Standard heterogeneous rate constant

Here, cp = (E —E ) is a dimensionless potential and rs = 1 cm is an auxiliary constant. Recall that in units of cm s is heterogeneous standard rate constant typical for all electrode processes of dissolved redox couples (Sect. 2.2 to 2.4), whereas the standard rate constant ur in units of s is typical for surface electrode processes (Sect. 2.5). This results from the inherent nature of reaction (2.204) in which the reactant HgL(g) is present only immobilized on the electrode surface, whereas the product is dissolved in the solution. For these reasons the cathodic stripping reaction (2.204) is considered as an intermediate form between the electrode reaction of a dissolved redox couple and the genuine surface electrode reaction [135]. The same holds true for the cathodic stripping reaction of a second order (2.205). Using the standard rate constant in units of cms , the kinetic equation for reaction (2.205) has the following form ... 

The electrochemical rate constants of the Zn(II)/Zn(Hg) system obtained in propylene carbonate (PC), acetonitrile (AN), and HMPA with different concentrations of tetraethylammonium perchlorate (TEAP) decreased with increasing concentration of the electrolyte and were always lower in AN than in PC solution [72]. The mechanism of Zn(II) electroreduction was proposed in PC and AN the electroreduction process proceeds in one step. In HMPA, the Zn(II) electroreduction on the mercury electrode is very slow and proceeds according to the mechanism in which a chemical reaction was followed by charge transfer in two steps (CEE). The linear dependence of logarithm of heterogeneous standard rate constant on solvent DN was observed only for values corrected for the double-layer effect. 

Fig. 51.1. Schematic of the feedback method of measuring heterogeneous standard rate constant at titantium nitride thin film. The UME is poised at a potential where ferrocene methanol is oxidized at a diffusion-controlled rate (Et = 0.4 V). The substrate is biased at a potential so that it reduces the species being produced at the UME, thus controlling the feedback effect.

For a range of diffusion coefficients Table II shows values that may be expected for heterogeneous standard rate constants and apparent molecular weights. These numbers are presented as an illustration of their relative magnitudes and the manner in which they interrelate. It must be observed that the apparent molecular weight is a reaction parameter and does not, necessarily, express the molecular size. 

Activation volume — As in case of homogeneous chemical reactions, also the rate of heterogeneous electron transfer reactions at electrode interfaces can depend on pressure. The activation volume AVZ involved in electrochemical reactions can be determined by studying the pressure dependence of the heterogeneous -> standard rate constant ks AVa = -RT j (p is the molar - gas constant, T absolute temperature, and P the pressure inside the electrochemical cell). If AI4 is smaller than zero, i.e., when the volume of the activated complex is smaller than the volume of the reactant molecule, an increase of pressure will enhance the reaction rate and the opposite holds true when A14 is larger than zero. Refs. [i] Swaddle TW, Tregloan PA (1999) Coord Chem Rev 187 255 [ii] Dolidze TD, Khoshtariya DE, Waldeck DH, Macyk J, van Eldik R (2003) JPhys Chem B 107 7172... 

Activation volume — As in case of homogeneous chemical reactions, also the rate of heterogeneous electron transfer reactions at electrode interfaces can depend on pressure. The activation volume AV involved in electrochemical reactions can be determined by studying the pressure dependence of the heterogeneous standard rate constant ks AV = R is the molar... 

It was shown later that a mass transfer rate sufficiently high to measure the rate constant of potassium transfer [reaction (10a)] under steady-state conditions can be obtained using nanometer-sized pipettes (r < 250 nm) [8a]. Assuming uniform accessibility of the ITIES, the standard rate constant (k°) and transfer coefficient (a) were found by fitting the experimental data to Eq. (7) (Fig. 8). (Alternatively, the kinetic parameters of the interfacial reaction can be evaluated by the three-point method, i.e., the half-wave potential, iii/2, and two quartile potentials, and ii3/4 [8a,27].) A number of voltam-mograms obtained at 5-250 nm pipettes yielded similar values of kinetic parameters, = 1.3 0.6 cm/s, and a = 0.4 0.1. Importantly, no apparent correlation was found between the measured rate constant and the pipette size. The mass transfer coefficient for a 10 nm-radius pipette is > 10 cm/s (assuming D = 10 cm /s). Thus the upper limit for the determinable heterogeneous rate constant is at least 50 cm/s. 

In Ref. 30, the transfer of tetraethylammonium (TEA ) across nonpolarizable DCE-water interface was used as a model experimental system. No attempt to measure kinetics of the rapid TEA+ transfer was made because of the lack of suitable quantitative theory for IT feedback mode. Such theory must take into account both finite quasirever-sible IT kinetics at the ITIES and a small RG value for the pipette tip. The mass transfer rate for IT experiments by SECM is similar to that for heterogeneous ET measurements, and the standard rate constants of the order of 1 cm/s should be accessible. This technique should be most useful for probing IT rates in biological systems and polymer films. 

In electrochemical literature the standard rate constant fe is often designated as fes h or fe9, called the specific heterogeneous rate constant or the intrinsic rate constant. According to eqns. 3.5 and 3.6, we have... 

Thus we have succeeded in expressing the individual heterogeneous rate constants in terms of a standard rate constant and a potential-dependent exponential term. Since if = nFC0kf h and ib = nFCRkbh, the individual current densities can now be written as follows ... 

Similar SECM experiments can be performed using a simple (unassisted) IT process [41]. In this case, both the top and the bottom phases contain the same ion at equilibrium. The micropipet tip is used to deplete concentration of this common ion in the top solvent near the ITIES. The depletion results in the IT across the ITIES, which produces positive feedback. Any solid surface (or a liquid phase containing no specific ion) acts as an insulator in this experiment. The mass transfer rate for IT measurements by SECM is similar to that for heterogeneous ET measurements, and the standard rate constants in excess of 1 cms-1 should be measurable. 

For the LSV and CV techniques, the concept of reversibility/irreversibility is therefore very important. Electrochemists are responsible for some confusion about the term irreversible, since a reaction may be electrochemically irreversible, yet chemically reversible. In electrochemistry, the term irreversible is used in a double sense, to describe effects from both homogeneous and heterogeneous reactions. In both cases, the irreversible situation arises when deviations from the Nernst equation can be seen as fast changes in the electrode potential, E, are attempted and the apparent heterogeneous rate constants, /capp, for the O/R redox couple is relatively small. The heterogeneous rate constant can be split into two parts a constant factor in terms of the standard rate constant, k°, and an exponential function of the overpotential E - Eq), as expressed in Eq. 59, where only the reductive process is considered (see also Eq. 5). 

Studies on the electrochemical behavior of ferrocene encapsulated in the hemi-carcerands 61 and 62, indicated that encapsulation induces substantial changes in the oxidation behavior of the ferrocene subunit [98]. In particular, encapsulated ferrocene exhibits a positive shift of the oxidation potential of c. 120 mV, probably because of the poor solvation of ferrocenium inside the apolar guest cavity. Lower apparent standard rate constants were found for the heterogeneous electron transfer reactions, compared to those found in the uncomplexed ferrocene under identical experimental conditions. This effect may be due to two main contributions (i) the increased effective molecular mass of the electroactive species and (ii) the increased distance of maximum approach of the redox active center to the electrode surface. 

It has been shown very recently that l,3,5-tris(4-(ALphenyl-AL3-methylphenyl)phenyl) benzene (56), when studied in DMF and dichloromethane solutions, forms two and three reversible one-electron anodic waves, respectively92. The kinetics of heterogeneous electron transfers has recently been studied using the high speed microband channel electrode in solutions containing 0.1 M TBAP as electrolyte it was possible to find standard potentials E°, transfer coefficients a and, finally, standard rate constants k°. The obtained data, labeled by subscripts 1, 2 and 3 for the first, second and third electron transfer, respectively, are collected in Table 2. 

In this equation, and represent the surface concentrations of the oxidized and reduced forms of the electroactive species, respectively k° is the standard rate constant for the heterogeneous electron transfer process at the standard potential (cm/sec) and oc is the symmetry factor, a parameter characterizing the symmetry of the energy barrier that has to be surpassed during charge transfer. In Equation (1.2), E represents the applied potential and E° is the formal electrode potential, usually close to the standard electrode potential. The difference E-E° represents the overvoltage, a measure of the extra energy imparted to the electrode beyond the equilibrium potential for the reaction. Note that the Butler-Volmer equation reduces to the Nernst equation when the current is equal to zero (i.e., under equilibrium conditions) and when the reaction is very fast (i.e., when k° tends to approach oo). The latter is the condition of reversibility (Oldham and Myland, 1994 Rolison, 1995). 

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