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t = 0, x > 0: t > 0, x = 0:
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cB = cA/K D(dcB/dx) = kccB a steady state is (5.6.11)
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After a sufficiently long time interval (t l/k1K), established (dcB/dt = 0), giving D ^ + k1(cA-KcB) =0
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Figure 5.37 depicts the stationary distribution of the electroactive substance (the reaction layer) for &c < . The thickness of the reaction layer is defined in an analogous way as the effective diffusion layer thickness (Fig. 2.12). It equals the distance \i of the intersection of the tangent drawn to the concentration curve in the point x = 0 with the line c = cA/K, (5.6.13) The current density is given by the equation (5.6.14) which describes the polarization curve of a kinetic current, controlled by the electrode reaction and the steady state of the chemical volume reaction and diffusion. For kc ^JkxKDy the limiting current density is obtained:
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Fig. 5.37 Steady-state concentration distribution (reaction layer) in the case of a chemical volume reaction preceding an electrode reaction (q (Eq. 5 ( ) ) \ k x = 0.04 s"1, D = 10~5cm s \\ \i is the 004 1 (5.6.12)); K = 10\ k effective reaction layer thickness
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349 This relationship forms the basis for the method of determining the rate constants of fast chemical reactions from the kinetic current. A similar procedure can be employed for subsequent and parallel chemical reactions and for higher-order reactions. The reduction of formaldehyde at a mercury electrode is an example of a system in which a chemical reaction precedes the electrode reaction. Formaldehyde is present in aqueous solution as the hydrated form (as dihydroxymethane), which cannot be reduced at a mercury electrode. This form is in equilibrium with the carbonyl form
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According to R. Brdicka and K. Vesely the carbonyl form of formaldehyde is reduced and the limiting kinetic current is given by the rate of the chemical volume reaction of dehydration. An analogous situation occurs for the equilibria among complexes, metal ions and complexing agents if the rates of complex formation and decomposition are insufficient to preserve the equilibrium. A simple example is the deposition of cadmium at a mercury electrode from its complex with nitrilotriacetic H N + ^ C O O H = H3X. Here, a single X3~ species is bound in the COOH complex. According to J. Koryta, equilibrium in solution in the pH range from 3 to 5 can be described by the equation Cd2+ + HX2~ <=^ CdX" + H + (5.6.17)
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The symbols kf and kd are the rate constants for the formation and dissociation of the complex. The equilibrium constant of reaction (5.6.17) is given by the relationship KC6xK3 = ^
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where KCdX is the stability constant of the complex and K3 is the third dissociation constant of the acid. If only free metal ions react in the given potential range at the electrode and the rate of the electrode reaction of the complex is negligible, then the constants in Eq. (5.6.15) for the limiting kinetic current are given as k\ = /cdcH3o+ (5.6.19) where cHX2- is the concentration of the complexing agent, present in excess
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so that its concentration in the electrolyte does not change during the electrolysis. A chemical reaction subsequent to a fast (reversible) electrode reaction (Eq. 5.6.1, case b) can consume the product of the electrode reaction, whose concentration in solution thus decreases. This decreases the overpotential of the overall electrode process. This mechanism was proposed by R. Brdicka and D. H. M. Kern for the oxidation of ascorbic acid, converted by a fast electrode reaction at the mercury electrode to form dehydroascorbic acid. An equilibrium described by the Nernst equation is established at the electrode between the initial substance and this intermediate product. Dehydroascorbic acid is then deactivated by a fast chemical reaction with water to form diketogulonic acid, which is electroinactive. If a chemical reaction regenerates the initial substance completely or partially from the products of the electrode reaction, such case is termed a chemical reaction parallel to the electrode reaction (see Eq. 5.6.1, case c). An example of this process is the catalytic reduction of hydroxylamine in the presence of the oxalate complex of TiIV, found by A. Blazek and J. Koryta. At the electrode, the complex of tetravalent titanium is reduced to the complex of trivalent titanium, which is oxidized by the hydroxylamine during diffusion from the electrode, regenerating tetravalent titanium, which is again reduced. The electrode process obeys the equations (5.6.20)
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