CEJC 3(2) 2005 216–229
Kinetics of electrode reaction coupled to ion transfer
across the liquid/liquid interface
Šebojka Komorsky-Lovrić∗, Milivoj Lovrić
Centre for Marine and Environmental Research,
”Rudjer Bošković” Institute,
P.O. Box 180, Zagreb, HR-10002, Croatia
Received 13 October 2004; accepted 15 December 2004
Abstract: In the theoretical model it is assumed that a graphite disk electrode is covered by
a thin film of solution of decamethylferrocene (dmfc) and some electrolyte CX in nitrobenzene
and immersed in an aqueous solution of the electrolyte MX. Oxidation of dmfc is accompanied
by the transfer of anion X − from water into nitrobenzene since it is also assumed that cations
dmfc + and C + are insoluble in water and cation M + is insoluble in nitrobenzene. Kinetic
parameters of the electrode reaction can be determined if the total potential difference across
the nitrobenzene/water interface is maintained constant by adding the electrolytes CX and MX
in concentrations which are much higher than the initial concentration of dmfc in nitrobenzene.
c Central European Science Journals. All rights reserved.
Keywords: Cyclic voltammetry, ion transfer, electrode kinetics, liquid/liquid interface,
decamethylferrocene
1
Introduction
Ion transfer across the interface between two immiscible electrolyte solutions (ITIES) is
a phenomenon that is important for studies of phase transfer catalysis, liquid – liquid
extraction, electroanalysis, liquid state ion-selective electrodes, solvation, metal refining
and models of biological membranes [1-7]. Standard Gibbs energies of ion transfers across
the ITIES can be measured by the recently proposed three - electrode system [8-11]. A solution droplet of decamethylferrocene in nitrobenzene is attached to the surface of paraffin
– impregnated graphite electrode and immersed into an aqueous electrolyte. Oxidation of
decamethylferrocene in the droplet is accompanied by the transfer of anions from water
∗
E-mail: slovric@rudjer.irb.hr
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into nitrobenzene. The reaction starts near the three – phase boundary in the cuneiform region close to the edge of the drop, in which the initial conductivity is achieved
by the partition of the electrolyte between water and nitrobenzene under open – circuit
conditions [12-14]. This partition also explains similar experiments in which the entire
electrode surface is covered by a very thin film of the organic phase [15,16]. The flux of
decamethylferrocenium cations at the electrode surface and the flux of anions at the liquid/liquid interface are separated by the nitrobenzene layer, but the migration of cations
of the electrolyte maintains the electroneutrality in the nitrobenzene [17]. However, the
current depends mainly on the diffusion and migration of decamethylferrocenium cations
and anions of the aqueous electrolyte into the film.
The first theory of ion transfer across a liquid – liquid interface was based on the assumption that the Galvani potential difference between the two phases is located entirely
at the interface [18]. In this model, the current – voltage relationship follows the Butler
– Volmer equation [19- 22]. The theory was extended by assuming that the interface
consists of two diffuse double layers and a central compact layer of oriented solvent molecules [23]. In contrast to treating interfaces as molecularly sharp boundaries, a model of
a mixed solvent layer was suggested, in which the interfacial region was considered as an
inhomogeneous phase through which ions are transferred in the gradients of chemical and
electrical potentials [24]. In this model the transfer coefficient may be concentration and
potential dependent, the exchange current density depends on the value of the standard
transfer potential and the formal standard rate constant can be related to the diffusion
coefficient of the transferring ion and to the thickness of the interfacial region [25-29].
In this short communication a simple theoretical model of kinetically controlled electrode reaction coupled to ion transfer across the liquid/liquid interface is developed in
order to investigate whether the electrode kinetics can be distinguished from the ion
transfer kinetics, or not, and under which conditions the kinetic parameters can be determined.
2
The model
It is assumed that a stationary, planar, graphite electrode is covered by a thin film of
nitrobenzene in which decamethylferrocene (dmfc) and supporting electrolyte CX are
dissolved. The electrode is immersed in the aqueous solution of the electrolyte MX. On
anodic polarisation of the electrode, dmfc is oxidised in nitrobenzene while anions of the
supporting electrolyte are transferred across the water/nitrobenzene interface:
−
−
−
dmf c(nb) + X(w)
↔ dmf c+
(nb) + X(nb) + e
(1)
It is further assumed that there are no fluxes of dmfc, dmfc + , C + and M + across the
liquid/liquid interface.
The potential difference between the working electrode and the reference electrode in
water is a sum of the potential drop at the graphite/nitrobenzene interface and the total
potential drop across the nitrobenzene/water interface [8]:
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E = EC/nb + ∆nb
wφ
(2)
anb
dmf c+
RT
x=0
ln nb
F
admf c
0
EC/nb = Edmf
c+ /dmf c +
(3)
x=0
∆nb
w φ
=
0
∆nb
w φX −
anb
RT
X−
ln w x=L
+
F
(aX − )x=L
(4)
where L is the film thickness.
If it is assumed that activity coefficients of all species are equal, in the equilibrium
the redox reaction (1) can be described by a Nernst equation:
E = E0 +
where:
h
−
[dmf c+ ]x=0 X(nb)
RT
h
i
ln
−
F
[dmf c]x=0 X(w)
i
x=L
(5)
x=L
0
nb 0
E 0 = Edmf
c+ /dmf c + ∆w φX −
(6)
2.1 Kinetics of electrode reaction
If only the oxidation of dmfc is kinetically controlled, the current depends on the potential
drop at the graphite/nitrobenzene interface:
I = −F S ks exp(−αϕ1 )
h
dmf c+
i
x=0
− [dmf c]x=0 exp(ϕ1 )
(7)
where I is a current, ks is a standard reaction rate constant, α is a transfer coefficient, S
is the electrode surface area and:
F 0
EC/nb − Edmf
(8)
c+ /dmf c
RT
Using equations (2) and (4), the current can be calculated as a function of the electrode
potential E:
ϕ1 =
0
0
nb 0
EC/nb − Edmf
c+ /dmf c = E − Edmf c+ /dmf c − ∆w φX −
h
i
−
X(w)
RT
+
ln h − ix=L
F
X(nb)
(9)
x=L
−
X(nb)
i−1
exp (ϕ)
(10)
h
i−α
exp (−αϕ)
(11)
h
i
h
h
iα
−
exp (ϕ1 ) = X(w)
−
exp (−αϕ1 ) = X(nb)
x=L
x=L
−
X(w)
x=L
x=L
F E − E0
RT
So, equation (7) is transformed into:
ϕ=
(12)
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h
−
I = −F S ks exp (−αϕ) X(nb)
·
h
dmf c+
i
x=0
iα
x=L
h
h
−
X(w)
−
− [dmf c]x=0 X(w)
i
i−α
x=L
(13)
·
x=L
219
h
−
X(nb)
i−1
x=L
exp (ϕ)
2.2 Kinetics of anion transfer
The kinetics of interfacial ion transfer was interpreted by several mechanisms [1, 30], but
the Butler – Volmer equation is still widely used [30, 31]. The current depends on the
total potential difference across the liquid/liquid interface:
I = −F S k0 exp (−αit ε)
h
−
X(nb)
i
x=L
h
−
− X(w)
i
x=L
exp (ε)
(14)
F nb
0
∆w φ − ∆nb
φ
(15)
−
w X
RT
where k0 is an apparent standard rate constant that depends on the potential distribution
in the aqueous diffuse layer and the organic diffuse layer of the interface [30], and αit is a
transfer coefficient that may depend on the concentration of ions [1]. However, we assume
that both k0 and αit are constants and that double layer corrections can be neglected [31].
Using equations (2) and (3) the current can be expressed as a function of the electrode
potential E:
ε=
nb 0
nb 0
0
∆nb
w φ − ∆w φX − = E − ∆w φX − − Edmf c+ /dmf c +
ε = ϕ + ln
h
[dmf c]x=0
[dmf c+ ]x=0
I = −F S k0 exp (−αit ϕ) dmf c+
·
3
h
−
X(nb)
i
x=L
h
−
− X(w)
i
x=L
RT
[dmf c]x=0
ln
F
[dmf c+ ]x=0
iαit
x=0
(17)
it
[dmf c]−α
x=0 ·
h
[dmf c]x=0 dmf c+
(16)
(18)
i−1
x=0
exp (ϕ)
Results and discussion
−
Calculations are simplified by assuming that the transport of dmfc, dmfc + and X(nb)
in
the film can be neglected:
∗
[dmf c] = [dmf c] −
Zt
0
h
dmf c
+
i
=
Zt
0
I
dτ
F SL
I
dτ
F SL
(19)
(20)
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h
−
X(nb)
∗
h
i
=
i∗
−
X(nb)
h
+
Zt
0
I
dτ
F SL
(21)
i∗
−
−
where [dmf c] and X(nb)
are initial concentrations of dmfc and X(nb)
in the film. Also,
−
it can be assumed that the initial concentration of anion
Xi in water is much higher
h
− ∗
than the initial concentration of dmfc in nitrobenzene X(w) >> [dmf c]∗ , so that the
diffusion of X − in water can be also neglected:
h
i
h
−
−
X(w)
= X(w)
i∗
(22)
Under these conditions, equation (5) can be transformed into a dimensionless recursive
formula:
Φ2k + BΦk + C = 0
B=
c∗X,nb
+ exp (ϕf ) + 2
(23)
k−1
X
Φj
(24)
j=1
C = − exp (ϕf ) + c∗X,nb + exp (ϕf )
ϕf =
j=1
F
(E − Ef )
RT
Ef = E 0 −
where Φ =
h
i∗
Id
F SL[dmf c]∗
∗
k−1
X

Φj + 
RT ∗ ln cX,w
F
k−1
X
j=1
2
Φj 
(25)
(26)
(27)
h
i∗
−
is a dimensionless current, c∗X,nb = X(nb)
/ [dmf c]∗ and c∗X,w =
−
−
−
X(w)
/ [dmf c] are dimensionless concentrations of X(nb)
and X(w)
, respectively, d is a
time increment and k = 1, 2, 3 .... is the number of time increments in the certain moment
t. In cyclic voltammetry the time increment depends on the scan rate: d = ∆E/v, where
∆E = 0.0001 V is a constant potential increment.
Figure 1 shows the dependence of the dimensionless current Φ∗ = F SL[dmf c]I ∗ (F/RT )v
on the relative electrode potential E – Ef , for three different values of the initial dimensionless concentration of X − in nitrobenzene. If c∗X,nb ≤ 0.01, the peak potentials of both
oxidation and reduction branches of the cyclic voltammogram are Ep – Ef = -0.005 V.
This value corresponds
to − (RT /F ) ln (1.215), which means that the peaks appear when
h
i
−
+
[dmf c ]p = X(nb) = 0.651 × [dmf c]∗ . So, the peak potentials related to the aqueous
p
reference electrode are:
0
nb 0
Ep = Edmf
c+ /dmf c + ∆w φX − +
RT h − i∗ RT
RT
ln [dmf c]∗ −
ln X(w) −
ln (1.215)
F
F
F
(28)
Generally, the peak potentials depend on the logarithm of c∗X,nb , as can be
seen in
∗
∗
Fig. 2. If cX,nb ≥ 100, this relationship is linear, with the slope ∂Ep /∂ log cX,nb =
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221
Fig. 1 Cyclic voltammetry of reversible electrode reaction (1) under thin film conditions (eq.
∗
23). Φ∗ = I/F SL [dmf c] (F/RT ) v and c∗X,nb = 0.01 (1), 1 (2) and 100 (3)
Fig. 2 Dependence of peak potentials of cyclic voltammograms of reversible reaction (1) on the
logarithm of dimensionless initial concentration of anion X − in the film.
2.3 × RT /F . This can be explained by assuming that the transfer of anion X − from
−
water into nitrobenzene does not change the concentration of X(nb)
in the film significantly
h
−
(cX,nb = c∗X,nb and X(nb)
to the form:
i∗
>> [dmf c]∗ ). Under this assumption equation (23) is reduced
1−
Φk =
k−1
P
j=1
ϕ∗f =
!
Φj exp ϕ∗f −
1 + exp ϕ∗f
F E − Ef∗
RT
k−1
P
j=1
Φj
(29)
(30)
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Ef∗ = E 0 −
RT h − i∗ RT h − i∗
ln X(w) +
ln X(nb)
F
F
(31)
Fig. 3 Cyclic voltammogram of reversible reaction (1) in the superfluity of anion X − in the film
(eq. 29).
Fig. 3 shows the solution of equation (29). The peak potentials are Ep − Ef∗ = 0 V.
This is equal to Ep − Ef = 0.118 V, for c∗X,nb = 100, which is shown in Fig. 2.
Absolute values of anodic and cathodic peak currents
of
each particular voltammo ∗
gram shown in Figs. 1 and 3 are equal and change from Φp = 0.1716, for c∗X,nb < 0.01, to
∗
Φp =
0.25, for c∗X,nb > 100. The latter value is in agreement with the theory of surface
confined electrode reactions [32].
3.1 Kinetics of electrode reaction
General solutions of kinetic equations (13) and (18) cannot be obtained by the numerical
integration under the assumed
conditions
(19)–(22). However, considering equations (22),
h
i
− ∗
(26) and (27), the product X(w) exp (ϕ) appearing in eq. (13) can be substituted by the
term exp (ϕf ). So, the dependence of peak potentials of kinetically controlled electrode
reaction on the logarithm of the concentration of anion X − in aqueous electrolyte is:
h
−
∂Ep /∂ log X(w)
i∗
= −2.3 × RT /F
(32)
Equation (13) can be solved if it is assumed that cX,nb = c∗X,nb , i.e. that the initial
concentration of X − anion in the film is much higher than the initial concentration of
dmfc. This means that the potential drop across the liquid/liquid interface is constant
and that the change of the electrode potential E is identical to the change of the potential
EC/nb at the graphite/nitrobenzene interface. Under this assumption, eq. (13) can be
transformed into the recursive formula:
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ks∗
exp
Φk =
−αϕ∗f
"
exp
ϕ∗f
− 1 + exp
1 + ks∗ exp −αϕ∗f
h
ϕ∗f
1 + exp ϕ∗f
k−1
P
j=1
Φj
#
i
223
(33)
where ks∗ = ks ∆E/Lv is dimensionless standard electrode reaction rate constant.
Fig. 4 Cyclic voltammograms of quasireversible reaction (1) (eq. 33). The transfer coefficient
α = 0.5 and the dimensionless standard rate constant ks∗ = 10−3 (1) and 10−4 (2).
The solution of eq. (33) for rather slow electrode reactions is shown in Fig. 4, and the
dependences of anodic and cathodic peak currents and peak potentials on the logarithm
of dimensionless standard rate constant are shown in Fig. 5. The latter relationships
are linear if log(ks∗ ) < -3.5, with the slopes ∂Ep /∂ log (ks∗ ) = −2.3 × RT / (1 − α) F and
∂Ep /∂ log (ks∗ ) = 2.3 × RT /αF for the anodic and cathodic reactions, respectively. If the
reaction appears fast and reversible (log(ks∗) > -1), the peak potentials are independent
of the dimensionless rate constant (Ep − Ef∗ = 0 V).
If the electrooxidation of dmfc is totally irreversible, and cX,nb = c∗X,nb , equation (33)
is reduced to the form:
exp (1 −
Φk =
1−
k−1
P
j=1
1 + exp (1 − α) ϕ∗∗
f
where:
ϕ∗∗
f =
Ef∗∗ = E 0 −
α) ϕ∗∗
f
Φj
!
(34)
F E − Ef∗∗
RT
(35)
RT h − i∗ RT h − i∗
RT
ln X(w) +
ln X(nb) −
ln (ks∗ )
F
F
(1 − α) F
(36)
The solution of eq. (34) is shown in Fig. 6. The peak potential is Ep − Ef∗∗ = -0.321
V. This is identical to the value Ep − Ef∗ = 0.153 V corresponding to the log(ks∗ ) = -4
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Fig. 5 Dependence of dimensionless peak currents (A) and peak potentials (B) of cyclic voltammograms of quasireversible reaction (1) on the logarithm of dimensionless standard rate
constant, for α = 0.5.
that is shown in Fig. 5. So, the standard rate constant ks can be determined by the
variation of scan rate, if the concentration of the electrolyte CX in the film is hundred
times higher than the initial concentration of dmfc. At low scan rates the peak potentials
are defined by eq. (31), and at high scan rates they are defined as Ep = Ef∗∗ - 0.321 V.
So, the intersection of the extrapolations of these two linear relationships is defined by
the relation [32]:
ln (ks ) = ln (v0 ) − 0.321 ×
(1 − α) F
− ln (∆E/L)
RT
(37)
where v0 is the scan rate corresponding to the intersection.
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Fig. 6 Voltammetry of totally irreversible reaction (1); α = 0.5 (eq. 34).
3.2 Kinetics of anion transfer
Totally irreversible transfer of anion X − from water into nitrobenzene is defined by the
equations:
Φ = exp (1 − αit ) ϕitf (cdmf c )1−αit (cdmf c+ )αit −1
ϕitf =
Efit = E 0 −
F E − Efit
RT
(38)
(39)
RT
RT
ln (kit ) −
ln c∗X,w
(1 − αit ) F
(1 − αit ) F
(40)
where kit = k0 ∆E/Lv is the dimensionless rate constant. Although the relationship
between the current and the relative potential E − Efit cannot be solved, one can assume
that the solution should be in the form Ep − Efit = n V. If this is so, the relationship
between Ep and log(c∗X,w ) should be linear, with the slope ∂Ep /∂ log c∗X,w = −2.3 ×
RT / (1 − αit ) F . This is essentially different compared to the kinetics of the electrode
reaction (see eq. 32).
The determination of kinetic parameters of the anion transfer is prevented by the fact
that by changing the electrode potential E, both potential drops at the graphite/nitrobenzene
and nitrobenzene/water interfaces are changed. To maintain the constant potential at
the graphite/nitrobenzene interface, the concentrations of dmfc and a salt dmfcX in the
film should be much higher than the concentration of the electrolyte, MX, in water. This
means that an additional electrolyte, K2 Y , should be present in water to avoid an IR drop
between the working and the reference electrode. Usually the flux of anions of the type
Y 2− across the water/nitrobenzene interface is negligible [12]. Under these experimental
conditions a small change of the electrode potential E should be identical to the change
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of the potential drop across the liquid/liquid interface and the kinetics of anion transfer
across this interface should be measurable. This assumption needs further experimental
and theoretical investigations.
Previously we described experiments with a droplet of solution of decamethylferrocene in 1,2-dichloroethane which was attached to the surface of a paraffin-impregnated
graphite electrode and immersed into an aqueous electrolyte [33]. Square-wave voltammetric measurements indicated that the oxidation of dmfc in the droplet was kinetically
controlled, but the mechanism of this reaction could not be explained by a simple model
for the kinetics of the second order which was directly derived from equation (5):
I = −F S k2 exp(−αϕ)
h
dmf c+
i
x=0
h
−
X(nb)
i
x=L
h
−
− [dmf c]x=0 X(w)
i
x=L
exp(ϕ)
(41)
where k2 is an overall rate constant and α is an average transfer coefficient of the reaction
(1). The model proposed in this paper could be used for better understanding of these
experiments.
4
Conclusions
The complex redox reaction (1), occuring in a thin film of nitrobenzene interposed between
a graphite electrode surface and an aqueous electrolyte, depends on two potential drops
defined by equation (2). The reaction (1) consists of two charge transfers that occur
simultaneously: the electron transfer at the electrode surface and the transfer of anion
across the liquid/liquid interface. If both transfers are fast and reversible, the peak
potentials in cyclic voltammetry depend on initial concentrations of dmfc and X − in
nitrobenzene and X − in water (see equations 28 and 31). Both charge transfers can be
kinetically controlled. In this paper two limiting cases are considered. If the oxidation
of dmfc is a rate determining step, the current is defined by equation (13), while if
the interfacial ion transfer is the slowest step the current is defined by equation (18).
The kinetics of the electrode reaction can be measured if the potential drop across the
liquid/liquid interface is constant. The standard rate constant can be determined by the
variation of scan rate (see equation 37). The kinetic parameters of the anion transfer
can not be measured if both potential drops are changed simultaneously. This can be
prevented by using high concentrations of dmfc and a salt dmfcX in the film, and a
much smaller concentration of the electrolyte, MX in water. If the transfer of anion X −
from water into nitrobenzene is totally irreversible, the peak potential depends linearly
on the logarithm of the initial concentration of the anion in water, and the slope of this
relationship depends on the transfer coefficient. In the case of slow electron transfer, the
slope of this linear dependency is independent of the transfer coefficient (see equation 32).
This difference can be used to distinguish the electrode kinetics from the ion transfer
kinetics. Compared to the model proposed previously (see equation 41), the physical
meanings of the kinetic parameters in the present model are better defined.
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