Diffusion semiinfinite-linear

This relationship holds for any electrochemical process that involves semiinfinite linear diffusion and is the basis for a variety of electrochemical methods (e.g., polarography, voltammetry, and controlled-potential electrolysis). Equation (3.6) is the basic relationship used for solid-electrode voltammetry with a preset initial potential on a plateau region of the current-voltage curve. Its application requires that the electrode configuration be such that semiinfinite linear diffusion is the controlling condition for the mass-transfer process. 

This equation is valid for only semiinfinite linear diffusion to a shielded planar electrode the effects of electrode geometry (upward or downward diffusion) and electrode shielding on the transition time have been discussed.94... 

Buckinghams 7r-theorem [i] predicts the number of -> dimensionless parameters that are required to characterize a given physical system. A relationship between m different physical parameters (e.g., flux, - diffusion coefficient, time, concentration) can be expressed in terms of m-n dimensionless parameters (which Buckingham dubbed n groups ), where n is the total number of fundamental units (such as m, s, mol) required to express the variables. For an electrochemical system with semiinfinite linear geometry involving a diffusion coefficient (D, units cm2 s 1), flux at x = 0 (fx=o> units moles cm-2 s 1), bulk concentration (coo> units moles cm-3) and time (f, units s), m = 4 (D, fx=0, c, t) and n - 3 (cm, s, moles). Thus m-n - 1 therefore only one dimensionless parameter can be constructed and that is fx=o (t/Dy /coo. Dimensional analysis is a powerful tool for characterizing the behavior of complex physical systems and in many cases can define relationships... 

Dimensionless analysis — Use of dimensionless parameters (-> dimensionless parameters) to characterize the behavior of a system (- Buckinghams n-theorem and dimensional analysis). For example, the chronoampero-metric experiment (-> chronoamperometry) with semiinfinite linear geometry relates flux at x = 0 (fx=o, units moles cm-2 s-1), time (t, units s-1), diffusion coefficient (D, units cm2 s-1), and concentration at x = oo (coo, units moles cm-3). Only one dimensionless parameter can be created from these variables (-> Buckingham s n-theorem and dimensional analysis) and that is fx=o (t/D)1/2/c0C thereby predicting that fx=ot1 2 will be a constant proportional to D1/,2c0O) a conclusion reached without any additional mathematical analysis. Determining that the numerical value of fx=o (f/D) 2/coo is 1/7T1/2 or the concentration profile as a function of x and t does require mathematical analysis [i]. 

Figure 3.4 Cyclic voltammograms (semiinfinite-linear diffusion conditions) in systems Ag(lll)/ 5 X 10-2 pb(C104)2 + 5 X lO l M NaC104 + 5 x 10 2 M HCIO4 with IdE/d/l = 10 mV s at T= 298 K...

Figure 3.5 Cyclic voltammogram (semiinfinite-linear diffusion conditions) in the system (poly)/ 5X 10 2MPb(C104)2 + 5X lO MNaC104 + 5xlO MHCIO4with IdE/dtl = 10 mVs at T= 298K.

Figure 3.25 Cyclic voltamtnograin measured under semiinfinite-linear diffusion conditions in the system Au(lll)/5 x 10 M CuS04 + 9 x 10 M H2SO4 with IdE/dfl = 10 mV s" at T = 298 K [3.301]. A and D with = 1,2 denote cathodic adsorption and anodic desorption peaks, respectively.

The results strongly depend on the crystallographic orientation of the substrate and on the crystal imperfection density. The time-dependence of 3D Me-S bulk alloy formation obeys a parabolic rate law (Fig. 3.65) as found for many other systems. The results were discussed in terms of a semiinfinite-linear diffusion model assuming mutual diffusion of Me and S and reversible 2D Meads overlayer formation. The following time-dependence of q(,AE,i) was derived... 

First, at short times, where the diffusion-layer thickness is small compared to the critical dimension, the current at any UME follows the Cottrell equation, (5.2.11), and semiinfinite linear diffusion applies. 

Figure 7.3.16 contains data for a system involving the oxidation of Fe(CN)5 in a positive-going scan at a small Pt disk. The results have been treated theoretically by assuming reversibility at all frequencies and by adjusting two parameters, the radius of the disk, ro, and Ei 2 = E + RTInF) ln(DR/Do), to provide the best fit. The change in behavior with frequency is rooted in the fact that the diffusion pattern at a UME can deviate from the semiinfinite linear case, as discussed in Section 5.3. The validity of the model is supported by the consistency of these parameters for runs at different frequencies and by the quality of the fit. This example illustrates the typical manner of comparing SWV results with theory. 

Under such assumptions ( reversible system), diffusion is the main mode of transport and semiinfinite linear diffusion conditions prevail. Thus, Pick s 2nd law describes the variation of concentrations with t and the space coordinate x (extending from the electrode surface, x = 0, into... 

Impedance in Presence of Redox Process in Semiinfinite Linear Diffusion Determination of Parameters... 

For any transient electrochemical technique under conditions of semiinfinite linear diffusion, it can be shown that solution of the diffusion equations, when only 0 is initially present, yields, irrespective of the reaction mechanism, the following expression for the time dependent surface concentration of O. 

In the case of semiinfinite linear diffusion the semiintegration coincides with the simplest case of convolutive potential sweep voltammetry ... 

Let us now consider a semiinfinite linear diffusion of charged particles from and to the electrode. The Faraday impedance is defined as the sum of the charge transfer resistance R and the Warburg impedance W corresponding to the semiinfinite diffusion of the charged particles... 

Semi-infinite linear diffusion conditions The rate of an electrochemical process depends not only on electrode kinetics but also on the transport of species to/from the bulk solution. Mass transport can occur by diffusion, convection or migration. Generally, in a spectroeiectrochemicai experiment, conditions are chosen in which migration and convection effects are negligible. The solution of diffusion equations, that is the discovery of an equation for the calculation of oxidized form [O] and reduced form [R] concentrations as functions of distance from electrode and time, requires boundary conditions to be assumed. Usually the electrochemical cell is so large relative to the length of the diffusion path that effects at walls of the cell are not felt at the electrode. For semiinfinite linear diffusion boundary conditions, one can assume that at large distances from the electrode the concentration reaches a constant value. 

For a reversible redox couple, Ramaley and Krause (1969) and Christie, Turner Osteryoung (1977) have presented a general analytical solution under conditions of semiinfinite linear diffusion. Dimensionless currents (T) are defined (Osteryoung O Dea, 1987) as ... 

Feldberg, S. W. Implications of extended heterogeneous electron transfer Part I. Coupling of semiinfinite linear diffusion and heterogeneous electron transfer with a decaying exponential dependence upon distance from the outer hehnholtz plane. J Electroanal Chem 1986,198,1-18. 

Thus, in a spherical field of diffusion (which is achieved for a microelectrode after a time determined by its radius), one obtains an equation similar to that given for semiinfinite linear diffusion, except that the radius of the electrode plays the role of the Nernst diffusion-layer thickness and the limiting current density is independent of time. The validity of Eq. (14.47) is one of the incentives for fabricating... 

At low frequencies the impedance is dominated by diffusion. Two regions may be identified in the complex impedance, a linear region with a phase angle of ti/4 corresponding to semiinfinite diffusion and... 

At a planar electrode, the equation to be solved under conditions of linear semiinfinite diffusion is the same as for a potential step. The difference is the third boundary condition, which instead of defining the diffusion-limited current expresses the concentration gradient resulting from the applied current at the electrode surface. 

---