406
Langmuir 2003, 19, 406-415
Cyclic Reciprocal Derivative Chronopotentiometry with
Power Time Currents Applied to Electrodes Coated with
Electroactive Molecular Films. Influence of the
Reversibility
Angela Molina* and Joaquin Gonzalez
Departamento de Quı́mica Fı́sica, Facultad de Quı́mica, Universidad de Murcia,
Espinardo, 30100, Murcia, Spain
Received April 17, 2002. In Final Form: August 14, 2002
The theory of reversible, quasi-irreversible, and totally irreversible charge-transfer reactions between
adsorbed molecules in cyclic reciprocal derivative chronopotentiometry with power currents of the form
I(t) ) (I0tu (CRDPC) is presented. For certain values of u, it is possible to set the peak heights to increase
when the reversibility of the process decreases. This possibility represents an enormous advantage of this
electrochemical technique in the characterization of nonreversible processes, since it is the only technique
for which this behavior is observed. The use of programmed currents is advantageous versus constant
currents (used in the usual reciprocal derivative chronopotentiometry) since in this last case peaks are
not obtained for totally irreversible processes. CRDPC presents, as compared to cyclic voltammetry, higher
and narrower peaks. Moreover, the signals obtained in CRDPC are also narrower than those corresponding
to cyclic reciprocal derivative exponential chronopotentiometry. The theoretical predictions have been
proved by applying CRDPC to the study of adsorption of quinizarine (reversible behavior) and azobenzene
(irreversible behavior) systems on mercury in aqueous media.
Introduction
Reciprocal derivative chronopotentiometry (RDC) with
constant current, introduced by Jagner,1,2 is a derivative
technique which has enjoyed wide use in recent years for
the study of different electrode processes. This is because
the technique, which is based on the plotting of the
reciprocal of the time derivative of the chronopotentiogram
versus the measured potential (dt/dE vs E), presents a
characteristic peak that is quantitatively related to the
kinetic and thermodynamic parameters of the electrode
process which remain almost unaffected by the capacitive
effects.3,4 At present, this technique, also known as
potentiometric stripping analysis (PSA),5 is widely used
in the collection of analytically useful data and in the
diagnosis of electrochemical reactions,3,4,6-12 including the
study of electrodes coated with molecular films.13
In a previous paper, we introduced cyclic reciprocal
derivative exponential chronopotentiometry (CRDEC), in
which exponential currents of the form I(t) ) (I0eωt are
applied instead of constant currents.14 This new technique
* To whom correspondence should be addressed. E-mail:
[email protected]. Tel: 34 968 36 75 24. Fax: 34 968 36 41 48.
(1) Anderson, L.; Jagner, D.; Josefson, M. Anal. Chem. 1982, 54,
1371.
(2) Jagner, D. Trends Anal. Chem. 1983, 2, 53.
(3) Molina, A.; Gonzalez, J.; Saavedra, F.; Abrantes, L. M. Electrochim.
Acta 1999, 45, 761.
(4) Gonzalez, J.; Molina, A.; Lopez-Tenes, M.; Serna, C. J. Electrochem. Soc. 2000, 147, 3429.
(5) Fogg, A.; Wang, J. Pure Appl. Chem. 1999, 71, 891.
(6) Wang, J.; Tian, B. Anal. Chem. 2000, 72, 3241.
(7) Wang, J.; Rivas, G.; Jiang, M.; Zhang, H. Langmuir 1999, 15,
6541.
(8) Wang, J.; Cai, X.; Wang, J.; Jonsson, C. Anal. Chem. 1995, 67,
4065.
(9) Tomschik, M.; Jelen, F.; Havran, L.; Trnková, L.; Nielsen, P. E.;
Paleček, E. J. J. Electroanal. Chem. 1999, 476, 71.
(10) Bi, S.; Yu, J. J. Electroanal. Chem. 1996, 405, 51.
(11) Abrantes, L. M.; Molina, A.; Gonzalez, J.; Saavedra, F. Electrochim. Acta 1999, 45, 457.
(12) Kizek, R.; Trnková, L.; Paleček, E. Anal. Chem. 2001, 73, 4801.
(13) Honeychurch, M. J. J. Electroanal. Chem. 1998, 445, 63.
greatly improves RDC with constant current in the study
of adsorbed molecules exhibiting irreversible behavior,
since the response in CRDEC (the deωt/dE vs E curve)
presents peaks whatever the reversibility of the process,
whereas in RDC with constant current the responses do
not present peaks when totally irreversible systems are
analyzed.14
In this paper, we study the use of successive current
time functions which vary with a power of time u of the
form I(t) ) (I0tu in order to introduce cyclic reciprocal
derivative power chronopotentiometry (CRDPC). This
technique greatly improves the results obtained in CRDEC
when studying the response of electrodes coated with
electroactive irreversible molecular films. This is because
a comparison of the sensitivity and resolution of the signals
obtained in reversible and irreversible electrode processes
in any electrochemical technique (with potential controlled14,15 or current controlled included CRDEC3,4,11,14-17)
reveals that the signal for a totally irreversible process
always presents less sensitivity and resolution than that
obtained for a reversible one. However, whenever CRDPC
is applied in the study of irreversible processes, we observe
that there are values of the power of time u in the applied
current for which the peaks corresponding to irreversible
processes are higher and narrower than those corresponding to reversible processes. Such a possibility represents, without doubt, an enormous advantage of this
technique over other electrochemical techniques, and as
far as we know, it is the only technique for which this type
of behavior is observed.
The comparison between the CRDPC responses and the
cyclic voltammetry (CV) currents for nonreversible pro(14) Gonzalez, J.; Molina, A. Langmuir 2001, 17, 5520.
(15) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.;
Wiley: New York, 2001.
(16) Galus, Z. Fundamentals of electrochemical analysis, 2nd ed.;
Ellis Horwood: New York, 1994.
(17) Molina, A.; Gonzalez, J.; Moreno, M. M. Electroanalysis 2002,
14, 281.
10.1021/la020369x CCC: $25.00 © 2003 American Chemical Society
Published on Web 12/12/2002
Cyclic Reciprocal Derivative Chronopotentiometry
Langmuir, Vol. 19, No. 2, 2003 407
cesses reveals higher and better defined peaks in CRDPC.
These peaks are also narrower and higher than those
obtained in CRDEC.14
We have analyzed the influence of the exponent u in
the applied current and of the kinetic parameters k0 (rate
constant of the surface electrochemical reaction, in s-1)
and R and (1 - R) (charge-transfer coefficients of the
cathodic and anodic surface reactions, respectively) on
the CRDPC curves, and we have proposed methods for
calculating kinetic parameters of the charge-transfer
reaction k0 and R and the interfacial standard potential
for the Langmuir isotherm E0.
The above methods have been applied to the study of
the adsorption of quinizarine (reversible) and azobenzene
(irreversible) systems on mercury in aqueous media. From
the measurement of the peak potentials and peak heights,
we have obtained accurate values for the kinetic and
thermodynamic parameters of both systems. In the case
of the azobenzene system, for a u value of 0.075 we obtain
CRDPC peaks which are =1.7 times higher and half peak
widths that are =50% smaller than those of CV.
The greater sensitivity and resolution of the signals
obtained, the lower influence of capacitive and ohmic drop
effects,3,4,14,17,18 and its higher versatility as compared to
that of CV make CRDPC one of the best electrochemical
methods both in the detection of trace levels and in the
characterization of the electrochemical behavior of molecular films.
Peak potentials and peak heights in CRDPC were measured
from the fitted differentiated curves without further analysis.
All the kinetic and thermodynamic values obtained for the
quinizarine and azobenzene systems correspond to series of five
essays. The results obtained are the mean of the five experimental
values, while the errors correspond to the standard deviation.
Chemical Reagents. K2HPO4, KH2PO4, KNO3, HClO4, ethanol (Merck, reagent grade), quinizarine, and azobenzene (Aldrich,
reagent grade) were used as received.
Azobenzene and quinizarine were dissolved in ethanol and
then diluted with water until the final proportion of ethanol was
1% (azobenzene and quinizarine concentrations were 3 × 10-5
and 10-5 M, respectively). In the case of azobenzene solutions,
K2HPO4 and KH2PO4 were used to fix the pH at 6.74 (with
KNO3 0.1 M as the supporting electrolyte). Azobenzene and
quinizarine were adsorbed at the mercury surface at a potential of -0.60 and -0.40 V, respectively, for 30 s prior to measurements.
Water was bidistilled, and nitrogen gas was passed through
solutions for deaeration for 15 min prior to measurements.
Experimental Section
To obtain the equations for the response corresponding
to CRDPC, we assume for the adsorbate monolayer that
the Langmuir isotherm is obeyed. The adsorption coefficients of both electroactive species and the maximum
surface coverage are independent of the potential.
In a chronopotentiometric experiment, the potential
time response (E vs t curve) is given by14,22
Apparatus. A computer-driven potentiostat-galvanostat was
designed and constructed by QUICELTRON (Spain).
Pulse and waveform generation and data acquisition were
performed using i-SBXDD4 and DAS16-330i (ComputerBoards,
USA) boards, respectively. All computer programs were written
in our laboratory.
In the cyclic chronopotentiometric experiments, the current
switch was performed when the potential attained a prefixed
value (cathodic or anodic) at which we supposed that the
transition time had already been reached. The necessary
comparison was carried out by means of an interrupt service
routine using the clock of the personal computer.
Electrodes. A three-electrode cell was employed in the
experiments. A static mercury drop electrode (SMDE) served as
working electrode. The SMDE was constructed using a dropping
mercury electrode (DME), EA 1019-1 (Metrohm), to which a
homemade valve was sealed. The electrode radius of the
SMDE was determined by weighing a large number of drops.
The counter electrode was a Pt foil, and the reference electrode
was a Ag/AgCl, KCl 1.0 M electrode.
Signal Processing. In the experimental measurements of
the potential-time curves, we have used different digital noise
filters of the instrument supported software. The experimental
E versus t curves were then smoothed by applying the moving
average smoothing procedure proposed by Savitzky and Golay19
and transformed to the corresponding tu+1 versus E curves. The
experimental tu+1 versus E curves obtained for the quinizarine
system were fitted with a sigmoidal regression. In the case of the
curves corresponding to the azobenzene solution, we programmed
theoretical equations for the totally irreversible tu+1 versus E
curves in order to carry out the regressions. In both cases, we
used the SigmaPlot program,20 and the fitted curves obtained
were numerically differentiated by using a finite differences
formula of the fifth degree21 so as to obtain the dtu+1/dE versus
E curves (CRDPC).
(18) Smith, C. P.; White, H. S. Anal. Chem. 1992, 64, 2398.
(19) Savitzky, A.; Golay, M. J. E. Anal. Chem. 1964, 36, 1627.
(20) Sigma Plot for Windows, version 5.0; Jandel Scientific: Corte
Madera, CA, 1999.
(21) Maron, M. J. Numerical Analysis: A Practical Approach; Collier
McMillan: London, 1982.
Theory
We will consider the subsequent reductions and oxidations between adsorbates that take place when several
successive programmed power time currents of the form
I(ti) ( I0 tiu (with u > -1) are applied. The surface electrode
process considered is given by
kred
O(absorbed) + ne- {\
} R(absorbed)
k
ox
I0tu
) ( (kredΓA(t) - koxΓB(t))
nFA
(1)
with the sign “+” referring to cathodic currents and the
sign “-” to anodic ones and
kred ) k0η-R
(2)
kox ) k0η1-R
(3)
η ) exp
nF
(E - E )]
[RT
0
(4)
Taking into account that the total adsorbate excess, ΓT,
is constant during the experiment,
ΓT ) ΓA,0 + ΓB,0 ) ΓA(t) ) ΓB(t)
(5)
we deduce the expressions of the surface excesses of both
species, and by imposing the condition Γi(t ) τ) ) 0 (with
i ) A for cathodic currents and i ) B for anodic ones) we
obtain the following expressions for the transition times:3
τu+1 )
(u + 1)nFAΓA,0
I0
for the first power current applied (6)
(22) Laviron, E. In Electroanalytical Chemistry; Bard, A. J., Ed.;
Marcel Dekker: New York, 1982; Vol. 12.
408
Langmuir, Vol. 19, No. 2, 2003
Molina and Gonzalez
(u + 1)nFAΓT
I0
for the second, third, etc. power currents applied
(7)
τu+1 )
In eqs 1-7, ΓA,0 and ΓB,0 are the initial values of the
surface excesses of the oxidized and reduced species,
respectively, A is the electrode area (cm2), and n, F, R, and
T have their usual meanings.
By introducing eqs 2-7 in eq 1 and by making the
following change of variable,
T ) (t/τ)u+1
(8)
we deduce
u + 1 u/(u+1) R
T
η ) 1 - T - η(ΓB,0/ΓA,0) - ηT
τk0
first power time current applied (9)
of successive current steps (u ) 0), eqs 14 take the following
explicit form:
dT
dE
)
dT
dE
)
Cyclic Reciprocal Derivative Power Chronopotentiometry for Totally Irreversible Processes. For
a totally irreversible process (k0 , 1 s-1), eqs 9-11 are
simplified to the following form:14-17
1-T
RT
RT
ln(τk0) +
ln
RnF
RnF (u + 1)T(u/u+1)
cathodic (12)
E - E0 )
RT
ln(τk0) E-E )(1 - R)nF
1-T
RT
ln
(1 - R)nF (u + 1)T(u/u+1)
,u)0 )
anodic
Note that in these equations, the E versus t response
given by eq 1 has been transformed into the E versus T
response (with T being the dimensionless time given by
eq 8). This presents interesting advantages, as will be
shown in the following sections.
By differentiating eqs 12 and 13 with respect to the
potential, we deduce the analytical expressions for the
cathodic and anodic dT/dE versus E curves (CRDPC
responses) corresponding to an irreversible cathodic,
dT/dE)cathodic, or anodic, dT/dE)anodic, process for any value
of u,
dT
dE
)
dT
dE
)
2
)cathodic
)
anodic
(2u+1)/(u+1) R
η
RnF (u + 1) T
0
RT
T
+
u
τk
(14)
(1 - R)nF (u + 1)2 T(2u+1)/(u+1)η-(1-R)
RT
T+u
τk0
for u * 0
Equations 14 are not explicit expressions of the potential
E(t). However, for the particular case of the application
(15)
(1 - R)nF 1 -(1-R)
η
RT
τk0
Peak Parameters. The expressions for the peak
heights and peak potentials corresponding to the dT/dE
versus E curve can be easily obtained from eqs 14 by
carrying out the second derivative of these equations with
respect to the potential and by making d2T/dE2 ) 0. We
must also take into account that the power of time u must
fulfill u > 0, since for u e 0, peaks are not obtained. Hence,
we deduce
dT
dE
)
dT
dE
)c,peak
)
)
a,peak
RnF(u + 1)
Gu
RT
(16)
(1 - R)nF(u + 1)
Gu
RT
for u > 0
for the cathodic and anodic peak heights and
Ec,peak ) E0 +
Ea,peak ) E0 -
RT
RT
ln(τk0) +
F
RnF
RnF u
(17)
RT
RT
ln(τk0) F
(1 - R)nF
(1 - R)nF u
for u > 0
for the cathodic and anodic peak potentials, with
Gu )
0
anodic (13)
RnF 1 R
η
RT τk0
for u ) 0
u + 1 u/(u+1) R
T
η ) 1 - T - ηT
τk0
third, fifth, etc. power time currents applied
(10)
u + 1 u/(u+1) R
T
η ) -T + η(1 - T)
τk0
second, fourth, etc. power time currents applied
(11)
,u)0 ) -
cathodic
Fu ) ln
Hu - Hu2
u + Hu
(18)
1 - Hu
(u + 1)Hu(u/u+1)
Hu ) xu(u + 1) - u
From eqs 16 and 17, we can immediately obtain the
ratio between peak heights, |dT/dE)c,peak/dT/dE)a,peak|, and
the difference between peak potentials, ∆Epeak ) |Ec,peak
- Ea,peak|, given by
|
dT/dE)c,peak
|
dT/dE)a,peak
∆Epeak )
)
R
1-R
RT
1
(ln(τk0) + Fu)
nF
R(1 - R)
(19)
(20)
Equations 16-20 for the peak parameters are valid for
u > 0 since for the case u ) 0, peaks are not obtained from
eqs 15.
Another interesting feature of the response of the
CRDPC technique is the half peak width of the dT/dE
versus E curves, defined as the difference between the
potentials at which a value of the CRDPC curve equal to
the half peak height (dT/dE)peak/2) is obtained. We have
found that the half peak width for this technique is a
Cyclic Reciprocal Derivative Chronopotentiometry
Langmuir, Vol. 19, No. 2, 2003 409
function of the power of time u and is given by (see eqs
14 and 16)
Wc1/2 )
u/(u+1)
RT (1 - T2)T1
ln
RnF (1 - T )T u/(u+1)
1
Wa1/2 )
(21)
2
(1 - T2)T1u/(u+1)
RT
ln
(1 - R)nF (1 - T1)T2u/(u+1)
with T1 and T2 being
-(Gu - 2) + x(Gu - 2)2 - 8uGu
T1 )
4
(22)
-(Gu - 2) - x(Gu - 2)2 - 8uGu
T2 )
4
and Gu given in eqs 18.
Equations 16-22 have been obtained from eqs 14 corresponding to totally irreversible processes. In the case of
quasi-irreversible processes, it is not possible to use any
simplification in the theoretical equations for the potential
time response (eqs 9-11) and their derivatives. The kinetic
parameters of the surface electrochemical process in this
last case can be obtained by fitting the experimental results
with theoretical curves calculated for different sets of
values of the rate constant, the charge-transfer coefficients,
and the interfacial standard potential.
Cyclic Reciprocal Derivative Power Chronopotentiometry for Reversible Processes. For a reversible
process (k0 f ∞), eqs 9-11 are transformed into23
E(t) ) E0 +
1-T
RT
ln
nF ΓB,0/ΓA,0 - T
first power time current applied (23)
RT 1 - T
ln
nF
T
second, third, etc. power time currents applied
E(t) ) E0 (
with T given in eq 8.
By differentiating eqs 23 with respect to the potential,
we obtain the analytical expressions for the dT/dE versus
E curves (CRDPC responses) corresponding to a reversible
electrode process,23
| |
nF
η
dT
)dE
RT (1 + η)2
cathodic and anodic
(24)
From eqs 23 and 24, it can be deduced immediately
from this model that the cathodic and anodic responses
corresponding to a reversible process possess the same
peak potential; this, ideally, will be equal to E0 (i.e., Ec,peak
) Ea,peak ) E0), and therefore ∆Epeak ) |Ec,peak - Ea,peak| is
equal to zero.23 Moreover, the peak heights, dT/dE)peak,
and the ratio dT/dE)/dT/dE)peak are given by
dT
dE
)
peak
nF
)4RT
(25)
dT/dE)
η
)4
dT/dE)a,peak
(1 + η)2
cathodic and anodic
Thus, in this case the ratio dT/dE)/dT/dE)peak is only
dependent on η and is identical to that corresponding to
Figure 1. Theoretical normalized (E - E0) vs T (a) and dT/dE
vs (E - E0) curves (b) for a reversible surface charge-transfer
process obtained from eqs 23 and 24, respectively. T )
(t/τ)u+1. I(t) ) I0tu . I0/(nFAΓA,0) ) 1.0 s-(u+1), n ) 1, T ) 298 K.
The values of the power of time u in the applied current are on
the curves.
cyclic voltammetry (see eq 68 in ref 22). Moreover, it is
fulfilled that |dT/dE)c,peak/dT/dE)a,peak| ) 1.
Finally, the half width of reversible peaks can be
obtained in the same way as described above for an
irreversible process. Thus we obtain the following value
for cathodic and anodic curves: W1/2 = 3.53 RT/(nF), which
is not dependent on the experimental conditions and is
identical to the half width value obtained in cyclic
voltammetry for a reversible surface process.22
Results and Discussion
Theoretical Results. In Figure 1, we have plotted the
normalized E versus T curves with T ) (t/τ)u+1 (Figure 1a)
and the dT/dE versus E curves (Figure 1b), corresponding
to the application of a programmed current I0tu to a
reversible process for three different values of u. As can
be deduced from these figures, both plots give rise to a
single response which is independent of the experimental
conditions.
The behavior of the d(t/τ)u+1/dE versus E curves shown
in Figure 1b presents great advantages over the signal
usually obtained in reciprocal derivative chronopotentiometry, that is, the d(t/τ)/dE versus E curve, since in this
last case the reversible response depends on the value of
the power of time u (see Table 2 in ref 3) and, therefore,
we do not obtain a unique curve as in the case of the
d(t/τ)u+1/dE versus E curves.
410
Langmuir, Vol. 19, No. 2, 2003
Molina and Gonzalez
Table 1. Values of the Ratio between the Cathodic and
Anodic Peak Heights for Irreversible and Reversible
Processes, RCRDPC
and RCRDPC
, Given by Equations 26 and
c
a
27, and of the Cathodic and Anodic Half Peak Widths,
c
a
W1/2 and W1/2, Obtained from Equations 21 and 22a
u
RCRDPC
,
c
RCRDPC
a
RCRDPC
) RCRDPC
c
a
for R ) (1-R) ) 0.5
Wc1/2, Wa1/2
0.075
0.10
0.20
0.25
0.50
0.75
1.00
2.503ξ
2.361ξ
2.022ξ
1.910ξ
1.607ξ
1.461ξ
1.373ξ
1.251
1.180
1.011
0.955
0.804
0.730
0.686
1.31RT/(ξnF)
1.40RT/(ξnF)
1.68RT/(ξnF)
1.79RT/(ξnF)
2.16RT/(ξnF)
2.39RT/(ξnF)
2.55RT/(ξnF)
a ξ ) R for a cathodic (c) process, and ξ ) 1 - R for an anodic (a)
one.
of this technique due to the fact that, as far as we know,
this is the only electrochemical technique for which such
behavior is observed. The reason for this higher sensitivity
for irreversible processes lies in the fact that peak heights
obtained in CRDPC are functions of the value of u. Hence,
the ratio between peak heights corresponding to a totally
irreversible process and a reversible one for a cathodic or
anodic applied current is given by (see eqs 16 and 25)
RCRDPC
)
c
irrev
dT/dE)c,peak
rev
dT/dE) c,peak
) 4R(u + 1)Gu
cathodic peaks (26)
and
Figure 2. Theoretical normalized (E - E0) vs T (a) and dT/dE
vs (E - E0) curves (b) for a totally irreversible surface chargetransfer process obtained from eqs 12-13 and 14, respectively.
T ) (t/τ)u+1 . I(t) ) I0tu . k0 ) 0.1 s-1, R ) 0.5. The values of the
power of time u in the applied current are on the curves. Other
conditions are as in Figure 1.
When a totally irreversible process is considered (see
Figure 2), the temporal dependence of the response cannot
be eliminated, and it is not possible to obtain a single E
versus T curve (Figure 2a). As a consequence, the dT/dE
()d(t/τ)u+1/dE) curves depend on the value of u (Figure
2b). This dependence and the absence of peaks in these
curves for u e 0 can be considered as criteria for identifying
a totally irreversible process.
In general, when we compare the signal obtained in
reversible and irreversible electrode processes in electrochemical techniques with potential controlled or current
controlled, it is observed that the signal corresponding to
a totally irreversible process presents a lower sensitivity
than that obtained for a reversible one.14,15 Thus, in
differential pulse voltammetry (DPV),24 CV,25 derivative
voltammetry (DV),26 or even in CRDEC,14 it is observed
that the peak heights always decrease when the reversibility of the process decreases.
However, when we apply power currents of the form
(I0tu and the reciprocal derivative of the E versus T curve
is obtained, there are values of the power of time u for
which the peak heights corresponding to irreversible
processes are higher than those corresponding to reversible
ones. This possibility represents an enormous advantage
(23) Molina, A.; Gonzalez, J. J. Electroanal. Chem. 2000, 493, 117.
(24) Birke, R.; Kim, M. H.; Strassfeld, M. Anal. Chem. 1981, 53, 852.
(25) Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 36, 706.
(26) Kim, M. H.; Smith, V. P.; Hong, T. K. J. Electrochem. Soc. 1993,
140, 712.
RCRDPC
a
)
irrev
dT/dE)a,peak
rev
dT/dE) a,peak
) 4(1 - R)(u + 1)Gu
anodic peaks (27)
with Gu given in eqs 18, in such a way that for a value of
R ) (1 - R) ) 0.5 it is fulfilled that 4R(u + 1)Gu > 1 for
u < 0.20, whereas 4R(u + 1)Gu < 1 for u g 0.20; that is,
the peak heights of the cathodic or anodic curves corresponding to an irreversible process are greater than
those corresponding to a reversible one if u < 0.20 (see
Table 1).
To illustrate the behavior discussed above, Figure 3a,b
shows the influence of the surface rate constant k0 on the
normalized dT/dE versus E curves deduced from eqs 9-11
for two different values of the power of time u in the applied
currents I(t) ) (I0tu (u ) 0.50, Figure 3a, and u ) 0.10,
Figure 3b). From these figures, it can be observed that as
k0 decreases, the cathodic and anodic curves are shifted
toward negative and positive directions, respectively, thus
increasing the difference between both peak potentials.
In relation to the peak heights, Figure 3a, corresponding
to u ) 0.5, shows a typical electrochemical behavior. Both
cathodic and anodic peak heights decrease in absolute
value when k0 decreases, until the limit values given in
eqs 16 are reached. However, in Figure 3b, obtained for
u ) 0.1, we can observe that the heights of both peaks
increase in absolute value as k0 decreases, until the limit
values given in eqs 16 are reached.
We can conclude that the interest of the dT/dE versus
E curves for values of u e 0.20 should be apparent since,
as Figure 3b shows, irreversible processes provide, in this
case, more sensitive signals than do reversible ones.
As compared to cyclic voltammetry, CRDPC is much
more advantageous for the characterization of irreversible charge-transfer reactions of adsorbed molecules due to the lower influence of capacitive and ohmic
Cyclic Reciprocal Derivative Chronopotentiometry
Langmuir, Vol. 19, No. 2, 2003 411
Figure 3. Theoretical dT/dE vs (E - E0) curves obtained from
eqs 9-11 corresponding to the application of two successive
power time functions. I(t) ) I0tu and I(t) ) -I0tu. T ) (t/τ)u+1.
The values of the power of time u in the applied currents are
u ) 0.5 (a) and u ) 0.1 (b). I0/(nFAΓA,0) ) 2.0 s-(u+1), R ) 0.5.
The values of k0 (in s-1) are on the curves. Other conditions are
as in Figure 1.
drop effects3,4,14,17,18 and also because the peaks obtained
in CRDCP are much higher and narrower than those
obtained in CV. For comparison, both voltammetric and
chronopotentiometric signals should be given in V-1; hence
the voltammetric current should be converted to V-1 by
dividing it by (nFAΓA,0v). Thus, the ratio between peak
heights for totally irreversible processes obtained in
CRDPC and CV is given by (see eq 16 of this paper and
eqs 18 and 19 in ref 27)
dT/dE)peak
ψCV
peak
) 2.72(u + 1)Gu
cathodic and anodic
(28)
-1
where the peak current, ψCV
peak (in V ), and the half peak
27
,
for
CV
are
given
by
width, WCV
1/2
WCV
1/2 = 2.43RT/(ξnF)
(29)
CV
|ΨCV
peak| ) |Ipeak/(nFAΓA,0v)| ) ξnF/(2.72RT)
ξ ) R or ξ ) 1 - R
From eqs 28 and 29, we deduce that for values of the
(27) Laviron, E. J. Electroanal. Chem. 1979, 101, 19.
Figure 4. Experimental E vs t (a,b) and (t/τ)u+1 vs E curves
(c) for the quinizarine 10 µM/HClO4 1.0 M system on a SMDE.
I(t) ) (I0tu. Γquinizarine,0 ) (8.29 ( 0.2) × 10-11 mol cm-2, r0 )
0.035 cm, n ) 2, T ) 295 K. The values of the power of time
u in the applied currents are on the figure.
power of time u < 0.75, the peaks obtained in CRDPC are
higher and narrower (see Table 1) than those obtained in
CV (see also experimental results). For example, for u )
0.1 the peaks obtained in CRDPC are 1.6 times higher
and =47% narrower than those obtained in CV. Moreover,
CRDPC peaks are also narrower than those obtained when
an exponential current of the form I(t) ) -I0eωt is used
(CRDEC14).
Equations 12 and 13 for the potential time curves of a
totally irreversible process are also valid for u e 0, and
specifically for -1 e u e 0. But this last interval of values
is not advisable for determining kinetic parameters of the
charge transfer, since, in these conditions, the anodic and
cathodic normalized dT/dE versus E curves do not present
peaks. Nevertheless, due to the fact that this is a typical
characteristic of irreversible processes, this behavior can
be used for detecting its presence.
Experimental Results. Reversible Behavior: Quinizarine in Aqueous Acidic Media. In the first place, we
have applied the CRDPC technique to the study of the
redox behavior of quinizarine (1,4-dihydroxyanthraquinone) in aqueous acidic solutions. This system is strongly
adsorbed on mercury and behaves as reversible.28
412
Langmuir, Vol. 19, No. 2, 2003
Molina and Gonzalez
Figure 6. (Lines with symbols) Experimental dT/dE vs E
curves for the quinizarine 10 µM/HClO4 1.0 M system on a
SMDE. T ) (t/τ)u+1. I(t) ) I0tu and I(t) ) -I0tu. I0 ) 6.0 µA s-u.
The values of u and τ (in ms) are as follows: (black circles)
0.075, 45.4; (white triangles) 0.10, 42.9. (Solid lines) Experimental cyclic voltammograms (ICV/(nFAΓquinizarine,0v) vs E curves)
for the quinizarine 10 µM/HClO4 1.0 M system on a SMDE.
Sweep rate v ) 5.0 V s-1. Γquinizarine,0 ) (8.29 ( 0.2) × 10-11 mol
cm-2. Other conditions are as in Figure 4.
Figure 5. Experimental dt1.075/dE vs E (a) and dT/dE vs E (b)
curves for the quinizarine 10 µM/HClO4 1.0 M system on a
SMDE. T ) (t/τ)1.075. I(t) ) I0t0.075, u ) 0.075. The values of I0
(in µA s-0.075) and τ (in ms) are as follows: 5, 59.9; 6, 48.8; 7,
42.2; 8, 36.3. Other conditions are as in Figure 4.
Figure 4 shows the chronopotentiometric E versus t
curves (Figure 4a,b) and their corresponding (t/τ)u+1 vs E
curves (Figure 4c), obtained for the quinizarine 10 µM/
HClO4 1.0 M system in an aqueous solution with 1% of
ethanol. All these curves have been obtained for the
application of 16 programmed currents of the form I(t) )
(I0tu to a SMDE and for two different values of the
exponent of time u (0.075 and 0.20). From the E versus
t curves, it can be deduced that the transition times for
the reduction and oxidation are the same (i.e., anodic and
cathodic E vs t curves are specular images, with the same
transition time). It can be also observed from Figure 4c
how the I(t) ) (I0tu versus E curves remain practically
independent of the value of u, in accordance with
theoretical predictions (eqs 23).
From the chronopotentiometric curves shown in Figure
4, we have obtained the non-normalized reciprocal derivative curves (dtu+1/dE vs E curves, see Figure 5a) and
the normalized derivative ones (dT/dE or d(t/τ)u+1/dE vs
E curves, see Figure 5b) shown in Figure 5 for u ) 0.075
and for four different values of the current amplitude I0.
From the non-normalized curves dtu+1/dE versus E
curves in Figure 5a, it can be seen that for this system we
obtain symmetrical curves for the anodic and cathodic
processes in which Ec,peak ) Ea,peak ) E0, with E0 being the
interfacial standard potential for the Langmuir isotherm.23
From the peak potentials in Figure 5a, we have obtained
(28) Forster, R. J. Analyst 1996, 121, 733.
the following values for E0 from the cathodic and anodic
curves, respectively: (-0.198 ( 0.001) V and (-0.197 (
0.001) V versus Ag/AgCl, KCl 1.0 M. Moreover, the distance
between the cathodic and anodic peak potentials of these
curves is ∆Epeak = 0 mV with a maximum error of 2 mV
in all the cases. These curves also fulfill that the ratio
between cathodic and anodic peak heights is |dT/dE)c,peak/
dT/dE)a,peak| = 1.0 for any value of the current amplitude
and with an error margin of less than 5%. The results for
both peak potentials and peak heights indicate that this
process behaves as a reversible surface charge transfer.23
It can also be observed in Figure 5a that the cathodic
and anodic peak heights decrease in absolute value as I0
increases. The dependence of the peak heights on the
current amplitude is given by the following equations:
)
dtu+1
dE
)
dtu+1
dE
)c,peak
a,peak
)
n2F2AΓquinizarine,0 u + 1
nF u+1
τ
)4RT
4RT
I0
(30)
2 2
nF u+1 n F AΓquinizarine,0 u + 1
τ
)
(31)
4RT
4RT
I0
with Γquinizarine,0 being the initial excess of quinizarine on
mercury.
Thus, by plotting the cathodic and anodic peak heights
versus (u + 1)/I0, a linear dependence is found and we can
easily obtain the initial surface excess of quinizarine. From
the data shown in Figure 5a, we have obtained Γquinizarine,0
) (8.29 ( 0.2) × 10-11 mol cm-2, which is in agreement
with the value reported in ref 28, which is Γquinizarine,0 )
1.1 × 10-10 mol cm-2 (obtained in a pure aqueous medium).
Figure 5b shows the normalized dT/dE ()d(t/τ)u+1/dE)
versus E curves obtained by dividing curves in Figure 5a
Cyclic Reciprocal Derivative Chronopotentiometry
Langmuir, Vol. 19, No. 2, 2003 413
Figure 8. Experimental dt1.1/dE vs E (a) and dT/dE vs E (b)
curves for the azobenzene 30 µM in KNO3 0.1 M (pH ) 6.74)
system, on a SMDE. T ) (t/τ)1.1. I(t) ) I0t0.1, u ) 0.10. The values
of I0 (in µA s-0.10) and τ (in ms) are as follows: 6, 114.8; 8, 88.0;
10, 70.1; 12, 59.2. Other conditions are as in Figure 7.
Figure 7. Experimental E vs t curves (a,b) and (t/τ)u+1 vs E
curves (c), for the azobenzene 30 µM in KNO3 0.1 M (pH ) 6.74)
system, on a SMDE. I(t) ) (I0tu. Γquinizarine,0 ) (1.81 ( 0.2) ×
10-10 mol cm-2, r0 ) 0.035 cm, n ) 2, T ) 295 K. The values
of the power of time u in the applied currents are on the figure.
by τu+1. According to the theoretical results discussed
above, in the case of a reversible surface electrode process
we obtain a single response for the CRDPC curves (see
eqs 23 for the potential time curves and eq 24 for the
reciprocal derivative ones).
Finally, we have demonstrated in a previous work that
the responses obtained in CRDPC and CV techniques are
identical for the case of an electrode coated with a
molecular film of a reversible electroactive couple.23 In
Figure 6, we have plotted the response obtained for the
quinizarine system in cyclic voltammetry for a sweep rate
of v ) 5 V s-1 (solid line) for which the capacitive effects
have been corrected and in CRDPC for the application of
two successive cathodic and anodic programmed currents,
I(t) ) I0tu and I(t) ) -I0tu, respectively, with u ) 0.075
(black circles) and 0.10 (white triangles), with both
voltammetric and chronopotentiometric signals being
given in V-1. The curves in this figure clearly show the
total analogy between CRDPC (d(t/τ)u+1/dE vs E) and CV
(ICV/(nFAΓquinizarine,0v) vs E) techniques. The reason for this
analogy has been extensively discussed in ref 23.
Irreversible Behavior: Azobenzene in Aqueous Solution.
We have also applied the CRDPC technique to the study
of the experimental system azobenzene 30 µM in KNO3
0.1 M (pH ) 6.74), which behaves as quasi-irreversible as
a function of the pH.14,29,30
Figure 7 shows the experimental chronopotentiometric
E versus t curves (Figure 7a,b) and their corresponding
(t/τ)u+1 versus E curves (Figure 7c) obtained for this system
by the application of six programmed currents of the form
I(t) ) (I0tu to a SMDE with I0 ) 8.0 µA s-u and for two
different values of the exponent of time u (0.075 and 0.20).
These curves show the great asymmetry between reduction E versus t curves (which are shifted toward negative
potential values) and oxidation E versus t ones (which are
shifted toward positive potential values). Transition times,
logically, remain unaffected by the reversibility of the
surface process and are approximately the same in both
cathodic and anodic responses. Figure 7c also shows that
the (t/τ)u+1 versus E curves are dependent on the value of
u, contrary to those obtained for a reversible process (see
Figure 4c).
Figure 8 shows the influence of the current amplitude
I0 on the non-normalized dtu+1/dE versus E curves (Figure
8a) and on the normalized dT/dE ()d(t/τ)u+1/dE) versus E
ones (Figure 8b) obtained for the azobenzene system for
a fixed value of the power of time u ) 0.1 and for four
values of I0. From these figures, we can conclude that
when the value of I0 increases, the cathodic and anodic
(29) Komorsky-Lovric, S.; Lovric, M. Electrochim. Acta 1995, 40,
1781.
(30) Laviron, E.; Mugnier, Y. J. Electroanal. Chem. 1980, 111, 337.
414
Langmuir, Vol. 19, No. 2, 2003
Molina and Gonzalez
Figure 10. (Solid lines) Experimental dT/dE vs E curves for
the azobenzene 30 µM in KNO3 0.1 M (pH ) 6.74) system, on
a SMDE. T ) (t/τ)u+1. I(t) ) I0tu and I(t) ) -I0tu . I0 ) 12.0 µA
s-u. The values of u are on the curves. The values of τ (in ms)
are as follows: u ) 0.075, 54.7; u ) 0.10, 60.4; u ) 0.20, 89.8.
(Dotted line) Experimental deωt/dE vs E curves for the azobenzene 30 µM in KNO3 0.1 M (pH ) 6.74) system, on a SMDE.
I0 ) 1.5 µA, ω ) 10 s-1, τ ) 0.136 s. (Broken lines) Experimental cyclic voltammograms (ICV/(nFAΓA,0v) vs E curves) for
the system azobenzene 30 µM in NO3K 0.1 M (pH ) 6.74), on
a SMDE. Sweep rate v ) 2.0 V s-1. Other conditions are as in
Figure 7.
Figure 9. Experimental dependence of Ec,peak and Ea,peak with
ln(τ/s) (a) and of ∆Epeak with ln(τ/s) (b) for the azobenzene 30
µM in KNO3 0.1 M (pH ) 6.74) system, on a SMDE. I(t) ) I0t0.1,
u ) 0.10. Other conditions are as in Figure 8.
curves are shifted toward more negative and positive
potential values, respectively, and this causes the difference between peak potentials, ∆Epeak, to increase. The
variation of the peak potentials, Ec,peak and Ea,peak, and
also of ∆Epeak with the logarithm of transition time, is
linear, as is shown in Figure 9. According to eqs 17 and
20, the linear dependence of Ec,peak, Ea,peak and ∆Epeak with
ln τ is a typical feature of totally irreversible processes.
Hence we can conclude that in these conditions, azobenzene behaves as totally irreversible and we can calculate
the values of the kinetic and thermodynamic parameters
of the charge transfer by using eqs 16-20.
From non-normalized dtu+1/dE versus E curves in Figure
8a, we can conclude that, analogously to the behavior
observed in the case of a reversible system, cathodic and
anodic peaks heights decrease in absolute value as I0
increases.
If we compare the normalized curves of Figure 8b with
those obtained for a reversible system (Figure 5b), we can
see that in the case of irreversible behavior we do not
obtain a single cathodic and a single anodic curve, as we
do in the reversible case. Note that in this figure all the
anodic and cathodic peaks possess the same respective
heights, since peak heights for totally irreversible processes are only dependent on the value of the exponent
u and on the cathodic and anodic charge-transfer coefficients (eq 16). Therefore, from the measurement of
cathodic and anodic peak heights, R and (1 - R) can be
obtained immediately. Thus, from the curves in Figure 8b
we have obtained R ) (0.69 ( 0.02) and (1 - R) ) (0.28
( 0.01). Moreover, once the charge-transfer coefficients
are known, we can obtain the value of the rate constant
k0 from the intercept of the linear plots of Figure 9. Thus,
from Figure 9b (eq 20), we obtain the following value:
log(k0/s-1) ) (0.20 ( 0.02); from Figure 9a, we deduce the
following value of the interfacial standard potential E0:
(-0.344 ( 0.002) V versus Ag/AgCl, KCl 1.0 M. These
values are in line with those previously obtained by us
with CRDEC in ref 14 (log(k0/s-1) ) (0.32 ( 0.02), R )
(0.63 ( 0.02), (1 - R) ) (0.35 ( 0.02), and E0 ) (0.336 (
0.002) V versus Ag/AgCl, KCl 1.0 M). The difference
between both sets of data is related to the difference in
pH (6.60 in ref 14, 6.74 in the present work) and is in good
agreement with data in the literature.30
The initial excess of azobenzene on mercury has also
been determined from measurements of peak heights of
the CRDPC curves corresponding to u ) 0.10. This value
has been obtained by plotting cathodic and anodic peak
heights of the dtu+1/dE curves in Figure 8a versus (u +
1)/I0 for different values of the current amplitude I0 (see
eqs 6, 7, and 16). Thus, we have obtained Γazobenzene,0 )
(1.81 ( 0.02) × 10-10 mol cm-2, which is in agreement
with the value Γazobenzene,0 ) 10-10 mol cm-2 reported in ref
31 by Wopschall and Shain and obtained with linear sweep
voltammetry.
Finally, Figure 10 shows the comparison between the
experimental normalized CRDPC responses for the azobenzene system obtained for three values of the u exponent
(0.075, 0.100, and 0.200) and a fixed value of the current
amplitude I0 ) 12 µA s-u, the normalized cyclic voltammograms of this system obtained in the same conditions
for a sweep rate of v ) 2.0 V s-1, and the CRDEC curves
for the azobenzene system corresponding to the application
(31) Wopschall, R. H.; Shain, J. Anal. Chem. 1965, 35, 1535.
Cyclic Reciprocal Derivative Chronopotentiometry
of exponential currents of the form I(t) ) (I0eωt with ω )
10 s-1 and ) I0 ) 1.5 µA. Voltammetric and chronopotentiometric curves are given in V-1 for comparison.
It can be clearly observed from this figure that the
signals corresponding to CRDPC present both higher and
narrower peaks than those corresponding to CV or to
CRDEC. This behavior is in agreement with theoretical predictions (see Table 1 in ref 14 and eq 28 and
Table 1 in this work). Thus, according to eq 28, for u )
0.1 the ratio dT/dE)peak/ψCV
peak ) 1.60, and for u ) 0.075,
dT/dE)peak/ψCV
peak ) 1.70. From data in Figure 10, we obtain
that the values for the ratio between peak heights obtained
in CRDPC and CV are similar to theoretical values for the
anodic peaks. In the case of the cathodic peaks, the
CV
experimental ratio dT/dE)peak/ψpeak
is even greater than
the values theoretically expected because of a strong
distortion in the cyclic voltammograms, which is probably
due to capacitive effects. This distortion mainly affects
the voltammetric cathodic peak and strongly increases
with the sweep rate.18 Nevertheless, CRDPC curves are
Langmuir, Vol. 19, No. 2, 2003 415
highly reproducible since the capacitive effects are negligible under these conditions.
We have also measured the cathodic and anodic half
peak widths in CRDPC curves and obtained the following
values for the cathodic peaks: Wc1/2(u ) 0.075) ) 25 mV
and Wc1/2(u ) 0.10) ) 27 mV, which are in good agreement
with the theoretical ones (see Table 1). These values are,
in all the cases, lower than those corresponding to CRDEC.
From these results, we can conclude that, as compared
to CV, the greater sensitivity, the lower values of the half
peak widths, and the smaller distortion of the signal make
CRDPC one of the best electrochemical methods for dealing
with surface processes.
Acknowledgment. The authors greatly appreciate the
financial support provided by the Dirección General de
Investigación Cientı́fica y Técnica (Project Number
BQU2000-0231) and to the Fundación SENECA (Expedient Number 00696/CV/99).
LA020369X