##
1Hydrogen diffusion effects on the kinetics of the hydrogen electrode reactionPart II.^{} Evaluation of kinetic parameters
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Hydrogen diffusion effects on the kinetics of the hydrogen electrode reactionPart II.

^{}Evaluation of kinetic parameters
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The present work proposes a methodology for the evaluation of the kinetic parameters of the hydrogen electrode reaction for the Tafel–Heyrovsky–Volmer mechanism from experimental results corresponding to the study of the hydrogen oxidation on a rotating disc electrode.

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It is based on the correlation of the experimental dependence

*j*(*η,ω*) with a theoretical expression together with some constraints that involve the use of the equilibrium polarisation resistance, which reduces the number of adjustable parameters.
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This methodology was applied to the hydrogen oxidation reaction on a platinum electrode, which was studied in the range 900 ≤

*ω*/rpm ≤ 8100 and −0.05 ≤*η*/V ≤ 0.40 in 0.5 M H_{2}SO_{4}.
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The results obtained have been discussed in the light of the theoretical treatment described in Part I of this work.

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## Introduction

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Diffusion plays an important role in the hydrogen electrode reaction (HER), especially when it is taking place in the anodic direction, the hydrogen oxidation reaction (hor).

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Consequently, the dependence of the current density on overpotential

*j*(*η*) was described in many cases as a purely diffusional process, commonly named a ‘reversible reaction’.^{1–3}
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On the other hand, more recent studies consider the hor a mixed controlled reaction, though with a strong diffusional contribution.

^{4–7}
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In this sense, the theoretical analysis described in the first part

^{}of the present work indicated that it is necessary for the application of diagnostic criteria and/or appropriate methodologies for the evaluation of the kinetic parameters characteristics of the HER from the analysis of the experimental results corresponding to the hydrogen oxidation.^{8}
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It should be taken into account that the kinetic mechanism for the HER must be unique, independently if the reaction proceeds in the anodic (hor) or cathodic (hydrogen evolution reaction, her) direction and therefore, the set of kinetic parameters must be valid for both cases.

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Besides, the correct interpretation of the experimental results needs the appropriate description of the behaviour of the reaction intermediate

*H*_{(a)}and/or those of other adsorbed species.
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These two aspects should lead to a successful description of the HER in the whole range of overpotentials.

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The present work proposes a methodology for the evaluation of the kinetic parameters of the HER for the Tafel–Heyrovsky–Volmer mechanism from experimental results.

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It is based on the correlation of the experimental dependence

*j*(*η,ω*) with a theoretical expression together with some constraints that involve the use of the equilibrium polarisation resistance, which theoretical dependence on the rotation rate is also derived.
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The proposed methodology is applied for the study of the hydrogen oxidation on a rotating disc electrode (RDE) of polycrystalline platinum.

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## Evaluation of the kinetic parameters

16

Theoretical expressions of the current density (

*j*) and the surface coverage (*θ*) as a function of overpotential (*η*) and limiting diffusional current density (*j*_{L}) are necessary in order to evaluate the kinetic parameters of the hydrogen electrode reaction through the correlation of the experimental dependence*j*(*η*) of the hydrogen oxidation on a rotating disc electrode at different rotation rates (*ω*).
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The expressions

*j*(*η*,*j*_{L}) and*θ*(*η*,*j*_{L}) corresponding to the simultaneous occurrence of the Tafel, Heyrovsky and Volmer steps were derived in Part I:^{8}being*α*the symmetry factor,*f*=*F*/*RT*,*θ*^{e}the equilibrium surface coverage and*v*_{i}^{e}the equilibrium reaction rate of the step*i*(*i*= Volmer, Heyrovsky, Tafel).
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As the current density generally reaches its maximum value at low overpotentials,

*j*values near the equilibrium condition have only very little influence on the correlation of the experimental results obtained in a wide overpotentials region.
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Therefore, the use of the equilibrium polarisation resistance [

*R*_{p}= d*η*/d*j*∣*] is proposed in order to improve the evaluation of the kinetic parameters.*_{η = 0}
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The theoretical expression of the dependence

*R*_{p}=*R*_{p}(*v*_{T}^{e},*v*_{H}^{e},*v*_{V}^{e},*θ*^{e},*ω*) can be obtained from the derivation of the first equality of eqn. (1) with respect to*η*.
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21

The resulting equation, for

*η*→ 0 is:The corresponding expression for d*θ*/d*η*∣*was obtained from the derivation of eqn. (2) and the application of the condition*_{η = 0}*η*→ 0.
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After reordering, the equation can be written as follows:

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Substituting eqn. (4) into eqn. (3) and rearranging, the expression for

*R*_{p}(*ω*) is obtained: where*j*_{L}was substituted by the following relationship:^{9}*j*_{L}=*Bω*^{1/2}Furthermore, the corresponding expression for the origin ordinate*R*_{p}^{o}of the linear dependence*R*_{p}=*f*(*ω*^{−1/2}), which can be defined as the equilibrium polarisation resistance free from diffusion effects, is:
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This equation is coincident to that previously derived for the hydrogen evolution reaction taking place under activated control.

^{10}
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Consequently, if

*R*_{p}^{o}is known, a relationship between the parameters*v*_{T}^{e},*v*_{H}^{e}and*v*_{V}^{e}can be obtained from eqn. (7), which can be written as:
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From the use of eqn. (8), the behaviour of the dependence

*j*(*η*) at low overpotentials can be included in the determination of the kinetic parameters of the HER.
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Although the parameters

*v*_{T}^{e},*v*_{H}^{e},*θ*^{e}and*B*can be evaluated from the correlation of the experimental results with the simultaneous use of eqns. (1), (2) and (8), the theoretical analysis described in the first part of the present work indicated that two possible situations can be distinguished.^{8}
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They are determined for the value of the origin ordinate of the linear dependence

*j*_{max}^{−1}=*f*(*ω*^{−1/2}).*j*_{max}is the maximum current density observed experimentally in the usual overpotentials range (*η*< 0.6 V), which is achieved when*θ*(*η*) = 0.
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From eqns. (17–19) of Part I

^{8}and eqn. (6), such dependence can be written as follows:This equation shows that for*η*< 0.6 V, the origin ordinate is zero when the Tafel, Heyrovsky and Volmer steps are taking place simultaneously, with*v*_{H}^{e}> 10^{−14}mol s^{−1}cm^{−2}.
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The other alternative is an origin ordinate with a positive value and corresponds to

*v*_{H}^{e}< 10^{−14}mol s^{−1}cm^{−2}.
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In this case, eqn. (9) is reduced to eqn. (14) of Part I.

^{8}
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### (a) *v*_{H}^{e} > 10^{−14} mol s^{−1} cm^{−2}

32

An origin ordinate of

*j*_{max}^{−1}=*f*(*ω*^{−1/2}) equal to zero implies that in eqn. (9)*v*_{H}^{e}*e*^{αfη}≫*v*_{T}^{e}/(1 −*θ*^{e}).
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In this case the kinetic parameters can be evaluated from the correlation of the experimental dependence

*j*(*η*,*ω*) through eqns. (1), (2) and (8), besides the use of the constants*R*_{p}^{o}and*B*.
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The value of

*R*_{p}^{o}is obtained from the extrapolation of the experimental linear dependence*R*_{p}=*f*(*ω*^{−1/2}), while*B*can be obtained from the slope of the straight lines*j*_{max}^{−1}=*f*(*ω*^{−1/2}) or*R*_{p}=*f*(*ω*^{−1/2}).
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The adjustable parameters are

*v*_{T}^{e},*v*_{H}^{e}and*θ*^{e}, being the others (*v*_{V}^{e}and*j*_{L}) evaluated from those obtained in the regression.
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### (b) *v*_{H}^{e} < 10^{−14} mol s^{−1} cm^{−2}

36

A positive value of the origin ordinate of

*j*_{max}^{−1}=*f*(*ω*^{−1/2}) implies that in eqn. (9)*v*_{H}^{e}*e*^{αfη}≪*v*_{T}^{e}/(1 −*θ*^{e}), for*η*< 0.6 V. The inverse of this value is the maximum kinetic current density*j*_{max}^{kin}, which can be written as:Besides, neglecting*v*_{H}^{e}, eqn. (8) turns to be:
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The dependences given by eqns. (1) and (2), with

*v*_{H}^{e}= 0, can be used together with eqn. (11) in the correlation of the experimental results.
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Therefore, the kinetic parameters to be obtained from the correlation of

*j*(*η*,*ω*) and the use of the constants*R*_{p}^{o}and*B*are*v*_{T}^{e}and*θ*^{e}, the other parameters being evaluated from them.
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The calculated values of the parameters

*v*_{T}^{e}and*θ*^{e}can be used for the evaluation of the limiting kinetic current density, through eqn. (10).
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This value should be equal to that obtained from the extrapolation of the experimental dependence

*j*_{max}^{−1}=*f*(*ω*^{−1/2}).
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This is a self-consistency test for the evaluated parameters.

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#### (b1) Approximated method

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An approximated method can be developed for the case described in the previous item when the equilibrium surface coverage of the adsorbed intermediate of the HER is very low and therefore it can be considered (1 −

*θ*^{e}) ≅ 1.
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Using this approximation, the parameter

*v*_{T}^{e}can be estimated from eqn. (10) and then*v*_{V}^{e}from eqn. (11).
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Therefore, approximated values of the kinetic parameters can be obtained in this way.

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### (c) Levich–Koutecky method

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The Levich–Koutecky method is usually employed for the evaluation of the dependence

*j*(*η*) of the hor under activated control.^{11}
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It consists in plotting the relationships

*j*(*η*,*ω*)^{−1}=*f*(*ω*^{−1/2}) at different overpotentials, which is supposed to be linear.
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The extrapolation of each straight line should give at the origin ordinate the inverse of

*j*at such*η*, value that corresponds to the faradaic process.
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Once the dependence

*j*(*η*) free from diffusion contribution is known, the usual procedure is applied for the evaluation of the kinetic parameters of the HER.
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Nevertheless, as has been demonstrated in Part I,

^{8}*j*(*η*,*ω*)^{−1}=*f*(*ω*^{−1/2}) is not necessarily linear, with the exception of the overpotentials range in which*j*=*j*_{max}.
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Consequently, the linear extrapolation is only valid if the condition given in eqn. (20) of Part I is satisfied.

^{8}
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As the values of the kinetic parameters

*v*_{T}^{e},*v*_{H}^{e},*v*_{V}^{e}and*θ*^{e}are unknown, it is impossible to verify*a priori*the applicability of the Levich–Koutecky method and therefore, its use is not recommended.
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## Evaluation of the kinetic parameters on platinum

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The methodology proposed above is applied to the evaluation of the kinetic parameters of the HER from experimental results obtained for the molecular hydrogen oxidation on a platinum RDE.

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### (a) Experimental details

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The experimental determination of the dependence

*j*(*η*) for the hydrogen oxidation reaction on steady state was carried out on a rotating disc electrode made of polycrystalline smooth platinum (0.07 cm^{2}geometric area).
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The rotation rate was varied in the range 900 ≤

*ω*/rpm ≤ 8100, using a Tacussel rotating disc equipment.
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The electrolyte solution was 0.5 M H

_{2}SO_{4}, which was subjected to a continuous and efficient hydrogen bubbling at*P*= 1 atm.
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The counterelectrode was a large area Pt electrode and the working temperature was fixed at 30 °C.

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The applied overpotential was varied in the range −0.05 ≤

*η*/V ≤ 0.40, controlled against a hydrogen electrode reference in the same electrolyte solution.
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Measurements were made using a potentiostat–galvanostat Radiometer, controlled by the generation/acquisition software Voltamaster.

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Special attention was paid to the purity of the electrolyte solution as well as to ensure its saturation with molecular hydrogen during the experimental runs.

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These two aspects are critical in order to obtain reproducible results.

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The verification of the solution purity was made through the application of two consecutive steps of potentiostatic adsorption under nitrogen bubbling at 0.4 V for 5 min and 0.0 V for another 5 min, respectively.

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Then, without interrupting the electric circuit, a potentiodynamic sweep in the anodic direction (up to 1.5 V) was applied, returning to 0.0 V and after that applying repetitive cycles between these limiting potential values.

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The purity of the solution is determined by the absence of voltammetric peaks due to electro-oxidation processes different from those characteristics of the Pt electrode and/or the inhibition of the peaks related to the adsorption/desorption of hydrogen.

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In order to ensure reproducible results in the evaluation of the experimental dependence

*j*(*η*), the potential program shown in Fig. 1a was applied.
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The electrode potential was held at 1.50 V for 4 s in order to oxidise any adsorbed potentially polluting substance, ensuring a reproducible initial condition.

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Then, a step was applied to the desired overpotential value, which was maintained for 30 s.

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During this period, readings of the current value were made every 2 s and the mean value of the last 20 s was assigned to this overpotential.

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After each measuring step, the corresponding oxidation step was applied.

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### (b) Results

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The current (

*I*) values obtained from the application of the potential program described in the previous paragraph can be observed in Fig. 1b for the case corresponding to a rotation rate of 1600 rpm.
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These results exhibit an excellent reproducibility when the solution is appropriately purified and the condition of hydrogen saturation is maintained during the whole run.

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The dependences

*j*(*η*) are shown in Fig. 2a, where different types of symbols were used for the different*ω*values.
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According to the methodology proposed in the present work, the value of the origin ordinate of the dependence

*j*_{max}^{−1}=*f*(*ω*^{−1/2}) should be determined.
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This dependence was evaluated from the experimental results shown in Fig. 2a and is illustrated in Fig. 3.

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An origin ordinate definitively finite, equal to

*j*_{max}^{kin−1}= 4.15 A^{−1}cm^{2}, was calculated from the linear regression.
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This result indicates that the HER operates through the Tafel–Volmer route, with a negligible contribution of the Heyrovsky step in the domain of overpotentials analysed.

^{8}
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Besides, the value

*B*= 0.068 mA cm^{−2}rpm^{−1/2}was obtained from the slope of the straight line of Fig. 3.
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Furthermore, starting from the values of the current density measured near the equilibrium potential (−0.05 ≤

*η*/V ≤ 0.05), the equilibrium polarisation resistance [*R*_{p}= d*η*/d*j*∣_{η=0}] was evaluated for each rotating rate.
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In order to obtain it, the dependence

*j*(*η*) at a given*ω*was adjusted with a third-order polynomial, the first order coefficient being the*R*_{p}value.
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It can be observed in Fig. 4 that the dependence

*R*_{p}=*f*(*ω*^{−1/2}) has the linear variation predicted by eqn. (5).
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The values

*R*_{p}^{o}= 0.546 Ω cm^{2}and*B*= 0.063 mA cm^{−2}rpm^{−1/2}were obtained from the origin ordinate and the slope, respectively, of the corresponding linear regression.
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It can be noted that there is an agreement between this

*B*value and the one evaluated from the slope of Fig. 3, corresponding to two clearly different and independent experimental determinations.
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82

Fig. 3 corresponds to data measured for

*η*> 0.2 V while Fig. 4 was obtained from data measured near the equilibrium potential.
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The experimental dependence

*j*(*η*,*ω*) was correlated with eqns. (1), (2) and (8), using the constant values*B*= 0.068 mA cm^{−2}rpm^{−1/2}and*R*_{p}^{o}= 0.546 Ω cm^{2}obtained from Figs. 3 and 4, respectively.
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84

It should be noted that the fitting was made with only two adjustable parameters,

*v*_{T}^{e}and*θ*^{e}, the other kinetic parameters (*v*_{V}^{e}and*j*_{L}) being evaluated from them.
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The results obtained are summarised in Table 1, for each value of the rotation rate.

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The values of

*j*_{L}obtained from eqn. (6) have also been included.
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87

It can be appreciated that the resulting values of

*v*_{V}^{e},*v*_{T}^{e}and*θ*^{e}for the different*ω*, are quite similar.
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88

In this sense, it should be borne in mind that the fittings were made on data obtained from independent experimental runs corresponding to seven different rotation rate values.

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89

The descriptive capability of the evaluated kinetic parameters is illustrated in Fig. 2a (continuous lines).

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90

It should be noted that these lines were calculated using only one set of parameters, corresponding to the average of the seven values obtained from each fitting (Table 1).

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91

These results demonstrate that the description of the hydrogen electrode reaction made through the kinetic study of the hydrogen oxidation reaction was adequate and self-consistent.

Type: Conclusion |
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92

Fig. 2b shows the dependence

*θ*(*η*) calculated with this set of parameters for the less and the greater experimental*ω*values and for the limiting case of activated control corresponding to*ω*= ∞ (continuous line).
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93

A large variation of the surface coverage can be observed with the rotation rate at a given overpotential.

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94

According to the results described above and taking into account that for platinum it can be considered (1 −

*θ*^{e}) ≅ 1, the approximated method can also be used.^{12}
Type: Conclusion |
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As was indicated in subsection (b1),

*v*_{T}^{e}can be estimated from eqn. (10) and then*v*_{V}^{e}can be evaluated from eqn. (11).
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96

The constant values

*j*_{max}^{kin−1}and*R*_{p}^{o}needed for the calculations were obtained from Figs. 3 and 4, respectively.
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97

The corresponding values of

*v*_{T}^{e}and*v*_{V}^{e}obtained in this way are shown in Table 1 (second column).
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98

Finally, the applicability of the Levich–Koutecky method was analysed.

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It has been demonstrated that this alternative is limited by the fulfilment of the condition given in eqn. (20) of Part I.

^{8}
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100

In the present case, as

*v*_{H}^{e}≅ 0, the condition is the following:Taking the most favourable value of*j*_{L}(0.01 A cm^{−2}), the condition of eqn. (11) is not fulfilled.
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101

Consequently, the Levich–Koutecky method is not applicable for this case.

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102

This result is in agreement with the marked effect of the rotation rate on the surface coverage at constant potential, which is shown in Fig. 2b.

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103

As has already been demonstrated, Levich–Koutecky plot can be applied only if

*θ*(*η*) is invariant with*j*_{L}^{8}.
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## Discussion

104

Starting from the kinetic expressions derived in the first part of this work, a methodology for the evaluation of the kinetic parameters of the hydrogen electrode reaction for the Tafel–Heyrovsky–Volmer mechanism from experimental data of the hydrogen oxidation on a rotating disc electrode has been proposed.

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105

Although the method consisted basically in the correlation of the experimental dependence

*j*(*η*,*ω*), other relationships were used in order to decrease the number of adjustable parameters as well as to improve the evaluation.
Type: Method |
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ConceptID: Met8

106

The first one is the relationship between the limiting diffusional current density and the rotation rate (eqn. (6)).

Type: Method |
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ConceptID: Met8

107

The second one is the relationship between

*R*_{p}^{o}and the equilibrium reaction rates of the elementary steps (eqn. (7)).
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108

Thus, the adjustable parameters are three (

*v*_{T}^{e},*v*_{H}^{e}and*θ*^{e}) and the others are obtained from them.
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109

Finally, in the case when the origin ordinate of

*j*_{max}^{−1}=*f*(*ω*^{−1/2}) is positive, a third constraint is generated (*v*_{H}^{e}≅ 0).
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110

In this case the adjustable parameters are reduced to two (

*v*_{T}^{e}and*θ*^{e}).
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111

This methodology was applied for the analysis of the experimental results obtained for the hydrogen oxidation reaction on a platinum RDE.

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112

The dependence

*j*_{max}^{−1}=*f*(*ω*^{−1/2}) has a finite origin ordinate for this electrode, which means that the maximum kinetic current density*j*_{max}^{kin}for the hor is finite.
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113

This result indicates that the hydrogen oxidation is verified under the Tafel–Volmer route (

*v*_{H}^{e}< 10^{−14}mol s^{−1}cm^{−2}).
Type: Conclusion |
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ConceptID: Con5

Type: Background |
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115

Furthermore, a finite value of

*j*_{max}^{kin}implies that, at least in the overpotentials range analysed, the maximum current density is different from the limiting diffusional current density, although in the present case the difference is small.
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116

On the other hand,

*j*_{max}^{kin}can also be evaluated from eqn. (10) using the mean value of*v*_{T}^{e}illustrated in Table 1, the resulting value is*j*_{max}^{kin}= 0.264 A cm^{−2}, which is quite similar to that (*j*_{max}^{kin}= 0.241 A cm^{−2}) obtained from the origin ordinate of Fig. 3.
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117

An attempt was made in order to show by a different method the finite value of

*j*_{max}^{kin}.
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118

Two alternatives are proposed, which correspond to different rearranging of eqn. (9), with

*v*_{H}^{e}≅ 0 and taking into account eqn. (10):The parameter*j*_{max}^{kin}can be evaluated from both eqns. (13) and (14).
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119

Taking into account that the equality

*j*_{max}(*ω*) =*j*_{L}(*ω*) is fulfilled only if*j*_{max}^{kin}→ ∞ (or*v*_{T}^{e}→ ∞), then in this case the slope obtained for the application of eqn. (13) should be null and that corresponding to the application of eqn. (14) should be infinite.
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120

Figs. 5 and 6 illustrate the experimental dependences corresponding to eqns. (13) and (14), respectively.

Type: Observation |
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121

It can be observed that, within the dispersion of the experimental points, the value of

*j*_{max}^{kin}is finite and the corresponding slopes are different from zero and infinite, respectively.
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122

Besides, it should be noted that the straight lines shown in both Figs. 5 and 6 are not the corresponding linear regressions.

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123

They are the result of the application of eqns. (13) and (14), respectively, using the values of

*j*_{max}^{kin}and*B*obtained from the linear regression illustrated in Fig. 3.
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124

These results should contribute to demonstrate that the relationship

*j*_{max}^{−1}=*f*(*ω*^{−1/2}) has a finite origin ordinate.
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125

All the results described above confirm that the Tafel–Volmer route determines the behaviour of the HER on platinum electrodes, with

*v*_{H}^{e}< 10^{−14}mol s^{−1}cm^{−2}.
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126

It should also be noted that the dependence

*θ*(*η*) has a unique solution, a behaviour that is not observed in the HER under activated control.^{18}
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127

Besides, the surface coverage decreases monotonically from

*θ*^{e}and takes very low values.
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128

This behaviour justifies the description of the adsorbed intermediate through the Langmuir isotherm.

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129

Finally, it is known that the hydrogen oxidation on platinum is produced on the electrode surface covered by adsorbed hydrogen of the UPD (under potential deposition) type.

^{19}
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130

Nevertheless, the experimental results indicate that this type of adsorbed hydrogen does not affect the maximum current density on steady state.

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131

Therefore, it can be inferred that the type of adsorbed hydrogen that participates in the hor as the reaction intermediate is different from the H(UPD).

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