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

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  • Kinetics of hydrogen evolution reaction with Frumkinadsorption: re-examination of the VolmerHeyrovsky and

    VolmerTafel routes

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

    Programa de Electroqumica Aplicada e Ingeniera Electroqumica (PRELINE), Facultad de Ingeniera Qumica, Universidad

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

    Received 3 April 1998


    A re-examination of the basic kinetic derivations of the VolmerHeyrovsky and VolmerTafel 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 dierent 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 dierences 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 VolmerHeyrovsky (VH) and the VolmerTafel

    (VT) routes. Using the approximation of the rate

    determining step (rds), diagnostic criteria were estab-

    lished and widely used for both routes [17].

    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

    [13, 5, 6]. On this range the relationship y/(1 y) hasbeen approximated to unity and in the resulting ex-

    pressions of the reaction rate, the dependence of y onoverpotential (Z) in the pre-exponential factor has beenneglected [3, 6]. As a result of these approximations, it

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

    overpotentials region should be influenced for the reac-

    tion symmetry factor (a) and for the adsorption sym-metry factor (l) as well. Nevertheless, from theanalysis 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 ofthe 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 VolmerHeyrovskyTafel 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) 841851

    0013-4686/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved.PII: S0013-4686(98 )00233-3


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

  • 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 fitted

    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-fied. 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 defined 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 fulfilled, 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 2e $ H2 g 2OH , 1

    will be derived in steady state for both VolmerHeyrovsky and VolmerTafel 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. VolmerHeyrovsky route

    The elementary steps are

    H2O e $ Ha OH Volmer, 3

    H2OHa e $ H2 g OH Heyrovsky: 4

    The equations for the rate of the dierent reactionsteps with a Frumkian behavior of the adsorbed

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

    vV veV1 y1 ye s

    le1a fZ, 5

    vV veVyyes1leafZ, 6

    vH veHyyes1le1a fZ, 7

    vH veH1 y1 ye s

    leafZ, 8


    s euyye, 9

    u the interaction parameter between the adsorbedhydrogen atoms, v + i and v 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 V1). 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 0:5vV vH, vV vV vV,vH vH vH:


    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) 841851842

  • VveV 1 y1 ye s

    le1a fZ yyes1leafZ

    mHyyes1le1a fZ 1 y

    1 ye sleafZ


    1 y1 ye s

    le1a fZ yyes1leafZ

    mHyyes1le1a fZ 1 y

    1 ye sleafZ

    , 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 yes1mH efZ

    1 ye1mH efZ yes1mH efZ : 12

    The amounts y/y e and (1 y)/(1 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,


    j0V 2mHs

    le2a fZ eafZ1 ye1mH efZ yes1mH efZ , 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 VolmerHeyrovsky 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 ye values, a linear variation can be

    obtained. The cases in which two Tafelian domainscan be distinguished will be analyzed first,

    (a) For y e

  • 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 termsof current density is reduced to


    j0V 2mH y*ye s*

    1l e1a fZ

    2 1 y*1 ye s*

    le1a fZ, 17

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

    ext) is

    jexth 2mHj0Vy*ye

    s*1l 2mHj0Vs*1l

    ye 1 yemHs* : 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

    *(1 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 distinguishedthe 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 isnecessary that the condition y(Z)3y * be fulfilled at anvZv value suciently low such that the first Tafeliandomain cannot be developed. In order to determine

    the y e and mH values that obey such condition, anoverpotential Z # such that y(Z #) y *=102 wasdefined 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 e1a fZ, jexth 2mHj


    ye 1 yemH : 19

    Fig. 4 shows the dependence Z # vs. log[y e/(1 y e)] fordierent mH. It can be easily established that fory e3 103 or for y e>0.5 andmH

  • a= l= 0.5, u= 10 and mH=104. In the range 0.05

    VR vZvR 0.20 V, where 0.2083R y(Z)R 0.5976, a slightlycurved dependence is observed. The linear regression insuch range gives a slope equal to 57.6 mV dec1 and anorigin ordinate equal to 8.2218. These results will beanalyzed 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

  • vT veT1 y21 ye2 s

    2l, 25

    where v +T and v T 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 vT, vV vV vV,vT vT vT:


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


    veV 0:5

    1 y1 ye s

    l e1a fZ yyes1l eafZ


    ye2s21l 1 y


    1 ye2 s2l

    1 y1 ye s

    l e1a fZ yyes1l eafZ


    ye2s21l 1 y


    1 ye2 s2l, 27

    where mT= v Te /vV

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

    ayy2 by, Zy cy, Z 0, 28where

    ay 2mTs21l

    ye2 s


    1 ye2, 28a

    by, Z 4mTs2l

    1 ye2 eafZefZsl

    1 ye s1l


    , 28b

    cy, Z 2mTs2l

    1 ye2 e1a fZsl

    1 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 VolmerTafel 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 verified 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

  • Consequently, the VolmerTafel route at overpoten-tials suciently high always defines, 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 VolmerTafel route [3, 6], that cannot be justified 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=105, y e=101, l= a= 0.5

    and 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 dec1 and an origin ordinate equal to10.3427. These results will also be analyzed in detailin Section 4.Furthermore, the simulations shown in Fig. 6

    allowed finding 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 fittedlines at Z= 0 does not give an amount related to theparameter 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 equalto 168.6 mV dec1. The same for line b in the range0.30 VR vZvR 0.65 V, where the Tafel slope is equal to136.7 mV dec1. 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 dierent 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/(1 a)F andj 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=105; a= l= 0.5; u= 5; mT=(a) 107, (b) 105, (c)

    103, (d) 101, (e) 10. ln( j lext/jV

    0 ): (*) Eq. (31), (w) Eq. (33);(q) ln( jTlim/jV0 ).

    Fig. 7. Dependence of ln( j/jV0 ) and y on Z for the VT route.

    y e=101; a= l= 0.5; u= 10; mT=105. ln(j l

    ext/jV0 ): (W)

    linear fitting; (r) Eq. (41).

    M. Gennero de Chialvo, A. Chialvo / Electrochimica Acta 44 (1998) 841851 847

  • 4. Discussion

    The dependencies of current density and surface cov-

    erage on overpotent...


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