Potential independent constants

Here, k and are potential independent constants characteristic of the two reactions. 

In the reversible domain, the peak potential is close to the formal potential independent of the scan rate (Section 1.2). As soon as the system ceases to be reversible, the cathodic peak shifts to negative potentials the more so the smaller the rate constant and the higher the scan rate according to... 

Under open circuit conditions, the electric current /= J) Zj-F-Jj vanishes. As long as tQ2- = 1 this means thaty o2- = To2- 7o2- - 0. or equally V /02- = Zj-F-Vtp = 0. This is true since oxygen ions are the mobile majority species with a constant chemical potential independent of any variation in the oxygen potential. It follows that the electrical potential in the oxide electrolyte of a galvanic cell is constant under open circuit conditions, despite the different oxygen potentials at the two electrodes. 

The equilibrium at an electrode is dynamic, the potential determining ions moving in opposite directions at equal rates. If the process is reversible, the ions move smoothly from one phase to the other via the phase boundary and perform no work. The electrode potential remains constant and independent of current. 

The above discussion has outlined the theoretical approach to the determination of the fundamental vibration frequencies of a molecule. The practical solution of the problem as formulated above presents, however, certain more or less serious difficulties. For example, the completely general potential function of equation (4) is generally not usable even for small molecules, because it contains more independent constants than can be determined from the experimental data. However, by making certain assumptions about the nature of the force field in the molecule, the number of constants can be reduced. One assumption which often works quite well in practice is that of a valence force field [Herzberg 76)]. This assumes that contributions to the potential energy... 

It follows from this expressioin that the electrophoretic mobility of a non-conducting particle for which Ka is large at all points on the surface should be independent of its size and shape provided that the zeta potential is constant. 

The method is based on determining the potential difference between an electrode pair, consisting of a glass electrode sensitive to the difference in the hydrogen ion activity in the sample solution and the internal filling solution, and a reference electrode, which is supposed to have a constant potential independent of the immersing solution. 

With these relations, zeta potentials can be calculated for many practical systems. Note that within each set of limiting conditions the electrophoretic mobility is independent of particle size and shape as long as the zeta potential is constant. For intermediate values of Ka another equation, the Henry equation (4.10) can be used other such equations are available in the literature as well [81,253,264],... 

Potentiometric measurements involve the determination of the electrical potential between two electrodes at zero current flow. A reference electrode has to establish a constant potential independent of the electrolyte used. This is not easily performed even in a conventional measuring set-up and therefore various constructions have been investigated using different electrolyte chambers and porous plugs [33]. For microelectronic realization this problem becomes very severe [34]. 

The remaining two independent constants, a and a2 can be determined using the boundary conditions for the electrical potential and polarization at the surfaces. 

Peter et al. [71] have given a detailed analysis of mechanisms I and II. It can be shown that apparent rate constants ktr and krec derived from IMPS data are in fact functions of the rate constants for all the steps in the reaction. For both mechanisms, the analysis shows that ktr is no longer potential independent because it contains a term associated with recombination via the intermediate. Similarly krcc is determined by the all three rate constants, and no longer follows a simple Boltzmann dependence on potential. Figures 8.13a and 8.13b illustrate the quite complicated potential dependence of the apparent rate constants kn and k[ec predicted for mechanism I when dynamic surface charging is taken into account. 

The physical meaning of the term /i jj, is the current density achieved if all surface reactions are exergonic (i.e. the highest possible turnover frequency per site). The term jxxmii is dependent on the number of active sites per area and potential independent surface reactions. This means that / miii is dependent on the catalyst material. However, if similar surfaces are compared (e.g. a set of 110 rutile oxide surfaces), the number of active sites per area only varies with a few presents, as the lattice constants are very similar, and y limit can effectively be considered material independent, unlike exchange current density. In that case, trends in fy°ER should correlate with trends in activity, ya. 

We may collect all the potential-independent terms in the standard rate constant (fcs) ... 

The e.m.f. of an electrochemical cell can be regarded as the absolute value of the difference of the electrode potentials of the two electrodes. The two electrodes applied in building the electrochemical cell have different roles in the measurement, and must be chosen adequately. One of the electrodes, termed the indicator electrode acquires a potential which depends on the pH of the solution. In practice the glass electrode is used as the indicator electrode. The second electrode, on the other hand, has to have a constant potential, independent of the pH of the solution, to which the potential of the indicator electrode therefore can be compared in various solutions, hence the term reference electrode is applied for this second electrode. In pH measurements the (saturated) calomel electrode is applied as an indicator electrode. 

In the case of semiconductor electrodes, the situation is different insofar as the carrier density is usually much smaller and varies with the electrode potential. Accordingly, it is useful to define the rate in terms of the product of a potential-independent rate constant and of carrier density at the semiconductor surface, as it has already been introduced in Eqs. (25)-(32). The maximum second order rate constant k , k etc. given in units of cm s have recently been estimated by Lewis [129] on the basis of various models [130, 131, 7]. He has obtained k" = 10 — W cm s L Applying this value also to metal electrodes, one has... 

As in the case of two interacting soft plates, when the thicknesses of the surface charge layers on soft spheres 1 and 2 are very large compared with the Debye length 1/k, the potential deep inside the surface charge layer is practically equal to the Donnan potential (Eqs. (15.51) and (15.52)), independent of the particle separation H. In contrast to the usual electrostatic interaction models assuming constant surface potential or constant surface... 

Here, V is the full coupled potential and the summation runs over all MD trajectories. The weight w" of each MD trajectory is proportional to the square of the initial wavepacket at the initial position of the given trajectory. Energy transfer between the modes is implicitly accounted for in the CSP method through the time dependence of the effective separable potentials. The coordinate-independence constant V... 

In certain cases encountered experimentally, for example, for the HER at Ni or Ni-Mo alloys (75), the electrochemical barrier symmetry factor for the initial proton-discharge step [Eq. (4)] may be close to that for the electrochemical desorption step [Eq. (5)] then a limiting coverage ([Pg.42]

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