Effect of Concentration Gradients

The effect of concentration gradients in electrode reactions is really not a problem of mechanism but rather a troublesome source of possible systematic ambiguity in the interpretation of the product distributions observed, one of the tasks that lies close to the heart of the organic chemist. To see how this comes about, it is instructive to make the mental experiment that we generate acetoxy radicals by the Kolbe reaction of acetate ion in acetic acid [eqn (52)] at an electrode of 1 cm2 surface area, passing a current of 1 A during 

10-6 s. This generates 10-11 moles of CH3COO, and if we now for simplicity assume—remember it is a mental construction—that we can arrest the further reactions of CI COO and distribute it evenly in the first 100 A layer, it is easy to calculate that [CH3COO ] would 

neglecting adsorption effects in the El for a while, we can conclude that concentration gradients (see Fig. 4) in electrochemical experiments will have the following consequences for mechanistic studies they will 

If we also consider the role of adsorption on the distribution of products, then we should note that surface concentrations of substrate and intermediate(s) must be taken into account, i.e. their concentrations in the inner Helmholtz layer (cf. for example, Wendt, 1973). One effect of this would possibly be to 

The very success of electrochemical reactions that give products resulting from coupling between two fragments originating from two molecules is an indication that locally high concentrations of intermediates must play an important role. Owing to the difficulty of 

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The release of soluble species contained in a porous medium in contact with water is the result of complex and coupled phenomena. This includes water transfer in the porous medium up to saturation dissolution of the species in the pore water according to the local chemical context and transport of species in solution due to the effect of concentration gradients. 

Temperature gradients in endothermal reactions amplify the effects of concentration gradients. In exothermal reactions the thermal effects can compensate the concentration effects or dominate completely. In the latter cases the apparent activation energy can become higher than the true one due to a kind of ignition [5], Also in this case at the highest temperatures the apparent Ea approaches zero. 

Nicas, M. (1998). The effect of concentration gradients on deducing a contaminant generation rate function. Am. Ind. Hyg. Assoc. J., 59, 680-688. 

M. E. Breuer and K. Kammermeyer, Effects of Concentration Gradients in Barrier Separation... 

For our purposes we need to use these equations to find the effect of concentration gradients on the free energy. To do this, note that we can expand the Debye function at small q as... 

The quantity k is the so-called gradient-energy coefficient. The set of terms in the summation in Eq. (A.6) is the added term to Tick s first law of diffusion (see Section 2.2), which provides the effect of concentration gradient to the diffusional flux of material. For a purely diffusive process, Eq. (A.6) is combined with the differential component balance expression to give... 

The governing equations for the combined effect of concentration and temperature gradient are ... 

The drop in current that occurs in coulometric experiments may arise not only from the decrease in bulk concentration of the substance being analyzed, but also from a decrease in its surface concentration caused by the development of concentration gradients (see Section 11.2.1). Low values of current density and strong solution stirring are used to avoid the interference of such effects. Thin-layer cells where the electrodes are very close together (tens of micrometers) and the parameter ratio SIV is high, are often used to shorten the experiments. 

Thus, the ideas above do not suffice for an interpretation of all experimental results. These ideas include the assumption that the ions move in the membrane only under the effect of concentration and potential gradients (diffusion and migration), and that transport of one sort of ions is independent of the transport of other sorts of ions. This transport of ions under the effect of external forces has been named passive ionic transport. 

Very much more is known about the theory of concentration gradients at electrodes than has been mentioned in this brief account. Experimental methods for observing them have also been devised, based on the dependence of refractive index on concentration (the Schlieren method) by means of interferometry (O Brien, 1986). Nevertheless, the basic concept of an effective diffusion-layer thickness, treated here as varying in thickness with fi until the onset of natural convection and as constant with time after convection sets in (though decreasing in value with the degree of disturbance, Table 7.10), is a useful aid to the simple and approximate analysis of many transport-controlled electrodic situations. A few of the uses of the concept of 8 will now be outlined. 

Unless carried out very carefully, data from flow reactors may be influenced by experimental uncertainties. Potential problems with the flow reactor technique include imperfect mixing of reactants, radial gradients of concentration and temperature, and catalytic effects on reactor walls. Uncertainties in induction times, introduced by finite rate mixing of reactants, presence of impurities, or catalytic effects, may require interpretation of the data in terms of concentration gradients, rather than just exhaust composition [442]. 

The structure of the double layer can be altered if there is interaction of concentration gradients, due to chemical reactions or diffusion processes, and the diffuse ionic double layer. These effects may be important in very fast reactions where relaxation techniques are used and high current densities flow through the interface. From the work of Levich, only in very dilute solutions and at electrode potentials far from the pzc are superposition of concentration gradients due to diffuse double layer and diffusion expected [25]. It has been found that, even at high current densities, no difficulties arise in the use of the equilibrium double layer conditions in the analysis of electrode kinetics, as will be discussed in Sect. 3.5. 

The transport (or transference) numbers (/ ) represent the fractions of current carried out by such specific ions in the absence of concentration gradients. In dilute solutions can be estimated by dividing the equivalent ion conductance (Hq ) by A0. When concentration gradients are present, some of the current arises from diffusion and this affects the effective numbers. Generally, the transference numbers are weak functions of concentration and temperature and may be regarded as practically independent of salt concentration (Prentice, 1991 Robinson and Stokes, 2002). 

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