Noise study of hydrogen evolution process on Cu and Ag microelectrodes in sulphuric acid solution

Article abstract of DOI:10.1016/j.electacta.2007.01.013

Recently we reported a method to measure noise and impedance quasi simultaneously. The method was demonstrated in the determination of the symmetry factor of a single electron charge transfer process. In the present work we applied the noise + impedance measurement to a complex electrochemical reaction, namely to the hydrogen evolution process, which, in acidic media is supposed to take place via the Volmer-Heyrovsky mechanism. The theory of the noise corresponding to this mechanism has been derived by Tyagai in the early 1970s, however, for the verification no experiments have been published so far. Our investigation was based on the apparent electron number that we calculated from the current noise spectra with certain corrections. We found apparent electron number values between 1 and 2, depending on the electrode material and the applied external current. The results could be rationalized on the basis of Tyagai's theory.

Full text of DOI:10.1016/j.electacta.2007.01.013

Electrochimica Acta 52 (2007) 4752–4759
Noise study of hydrogen evolution process on Cu and Ag
microelectrodes in sulphuric acid solution
∗
´
´
´ ´
´
Ildiko Szenes , Gabor Meszaros, Bela Lengyel
Institute of Materials and Environmental Chemistry, Chemical Research Center Hung. Acad. Sci.,
H-1025 Budapest, Pusztaszeri u´t 59-67, Hungary
Received 11 September 2006; received in revised form 11 January 2007; accepted 14 January 2007
Available online 26 January 2007
Abstract
Recently we reported a method to measure noise and impedance quasi simultaneously. The method was demonstrated in the determination of
the symmetry factor of a single electron charge transfer process. In the present work we applied the noise + impedance measurement to a complex
electrochemical reaction, namely to the hydrogen evolution process, which, in acidic media is supposed to take place via the Volmer–Heyrovsky´
mechanism. The theory of the noise corresponding to this mechanism has been derived by Tyagai in the early 1970s, however, for the verification
no experiments have been published so far. Our investigation was based on the apparent electron number that we calculated from the current noise
spectra with certain corrections. We found apparent electron number values between 1 and 2, depending on the electrode material and the applied
external current. The results could be rationalized on the basis of Tyagai’s theory.
© 2007 Elsevier Ltd. All rights reserved.
Keywords: Hydrogen reduction; Apparent electron number; Electrochemical noise; Electrochemical impedance
1. Introduction
factor is calculated from consecutive noise and impedance mea-
surements – in contrast to any other technique – at one single
potential hence making the symmetry factor a “local” charac-
ter of the polarization curve. In the present work we intended
to develop the combined noise/impedance method for the case
The kinetics of hydrogen evolution on different electrode
materials is an evergreen and extensively studied topic of elec-
trochemistry. The classical kinetic models assume three basic
reactions, namely the electrochemical adsorption of hydrogen
(Volmer reaction) as a first step, and electrochemical desorption
´
of the Volmer–Heyrovsky mechanism and also carry out experi-
ments, concentrating on the apparent electron numbers involved
in the elementary reaction step.
´
(Heyrovsky reaction) or the chemical desorption (Tafel reac-
tion) as the second step [1]. Though careful electrochemical
impedancestudieshavealsobeenpublishedinthisfield[2,3], the
distinction between the above reaction schemes is made mainly
on the basis of charge transfer coefficients obtained from dc
polarization curves. Tyagai derived the formulas describing the
The hydrogen evolution experiments on polycrystalline Ag
and Cu electrodes in acidic media [6–9] showed quite high vari-
ety in exchange current density and Tafel slopes as well. The
reaction rate was found to be sensitive also to the surface state
of the electrode in the case of Au, Ag and Cu [10,11]. Eber-
hardt, Santos and Schmickler [12] found different behaviour for
Ag single crystals with different orientations.
From the point of view of measuring technique the
hydrogen evolution is an attractive choice for the combined
noise/impedance study. In our previous paper [5] we showed
that the shot noise of the charge transfer process can be deter-
mined in the case of relatively high reaction rates only, otherwise
it becomes completely shunted by the double layer capacity.
The measurement should be performed at a potential where
the charge transfer overpotential has a considerable contribution
´
noise of the Volmer–Heyrovsky mechanism in the early 1970s
[4], however, no experiment has been carried out so far in order
to verify the theory. Recently we reported a method based on
the combination of electrochemical impedance and noise mea-
surement which is capable to determine the symmetry factor of
a single charge transfer process [5]. In that case the symmetry
∗
Corresponding author. Fax: +36 1 438 1147.
E-mail address: szenes.ildiko@kvk.bmf.hu (I. Szenes).
0013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.electacta.2007.01.013

I. Szenes et al. / Electrochimica Acta 52 (2007) 4752–4759
4753
(∼4 kT/e) to the total overpotential, i.e. one of the anodic and
cathodic partial currents must be remarkably higher in absolute
value than the other one. (A quasi-reversible reaction is required
with a relatively low exchange current density.) The preferred
cell arrangement is a two electrode one with a microelectrode as
working electrode and with a counter/reference electrode having
much larger area than that of the working electrode so that the
noise of the counter electrode be neglected.
The above requirements can easily be fulfilled in the study
of hydrogen evolution process. A relatively large area hydrogen
filled Pd electrode can serve as an ideal counter/reference elec-
trode. The exchange current density of the hydrogen evolution
process is rather low on many electrode materials therefore it
can behave as a quasi-reversible electrode process.
The measurement of the impedance simultaneously with that
of the noise permits the elimination of noise of the solution
resistance and that of the constant phase element to a certain
extent. The solution resistance (Rs) exhibit white noise having
the power spectral density
SU(ω) = 4kTRs,
(3)
where k is the Boltzmann constant and T is the absolute temper-
ature. On the noise of the constant phase element we assume that
– similarly to an arbitrary electric two-pole in thermodynamic
equilibrium – it provides a current noise having a power spectral
density proportional to the real part of its admittance (YCꢀ PE):
Si(ω) = 4kTYCꢀ PE(ω).
(4)
Based on Eqs. (3) and (4) the following expression will be
referred to as corrected apparent electron number change:
2. Apparent electron number
ꢀ
ꢁ
1
SU,m(ω) − 4kTRs
A simple charge transfer process, having ia anodic and ic
cathodic partial current exhibits a white current noise with a
power spectral density of
napp,corr(ω) =
− 4kTYCꢀ PE(ω) ,
(5)
2
2ei |Zm(ω) − Rs|
where Zm and SU,m mean the measured impedance and volt-
age noise spectra of the eꢀlectrochemical cell, respectively, i the
netto electrode current, YCPE is the determined by analyzing the
impedance data.
Si(ω) = 2ne(|ia| + |ic|),
(1)
where e means the electron charge, and the elementary charge
transfer step involves n electrons. Once the process is far from
equilibrium, ie the one of the partial currents is negligible, one
can calculate the n value from the external current (i) and the
measured power spectral density of the current:
3. Noise of the Volmer–Heyrovsky´ mechanism
In strongly acidic media the hydrogen evolution reaction
´
is usually considered to take place via the Volmer–Heyrovsky
Si(ω)
n =
.
(2)
mechanism. In the first step (Volmer reaction) the hydrogen ions
are adsorbed in an electrochemical step:
2ei
Considering a complex electrode reaction, the above ratio
can be frequency dependent and may not correspond to any real
electron number changes. In this case Eq. (2) yields an appar-
ent electron number (napp). Here we would like to emphasize
that napp can be really informative only considerably far from
thermodynamic equilibrium.
H+ + e ↔ Hads
.
(R1)
´
In the second step (Heyrovsky reaction) the absorbed hydro-
gen, together with a hydrogen ion forms a hydrogen molecule:
Hads + H+ + e ↔ H2.
(R2)
In order to be able to determine napp of a charge transfer
process the conditions in the cell should be chosen in such a
way that there be a frequency region where the charge trans-
fer is the main noise generation process and also it determines
dominantly the real part of the admittance. In a stochastically
behaving electrochemical cell with a charge transfer process five
kinds of noise generation should be considered, namely (i) the
resistance noise of the solution, (ii) the noise corresponding the
usually present constant phase element (CPE, which is often
attributed to the double layer), (iii) the shot noise of the charge
transfer process, (iv) noise due to diffusion, and (v) the (usually
of unknown origin) flicker noise. The high frequency noise is
determined by (i) and (ii), while the low frequency region by (iv)
and (v); under properly selected conditions at medium frequen-
cies one can detect the noise corresponding to (iii), which is the
target of the present work. The effect of the solution resistance
and the diffusion can be minimized by applying a microelec-
trode. (Considering the high diffusion coefficient of H+ in the
present work we disregarded the diffusional effects and thus the
diffusion contribution to noise.)
The simplest form of the corresponding kinetic equation can
be written as follows [2]:
i = FA(vV + vH ),
dΘ
(6)
(7)
ꢃ
Γmax
= −vV + vH ,
dt
ꢂ
ꢃ
ꢂ
Θ
V ΘEq
βV e
kT
1−Θ
(1 − βV )e
vV =kꢀ
exp
η −kꢀ
exp
−
η
,
V 1−ΘEq
kT
(8)
ꢂ
ꢃ
ꢂ
ꢃ
1−Θ
βH e
kT
Θ
(1 − βH )e
vH =kꢀ
exp
η −kHꢀ
exp −
η ,
H 1−ΘEq
ΘEq
kT
(9)
where i is the net electrode current, vV, vH, kVꢀ , kHꢀ , βV and βH
are the actual rates, the apparent rate constants and the symme-
´
try factors of the Volmer and Heyrovsky reactions, respectively,
Θ, ΘEq and Γ max represent the actual coverage, the coverage

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I. Szenes et al. / Electrochimica Acta 52 (2007) 4752–4759
ꢂ
ꢃ
corresponding to the equilibrium potential (η = 0) and the max-
imum surface concentration of the adsorbed hydrogen, η stands
for the overpotential.
1 − Θ0
βH e
kT
iH,a = FA kꢀ
exp
η
,
(17)
(18)
H 1 − ΘEq
Θ0
ꢂ
ꢃ
(1 − βH )e
The above kinetic equations contain some simplifications.
The interfacial concentrations of both H+ and H2 (involved in
the apparent rate constants) are considered to be constant and
the adsorption energy of hydrogen is also regarded as to be inde-
pendent of coverage (Langmuir type adsorption). In spite of the
simplifications the equations still contain 7 parameters, namely
A, kVꢀ , kHꢀ , βV, βH, ΘEq and Γ max. However, if we represent the
polarization curve with the apparent charge transfer coefficient
(αapp) calculated formally in the following way:
iH,c = FA kꢀ
exp
−
η
.
H ΘEq
kT
The vV,η, vH,η, vV,Θ, vH,Θ constants represent the values of the
following partial derivatives at η = η0 and Θ = Θ0:
ꢄ
ꢄ
ꢄ
ꢄ
∂vV
∂η
1
FA kT
e
vV,η
=
=
=
=
=
(βV iV,a + (1 − βV )iV,c),
η = η0
Θ = Θ0
(19)
(20)
ꢄ
d(ln i) kT
1 di kT
i dη e
1 kT
i e
αapp
=
=
=
lim Yꢀ(ω)
(10)
ꢄ
ꢄ
ꢄ
∂vH
∂η
1
FA kT
e
vH,η
vV,Θ
vH,Θ
=
(βH iH,a + (1 − βH )iH,c),
ω→0
dη
e
η = η0
Θ = Θ0
then the surface are (A) will be eliminated and also only the ratio
of the two apparent rate constants will count. (Here we stress
that while symmetry factors βV and βH belong to the anodic
and 1-βV and 1-βH to the cathodic partial processes, based on
Eq. (10) we consider αapp to be simply a function of potential.)
The value of Γ max influences only the τ time constant of the
hydrogen adsorption dividing the noise and impedance into two
frequency region. The remaining four parameters to be taken
ꢄ
ꢄ
ꢄ
ꢄ
ꢂ
ꢃ
∂vV
∂Θ
1
FA Θ0
1
1
=
iV,a
+
iV,c
,
η = η0
Θ = Θ0
1 − Θ0
(21)
.
ꢄ
ꢂ
ꢃ
into account are kVꢀ /kHꢀ , βV, βH and ΘEq
Although the description of the noise of the Volmer–
.
ꢄ
ꢄ
ꢄ
∂vH
∂Θ
1
1
1
Θ0
= −
iH,a
+
iH,c
η = η0
Θ = Θ0
FA 1 − Θ0
´
Heyrovsky mechanism has already been published by Tyagai
[4] in the early 1970s, here we repeat the Langevin equations
[13] corresponding to Eqs. (6)–(9) and also the derived results
with our notations. At a stationary state characterized by η0, Θ0
and i0 those equations are as follows:
(22)
The shortcut noise current (δiN) can be derived with substi-
tuting δη = 0:
ꢂ
ꢃ
vV,Θ +vH,Θ
1
δi = FA((vV,η + vH,η)δη + (vV,Θ + vH,Θ)δΘ) + δiN,V + δiN,H
δiN
=
1−
ꢂ
δiN,V
vV,Θ − vH,Θ 1 + iωτ
(11)
ꢃ
vV,Θ +vH,Θ
1
Γmax
+
1+
δiN,H , τ =
,
vV,Θ − vH,Θ 1 + iωτ
vV,Θ −vH,Θ
iωΓmaxδΘ = (−vV,η + vH,η)δη + (−vV,Θ + vH,Θ)δΘ
(23)
1
FA
+
(−δiN,V + δiN,H
)
(12)
and its power spectral density is given in the form:
ꢅꢆ
4v2H,Θ
whereδi, δηandδΘ denotetotheFouriertransformsof i–i0, η–η0
and Θ–Θ0, respectively, i is the imaginary unit. δiN,V and δiN,H
symbolize the independent white noise sources corresponding
1
Si(ω) =
+ ω2τ2 Si,V
)
2
1 + ω2τ2
(vV,Θ − vH,Θ
ꢆ
ꢇ
4v2V,Θ
´
to the Volmer and to the Heyrovsky reactions, having the power
+
+ ω2τ2 Si,H
.
(24)
spectral densities (Si,V, Si,H):
2
(vV,Θ − vH,Θ
)
Si,V = 2e(iV,a + iV,c
Si,H = 2e(iH,a + iH,c),
)
and
(13)
(14)
The τ time constant in the above equation is equivalent to
what is present in the admittance of the Volmer–Heyrovsky
´
mechanism:
where iV,a , iV,c , iH,a and iH,c denote to the absolute value of the
steady-state anodic and cathodic partial currents corresponding
ꢂ
ꢃ
(vH,η − vV,η)(vV,Θ + vH,Θ
vV,Θ − vH,Θ(1 + iωτ)
)
Y(ω) = FA vV,η + vH,η
+
.
´
totheVolmerandtheHeyrovskyreactions, respectively, atη = η0
and Θ = Θ0:
(25)
ꢂ
ꢃ
Θ0
V ΘEq
βV e
The equivalence is natural taking into account that also Eq.
(25) follows from Eqs. (11) and (12), just by eliminating the
noise terms.
The high frequency noise equals to the direct sum while at
low frequencies it equals to a weighted sum of the shot noise of
iV,a = FAkꢀ
exp
η0
,
(15)
(16)
kT
ꢂ
ꢃ
1 − Θ0
1 − βV )e
iV,c = FAkꢀ
exp
−
η0
,
V 1 − ΘEq
kT

I. Szenes et al. / Electrochimica Acta 52 (2007) 4752–4759
4755
´
microelectrode directly before the experiments, from solu-
tions containing 0.5 mol/dm3 H2SO4 + 10−2 mol/dm3 CuSO4
and 28 g/dm3 Ag2CO3 + 370 g/dm3 Na2S2O3 × 5H2O, respec-
tively. In the case of Cu the deposition took place by cycling
with 100 mV/s sweeprate between −160 mV and +30 mV using
a Cu counter electrode. In the case of Ag a −25 mV deposition
potential superimposed with a 1 kHz, 0.1 V amplitude sinu-
soidal signal was applied using an Ag counter electrode. The
compactness of the deposited layer was tested with an optical
microscope.
The preparation of the H2 filled Pd counter electrode involved
the following steps. The Pd electrode was first mildly annealed
by a butane flame, than immersed to 0.5 M H2SO4 solution
and polarized with −2.3 V with respect to a Pt electrode.
The cathodic polarization was stopped as the first H2 bubbles
appeared on the edges of the Pd electrode. (According to our
experience a completely saturated Pd electrode exhibited too
much noise during the experiments.) The oxide film formed by
flame annealing did not disappear completely during the hydro-
gen filling process. In order to remove the oxide layer we kept
the hydrogen filled Pd electrode in 0.5 M H2SO4 for at least 12 h
to let the oxide layer reduce or dissolve completely. After this
preparation the Pd electrode retained the potential for about 3
days within 10 mV.
The metal depositions and recording of cycling voltammo-
grams were carried out using a laboratory-made potentiostat.
The combined impedance/noise measurements were carried out
with the setup described in our previous communication [5].
Here we need to clarify that although the investigation concerns
the power spectral density of the current noise, from the prac-
tical point of view we preferred to measure the voltage noise
and impedance of the electrode, and calculated the current noise
spectra from those data sets. The reason is that while during
voltage noise measurement the instrumentation noise can be
considered to be additive to the measured noise, in the case of
current noise measurement the instrumentation noise depends
also on the impedance of the cell [14]. Thus the knowledge of
the electrode impedance is required in both cases; however, the
correction of the measured current noise spectra is more difficult
than that of the voltage noise spectra.
the Volmer and the Heyrovsky reactions:
Si(ω) = Si,V + Si,H = 2e(iV,a + iV,c + iH,a + iH,c),
ꢂ
ꢃ
1
ω ꢁ
and
(26)
τ
ꢆ
ꢂ
ꢂ
ꢃ
2
2vH,Θ
Si(ω) = 2e
(iV,a + iV,c
)
vV,Θ − vH,Θ
ꢃ
ꢂ
ꢃ
2
2vV,Θ
1
+
(iH,a + iH,c
)
,
ω ꢂ
.
vV,Θ − vH,Θ
τ
(27)
At such overpotentials where any of the reactions is reversible
(iH,a ≈ iH,c ꢁ i or iV,a ≈ iV,c ꢁ i) the value of napp obtained from
high frequency data will be a huge, meaningless value. In con-
trast, napp obtained from low frequency data can show interesting
behaviour. If only one of the reactions is close to reversibility
(iH,a ≈ iH,c ꢁ iV,c ≈ i/2 or iV,a ≈ iV,c ꢁ iH,c ≈ i/2) Eq. (27) will
reduce to
Si(ω) = 4ei,
(28)
which implies 2 as the value of napp. According Tyagai [4] the
explanation of the apparent two electron steps is due to the fact
that the two reactions are coupled through the hydrogen surface
coverage. An elementary charge transfer event of the rate deter-
mining process changes the hydrogen coverage of the surface.
Because of that change an elementary charge transfer step of
the other reaction will take place within τ with high probability.
Thus at low frequencies (ω ꢂ 1/τ) strong correlation can be
observed between the two reactions while at high frequencies
(ω ꢁ 1/τ) they appear to be completely independent processes.
We also investigated the case with such high overpoten-
tials, where both reactions are far from equilibrium (iV,a ꢂ iV,c
and iH,a ꢂ iH,c). Under these conditions the napp value calcu-
lated from high frequency data equals to unity. Considering
the low frequency napp value, substituting iV,a = iH,a = 0 and
iV,a = iH,a = i/2 into Eq. (27) the expression of Si corresponding
to low frequencies reduces to the following form:
Si(ω) = 4ei((1 − Θ)2 + Θ2).
(29)
All the experiments have been carried out at room temper-
ature around 25 ◦C. The temperature of each experiment was
separately measured and recorded.
The above noise expression yields an napp value between 1 and 2,
depending on the hydrogen coverage. Thus based on the appar-
ent electron number change one can conclude on the stationary
hydrogen coverage.
5. Results and discussion
4. Experimental
The current–voltage characteristics recorded with low speed
cyclic voltammetry are given in Figs. 1 and 2. In both cases we
carried out the CV experiments starting from 0 V (versus the Pd
electrode) towards cathodic direction. The current of the first
and that of the subsequent cathodic scans are of similar shape
while the backward scans are different. The reason is that during
the cathodic scans H2 gas is produced which partly remains in
the vicinity of the electrode, and during the anodic scans the
oxidation of H2 also takes place. In the vicinity of zero potential
(versus Pd) major part of that H2 becomes oxidized. By inserting
a waiting time of a few seconds at zero potential between the
All the measurements have been carried out in a two elec-
trode cell with a d = 50 ␮m embedded disk microelectrode as
working electrode and with a surface area of ∼4 cm2, H2 filled
Pd foil with a surface area of ∼4 cm2 as counter/reference
electrode. The microelectrode surface was set to be vertical
so that the formed H2 bubbles could easily get away. The
Pt electrode was freshly polished (down to 0.25 ␮m diamond
paste) before each experiment. The Cu and Ag electrodes
were prepared by electrochemical deposition on the same Pt

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I. Szenes et al. / Electrochimica Acta 52 (2007) 4752–4759
Fig. 1. Potentiodynamic current–voltage curves on Ag microelectrode
(d = 50 ␮m) in 0.5 M H2SO4 solution with 5 mV/s sweep rate.
Fig. 3. Bode plots of the measured impedance data with Ag microelectrodes at
various polarizing currents. The solid symbols correspond to the absolute value
while the hollow ones to the phase angle of the impedance.
individual scans during each cathodic scan the first one can be
reproduced. On the other hand, by applying a higher sweep rate
only minor part of the H2 gas in the vicinity of the electrode is
oxidized and the curve of the cathodic scan overlaps with that
of the anodic one. As the shape of the curve during the cathodic
scan is highly influenced by the formation of H2 gas we used
curves taken during the anodic (backward) scans. In the case
of the Ag electrode in accordance to literature data [8,9] two
regions can be distinguished. At small current densities (up to
1 mA/cm2) the estimated Tafel slope is about 60 mV while for
higher current densities it is around 120 mV. In the case of the
Cu electrode the first region with lower Tafel slope is well seen
only on the cathodic scan. The Tafel slope of the second region
– in contrast to the case of the Ag electrode – is about 200 mV.
Fig. 3 shows the impedance data recorded on Ag microelec-
trode applying three different external dc currents. We carried
out parameter fitting with the following model:
parameters are listed in Table 1. The quality of the fit proved
to be excellent without considerable systematic deviation. The
values for the solution resistance (Rsol) were reasonable for a
d = 50 ␮m microelectrode in 0.5 M H2SO4 solution. The αapp
values calculated from Gp (using Eq. (10)) also showed good
agreement with the potentiodynamic curve in Fig. 1, with the
indication that the three measurements were taken around the
intersection of the two Tafel-regions. On the other hand, we
obtained unrealistically low double layer capacitance values.
Considering that the λ exponents of the assumed CPE ele-
ments scattered around the value of 0.5 we also tried to fit the
following model to the measured data:
1
Zfit(ω) = Rsol
+
.
(31)
1
Gp + iωCdl +
1
iωC
R +RW(iωτ )−1/2
+
a
0
a
Rsol, Gp and Cdl showed similar values to those calculated with
the former model. In spite that the latter model has an additional
parameter with respect to the previous one we obtained a some-
what poorer fit and even a small systematic deviation. For this
reason we applied Eq. (30) as a model in the case of the Ag
electrode.
1
Zfit(ω) = Rsol
+
,
(30)
Gp + iωCdl + GCPE(iωτ0)λ
where Rsol, Gp, Cdl, GCPE and λ are the free parameters with the
usual meaning, the constant τ0 = 1s is used in order to achieve
dimensionally correct expression. The calculated values of the
Table 1
Parameters obtained by fitting Eq. (30) to the impedance data measured on Ag
electrode
i
−9.5 nA
−20 nA
−50 nA
(−0.48 mA/cm2)
(−1 mA/cm2)
(−2.5 mA/cm2)
Gp
369 nS
665 pF
63.1 nS
0.521
701 nS
611 pF
37.8 nS
0.580
1.25 ␮S
626 pF
34.4 nS
0.603
Cdl
GCPE
λ
Rsol
χ2
391 ꢀ
346 ꢀ
380 ꢀ
2.47 × 10−5
4.77 × 10−5
5.86 × 10−5
αapp
0.971
0.876
0.625
χ2 indicate the value of the target function at the optimum:
ꢈ
N
2
2
χ2 = 1/N j=1|Zmeas(ωj) − Zfit(ωj)| /|Zfit(ωj)| . αapp was calculated with
Fig. 2. Potentiodynamic current–voltage curves on Cu microelectrode
(d = 50 ␮m) in 0.5 M H2SO4 solution with 5 mV/s sweep rate.
Eq. (10), using the values of the Gp parameter in place of Yꢀ.

I. Szenes et al. / Electrochimica Acta 52 (2007) 4752–4759
4757
thermore we introduced the rV and rH parameters that we call the
´
reversibility index of the Volmer and the Heyrovsky reactions,
respectively, defined in the following way:
|ix,a − ix,c
ix,a + ix,c
|
.
rx = 1 −
(32)
The unity value of the reversibility index indicates complete
reversibility of the corresponding reaction at the applied poten-
tial while the zero value indicates complete irreversibility. This
´
way we can characterize the Volmer–Heyrovsky model with five
dimensionless numbers (αapp, rV, rH, Θ and napp(ω ꢂ 1/τ)) ver-
sus overpotential. The approach has the advantage that all those
numberstakeplacewithinthesameorderofmagnitude. Inaccor-
dance to the classical literature data [1,15] we found that the
model provides polarization curves with two Tafel regions only
in two cases: (a) ΘEq ≈ 0 and kV ꢁ kH; (b) ΘEq ≈ 1 and kV ꢂ kH,
more or less independently of the βV and βH values.
In the Tafel region at low overpotentials – exhibiting a low
Tafel slope – the rate determining process shows irreversible
while the other one shows reversible behaviour. The latter case
is one of those previously discussed ones which provide napp = 2
in the frequency region ω ꢂ 1/τ, exactly what we observed in
the calculated napp,corr curves.
In the second Tafel region at higher overpotentials – exhibit-
ing a relatively high Tafel slope – both reactions become
irreversible. The value of αapp will be equal to the 1 − β value of
the rate determining process. As we already discussed before,
in this region napp can vary between 1 and 2, depending on the
value of Θ.
A typical case with two Tafel regions is shown in Fig. 5.
Though our data are not sufficient to carry out a reliable fitting
procedure, a qualitative agreement can be seen between the cal-
culated curves and the measured values of αapp and napp. On the
other hand, we can clearly state that we could extract the cur-
Fig. 4. Apparent electron change number (napp, grey curve) and corrected
apparent electron change number (napp,corr, light grey curve) in the case of
Ag electrode, (a) i = − 50 nA (−2.5 mA/cm2) E = − 176 mV; (b) i = − 20 nA
(−1 mA/cm2) E = − 143 mV; (c) i = − 9.5 nA (−0.48 mA/cm2) E = − 132 mV.
(The potentials were measured vs. the hydrogen filled Pd electrode.) The black
curves indicate napp,corr filtered with a 100 points moving average window.
Though the complete immitance model of the Volmer–
´
Heyrovsky mechanism containes also an adsorption correlated
time constant, we did not need to introduce it into the model,
indicating that the time constant is either very small or rather
large. At the applied external current densities the adsorption
time constant is usually considered to be very small (cf. Eq.
(23)). This assumption is also supported by the fact that the cal-
culated Gp values show good agreement with the Tafel-slopes
obtained from the potantiodynamic curves. Consequently in the
investigated frequency range the current noise spectra of to the
elementary charge transfer steps should be a white noise with
the spectral density given by Eq. (27).
The napp and napp,corr values obtained from the noise and
impedance data are plotted in Fig. 4 for the three different exter-
nal currents. The uncorrected curves show a clear minimum at
100 Hz, 200 Hz and 2 kHz, respectively, however, the napp at
the minima are still too high as compared to a realistic electron
change number. On the other hand, the corrected curves show
a constant value around 1.5, . . ., 2 in a significantly wide fre-
quency region that can be considered to be reasonable apparent
electron change number values.
Fig. 5. Characteristics of the Volmer–Heyrovsky´ mechanism in a typical case
showing two Tafel regions. The grey lines with symbols correspond to the cal-
culated napp (ꢀ), αapp (᭹), Θ (ꢁ), rV (ꢂ) and rH (ꢃ) values, in the case of
ΘEq = 0.01, kVꢀ /kHꢀ = 50, βV = βH = 0.5. The napp (+) and the αapp (×) val-
ues obtained from noise/impedance measurements for Ag electrode are also
indicated.
In order to explain the obtained results we carried out simple
numerical calculations concerning the apparent charge transfer
coefficient (from Eq. (10)), the hydrogen coverage and the napp
´
values on the basis of the Volmer–Heyrovsky mechanism. Fur-

4758
I. Szenes et al. / Electrochimica Acta 52 (2007) 4752–4759
must be highly reversible. In the view of the above considera-
tion we think that the term containing Ga and τ by no means
corresponds to the hydrogen adsorption process. We handle that
term – similarly to the presence of the constant phase element
– as a necessary term to obtain good fit without any physical
explanation. Consequently in the calculation of napp,corr we sup-
plemented Eq. (5) with a term compensating for Ga and τ in the
following way:
ꢂ
1
SU,m(ω) − 4kTRs
napp,corr(ω) =
2
2ei
|Zm(ω) − Rs|
ꢂ
ꢃꢃ
ω2τ2
− 4kT YCꢀ PE(ω) + Ga
.
(34)
1 + ω2τ2
Fig. 6. Bode plots of the measured impedance data with Cu microelectrodes at
various polarizing currents. The solid symbols correspond to the absolute value
while the hollow ones to the phase angle of the impedance.
The napp and the napp,corr (calculated with both Eqs. (5) and
(34)) values are plotted in Fig. 7 for three different external cur-
rents. Theminimaoftheuncorrectedcurvesshowvaluesatabout
3 which are considerably lower than those in case of the Ag elec-
trode. Furthermore, thisvalueisclosetoarealisticlowfrequency
napp. The correction based on Eq. (5) leads to an implausible
curve(turningtonegativevaluesathighfrequencies)forthelow-
est external current while it does not result a qualitative change
in the curves corresponding to the two higher external currents.
However, in the case of i = − 100 nA the napp,corr calculated by
Eq. (34) shows a constant value of about 1.2 in the 2 kHz–20 kHz
frequency range. In the case of i = − 200 nA the same correction
rent noise spectra of the individual hydrogen reduction charge
transfer steps.
The impedance data in the case of the Cu electrode are shown
in Fig. 6. In these cases we also had to involve one more time
constant with respect to Eq. (30):
1
Zfit(ω) = Rsol
+
,
iωτ
Gp + iωCdl + GCPE(iωτ0)λ + Ga 1+iωτ
(33)
where, accordingtoEq. (25)Ga andτ aresupposedtocorrespond
to the hydrogen adsorption process. The obtained parameters are
listed in Table 2. The fit was found to be good also in these cases,
however, a small systematic error was observed. The appar-
ent charge transfer coefficients calculated from the Gp values
showed good agreement with the potentiodinamic curve of Fig. 2
while the Cdl values are again unreasonably low. According to
´
the Volmer–Heyrovsky mechanism, a single Tafel slope can be
achieved only if both reactions are rather far from reversibility.
On the other hand, the high values of the Ga parameter in all
three cases – assuming that it really corresponds to the hydro-
gen adsorption process – indicate that one of the two reactions
Table 2
Parameters obtained by fitting Eq. (33) to the impedance data measured on Cu
electrode
i
−50 nA
−100 nA
−200 nA
(−2.5 mA/cm2)
(−5 mA/cm2)
(−10 mA/cm2)
Gp
Cdl
GCPE
λ
500 nS
175 pF
4.19 nS
0.873
1.03 ␮S
0.97 pF
4.04 nS
0.869
1.86 ␮S
2.1 pF
3.85 nS
0.872
Rsol
Ga
τ
χ2
390 ꢀ*
133 ␮S
4.78 ␮s
6.65 × 10−5
0.25
390 ꢀ*
62.2 ␮S
6.8 ␮s
390 ꢀ*
29.7 ␮S
10 ␮s
Fig. 7. Apparent electron number (napp, dark grey curve), corrected apparent
electron number corrected using Eq. (5) (grey curve) and corrected apparent
electron number corrected using Eq. (34) (light grey curve; the black curve
inside indicates its filtered version with a 100 point moving average) in the case
of Cu electrode, (a) i = − 200 nA (−10 mA/cm2) E = − 382 mV; (b) i = − 100 nA
(−5 mA/cm2) E = − 359 mV; (c) i = − 50 nA (−2.5 mA/cm2) E = − 313 mV.
9.1 × 10−5
1.59 × 10−4
αapp
0.258
0.235
The Rsol values labeled in the table with (*) were kept constant.

I. Szenes et al. / Electrochimica Acta 52 (2007) 4752–4759
4759
leads to a minimum with a value of about 1.7 which seems to
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the value of Θ is close to zero or unity, depending on whether
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´
the Heyrovsky or the Volmer reaction is the rate determining
one. Here we need to emphasise that due to the weakly justified
use of Eq. (34) one needs to handle the above conclusions with
care.
We also need to mention that beyond a certain external cur-
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Acknowledgement
Financial support of the Hungarian Scientific Research Fund
(OTKA T 029727 and T 42452) is gratefully acknowledged.
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