Nernstian diffusion

When the concentration boundary layer is sufficiently thin the mass transport problem can be solved under the approximation that the solution velocity within the concentration boundary layer varies linearly with distance away from the surface. This is called the L6v que approximation (8, 9] and is satisfactory under conditions where convection is efficient compared with diffusion. More accurate treatments of mass transfer taking account of the full velocity profile can be obtained numerically [10, 11] but the Ldveque approximation has been shown to be valid for most practical electrodes and solution velocities. Using the L vSque approximation, the local value of the concentration boundary layer thickness, 8k, (determined by equating the calculated flux to the flux that would be obtained according to a Nernstian diffusion layer approximation that is with a linear variation of concentration across the boundary layer) is given by equation (10.6) [12]. 

Electropolish in g, when high points are removed selectively by anodic dissolution via a viscous Nernstian diffusion layer... 

X18 The eRect of electrode surface profile on the tertiary current d tHbutioil showing the size of the surface roughness x compared to the Nernstian diffusion layer thickness and the relative Umiting-cuirent density /l on the peaks and valleys. 

The explicit mathematical treatment for such stationary-state situations at certain ion-selective membranes was performed by Iljuschenko and Mirkin 106). As the publication is in Russian and in a not widely distributed journal, their work will be cited in the appendix. The authors obtain an equation (s. (34) on page 28) similar to the one developed by Eisenman et al. 6) for glass membranes using the three-segment potential approach. However, the mobilities used in the stationary-state treatment are those which describe the ion migration in an electric field through a diffusion layer at the phase boundary. A diffusion process through the entire membrane with constant ion mobilities does not have to be assumed. The non-Nernstian behavior of extremely thin layers (i.e., ISFET) can therefore also be described, as well as the role of an electron transfer at solid-state membranes. 

FIGURE 1.11. Convolution of the cyclic voltammetric current with the function I j Jnt, characteristic of transient linear and semi-infinite diffusion. Application to the correction of ohmic drop, a —, Nernstian voltammogram distorted by ohmic drop , ideal Nernstian voltammogram. b Convoluted current vs. the applied potential, E. c Correction of the potential scale, d Logarithmic analysis. 

In addition to this, and in contrast with the homogeneous case discussed in Section 5.2.2, the diffusion of P and Q is therefore not perturbed by any homogeneous reaction. If, furthermore, the P/Q electron transfer at the electrode is fast and thus obeys Nernst s law, the diffusive contribution to the current in equations (5.11) and (5.12) is simply equal to the reversible diffusion-controlled Nernstian response, idif, discussed in Section 1.2. The mutual independence of the diffusive and catalytic contributions to the current, expressed as... 

Figure 10. Kleitz s reaction pathway model for solid-state gas-diffusion electrodes. Traditionally, losses in reversible work at an electrochemical interface can be described as a series of contiguous drops in electrical state along a current pathway, for example. A—E—B. However, if charge transfer at point E is limited by the availability of a neutral electroactive intermediate (in this case ad (b) sorbed oxygen at the interface), a thermodynamic (Nernstian) step in electrical state [d/j) develops, related to the displacement in concentration of that intermediate from equilibrium. In this way it is possible for irreversibilities along a current-independent pathway (in this case formation and transport of electroactive oxygen) to manifest themselves as electrical resistance. This type of chemical valve , as Kleitz calls it, may also involve a significant reservoir of intermediates that appears as a capacitance in transient measurements such as impedance. Portions of this image are adapted from ref 46. (Adapted with permission from ref 46. Copyright 1993 Rise National Laboratory, Denmark.)...

Therefore, the ISE potential depends on the CO2 partial pressure with Nernstian slope. Contemporary microporous hydrophobic membranes permitted the construction of a number of gas probes, developed mainly by the Orion Research Company (for a survey see [143]. The most important among these sensors is the ammonia electrode, in which ammonia diffusing through the membrane affects the pH at a glass electrode. Other electrodes based on similar principles respond to SO2, HCN, H2S (with an internal S ISE), etc. The ammonia probe has a better detection limit than the ammonium ion ISE based on the non-actin ionophore. The response time of gas probes depends mostly on the rate of diffusion of the test gas through the microporous medium [77,143]. 

The simulation of other electrochemical experiments will require different electrode boundary conditions. The simulation of potential-step Nernstian behavior will require that the ratio of reactant and product concentrations at the electrode surface be a fixed function of electrode potential. In the simulation of voltammetry, this ratio is no longer fixed it is a function of time. Chrono-potentiometry may be simulated by fixing the slope of the concentration profile in the vicinity of the electrode surface according to the magnitude of the constant current passed. These other techniques are discussed later a model for diffusion-limited semi-infinite linear diffusion is developed immediately. 

Ohmic effects render Epc more negative, Epa more positive, AEp and 8Ep larger, and X smaller than the true values. Since experimental approaches to elimination of iRu errors are not foolproof (see Chap. 7), the presence of ohmic distortions should be tested by measurements on a Nernstian couple such as ferrocene/ferrocenium under conditions identical to those used to probe the test compound. In principle, errors in the measured CV parameters for a test compound can be eliminated by referencing its responses to those of the Nernstian standard. Note that this approach is accurate only if the current level of the standard, rather than its concentration, is equal to that of the test compound, since the diffusion coefficients of the two species may appreciably differ. 

Recall that a Nernstian behavior of diffusing species yields a vm dependence. In practice, the ideal behavior is approached for relatively slow scan rates, and for an adsorbed layer that shows no intermolecular interactions and fast electron transfers. 

If the two species diffusion coefficients are assumed equal (d = 1), the above equations simplify in an obvious way. In fact, then the problem is mathematically equivalent to the simple Cottrell case. Cottrell pointed out [181] that then, initially the concentrations at the electrode of the two species will instantly change to their Nernstian values and remain there after that. 

Our modeling approach was first used to describe the EDL properties of well-characterized, crystalline oxides ( 1). It was shown that the model accounts for many of the experimentally observed phenomena reported in the literature, e.g. the effect of supporting electrolyte on the development of surface charge, estimates of differential capacity for oxide surfaces, and measurements of diffuse layer potential. It is important to note that a Nernstian dependence of surface potential (iIJq) as a function of pH was not assumed. The interfacial potentials (4>q9 4> 9 in Figure 1) are... 

Let us consider, for example, the simple nernstian reduction reaction in Eq. (221) and a solution containing initially only the reactant R. Before any electrochemical perturbation the electrode rest potential Ej is made largely positive to E . At time zero the potential is stepped to a value E2, sufficiently negative to E , so that the concentration of R is close to zero at the electrode surface. After a time 6, the electrode potential is stepped back to El, so that the concentration of P at the electrode surface becomes zero. When this potentiostatic perturbation, represented in Fig. 21a, is applied in a steady-state method, the R and P concentration profiles are linear and depend only on the electrode potential but not on time, as shown in Fig. 20a (for k 0). Yet when the same perturbation is applied in transient methods, the concentration profiles are curved and time dependent, as evidenced in Fig. 21b. Thus it is seen from this figure that a step duration at Ei, much longer than the step duration 0 at E2, is needed for the initial concentration profiles to be restored. This hysterisis corresponds to the propagation of the diffusion perturbation within the solution, which then keeps a memory of the past perturbation. This information is stored via the structuring of the concentrations in the space near the electrode as a function of the elapsed time. 

The COT and azocinyl dianions evinced contrasting electrochemical behavior. Whereas COT underwent two nearly Nernstian one-electron reductions, the azocines gave a single reduction wave, the diffusion current indicating an overall two-electron transfer. 

Van Hal et al. [48] used the 2-pK and MUSIC models combined with diffuse layer and Stern electrostatic models (with pre-assumed site-density and surface acidity constants) to calculate the surface potential, the intrinsic buffer capacity -(d(To/dpHs)/e where pHs is the pH at the surface, the sensitivity factor -(d o/dpH) x [e/(kTln 10)], which equals unity for Nernstian response, and the differential capacitance for three ionic strengths as a function of pH. The calculated surface potentials were compared with the experimentally measured ISFET response. 

Diffusion-Controlled and Nernstian Steady-State Processes... 

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