatic step was transformed from the time domain to the frequency domain. The equivalent circuit which was used to fit the resultant impedance data contained an element which described the finite-length diffusion of O into the sample. Other elements which were included were the gas-phase capacitance, and the sum of the gas-phase diffusion resistance and that which was associated with the limited surface exchange kinetics of the sample. The chemical diffusion coefficient of the perovskite, Laq gSrq 2C0O3,... 

Pilot plant fixed-bed reactors are traditionally designed at space velocities equivalent to commercial scale fixed-bed reactors. Thus the film diffusion resistance of the process at the two scales is different. In general, pilot plant fixed-bed reactors are film diffusion rate limited, whereas commercial-size fixed-bed reactors are either pore diffusion rate or, more rarely, reaction rate limited. This shift from film diffusion rate limited to, more generally, pore diffusion rate limited occurs due to the high volumetric fluid flow through the catalyst mass in a commercial-size fixed-bed reactor. Thus reactant consumption or product formation is faster in the commercial-size fixed-bed reactor than in the pilot plant fixed-bed reactor. 

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