Concentration gradients, electrochemistry

Processes involving transport of a substance and charge in a streaming electrolyte are very important in electrochemistry, particularly in the study of the kinetics of electrode processes and in technology. If no concentration gradient is formed, transport is controlled by migration alone convection... 

For chemists, the second important application of electrochemistry (beyond potentiometry) is the measurement of species-specific [e.g., iron(III) and iron(II)] concentrations. This is accomplished by an experiment in which the electrolysis current for a specific species is independent of applied potential (within narrow limits) and controlled by mass transfer across a concentration gradient, such that it is directly proportional to concentration (/ = kC). Although the contemporary methodology of choice is cyclic voltammetry, the foundation for all voltammetric techniques is polarography (discovered in 1922 by Professor Jaroslov Heyrovsky awarded the Nobel Prize for Chemistry in 1959). Hence, we have adopted a historical approach with a recognition that cyclic voltammetry will be the primary methodology for most chemists. 

Boundary value — A boundary value is the value of a parameter in a differential equation at a particular location and/or time. In electrochemistry a boundary value could refer to a concentration or concentration gradient at x = 0 and/or x = oo or to the concentration or to the time derivative of the concentration at l = oo (for example, the steady-state boundary condition requires that (dc/dt)t=oo = 0). Some examples (dc/ dx)x=o = 0 for any species that is not consumed or produced at the electrode surface (dc/dx)x=o = -fx=0/D where fx=o is the flux of the species, perhaps defined by application of a constant current (-> von Neumann boundary condition) and D is its diffusion coefficient cx=o is defined by the electrode potential (-> Dirichlet boundary condition) cx=oo, the concentration at x = oo (commonly referred to as the bulk concentration) is a constant. 

The mathematical methods and the derivation can be found in several electrochemistry books [1 ] and reviews [5]. Herein, we present the most important considerations and formulae for steady-state electrolysis conditions. It is assumed that the solution is well stirred (the concentration gradient at the electrode surface is constant) and that both the reactant and product molecules are soluble. By combining Eqs. (1.3.28), (1.3.29), and (1.3.30) and considering the respective initial and boundary conditions [Eqs. (1.3.31), (1.3.32), and (1.3.36)], we obtain... 

In electrochemistry the concentration gradients always occur in the boundary layer close to the electrode surface. 

Using the mass transport equations and the boundary conditions, the variation of concentration gradient with time and distance could be obtained. Therefore, the calculation could use the methods developed previously in electrochemistry to calculate the concentration gradients in front of the electrode. In general, to obtain the equations to simulate the PBD response, the following steps have to be carried out (i) obtain the concentration profile at distance x, for the technique under study, C(x, t) (ii) differentiate with respect to x, to obtain dC(xd)ldx (iii) combine with Eq. 1 to obtain the dependence of deflection with time and space 0(x, t). 

Coen et al (1987) report an approach, new to electrochemistry, to reduce the computation time required for a 2-D system (the current at a micro-band electrode). The diffusion equation is Laplace transformed, converted to an integral equation and solved, still in Laplace space, for the concentration gradient at the boundaries. They developed an efficient algorithm for the inverse Laplace transformation, which then yields the current as a function of time. Clearly, this is not for everyone at least one of the authors is a mathematician. The method has been used previously (Rizzo and Shippy, 1970) to simulate heat conduction. 

In this way, the coefficients for any y - ri) can be calculated. Table A.l in Appendix A shows a number of these, as whole numbers mPu where m is the multiplier mentioned above. For each n, the table shows forward differences (at index 1), backward derivatives (at index n) and derivatives applying at points between the two ends. In case the reader wonders why all this is of interest the forms ( ) will be used to approximate the current or, in general, the concentration gradient, in simulations (see the next section) the backward forms y (n) will be used in the section on the BDF method in Chaps. 4 and 9, and the intermediate forms shown in the table will be used for the Kimble and White (high-order) start of the BDF method, aiso described in these chapters. The coefficients have a long history. Collatz [1] derived some of them in 1935 and presents more of them in [2]. Bickley tabulated a number of them in 1941 [3]. The three-point current approximation, essentiaily 34(3) in the present notation, was first used in electrochemistry by Randles [4] (preempted by 2 years by Eyres et al. [5] for heat flow simulations), then by Heinze et al. [6] Newman [7, p. 554] used a five-point current approximation, and schemes of up to seven-point were provided in [8]. 

Spoor, P.B., Koene, L. Janssen, L.J.J. (2002b) Potential and concentration gradients in a hybrid ion-exchange/electrodialysis cell. Journal of Applied Electrochemistry, 32, 369-377. 

The situation inside an electrolyte—the ionic aspect of electrochemistry—has been considered in the first volume of this text. The basic phenomena involve— ion—solvent interactions (Chapter 2), ion—ion interactions (Chapter 3), and the random walk of ions, which becomes a drift in a preferred direction under the influence of a concentration or a potential gradient (Chapter 4). In what way is the situation at the electrode/electrolyte interface any different from that in the bulk of the electrolyte To answer this question, one must treat quiescent (equilibrium) and active (nonequilibrium) interfaces, the structural and electrical characteristics of the interface, the rates and mechanism of changeover from ionic to electronic conduction, etc. In short, one is led into electrodics, the newest and most exciting part of electrochemistry. 

M. Keddam, C. Rakotomavo, and H. Takenouti, "Impedance of a Porous Electrode with an Axial Gradient of Concentration," Journal of Applied Electrochemistry, 14 (1984) 437-448. 

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