Kinetics of hydrogen evolution reaction with Frumkin adsorption: re-examination of the Volmer–Heyrovsky and Volmer–Tafel routes

Kinetics of hydrogen evolution reaction with Frumkinadsorption: re-examination of the Volmer±Heyrovsky and

Volmer±Tafel routes

M.R. Gennero de Chialvo, A.C. Chialvo *

Programa de ElectroquõÂmica Aplicada e IngenierõÂa ElectroquõÂmica (PRELINE), Facultad de IngenierõÂa QuõÂmica, Universidad

Nacional del Litoral, Santiago del Estero 2829, 3000 Santa Fe, Argentina

Received 3 April 1998

Abstract

A re-examination of the basic kinetic derivations of the Volmer±Heyrovsky and Volmer±Tafel routes for the

hydrogen evolution reaction with a Frumkin adsorption of the intermediate was carried out. Expressions for thedependence of the surface coverage and current density on overpotential were derived for both routes withoutkinetic approximations. On the basis of these dependencies, the kinetic behavior was simulated for di�erent values

of the parameters involved at 298.16 K. Conditions for the existence of Tafelian domains were discussed and theindependence of the Tafel slopes on the type of adsorption of the reaction intermediate was demonstrated. Theresults obtained were critically compared with those derived from approximated expressions customarily used and

the di�erences with them were pointed out. # 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Hydrogen evolution reaction; Kinetic analysis; Frumkin adsorption; Tafel slopes; Exchange current densities

1. Introduction

The discharge of proton or water through the

Volmer step with the formation of adsorbed hydrogen

H(a), the electrochemical desorption (Heyrovsky step)

and the recombination of the H(a) (Tafel step) are gen-

erally accepted as the steps of the kinetic mechanism

of the hydrogen evolution reaction (HER). The analy-

sis of the kinetic behavior is usually done on the basis

of the Volmer±Heyrovsky (VH) and the Volmer±Tafel

(VT) routes. Using the approximation of the rate

determining step (rds), diagnostic criteria were estab-

lished and widely used for both routes [1±7].

Furthermore, it has been considered that the adsorp-

tion process of the reaction intermediate can be

described through the Frumkin isotherm in the domain

of surface coverage (y) ranging between 0.2 and 0.8

[1±3, 5, 6]. On this range the relationship y/(1ÿ y) has

been approximated to unity and in the resulting ex-

pressions of the reaction rate, the dependence of y on

overpotential (Z) in the pre-exponential factor has been

neglected [3, 6]. As a result of these approximations, it

has been concluded that the Tafel slope (b) in the low

overpotentials region should be in¯uenced for the reac-

tion symmetry factor (a) and for the adsorption sym-

metry factor (l) as well. Nevertheless, from the

analysis of the results obtained in the study of the

HER without kinetic approximations due to

Enyo [8, 9], it can be arrived to the conclusion that b

depends only on a. Consequently, a re-examination of

the basic kinetic concepts is worthwhile in order to elu-

cidate this apparent controversy.

The present work deals with a kinetic study of the

HER under the Volmer±Heyrovsky±Tafel mechanism

with a Frumkin adsorption and without kinetic ap-

proximations. The expressions for the variation of the

current density and the surface coverage on overpoten-

tial will be derived for the VH and VT routes. The

Electrochimica Acta 44 (1998) 841±851

0013-4686/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved.

PII: S0013-4686(98 )00233-3

PERGAMON

* Corresponding author. Fax: +54-42571162; E-mail:

achialvo@®qus.unl.edu.ar

conditions for the existence of Tafelian domains willbe obtained for both routes and the relation between

the amounts resulting from the extrapolation of theselinear regions and the kinetic parameters will be deter-mined.

2. Preliminary considerations

The Tafel slope is one of the experimental kinetic

parameters often used for the characterization of anelectrochemical reaction, but its determination is inmany cases questionable. It is common to observeTafel lines drawn over slightly curved but non-linear

experimental dependencies, which we will call pseudo-Tafelian behaviors. Moreover, an exchange currentdensity is obtained by extrapolation of the ®tted

straight line. Such experimental kinetic parameters,with arbitrary values, could lead to a wrong interpret-ation of the behavior of the system under study.

The explanation for the linear dependence of thelogarithm of the current density ( j) on Z is based ontheoretical dependencies resulting from kinetic and

also mathematical approximations, such as the con-sideration of quasi equilibrium steps, neglect of thevariation of y(Z), etc. Nevertheless, the consistency ofsuch approximations with the rigorous solution of the

corresponding kinetic mechanism is usually not veri-®ed. It should be noticed that the usefulness of the ex-perimental kinetic parameters lies on their relationship

with the kinetic parameters of the elementary steps ofthe reaction mechanism. Consequently, the correct in-terpretation of a Tafelian domain de®ned as the region

where the dependence log j vs. Z is linear and thereforedZ/d log j is constant, depends on the previous clearknowledge of the descriptive capability of a given kin-etic mechanism. Only if this condition is ful®lled, the

Tafel slope and the exchange current density can bequantitatively related with the corresponding kineticparameters of the elementary steps.

3. Theoretical analysis

The expressions of the reaction rate (V) of the

hydrogen evolution reaction, corresponding to the fol-lowing stoichiometry:

H2O� 2eÿ $ H2� g� � 2OH ÿ, �1�

will be derived in steady state for both Volmer±Heyrovsky and Volmer±Tafel routes. From them and

taking into account the relationship of V with the cur-rent density,

j � 2FV, �2�

the dependence of j on the overpotential will be simu-lated by computational calculations at T= 298.16 K.

3.1. Volmer±Heyrovsky route

The elementary steps are

H2O� eÿ $ H�a� �OH ÿ �Volmer�, �3�

H2O�H�a� � eÿ $ H2� g� �OH ÿ �Heyrovsky�: �4�

The equations for the rate of the di�erent reactionsteps with a Frumkian behavior of the adsorbed

hydrogen, following the treatment given by Enyo [8],can be written as

v�V � veV1ÿ y1ÿ ye

sÿleÿ�1ÿa� fZ, �5�

vÿV � veVyye

s�1ÿl�eafZ, �6�

v�H � veHyye

s�1ÿl�eÿ�1ÿa� fZ, �7�

vÿH � veH1ÿ y1ÿ ye

sÿleafZ, �8�

where

s � eu�yÿye�, �9�

u the interaction parameter between the adsorbedhydrogen atoms, v + i and v ÿ i are the forward andbackward reaction rates of step i (i= V, H), respect-

ively, v ie is the equilibrium reaction rate of step i, y e is

the equilibrium surface coverage and f= F/RT(38.92039 Vÿ1). Furthermore, a and l are the reactionand adsorption symmetry factors, respectively, and

they are considered to be equal for all elementarysteps.On steady state, the rate of Eq. (1) and those of

Eqs. (3) and (4) are related by [10]

V � vV � vH � 0:5�vV � vH�, vV � v�V ÿ vÿV,

vH � v�H ÿ vÿH:�10�

Substituting the expressions of the reaction rate of thecorresponding steps (Eqs. (5)±(8)) and dividing by vV

e ,we obtain

M. Gennero de Chialvo, A. Chialvo / Electrochimica Acta 44 (1998) 841±851842

V

veV� 1ÿ y

1ÿ yesÿleÿ�1ÿa� fZ ÿ y

yes1ÿleafZ

� mH

�yye

s1ÿleÿ�1ÿa� fZ ÿ 1ÿ y1ÿ ye

sÿleafZ�

� 1

2

�1ÿ y1ÿ ye

sÿleÿ�1ÿa� fZ ÿ yye

s1ÿleafZ

�mH

�yye

s1ÿleÿ�1ÿa� fZ ÿ 1ÿ y1ÿ ye

sÿleafZ��

, �11�

where mH= vHe /vV

e . From Eq. (11), the following im-plicit function y= f(Z, y, mH, y

e, u) can be obtained:

y � yesÿ1�mH � eÿfZ��1ÿ ye��1�mH eÿfZ� � yesÿ1�mH � eÿfZ� : �12�

The amounts y/y e and (1ÿ y)/(1ÿ y e) can be evalu-

ated from Eq. (12). Substituting them on Eq. (11) andtaking into account Eq. (2), the general equation ofthe dependence of the current density on overpotential

is obtained,

j

j0V� 2mHs

ÿl�eÿ�2ÿa� fZ ÿ eafZ��1ÿ ye��1�mH eÿfZ� � yesÿ1�mH � eÿfZ� , �13�

where jV0 is the exchange current density of the Volmer

step and s= s[y(Z)] is given by Eq. (9).The complete description of the kinetics of the HER

when the Volmer±Heyrovsky route is applicable can

be obtained from the simultaneous resolution ofEqs. (12) and (13).

3.1.1. Tafelian domainsThe existence of overpotential domains where there

is a linear dependence of the logarithm of j on Z is notclearly inferred from Eq. (13) and it will depend on thevalues of the parameters. Nevertheless, for certain

domains of mH and y e values, a linear variation can beobtained. The cases in which two Tafelian domainscan be distinguished will be analyzed ®rst,

(a) For y e<<1 (y e<10ÿ4) and mH<<1 (mH<10ÿ3),in the low overpotentials region it is veri®ed thatmH eÿ fZ<<1. On these conditions, yÿ y e<<1 and s31

and therefore Eq. (13) can be written as follows whenvZv>RT/F (it should be taken into account that Z is anegative value):

j � jextl eÿ�2ÿa� fZ, jextl � 2j0VmH: �14�The corresponding Tafel slope (bl) is equal to

2.3026RT/(2ÿ a)F and from the extrapolation atZ= 0 the pre-exponential factor j l

ext (extrapolated cur-rent density at low vZv) can be obtained. Fig. 1 shows

the dependencies of ln( j/jV0 ) and y on Z rigorously

simulated through Eqs. (12) and (13) when y e=10ÿ6,a= l= 0.5, u= 5 and 10ÿ5RmHR102. Solid circles

(lines a±c) correspond to ln( j lext/jV

0 )= ln(2mH) values,

which veri®es the validity of Eq. (14).(b) For y e31 (1ÿ y e<10ÿ4) and mH>>1

(mH>103), there is a low overpotentials region where

it can be considered that mH>>eÿ fZ, yÿ y e<<1 and

s31. On these conditions and for vZvr RT/F, Eq. (13)

turns to

j � jextl eÿ�2ÿa� fZ, jextl � 2j0V �15�and therefore the Tafel slope is equal to case (a). Thesimulation of Eqs. (12) and (13) with (1ÿ y e)= 10ÿ5,

a= l= 0.5, u= 5 and 1R mHR105 is illustrated inFig. 2. It can be observed clearly in the ln( j/jV

0 ) depen-dence the overpotentials domain (vZv< 0.3 V) where

Eq. (15) is accomplished. The solid circle correspondsto ln( j l

ext/jV0 )= ln 2= 0.69315.

(c) A Tafelian domain appears always at high over-potentials. On such conditions, it can be concludedfrom Eq. (12) that the surface coverage reaches a limit-

ing value (y *) given by the following implicit equation:

y* ��1�mH

1ÿ ye

yes*

�ÿ1, s* � eu�y*ÿy

e�: �16�

The dependence y *= y*(y e, mH) is illustrated in Fig. 3for u= 5 and 10ÿ5RmHR105. It should be noticed

that if u= 0, s*=1 and Eq. (16) is equal to that

Fig. 1. Dependence of ln( j/jV0 ) and y on Z for the VH route.

y e=10ÿ6; a= l= 0.5; u= 5; mH=(a) 10ÿ6, (b) 10ÿ5, (c)

10ÿ4, (d) 10ÿ3, (e) 10ÿ2, (f) 1. (.) ln( j lext/jV

0 ); (w) ln( j hext/jV

0 ).

M. Gennero de Chialvo, A. Chialvo / Electrochimica Acta 44 (1998) 841±851 843

corresponding to a Langmuir adsorption [8, 11].

Taking into account that at high vZv values v i3v + i

(i= V, H), when y(Z)= y * Eq. (11) written on terms

of current density is reduced to

j

j0V� 2mH

y*

yes*�1ÿl� eÿ�1ÿa� fZ

� 21ÿ y*

1ÿ yes*ÿleÿ�1ÿa� fZ, �17�

which shows the existence of a Tafelian domain athigh overpotential values with a slope bh=2.3026RT/(1ÿ a)F. The exchange current density at high vZvobtained by extrapolation ( j h

ext) is

jexth � 2mHj0V

y*

yes*�1ÿl� � 2mHj

0Vs*�1ÿl�

ye � �1ÿ ye�mHs*: �18�

This behavior is clearly illustrated in Figs. 1 and 2. Inthe dependence ln( j/jV

0 ) vs. Z, the values of ln( j hext/

jV0 )= ln 2mHs

*(1ÿ l)y */y e are shown as open circles,evaluated on each case by Eq. (18). Furthermore, inthe y vs. Z relationship, it can be clearly distinguished

the overpotentials range where y(Z)= y *.(d) The existence of a unique linear dependence of

ln j in the whole range of overpotentials is possible forcertain values of the parameters y e and mH. It is

necessary that the condition y(Z)3y * be ful®lled at anvZv value su�ciently low such that the ®rst Tafeliandomain cannot be developed. In order to determine

the y e and mH values that obey such condition, anoverpotential Z # such that y(Z #)ÿ y *=10ÿ2 wasde®ned and calculated from Eqs. (12) and (16), with

u= 5. Such vZ #v, which denotes the beginning of thelinear domain corresponding to the high overpoten-tials, should be less than 0.2 V. In this case the corre-sponding Tafelian dependence follows Eq. (17) with

s *31,

j � jexth eÿ�1ÿa� fZ, jexth �2mHj

0V

ye � �1ÿ ye�mH: �19�

Fig. 4 shows the dependence Z # vs. log[y e/(1ÿ y e)] fordi�erent mH. It can be easily established that fory e<0.5 and mH>3� 10ÿ3 or for y e>0.5 and

mH<3� 102, a unique linear domain is obtained.Finally, it should be noticed that for mH=1,

Eq. (12) gives y(Z)= y e and the widest linear domainis obtained (e.g. Fig. 2, line a). In this case the solid

circle, evaluated from Eq. (19), coincides with the openone calculated from Eq. (14) because y e31.

3.1.2. Pseudo-Tafelian dependencies

The existence of a Tafelian domain at low vZv with aslope equal to 2.3026RT/F has been proposed on thebasis of approximated kinetic analysis [3, 6]. Never-

theless, it cannot be justi®ed starting from Eq. (12).Furthermore, the results of many simulations donewith di�erent parameter values were non-linear log j

vs. Z dependencies, although with slight curvatures,which are de®ned as pseudo-Tafelian domains. An ex-ample of such behavior is given in Fig. 5, for y e=10ÿ1,


1ÿ y e=10ÿ5; a= l= 0.5; u= 5; mH=(a) 1, (b) 10, (c)

102, (d) 103, (e) 104, (f) 105. (.) ln( j lext/jV


0 ).

Fig. 3. Dependence of y* on y e for di�erent values of mH

(indicated in the ®gure) for the VH route. u= 5.


a= l= 0.5, u= 10 and mH=10ÿ4. In the range 0.05VR vZvR 0.20 V, where 0.2083R y(Z)R 0.5976, a slightly

curved dependence is observed. The linear regression insuch range gives a slope equal to 57.6 mV decÿ1 and anorigin ordinate equal to ÿ8.2218. These results will be

analyzed in detail in Section 4.

3.1.3. Interpretation of measurable quantitiesFrom the experimental determination of the depen-

dence of current density on a wide range of overpoten-tial, the existence of Tafelian domains can beestablished and the values j l

ext, j hext, bl and bh, or some

of them, can be calculated.

The relationship between the extrapolated currentdensities and the kinetic parameters y e, mH, u and lcan be evaluated through Eqs. (14), (15) and (18). It

can be observed that for j lext there are two alternatives

(Eq. (14) for y e<<1 and Eq. (15) for y e31). The correctone should be determined by the experimental evalu-

ation of the dependence of the surface coverage onoverpotential, from which the values y e and y * couldbe obtained. Unfortunately, this determination is di�-

cult to carry out and we have found only one measurereported in the literature [12].It should be useful to take into account that, if the

existence of two Tafelian domains is veri®ed exper-

imentally, the following relationship between the corre-sponding Tafel slopes should be ful®lled:

bhbl� 2ÿ a

1ÿ a: �20�

Besides, the following relationships between the corre-

sponding j ext values should be accomplished:

jexth

jextl

� s*�1ÿl�

ye � �1ÿ ye�mHs*, ye ÿ40, �21�

jexth

jextl

� mHs*�1ÿl�

ye � �1ÿ ye�mHs*, ye ÿ41: �22�

Finally, a non-linear regression of all the experimen-tal points should be done when a clear Tafelian

domain at low vZv is not observed. On this basis, kin-etic information can be obtained from this region.

3.2. Volmer±Tafel route

The elementary steps that describe the Volmer±Tafelroute are given by Eq. (3) and

2H�a� $ H2� g� �Tafel �: �23�The corresponding expressions for the reaction rate ofeach step are given in Eqs. (5) and (6) and [8]

v�T � veTy2

ye2s2�1ÿl�, �24�

Fig. 4. Dependence of Z # (being y(Z #)ÿ y *=10ÿ2) on

log[y e/(1ÿ y e)] for di�erent values of mH (indicated in the

®gure). a= l= 0.5; u= 5.


y e=10ÿ1; a= l= 0.5; u= 10; mH=10ÿ4. ln( j lext/jV

0 ): (W)

linear ®tting; (r) Eq. (39).


vÿT � veT�1ÿ y�2�1ÿ ye�2 s

ÿ2l, �25�

where v +T and v ÿT are the forward and backwardreaction rates of the Tafel step and v T

e is the equili-brium reaction rate of such step. On steady state, therate of Eq. (1) is given by [10]

V � 0:5vV � vT � vV ÿ vT, vV � v�V ÿ vÿV,

vT � v�T ÿ vÿT:�26�

Substituting Eqs. (5), (6), (24) and (25) in Eq. (26)gives

V

veV� 0:5

�1ÿ y1ÿ ye

sÿl eÿ�1ÿa� fZ ÿ yye

s1ÿl eafZ�

� mT

�y2

ye2s2�1ÿl� ÿ �1ÿ y�2

�1ÿ ye�2 sÿ2l�

��

1ÿ y1ÿ ye

sÿl eÿ�1ÿa� fZ ÿ yye

s1ÿl eafZ�

ÿmT

�y2

ye2s2�1ÿl� ÿ �1ÿ y�2

�1ÿ ye�2 sÿ2l�, �27�

where mT= v Te /vV

e . Reordering the last two terms ofEq. (27), the following implicit function of y can bede®ned:

a�y�y2 � b�y, Z�y� c�y, Z� � 0, �28�where

a�y� � 2mT

�s2�1ÿl�

ye2ÿ sÿ2l

�1ÿ ye�2�, �28a�

b�y, Z� � 4mTsÿ2l

�1ÿ ye�2 � eafZ�eÿfZsÿl

1ÿ ye� s1ÿl

ye

�, �28b�

c�y, Z� � ÿ 2mTsÿ2l

�1ÿ ye�2 ÿeÿ�1ÿa� fZsÿl

1ÿ ye, �28c�

and s= s(y(Z)) is given by Eq. (9).Eqs. (27), (28) and (28a)±(c), together with Eq. (2),

describe completely the dependence of the current den-sity on overpotential for the Volmer±Tafel route with-out kinetic approximations.

3.2.1. Tafelian domainsAs in the previous case, the existence of overpoten-

tial domains where a linear dependence is veri®ed is

not straightforward. However, for certain ranges of y e

and mT values, Tafelian domains can be found.(a) Considering the case in which mT<<1

(mT<10ÿ3), the following limiting implicit functiony= f(Z, y, y e, u) can be obtained from Eqs. (28) and(28a)±(c):

y � yesÿ1eÿfZ

�1ÿ ye� � yesÿ1eÿfZ: �29�

Evaluating the amounts (y/y e)2 and [(1ÿ y)/(1ÿ y e)]2

needed in Eq. (27) and taking into account Eq. (2), the

dependence of current density on overpotential in this

case can be written as follows:

j

j0V� 2mTs

ÿ2l�eÿ2fZ ÿ 1��1ÿ ye� � yesÿ1eÿfZ�2 : �30�

This expression leads to a linear dependence only if

y e<<1 (y e<10ÿ4). In this case there is an overpoten-

tials region where y eeÿ fZ<<1. From Eq. (29) it follows

that yÿ y e<<1, therefore s31 and consequently the

following equation is performed:

j � jextl eÿ2fZ, jextl � 2mTj0V: �31�

In this domain of overpotentials the Tafel slope (bl) is

equal to 2.3026RT/2F and j lext can be obtained by

extrapolation. Fig. 6 illustrates the dependence of ln( j/

jV0 ) and y on Z with the following values of the par-

ameters involved: y e=10ÿ5, l= a= 0.5, u= 5 and

10ÿ7RmTR10. Solid circles correspond to the ln( j lext/

jV0 )= ln(2mT) values of lines a and b.

(b) When the condition mT>1 is applied, it can be

found a range of low overpotentials where the terms

that do not contain mT can be neglected. In this case

Eq. (28) can be written as� �1ÿ ye�2ye2

s2 ÿ 1

�y2 � 2yÿ 1 � 0, �32�

which solution is y= y e. In the overpotential range

where eÿ (1ÿ a)fZ>>eafZ, the following linear expression

is obtained from Eq. (27):

j � jextl eÿ�1ÿa� fZ, jextl � j0V: �33�Line e in the ln( j/jV

0 ) vs. Z dependence of Fig. 6 shows

this behavior, which is characterized by a Tafel slope

(bl) equal to 2.3026RT/(1ÿ a)F. Open circle corre-

sponds to ln( j lext/jV

0 )= 0.

(c) Considering the opposite of case (b), that is over-

potentials high enough so as the terms containing mT

can be neglected in Eqs. (28) and (28a)±(c), the follow-

ing expression is obtained:

y1ÿ y

� ye

1ÿ yesÿ1 eÿfZ: �34�

A limiting surface coverage equal to 1 is obtained for

Z 4ÿ 1. This behavior leads to a limiting current

density of kinetic origin, which from Eq. (30) can be

written as

jlimT �2j0VmTs

*2�1ÿl�T

ye2, s*T � eu�1ÿy

e�, �35�

which implies an in®nite Tafel slope (bh=1).


Consequently, the Volmer±Tafel route at overpoten-tials su�ciently high always de®nes, as it is wellknown, a limiting kinetic current density independentlyof the behavior in the low overpotentials region. Open

squares in Fig. 6 illustrate the values of ln( jTlim/jV

0 ) cor-responding to lines a and b.

3.2.2. Pseudo-Tafelian dependence

The existence of Tafel lines with slopes near2.3026RT/F has been also proposed for the Volmer±Tafel route [3, 6], that cannot be justi®ed from Eqs. (2)

and (28), as in the previous case. Nevertheless, pseudo-Tafelian behaviors can be observed at low Z, as it is il-lustrated in Fig. 7, where the following parameters

values were used: mT=10ÿ5, y e=10ÿ1, l= a= 0.5and u= 10. In the range 0.05 VR vZvR 0.20 V, where0.2084R y(Z)R 0.6077, a slightly curved dependence isobserved. The least squares linear regression in such

range, also shown in Fig. 7, gave a slope equal to54.74 mV decÿ1 and an origin ordinate equal toÿ10.3427. These results will also be analyzed in detail

in Section 4.Furthermore, the simulations shown in Fig. 6

allowed ®nding another overpotential range where a

slight curvature of ln j vs. Z dependence is observed.They can also be considered as a pseudo-Tafeliandomain. On these cases, Eq. (30) cannot be reduced to

a linear expression and the extrapolation of the ®ttedlines at Z= 0 does not give an amount related to the

parameter y e, mH, etc. For example, for line a inFig. 6, a pseudo-linear domain can be found in therange 0.35 VR vZvR 0.65 V, with a Tafel slope equal

to 168.6 mV decÿ1. The same for line b in the range0.30 VR vZvR 0.65 V, where the Tafel slope is equal to136.7 mV decÿ1. It should be noticed that as |Z|increases, the limiting current will be achieved, accord-ing to Eq. (35).

3.2.3. Interpretation of measurable quantitiesThe experimental determination of the dependence

of current density on a wide range of overpotentialsallows in this case the evaluation of j l

ext, jTlim and bl.

Two di�erent Tafelian behaviors can be obtained at

low overpotentials. One of them is characterized bybl=2.3026RT/2F and a j l

ext value evaluated byEq. (31). The other has bl=2.3026RT/(1ÿ a)F and

j lext follows Eq. (33). On the other hand, the determi-nation of j T

lim allows infer that the HER is taking placethrough the VT route. This limiting current densitycontributes also to the calculation of the kinetic par-

ameters. However, its determination is not alwayspossible and rather unusual [13, 14].Finally, a non-linear regression of all the experimen-

tal points should be done when a clear Tafeliandomain at low Z is not observed.

Fig. 6. Dependence of ln( j/jV0 ) and y on Z for the VT route.

y e=10ÿ5; a= l= 0.5; u= 5; mT=(a) 10ÿ7, (b) 10ÿ5, (c)

10ÿ3, (d) 10ÿ1, (e) 10. ln( j lext/jV

0 ): (*) Eq. (31), (w) Eq. (33);

(q) ln( jTlim/jV

0 ).


y e=10ÿ1; a= l= 0.5; u= 10; mT=105. ln(j lext/jV

0 ): (W)

linear ®tting; (r) Eq. (41).


4. Discussion

The dependencies of current density and surface cov-

erage on overpotential for the HER, when it takes

place through the Volmer±Heyrovsky or the Volmer±

Tafel routes and the adsorbed intermediate is modelled

by the Frumkin isotherm, have been simulated through

the resolution of implicit equations and the conditions

for the existence of Tafelian domains were determined.Such behaviors can be used for the interpretation of

the experimental results when the double layer e�ects

are considered virtually eliminated by the presence of

an excess of supporting electrolytes [15].

At ®rst, the pseudo-Tafelian domains at low overpo-tentials values will be analyzed. For the Volmer±

Heyrovsky route, slightly curved dependencies can be

observed. When they are ®tted by linear regression,

pseudo-Tafel slopes with a value near 2.3026RT/F are

obtained, as well as the corresponding origin ordinate

( j ext). It has been also found that the surface coverage

varies strongly with overpotential in such domains.

This behavior is usually interpreted with an approxi-mated kinetic analysis, which assumes that the

Heyrovsky step is the rds [3, 6] and on this context the

current density becomes equal to

j � 2Fv�H � 2FveHyye

s1ÿl eÿ�1ÿa� fZ: �36�

On this approximated analysis, the dependence of yon Z is derived applying the quasi-equilibrium con-

dition to the Volmer step, resulting from Eqs. (5) and

(6),

y1ÿ y

� ye

1ÿ yesÿ1 eÿfZ: �37�

In the range 0.2R yR 0.8, another approximation isapplied to Eq. (37), which consists in considering y/(1ÿ y)31. On these conditions, obtaining s from

Eq. (37), substituting it in Eq. (36) and taking logar-

ithm, the following expression is obtained:

ln j � ln�2FveH�ye�ÿl�1ÿ ye�lÿ1� � ln yÿ �2ÿ �a

� l��fZ: �38�

This expression does not de®ne a Tafelian domain, as

the slope depends on Z. Nevertheless, trying to justify

a domain such that the variation of ln y on Z is neg-

lected. On this basis, an slope equal to 2.3026RT/

(2ÿ aÿ l)F is obtained [3, 6]. If this argumentation

would be correct, then the origin ordinate would be

given by

ln jextl � ln�2jeVmH�ye�ÿl�1ÿ ye�lÿ1�: �39�

Consequently, the linear ®tting of the rigorously

simulated results should give an origin ordinate coinci-

dent with that of Eq. (39). Nevertheless, the simu-

lations lead to the conclusion that there is no

agreement between the value obtained by extrapolation

and that calculated by Eq. (39). For example, the

values corresponding to the case illustrated in Fig. 5

are ln( j lext/jV

0 )= ÿ 8.2218 (extrapolation) and ln( j lext/

jV0 )= ÿ 6.1093 (Eq. (39)). Besides, the slope obtained

from the linear regression (57.6 mV decÿ1) is slightly

less than that resulting of considering l= a= 0.5 in

2.3026RT/(2ÿ aÿ l)F (59.16 mV decÿ1).

Consequently, the approximation consisting of neglect-

ing the term ln y in Eq. (38) is incorrect.

A similar situation can be found for the Volmer±

Tafel route at low vZv if the Tafel step is considered as

the rds and the Volmer step at equilibrium. The ex-

pression of the current density is in this case

ln j � ln�2jeVmT��ye�ÿ2l�1ÿ ye�2�lÿ1�� 2 ln yÿ 2�1

ÿ l� fZ, �40�

where a Tafelian domain cannot be de®ned and, as in

the previous case, the variation of ln y on Z was neg-

lected in order to justify a Tafel slope equal to

2.3026RT/2(1ÿ l)F [3, 6]. The corresponding origin

ordinate of this linear approximation is

ln jextl � ln�2jeVmT�ye�ÿ2l�1ÿ ye�2�lÿ1��: �41�

As in the previous case, the amount resulting from

the application of Eq. (41) is far from the origin ordi-

nate obtained from the extrapolation of the linear re-

gression of the pseudo-Tafelian domain, as is

illustrated in Fig. 7. These values are ln( j lext/

jV0 )= ÿ 10.0641 and ln( j l

ext/jV0 )= ÿ 8.4118 for the

extrapolation and Eq. (41), respectively. Besides, the

slope obtained from the linear regression (54.74 mV

decÿ1) di�ers from that resulting of considering

l= 0.5 in 2.3026RT/2(1ÿ l)F (59.16 mV decÿ1).

The results described above demonstrate that the

regions usually considered linear with slopes equal to

2.3026RT/(2ÿ aÿ l)F and 2.3026RT/2(1ÿ l)F are

pseudo-Tafelian domains. Notwithstanding, these

domains are useful because they are only possible

when ur 5 and therefore they are a clear indication of

a Frumkian behavior, in spite of that the kinetic par-

ameters of the elementary reaction steps cannot be

obtained, as in the case of real Tafelian domains.

The results obtained in the present work allow to

conclude that Tafelian domains at low vZv can be found

only if y is constant or else if it has a small variation

on overpotentials. In this case, the Tafel slope can take

the value 2.3026RT/(2ÿ a)F or 2.3026RT/(1ÿ a)F for

the Volmer±Heyrovsky route and the values 2.3026RT/

2F or 2.3026RT/(1ÿ a)F for the Volmer±Tafel route.

Therefore, the values that can take the Tafel slope are

independent of the adsorptive characteristics of the


substrate, due to the constancy of the surface coverage

in such domain. A similar behavior was found for the

high vZv region, where y= y* and b= 2.3026RT/(1ÿ a)F for the Volmer±Heyrovsky route and y= 1

and b= 1 for the Volmer±Tafel route. Besides, a re-

lationship between the extrapolated current density atZ= 0 and the kinetic parameters of the elementary

steps can be established for each case. It should be

noticed that the same Tafel slope value for substrateswith di�erent electrosorption characteristics does not

imply that the corresponding kinetic behaviors are also

equal. For example, the extrapolated current density(Eq. (18)), the limiting values y* (Eq. (16)) and j lim

(Eq. (35)), etc. demonstrate the existence of very di�er-

ent behaviors, in spite of the same Tafel slope value.

A second aspect to be analyzed is related to the in-

¯uence of the interaction parameter u on the behaviorof the ln j vs. Z relationship. It should be particularly

interesting to analyze the in¯uence on the low vZvregion. Fig. 8 shows the simulation corresponding tothe Volmer±Heyrovsky route with the parameters

y e=10ÿ3, l= a= 0.5, mH=10ÿ7 and 0R uR 10. A

small range of overpotentials can be observed wherey(Z)3y e, which constitutes a Tafelian domain with a

slope equal to 2.3026RT/(2ÿ a)F and a j lext value

given by Eq. (14), shown as a solid circle in Fig. 8. Inthe following Z range, a slight curvature can be

observed, directly related to the sharp increase of y(Z),generating a pseudo-Tafelian domain, which widely

increases as u increases. The in¯uence of u on the highvZv region is also illustrated in Fig. 8, where the opencircles indicate the values of ln( j h

ext/jV0 ) for the di�erent

lines. A similar behavior can be found in the case of

the Volmer±Tafel route. The in¯uence of the inter-

action parameter on the ln( j/jV0 ) and y vs. Z dependen-

cies can be appreciated in Fig. 9, where the following

values of the parameters were used: y e=10ÿ3,

mT=10ÿ6, l= a= 0.5 and 0R uR 10. At low vZvthere is a small Tafelian domain with a slope equal to

2.3026RT/2F and a j lext value given by Eq. (31),

shown as a solid circle in Fig. 9. In the following Zrange there is a pseudo-Tafelian domain, which

widely increases as u increases. Finally, the in¯uence

of u on the high vZv region is illustrated in Fig. 9,

where the open squares indicate the values ofln( jT

lim/jV0 ) for the di�erent lines calculated by

Eq. (35). These examples demonstrate the origin of

the pseudo-Tafelian behaviors at low overpotentials

previously discussed.

Another aspect that should be useful to analyze

is the distinction between the Frumkian and

Langmuirian behavior for the Volmer±Heyrovskyroute. On this sense, for certain values of the par-

ameters of the elementary steps, the HER becomes


y e=10ÿ3; a= l= 0.5; mH=10ÿ7; u= (a) 0, (b) 2.5, (c) 5,

(d) 10. (.) ln( j lext/jV


0 ).


y e=10ÿ3; a= l= 0.5; mT=10ÿ6; u= (a) 0, (b) 2.5, (c) 5,

(d) 10. (.) ln( j lext/jV

0 ); (q) ln( jTlim/jV

0 ).


independent of the behavior of the adsorbed inter-

mediate. It should be taken into account that the

di�erence between the expressions of the reaction

rate of the elementary steps when the Frumkin or

Langmuir adsorption are considered is the factor s,

which at su�ciently high vZv reaches the value s *.

Therefore, if s *31 for certain values of the par-

ameters involved, the behavior will be similar to

that corresponding to a Langmuirian adsorption, in

spite of the di�erent adsorption properties (u$ 0).

This case corresponds to vy *ÿy ev<<1(R10ÿ3). For

the Volmer±Heyrovsky route, the domain of the

parameters mH and y e for which vy *ÿy evR 10ÿ3

can be evaluated from Eq. (16). Fig. 10 illustrates

this domain (shading area), which is practically

insensible to the variations of the parameter u

(0R uR 10). It can be concluded that for mH<1

and y e/1ÿ y e3y e<10ÿ3mH or 1ÿ y e<10ÿ3,

as well as for mH>1 and y e<10ÿ3 or y e/

1ÿ y e3(1ÿ y e)ÿ1>103mH, s*<1.01 and conse-

quently the ln j and y on Z dependencies coincide

with those corresponding to a Langmuir adsorp-

tion.

Finally, it should be mentioned that non-steady elec-

trochemical techniques should be used in order to ob-

serve the e�ect of the interaction parameter u on the

HER and therefore the di�erence with a Langmuirian

behavior, by producing a signi®cant variation of yduring the transient. This results could be obtained

with, for example, the open circuit overpotential decaytechnique [16, 17].

5. Conclusions

The kinetics of the hydrogen evolution reactionthrough the Volmer±Heyrovsky and Volmer±Tafel

routes when the Frumkin adsorption model describesthe behavior of the adsorbed intermediate has beendiscussed. It has been demonstrated that the existenceof Tafelian domains is a consequence of the constancy

or a very slight variation of the surface coverage onoverpotential. This can explain the invariability of theTafel slopes with the type of adsorption (Langmuir or

Frumkin) used to describe the adsorbed hydrogen.Relationships between the quantities obtained by

extrapolation of the Tafelian domains with the kinetic

parameters of each route has been found. It has beenalso veri®ed that the usually de®ned as the character-istic behavior for a Frumkian adsorption at low over-potentials is actually a pseudo-Tafelian domain.

Therefore, the attainment of Tafel slopes in this case isarbitrary and the determination of the kinetic par-ameters of the elementary reaction steps from the ori-

gin ordinate is not possible.

Acknowledgements

This work was supported by the Consejo Nacionalde Investigaciones CientõÂ ®cas y TeÂ cnicas (CONICET,

Argentina) and the Universidad Nacional del Litoral(UNL, Santa Fe, Argentina).

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Documents

Kinetics of hydrogen evolution reaction with Frumkin adsorption: re-examination of the Volmer–Heyrovsky and Volmer–Tafel routes