ELECTIZOI)E PROCESSES WITH ComEcuww: CHARGE TIZAMFER STEPS
623
Current-Potential Characteristics for Electrode Processes Iinvolving Consecutive Charge Transfer Steps with Nonspecific Adsorption of Reactants and Products'
by David M. Mohilner Department o j Chemistry, Cniwrsity o j Piltsburoh, I'iltsburgh 13, I'ennsylaania
(Receiwd .June 13, 1BL;S)
Current-potential equations are derived for electrode processes involving consecutive charge transfer steps with nonspecific adsorption of reactants and products. When the rate of the over-all process is controlled by one of the consecutive charge transfer steps, the current--potential equation, including double layer correction, may always be put' i n the form of the classical Frumkin equation for an electrode process involving a single charge transfer step, provided that an o v e w d l transfer coegicient is defined. Therefore. in the region of potentials over which one charge transfer step remains rate controlling. it is not, in general, possible to determine from the current-potential characteristic alone whether or not an electrode process involves consecutive charge transfer steps. For the caw of mixed control by several charge transfer steps, the current-potential equation may be expressed as a product of the current density which would be obser\.ed if one charge transfer step were rate controlling (Frumkin equation) and a correction factor which depends only on the relative heights of the various standard electrochemical free energy barriers. This equation for mixed control predicts the existence of the so-called "pseudo-Tafel" regions discussed recently and explains under what conditions such regions may be expected.
'Theoretical discussions of current-potential characixristics for electrode processes involving consecutive charge traiisfer steps have appeared previously in the litc:rature.*-j These treatments do not' consider explicitly the influence of the double layer structure on tht: kiiictics of the electrode process. In the preseiit pa,per, the explicit form of the double layer correction in thc absence of specific adsorption of reactants arid products will be derived. Particular attention will be callcd to the special case in which one of the charge transfer steps is rate controlling. I{'oi*this case, it will be s h o w tliat tlic form of t8he electrochemical rat'(> equation may always hc reduced to t,liat of the classical lcriimkiti cxluation,fiprovided that the symmetry ,factor7 for t h r highest standard electrochemical free energy barrier d o n g thc reactioii path is replaced by a11 over-all biansjer~coeficienl, properly defined. I t will be shown further that i n the general case of mixed control by two or more charge transfer st,eps the rate equation may
always be expressed as a product of a I'rumkin rate multiplied by a correction factor which depends only 011 the relatii-e heights of the various standard electrochemical free energy barriers.
Control by One Charge Transfer Step We follow the method of I'arso~is.~1,et the net electrode reaction be givcn by
DAVIDAI. NOBILNER
6 24
Cv,OtLt
+ ne --+C I
1
U ~ R ~ (1) ~ ~
Here Otz*is the cheniica,l symbol for the ith reactant of ionic charge zt, and u t is the absolute value of its stoichiometric coefficient in the DcIIonder s e n s e . 8 Rjz, is the chemical symbol for the j t h product of ionic charge z,, and u j is the absolute value of it>sstoichiometric c~efficient.~The total number of electrons involved in the over-all process as determined by equilibrium measurements is n. I& the stoichiometric number’ be u . Then, when one mole of the act,ivat,ed complex corresponding to the highest, standard electrochemical free energy barrier along the reaction path is formed and decomposed, reaction 1 occurs thus
(l/u)Cv,O,z‘ i
+ (n/u)e -+-(l/v)Cv,R,‘i
(2)
I
Assume that one or more charge transfer steps occurs before t,he rate controlling step. I& np be the total number of electrons involved in t,hesr, preceding charge transfer steps. Then the net reaction required to transfer the system from the initial state to the state immediately preceding the rate controlling step is (l/u)Cu,Otzi t
+ n,e
(4)
+ n,e -+ ( l / / v ) C v j R j Z ~
+
[(n/v) - np]e (electrode) 111. State Immedaately Folloiuing Rate Controlling Step
z~,,J’,,~m
+ n,c (elcctrode)
(outcr Ilclmholtz plane)
m
11‘. Fanal &’%ate
v j R j 2(bulk ~ of solution)
(1/ u ) 7
The corresponding standard clectrochemical free energies of the system in each of these states ale -_ GI’ = ( I / v ) C v p p t o ( n / u ) i i c
+
1
CvliiLo + k
_I_
=
GI’
-
--
=
GIIIO
[ ( n l u ) - n,]p,
(6)
Cvmpm0 + n3p, m
C V ~ G ) ~(3)‘ ~
Here P m z m is the mth member of *the set of intermcdiat,es formed by the step, zm is its ionic charge, and urn is the absolute value of its stoichiometric coefficient. Finally, following the rate controlling step, one or more charge transfer steps may occur. I A n, be the tot.al number of elect,rons involved in these succeeding steps. Then the net reaction required to transfer the system from the state immediat8ely following the rate controlling charge transfer step to the final state is
--
= (liu)zv3p,o
GIY’
I
Here po is the standard electrochemical potential of thc co~istituents,and pr is the standard electrochtmical potential of the electrons in the elcctrode. (We shall designate the standard “chemical” potentials by the same symbols without the bar.) IRt $z and $’I rcprcsent thr inner potciitials (with rrspect to thc bulk of the solution) a t the outcr Helmholtz plane arid in the metal clcctrode. respectively. Thcii the “chemical” arid “clcctrical” parts of the standard elcctrochemical free cnergy of thc system in these four states arc Chemical Parts GIo
+
= (1iv)CvrFt’ 1
G I o
= CvapnO k
Qrrr’
=
GI””
+
(n/v)Pr
-
[(77,/~)
CUmPm0 + m
npIkr
(‘7)
%Po
= (l/v)CvlF1’ I
Electrical Parts
(5)
3
We consider nom the following four st~atesof the system :
I. In,itial State
( ~ / u ) C V(bulk ~ Oof~solut,ion) ~ ’ + (n/u)e (elect,rode) i
(outer IIeImholta plane)
--+ k
C V ~+ Q n,e ~ -+ ~CV,P,~~ ~ k m
m
uXQILk
li
~
Here Q c z k is the chemical symbol for the kth member of the set of intermediates formed immediately before the rate controlling charge transfer step, zli is the ionic charge of this constituent, arid v k is the absolute value of its stoichiometric coefficient. Let na be the number of electrons involved in the rate controlling charge transfer step. Then the rate controlling reaction is
Cv,~’,zm
11. State Immedaatelg Precedang Rate Controllang Slep
618
= -(?abJ)F$’l
~~~~
(8) C‘f. I. I’rigopine and IC, Defay, ”Therniodynaniiqiie Chirniqrie.” Editions Desoer. Liege, 1950, p. 10. (9) We define the set,* of iiitearul mhwripts [ i } , ( j l , (kl. tlnd ( m ) as noniritersertinp. Thus specification of any variable by one of these sutxwripts is uiianibig~ious.
625
ELECTRODE PROCESSES WITH CONSECUTIVE CHARGE TRANSFER STEPS
-
Cv,iz, = C v m z m + nn k m
G I I I ~= ( z v m z r n ) F + E - n,F+"
Give
=
Cvmzm=
0
m
(In eq. 8 F is the value of the faraday.) The standard electrochemical free energy of the system in each of the four states is the sum of the corresponding chemical and electrical parts. Let G* O be the standard electrochemical free energy of i,he system in the transition state corresponding to the summit of the standard electrochLemica1 free energy barrier for the rate controlling charge transfer step. Let G*" and G*e be the corresponding chemical and electrical parts of the standard electrochemical free energy of the system in this activated state. Then the syncnaetry factor for the highest standard electrochemical free energy barrier @ (0 < @ < l),is defined' by Gae - G I I ~= B(GIII~- G I I ~ )
(9)
Equations 8 and 9 give
G+e - G I I ~= -PnaF+?
+ Pn,F#"
G*e - G I I I ~= (1 - P)%F& - (1 - P)n,F+hf (11)
aG*"
Let = G T - G z be the standard electrochemical free energy of activation in the forward (cathodic) u
direction, and let AG1;O = G * O - GI' and AG*e = G*" - G I e be its chemical and electrical parts. Similarly, __
L__
~
define AG*O = G*O -- GIvO, AG*" = G * O - GIvO, and A G e = G*e - G I V for ~ the backward (anodic)
__
-.
AGee
=
+
L -
(FVA- Bna)F& +
By conservation of charge we have (cf. eq. 2-5) =
np
(17)
Using eq. 15 and 16, eq. 12 and 13 become _ A
AG*~ =
[ ( I / V ) C-Wnp ~ - pn,lF+?+ 2
(pnR
+
nP)F@M
(l8)
LA
A G * ~= [ ( i / v ) C ~ ~-znp ~ - pnaIF+2 2
[(I - P)na
+ n s I ~ +(19) ~
We recall that the corresponding equations for a simple electrode process involving only a single charge transfer step (with n/v electrons) and the same reactants and products would be _ A
AG*e = ( T ~ z i Bn)Fh/v 4- pnFch'/v = (Fvtz,
(18')
- pn F&/v - (I - /?)nF@M/v
)
(19') It has been stated frequently that current-potential characteristics for electrode processes with consecutive charge transfer steps, but for which one of these steps (involving na electrons) is rate controlling, depend on the electrode potential in exactly the same way as do electrode processes with a single charge transfer step except that the factor pn/v in the exponential involving overpotential is replaced by pn,. lo The implication of such statements is that one can thereby determine the number of electrons involved in the rate determining step, or a t least the product Pna,by analysis of the current-potential curves. However, comparison of eq. 18 and 19 with eq. 18' and 19' shows that this statement cannot, in general, be valid because of the appearance of the terms n p and n, in the former pair of equations. What is determinable from current-potential characteristics for such electrode processes may be seen in the following manner. We define a new kinetic parameter cy' which Jye shall call the over-all transfer coeficient. Let a' =
n/v
+ nu
I
AG*", and direction. By definition AG*O = AG*O -__AG*" = AG*' AG*". The electrical parts of the standard electrochemjcal free energies of activation for the forward and backward directions are then given by (cf.eq. 8, 10, and 11)
+
?
L -
(10)
Likiemise
__
(l/y)Cvgz,
(16)
+ n, + n,
(14)
(pna
+ np)/(n/v!
(20)
(10) Csually, instead of the symbol P , the symbol a called the "transfer coefficient" is employed. However, t h e meaning of a In such statements IS clearly identical with t h a t of the symmetry factor 8 as defined above. Such transfer coefficients should not he confused with the over-ail transfer coefficient a' defined below,
Volume 68,Number S
March, 10s.4
DAVIDM. MOHILNER
626
Note that a’, like p, is also a number lying between zero and unity (0 < 01‘ < 1). Substitution of eq. 20 into eq. 18 and 19 yields
(1 - a’)nF+M/v (19”) The last pair of equations are now of precisely the same form, including the double layer correction, as eq. 18’ and 19’ which pertain to a n electrode process involving only a single charge transfer step. However, the significance of the two pairs of equations is quite different. In eq. 18’ and 19’ P, being a true symmetry factor, gives direct information about the nature of the standard electrochemical free energy barrier for the rate controlling charge transfer step. On the other hand, a’, the over-all transfer coefficient which appears in eq. 18” and 19”, is not a simple characteristic of the standard electrochemical free energy barrier for the rate controlling step. Rather, a’ depends on both the true symmetry factor p and on the number of electrons involved in other (fast) charge transfer steps which occur in the sequence of the over-all reaction. The value of a’ is therefore necessarily less informative about the nature of the rate controlling charge transfer step than the valae of would be, were it determinable. Unfortunately, eq. 18’’ and 19” show that neither p nor n, are, in general, determinable from an analysis of the current-potential curves. All that such an analysis (for example, Tafel slopes) can yield are the number of electrons involved in the over-all process (nlu) and the over-all transfer coefficient, a’. Exceptions to this last statement will occur only in the particular cases when either the first charge transfer step in the cathodic direction 01 the first charge transfer step in the anodic direction happens to be rate controlling. I n the first case (n, = 0) the slope of the corrected cathodic Tafel line11 will yield the product an, = a’nlv, and it will then be possible to assign a lower limit to n, and an upper limit to p. I n the second case (n, = 0) the slope of the corrected anodic Tafel line will yield the product (I - p)n, = (1 - a’)n/u. Such determinations, however, require independent study of intermediates in order to ensure that the first charge transfer step is indeed rate controlling.12 The current density-overpotential equation is easily derived from eq. IS” and 19” using standard methods.7 In the absence of mass transfer polarization, this equation is T h e ’ J o w a l of Physical Chemistry
I = I,O{exp(-a‘nFq/vRT) exp[(1 - a ’ ) n ~ q / u ~x~ ] l [exp(a’n - ~ zv + z ~ ) F A + ~ / v R(21) T] Here I is the current density (net cathodic current considered positive), I 2 is the apparent exchange current density, ?I is the overpotential, and A+2 = +2(q) +Z(q = 0 ) . This equation for an electrode process with consecutive charge transfer steps is identical in form with the Frumkin equation for an electrode process with only a single charge transfer step, except that a’ replaces p. The apparent exchange current density I 2 , which includes a double layer correction term, is related to the true exchange current density I o by the equation
I,O = Ioexp(a’n
[
-
CU~Z~= ) FO)/uRT] +~(~ z
(22)
and
I o = (n/u)k”F(lp,‘j)(’
-
a’)/v(qalY,)a’/U
(23)
I n the last equation ko is the standard rate constant for the over-all electrode process, and ut and ajrare the activities of the corresponding constituents in the bulk of the solution. Clearly, neither the exchange current density I o nor the standard rate constant ICo for electrode processes with consecutive charge transfer steps is so directly related to the energetics of the rate controlling charge transfer step as are the corresponding quantities for an electrode process involving only one charge transfer step. Rather, these parameters, like a’, pertain, in a rather complicated way, to the over-all process. Thus considerable caution should be exercised in assigning mechanistic significance to the experimental values of these parameters. I n the case of mixed control by mass transfer and charge transfer eq. 21 becomes
I
=
L
I,O II(iii/a,)”’/”exp(- a’nFv/vRT) -
+ a’)nFq/vRT 1 [exp(a’n) -
rI(dj/aJPf’/”exp(l 3
\
Here tii and dl are the activities of constituents O t l and Rjzl just outside the diffuse double layer,13 and the other quantities are defined as above. (11) For information on the procedure for plotting Tafel lines corrected for double layer structure in the absence of specific adsorption see K. dsada, P. Delahay, and A. K. Sundaram, J . ilm. Chem. SOC., 83, 3396 (1961). (12) See for example, D. M. Sfohilner, R. N. Adams, and m. J. Argersinger, Jr., ibid., 84, 3618 (1962).
I’hECTRODE PROCESSES WITH COSSECUTIVE C H A R G E
TRANSFER STFWS
Mixed Control by Several Charge Transfer Steps
(I/’.)
Again, let reaction 1 be the net electrochemical reaction as determined by equilibrium measurements. A~sunieIIOW, however, that no single charge transfer stcap is rate controllirig; rather there is mixed control by two or more of the corisccutive charge transfer stcps. .Issume further that a steady state is achieved foi the production of all intermediates. [,et the total number of charge transfer steps in the rcstctiori sequence be r . At the equilibrium potential roi*rcsponding to a given solution composition we have for one step, say the mth p =
-
1, . . . , T
(25)
627
+
C v r O t Z(outer T-TeIrnholtz plane) 1
nle (elcctrode) 1-II I. Stat e I mmcd ia t ely Following Tra nsi t i on State
C V ~ (outcr ~ X IIclriiho~tz ~ ~ A planr) k
plh Charge Tramjer S t e p (1
< p < r)
Stat e Imriiedia tely I’receding
p-I. State
Transit ion
cuk,- 1XA2k(outer Helmholtz plane) + k
n,e (rlrctrodt)
--
where G*,O is the standard electrochemical _free _ energy of thc transition state for the mth, and G*,” is the standard electrochemical free energy of the transition state for the pth charge trarisfcr step. We choose the with ch:irge transfer step as the refercrice reaction and define vas the stoichiometric number with respect to that step. l‘his rncaiis that, for steady state, every time the overall rcaction goes onrc, the mth reaction goes v times. Then the over-all reaction which occurs every time the nit 7 rcactioii goes once is given by eq. 2 . ‘I‘hc set of consecutivc charge transfer steps making up thc sequcrice inay hc written as follows
+
( 1 / u ) C v t O t 2 ~ nle a
--+
CvllXxzk k
p-11. State
State Imrncdiately Following Transition v L p X I L k(outer Hrlmholtz planc) k
rth C h a q e Transfer Step r-I. State
Xuk k
State Immediatrly l’rcccdjng
r- lXLzk
r-I1 State
(outer Hclmholtz planr)
Trarisition
+ n,r (clcrtrode )
State Inimediately E’ollo\ving ’I’ransi Lion (1,’v)C v1IijZ~(outer IIclmtioltz plane) I
k ~ V X , -.
T-111. Final State
k
1XAZkf n,c --+ (l/u)CvjRjzl
k
3
1Tri.e the sets of rm,ctarits, Oi“1, and products, R,”, anc thr corrcsponding sets ofstoichiometric coefficients, v i and v,, ai*cdefined as before. The set of X,’h is the set of all intermediate constituents, zk is the ionic charge of the iith intwmediate, and Vkp designates the nunher of moles of this coiistituerit produced in the pth and consumed in the ( p 1)th chaige transfer
+
Wc now establish forward (cathodic) and backward (anodic) tlectrochernical rate constants for each of the r re:ictions by the method of I’arsoris.’ For this purpose con4dcr the following states of the system: 1.d (’harge ‘I’ransjw Step
Initial State
1-1 (1
v ) C v , 0 L 2(bulk ~ of solution) + nle (electrode) 1
1-11 State Immed iately Preceding Transition State
(I ,’v)c
(bulk of solution)
3
Standard clcctm~hemical and standard chcmical free energies of activation are defiiied for cach of the r chargr trailsfel. steps in tcrnis of thcb states listed above. The electrical parts of thc standard tlectrochcrnica1 free ericrgics of the systrni i i i the various statcs are designated by a sct of C valucs. ‘i’hcn thc symmetry factors (p) of the standard electrocli~~~nical f i w energy barriers for each of thc r charge tra~isfcrsteps arc dcfined by the equations
wherci I , is given by cq. 21 with a’ = ani’ and 1,” = I.,%‘]. III the casc of inabb trarisfer polarization eq. 3 will remain valid (provided thcw is still a s t d y state with rcbpcct to the production of intcrmcdiatcs), G*rP- G - i P = /~,(G-II‘’ - G r - ~ r ) but Irnwill thrn be given t)y cq. 24. The particular form of rcl. i1.1 is useful 111 euplaiiiing Using these symmetry factors, clcctrochcmical rate how thc control in electrode processes involving conconstarits for the forward (cattiodir) anti t)ackward secuti\,e rliarge transfer steps may shift with over(anodic) for the r charge transfer steps are dcrircd. potcntial and in rxplaining the csktcnce of the so‘l’hese rate constarits are given hy a set of li values arid called “pscudo-Tafel lines” discussed by EIurd.3 Suppose, for exaniplc, that a t the cqiiilibrium potcntial a set of k values. Applying the steady state condicorrcsponding to a given solutiori composition thr diftion, wc obtain the following equation for the nct ferences i n the heights of the stariclard c~lcctrochrmical current density. frrr cnrrgy barriers t)rtwceri t h c uzth and t hc. otlicr ,--rm - 1 i charge transfer steps arc sufficiently gwat to make Jc neai-ly equal to unity. l’hw the rict curreiit density i n a crrtain neighborhood of o\-crpotentials around the cqui1it)rium potential will b(1 given csscntially by I,, p = t n + l i . e . , eq. 21 will be valid, and the rate of thc over-all reaction will bc controlled by the ?nth charge trarisfcr step. For cxamplr, if therr are only two trans_ _ charge fer steps i n the scqucnce, and if ((:*1O - c‘ r*L ”) = .5.3RT, thc rrlativc error in approxirnating I hy cq. 21 with the first chargr transfer step rate controlling In analogy to the case of control t)y one charge transwould be only about 0.3%. The deprnderict of the fer step, we now define an ocer-all lransler coejicwnt corrcction factor J? on the corrected ovcrpotcritialll with Tespect lo the inth ciiargr transfer step am’, Let (7 - Ad2), is casily obtained. In the case of two m - I consecutive charge transfer steps (with V L = I ) this p = l dependrnce is given by the term, cxp( [ ( l - P 1 ) ? ~ 1 ,3,n,]P(q - Ad2) R T } . (’onsidcr now onc of thr cases Substituting cq. 20 into eq. 28 it is seen that the prodcited by Hurdd-lo.p1 = pL = 0.5, n1 = n2 = 1, and uct of (n/v)P’rnultiplicd by the first term in braces on ( n V ) = 2 . 111 this case the dcpendcricr of j c on the the right-hand side of cq. 28 is identical with the current d ovcrpotciritial is giveii by exp[(q - A&)I“/ density which would lie observed if the mth rhargc transfer step were rate controlling (cj. eq. 21). We R T ] . If (G‘*lo - (;*20) = 5.:3RT a t the equi1it)rium shall designate this current density by I , and the corpotential we firid that, for l ( q - A&) ’ < 118 mv. the responding apparcnt exchange current density by lamo. rclativc error in approximating I by eq. 21 with the That is, Inmo is the exchange current density which first chargr transfer step rate controlling will still be would be ohserved if the mth reaction were rate cononly about 4%. Tlie corresponding error i n log 1 will he less than 2y6 of one logarithmic cycle. In the trolling (6. eq, ’22). The remaining tcrm on t h r righthand side of cq. 28 is thus the corrcction factor whirh region 00 niv. < ( q - A&) < 118 mv. eq. 21 shons that takes account of thc fact that there is mixed control. a well drfinrd anodic ‘l’afel line would be observed. This type of Tafel line has been called a [‘pseudoThis correction factor is casily shown to depend only on thr diffcrcnres in thr hcights of the standard elcctro‘t’afel line” on the grounds that its slope does not yield the correct valuc for thr symmrtry factor p, for t h e chemical f r w energy barriers brtwceii the iiztli aiid the sccond chargr transfor step. However, the slopc of such rmiaining chargc transfer steps. I d J, designate the corrrction factor, thrn a ‘rafcl line does give the corrcct value for thc over-all transfer coefficient a’. (In this casc a’ = 0.25.) -1 Iurtliermore, the sum of the dopes of the anodic and P - 1 cathodic ’I’afcl lines cquals (n ’ v ) ) ~2.XRY‘ which is thc T h e current density which will be obsrrvcd in thc case usual criterion for tfeciding that one is deuling with of mixed control is thrn given by “tru(”’ Tafcl lines for ail electrode procrss involving only oiic charge transfer step. For all values of the rquilibA
L
--
+
ELECTRODE PROCESSES WITH COSSECUTIVE CHARGETRASSFER STEPS
riuirn potential (solution compositilon) for which the difl erence in the heights of the standard electrochemical free energy barriers remains large (enough such Tafel lines will be observed, and they will be parallel as Hurd’s computer calculations indicated. At overpotentials greater than a certain upper limit (118 mv. in the example cited), the approximation of I by eq. 21 will begin to fail noticeably, and there will be a negative deviation from the extirapolated “pseudoTa’el” line, En this region there is mixed control by two, or more, of the consecutive charge transfer steps. The value of the correction factor j c is then given by the ratio of the current on the extrapolated “pseudoTafel” line divided by the observid current. This region corresponds to the curved portions of Hurd’s graphs. Eventually, at still higher overpotentials, control may shift entirely to another charge transfer step and a second Tafel line (called 1% “true Tafel line” by Hurd) will be observed. However, the slope of this second Tafel line mill not necessarily yield the symme Lry factor of‘ the standard electrochemical free energy barrier for the new rate controlling step: this would be true only if the new rate controlling step happens to be r,he first (or last) charge transfer step in the sequence (cf. discussion on control by one charge transfer step above). Whether or not such shifts in control can actially be observed will depend on whether or not maw transfer polarization can be eliminated, or corrected for, at high overpotentials.
Coriclusion Current-potential equations for electrode processes involving consecutive transfer steps with nonspecific adsorption of reactants and products have been derived for the cases of control by one charge transfer step and mixed control by several charge transfer
629
steps. In the case of control by one charge transfer step the current-potential characteristic, including double layer correction, may always be put in the form of the classical Frumkin equation provided that an over-all transfer coefficient a’ for the process is properly defined. In the region of electrode potentials over which one charge transfer step remains rate controlling it is therefoie impossible to ascertain, from the currentpotential characteristic alone, whether or not the electrode process involves one, or several consecutive, charge transfer step(s) . In the case of mixed control by two or more consecutive charge transfer steps, the current density is given by the product IJ,, where I , is the current density which would be observed if the inth charge transfer step were rate controlling, and f c is a correction factor depending only on the differences in height between the standard electrochemical free energy barriers for the mth and the other charge transfer steps. If one charge transfer step is rate controlling at the equilibrium potential, and in a sufficiently wide neighborhood of overpotentials around the equilibrium potential, Tafel lines of the type which have been termed “pseudoTafel lines” will be observed. However, these Tafel lines are “pseudo-” only in the sense that their slopes give correct values for the over-all transfer coefficent and not for the symmetry factor for the rate controlling step. At sufficiently high overpotentials, provided that control is not taken over completely by mass transfer, negative deviations from such Tafel lines indicate mixed control, and the correction factor f c in the current-potential equation may be evaluated. Acknowledgment. The author is indebted to Professor Joseph Jordan of the Pennsylvania State University for his interest and discussion of this work.
Volume 6‘8, .\-timber
S
X a r c h , 1964
