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Back | Show only these items: | Hydrogen diffusion effects on the kinetics of the hydrogen electrode reactionPart II. Evaluation of kinetic parameters

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

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 H₂SO₄.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

The results obtained have been discussed in the light of the theoretical treatment described in Part I of this work.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

Introduction

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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

On the other hand, more recent studies consider the hor a mixed controlled reaction, though with a strong diffusional contribution.^4–7

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

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

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

These two aspects should lead to a successful description of the HER in the whole range of overpotentials.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

The proposed methodology is applied for the study of the hydrogen oxidation on a rotating disc electrode (RDE) of polycrystalline platinum.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met1

Evaluation of the kinetic parameters

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

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:⁸being 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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

Therefore, the use of the equilibrium polarisation resistance [R_p = dη/dj∣_{η = 0}] is proposed in order to improve the evaluation of the kinetic parameters.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

The resulting equation, for η → 0 is:The corresponding expression for dθ/dη∣_{η = 0} was obtained from the derivation of eqn. (2) and the application of the condition η → 0.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

After reordering, the equation can be written as follows:

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

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:⁹j_L = Bω^1/2Furthermore, 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:

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

This equation is coincident to that previously derived for the hydrogen evolution reaction taking place under activated control.¹⁰

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

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:

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod8

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.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met2

From eqns. (17–19) of Part I⁸ 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.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met2

The other alternative is an origin ordinate with a positive value and corresponds to v_H^e < 10⁻¹⁴ mol s⁻¹ cm⁻².

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

In this case, eqn. (9) is reduced to eqn. (14) of Part I.⁸

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met3

(a) v_H^e > 10⁻¹⁴ mol s⁻¹ cm⁻²

An origin ordinate of j_max⁻¹ = f(ω^−1/2) equal to zero implies that in eqn. (9) v_H^ee^αfη ≫ v_T^e/(1 − θ^e).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

(b) v_H^e < 10⁻¹⁴ mol s⁻¹ cm⁻²

A positive value of the origin ordinate of j_max⁻¹ = f(ω^−1/2) implies that in eqn. (9) v_H^ee^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:

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

This value should be equal to that obtained from the extrapolation of the experimental dependence j_max⁻¹ = f(ω^−1/2).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

This is a self-consistency test for the evaluated parameters.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

(b1) Approximated method

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

Using this approximation, the parameter v_T^e can be estimated from eqn. (10) and then v_V^e from eqn. (11).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

Therefore, approximated values of the kinetic parameters can be obtained in this way.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

(c) Levich–Koutecky method

The Levich–Koutecky method is usually employed for the evaluation of the dependence j(η) of the hor under activated control.¹¹

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

It consists in plotting the relationships j(η,ω)⁻¹ = f(ω^−1/2) at different overpotentials, which is supposed to be linear.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

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.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

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.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

Nevertheless, as has been demonstrated in Part I,⁸j(η,ω)⁻¹ = f(ω^−1/2) is not necessarily linear, with the exception of the overpotentials range in which j = j_max.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

Consequently, the linear extrapolation is only valid if the condition given in eqn. (20) of Part I is satisfied.⁸

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

Evaluation of the kinetic parameters on platinum

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj2

(a) Experimental details

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 cm² geometric area).

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

The rotation rate was varied in the range 900 ≤ ω/rpm ≤ 8100, using a Tacussel rotating disc equipment.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

The electrolyte solution was 0.5 M H₂SO₄, which was subjected to a continuous and efficient hydrogen bubbling at P = 1 atm.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

The counterelectrode was a large area Pt electrode and the working temperature was fixed at 30 °C.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

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.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

Measurements were made using a potentiostat–galvanostat Radiometer, controlled by the generation/acquisition software Voltamaster.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met5

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

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.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp3

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.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp3

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.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp3

In order to ensure reproducible results in the evaluation of the experimental dependence j(η), the potential program shown in Fig. 1a was applied.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

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.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

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

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

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.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

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

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp4

(b) Results

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

These results exhibit an excellent reproducibility when the solution is appropriately purified and the condition of hydrogen saturation is maintained during the whole run.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

The dependences j(η) are shown in Fig. 2a, where different types of symbols were used for the different ω values.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj3

This dependence was evaluated from the experimental results shown in Fig. 2a and is illustrated in Fig. 3.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

An origin ordinate definitively finite, equal to j_max^kin⁻¹ = 4.15 A⁻¹ cm², was calculated from the linear regression.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

Besides, the value B = 0.068 mA cm⁻² rpm^−1/2 was obtained from the slope of the straight line of Fig. 3.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

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 = dη/dj∣_η=0] was evaluated for each rotating rate.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met6

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod13

It can be observed in Fig. 4 that the dependence R_p = f(ω^−1/2) has the linear variation predicted by eqn. (5).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

The values R_p^o = 0.546 Ω cm² 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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

Fig. 3 corresponds to data measured for η > 0.2 V while Fig. 4 was obtained from data measured near the equilibrium potential.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

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 cm² obtained from Figs. 3 and 4, respectively.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac9

The results obtained are summarised in Table 1, for each value of the rotation rate.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

The values of j_L obtained from eqn. (6) have also been included.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs7

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs8

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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

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 | Advantage: None | Novelty: None | ConceptID: Con2

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).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

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 | Advantage: None | Novelty: None | ConceptID: Con3

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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

The constant values j_max^kin⁻¹ and R_p^o needed for the calculations were obtained from Figs. 3 and 4, respectively.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met7

The corresponding values of v_T^e and v_V^e obtained in this way are shown in Table 1 (second column).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs10

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj4

It has been demonstrated that this alternative is limited by the fulfilment of the condition given in eqn. (20) of Part I.⁸

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac13

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

101

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4

103

As has already been demonstrated, Levich–Koutecky plot can be applied only if θ(η) is invariant with j_L⁸.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac14

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj5

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.