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Electrochemical reaction rates in terms of electrochemical affinities and potentials-case of hydrogen overvoltage
Electrochimica Acta, 1963, Vol. 8, pp. 583 to 588. Pergamon Press Ltd. Printed in Northern Ireland
ELECTROCHEMICAL REACTION RATES IN TERMS OF ELECTROCHEMICAL AFFINITIES A N D POTENTIALS-CASE OF H Y D R O G E N OVERVOLTAGE* P. VAN RYSSELBERGHE Departments of Chemistry and Chemical Engineering, Stanford University, Stanford, California, U.S.A. Abstract----On the basis of previously presented ideas, rate expressions are written for the various possible rate-determining steps of the hydrogen overvoltage mechanism in terms of the corresponding electrochemical affinities and potentials. This is done for two types of cases, according to whether the overall electrochemical affinity is concentrated in one step or distributed between two or more steps. The equality of the anodie and cathodic transfer coefficients is shown to imply the local equilibrium Ha ~ H + H + + e- at the actual reaction sites, this step being otherwise the rate-determining one of the Heyrovsky-Horiuti mechanism. The cases of both transfer coefficients equal to 1 or to 1[2 are examined. The presentation is to be regarded as covering only some aspects of a general treatment characterized by the use of Marcelin-De Donder or M D formulae. Rgmma6--Sur la base d'id6es pr6sent6es pr6c6demment on 6crit les vitesses r6actionnelles des diff6rentes 6tapes r6gulatrices possibles do m6canisme de la surtension d'hydrog6ne en fonction des affinit6s et potentiels 61ectrochimiques correspondants. On examine deux esp6ces de cas, selon que raffinit6 61ectrochimique globale se concentre en lane 6tape ou se distribue entre deux ou plusieurs 6tapes. On montre que l'6galit6 des coefficients de transfert anodique et cathodique implique un 6quilibre local Ha ~ H + H + + e- dans la couche interphase, r6action qui d'autre part est l'6tape r6gulatrice du m6canisme de Heyrovsky et Horiuti. On examine les cas off les coefficients de transfert sont tous deux 6gaux /t 1 ou ~t 1/2. On ne doit consid6rer la pr6sentation que comme couvrant seulement certains aspects d'un traitement g6n6ral caract6ris6 par l'emploi de formules de Marceiin-De Donder ou formules MD. Zusammenfassung--Auf der Grundlage yon friiher entwickelten Vorstellungen werden die Gesehwindigkeiten der versehiedenen miSglichen geschwindigkeitsbestimmenden Teilschritte des Mechanismus der Wasserstoffiibersparmung in Abh~agigkeit von den Affmit~ten mad yon den entsprechenden elektrochemischen Potentialen angeschrieben. Man betrachtet zwei Arten yon F~llen, je nachdem die globale elektrochemische Affmiffit sich auf einen Teilschritt konzentriert oder sich auf zwei oder mehrere Teilschritte verteilt. Man zeigt, dass die Gleichheit der anodischen und kathodischen Durchtrittsfaktoren ein ~rtliches Gleichgewicht H2 ~ H + H + + e- in der Phasengrenzschicht voraussetzt, eine Reaktion, die anderseits der geschwindigkeitsbestimmende Teilschritt des Mechanismus von Heyrovsky-Horiuti ist. Es werden die F/ille, in denen die Durchtrittsfaktoren gleich 1 oder 1/2 sind, betrachtet. Die DarsteUung erstreckt sich nur auf einige Aspekte einer allgemeinen Methode, welche durch die Verwendung der Formein yon Marcelin-De Donder (Formelu MD) charakterisiert ist. IN several earlier writings t we have called a t t e n t i o n to t h e desirability o f writing electrochemical rate expressions in terms o f t h e electrochemical p o t e n t i a l s o f t h e reacting species at t h e a c t u a l r e a c t i o n sites in t h e i n t e r p h a s e regions. W e have also s h o w n z h o w a general expression f o r t h e rate o f a n e l e m e n t a r y reaction, such as one o f t h e steps in a steady-state m e c h a n i s m , c a n be written in t e r m s o f t h e f o r w a r d a n d reverse affinities o f t h e e l e m e n t a r y reaction, this general expression yielding t h e two limiting types o f rate expressions: (1) P r o p o r t i o n a l i t y to a n e x p o n e n t i a l f u n c t i o n o f t h e excess o f t h e f o r w a r d affinity o f t h e e l e m e n t a r y step over its e q u i l i b r i u m value, w h e n * Manuscript received 25 February 1963. 583
584
P. VAN RYSSELBERGHE
the reverse rate is negligible. (2) Proportionality to the small overall affinity of a socalled equilibrium step. If the mechanism involves at least one step of each type we have, for the overall rate v (v~. . . . are the stoiehiometric numbers of these steps; 2~. . . . are functions of temperature and pressure ~but not_. of composition; A~ and A~. . . . are the forward and reverse affinities, with A~ = A~ -- A~; A~ = Ai, . . . . are the equilibrium values of these affinities; ~ = t~ . . . . are the forward and reverse rates at equilibrium; R is the molar gas constant; T is the absolute temperature),
v = vdv ~ = (1/vi2i)" [exp ( A d R T ) -- exp CAJRT)] = (~,dv,) "{exp [ ( A ~ - A , , ) / R T l -- exp [(A, -- A,,)/RT]}
= vj/v, = (1/v,2~) • exp (A~/RT) = (/~,b'j) " exp [(X, -- ~ , ) [ R T ] =
vd,
=
(AdRr).
(1)
Recently8 we have discussed current Tafel-like expressions for the rates of electrode processes and have shown how formulae of the above type (to which we give the general name Marcelin-De Donder or M D formulae) can be used, with electrochemical affinities replacing chemical ones, as first steps in the construction of a coherent theory of electrode kinetics. The pertinence of M D kinetics in electrochemistry is immediately evident when one recalls that electrochemical affinities ~, are equal to corresponding overvoltages ~ multiplied by the reaction charge zF 4, J, = zF,7.
(2)
A detailed treatment of electrode kinetics based upon the use of M D formulae is in preparation but we wish to present at this time two aspects of its application to the typical problem of hydrogen overvoltage: (1) Rate expressions when the overall electrochemical affinity is "concentrated" in a single step of the mechanism, all other steps having negligible electrochemical affinities (2) M D interpretation of the cases in which the anodic and cathodic parameters ~ and/~, in a Tafel expression are equal to each other and, particularly, equal to 1 or to 1[2. 1. M D
A N D T A F E L E X P R E S S I O N S F O R STEPS C O N C E N T R A T I N G THE OVERALL ELECTROCHEMICAL AFFINITY
We shah consider cases for which it is experimentally found that the net current density I, for a given electrode metal and a given solution composition, is related to the overvoltage ~7by the Tafel expression I = I0[exp (~aF~7/RT) -- exp (--~eF~7[RT)],
(3)
with I > 0 and ~7 > 0 when the net current is anodic, I < 0 and ~7 < 0 when it is cathodic. The exchange current I o may contain a composition dependence which we shall not look into here. The parameters fla and fl~ are related to the transfer coefficients c% and 0% the charge number z and the stoichiometric number of the ratedetermining step i, as follows, =
zl,,,,
Fo =
zl,,.
(4)
Electrochemical affinitiesand potentials--ca~ of hydrogen overvoltage
585
Calling I the bulk of the metal, I ' the gaseous hydrogen phase and I I the bulk of the solution, the overall reaction is H2(I') --~ 2 H + ( I I ) + 2 e - ( I ) .
(5)
The reaction charge is 2F. The overall electrochemical affinity and the overvoltage are related by .~ = p r _ 2fi~+ -- 2fi[- = 2F~,
(6)
the p's and fi's being chemical and electrochemical potentials. At equilibrium (indicated by the subscript e) we have, the 9's being inner electric potentials, •g- = 0
and
(9 ~ -- 9~)~ = (1/2F)-(2p~+ + 2p~- -- p~,)
(7)
With I different from zero we have 91 _ 9u = (~i _ 9n), + ~/.
(8)
The mechanism is, essentially, the following one (hydration, hydride formation, etc. are regarded as complete equilibrium steps implicitly taken into account) in which the absence of phase specification indicates that the particular reaction site is being considered: 1.
H2(/')-* H 2
v1 :
2. 3. 4. 5.
H~--+2H H - + H + + eH + -+ H~-(II) e---~e-(/)
~'~ = 1 va=2 v4 = 2 v5 = 2.
1
(9)
Let us assume that one of these steps, say i, is such that *,i'Ai=A
with
,~=0(j:#i).
(10)
Actually the Al's are very small and such that (see formula (1)) (v~,/~,j) " ( A ~ / R T )
=
(ii)
v.
We have I = 2F" vffv,,
(12)
I o = 2F" ~ , / , ,
and, identifying the M D formula with (3), we obtain A,--A,6=fl..F~/
and
A,--A,~=--flc.
Fr/.
03)
It follows that •g'i = A~ -- A, = (ft. + flc).F~ -----(o% + 0%).A/v,
(14)
and hence that fl.+flc=2/vi
and
0 % + 0 % = 1.
(15)
Let us note, however, that these additivity properties do not necessarily hold when
586
P. VAN RYSSSELBEKGHE
part o f the overall electrochemical affinity resides in steps other t h a n i. It will now easily be found that: 1. W h e n A1 = -~ we have: fl~ = ~ = 0, fl~ ---- 2, 0~ = 1. 2. W h e n ~-z =
we have the same situation.
3. W h e n -~3 = .A/2 we have: fla = ~a---- 0, fl~ = ~ ---- 1. 4. W h e n -~a = A/2 we have: fl~ = ~ -----1, fl~ = ~ = 0. 5. W h e n -~5 = A/2 we have: fl~ = ~ = 0, fit = 0c, = 1. W e see that, in cases 1, 2, 3 a n d 5, the anodic c o m p o n e n t of the current is independent of the overvoltage and remains equal to the exchange current, while in case 4 it is the cathodic c o m p o n e n t which remains constant and equal to the exchange current. Case 1 does not a p p e a r likely to occur. Case 2 would correspond to a pure Tafel m e c h a n i s m in the cathodic direction. Case 3 would be o f the Volmer type with the whole overvoltage involved in the cathodic direction. Case 4 is unlikely to occur as a pure case but a non-zero value of the electrochemical affinity o f this step m a y well have to be considered in other mechanisms. Case 5 is interesting: it would m a k e the whole overvoltage correspond to a w o r k of extraction of the electron f r o m the metal to the reacting site or vice-versa. T h a t a portion of the overvoltage might correspond to such a step in other m e c h a n i s m s seems very probable. T h e M D f o r m u l a for case 5 is vs = v~,. {exp [(ge- -- ge-,)l R T ] -- exp [(g~- -- gr~-,)IRT] }, (16) and we see that, since biot- - - fi~-, ---- - - F . (90r -- ~0J) = --Fr/,
(17)
we have fie- -= fie-e : the electrochemical potential o f the electron at the reaction site remains constant as the overvoltage varies. W e shall n o w show that it is possible to have b o t h fla and fl, different f r o m zero in f o r m u l a (3) if, instead o f being concentrated in a single step o f the mechanism, the overall electrochemical affinity is distributed between two or m o r e steps. 2. M D A N D T A F E L E X P R E S S I O N S F O R STEPS I N V O L V I N G ELECTROCHEMICAL AFFINITIES DIFFERENT FROM THE O V E R A L L ONE
F o r m u l a e (13), (14) and (15) already give the essential information pertaining to this type of case: (1 3) and (1 4) r e m a i n true if the Tafel-like behaviour represented b y f o r m u l a (3) is actually observed. O n the other h a n d agreement with the two formulae (15) is no m o r e required. Let us n o w take the particular case fla = fl~ = fl and write the M D rate expression for step 3, va : ~3e" (exp [ ~ H - - / t H e ) / R T ] - - exp [(/~H+ + /~e- --/~a+, -- fte-e)/RT]},
(18)
with
~H÷ + ~e- --/~H+, --,/Te-, = /TH+ + /7~- --/ ~r / 2 = --tiFf I = --flA/2.
(19)
Electrochemical affinitiesand potentials--caseof hydrogen overvoltage
587
It follows that / ~ , = / Z a + fin+ + fie-.
(20)
This i m p l i e s / ~ =/~H~ and we thus have, at the reaction sites, the electrochemical equilibrium H 2 ~ H + H + + e-,
(21)
a most interesting implication of the equality of fl~ and/3,. Let us note also that reaction (21) is the sum of steps 2 and 3 of the mechanism (9). It is also the ratedetermining step in the Heyrovsky-Horiuti mechanism. We shall come back to this point below. Since ~,s=Aa--A3=flA
and
As+A3=0
(22)
we have A~ = " f l A ,
A2 + 2A3 -- flA, A1 + 2.~4 + 2.~5 = (1 -- p)..~.
(23)
If fl = 1 the overall electrochemical affinity is concentrated in steps 2 and 3 and the sum ~'1 + 2A~ + 2A5 is equal to zero, each term being zero or, less likely, the indicated compensation occurring. If, following a rather frequent practice or having obtained, in some hypothetical experimental case, the corresponding evidence, we were to put fla = fl, -- 1/2 (and thus also in this case % = 0% = 1/2), the equilibrium condition (20) would of course still hold and we would have, From (22) and (23)
A3 = ---A2 = A/2,
-~2 + 2A3 = A/2,
,~1 + 2-~4 + 2,~5 = A/2.
(24)
Let us assume ,~1 and A4 equal to zero. We then have ~'5 = fie- -- fi~e- = A / 4 = F~I/2.
(25)
We submit that a very attractive interpretation of the % = 0% = 1/2 possibility has hereby been obtained. Let us, finally, write the M D expression for the rate v2' of reaction (21) in the Heyrovsky-Horiuti mechanism v2' = 3,,'. {exp [ ~ n , -- ~H~,)/RT] -- exp {[(#H + fia+ + fie-) -- (,IXH, + fia+e + fio-,)]/RT)}.
(26)
If steps 1, 3, 4 and 5 (identical to the corresponding ones of mechanism (9)) are assumed to be at equilibrium we have /)2 t
= ~2e '- {1 -- exp [(2/xH -- tz~,)/RT]} = v~," {1 -- exp (--2F~1/RT)}
(27)
and we are led back to the pure Tafel mechanism considered in Section 1. This, however, would not occur if a part of the overall affinity A = 2F~ is assigned to one or several other steps. The M D method should be a safe and straightforward guide in the exploration of such possible mixed mechanisms.
588
P. VAN RYSSmJSERGHE
Our previous writings on the M D method and the present discussion of a somewhat idealized hydrogen overvoltage case will show, we hope, that this method presents distinct advantages over the empirical or semi-empirical adoption of certain Tafel-like equations. Beyond the level of inquiry at which the foregoing paper has been kept the method is capable of handling considerable refinements. The composition dependence of the exchange current, the role of activation affinities, the significance of transfer coefficients, the examination of cases for which no constant transfer coefficients exist but M D formulae are available, etc. are allproblems for which solutions can be provided or at least suggested. The main advantage of the method is its close connection with thermodynamics, in both its reversible and irreversible aspects. It is necessarily phenomenological and somewhat formal but it establishes connections with mechanistic and molecular models and assumptions at points in the developments at which the introduction of such devices involves a minimum of danger. REFERENCES 1. P. VAN RYSSELIIERGHE,J. Chem. Phys. 17, 1229 (1949); 20, 1522 (1952); 3". chim. phys. 49, C47 (1952); Proc. 8th Meeting Int. Comm. Electrochem. Thermodynam. and Kinet. (Madrid 1956) p. 405. Butterworths, London (1958). 2. P. VAN RYSS~n~ZOI~,J. Chem. Phys. 29, 640 (1958). 3. P. VAN RYSLqm.nERoI-IE,R. C. Accad. Lincei, 31, 391 (1961). 4. P. VAN RYSS~n~ZOHE,Electrochemical A~inity. Hermann, Paris (1955).
ELECTROCHEMICAL REACTION RATES IN TERMS OF ELECTROCHEMICAL AFFINITIES A N D POTENTIALS-CASE OF H Y D R O G E N OVERVOLTAGE* P. VAN RYSSELBERGHE Departments of Chemistry and Chemical Engineering, Stanford University, Stanford, California, U.S.A. Abstract----On the basis of previously presented ideas, rate expressions are written for the various possible rate-determining steps of the hydrogen overvoltage mechanism in terms of the corresponding electrochemical affinities and potentials. This is done for two types of cases, according to whether the overall electrochemical affinity is concentrated in one step or distributed between two or more steps. The equality of the anodie and cathodic transfer coefficients is shown to imply the local equilibrium Ha ~ H + H + + e- at the actual reaction sites, this step being otherwise the rate-determining one of the Heyrovsky-Horiuti mechanism. The cases of both transfer coefficients equal to 1 or to 1[2 are examined. The presentation is to be regarded as covering only some aspects of a general treatment characterized by the use of Marcelin-De Donder or M D formulae. Rgmma6--Sur la base d'id6es pr6sent6es pr6c6demment on 6crit les vitesses r6actionnelles des diff6rentes 6tapes r6gulatrices possibles do m6canisme de la surtension d'hydrog6ne en fonction des affinit6s et potentiels 61ectrochimiques correspondants. On examine deux esp6ces de cas, selon que raffinit6 61ectrochimique globale se concentre en lane 6tape ou se distribue entre deux ou plusieurs 6tapes. On montre que l'6galit6 des coefficients de transfert anodique et cathodique implique un 6quilibre local Ha ~ H + H + + e- dans la couche interphase, r6action qui d'autre part est l'6tape r6gulatrice du m6canisme de Heyrovsky et Horiuti. On examine les cas off les coefficients de transfert sont tous deux 6gaux /t 1 ou ~t 1/2. On ne doit consid6rer la pr6sentation que comme couvrant seulement certains aspects d'un traitement g6n6ral caract6ris6 par l'emploi de formules de Marceiin-De Donder ou formules MD. Zusammenfassung--Auf der Grundlage yon friiher entwickelten Vorstellungen werden die Gesehwindigkeiten der versehiedenen miSglichen geschwindigkeitsbestimmenden Teilschritte des Mechanismus der Wasserstoffiibersparmung in Abh~agigkeit von den Affmit~ten mad yon den entsprechenden elektrochemischen Potentialen angeschrieben. Man betrachtet zwei Arten yon F~llen, je nachdem die globale elektrochemische Affmiffit sich auf einen Teilschritt konzentriert oder sich auf zwei oder mehrere Teilschritte verteilt. Man zeigt, dass die Gleichheit der anodischen und kathodischen Durchtrittsfaktoren ein ~rtliches Gleichgewicht H2 ~ H + H + + e- in der Phasengrenzschicht voraussetzt, eine Reaktion, die anderseits der geschwindigkeitsbestimmende Teilschritt des Mechanismus von Heyrovsky-Horiuti ist. Es werden die F/ille, in denen die Durchtrittsfaktoren gleich 1 oder 1/2 sind, betrachtet. Die DarsteUung erstreckt sich nur auf einige Aspekte einer allgemeinen Methode, welche durch die Verwendung der Formein yon Marcelin-De Donder (Formelu MD) charakterisiert ist. IN several earlier writings t we have called a t t e n t i o n to t h e desirability o f writing electrochemical rate expressions in terms o f t h e electrochemical p o t e n t i a l s o f t h e reacting species at t h e a c t u a l r e a c t i o n sites in t h e i n t e r p h a s e regions. W e have also s h o w n z h o w a general expression f o r t h e rate o f a n e l e m e n t a r y reaction, such as one o f t h e steps in a steady-state m e c h a n i s m , c a n be written in t e r m s o f t h e f o r w a r d a n d reverse affinities o f t h e e l e m e n t a r y reaction, this general expression yielding t h e two limiting types o f rate expressions: (1) P r o p o r t i o n a l i t y to a n e x p o n e n t i a l f u n c t i o n o f t h e excess o f t h e f o r w a r d affinity o f t h e e l e m e n t a r y step over its e q u i l i b r i u m value, w h e n * Manuscript received 25 February 1963. 583
584
P. VAN RYSSELBERGHE
the reverse rate is negligible. (2) Proportionality to the small overall affinity of a socalled equilibrium step. If the mechanism involves at least one step of each type we have, for the overall rate v (v~. . . . are the stoiehiometric numbers of these steps; 2~. . . . are functions of temperature and pressure ~but not_. of composition; A~ and A~. . . . are the forward and reverse affinities, with A~ = A~ -- A~; A~ = Ai, . . . . are the equilibrium values of these affinities; ~ = t~ . . . . are the forward and reverse rates at equilibrium; R is the molar gas constant; T is the absolute temperature),
v = vdv ~ = (1/vi2i)" [exp ( A d R T ) -- exp CAJRT)] = (~,dv,) "{exp [ ( A ~ - A , , ) / R T l -- exp [(A, -- A,,)/RT]}
= vj/v, = (1/v,2~) • exp (A~/RT) = (/~,b'j) " exp [(X, -- ~ , ) [ R T ] =
vd,
=
(AdRr).
(1)
Recently8 we have discussed current Tafel-like expressions for the rates of electrode processes and have shown how formulae of the above type (to which we give the general name Marcelin-De Donder or M D formulae) can be used, with electrochemical affinities replacing chemical ones, as first steps in the construction of a coherent theory of electrode kinetics. The pertinence of M D kinetics in electrochemistry is immediately evident when one recalls that electrochemical affinities ~, are equal to corresponding overvoltages ~ multiplied by the reaction charge zF 4, J, = zF,7.
(2)
A detailed treatment of electrode kinetics based upon the use of M D formulae is in preparation but we wish to present at this time two aspects of its application to the typical problem of hydrogen overvoltage: (1) Rate expressions when the overall electrochemical affinity is "concentrated" in a single step of the mechanism, all other steps having negligible electrochemical affinities (2) M D interpretation of the cases in which the anodic and cathodic parameters ~ and/~, in a Tafel expression are equal to each other and, particularly, equal to 1 or to 1[2. 1. M D
A N D T A F E L E X P R E S S I O N S F O R STEPS C O N C E N T R A T I N G THE OVERALL ELECTROCHEMICAL AFFINITY
We shah consider cases for which it is experimentally found that the net current density I, for a given electrode metal and a given solution composition, is related to the overvoltage ~7by the Tafel expression I = I0[exp (~aF~7/RT) -- exp (--~eF~7[RT)],
(3)
with I > 0 and ~7 > 0 when the net current is anodic, I < 0 and ~7 < 0 when it is cathodic. The exchange current I o may contain a composition dependence which we shall not look into here. The parameters fla and fl~ are related to the transfer coefficients c% and 0% the charge number z and the stoichiometric number of the ratedetermining step i, as follows, =
zl,,,,
Fo =
zl,,.
(4)
Electrochemical affinitiesand potentials--ca~ of hydrogen overvoltage
585
Calling I the bulk of the metal, I ' the gaseous hydrogen phase and I I the bulk of the solution, the overall reaction is H2(I') --~ 2 H + ( I I ) + 2 e - ( I ) .
(5)
The reaction charge is 2F. The overall electrochemical affinity and the overvoltage are related by .~ = p r _ 2fi~+ -- 2fi[- = 2F~,
(6)
the p's and fi's being chemical and electrochemical potentials. At equilibrium (indicated by the subscript e) we have, the 9's being inner electric potentials, •g- = 0
and
(9 ~ -- 9~)~ = (1/2F)-(2p~+ + 2p~- -- p~,)
(7)
With I different from zero we have 91 _ 9u = (~i _ 9n), + ~/.
(8)
The mechanism is, essentially, the following one (hydration, hydride formation, etc. are regarded as complete equilibrium steps implicitly taken into account) in which the absence of phase specification indicates that the particular reaction site is being considered: 1.
H2(/')-* H 2
v1 :
2. 3. 4. 5.
H~--+2H H - + H + + eH + -+ H~-(II) e---~e-(/)
~'~ = 1 va=2 v4 = 2 v5 = 2.
1
(9)
Let us assume that one of these steps, say i, is such that *,i'Ai=A
with
,~=0(j:#i).
(10)
Actually the Al's are very small and such that (see formula (1)) (v~,/~,j) " ( A ~ / R T )
=
(ii)
v.
We have I = 2F" vffv,,
(12)
I o = 2F" ~ , / , ,
and, identifying the M D formula with (3), we obtain A,--A,6=fl..F~/
and
A,--A,~=--flc.
Fr/.
03)
It follows that •g'i = A~ -- A, = (ft. + flc).F~ -----(o% + 0%).A/v,
(14)
and hence that fl.+flc=2/vi
and
0 % + 0 % = 1.
(15)
Let us note, however, that these additivity properties do not necessarily hold when
586
P. VAN RYSSSELBEKGHE
part o f the overall electrochemical affinity resides in steps other t h a n i. It will now easily be found that: 1. W h e n A1 = -~ we have: fl~ = ~ = 0, fl~ ---- 2, 0~ = 1. 2. W h e n ~-z =
we have the same situation.
3. W h e n -~3 = .A/2 we have: fla = ~a---- 0, fl~ = ~ ---- 1. 4. W h e n -~a = A/2 we have: fl~ = ~ -----1, fl~ = ~ = 0. 5. W h e n -~5 = A/2 we have: fl~ = ~ = 0, fit = 0c, = 1. W e see that, in cases 1, 2, 3 a n d 5, the anodic c o m p o n e n t of the current is independent of the overvoltage and remains equal to the exchange current, while in case 4 it is the cathodic c o m p o n e n t which remains constant and equal to the exchange current. Case 1 does not a p p e a r likely to occur. Case 2 would correspond to a pure Tafel m e c h a n i s m in the cathodic direction. Case 3 would be o f the Volmer type with the whole overvoltage involved in the cathodic direction. Case 4 is unlikely to occur as a pure case but a non-zero value of the electrochemical affinity o f this step m a y well have to be considered in other mechanisms. Case 5 is interesting: it would m a k e the whole overvoltage correspond to a w o r k of extraction of the electron f r o m the metal to the reacting site or vice-versa. T h a t a portion of the overvoltage might correspond to such a step in other m e c h a n i s m s seems very probable. T h e M D f o r m u l a for case 5 is vs = v~,. {exp [(ge- -- ge-,)l R T ] -- exp [(g~- -- gr~-,)IRT] }, (16) and we see that, since biot- - - fi~-, ---- - - F . (90r -- ~0J) = --Fr/,
(17)
we have fie- -= fie-e : the electrochemical potential o f the electron at the reaction site remains constant as the overvoltage varies. W e shall n o w show that it is possible to have b o t h fla and fl, different f r o m zero in f o r m u l a (3) if, instead o f being concentrated in a single step o f the mechanism, the overall electrochemical affinity is distributed between two or m o r e steps. 2. M D A N D T A F E L E X P R E S S I O N S F O R STEPS I N V O L V I N G ELECTROCHEMICAL AFFINITIES DIFFERENT FROM THE O V E R A L L ONE
F o r m u l a e (13), (14) and (15) already give the essential information pertaining to this type of case: (1 3) and (1 4) r e m a i n true if the Tafel-like behaviour represented b y f o r m u l a (3) is actually observed. O n the other h a n d agreement with the two formulae (15) is no m o r e required. Let us n o w take the particular case fla = fl~ = fl and write the M D rate expression for step 3, va : ~3e" (exp [ ~ H - - / t H e ) / R T ] - - exp [(/~H+ + /~e- --/~a+, -- fte-e)/RT]},
(18)
with
~H÷ + ~e- --/~H+, --,/Te-, = /TH+ + /7~- --/ ~r / 2 = --tiFf I = --flA/2.
(19)
Electrochemical affinitiesand potentials--caseof hydrogen overvoltage
587
It follows that / ~ , = / Z a + fin+ + fie-.
(20)
This i m p l i e s / ~ =/~H~ and we thus have, at the reaction sites, the electrochemical equilibrium H 2 ~ H + H + + e-,
(21)
a most interesting implication of the equality of fl~ and/3,. Let us note also that reaction (21) is the sum of steps 2 and 3 of the mechanism (9). It is also the ratedetermining step in the Heyrovsky-Horiuti mechanism. We shall come back to this point below. Since ~,s=Aa--A3=flA
and
As+A3=0
(22)
we have A~ = " f l A ,
A2 + 2A3 -- flA, A1 + 2.~4 + 2.~5 = (1 -- p)..~.
(23)
If fl = 1 the overall electrochemical affinity is concentrated in steps 2 and 3 and the sum ~'1 + 2A~ + 2A5 is equal to zero, each term being zero or, less likely, the indicated compensation occurring. If, following a rather frequent practice or having obtained, in some hypothetical experimental case, the corresponding evidence, we were to put fla = fl, -- 1/2 (and thus also in this case % = 0% = 1/2), the equilibrium condition (20) would of course still hold and we would have, From (22) and (23)
A3 = ---A2 = A/2,
-~2 + 2A3 = A/2,
,~1 + 2-~4 + 2,~5 = A/2.
(24)
Let us assume ,~1 and A4 equal to zero. We then have ~'5 = fie- -- fi~e- = A / 4 = F~I/2.
(25)
We submit that a very attractive interpretation of the % = 0% = 1/2 possibility has hereby been obtained. Let us, finally, write the M D expression for the rate v2' of reaction (21) in the Heyrovsky-Horiuti mechanism v2' = 3,,'. {exp [ ~ n , -- ~H~,)/RT] -- exp {[(#H + fia+ + fie-) -- (,IXH, + fia+e + fio-,)]/RT)}.
(26)
If steps 1, 3, 4 and 5 (identical to the corresponding ones of mechanism (9)) are assumed to be at equilibrium we have /)2 t
= ~2e '- {1 -- exp [(2/xH -- tz~,)/RT]} = v~," {1 -- exp (--2F~1/RT)}
(27)
and we are led back to the pure Tafel mechanism considered in Section 1. This, however, would not occur if a part of the overall affinity A = 2F~ is assigned to one or several other steps. The M D method should be a safe and straightforward guide in the exploration of such possible mixed mechanisms.
588
P. VAN RYSSmJSERGHE
Our previous writings on the M D method and the present discussion of a somewhat idealized hydrogen overvoltage case will show, we hope, that this method presents distinct advantages over the empirical or semi-empirical adoption of certain Tafel-like equations. Beyond the level of inquiry at which the foregoing paper has been kept the method is capable of handling considerable refinements. The composition dependence of the exchange current, the role of activation affinities, the significance of transfer coefficients, the examination of cases for which no constant transfer coefficients exist but M D formulae are available, etc. are allproblems for which solutions can be provided or at least suggested. The main advantage of the method is its close connection with thermodynamics, in both its reversible and irreversible aspects. It is necessarily phenomenological and somewhat formal but it establishes connections with mechanistic and molecular models and assumptions at points in the developments at which the introduction of such devices involves a minimum of danger. REFERENCES 1. P. VAN RYSSELIIERGHE,J. Chem. Phys. 17, 1229 (1949); 20, 1522 (1952); 3". chim. phys. 49, C47 (1952); Proc. 8th Meeting Int. Comm. Electrochem. Thermodynam. and Kinet. (Madrid 1956) p. 405. Butterworths, London (1958). 2. P. VAN RYSS~n~ZOI~,J. Chem. Phys. 29, 640 (1958). 3. P. VAN RYSLqm.nERoI-IE,R. C. Accad. Lincei, 31, 391 (1961). 4. P. VAN RYSS~n~ZOHE,Electrochemical A~inity. Hermann, Paris (1955).