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nitrobenzene interface
Electrochimica
Pergamon
Acto, Vol. 40, No. 18, pp. 297 1-2977. 1995 Copyright 0 1995 Elsevier Science Ltd. Printed in Great Britain. All rights reserved 0013-4686/95 $9.50 + 0.00
0013~468fi(!6)00230-8
DOUBLE-LAYER EFFECTS ON THE Cs+ ION TRANSFER KINETICS AT THE WATER/NITROBENZENE INTERFACE ZDENEK SAMEC,*? TAKASHI KAKIUCHI~ and MITSUGI SENDAS Department
of Agricultural
Chemistry,
Faculty
(Received
of Agriculture, Japan 18 Nouember
Kyoto
University,
Sakyo-ku,
Kyoto
606,
1994)
Abstract-Double layer effects on the kinetics of the Cs’ Ion transfer across the water/nitrobenzene interface were studied by an ac polarographic technique. Apparent kinetic parameters agree well with those reported previously. By applying the Frumkin-type correction based on the Gouy-Chapman theory, the non-linear Tafel plots were inferred indicating some dependence of the charge transfer coeflicient on the interfacial potential difference. The latter effect can weaken the contribution of the diffuselayer potentials to the apparent kinetic parameters, thereby accounting in part for the observed negligible
change of these parameters with the aqueous electrolyte concentration. The Frumkin-type correction failure at low electrolyte concentrations and high surface charge densities is due to an overestimation of the diffuse/layer potential by the Gouy-Chapman of the diffuse double layer. Key words:
immiscible
liquid electrolytes,
theory,
ion transfer
INTRODUCTION Experimental approaches to the kinetics of ion transfer across the interface between two immiscible electrolyte solutions (ITIES) have relied mostly on classical techniques,
galvanostatic[l]
or
potentiostatic[2]
such as chronopotentiometry[3, 43, chronocoulometry[S], cyclic voltammetry[2], convolution potential sweep voltammetry[6], phase selective UC voltammetry[7] or equilibrium impedance measurements[8]. These techniques were applied to liquid/liquid interfaces with a macroscopic area, typically around 0.1 cm’. Later on, the microelectrode methodology was introduced as a novel electroanalytical tool[9, lo], which has been further developed for kinetic measurements based on the fluctuation[l l] or impedance analysis[12]. Systematic kinetic analysis based on UC polarographic[l3-161 or equilibrium impedance measurements[8, 17, IS], yielded k”,around 0.1 cm s- ‘. Owing to a good agreement between kinetic data inferred for the same ion transfer reactionsC13, 17, 181, these results can be taken as the reliable reference, at least for macroscopic interfaces. In order to account for the effect of the electrical double layer on the apparent charge transfer coeflicient, the Frumkin-type correction[19] was applied to kinetic data in the earliest kinetic analysesC6, 201. * Author to whom correspondence should be addressed. t On leave from The J. Heyrovskq Institute of Physical Chemistry, DolejSkova 3, 182 23 Prague 8, Czech Republic 1 Present address: Department of Physical Chemistry, Yokohama National University, Tokiwadai 156, Hodogaya-ku, Yokohama 240, Japan 0 Present address: Department of Bioscience, Fukui Prefectural University, Matsuoka-chu, Fukui 910-l 1, Japan 297
rather
than due to the dynamic
kinetics, Frumkin
(Levich) effect
correction.
The corrected rate constants were found to correlate with the standard ion transfer potential ATcpOfor the transfer of a series of monovalent ions[3, 17, 18, 211, and with the inner layer potential difference 8::~ for the picrate ion transfer[8,22]. On the other hand, the effect of the base electrolyte concentration on the ion transfer rate casts some doubts on the validity of the classical Frumkin correction[l4, 223. Such an unexpected effect was illustrated for both the corrected rate constant of the picrate ion transfer[22] and the apparent standard rate constant k”, for a series of monovalent ions[14]. The latter constant should vary with the base electrolyte concentration in a way that is opposite for cations and anions, and opposite for ions of the same sign but with the standard potential difference on the positive and negative side relative to the pzc[14]. In contrast, experimental rate constants for various ions show a tendency to increase slightly. However, such a disagreement may arise when a comparison is made with the theoretical plots derived on assuming a constant charge transfer coefficient, ie for the Butler-Volmer type currentpotential characteristicsC141. Upon relaxing this condition, a change in the charge transfer coefficient can, in part, compensate for the effect of the electrical double layer on the apparent rate constant. Indeed, the theoretical predictions[14] for the case of the Goldman-type characteristics, which comprises a variation of the apparent charge transfer coefficient, appears to be in a better agreement with experiment. In this work, the electrolyte effect on the kinetics of the Cs+ ion transfer is examined by using an UC polarographic technique, so as to check both kinetic data previously reported[2,6] and the validity of the Frumkin-type correction. Because of the high value of its standard ion transfer potential, this ion transfer 1
2912
z.
SAMEC
occurs relatively far from the potential of zero charge, so that the effect of the electrical double layer is likely to be rather pronounced.
et d.
in equation (7). The following relationship between dr and a can be derived, 2 = CZ(~A~~~/~A~~)+ (aa/aAycp)
THEORETICAL
x (A,“zzq- A:&‘) - (aA;‘c@A;cp)
The transfer of an ion X’ with the charge number z from the aqueous phase w to the organic solvent phase o XL(W)P X2(0) is currently law
(1)
supposed to follow the first-order rate IfzFA = Lcw - iic”
(2)
where I is the current, A is the interfacial area, and c’” or co are the ion concentrations in the phase w or o, respectively. In general, the forward and backward heterogeneous rate constants k’ and fEdepend on the interfacial potential difference A,“cp,being related to each other by[23], I& = exp[zF(A:cp - AJJcpo)/RT]
(3)
where A,W(po is the standard ion transfer potential for the reaction (1). For the sake of comparison, two apparent kinetic parameters are usually introduced. First one is the standard rate constant kb at E = E”, k”, = l(A,wcp= Arc$‘) = &(A,“rp= A:&“). Second parameter is the apparent charge transfer coefficient B (sometimes denoted as uapp), which characterizes the potential dependence-of the forward rate constant, L?= (RT/zF) (aln k/aA,“cp). Its value at the standard ion transfer potential, A,Wpo, will be denoted as ho. The Frumkin-type correction of the apparent rat: constant k consists in calculating the rate constant k, from the equation k’= /&w2/cw) = i, exp( -zF
A;‘q/RT)
(4)
where c’“* is the concentration of the transferred ion at the outer Helmholtz plane (oHp) and Az”p is the potential difference between oHp and the bulk aqueous phase (the diffuse layer). This potential difference is a part of the interfacial potential difference, A,“cp,which splits into the inner-layer potential difference, Arzq, and the diffuse-layer potential differences in the aqueous and the organic solvent phase, Ac2p and Az2c$24] A,“q = A,w,z(p + A;“p - A;“p
(5)
Alternatively, by assuming that R, _ exp[azF(A,“,Zq - A,W&/RT] (a Bronsted type relationship),_ an _. . equation can be herived[23], __ k’ = k’ exp[azF(Azcp - Arq$)/RT]
(6)
x = L*/L - zF(A,“,2rp- A,“rp;)/2M(e~*L)~ (9) where L* or L are the equivalent jump lengths between the initial (~2) and transition (*) or the initial (~2) and final (02) locations, respectively M is the molar mass and o* is of the angular frequency of the ion motion in the transition state (harmonic approximation). It follows from equation (8), that the apparent charge transfer ?i is not a constant, ie the non-linear Tafel plots are predicted in this model. In another approach[27], the Nernst-Planck equation was integrated between two locations a and b upon assuming a constant gradient of the electrochemical potential in this region, which was not supposed to be necessarily the same entity as the ion-free inner layer at the interface. In the absence of an activation barrier at the interface, the equation for the rate constant i, in equation (4) can be written by using the present notation as k, = k, yeY/sinh y
(10)
y = (zF/2RT)(A;(p - A,“&
(11)
k, = D/L”
(12)
with
The non-linear character of equation (10) is essenthe tially same as the Goldman-type rectification[28]. Besides the prediction of the curved Tafel plots, the charge transfer coefficient at the standard ion transfer potential should always equal 0.5. EXPERIMENTAL The procedures used for the preparation and purification of reagents have been described elsewhere[29]. CsCl (Wako, 99.9%) was used as received. The ac impedance measurements were performed in a thermostated two-electrode electrochemical cell[13], in which the flat water/ nitrobenzene interface was formed having an area of 0.124cm’. The cell can be represented by Scheme I Ag AgCl
a M LiCl
0.05 M TPeATFPB
+ 1 mM CsCl
where k” = k, exp{ -zF[(l
(8)
A dependence of the charge transfer coefficient, a, on the inner-layer potential difference, A,;‘cp, is predicted by the stochastic model based on the integration of the Fokker-Planck equation[25,26]
- u)A;‘cp + czA~‘~)/RT]
(7)
and k, is a constant. The parameters k” and a are not identical with the apparent rate constant, k”, = k”(A,“cp= Ar(po), and the apparent charge transfer coefficient, h, respectively. The parameter k” can obviously depend on the potential difference, A,“cp, indirectly through the exponential term with the potential differences across the space charge regions
(w)
(0) 0.01 M MgCl, AgClI Ag’ + +5 mM TPeACl (w’) Scheme I
2913
Double layer effects
where (w) or (w’) denote the aqueous phases and (0) the nitrobenzene phase, TPeATFPB and TPeACl represent tetrapentylammonium tetrakis[3,5bis(triand tetrapentylfluoromethyl)-phenyllborate ammonium chloride, respectively and the LiCl concentration II = 0.01,0.02,0.05,0.1, 0.2 or 0.5. The potential E of the cell (I) can be written as the difference E = A,“rp- Eref
(13)
where the reference potential Ercf comprises the contributions of the two reference Ag/AgCl electrodes and the w’/o interface. Since the potential of the aqueous Ag/AgCl reference electrode changes with the concentration of LiCl in the cell (I), it is more convenient to use the potential scale based on the potential difference A,“cprather then on the potential E. Therefore, at various concentrations of LiCl, the reversible half-wave potential E;;i of the tetraethylammonium ion transfer was determined as the mid-potential of the dc cyclic voltamogram recorded at a low scan rate (5mVs-‘)[13]. Eref was then inferred by using the equation
6
c:
”
4
r c 2
0
AoW+ v
Ey2 = A,Wc$’+ (RT/2F) ln(D”/D”)
+ (R T/F) ln(rW”) - Srcl
(14)
where the standard ion transfer potential A,W(po = -O.O66V, the ratio of the diffusion coefficients D”/ D” = 2, and the activity coefficients y“ and y” are estimated by the Debye-Hhckel theory[30]. In the ac polarographic measurements of Cs+ ion transfer, an ac voltage of 5mV amplitude (peak-topeak) and frequency of 10, 20, 50, 100 and 2OOHz was superimposed on the dc voltage ramp (0.5 mV s- ‘). The solution resistance was compensated for by the positive feedback techniqueC13). The capacitance measurements were carried out in the same measuring cell in the absence of Cs+ ion, without introducing the positive feedback. The applied potential was changed stepwise (20mV per step) over a potential range, and the impedance spectrum was measured at each step of sweeping the frequency (l-1000 Hz). The spectrum was then analyzed with the help of the commercially available program for the non-linear least-squares fit (NLLSF) of immitance data[31]. The eletrochemical system was represented as a parallel combination of the double-layer capacitance and faradaic impedance, with the solution resistance in series. All measurements were made at 25°C.
RESULTS
AND DISCUSSION
Kinetic parameters
The real and imaginary parts of the faradaic admittance, Y’ and Y”, of the Cs+ ion transfer were obtained by subtracting the corresponding admittances measured in the presence and absence of the transferred ion in the aqueous phase, c5 Fig. 1. The forward rate constant k was evaluated by using the equation[ 131 Y’/y” = cot e = 1 + (2w)“2/1
(15)
7
c
0.0
0.1
0.2
0.3
now9 : ‘I Fig. 1. Real (A) and imaginary (B) admittance of the interface between 0.05M LiCl in water and 0.05M TPeATFPB in nitrobenzene in the presence (0) or absence (0) of 1 mM CsCl in the aqueous phase, and the contribution due to the Cs+ ion transfer (V) at 100 Hz.
where w is the angular frequence of the ac voltage and the parameter y is given by i = &(D’“)-l”[l + exp( - <)I
(16)
5 = (zF/RT)(E - E;‘;;)
(17)
The value of the diffusion coefficient of Cs+ ion in the aqueous phase was taken from literature: D” = 2 x lo-’ cm2 SK '[Z]. The reversible half-wave potential EyG was assumed to be given by the peak potential of the ac voltamogram, cf Fig. 1. The corresponding standard ion transfer potential Arqy, as evaluated by equation(l4), are all around the value of 0.159V which was derived from the ion partition measurements[32], c$ Table 1. In an agreement with equation (15), the plot of cot 6 vs. cui/* for Cs+ ion transfer at various LiCl concentrations (Fig. 2) is linear, with a slope indicating that the apparent rate constant k’, < 0.1 cm s- ‘, cf: the dashed line in Fig. 2. Figure 3 shows the effect of the potential difference on the apparent rate constant k at various concentrations of LiCl in the aqueous phase. As it has been
z. SAMEC
2974
et al. Table 1. Standard ion transfer potential A:(pp and the apparent kinetic parameters & and 8, of the Cs+ ion transfer across the interface between the nitrobenzene solution of 0.05 M TPeATFPB and the aqueous solution of LiCl of a concentration c. Values in parentheses show the standard deviation of the mean of three independent measurements.
c
0
10
20 w
‘, 2
Y
30
40
-l/2 _'
5
Fig. 2. Plot of cot tI vs. o ‘I2 for the Cs’ ion transfer at the standard potential difference Ar(po across the interface between the nitrobenzene solution of 0.05M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (O), 0.02 (e), 0.05 (V), 0.10 (v), 0.2 (0) and 0.5 (m). Dashed line shows the slope for rP,= 0.1cm-’ and D” = 2 x 10-5cms-‘. observed previously[6], the Tafel plots are curved lines, ie the charge transfer coefficient h is a function of the potential. The values of the standard rate constant kt and the apparent charge transfer coefficient &, at the standard ion transfer potential Arqy are summarized in Table 1. The mean values and the standard deviations were calculated from three independent measurements at a LiCl concentration. Relatively large error in both parameters is probably due to the inaccurate subtraction of the increasing admittance of the base electrolyte ion transfer, cc
c/M
A:cp”lv
lO%“,/cm s - 1
a0
0.01 0.02 0.05 0.10 0.20 0.50
0.150 0.155 0.161 0.146 0.136 0.134
0.67 (0.06) 0.89 (0.19) 0.66 (0.12) 0.77 (0.19) 1.10 (0.29) 1.20 (0.30)
0.47 (0.01) 0.59 (0.03) 0.56 (0.04) 0.45 (0.04) 0.37 (0.07) 0.33 (0.10)
1. These kinetic data agree well with those obtained by convolution potential sweep voltatmmetry at the concentration of both electrolytes 0.5moldm-3: k”, z 5 x 10-2cms-1 and 8, x 0.5[2, Fig.
61.
The theory of the Frumkin-type double-layer effect based on the Butler-Volmer or Goldman current-potential characteristics predicts a systematic variation of both the standard rate constant k”, and the apparent charge transfer coefftcient go with the base electrolyte concentration[14]. Namely, when the standard ion transfer potential A,W(po for a cation transfer is more positive than the zero-charge potential difference, both kb and h, are predicted to decrease with the increasing aqueous electrolyte concentration[14]. As is seen from Fig. 3 and Table
1, the experimental data for Cs+ are at variance with this prediction. While &, exhibits a tendency to decrease, k”, increases. Hence, a conclusion could have been drawn that, like for other ions[14], the contribution of the Frumkin-type double-layer effect is insignificant. However, such a behaviour can also be of another origin, vide infia. Distribution of the electrical potential
The plots of the double-layer capacitance C vs. the potential difference A,“cp at the various electrolyte concentrations in the aqueous phase exhibit a flat minimum at A,“cpz OV, cf: Fig. 4. This potential difference was supposed to be practically identical with the zero-charge potential difference A,“cp,,, , as in the case of the interface between LiCl in water and tetranitrotetraphenylborate in butylammonium benzene[33]. Hence, the capacitance curves shown in Fig. 4 were integrated with respect to this potential difference to yield the surface charge density trw on the aqueous side of the interface. The diffuse-layer components of the potential difference AFcp in equation (5) were calculated with the help of the GouyChapman (CC) theory by using the equation -0.1
0.0
0.1
AoWpl”w#,o v Fig. 3. Logarithm of the apparent rate constant k vs. the potential difference Azcp relative to the standard potential difference A:cpp for the Cs+ ion transfer across the interface between the nitrobenzene solution of 0.05M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (O), 0.02 (a), 0.05 (V), 0.10 (W), 0.2 (r) and 0.5 (0). Error bars (S.D.) are shown only for 0.1 M LiCl.
cs = -2(2RT&,,
co, ‘)l” sinh(FAi’q/RT)
(18)
where s = w or o, es is the solvent dielectric permitivity, co is the permitivity of vacuum and co*’ is the electrolyte concentration. The inner-layer potential difference Az:;‘cp,as inferred from equation (5), is plotted vs. the surface charge density 0’” in Fig. 5. The points at various electrolyte concentration show small deviations from a single line (cf dashed line in Fig. 5), ie the state of the inner layer is controlled
2915
Double layer effects
the aqueous phase is less significant[34], the application of the Gouy-Chapman theory to an analysis of the effect of the aqueous electrolyte concentration on the ion transfer kinetics seems to be adequate, at least as a first approximation. Indeed, the difference between the potential predicted for an 0.05 M nitrobenzene solution of an electrolyte by the GC and MPB theories does not exceed approximately 15 mV when 8” < 4&cm-‘[34].
40 ?I E c. i =. 0
20 Frumkin-type
0 -0.2
0.0 IoW$7
0.2 v
Fig. 4. Interfacial capacitance C vs. the potential difference Arcp for the interface between the nitrobenzene solution of 0.05 M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (0) 0.02 (a). 0.05 (O), 0.10 (W), 0.2 (V) and 0.5 (v). mainly by the surface charge density. Obviously, the inner-laryer potential difference represents here a considerably larger part of A,“cp, compared to the system comprising tetrabutylammonium tetraphenylborate in the nitrobenzene phase[33]. We are aware of the fact that the GC theory can overestimate the diffuse-layer potential difference, in particular in the organic solvent phase[34]. Unfortunately, the modified Poisson-Boltzmann equation (MPB)[35], which predicts a more realistic diffuselayer potential, was used in a limited range of concentrations[34]. On the other hand, because the organic electrolyte concentration was kept constant, and the correction to the diffuse-layer potential in
correction
Figure 6 shows the corrected rate constant ,$ of the Csf ion transfer, which was calculated from the apparent rate constant by using equation (4), as a function of the inner-layer potential difference A$q relative to the standard ion transfer potential A,WpF (the correct_ed standard Gibbs energy change). In an idea1 case, k, should be independent of the base electrolyte concentration and, when the charge transfer coefficient a is a constant (the Butler-Volmer characteristics), the plot of log l, vs. A$q - ArpF (the corrected Tafel plot) should be a straight line. Apparently, at concentrations of LiCl higher than 0.02mol dmm3, the values of &, tends to fall on a single Tafel plot, which is non-linear though, CJ the full line in Fig. 6. Towards lower LiCl oncentrations, there is a considerable drop in &, when the innerlayer potential difference increases. A similar effect was observed for the picrate ion transfer[22], the corrected rate constant of which shows smaller deviations from a single Tafel plot at LiCl concentrations higher than 0.02 mol dm- 3. However, the electrolyte effect is reversed in that k, tends to rise when the LiCl concentration decreases. What seems to be common to both ion transfer reactions, which occur at potentials
positive
to the potential
of zero charge,
-1
r
0.1
-2 i
Y
k
a
0.0
1’ _3
-0.1
Fig. 5. Inner-layer potential difference Ari;‘cpvs. the surface charge density u”’ calculated by using the Gouy-Chapman theory from the capacitance data for the interface between the nitrobenzene solution of 0.05M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (O), 0.02(m), 0.05(O), 0.10(m), 0.2(V) and 0.5 (A).
i -0.16
-0.12
-0.08
-0.04
Fig. 6. Logarithm of the corrected rate constant & vs. the inner-layer potential difference A~&I relative to standard potential difference (corrected Tafel plots) for the Cs+ ion transfer between the nitrobenzene solution of 0.05M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (O), 0.02 (a), 0.05 (V), 0.10 p), 0.2 (0) and 0.5 (B). Vertical bars indicate the standard deviation for 0.1 M LiCI, the dashed line corresponds to OL = 0.5.
2. SAMEc et al.
2916
overestimation (in absolute value) of the diffuse-layer potential Arp at lower LiCl concentrations. As a result, the Gouy-Chapman theory predicts too high concentration of Cs+ and too low concentrations of the picrate anion at the outer Helmholtz plane in the aqueous phase. Note that such an overestimation is indicated by the analysis based on the MPB theory[34]. In a way, the effect of the base electrolyte concentration seems to be linked to the failure, though surprising, of the GouyChapman theory to describe the diffuse-layer capacitance at low electrolyte concentrationsC361. The Frumkin-type correction to the apparent ion transfer kinetics can be generalized by considering the migration component of the ion transport across the diffuse layer[37, 381, giving rise to the dynamic (Levich) effect[38]. For ions attracted into the double layer, which is the case of Cs+, the ion tranfer is accelerated, while for ions such as picrate, which are repulsed from the double layer, the ion transfer is decelerated relative to the Frumkin effect, when the absolute value of the diffuse-layer potential difference increases[37]. The dynamic effect becomes more pronounced with the decreasing base electrolyte concentration[37]. Since the behaviour of both Cs+ and picrate ions is at variance with these predictions, the role of the dynamic effect is rather negligible. The non-linear shape of the corrected Tafel plots indicates a dependence of the charge transfer coetlicient CLon the inner layer potential difference Arz;“cp. Such a dependence is predicted by the stochastic approaches based on the Fokker-Planck[25, 261 or the Nernst-Planck[27] equation. In the former case, the denominator 2M (w*L)’ in equation (9) can be comparable with zF(Arirp - A:&. For M x 100 and a characteristic transition time and length of lops (ie, w* Z 10” s-‘) and 1 nm[39, 401, respectively, the former term would have the value around 2 kJ mol- ‘. Although this is a very rough estimate, it can explain the curvature of the corrected Tafel plots. It should be noted that the change in the inner-layer potential difference, A,“z’cp,results in a change in tl, which in turn can weaken the effect of diffuse-layer potential differences, Ar2q and Ar”p in equation (7). As a result, the apparent standard rate constant, k”,, can be less dependent on the base electrolyte concentration than in the case the charge transfer coefficient is a constant. is an
CONCLUSIONS Kinetic parameters of the Cs+ ion transfer across the water/nitrobenzene interface obtained by an UC polarographic technique agree well with those reported previously[2, 63. A weak effect of the aqueous electrolyte concentration on the kinetic parameters is analogous to that observed for the transfer of the picrate[22], as well as the other univalent[14] ions. In view of the unequivocal existence of the diffuse double layer at ITIES, the Frumkin-type correction represents a plausible working hypothesis. However, the Gouy-Chapman theory seems to predict too high diffuse-layer poten-
tial, in particular at low electrolyte concentrations. The non-linearity of the corrected Tafel plots points to a potential dependence of the charge transfer coefficient. Its change can weaken the contribution of the diffuse-layer potentials to the apparent kinetic parameters, thereby accounting for their small changes with the base electrolyte concentration. Dynamic effect of the diffuse double layer, which is linked to the migration component of the ion transport, seems to play an insignificant role in both the Cs+ and picrate ion transfer. Acknowledgements-Z.S.
is the recipient of a visiting fellowship to Japan provided by Japan Society for Promotion of Science; partial support for this work from the Grant Agency of the Academy of Sciences of the Czech Republic (Grant 440411) is also gratefully acknowledged.
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2911
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Pergamon
Acto, Vol. 40, No. 18, pp. 297 1-2977. 1995 Copyright 0 1995 Elsevier Science Ltd. Printed in Great Britain. All rights reserved 0013-4686/95 $9.50 + 0.00
0013~468fi(!6)00230-8
DOUBLE-LAYER EFFECTS ON THE Cs+ ION TRANSFER KINETICS AT THE WATER/NITROBENZENE INTERFACE ZDENEK SAMEC,*? TAKASHI KAKIUCHI~ and MITSUGI SENDAS Department
of Agricultural
Chemistry,
Faculty
(Received
of Agriculture, Japan 18 Nouember
Kyoto
University,
Sakyo-ku,
Kyoto
606,
1994)
Abstract-Double layer effects on the kinetics of the Cs’ Ion transfer across the water/nitrobenzene interface were studied by an ac polarographic technique. Apparent kinetic parameters agree well with those reported previously. By applying the Frumkin-type correction based on the Gouy-Chapman theory, the non-linear Tafel plots were inferred indicating some dependence of the charge transfer coeflicient on the interfacial potential difference. The latter effect can weaken the contribution of the diffuselayer potentials to the apparent kinetic parameters, thereby accounting in part for the observed negligible
change of these parameters with the aqueous electrolyte concentration. The Frumkin-type correction failure at low electrolyte concentrations and high surface charge densities is due to an overestimation of the diffuse/layer potential by the Gouy-Chapman of the diffuse double layer. Key words:
immiscible
liquid electrolytes,
theory,
ion transfer
INTRODUCTION Experimental approaches to the kinetics of ion transfer across the interface between two immiscible electrolyte solutions (ITIES) have relied mostly on classical techniques,
galvanostatic[l]
or
potentiostatic[2]
such as chronopotentiometry[3, 43, chronocoulometry[S], cyclic voltammetry[2], convolution potential sweep voltammetry[6], phase selective UC voltammetry[7] or equilibrium impedance measurements[8]. These techniques were applied to liquid/liquid interfaces with a macroscopic area, typically around 0.1 cm’. Later on, the microelectrode methodology was introduced as a novel electroanalytical tool[9, lo], which has been further developed for kinetic measurements based on the fluctuation[l l] or impedance analysis[12]. Systematic kinetic analysis based on UC polarographic[l3-161 or equilibrium impedance measurements[8, 17, IS], yielded k”,around 0.1 cm s- ‘. Owing to a good agreement between kinetic data inferred for the same ion transfer reactionsC13, 17, 181, these results can be taken as the reliable reference, at least for macroscopic interfaces. In order to account for the effect of the electrical double layer on the apparent charge transfer coeflicient, the Frumkin-type correction[19] was applied to kinetic data in the earliest kinetic analysesC6, 201. * Author to whom correspondence should be addressed. t On leave from The J. Heyrovskq Institute of Physical Chemistry, DolejSkova 3, 182 23 Prague 8, Czech Republic 1 Present address: Department of Physical Chemistry, Yokohama National University, Tokiwadai 156, Hodogaya-ku, Yokohama 240, Japan 0 Present address: Department of Bioscience, Fukui Prefectural University, Matsuoka-chu, Fukui 910-l 1, Japan 297
rather
than due to the dynamic
kinetics, Frumkin
(Levich) effect
correction.
The corrected rate constants were found to correlate with the standard ion transfer potential ATcpOfor the transfer of a series of monovalent ions[3, 17, 18, 211, and with the inner layer potential difference 8::~ for the picrate ion transfer[8,22]. On the other hand, the effect of the base electrolyte concentration on the ion transfer rate casts some doubts on the validity of the classical Frumkin correction[l4, 223. Such an unexpected effect was illustrated for both the corrected rate constant of the picrate ion transfer[22] and the apparent standard rate constant k”, for a series of monovalent ions[14]. The latter constant should vary with the base electrolyte concentration in a way that is opposite for cations and anions, and opposite for ions of the same sign but with the standard potential difference on the positive and negative side relative to the pzc[14]. In contrast, experimental rate constants for various ions show a tendency to increase slightly. However, such a disagreement may arise when a comparison is made with the theoretical plots derived on assuming a constant charge transfer coefficient, ie for the Butler-Volmer type currentpotential characteristicsC141. Upon relaxing this condition, a change in the charge transfer coefficient can, in part, compensate for the effect of the electrical double layer on the apparent rate constant. Indeed, the theoretical predictions[14] for the case of the Goldman-type characteristics, which comprises a variation of the apparent charge transfer coefficient, appears to be in a better agreement with experiment. In this work, the electrolyte effect on the kinetics of the Cs+ ion transfer is examined by using an UC polarographic technique, so as to check both kinetic data previously reported[2,6] and the validity of the Frumkin-type correction. Because of the high value of its standard ion transfer potential, this ion transfer 1
2912
z.
SAMEC
occurs relatively far from the potential of zero charge, so that the effect of the electrical double layer is likely to be rather pronounced.
et d.
in equation (7). The following relationship between dr and a can be derived, 2 = CZ(~A~~~/~A~~)+ (aa/aAycp)
THEORETICAL
x (A,“zzq- A:&‘) - (aA;‘c@A;cp)
The transfer of an ion X’ with the charge number z from the aqueous phase w to the organic solvent phase o XL(W)P X2(0) is currently law
(1)
supposed to follow the first-order rate IfzFA = Lcw - iic”
(2)
where I is the current, A is the interfacial area, and c’” or co are the ion concentrations in the phase w or o, respectively. In general, the forward and backward heterogeneous rate constants k’ and fEdepend on the interfacial potential difference A,“cp,being related to each other by[23], I& = exp[zF(A:cp - AJJcpo)/RT]
(3)
where A,W(po is the standard ion transfer potential for the reaction (1). For the sake of comparison, two apparent kinetic parameters are usually introduced. First one is the standard rate constant kb at E = E”, k”, = l(A,wcp= Arc$‘) = &(A,“rp= A:&“). Second parameter is the apparent charge transfer coefficient B (sometimes denoted as uapp), which characterizes the potential dependence-of the forward rate constant, L?= (RT/zF) (aln k/aA,“cp). Its value at the standard ion transfer potential, A,Wpo, will be denoted as ho. The Frumkin-type correction of the apparent rat: constant k consists in calculating the rate constant k, from the equation k’= /&w2/cw) = i, exp( -zF
A;‘q/RT)
(4)
where c’“* is the concentration of the transferred ion at the outer Helmholtz plane (oHp) and Az”p is the potential difference between oHp and the bulk aqueous phase (the diffuse layer). This potential difference is a part of the interfacial potential difference, A,“cp,which splits into the inner-layer potential difference, Arzq, and the diffuse-layer potential differences in the aqueous and the organic solvent phase, Ac2p and Az2c$24] A,“q = A,w,z(p + A;“p - A;“p
(5)
Alternatively, by assuming that R, _ exp[azF(A,“,Zq - A,W&/RT] (a Bronsted type relationship),_ an _. . equation can be herived[23], __ k’ = k’ exp[azF(Azcp - Arq$)/RT]
(6)
x = L*/L - zF(A,“,2rp- A,“rp;)/2M(e~*L)~ (9) where L* or L are the equivalent jump lengths between the initial (~2) and transition (*) or the initial (~2) and final (02) locations, respectively M is the molar mass and o* is of the angular frequency of the ion motion in the transition state (harmonic approximation). It follows from equation (8), that the apparent charge transfer ?i is not a constant, ie the non-linear Tafel plots are predicted in this model. In another approach[27], the Nernst-Planck equation was integrated between two locations a and b upon assuming a constant gradient of the electrochemical potential in this region, which was not supposed to be necessarily the same entity as the ion-free inner layer at the interface. In the absence of an activation barrier at the interface, the equation for the rate constant i, in equation (4) can be written by using the present notation as k, = k, yeY/sinh y
(10)
y = (zF/2RT)(A;(p - A,“&
(11)
k, = D/L”
(12)
with
The non-linear character of equation (10) is essenthe tially same as the Goldman-type rectification[28]. Besides the prediction of the curved Tafel plots, the charge transfer coefficient at the standard ion transfer potential should always equal 0.5. EXPERIMENTAL The procedures used for the preparation and purification of reagents have been described elsewhere[29]. CsCl (Wako, 99.9%) was used as received. The ac impedance measurements were performed in a thermostated two-electrode electrochemical cell[13], in which the flat water/ nitrobenzene interface was formed having an area of 0.124cm’. The cell can be represented by Scheme I Ag AgCl
a M LiCl
0.05 M TPeATFPB
+ 1 mM CsCl
where k” = k, exp{ -zF[(l
(8)
A dependence of the charge transfer coefficient, a, on the inner-layer potential difference, A,;‘cp, is predicted by the stochastic model based on the integration of the Fokker-Planck equation[25,26]
- u)A;‘cp + czA~‘~)/RT]
(7)
and k, is a constant. The parameters k” and a are not identical with the apparent rate constant, k”, = k”(A,“cp= Ar(po), and the apparent charge transfer coefficient, h, respectively. The parameter k” can obviously depend on the potential difference, A,“cp, indirectly through the exponential term with the potential differences across the space charge regions
(w)
(0) 0.01 M MgCl, AgClI Ag’ + +5 mM TPeACl (w’) Scheme I
2913
Double layer effects
where (w) or (w’) denote the aqueous phases and (0) the nitrobenzene phase, TPeATFPB and TPeACl represent tetrapentylammonium tetrakis[3,5bis(triand tetrapentylfluoromethyl)-phenyllborate ammonium chloride, respectively and the LiCl concentration II = 0.01,0.02,0.05,0.1, 0.2 or 0.5. The potential E of the cell (I) can be written as the difference E = A,“rp- Eref
(13)
where the reference potential Ercf comprises the contributions of the two reference Ag/AgCl electrodes and the w’/o interface. Since the potential of the aqueous Ag/AgCl reference electrode changes with the concentration of LiCl in the cell (I), it is more convenient to use the potential scale based on the potential difference A,“cprather then on the potential E. Therefore, at various concentrations of LiCl, the reversible half-wave potential E;;i of the tetraethylammonium ion transfer was determined as the mid-potential of the dc cyclic voltamogram recorded at a low scan rate (5mVs-‘)[13]. Eref was then inferred by using the equation
6
c:
”
4
r c 2
0
AoW+ v
Ey2 = A,Wc$’+ (RT/2F) ln(D”/D”)
+ (R T/F) ln(rW”) - Srcl
(14)
where the standard ion transfer potential A,W(po = -O.O66V, the ratio of the diffusion coefficients D”/ D” = 2, and the activity coefficients y“ and y” are estimated by the Debye-Hhckel theory[30]. In the ac polarographic measurements of Cs+ ion transfer, an ac voltage of 5mV amplitude (peak-topeak) and frequency of 10, 20, 50, 100 and 2OOHz was superimposed on the dc voltage ramp (0.5 mV s- ‘). The solution resistance was compensated for by the positive feedback techniqueC13). The capacitance measurements were carried out in the same measuring cell in the absence of Cs+ ion, without introducing the positive feedback. The applied potential was changed stepwise (20mV per step) over a potential range, and the impedance spectrum was measured at each step of sweeping the frequency (l-1000 Hz). The spectrum was then analyzed with the help of the commercially available program for the non-linear least-squares fit (NLLSF) of immitance data[31]. The eletrochemical system was represented as a parallel combination of the double-layer capacitance and faradaic impedance, with the solution resistance in series. All measurements were made at 25°C.
RESULTS
AND DISCUSSION
Kinetic parameters
The real and imaginary parts of the faradaic admittance, Y’ and Y”, of the Cs+ ion transfer were obtained by subtracting the corresponding admittances measured in the presence and absence of the transferred ion in the aqueous phase, c5 Fig. 1. The forward rate constant k was evaluated by using the equation[ 131 Y’/y” = cot e = 1 + (2w)“2/1
(15)
7
c
0.0
0.1
0.2
0.3
now9 : ‘I Fig. 1. Real (A) and imaginary (B) admittance of the interface between 0.05M LiCl in water and 0.05M TPeATFPB in nitrobenzene in the presence (0) or absence (0) of 1 mM CsCl in the aqueous phase, and the contribution due to the Cs+ ion transfer (V) at 100 Hz.
where w is the angular frequence of the ac voltage and the parameter y is given by i = &(D’“)-l”[l + exp( - <)I
(16)
5 = (zF/RT)(E - E;‘;;)
(17)
The value of the diffusion coefficient of Cs+ ion in the aqueous phase was taken from literature: D” = 2 x lo-’ cm2 SK '[Z]. The reversible half-wave potential EyG was assumed to be given by the peak potential of the ac voltamogram, cf Fig. 1. The corresponding standard ion transfer potential Arqy, as evaluated by equation(l4), are all around the value of 0.159V which was derived from the ion partition measurements[32], c$ Table 1. In an agreement with equation (15), the plot of cot 6 vs. cui/* for Cs+ ion transfer at various LiCl concentrations (Fig. 2) is linear, with a slope indicating that the apparent rate constant k’, < 0.1 cm s- ‘, cf: the dashed line in Fig. 2. Figure 3 shows the effect of the potential difference on the apparent rate constant k at various concentrations of LiCl in the aqueous phase. As it has been
z. SAMEC
2974
et al. Table 1. Standard ion transfer potential A:(pp and the apparent kinetic parameters & and 8, of the Cs+ ion transfer across the interface between the nitrobenzene solution of 0.05 M TPeATFPB and the aqueous solution of LiCl of a concentration c. Values in parentheses show the standard deviation of the mean of three independent measurements.
c
0
10
20 w
‘, 2
Y
30
40
-l/2 _'
5
Fig. 2. Plot of cot tI vs. o ‘I2 for the Cs’ ion transfer at the standard potential difference Ar(po across the interface between the nitrobenzene solution of 0.05M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (O), 0.02 (e), 0.05 (V), 0.10 (v), 0.2 (0) and 0.5 (m). Dashed line shows the slope for rP,= 0.1cm-’ and D” = 2 x 10-5cms-‘. observed previously[6], the Tafel plots are curved lines, ie the charge transfer coefficient h is a function of the potential. The values of the standard rate constant kt and the apparent charge transfer coefficient &, at the standard ion transfer potential Arqy are summarized in Table 1. The mean values and the standard deviations were calculated from three independent measurements at a LiCl concentration. Relatively large error in both parameters is probably due to the inaccurate subtraction of the increasing admittance of the base electrolyte ion transfer, cc
c/M
A:cp”lv
lO%“,/cm s - 1
a0
0.01 0.02 0.05 0.10 0.20 0.50
0.150 0.155 0.161 0.146 0.136 0.134
0.67 (0.06) 0.89 (0.19) 0.66 (0.12) 0.77 (0.19) 1.10 (0.29) 1.20 (0.30)
0.47 (0.01) 0.59 (0.03) 0.56 (0.04) 0.45 (0.04) 0.37 (0.07) 0.33 (0.10)
1. These kinetic data agree well with those obtained by convolution potential sweep voltatmmetry at the concentration of both electrolytes 0.5moldm-3: k”, z 5 x 10-2cms-1 and 8, x 0.5[2, Fig.
61.
The theory of the Frumkin-type double-layer effect based on the Butler-Volmer or Goldman current-potential characteristics predicts a systematic variation of both the standard rate constant k”, and the apparent charge transfer coefftcient go with the base electrolyte concentration[14]. Namely, when the standard ion transfer potential A,W(po for a cation transfer is more positive than the zero-charge potential difference, both kb and h, are predicted to decrease with the increasing aqueous electrolyte concentration[14]. As is seen from Fig. 3 and Table
1, the experimental data for Cs+ are at variance with this prediction. While &, exhibits a tendency to decrease, k”, increases. Hence, a conclusion could have been drawn that, like for other ions[14], the contribution of the Frumkin-type double-layer effect is insignificant. However, such a behaviour can also be of another origin, vide infia. Distribution of the electrical potential
The plots of the double-layer capacitance C vs. the potential difference A,“cp at the various electrolyte concentrations in the aqueous phase exhibit a flat minimum at A,“cpz OV, cf: Fig. 4. This potential difference was supposed to be practically identical with the zero-charge potential difference A,“cp,,, , as in the case of the interface between LiCl in water and tetranitrotetraphenylborate in butylammonium benzene[33]. Hence, the capacitance curves shown in Fig. 4 were integrated with respect to this potential difference to yield the surface charge density trw on the aqueous side of the interface. The diffuse-layer components of the potential difference AFcp in equation (5) were calculated with the help of the GouyChapman (CC) theory by using the equation -0.1
0.0
0.1
AoWpl”w#,o v Fig. 3. Logarithm of the apparent rate constant k vs. the potential difference Azcp relative to the standard potential difference A:cpp for the Cs+ ion transfer across the interface between the nitrobenzene solution of 0.05M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (O), 0.02 (a), 0.05 (V), 0.10 (W), 0.2 (r) and 0.5 (0). Error bars (S.D.) are shown only for 0.1 M LiCl.
cs = -2(2RT&,,
co, ‘)l” sinh(FAi’q/RT)
(18)
where s = w or o, es is the solvent dielectric permitivity, co is the permitivity of vacuum and co*’ is the electrolyte concentration. The inner-layer potential difference Az:;‘cp,as inferred from equation (5), is plotted vs. the surface charge density 0’” in Fig. 5. The points at various electrolyte concentration show small deviations from a single line (cf dashed line in Fig. 5), ie the state of the inner layer is controlled
2915
Double layer effects
the aqueous phase is less significant[34], the application of the Gouy-Chapman theory to an analysis of the effect of the aqueous electrolyte concentration on the ion transfer kinetics seems to be adequate, at least as a first approximation. Indeed, the difference between the potential predicted for an 0.05 M nitrobenzene solution of an electrolyte by the GC and MPB theories does not exceed approximately 15 mV when 8” < 4&cm-‘[34].
40 ?I E c. i =. 0
20 Frumkin-type
0 -0.2
0.0 IoW$7
0.2 v
Fig. 4. Interfacial capacitance C vs. the potential difference Arcp for the interface between the nitrobenzene solution of 0.05 M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (0) 0.02 (a). 0.05 (O), 0.10 (W), 0.2 (V) and 0.5 (v). mainly by the surface charge density. Obviously, the inner-laryer potential difference represents here a considerably larger part of A,“cp, compared to the system comprising tetrabutylammonium tetraphenylborate in the nitrobenzene phase[33]. We are aware of the fact that the GC theory can overestimate the diffuse-layer potential difference, in particular in the organic solvent phase[34]. Unfortunately, the modified Poisson-Boltzmann equation (MPB)[35], which predicts a more realistic diffuselayer potential, was used in a limited range of concentrations[34]. On the other hand, because the organic electrolyte concentration was kept constant, and the correction to the diffuse-layer potential in
correction
Figure 6 shows the corrected rate constant ,$ of the Csf ion transfer, which was calculated from the apparent rate constant by using equation (4), as a function of the inner-layer potential difference A$q relative to the standard ion transfer potential A,WpF (the correct_ed standard Gibbs energy change). In an idea1 case, k, should be independent of the base electrolyte concentration and, when the charge transfer coefficient a is a constant (the Butler-Volmer characteristics), the plot of log l, vs. A$q - ArpF (the corrected Tafel plot) should be a straight line. Apparently, at concentrations of LiCl higher than 0.02mol dmm3, the values of &, tends to fall on a single Tafel plot, which is non-linear though, CJ the full line in Fig. 6. Towards lower LiCl oncentrations, there is a considerable drop in &, when the innerlayer potential difference increases. A similar effect was observed for the picrate ion transfer[22], the corrected rate constant of which shows smaller deviations from a single Tafel plot at LiCl concentrations higher than 0.02 mol dm- 3. However, the electrolyte effect is reversed in that k, tends to rise when the LiCl concentration decreases. What seems to be common to both ion transfer reactions, which occur at potentials
positive
to the potential
of zero charge,
-1
r
0.1
-2 i
Y
k
a
0.0
1’ _3
-0.1
Fig. 5. Inner-layer potential difference Ari;‘cpvs. the surface charge density u”’ calculated by using the Gouy-Chapman theory from the capacitance data for the interface between the nitrobenzene solution of 0.05M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (O), 0.02(m), 0.05(O), 0.10(m), 0.2(V) and 0.5 (A).
i -0.16
-0.12
-0.08
-0.04
Fig. 6. Logarithm of the corrected rate constant & vs. the inner-layer potential difference A~&I relative to standard potential difference (corrected Tafel plots) for the Cs+ ion transfer between the nitrobenzene solution of 0.05M TPeATFPB and the aqueous solution of LiCl of the concentration (M): 0.01 (O), 0.02 (a), 0.05 (V), 0.10 p), 0.2 (0) and 0.5 (B). Vertical bars indicate the standard deviation for 0.1 M LiCI, the dashed line corresponds to OL = 0.5.
2. SAMEc et al.
2916
overestimation (in absolute value) of the diffuse-layer potential Arp at lower LiCl concentrations. As a result, the Gouy-Chapman theory predicts too high concentration of Cs+ and too low concentrations of the picrate anion at the outer Helmholtz plane in the aqueous phase. Note that such an overestimation is indicated by the analysis based on the MPB theory[34]. In a way, the effect of the base electrolyte concentration seems to be linked to the failure, though surprising, of the GouyChapman theory to describe the diffuse-layer capacitance at low electrolyte concentrationsC361. The Frumkin-type correction to the apparent ion transfer kinetics can be generalized by considering the migration component of the ion transport across the diffuse layer[37, 381, giving rise to the dynamic (Levich) effect[38]. For ions attracted into the double layer, which is the case of Cs+, the ion tranfer is accelerated, while for ions such as picrate, which are repulsed from the double layer, the ion transfer is decelerated relative to the Frumkin effect, when the absolute value of the diffuse-layer potential difference increases[37]. The dynamic effect becomes more pronounced with the decreasing base electrolyte concentration[37]. Since the behaviour of both Cs+ and picrate ions is at variance with these predictions, the role of the dynamic effect is rather negligible. The non-linear shape of the corrected Tafel plots indicates a dependence of the charge transfer coetlicient CLon the inner layer potential difference Arz;“cp. Such a dependence is predicted by the stochastic approaches based on the Fokker-Planck[25, 261 or the Nernst-Planck[27] equation. In the former case, the denominator 2M (w*L)’ in equation (9) can be comparable with zF(Arirp - A:&. For M x 100 and a characteristic transition time and length of lops (ie, w* Z 10” s-‘) and 1 nm[39, 401, respectively, the former term would have the value around 2 kJ mol- ‘. Although this is a very rough estimate, it can explain the curvature of the corrected Tafel plots. It should be noted that the change in the inner-layer potential difference, A,“z’cp,results in a change in tl, which in turn can weaken the effect of diffuse-layer potential differences, Ar2q and Ar”p in equation (7). As a result, the apparent standard rate constant, k”,, can be less dependent on the base electrolyte concentration than in the case the charge transfer coefficient is a constant. is an
CONCLUSIONS Kinetic parameters of the Cs+ ion transfer across the water/nitrobenzene interface obtained by an UC polarographic technique agree well with those reported previously[2, 63. A weak effect of the aqueous electrolyte concentration on the kinetic parameters is analogous to that observed for the transfer of the picrate[22], as well as the other univalent[14] ions. In view of the unequivocal existence of the diffuse double layer at ITIES, the Frumkin-type correction represents a plausible working hypothesis. However, the Gouy-Chapman theory seems to predict too high diffuse-layer poten-
tial, in particular at low electrolyte concentrations. The non-linearity of the corrected Tafel plots points to a potential dependence of the charge transfer coefficient. Its change can weaken the contribution of the diffuse-layer potentials to the apparent kinetic parameters, thereby accounting for their small changes with the base electrolyte concentration. Dynamic effect of the diffuse double layer, which is linked to the migration component of the ion transport, seems to play an insignificant role in both the Cs+ and picrate ion transfer. Acknowledgements-Z.S.
is the recipient of a visiting fellowship to Japan provided by Japan Society for Promotion of Science; partial support for this work from the Grant Agency of the Academy of Sciences of the Czech Republic (Grant 440411) is also gratefully acknowledged.
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