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The equilibrium electrode potential is given by the Nernst equation (cf. 3.2.17), l ^ '+ lln ^ (5.2.17) ln + nF aRed nF cRed where EOr is the formal potential of the half-cell reaction (5.2.5) (cf. Section 3.1.5). These equations satisfy Eqs (5.2.8) and (5.2.9) when A# a = A/fa* - (1 - a)nFE (5.2.18) E E +
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Obviously 0 < a r < l . For the conditional (formal) electrode potential we obtain
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If the system is at equilibrium at the formal potential, cOx = ^Red (see Eq. 5.2.14) and thus ka = kc = k^ (5.2.20)
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The quantity k^ is termed the conditional (formal) rate constant of the electrode reaction. It follows from Eqs (5.2.8), (5.2.9), (5.2.10) and (5.2.18) that
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Substitution for Paexp(-AH JRT) and Pcexp(-AH JRT) from these equations into Eqs (5.2.8) and (5.2.9), considering Eqs (5.2.10) and (5.2.18), yields the equations for the dependence of the rate constant on the electrode potential:
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A further substitution from these equations into Eq. (5.2.13) yields the fundamental equation of electrochemical kinetics:
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(5.2.24) In addition to the thermodynamic quantity EOr, the electrode reaction is characterized by two kinetic quantities: the charge transfer coefficient a and the conditional rate constant fc"0". These quantities are often sufficient for a complete description of an electrode reaction, assuming that they are constant over the given potential range. Table 5.1 lists some examples of the constant A: . If the constant k^ is small, then the electrode reaction occurs only at potentials considerably removed from the standard potential. At these potential values practically only one of the pair of electrode reactions proceeds which is the case of an irreversible or one-way electrode reaction. As mentioned above, at equilibrium two opposing currents pass through the electrode, with absolute values termed the exchange current. The exchange current density j0 can be expressed at an arbitrary value of the equilibrium potential as a function of the concentrations of the oxidized and
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Table 5.1
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Conditional electrode reaction rate constants k^ and charge transfer coefficients a. (From R. Tamamushi) Electrolyte solution 1 M HCIO4 1 M HCIO4 1 M H2SO4
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Electrode Ag Hg C paste Hg Hg Pt C Pt Hg Pt Hg Hg Hg Hg Hg Hg 25 25 25 20 20 25 21 35 24 2 20 22 20 22 20 22 22 0.025 0.005 0.35 0.03 0.14 0.02 3.8 x 1 0 4 0.28 1.1 x l 0 ~ 2 2.1 x KT4 9 x 10~6 1.2 x l O ' 4 6.6 X10- 2 1.3 1.2 x H T 2 2.0 9 x 10- 3 1.8 3.2 x l O ' 3 5 x 10-2 3 x 10- 3 0.50 0.59 0.49 0.28 0.50 0.52 0.25-0.30 0.25-0.30
Method Chronopotentiometry Faradaic impedance Current-potential curve Faradaic impedance Faradaic impedance Current-potential curve Current-potential curve Faradaic rectification Faradaic rectification Voltammetry at rotating disk electrode Faradaic impedance Faradaic impedance Faradaic impedance Faradaic impedance Polarography Polarography
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259 reduced forms of the electroactive substance. Equation (5.2.24) gives
Substitution for E - EOr from the Nernst equation (3.2.14) or (5.2.17) yields
The charge transfer coefficient a can be found from the dependence of ; 0 on cOx or cRed. It is often useful to express the current density as a function of the overpotential rj = E Ee (cf. Eq. 5.1.11) and of the exchange current density. On substituting for k^ from Eq. (5.2.26) and for E - EOr, RT E - E0' = r, + In (cOx/cRed) into Eq. (5.2.24) we obtain the equation (5.2.27)
(see Fig. 5.2). The j rj dependence is termed the polarization curve (voltammogram). If the overpotential is small (r}<RT/nF), then, on expanding the exponential function and neglecting all the terms in the series except the first two, we have ^ (5-2.29)
The value of the exchange current determines the deviation of the electrode potential from the equilibrium value during the current flow. The greater the deviation, the slower the electrode reaction. It follows from Eq. (5.2.29) that
The reciprocal value of this differential quotient,