Evaluation of the kinetic parameters of the hydrogen electrode

reaction from the analysis of the equilibrium polarisation resistance

Jose L. Fernandez, Marıa R. Gennero de Chialvo and Abel C. Chialvo*

Programa de Electroquımica Aplicada e Ingenierıa Electroquımica (PRELINE), Facultad deIngenierıa Quımica, Universidad Nacional del Litoral, Santiago del Estero 2829, 3000, Santa Fe,Argentina. E-mail: achialvo@fiqus.unl.edu.ar

Received 20th March 2003, Accepted 15th May 2003First published as an Advance Article on the web 3rd June 2003

The theoretical expression of the equilibrium polarisation resistance as a function of the partial hydrogenpressure and the activity of protons has been established for the hydrogen electrode reaction operating throughthe Volmer–Heyrovsky–Tafel mechanism. The corresponding elementary kinetic parameters have beenevaluated using experimental results reported in the literature, obtained on smooth platinum electrodes. Theadvantage of the proposed method for the calculation of these parameters with respect to the conventionaldetermination of the polarisation curves at high overpotential values has been also demonstrated.

Introduction

The experimental determination of the kinetic parameters ofthe elementary steps involved in a given mechanism for thehydrogen electrode reaction (HER) is currently carried outthrough the evaluation of the dependence of the current den-sity ( j) on overpotential (Z), for high Z values (|Z| > 0.050V). This experimental dependence can be obtained for boththe hydrogen evolution reaction (her) and the hydrogen oxida-tion reaction (hor). On these conditions, the rate of a given ele-mentary step i (vi ¼ v+i� v�i) is controlled by the rate of theforward reaction (v+i), as the contribution of the backwardreaction (v�i) is less significant. As the kinetic parameters areindependent of the direction of the net process, the conven-tional evaluation leads to uncertain values of the kinetic para-meters corresponding to the backward reactions and in manycases they are completely ignored. However, a correct kineticcharacterisation of the reaction requires the knowledge of allthe parameters of the elementary steps involved in the mechan-ism. On the other hand, near the equilibrium potential the for-ward and backward reaction rates of an elementary step areboth significant, contributing with a similar weight to the j(Z)dependence and therefore all the kinetic parameters can bemore appropriately evaluated. It should be noticed that inmany cases this dependence cannot be experimentally deter-mined due to the existence of corrosion processes. Therefore,the equilibrium polarisation resistance can be evaluated onlyif the working electrode defines a reversible potential for theHER in the electrolyte solution used.The purpose of the present work is to obtain the kinetic

parameters for the HER from the variation of the equilibriumpolarisation resistance (Rp) on the hydrogen pressure (PH2

)and the activity of protons (aH+). The Rp values are obtainedfrom the experimental dependence jexp(Z,PH2

,aH+) obtainednear the equilibrium potential. In order to perform the evalua-tion of the parameters, the theoretical expression for Rp(PH2

,aH+) ¼@Z/@j)Z¼ 0 must be obtained. The analytical dependence of Rp

on the equilibrium reaction rates of the elementary steps (vei )for the Volmer–Heyrovsky–Tafel mechanism was alreadyderived. The expressions for the Volmer–Heyrovsky andVolmer–Tafel routes were obtained by Makarides1 and thatcorresponding to the simultaneous occurrence of both routes

was derived previously by the authors.2 Starting from thisequation, the dependence Rp(PH2

,aH+) was developed and thenapplied to experimental results obtained from the literature.

Theoretical analysis

(a) Derivation of the expression Rp(PH2,aH+)

The well-known Volmer–Heyrovsky–Tafel mechanism forthe hydrogen electrode reaction in acidic solutions can bewritten as:

Hþ þ e�)v�V

*vþV

HðaÞ ð1aÞ

Hþ þHðaÞ þ e�)v�H

*vþH

H2ðgÞ ð1bÞ

HðaÞ þHðaÞ)v�T

*vþT

H2ðgÞ ð1cÞ

The expression of the equilibrium polarisation resistance onterms of the equilibrium reaction rates is:2

Rp ¼ RT

4F2

4veT þ veH þ veVveVv

eH þ veVv

eT þ veHv

eT

� �ð2Þ

where vei is the equilibrium reaction rate of the elementary stepi (i ¼ V,H,T). The corresponding expressions of these reactionrates as a function of the hydrogen pressure (PH2

) and theactivity of protons (aH+) are given by:3

veV ¼ kþVð1� yeÞaHþe�ð1�aVÞfEe ¼ k�VyeeaVfEe ð3aÞ

veH ¼ kþHyeaHþe�ð1�aH ÞfEe ¼ k�Hð1� yeÞPH2eaHfE

e ð3bÞ

veT ¼ kþT ðyeÞ2 ¼ k�T ð1� yeÞ2 PH2ð3cÞ

where k+i and k�i are the forward and backward specific rateconstants of the elementary step i (i ¼ V,H,T) respectively,which are independent of both the hydrogen pressure andthe activity of protons. ai (i ¼ V,H) is the symmetry factorof the step i. ye is the equilibrium surface coverage of theadsorbed hydrogen H(a) . Ee is the equilibrium potential ofthe HER and f ¼ F/RT.

DOI: 10.1039/b303183g Phys. Chem. Chem. Phys., 2003, 5, 2875–2880 2875

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Both Ee and ye are functions of the partial hydrogen pres-sure and the activity of protons:

e fEe ¼ e fE

�aHþ

P1=2H2

ð4Þ

ye ¼P1=2H2

K1=2T þ P

1=2H2

ð5Þ

where E� is the standard equilibrium potential of the HER(E� ¼ 0.0) and KT is the equilibrium constant of the Tafel step(KT ¼ k+T/k�T). eqns. (3a)–(3c) can be rewritten as follows:

veV ¼kþVa

aVHþK

1=2T P

ð1�aVÞ=2H2

ðK1=2T þ P

1=2H2

Þð6aÞ

veH ¼kþHa

aHHþP

ð1�aH=2ÞH2

ðK1=2T þ P

1=2H2

Þð6bÞ

veT ¼ kþTPH2

ðK1=2T þ P

1=2H2

Þ2ð6cÞ

Substituting eqns. (6a)–(6c) into eqn. (2) an expression forRp(PH2

,aH+) is obtained:

Rp ¼ RT

4F2

K1=2T þ P

1=2H2

� �4kþTPH2

þ K1=2T þ P

1=2H2

� �nK

1=2T þ P

1=2H2

� �kþVkþHa

ðaVþaHÞHþ K

1=2T P

ð3�aV�aHÞ=2H2

n8<:

� kþVaaVHþK

1=2T P

ð1�aVÞ=2H2

þ kþHaaHHþP

ð2�aHÞ=2H2

h ioþkþHkþTa

aHHþP

ð4�aHÞ=2H2

þ kþVkþTaaVHþK

1=2T P

ð3�aVÞ=2H2

o9=;

ð7Þ

The set of independent kinetic parameters (k+V ,k+H ,k+T ,K

1=2T ,aV ,aH) can be obtained from the experimental data

for Rp(PH2,aH+) and eqn. (7). The other kinetic parameters

can be evaluated from the following relationships:

k�V ¼ K1=2T kþV ð8Þ

k�H ¼ kþH

K1=2T

ð9Þ

k�T ¼ kþT

KTð10Þ

which are easily derived from eqns. (3a)–(3c) and eqn. (4).

(b) Remarks about rate constants k�i

It should be noticed that the rate constants defined in eqns.(3a)–(3c) are different from those involved in the usual kineticderivation of the HER, where the corresponding expressions ofthe equilibrium reaction rates are: 4

veV ¼ k0þVð1� yeÞ ¼ k0�Vye ð11aÞ

veH ¼ k0þHye ¼ k0�Hð1� yeÞ ð11bÞ

veT ¼ k0þTðyeÞ2 ¼ k0�Tð1� yeÞ2 ð11cÞ

The rate constants k0�i depend on T, PH2and, with the

exception of the Tafel step, on aH+ , while k�i depends onlyon temperature. The relationships between both types of rateconstants are obtained from the comparison of eqns. (3a)–(3c) and (11a)–(11c):

k0þV ¼ kþVaaVHþP

ð1�aVÞ=2H2

; k0�V ¼ k�VaaVHþP

�aV=2H2

ð12aÞ

k0þH ¼ kþHaaHHþP

ð1�aHÞ=2H2

; k0�H ¼ k�HaaHHþP

ð1�aH=2ÞH2

ð12bÞ

k0þT ¼ kþT; k0�T ¼ k�TP�1H2

ð12cÞ

Furthermore, while two different sets of kinetic parameterscan be obtained from the constants k0�i that produce thesame dependence j(Z), 4 the rate constants k�i constitute aunique set.

Application to experimental data

In spite of being the HER the most studied electrode reaction,works dealing with the analysis of the dependence jexp(Z) nearthe equilibrium potential are rather scarce.5–13 Several of thesestudies evaluated the exchange current density ( j�) at atmo-spheric pressure, but not the influence of the hydrogen pressureand pH.8–13 Thus, the most complete experimental data relatedto the function jexp(Z,PH2

, aH+) near equilibrium were obtainedby Schuldiner at 25 �C on smooth platinum electrodes, where aspecial attention to the purity of the electrolyte solution waspaid.5,6 One of them reported the polarisation curves jexp(Z)in 1 M H2SO4 solution at different hydrogen pressure.5 Thecorresponding values of Rp as a function of PH2

were alsogiven. The other studied the dependence Rexp

p (aH+) at PH2¼

1 atm.6 Starting from these results, the calculation of thekinetic parameters of the HER by non-linear regression ofall these experimental dependences was carried out. In orderto verify the descriptive capability of this procedure, each setof kinetic parameters was used to simulate the other experi-mental results different from that used in the regression.Regarding to the symmetry factors, the value aV ¼ aH ¼ 0.5was used in all the calculations.

(a) Kinetic parameters from Rp(PH2)

Experimental values of the polarisation resistance on smoothplatinum electrodes in 1 M H2SO4 solution at 25 �C were mea-sured in the range of hydrogen pressure 0.05 atm � PH2

�1 atm.5 These results were used in order to obtain the kineticparameters for the HER. The activity of protons needed ineqn. (7) was evaluated from aH+ ¼ 10�pH. As pH(1M H2SO4) ¼0.0125,14 then aH+ ¼ 0.9716. The result obtained from thenon-linear regression of the reported data using eqn. (7) is illu-strated in Fig. 1 (continuous line). The resulting values of thekinetic parameters can be observed in Table 1.

Fig. 1 Rp(PH2) dependence for the HER on Pt in 1 M H2SO4 . (S)

Experimental values;5 (—) Regression curve; (� � �) Simulation curveusing the kinetic parameters obtained from Rexp

p (aH+); (---) Simulationcurves using the kinetic parameters obtained from log j(Z) at dif-ferent PH2

.

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Furthermore, starting from these parameter values, thedependence Rp(aH+) at PH2

¼ 1 atm was simulated and thecorresponding curve is shown in Fig. 2 (dot line). The polarisa-tion curves log j(Z) in 1 M H2SO4 solution at different hydro-gen pressure were also simulated with the kinetic parametersobtained from the regression of Rexp

p (PH2). The equations used

in this simulation are described in subsection (c), below, andthe results obtained are shown in Fig. 3 (dashed lines).

(b) Kinetic parameters from Rp(aH+)

Experimental values of the polarisation resistance on smoothplatinum electrodes at 25 �C and PH2

¼ 1 atm were determinedin the range of pH comprised between 0.5� pH� 2.7, which interms of proton activity is 0.002� aH+� 0.30.6 It should benoticed that these measures were carried out with the presenceof other cations (Na+, NH4

+), being the ionic force of the elec-trolyte solution of approximately one third of that correspond-ing to the solution used in the evaluation described in theprevious item. These experimental data were also used in orderto obtain the kinetic parameters for the HER using eqn. (7).The result obtained from the non-linear regression is illu-strated in Fig. 2 (continuous line) and the values of the kineticparameters can be observed in Table 1.The set of kinetic parameters obtained in the regression was

used for the simulation of the dependence Rexpp (PH2

) ataH+ ¼ 0.9716 and the corresponding curve is shown in Fig. 1

(dot line). The polarisation curves log j(Z) in 1 M H2SO4

solution at different hydrogen pressure were also simulatedand the results obtained are shown in Fig. 3 (dot lines).

(c) Kinetic parameters from log j(g)

The polarisation curves log j(Z) were also determined onsmooth platinum electrodes in 1 M H2SO4 solution at 25 �Cin the range of hydrogen pressure 0.05 atm�PH2

� 1 atm.5

Therefore, other sets of kinetic parameters for the HER wereobtained by applying non-linear regressions to these experi-mental curves. The theoretical expression of j(Z), when aV ¼aH ¼ a is considered, can be written as follows:4

jðZÞ ¼ 2Feaf Z

(veVe

�f Z � veH� �

ð1� yeÞ

�veVe

�f Z � veH� �

ð1� yeÞ þveV � veHe

�f Z� �

ye

� �yðZÞ

)ð13Þ

where vei are defined in eqns. (6a)–(6c) and y(Z) satisfies theequation:

ay2 þ byþ c ¼ 0 ð14Þ

a ¼ 2veTð1� 2yeÞye

2ð1� yeÞ2ð15aÞ

b ¼ 4veTð1� yeÞ2 þ eaf Z

veVe�f Z þ veH

ð1� yeÞ þ veV þ veHe�f Z

ye

� �ð15bÞ

c ¼ � 2veT

ð1� yeÞ2� eaf Z

veVe�f Z þ veH

ð1� yeÞ

� �ð15cÞ

The results obtained for the non-linear regression of the curvesj exp(Z) are shown in Fig. 3 (continuous curves) for the differentvalues of the hydrogen pressure, while the values of the corre-sponding kinetic parameters are illustrated in Table 1.The sets of kinetic parameters obtained in these regressions

were used for the simulation of the dependence Rexpp (PH2

) in 1M H2SO4 solution. All the simulation curves for the differentPH2

values are shown in Fig. 1 (dashed lines). Likewise, thedependence Rp(aH+) at PH2

¼ 1 atm was simulated and thecorresponding curves are shown in Fig. 2 (dashed lines).

Discussion

The theoretical expression of the dependence of the polarisa-tion resistance of the HER on the hydrogen pressure and theactivity of protons has been derived for the Volmer–Heyrovsky–Tafel mechanism. Its application for the evaluationof the elementary kinetic parameters through the correlation ofthe dependence Rexp

p (PH2, aH+) was carried out for the case

of a platinum electrode in acidic solutions. The proposedmethod needs that the working electrode in the solution usedcould get the reversible potential of the hydrogen electrode

Table 1 Kinetic parameters for the HER evaluated from the regression of Rexpp (PH2

), Rexpp (aH+) and log jexp(Z) at different PH2

Kinetic

parameters

Evaluated from

Rp(PH2) Rp(aH+) log j(Z)

PH2/atm 1.00 1.00 1.00 0.46 0.31 0.24 0.14 0.05

k+V 10�5 3� 10�7 2� 10�5 10�5 1.1� 10�5 1.1� 10�5 1.1� 10�5 1.13� 10�5

k+H 6.37� 10�8 6.8� 10�8 6.58� 10�8 6.37� 10�8 6.37� 10�8 6.38� 10�8 1.5� 10�8 6.37� 10�8

k+T 0.5254 0.7693 0.5291 0.5552 0.4592 0.4473 0.4085 0.5254

KT 9256 9149 8316 9255 9256 9286 9998 9425

k�V 0.09256 0.002745 0.16632 0.09255 0.09256 0.09286 0.09998 0.1065

k�H 6.88� 10�12 7.43� 10�12 7.91� 10�12 6.89� 10�12 6.88� 10�12 6.87� 10�12 1.50� 10�12 6.76� 10�12

k�T 6.13� 10�9 9.19� 10�9 7.65� 10�9 6.48� 10�9 5.36� 10�9 5.19� 10�9 4.09� 10�9 5.91� 10�9

Fig. 2 Rp(aH+) dependence for the HER on Pt at PH2¼ 1 atm. (S)

Experimental values;6 (—) Regression curve; (. . .) Simulation curveusing the kinetic parameters obtained from Rexp

p (PH2); (---) Simulation

curves using the kinetic parameters obtained from log j(Z) at dif-ferent PH2

.

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reaction. This is a limitation because corrosion processes cantake place at those potentials in certain materials such asiron.15 However, for systems without this limitation, the proce-dure is advantageous with respect to the traditional methodsbased on the analysis of the polarisation curves log j(Z). Inter-fering factors such as those originated in the electrode poros-ity,16 the bubbles coverage produced at high overpotentials,17

the superficial hydrogen supersaturation,18 the localised alkali-nisation,19 etc., which are characteristics of the experimentaldetermination of j(Z) at potentials far away from equilibrium,can be considered negligible in the experimental conditionswhere the equilibrium polarisation resistance is evaluated.The simulations carried out using the kinetic parameters

obtained from the non-linear regression of the dependenceRexp

p (PH2) demonstrate the capacity of the method for the eva-

luation of such parameters. It should be noticed that, forexample, the dependence Rp(aH+) could be predicted for pHvalues far from those corresponding to the experimental condi-tions where the comparison is made. On this respect, it can beobserved that the relationship Rexp

p (aH+) shows a quite smoothdependence in the region 0.03� aH+� 1 and a sharp increase inthe range 0.002� aH+� 0.03. This behaviour is perfectly fittedby the Volmer–Heyrovsky–Tafel mechanism (continuous linein Fig. 2), but it is apparently quite sensitive to the values ofthe kinetic parameters. On this sense it should be taken intoaccount that, as it was already mentioned, the experimentalconditions in which this dependence was determined are notquite adequate for this study due to the absence of a support-ing electrolyte. On these conditions, it is unlikely that the zetapotential remained constant when pH was varied. This couldbe an explanation for the dispersion on the experimentalresults (filled circles in Fig. 2) and for the difference observedin the value of the parameter k+V , when the data shown inTable 1 are compared.

(a) Variation of the equilibrium reaction rates on PH2and aH+

Starting from the set of parameters that fitted appropriatelythe dependence Rexp

p (PH2), the variation of the equilibrium

reaction rates of the three elementary steps on the hydrogenpressure and the activity of protons were evaluated and areillustrated in Figs. 4, 5 and 6, respectively. It can be observedthat an increase in PH2

produces an increase in the vei values.

Fig. 4 Variation of veV on PH2and aH+ . Kinetic parameters obtained

from Rexpp (PH2

).

Fig. 3 log j(Z) dependence for the her on Pt in 1 M H2SO4at different PH2(indicated in the figure). (S) Experimental values;5 (—) Regression curve;

(. . .) Simulation curve using the kinetic parameters obtained from Rexpp (aH+); (---) Simulation curves using the kinetic parameters obtained from

Rexpp (PH2

).

Fig. 5 Variation of veH on PH2and aH+ . Kinetic parameters obtained

from Rexpp (PH2

).

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Similar variations are obtained with aH+ , with the exception ofveT, which remains constant.

(b) Analysis of the hydrogen evolution reaction (her) on Ptelectrodes

The relationships log j(Z) and y(Z) were simulated on the catho-dic overpotentials region 0.005 V� |Z|� 0.40 V, using thekinetic parameters obtained in the regression of the depen-dence Rexp

p (PH2) (Table 1). The resulting simulated polarisation

curves are shown in Fig. 7, where an increase on the electroca-talytic activity when PH2

increases is observed. It can be clearlyappreciated the existence of two Tafelian regions. That corre-sponding to the lower overpotentials (0.025 V� |Z|� 0.125 V)has a Tafel slope equal to RT/2F and a very slight variationof y(Z), which is more evident as the hydrogen pressuredecreases. The other Tafelian domain (0.20 V� |Z|� 0.40 V)shows a slope equal to 2RT/F and a larger variation of y(Z)than the preceding case. On the basis of a rigorous kineticdescription of the her, it is possible to interpret the valuesresulting from the extrapolation at Z ¼ 0 of both Tafelianstraight lines. For the first case, taking into account that atlow Z values y(Z)� 1 and that veV � veT � veH (Table 2), fromeqn. (13) the following expression is obtained:

jðZÞ ffi 2FveTe�2f Z ð16Þ

being jextl ¼ 2FveT the extrapolated current density of the Tafelregion at low overpotentials, as it is illustrated in Fig. 7 (opensquares) for all hydrogen pressure values. At higher Z values,but in the region where y(Z) has not achieved its maximumvalue yet, the Tafelian region follows the expression:20

jðZÞ ffi FveVe�ð1�aÞf Z ð17Þ

The corresponding value of the extrapolated current den-sity of the Tafel region at high overpotentials is jexth ¼ 2FveV,which is also shown for each hydrogen pressure in Fig. 7 (opencircles).These results indicate that the her on Pt is verified mainly

through the Tafel step in the range of overpotentials analysedand consequently the Heyrovsky step needs higher overpoten-tial values (|Z| > 0.5 V) in order to compete with the Tafel one.Finally, it should be noticed that the proposed methodology

is ideal for the study of the HER on porous electrodes, wherethe presence of pores produces interferences (ohmic effect,local alkalinisation, diffusion, etc.) that make uncertain theevaluation of the kinetic parameters.

Acknowledgement

The financial support of Consejo Nacional de InvestigacionesCientıficas y Tecnicas (CONICET), Agencia Nacional de Pro-mocion Cientıfica y Tecnologica (ANPCYT) and UniversidadNacional del Litoral is gratefully acknowledged.

Fig. 6 Variation of veT on PH2and aH+ . Kinetic parameters obtained

from Rexpp (PH2

).

Fig. 7 Simulation of (A) log j(Z) and (B) y(Z) in the region 0.005V� |Z|� 0.40 V, using the kinetic parameters obtained from Rexp

p (PH2).

PH2: (a) 0.05; (b) 0.14; (c) 0.24; (d) 0.31; (e) 0.46; (f) 1 atm. (K) jextl ; (X)

jexth .

Table 2 Equilibrium reaction rates and equilibrium surface coverage values for the HER calculated with the kinetic parameters evaluated from the

regression of Rexpp (PH2

) at different PH2

Parameter Rp(PH2)

PH21.00 0.46 0.31 0.24 0.14 0.05

veV 9.86� 10�6 8.12� 10�6 7.35� 10�6 6.89� 10�6 6.03� 10�6 4.66� 10�6

veH 6.78� 10�12 3.79� 10�12 2.81� 10�12 2.32� 10�12 1.55� 10�13 7.17� 10�13

veT 6.13� 10�9 2.82� 10�9 1.90� 10�9 1.47� 10�9 8.58� 10�10 3.07� 10�10

ye 1.080� 10�4 7.326� 10�5 6.015� 10�5 5.292� 10�5 4.042� 10�5 2.416� 10�5

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