312 [CONTRIBUTION FROM
PAULDELAHAY, MITSUGISENDAAND CARLH. WEIS THE
VOl. 83
COATESCHEMICAL LABORATORY, LOCISIANA STATEUNIVERSITY, BATONROUGE,LOUISIANA]
Faradaic Rectification and Electrode Processes BY PAULDELAHAY, MITSUGIS E N D A AND ~ ~CARLH. WE IS^^ RECEIVED JCLY 14, 1960
A rather general theory of faradaic rectification is developed and previous theories are clarified and greatly extended. Theoretical results are substantiated by experiment. It is shown that two types of control must be considered simultaneously, namely, control of the mean and alternating components of the current or potential. In practice, the alternating current or potential is an essentially harmonic-free sinusoidal function of time, and either the mean current is equal t o zero (1= 0) or the mean potential is equal to the equilibrium potential (8= E e ) . The rectification voltage or the rectification current is the same regardless of control of the alternating component, and the second harmonic component of current or potential resulting from rectification is the same regardless of control of the mean component. A general equation is derived for the rectification voltage for control a t I = 0 without assumption about the form of the I-E characteristic. Particular forms of this equation are derived in terms of the resistive and capacitive components of the series equivalent circuit for the faradaic impedance for a simple charge transfer process and a charge transfer process with preceding chemical reaction. The time-dependence of the rectification voltage is derived by consideration of the double layer charging without and with rectification from the double layer. Influence of the applied voltage, frequency and reactant concentrations is discussed. Methods for the calculation of the exchange current density and transfer coefficient are outlined without and with correction for the double layer structure. Limitations imposed by the circuit resistance are also discussed. It is shown from the general equation that the mean rectification current for control of E a t E, is the same as in the voltage-step potentiostatic method. The influence of the circuit resistance is readily accounted for. Apparatus is described for measurement of rectification voltages by application of a short single pulse (a few milliseconds or longer) of alternating voltage up to 2 megacycles per sec. Experimental results for discharge of mercurous ion in perchloric acid on mercury and for Ti(1V)-Ti(II1) in tartaric acid medium a t a dropping mercury eIectrode are given. It is concluded that the kinetic study of very fast reactions should still be feasible when other methods fail because interference by the double layer capacity is avoided and the influence of the cell resistance is greatly minimized.
Introduction The current-potential characteristic for electrode processes is linearized in the theory of the faradaic impedance and is assumed to be symmetrical with respect to the equilibrium potential E,. Actually, current-potential curves are, in general, asymmetrical with respect to E e , and there results faradaic recti$cation. This rectification effect can be applied to the study of electrode kinetics since it depends on the current-potential characteristic. Faradaic rectification was discovered by Doss and AgarwalZa who designated it as “redoxokinetic” effect. The expression “faradaic rectification” was coined by Oldham.3 Work on theory which is briefly covered below was done by Doss and Agarwal,2h OldhamIs Barker, Faircloth and Gardne^-,^ Vdovin6 and Matsuda and Delahay.8 Application to electrode kinetics was discussed by these authors but Barker and co-workers4 were the first to realize and fully demonstrate in an extensive experimental study the potentialities of faradaic rectification in the study of inherently fast electrode processes (ie., processes with a large exchange current density), The work of Fournier’ who studied the distortion of polarographic waves upon superimposition of an alternating voltage is related to faradaic rectification. This author,
however, did not even allude to the possibility of application to electrode kinetics and limited herself to experimental observations. A sound basis for her observations was given by Tachi and coworkers,s but these authors did not consider faradaic rectification as discussed below. Application to kinetics was examined by Tachi and collaborators but for irreversible polarographic waves. Control of mean electrical variables versus control of alternating variables was not clearly analyzed by previous authors and theory is in need of clarification. A rather general theory in which this and other questions not considered before is discussed here and substantiated by experimental work. These results were announced in a preliminary note. A qualitative discussion, not previously reported, in which the matter of control is clarified will be presented first.
(1) (a) Postdoctoral research associate January 1960-March 1961; on leave from Department of Agricultural Chemistry, Kyoto University, Kyoto. (b) Predoctoral fellow 1959-1960; on leave from the Department of Physics and Chemistry, Technical Iustitute of Aeronautics, Sao Jose dos Campos, S . P., Brazil. (2) (a) K. S. G.Doss and H. P . Agarwal, J. Sci. Induslr. Res. I n d i a , 9B, 280 (1950). (b) K . S. G. Doss and H. P. Agarwal, Pvm. India Acad. Sci., 34A, 263 (1951); 36A, 45 (1952). (3) K . B . Oldham, Trans. Foradoy SOC.,68, 80 (1957). (4) (a) G. C. Barker, Anal. Chinr. A c f a , 18, 118 (1958). (b) G . C. Barker, R . L. Faircloth and A. W.Oardner, Nafure, 181, 247 (1958). (c) G. C. Barker, “Transactions of the Symposium on Electrode Processes, Philadelphia, 1959,” E. Yeager, editor, John Wiley and Sons, Inc., New York, N. Y.,in course of publication. (5) Iu. A. Vdovin, Dokladr Akad. Nauk S.S.S.R., 120, 554 (1958). (6) H. Matsuda and P. Delahay, T R x s JOURNAL, 82, 1547 (1960). (7) M. Pournier, Comp. ~ c n d . 282, . 1673 (1951).
where I is the current density, CO and CR are the concentrations at the electrode, Coo and CRO are the bulk concentrations, Iaois the apparent (not corrected for double layer effects) exchange current density for the concentrations Cooand CRO, LY is the transfer coefficient, q is the overvoltage (7 = E - E,; E potential in the European convention a t the current density I, E , equilibrium potential), and F, R, and T have their usual significance. A net cathodic current is positive in eq. 1 and a net
Qualitative Discussion Types of Control.-We consider an electrode ne = R, involving soluble species process, 0 which has the current density-potential characteristic
+
(8) M. Senda and I . Tachi, Buil. Chcm. SOC.Japan, 28, 632 (1955); see also four previous papers in this series. (9) P. Delahay, M. Senda and C. € Weis, I. J . Phys. Chem., 64, 960 (1960).
FARADAIC RECTIFICATION AND ELECTRODE PROCESSES
Jan. 20, 1961
anodic current is negative. The kinetic parameter I2 can be written aslo I.0
= nFka0 Coo
CRO~
-
I=O,AC CONTROL
313
-
I=0 , A V . C O N T R O L
(2)
where k,O is the apparent standard rate constant. This equation holds provided that the concentrations of reactants in the preelectrode state (Le., after correction for the double layer structure) are proportional to the concentrations outside the diffuse double layer. In the faradaic rectification method, current and potential vary periodically and two types of control must be considered: (a) contrql of the mean ccmponent of the current density (I)or potential ( E ) and (b) control of the alternating component of the current density (A.C. control ) or potential ( A . V*control). If the alternating current is a harmonic-free sinusoidal function of time, the alternating Component of the potential includes harmonics as a result of the asymmetry of the I-E curve about E,. Conversely, the alternating component of the current contains harmonics when the alternating component of potential is free of harmonics. Two particular cases of control of the mean component which are of interest, I = 0 and E = E,, will be discussed qualitatively for A.V. and A.C. control, the double layer capacity and circuit resistance being neglected. Complications resulting from the double layer capacity and cell resistance are discussed below in the quantitative treatment. Control of Mean Current _Density to Zero.When there is A.C.control and I = 0, the variations of E with time” are represented by the sum of a pure sinusoidal function a t the same frequency f as the current and a periodic component a t the frequency 2f (Fig. 1A). The latter component results from the asymmetry of the I-E curve about E,. The net result is a shift (toward Cnodic potentials in Fig. IA) of the mean potential E with respzct to E,, ;.e., there appears a rectijtcation voltage E -E,. Since = 0, there is no net chemical change and there is no mass transfer polarization for the mean components of the concentrations. Hence, the mean rectification voltage is time-independent when the double layer is neglected. When the alternating current is interrupted, the rectification voltage drops to zero immegiately. Rectification for I = 0 and A.V. control results in an alternating current component of frequency Zf (Fig. 1B). This-current component must be compensated since I = 0. Further, the alternating component of the potential must be free of harmonics because of A.V. control. The alternating component of the current a t the frequency 2f must therefore be compensated on a time average by an equivalent current of constant amplitude and opposite direction. This compensation requires the mean potential to shift with respect to E e (toward anodic potentials in Fig. 1B) and there appears (10) For a recent review, see Chapter 5 by P. Delahay in “Advances in Electrochemistry and Electrochemical Engineering,” P. Delahay and C. W. Tobias, editors, Interscience Publishing Co.,New York, N. Y.,in course of publication. (11) The alternating component cannot be determined by direct reading on the I-E characteristic of the current corresponding t o a given departure of potential from E . except in the limiting case of an infinite frequency. Mass transfer must be considered at finite frequencies, and the sinusoidal components of I and E are not in phase. It will be assumed here that the frequency is infinite.
E=E,, AC.C ONT R O L
-
E=E e, A V C O N T R O L
+ *E-
Fig. 1.-Diagram
of the qualitative discussion of faradaic rectification.
a rectification voltage E - E,. Since I = 0, the mean rectification voltage is time-independent if the double layer is not considered, just as for I = 0 and A. C. control. Finally, the rectification voltage is the same whether there is A.C. or A.V. control because of the reciprocity relationship between Figs. 1A and B. Control of the Mean Potential at the Equilibrium Potential.-Rectification for E = E, and A.C. control results in an alternating component of potential a t the frequency 2f (Fig. 1C) which must be compensated since E = E e . Further, the alternating current component must be harmonic-free because of A.C. control. The frequency 2f component of the potential therefore must be compensated on a time average by an equivalent shift of potential of constant amplitude and opposite direction. Compensation thus results in a net recti$cation current I r . The rectification current is accompanied by a net chemical change. Reactant concentrations change a t the electrode, and E e drifts as t 3 03 (toward cathodic potentials for Fig. IC) toward a limiting value E e f about which the I-E curve is symmetrical. Rectification vanishes for t 3- 03. The necessary experimental conditions, -however, should prove difficult to achieve since E must be maintained at the varying equilibrium potential E, and a low resistance path must be provided for the rectification current. A solution is to compensate exactly the mean rectification current by a current of opposite direction from an external source, when the alternating current is applied, so that E e remains unchanged. Faradaic rectification is then studied this compensating current. from measurement 01 A.V. control for E = E, results in current-time variations equivalent to a sine wave of frequency f and a periodic component of frequency 2f (Fig. 1D). The latter is equivalent to a mean rectification current which has the same magnitude and direction as for A.C. control (Fig. 1C). The best experi-
PAULDELAHAY, MITSUGISENDA AND CARLH. WEIS
314
mental procedure is to compensate the rectification current to achieve constant E e just as for A.C. control. To summarize the previous analysis : control of I a t zero results in the same rectification voltagg whether there is A.C. or A.V. control; control of E a t E e results in a rectification current I , which is independent of the type of control of the alternating component. Further, the component a t the frequency 2f appears in the potential when there is -1.C. control and in the current when there is A.V. control. The type of alternating component control is thus immaterial as far as the rectification voltage or current are concerned, and only the control of the mean component of current or potential must be considered. Conversely, the control of the mean component of current or potential is immaterial for the frequency 2f component, and only the type of alternating component control need be considered. Apparatus for I = 0 control must be such that the direct current resistance of the circuit across the cell be, in principle, infinite, ;.e., the cell must be connected to the alternating current generator through a leak-free capacitor. Rectification Voltage and Mean Rectification Current Density General Correlations.-Non-linearity of the I-E characteristic about E e for 17 I > &E, 61E >> 6&1 &CO, >> 80c0, 6 1 c O >> 62c0, &CR >> 60CR, 6lCR
>> 62CR.
or 6oE can be conOne of the mean component trolled, and one generally selects either 6oI = 0 (mean current equal to zero) or 6& = O (mean potential equal to the equilibrium potential). If 6J = 0, there is no net change in the mean concentrations of substances 0 and R, and consequently SCCo= &CR 0 independently of the control of the alternating component (A.C. or A.V. control). One then deduces from eq. 3 (5)
where - l I ( d I / d E ) is the charge transfer resistance.13 Since there is no net change in the mean concentrations of 0 and R, 60E should be independent of time a t least if the double layer capacity is not considered. If &E = 0, the mean rectification current density is given by eq. 3 as written for p = 0 in which the term in &E vanishes. The value of 6 0 1 can be ex-
Jan. 20, 1961
FARADAIC RECTIFICATION AND ELECTRODE PROCESSES
315
pressed in terms of the shift of mean potential obtained in the method with control of S d a t zero by combination of eq. 3 and 5. Thus
dm (= 6d3) is
The mean rectification current is the same as the current which would be observed in the potentiostatic method for a potential step -SoE. The result derived by Gerischer and Vielstich14 for the current density in the potentiostatic method, as written for the linearized I-E characteristic, thus applies here
where rS and y. are the real (resistive) and imaginary (capacitive) components of the series equivalent circuit for the total faradaic impedance per unit , j = (- 1)’I2); and YO, area Z (2 = YF - j y ~ with rR, yo, YR are the corresponding components for substances 0 and R for that part of the impedance corresponding to supply and removal of reactants by diffusion and possibly a heterogeneous or homogeneous chemical reaction. It is assumed in the case of a coupled chemical reaction that the kinetic parameters for this reaction can be regarded as potential-independent in the interval ofgotentials being considered. We use the symbol A E m because AE is a function of time when the double layer capacity is considered (see below) and the above value corresponds to t 4 a. One has
I,
=
nF &E RT -IaO exp (A’%) erfc (At1/%)
(7)
with
The time-independent factors in eq. 7 represent the value of I a t t = 0, ie., the value of I one calculates for r] = - &E. The time-dependent factors represent the effect of increasing mass transfer polarization (exp(X2t)erfc(ht”2) 4 0 for t --+ 00 ). Second Harmonic Terms.-Second harmonic terms are also readily analyzed on the basis of eq. 3. When there is A.C. control, the current is free of harmonics, L e . , 621=0, and consequently &CO= &CR=0. Hence eq. 5 , as written in terms of 82E and 6221, holds independently of the control of the mean component. Conversely, &E=O for A.V. control, and eq. 6, as written in terms of & I , &E, 62Co and & C R , gives the variations of current for the second harmonic. This equation is the same as eq. 3 as written for p = 1 (&21=0),for the fundamental frequency, except for the sign of &E. Consequently, formulas for elements of the equivalent circuit for the second harmonic are the same as for the fundamental frequency (see eq. 10, 11 and 22 below) except that 2w is substituted for W , and 8 2 1 is the current corresponding to the application of - &E to the equivalent circuit. Theory of the second harmonic terms for the faradaic impedance is thus available without further work. Measurements of second harmonic terms might be of limited interest in analytical applications as noted by Bauer .I5 Interference from the double layer capacity is minimized for second harmonic terms because rectification due to the double layer capacity (as a result of the non-linearity of the charge on the electrode versus potential relationship) is in general smaller than for the charge transfer reaction. Bauer’s suggestion could be extended to application of the faradaic impedence method to fast reactions but the limitation imposed by the cell resistance is the same for the fundamental frequency and the second harmonic terms. Further, measurement of second harmonic terms requires a generator whose output is essentially free of the second harmonic.
General Equation for the Rectification Voltage.Application of eq. 5 requires the derivation of the defined by eq. 4. explicit form of the quantity 60~1 The partial derivatives a t E , in the latter equation are readily written from the I-E characteristic, and only the 6’s must be derived. It is to be noted that the terms in b21/bCo2,d21/dC~2and b21/bCo ~ C vanish R when the I-E characteristic is of the first order with respect to CO and CR since these derivatives are then equal to zero. The quantities &E, GoCo and &CR can be expressed in terms of elements of the equivalent circuit for the electrode reaction as shown in Appendix. If the I-E characteristic of eq. 1 holds, the rectification voltage
ret
Ys
Ys
+ +
= YO+
10
YR
IR
(10) (11)
rct being the charge transfer resistance per unit area.13 Values of the r’s and y’s in eq. 9 can be taken directly from the classical theory of faradaic impedance. Equation 9 reduces to the result derived by Doss and Agarwal,2b Barker, Faircloth and G a r d ~ ~ and e r ~Vdovin6 ~ for a simple charge transfer reaction. These authors did not write their results in the general form of eq. 9 but used the rate constant a t the equilibrium potential (not the apparent standard rate constant). They assumed that the current density is I A sin wt and thus postulated implicitly that the mean current is equal to zero. They wrote the poA E m - VAsin ( u t - 0,) where A E , is the tential as E. rectification voltage and the minus sign for VA sin (wt - 0A) results from the convention of considering a cathodic current as positive. They finally introduced in eq. l these values of the current density and E and the values of the concentrations and Og taken from the classical theory of the faradaic impedance and expanded the exponentials, the expansion being limited to the first three terms. The resulting equation was solved for AE,. The mean rectification current was derived by Matsuda and Delahays before the general considerations which lead to eq. 6 had been put forward. They correlated the concentrations in eq. 1 for the I-E Characteristic to the current density byle
+
where the current density I ( t ) is some function of time and u is the integration variable. The F sign corresponds to substances 0 and R, respectively. The potential in eq. 1 for the I-E characteristic was set as E. - V Asin w t , and it thus was implicitly assumed that the mean potential is controlled a t E,. There resulted an integral equation in I j t ) which was solved by expansion in series of the exponentials. Expansion, limited to the f i s t degree terms, yielded two integral equations whose solution corresponded to the classical theory of faradaic impedance. Consideration of a n additional term in the expansion yielded another integral equation whose solution gave an equation identical to eq. 7 ( ~ O5EAi?, ). It is to be noted that a clear distinction between the two types of control, namely, simultaneous control of one of the mean components and one of the alternating components of the electrical variables was not made by these various investigators.
(14) H. Gerischer and W. Vielstich, 2. physik. Chem. ( F v a n k f w t ) ,8 , 16
(1956).
(15) H.
H.Bauer, J . Eleclroanal. Chem., 1, 266 (1960).
(18) H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” Oxford University Press, London, 1947, p. 57. eq. 9.
316
PAULDELAHAY, MITSUGI SENDA AND CARLH. WEIS
Influence of the Double Layer Capacity for Control of f a t Zero No Rectification from the Double Layer.Variation of the charge of the double layer was neglected above, and it was assumed that f = 0 for the faradaic process, whereas actually the total current density is controlled a t zero. The previous analysis must be modified accordingly. It will be assumed first that the differential capacity of the double layer c1 is independent of potential for a small departure from E,, i.e., rectification due to the non-linearity of the chargepotential relationship for the double layer will be neglected a t first. I t will also be assumed that c1 is solely determined by the supporting electrolyte and is not affected by variations of COand CR.
VoI. 83
where &-is given by eq. 9 and G,H =
(15)
Onesan shQw from eq. 14 that AE = 0 for t = 0 and AE + AE, for t + 03 (Fig. 2). Mass transfer polarization for the mean concentrations CO a-nd CR thus vanishes progressively, and the value AE-, as calculated for f = 0, is ultimately attained. Further, variations of AE/AE, with time are not a@cted by frequency. One also can show- that AE = AE, for t > 0 when CI = 0 and that AE = 0 for t 2 0 when CI + 03. Further, one deduces from eq. 14 after expansion of the error functions that & varies linearly with l / t 1 / 2for large arguments, namelv
I
(16)
The lowest values of t for which this approximation holds decrease with an increasing apparent standard rate cogstant kaOfor given eo's; and the limiting value A E m is practically reached in a short time (perhaps a few milliseconds) for inherently fast reaction and sufficiently concentrated solutions. 3 : 2 x IO-^ The time dependence of AI? for a given reaction and 4 : 4 XiO+ a given Coo 4- CR' also depends on the ratio COO/CRO 0 as one would expect from the for? of the parameters I_-_ __L. --_I-__ I G and H of eq. 15. Likewise, AEIAE, depends on 0 IO 20 30 40 50 Coo CRO for a given COO/CROall other conditions TIME (rnillisec) Fig. 2.-Plot of A E / A E , against time for different total being identical (Fig. 2). The foregoing analysis also holds for the decay CRO (expressed as the parameter concentrations Coo n2(Coo CRO)where the Co's are in mole l.-l) and for COO/CRO curve for f f i when the alternating-current is inter- 1, kao = 1 cm. see.-', Do = DR = 10 -6 cm.2 sec.-l, rupted after the limiting value A E m is practically attained. The right-hand member of eq. 14 is now CI = 20 microfarads an.-* equal to-AEm - AE. This conclusion holds only The variation of the mean component of the when AE- is reached at the time of interruption charge p on the electrode is 609 = (dq/dE)6& or since the previous analysis presupposes no mass 60g = c i 6oE. One also has 60p = SOIdt where transfer polarization for the mean concentrations variation of the alternating currezt. 601is the varying mean current supplied by the before The foregoing conclusion that AEIAE, is indecharge transfer reaction for charging of the double pendent of V A is ~ strikingly verified in Fig. 3 for layer over the interval &E. The mean charging the Ti(IV)-Ti(IKI) charge transfer process. l7 The current is such that (see eq. 3 and 5) shape of-the A E against t curve was not altered when AE- varied in the ratio 1 to 20. Further, the decay curves have the same shape as the build'I BOCR (13) up curves (upside down). Note the effect of noise 3 CR the highest sensitivity. - Vecification of the freThe changes in concentrations 60Coand SOCRcan at quency-independence of AE/AE, against t is given be correlated to the current density by eq. 12, and in Fig. 4 for the discharge of Hg(1) on mercury.'* an equation is obtained which can be solved for Thejnflupnce of the concentrations Cooand CROon 6d. Since Sd dt = CIS& the mean rectification the AEIAE, against t curve is shown in Fig.-5 for voltage AI? (= 6&) is obtained by integration of the Ti(1V)-Ti(II1) couple. Note that AE- is 6d. There follows after substitution of bI/bE, approached more slowly when (l/Coo Do'/I) bI/bCo and dI/bCR by the corresponding values ( 1 / & DR'/*) increases as one would expect from eq. 14 (see its simplified form in eq. 16 which is deduced from eq. 1 however not applicable here in all rigor).
Y
-1
+
+
+
$d
+
AE=
A E m {1
-
I exp(H2t)erfc(HP/s)]
1
(14)
(17) For a faradaic impedance study of this couple, see J. E. R Randles and K. W.Somerton, Trans. Faraday SOC.,48, 937 (1952). (18) For studies of this process by the double pulse galvanostatic method, see H. Gerischer and M. Krause, 2.physik. Chem. (Frankfurt), 14, 184 (1968): H. Matsuda, S. Oka and P. De?ahay, TEISJOURNAL. 81, 5077 (1959).
Jan. 20, 1961
FARADAIC RECTIFICATION AND ELECTRODE PROCESSES
317
TIME Fig. 3.-Tracings of oscillograms of Ahe uerms t for 1.4 0.5 M m M Ti(1V) plus 18.1 mM Ti(II1) in 0.88 M HZSO~, tartaric acid for different applied voltages VA. Ripples are due to hum and noise.
TIME Fig. 4.-Tracings of oscillograms of A e against t for discharge of 0.08 m M Hg(1) in 1.14 M HClO, for two frequencies. Ripples are due t o hum and noise.
Rectification from the Double Layer.-The foregoing treatment will now be modified to take into account rectification due to the double layer. As was shown by Barker,4c variations of the differential capacity CI of the double layer with potential results in a shift of the mean potential
when the mean non-faradaic current is equal to zero. There VA is the amplitude of the alternating voltage across cl. The shift A E i is frequency independent if (l/cl)(dcl/dE) does not depend on frequency. This seems to be so for the mercury electrode a t frequencies studied thus far, ;.e., u p t o 1-2 megacycles per sec. for the electrolytes studied by Barker. 4c If there is a charge transfer reaction, the net shift of mean potential is a cogbination of the shift of potential of eq. 14 and AEi of eq. 17. Th_e term between brackets in eq. 13 now includes AEi, and eq. 14 is replaced by A E = AEm
- (AEm -
It folloys from_ eq. 18 that and AE = AEmfor t -+ m .
=
AEi a t
t = 0,
The shift of mean potential upon_interruption of the alternating current when A E m is practically reached is -AEi. These conclusions fromJheory are_borne out experimentally (Fig. 4). AEi and A E m may have the same or opposite signs. Influence of Circuit Resistance for Control of E at E,.-
The analysis is similar to the treatment of the potentiostatic voltage-step method given by Vielstich and Delahay.lQ We now include the ohmic drop r & I ( r total resistance of Fell and circuit for the measurement of 1,)in writing eq. 6, %.e.
(19) W. Vielstich and P. Delahay, Tars
JOURNAL,
19, 1874 (1957)
5
0
IO
T I M E(mil1isec.) Fig. 5.-Tracings of oscillograms of A E / A E m against t for Ti(1V)-Ti(II1) in 0.88 Af H,SO( and 0.5 M tartaric acid Concentrations for curves 1 to 5: 6.2, 17.4, 1.3, 0.61, and 1.4 m M Ti(1V); 13.3, 2.1, 18.2, 1.35, and 0.55 mMTi(II1). (~/CRODR”~) in 108 sec.1’2 mole-’ Values of (I/CooDO1’2) cm.2 for curves 1 to 5: 1.1, 2.8, 4.0, 11.4, 12.0 for DT,(IV) ctm2 see.-’ A b and A E m were = DT,(III)= 0.41 X positive for curve 2 and negative for the other curves. Ripples due to hum and noise not represented.
+
The mean rectification current density is
1, = Y+--
1 RT I,O n F
exp ( X ’ z t ) erfc ( X ‘ L * / Z )
(20)
with
and previous discussionis of this result applies here. f, is primarily determined by the circuit resistance r and not by the kinetics of the charge transfer reaction when Y is appreciably larger than the charge transfer resistance ( R T / nF)(l/Iao). This is the case for processes with large exchange current densities, and consequently direct measurement of rectification current is of limited usefulness for fast reactions. The rectification voltage, however, can be determined as was shown by Barker‘ by the following method which finds
PAULDELAHAY, MITSUGISENDA AND CARLH. WEIS
318
though theory was developedjor values of VA not exceeding a few millivolts, AE, is proportional to V A for ~ values of V Aexceeding even 50 mv.
-lob
-
.--E P In
I
I I
-lLL lr !
.01 .02
.05
~~
TABLE I S'ARIATIONSOF A,??, WITH V A FOR 0.7 m M Ti(1V) PLUS 18.8 m M Ti(II1) IN 0.88 A[ H2SO4AXD 0.5 AI TARTARIC ACID AT DIFFERENT FREQUENCIES
0
~
IW
1
?Iz
1 -.011!-I.-.-
001
Vol. 83
L
A
0.1
I
0.01
15 28
0.42 1.3
1.93 1.65
1.79
.02
13 24 37
0.27 1.0 2.4
1.63 1.75 1.78
1.72
.05
14 22 36 67
n.30 0.80 2.3
1.67
7.0
1.32 1.74 1.83 1.57
.1
13 40 85
0.28 2.6 12
1.37 1.58 1.66
1.60
.2
8.8 26 40 63
0.20 1.0 2.8 6.4
2.57 1.51 1.71 1.63
1.86
.5
8.9 16 30 48
0.14 0.36 1.3 4.0
1.82 1.50 1.49 1.73
1.59
5.9 13 18
0.05 .24 .8
1.43 1.35 2.54
1.77
I
J ;
.k2-
IO
CELL VOLTAGE (v.)
Fig. G.-Taliations of A E , with cell voltage (log-log plot) for 0.7 m M Ti(TV) plus 18.8 mlll Ti(II1) in 0.88 HZSOI and 0.5 IIf tartaric acid a t different frequencies (in megacycles per sec.). Note that all curves have a slope of i.
its justification in eq. 21: the rectification current is compared at a sufficiently short time t with the current (at the same time t ) which results from the application of a voltagestep t o the cell in the absence of alternating current; the voltage-step is equal to the rectification voltage when the two currents are equal a t time t.
Rectification Voltage in the Absence of Coupled Chemical Reaction. Influence of Applied Voltage.-Values of rs and ys derived for the faradaic impedance will now be introduced in eq. 10 and 11. According to Warburg,20RandlesZ1and Ershler22one has (i =0 or R) where D 1 is the diffusion coefficient of species i. Further, yi = 1 lacs, cs being the capacity of the series equivalent circuit. Combination of eq. 9 to 11 and 22 then yield
where ea is the apparent phase shift (no double layer correction) between current and voltage, ;.e. ctn 3,
= w1',c,
(21)
It follows that AE, = 0 when C o 5 D ~ 1= / z (2:. D R 1 I 2 and a = 0.5, as one-would expect. One also infers from eq. 23 that AE, is proportional to V A ~ . If the cell resistance r is such that r >> Z and Y >> l/wcI, AE, is practically proportional to the square note that of_the cell voltage (Fig. 6 and Table AE, for a given ITA is virtually frequency-independent in Table I). It is worth noting that al(20) E. Warburg. A n n . Physik, 67, 493 (1899); 6, 125 (1901). (21) J. E. B. Randles, Discussions Favaday Soc., 1, 11 (1947). (22) (a) B. V. Ershler, ibid., 1, 269 (1917). (b) R. V. Ershler, Zizuv. fia. K h i m . , 22, 683 (1948). (c) IC. Rozental and D. V . Rrshler, ibid., 22, 1344 (1948). (23) It was assumed in t h e calculation of V A t h a t 1lwei Z,Z being t h e faradaic impedance. I n this instance, C I = 27.0 microfarads. em.-' as measured with a bridge a t 250 cycles per sec.
>Z is approached, and AEw /IA' should vary with l/w, as is indeed the case for the lower ~ pure diffusion frequencies in Fig. 8. Further, A E / I A for control should be independent of the Hg( I ) concentration according to eq. 2 5 . This is indeed the case for th_e three lowest concentrations in Fig. 8. EXtPdpOlatiOn of AE m / VA' to w = 0 for the three lowest concentrations gave the limiting value AE/VA' = 20 f 5 volt-' for an assumed differential cdpacitp c1 of 160 microfarad. ern.+. This is the limiting ~ nF/4RT for w = O and n = 2 for a diffusionvalue A E / V A = controlled process. No accurate verification was possible since c1 could not be measured in this case. As order of magnitude one can set c1 = 100 farad. crn.-' from double pulse galvanostatic measurements.'* The frequency dependence for processes nearly controlled by charge transfer was not studied here because the upper frequency of 2 megacycles per sec. of the available instrument was not sufficiently high for accurate study of Hg(1) discharge. This matter will be taken up when an instrument for higher frequencies now under construction is available.
- 5 5-__
6
7
L O G w (sac-')
Fig. %--Plot of log ( A E m / 1 1 * )against log w for the discharge of Hg(1) a t different concentrations (mM per liter) on mercury in 1.14 M HC104. A E , was negative for all concentrations.
CURVE I 2
3 4
5 6 7 8
_,i
a 1 .5 .2 .I
.OS
.oe
1
.01
L
2
4
0
-2
$Ee-
-4 E, J
Fig. 9.-Plot of (4RT/n17)(AEm/V~2) against (nF/RT). ( E o - E'/z) for different values of ~ , O / ( ~ U D ) D ' / ~being , the common value of D O and DR, and for CY = 0.25.
A E , / V A ~for w (coo Do'/?
+0
-
varies with
+
CnoDR1/2)/(CooDO'/' CnoDn1/2)
This quantity approaches =t1 for Ee+ =t and is equal to zero at_the polarographic half-wave potential. Hence A E , / V A ~= i n F / 4 R T f o r E , + f and w + Q Equation 25 applies for w + a , and one has A E w / V A ?. 0 for a-2 0.5. An intermediate curve having the lines aE,/VA3 = *nF/4RT as asymptotes is obtained for mixed control of the faradaic impedance by mass and charge transfer (Fig. 9). All curves intersect a t the point having the ordinate given by eq. 26 and the abscissa obeying the equation
1 nF Variations of AEm/V A with ~ the Ratio CO~/CR~.tanh [5 (E, = (2a - 1) (28) Barker4 examined the variations of a quantity which is proportional to A E , / V A ~ with the ratio The parameter CY thus can be computed in two ways Coo C i (i.e. with the equilibrium potential). from the coordinates of the intersection point. Two limiting cases for which eq. 25 and 26 apply, Experimental curves for the Ti(1V)-Ti(II1) respectively, will be considered: w + 0 and w + 0. couple in Fig. 10 have the same general shape as
PAULD E L A H A Y , MITSUGI SENDA AND CARLH. WEIS
320 -0.30
Ee ( V . V S . $ C.E 1 -0.25 -0.20~ -0.15~
~
! P
_
Vol. 83
called thickness of the diffuse double layer and ro and {R are parameters depending on the double layer structure _eq. 4~ (cf. and 21 in ref. 25b). The limit of this additional term should also be included in the value of A,!?m for CY m (see eq. 2 6 ) . (Note that Y ~ / ( Y * y.*) -+ ( n F / R T ) I , , for Q! -+ m .) Further analysis of this problem in this Labordtory has shown that the above correction holds only when (0 and are sufficiently small to satisfy the conditions stated in ref. 25s and 26b.
+
--f
r~
Rectification Voltage for a Charge Transfer Reaction with Preceding Chemical Reaction. This case, which was discussed by Barker4c on the basis of an absolutely equivalent circuit (by the use of transmission lines), is readily analyzed from the general equation 9. The values of r J s and y’s derived by Gerischer26for the faradaic impedance in this instance must now be used. We only consider the first order or pseudo-first order reaction X a 0 preceding charge transfer since equations for higher order reactions can be linearized for a small departure from equilibrium. The following term c T ~IV)/cT~(!II) (
Fig. IO.-Plot of A E , / v ~ * against C T ~ ( I V , / ~ T , ( Ifor I I ) must be added to Y., and 7 0 and the term 1 1 1 (w’ Z’)‘/Z - 1 Ti(1V) - Ti(II1) in 0.88 M HzSOd and 0.5 M tartaric acid ( yrp ) 0 - RT (nF)2 2‘12 Co0D1/zK ~2 12 for different frequencies (in megacycles per sec.). CT~(IV)O c T i ( I I I ) o = 19.5 rnM per liter. Experimental points a t the must be added to y s and yo. The values of 7R and common intersection point nearly coincide for all frequencies YR remain unchanged (cf. eq. 22) since R is not inand are not indicated for the sake of clarity. The same volved in a chemical reaction. Notations in eq. 29 remark also holds for some of the other measurements. and 30 are: D the common value of the diffusion coefficients of X and 0 supposed to be equal, K =
+
the theoretical curves of Fig. 9. The experimental curves also exhibit a single intersection point, from whose coordinates one calculates a! = 0.25 (abscissa) and a! = 0.46. No conclusive interpretation can be given for this discrepancy, but there are a t least two possible explanations : the influence of adsorbed tartaric acid which complicates the interpretation and the possibility of a chemical reaction preceding or following the charge transfer reaction (see below). Correction for the Double Layer Structure in the Absence of Specific Adsorption and beyond the Frumkin Theory.Values of the Y’S and y’s which were introduced in eq. 9 and which led to eq. 23 for AEw were taken from the classical theory of the faradaic impedance, L e . , the structure of the double layer was not considered in the boundary value problem in the derivation of AEm. The approximation is quite crude since the attenuation factor (equivalent to the diffusion layer thickness) for the concentration waves, (2D/w)’/z, is not extremely large in comparison with the diffuse double layer thickness, e.g., (2D/w)I/z ;3 3 X cm. for D = 10- cm.2 sec.-l andf = 1 megacycle per sec. This effect of the double layer, which is beyond the Frumkin 4 analyzed in correction for the double layer s t r u c t ~ r e , ~was this Laboratory.26 The correction for A E m was already discussed by Matsuda and Delahay,G and their results will be briefly reviewed in the light of the general theory presented here. Correction is made by introduction in eq. 9 of the values of Y’S and y’s derived by Matsuda.2bb Equation 23 is still valid but its right-hand member now includes as a first approximation the additional term vA2 Q!rO - __ _--
[
- --
;r
2n F K C O ~ D O ‘ : c , 0 2 R 1 1.2 y.2 where i band ya are given by eq. 10 and 11, 1 / ~ is the so(24) For a recent review see Chapter l by R. Parsons in ”Advances in Electrochemistry and Electrochemical Engineering,” P. Delahay and C . W. Tobias, editors, Interscience Publishing Co., New York, N.Y.,in course of publication. See also Chapter 2 by A. N. Frumkin (25) (a) H. Matsuda and P. Delahay. J . Phys. C l e m . . 64, 332 (1960). (b) H. Matsuda, ibid., 64,339 (1960).
- - - A -
{
+
+
+
CoO/CxOthe equilibrium constant, CQ = COO exo, and 1 = kr f k b , kf and k b being the rate 0. constants for X The reaction polarization resistance (rrp)O of eq. 29
is frequency independent in the limiting case corresponding to w > 1. Further, ( r r p ) ~and ( y r p ) o approach zero for 1 + 0. If the general eq. 9 or its particular form in eq. 23 is applied without consideration of a cheinical reaction, the measured transfer coefficient, as deduced may be in from eq. 26 by extrapolation with l,’~’’~, error. If the charge transfer resistance rct is such that rct >> ( r r p ) o a t the highest frequencies being used, the error on a! is negligible. Conversely, when rct < < (7rp)o at the highest frequenciesbeing investi(- 1/4) gated, one deduces from eq. 9 that AE, (nF/RT)V Afor ~ w + m , ;.e., a! appareiztly approaches zero for the upper limit of W’S being used when the effect of the chemical reaction is overlooked. This conclusion is in agreement with Barker’s observat i ~ that n ~ abnormally ~ low apparent transfer coefficients are obtained for discharge of certain metal ions. This discussion can readily be extended t o charge transfer processes preceded by a heterogeneous chemical reaction since there is a formal analogy between the treatment of this case and the above analysis. This extension is valid provided that steady state is reached for the heterogeneous chemical reaction. -f
Limitations Resulting from the Cell Resistance Limitations are imposed by two conditions for the time t a t which A E m is measured: t must be long enough to allow (26) H. Gerischer. Z. p h y s i k . Chem., 198, 286 (1951).
Jan.20, 1961
FARADAIC RECTIFICATION AND ELECTRODE PROCESSES
A & o t o be reached but 1 must be short enough to prevent undesirable local heating of the solution. Corrections could possibly be worked out when these two conditions are not satisfied, and the limit of measurable 1 - 0 3 could be increased. Attainment of A E m does not raise any problem for fast reactions since the charging of the double layer is then very rapid under proper conditions. Further, the alternating cell current must practically attain its steady-state value. Steady state is reached within 1% for t > 5 rc, 7 being the total resistance of the circuit connected to the alternating current generator and c, for control a t I = 0, the capacity equivalent to the series combination of the double layer differential capacity and the capacity in series with the cell (see the end of the Qualitative Discussion). The latter capacity which generally should not exceed 0.01 microfarad for a cell with a dropping mercury electrode is determinative, and the condition for steady state attainment for usual values of r ( ~= 100 ohms) is t > 5 microseconds. This is not a stringent condition. The upper limit of t is generally determined by the maximum allowable rise in temperature of the electrolyte near the electrode. The change in equilibrium potential due to temperature rise must indeed be small in comparison with A E m . Since I E A m l < ( ~ F / ~ R T ) V and A ~VA < 0.005 volt (this limit might possibly be raised), one has(AE-1 = 100n to 200%microvolts. Further IdEe/dtl < 1 millivolt deg.-l in general, and consequently one should have AT < 0.001 0.01 degree. A conservative estimate of the maximum allowable time for measurement is readily obtained on the assumption that there is no heat transfer in solution and between solution and the electrode. One has for an electrode with uniform current density d T/dt = 0.24 ( ZA'/~)( l/Ke&Ch) a t the electrode surface, K. being the specific conductance of the electrolyte, C h its specific heat and 60 its density. For instance, for a 1 molar aqueous solution of a strong acid deg. sec.-l one has a t room temperature dT/dt = 0.3 or for ZA = 1 amp. cm.a (an order of magnitude forf = 1 megacycle per sec.), dT/dt = 0.3 deg. sec.-l. Since one should have AT < 0.001 to 0.01 degree, A E m must be measured within 3 to 30 milliseconds in this particular case. Such measurements are made by pulse techniques. A correction could of course be made for the temperature change, but it would require the solution of the boundary value problem for heat transfer. The large ohmic drop for the alternating current is eliminated in the measurement of A E m by means of a low pass filter. Requirements for this filter become more stringent as the resistance increases as one can readily show. Thus, A B - is proportional to V A ~L e, . , to the square of the alternating voltage V applied to the cell provided that Y >>Z and r > > I/wcl, r being the cell resistance and CI the differential capacity of the double layer. The output alternating voltage of a one-section inductance-capacity filter Vout in its cut off range is proportional to the input voltage and inversely proportional to w m where usually m = 2. Hence AE/Vout is proportional to V and inversely proportional to I * .
Experimental Solutions.Solutions were prepared from analytical grade reagents (except for Ti(II1)) and bidistilled water (over KMnO,). The stock solution of Hg(1) in perchloric acid was standardized by electrogravimetry after reduction of traces of Hg(I1) by mercury. Traces of adsorbable organic impurities were removed by treatment with purified charcoal according to Barker.& Charcoal was purified by a two-week extraction with HC1 in a Soxhlet extractor (without cartridge) followed by a one-week extraction with frequently renewed distilled water. The Ti(1V)-Tl(II1) solutions were prepared from a 20% Tic18 aqueous solution (Matheson Co., Inc.). This solution contained an appreciable amount of zinc as was found by polarography but this impurity did not interfere. The titanium solution was analyzed as follows: aliquot sample was oxidized by H?Ol; chloride was removed by heating with HzSO, to fuming; Ti(1V) was finally reduced to Ti(111) and titrated with KMmO, in a nitrogen atmosphere. The Ti(IV)/Ti(III) ratio was changed by complete reduction with zinc amalgam followed by partial oxidation by bubbling of oxygen. Finer control was achieved by the mixing of completely oxidized and reduced solutions. The Ti(IV)/Ti(III) ratio was determined in situ in the cell by pokrography from the cathodic and anodic diffusion
32 1
currents. The ratio obtained in this fashion agreed well with the ratios calculated from equilibrium potentials. Cell.-A polarographic H-cell was used with dropping mercury electrode and mercury pool in the same arm. The other arm of the cell was filled with a 1 M solution of ammonium nitrate (for the Hgp++ study) or 0.88 M sulfuric acid (for the Ti study) and connected with a saturated KC1 bridge to a saturated calomel electrode. Precipitation of potassium perchlorate or potassium hydrogen tartrate was thus avoided. The cell was immersed in a water-bath, the water of which was made conducting by salt addition. This provided adequate shielding for the cell. The dropping mercury electrode was fitted with a magnetic hammer for synchronization of drop fall with oscilloscopic recordings. Current-Potential Curves .-A Sargent polarograph model XXI was used according to conventional practice. Recordings were made with the saturated calomel electrode (and not the Hg-pool) as reference electrode. Double Layer Differential Capacities.-Measurements were made according to Graha~ne.~' Rectification Voltage .-Rectification voltages were measured during a short single pulse for each consecutive drop a t the dropping mercury electrode. Oscilloscopic display of the rectification voltage was synchronized with the fall of the mercury drop by means of magnetic hammer and a timing device. The output of a Hewlett-Packard oscillator model 200 CD (up to 0.6 Mc) or General Radio moduoscillator model 1211-B (above 0.5 Mc) was 1 0 0 ~ o lated by a Tektronix preamplifier 53-54 C used as gate. The modulating pulse obtained from a Tektronix pulse generator model 163 was properly synchronized with the drop fall. The Tektronix preamplifier 53-54 C was inserted in a Tektronix oscilloscope model 536, and a 15 volt modulated signal was thus obtained. This signal was further amplified with a Tektronix amplifier model 112 to approximately a 100 volt signal. The latter was applied to a conventional power amplifier using two 6AS7G tubes in parallel. The maximum output was 5 volts across 50 ohms a t frequencies up to 2 megacycles per sec. Rectification voltages were measured with a Tektronix oscilloscope model 531 equipped with a preamplifier model 53-54 E. The maximum sensitivity was 50 microvolts per cm. The low pass filter between the cell and the input of the preamplifier was composed of General Radio adjustable decade capacitor and inductance boxes, model 1419-K and 1490-C, respectively. The settings on these boxes were ad'usted according to the frequency. In general, values of A d m was read on the screen of the oscilloscope. A number of photographs were also made. The alternating voltage across the cell was displayed during the single pulse measurement on a Tektronix oscilloscope model 535 with preamplifier model 53-54 G. Experimental Conditions.-For the Hgp++ study: cell resistance, 52 ohms (mostly due to the mercury column of d.m.e.); m for d.m.e., 1.01 mg. sec.-l; 7 without synchronization, = 6 sec.; 7 synchronized, 5.02 sec.; t at the beginning of r.f. pulse, 3.7 or 1.3 sec. (to provide areas and 1.03 X lo-* cm.2 in the ratio 2 : l ) . of 2.06 X For the Ti(1V)-Ti(II1) system: cell resistance, 56 ohms: m for d.m.e., 1.65 mg. sec.-l; T without synchronization = 6 sec.; T synchronized, 5.02 sec.; t a t the beginning of r.f. pulse, 3.7 sec. The temperature was 30 0.1' in all experiments.
Conclusion A general theory of faradaic rectification was developed, and it was shown that both the mean and the alternating components of potential and/or current must be controlled. Transients due to charging of the double layer and to rectification by the double layer are quite essential in the interpretation of rectification voltage measurements. The difficulties associated with the double layer capacity and ohmic drop in other relaxation methods are largely eliminated in the application of faradaic rectification to very fast reactions. The (27) (1848).
D. C. Graham+. THISJOURNAL, 68, 301 (1846); 71, 2876
322
PAUL
DELAHAY, &fITSUGI
SENDB AND
method has two unique features in that (a) conditions in which the cell alternating current is primarily controlled by the double layer capacity do and O( but in fact not prevent measurement of Iao are desirable (A.V. control), and (b) frequency discrimination by means of a low pass filter in the measurement of AE- eliminates the difficulty resulting from a large ohmic drop and allows the determination of Ia0and a even when the cell resistance is very much larger than the charge transfer resistance. An upper limit of the k2's which could be determined can be set only after further evaluation but it appears that values of kao up to 10 to 100 cm. sec.-l could be measured. The upper limit is based on the fairly reasonable assumption that frequencies up to 100 megacycles per sec. can be used (perhaps with some difficulties). The question arises a t very high frequencies whether formulas derived from Fick's equation for diffusion still apply since motion can no longer be regarded as completely randomized. Such considerations, however, need not be introduced below 100 megacycles per sec. (or perhaps even higher frequencies) since the average path of a diffusing particle (D = l o p 5 cm.2 for t = 10-8 sec. is still quite large in comparison with molecular dimensions. Acknowledgments.-This investigation was supported in part by the Office of Naval Research and the National Science Foundation. One of US (C. H. W.) is also indebted to the Brazilian Research Council and the Brazilian Commission for Improvement of Personnel with Higher Degrees for support which made possible his stay in this Laboratory and to the Technical Institute of Aeronautics of Sao Jose dos Campos, Brazil, for a leave of absence. We wish to thank Dr. G. C. Barker (Harwell, England) for several useful comments on this paper. Appendix Derivation of Equation 9.-As noted above, eq. 9 is deduced from eq. 5 in which 6oZIis derived from eq. 4. Further, the terms in a21/bCo2,~ ~ I / and C Rd ~ 2 1 / d C o d C ~in l the case here, that the eq. 4 vanish if one assumes, as w ~ lbe I-E characteristic is linear in the concentrations of 0 and R (cf. eq. 1). The problem thus is to express 6Co and &CR of eq. 4 in terms of the resistive and capacitive elements of the series equivalent circuit for the faradaic impedance. Quite generally one has with the conventions of sign or eq. 1
CARL H. WEIS
Vol. 83
where r4 and y~are the resistive and capacitive elements of the series equivalent circuit for the faradaic impedance (cf. eq. 10 and 11). Further, if one assumes that the kinetic parameters for the supply and/or removal of 0 and R (diffusion, homogeneous or heterogeneous chemical reaction, etc.) are independent of potential, one has for species i ( & O or R )
SICi
= 3=
[(h,)i
- j(h,)liS
(32)
I I
where the q= sign holds for 0 and R , respectively, and h, and hu are constants which depend only on the processes for supply and/or removal of 0 and R between the bulk of solution and the electrode. By introduction of SlCi from eq. 32 into
one obtains after re-arrangement of terms
(34)
or 611
= -
&E
ret
+ + yo TR
-~(YR
+
YO)
(35)
where r o t , y o , r R , Y R , yo are defined in conjunction with eq. 10 and 11. The values of the r's and y's are readily written by comparison of eq. 34 and 35. By introducing in eq. 32 the r's and y's thus obtained one derives for species i (i = 0 or R ) 61Ci = [ ( b I / B E ) / ( B I / d C i )(ri ]
- jyi)81I
(36)
or after combination with eq. 31.
A general equation for 6021 of eq. 5 can thus be written from eq. 4 and 37. Equation 9 was derived by substitution of these values corresponding to eq. 1 for the partial derivatives a t E,: b I / b E = -I,o(nF/RT), bI/dCo = Iao/Coo, d I / d c R E -Iao/Cno, bZI/bE2 = Iao[CY2- (1 - C Y ) " , T), = bzI/dEaCo = - ( I & Q / C O Q ) / ( a n F / Rd21/aEdCn ( I a " / C ~ " ) [(l a)nF/RT], -(aI/bE)/(aI/bCo) = (nF/ RT)CoO and - ( a I / b E ) / ( a I / a c R ) = -(~F/RT)CRO.
