Equilibrium electrode potentials approximation

At equilibrium at 298 K the electrode potential of the half-reaction for copper, given approximately by... 

According to Sato et al.,6,9 the barrier-layer thickness is about 1.5 to 1.8 nm V-1, and increases to 3 nm around the oxygen-evolution potential. In Fig. 5, the scale of the electrode potential, Vrhe, is that of the reversible hydrogen electrode (RHE) in the same solution. The electrode potentials extrapolated from the linear plots of the potentials against the film thickness suggested that the potential corresponding to the barrier thickness equal to zero is almost equal to 0.0 V on the RHE scale, independent of the pH of the solution, and approximately agrees with the equilibrium potential for the oxide film formation of Fe or Fe. Therefore it is concluded that the anodic overpotential AE applied from the equilibrium potential to form the oxide film is almost entirely loaded with the barrier portion. 

The galvanostatic and potentiodynamic charging curves of platinum electrodes shift approximately 60 mV in the negative direction when the solution pH is raised by 1 unit. This implies that when potentials which refer to the equilibrium potential of a hydrogen electrode in the same solution (RHE) are used, these curves remain practically at the same place within a wide range of solution pH. Hence, we shall use this scale while analyzing these curves. 

Equation (6.13), in fact, reflects the physical nature of the electrode process, consisting of the anode (the first term) and cathode (the second term) reactions. At equilibrium potential, E = Eq, the rates of both reactions are equal and the net current is zero, although both anode and cathode currents are nonzero and are equal to the exchange current f. With the variation of the electrode potential, the rate of one of these reactions increases, whereas that of the other decreases. At sufficiently large electrode polarization (i.e., deviation of the electrode potential from Eg), one of these processes dominates (depending on the sign of E - Eg) and the dependence of the net current on the potential is approximately exponential (Tafel equation). 

The current increases first exponentially and then levels off. The same dependence follows from Eq. (34.27). At not large deviations of the electrode potential from the equilibrium potential (i.e., at not large overpotentials, r = Eq - E), the approximate form of Eq. (34.27) is as follows ... 

If the rates of the electrode reactions are large and the system is fairly close to equilibrium (the electrode potential is quite close to the reversible electrode potential), then the right-hand sides of Eqs (5.4.3) and (5.4.4) correspond to the difference between two large numbers whose absolute values are much larger than those of the left-hand sides. The left-hand side can then be set approximately equal to zero, kcc0x — A acRed 0, and in view of Eqs (5.2.14) and (5.2.17),... 

Hydrogen electrodes are approximately non-polarizable, which implies that the solution and the interface are in equilibrium. This simplifies the task of maintaining a constant reference potential. In an ideally non-polarizable electrode, the electrode... 

ACTIVITY AND ACTIVITY COEFFICIENTS In our deduction of the law of mass action we used the concentrations of species as variables, and deduced that the value of the equilibrium constant is independent of the concentrations themselves. More thorough investigations however showed that this statement is only approximately true for dilute solutions (the approximation being the better, the more dilute are the solutions), and in more concentrated solutions it is not correct at all. Similar discrepancies arise when other thermodynamic quantities, notably electrode potentials or chemical free energies are dealt with. To overcome these difficulties, and still to retain the simple expressions derived for such quantities, G. N. Lewis introduced a new thermodynamic quantity, termed activity, which when applied instead of concentrations in these thermodynamic functions, provides an exact fit with experimental results. This quantity has the same dimensions as concentration. The activity, aA, of a species A is proportional to its actual concentration [A], and can be expressed as... 

Approximate Determination of Standard Potentials.—Many studies have been made of oxidation-reduction systems with which, for one reason or another, it is not possible to obtain accurate results this may be due to the difficulty of applying activity corrections, uncertainty as to the exact concentrations of the substances involved, or to the slowness of the establishment of equilibrium with the inert metal of the electrode. It is probable that whenever the difference in the number of electrons between the oxidized and reduced states, i.e., the value of n for the oxidation-reduction system, is relatively large the processes of oxidation and reduction occur in stages, one or more of which may be slow. In that event equilibrium between the system in the solution and the electrode will be established slowly, and the measured potential may be in error. To expedite the attainment of the equilibrium a potential mediator may be emploj cd this is a substance that undergoes reversible oxidation-reduction and rapidly reaches equilibrium with the electrode. 

A frequent complication is that several simultaneous equilibria must be considered (Section 3-1). Our objective is to simplify mathematical operations by suitable approximations, without loss of chemical precision. An experienced chemist with sound chemical instinct usually can handle several solution equilibria correctly. Frequently, the greatest uncertainty in equilibrium calculations is imposed not so much by the necessity to approximate as by the existence of equilibria that are unsuspected or for which quantitative data for equilibrium constants are not available. Many calculations can be based on concentrations rather than activities, a procedure justifiable on the practical grounds that values of equilibrium constants are obtained by determining equilibrium concentrations at finite ionic strengths and that extrapolated values at zero ionic strength are unavailable. Often the thermodynamic values based on activities may be less useful than the practical values determined under conditions comparable to those under which the values are used. Similarly, thermodynamically significant standard electrode potentials may be of less immediate value than formal potentials measured under actual conditions. 

Numerous applications of standard electrode potentials have been made in various aspects of electrochemistry and analytical chemistry, as well as in thermodynamics. Some of these applications will be considered here, and others will be mentioned later. Just as standard potentials which cannot be determined directly can be calculated from equilibrium constant and free energy data, so the procedure can be reversed and electrode potentials used for the evaluation, for example, of equilibrium constants which do not permit of direct experimental study. Some of the results are of analjrtical interest, as may be shown by the following illustration. Stannous salts have been employed for the reduction of ferric ions to ferrous ions in acid solution, and it is of interest to know how far this process goes toward completion. Although the solutions undoubtedly contain complex ions, particularly those involving tin, the reaction may be represented, approximately, by... 

It is further clear from Fig. 5.19 that the n-electrode has to be polarized cathodically with respect to the equilibrium potential, and the p-electrode anodically, in order to reach the corresponding flatband situation (see lower part of Fig. 5.19), provided that the positions of the energy bands at the surface are the same for the two types of electrodes. Keeping in mind that the electrode potential refers to the Fermi level of the electrode, then the difference of flatband potentials corresponds exactly to the difference of the two Fermi levels. Since the Fermi level in the bulk of a semiconductor with the usual doping (>1() cm ) is rather close to the corresponding band, the difference in the flatband potentials approximates the bandgap of the semiconductor as found with GaP. 

Table 4.3 shows, however, that redox potentials often differ greatly from electrode potentials. Ion activities are only qualitatively related to redox potentials, except in rare circumstances. One reason is that the Nemst equation applies only to equilibrium. Redox reactions in soils are noiiequilibria, though in some cases for highly reduced soils, a steady state may be reached approximating equilibrium. Then only a few redox couples in the soil affect the platinum electrode and the result may approach a pseudo-equilibrium. 

The redox active film of interest is assumed to undergo electrochemically reversible interconversion between oxidized and reduced states as a function of electrode potential as described by the Nernst equation. If so, then the plot of reflectance Rdc as a function of the electrode potential E can be represented schematically as Fig. 2.5. When AFac-C R77 appT. the linear response approximation is valid A sine wave potential modulation should give rise to a sinusoidal change of the reflectance, as shown in Fig. 2.5, where R is the gas constant, T is the absolute temperature, app is the apparent number of electrons involved in the redox equilibrium relationship, and F is the Faraday constant. 

The value of E is known as the standard electrode potential of the Fe /Fe " couple. This is the measured potential of the cell shown in Fig. 1.8 when the hydrogen electrode is standard (flH+ = U Fh, = 1 atm) and when all the chemical species contributing to the potential determining equilibrium are present at unity activity (or, approximately speaking, concentration) so that... 

Free Ions Versus Complexed Ions In discussing the ion-selective electrode, we noted that the membrane potential is influenced by the concentration of F , but not the concentration of HF. An analysis for fluoride, therefore, is pH-dependent. Below a pH of approximately 4, fluoride is present predominantly as HF, and a quantitative analysis for total fluoride is impossible. If the pH is increased to greater than 4, however, the equilibrium... 

It must not be assumed that the protection potential is numerically equal to the equilibrium potential for the iron/ferrous-ion electrode (E ). The standard equilibrium potential (E ) for iron/ferrous-ion is -0-440V (vs. the standard hydrogen electrode). If the interfacial ferrous ion concentration when corrosion ceases is approximately 10 g ions/1 then, according to the Nernst equation, the equilibrium potential (E ) is given by ... 

We then study experimentally the effect of an inert electrolyte solution and show that ion motion forces an applied electrical potential in the dark to drop near the substrate electrode, thus reinforcing the effects of the distributed resistance. Overall, the 2 conduction and valence bands (whose spatial gradients reflect the electric field) remain approximately flat both at equilibrium and under illumination therefore, charge transfer occurs primarily by diffusion rather than by field-induced drift [4,40-42]. Recent numerical simulations [43,44] and modeling of photogenerated trapped charges [45] show that in an illuminated DSSC there may be, in fact, a very small bulk electric field of about 0.1-3 mV/pm, but this is not expected to have much influence. 

It is therefore almost impossible to establish the equilibrium potential of the oxygen electrode in aqueous electrolytes (+1.23 V vs rev. hydrogen electrode) and only in a very careful experiment of Bockris and Hug (112) has this been achieved. Usually at zero current density a lower potential of approximately 0.95 V is obtained, which is established by mixed reactions, with the main reaction being the reduction of oxygen to hydroperoxide. 

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