- Email: [email protected]
The use of the mean spherical approximation in calculation of the double-layer capacitance for the interface between two immiscible electrolyte solutions
J. Electroanal.
Elsevier
Chem.,
Sequoia
170 (1984) 383-386 S.A., Lausanne - Printed
383 in The Netherlands
Preliminary note THE USE OF THE MEAN SPHERICAL APPROXIMATION IN CALCULATION OF THE DOUBLE-LAYER CAPACITANCE FOR THE INTERFACE BETWEEN TWO IMMISCIBLE ELECTROLYTE SOLUTIONS
Z. SAMEC,
V. MARECEK
(Received
and D. HOMOLKA
Institute of Physical Chemistry and Electrochemistry, Czechoslovak U tova’ren 254, 102 00 Prague 10, Hostiva’: (Czechoslovakia)
J. Heyrovsky of Sciences,
Academy
15th May 1984)
It has been recognized that the double-layer capacitance of a metal/electrolyte interface depends on both the metal and the electrolyte solution. From a theoretical point of view, the effect of the metal cannot be accounted for using a classical model with a sharp boundary between the two contacting phases [I] . Instead, a model was suggested in which the metal electrons spill over into the solution and the electrical field penetrates the metal [l--3] . However, a similar diffuse boundary probably arises at the interface between two immiscible electrolyte solutions (ITIES) [4] due to the interfacial mixing of the ions [5]. Recently, the double-layer capacitance was calculated for an ion and dipole mixture using the mean spherical approximation (MSA) [6,7]. Comparison of the MSA theory with the experimental data for the mercury/electrolyte interface indicated that quantitative agreement is achieved only after the effect of the metal is taken into account [ 81. Thus, at low electrolyte concentrations, the inverse capacitance of the double layer can be written as [8] : C-’
=
(~oEK)-’
+
(2~0~)~’
(Ui
+
Ud(E
-
1)/h) + A
(1)
where the first two terms represent the MSA result and A is the concentrationindependent contribution of the metal. In eqn. (l), E is the relative dielectric permittivity, e. is the permittivity of the vacuum, K-~ is the Debye length, okand Od are the diameters of the charged and dipolar hard spheres, respectively, and h is related to e through X2(1 + h)4 = 16~. The physical background for the parameter A consists in the spill-over of electrons into the solution giving rise to a surface dipole of high polarizability, the potential drop across which compensates in part for the dipolar potential drop and thereby reduces the inverse capacitance [13] . Hence, A is negative and its absolute value increases with the increasing distance over which the electronic density extends into the solution [l].For the mercury/aqueous NaH,P03 interface the empirical value of A = -2.6 m2 F-l [8] is close to that calculated theoretically for a jellium model of the metal [9] . In this communication we have compared the experimental capacitance for the system consisting of aqueous NaBr and a solution of tetrabutylammonium tetraphenylborate (TBATPB) in nitrobenzene with the theoretical capacitance 002%0728/84/$03.00
0 1984 Elsevier
Sequoia
S.A.
384
caclulated using the MSA approach. In particular, we have assumed that the ITIES consists of two ion and dipole mixtures facing each other which are separated by a hypothetical plane of contact. For this model, the inverse capacitance is presumably c-1
= C,’
+ C;’
+ a
(2)
where the inverse capacitance C,’
= (EgEWKW)-l +
of the diffuse
= (2eWt-J1
is
(Ei+°Ko)-l
and the concentration-independent C,’
double-layer
(UT + u;(ew
part of the inverse capacitance,
- 1)/X”) + (2PElJ1
(up + .;(P
C;',is
~ l)/hO)
(41
The superscripts w and o in eqns. (3) and (4) refer to water and the organic solvent, respectively. Here, the parameter a is assumed to have the analogous meaning as above, i.e. it accounts for the reduction of the dipolar potential due to inter-penetration of through the of contact the two solutions. For non-permeable plane sharp boundary), would be to zero. double-layer capacitance the water/nitrobenzene was evalfrom the galvanostatic pulse which have de-
CIFti’
0.4
0.3
0.1
0.1
-0.1
0
0.1
A:y'lV
Fig. 1. Plot of capacitance C of the water/nitrobenzene interface as a function of the potcntial difference LIZ* for 0.02 M NaBr in water and 0.02 M TBATPB in nitrobenzene. Capacitance data were evaluated from the slope of the galvanostatic transient (I = k 50 PA) at f = 5 ms. Temperature 25.0* O.l”C.
305
scribed in detail elsewhere [4,10]. The electrolyte concentrations in water and nitrobenzene were the same and in the range from 0.01 to 0.1 M. In Fig. 1 the capacitance C is plotted as a function of the potential differences across the water/nitrobenzene interface Arp = p(w) - p(o). The capacitance has a minimum at AF(p = 0, which can be reasonably considered as the zero-charge potential difference [4]. Figure 2 shows the plot of the inverse capacitance C-’ at Azp = 0. The MSA inverse capacitance C;’ + C,’ was calculated vs. c,’ from eqns. (3) and (4) by taking ew = 78.4, e” = 34.8, ur = 0.276 nm [S], CJ~= 0.555 nm [ll], UT = 0.425 nm and up = 0.850 nm. The fit of the theoretical capacitance (eqn. 2) for the experimental data is very good if the value of -20.2 m2 F-’ is used for A, cf. the straight line in Fig. 2. I
-
8-
Fig. 2. Inverse capacitance C’ of the interface between aqueous NaBr and a solution of TBATPB in nitrobenzene at zero-charge potential difference as a function of C,’ calculated from eqn. (3). The straight line was obtained from eqn. (2) for A = 20.2 m2 F-‘.
We conclude that the MSA result may also be applicable to the electrical double-layer at the ITIES, though this conclusion should be confirmed by the theoretical estimation of A. As compared with the mercury/solution interface, the absolute value of the parameter A for the waterlnitrobenzene interface is greater by almost one order of magnitude, which could be ascribed to more extensive ionic penetration into the opposite solution. In fact, the restraints imposed on the ionic movement across the ITIES are not as strict as those for a metal/electrolyte interface.
REFERENCES A. Kornyshev, W. Schmickler and M. Vorotyntsev, Phys. Rev. B, 25 (1982) 5244. J.P. Badiali, J. Goodisman and M. Rosinberg, J. Electroanal. Chem., 130 (1981) 31; 143 (1983) 73. W. Schmickler, J. Electroanal. Chem., 150 (1983) 19. Z. Samec, V. MareEek and D. Homolka, Faraday Discuss. Chem. Sot., 77 (1984) paper 10. H.H. Girault and D.J. Schiffrin, J. Electroanal. Chem., 150 (1983) 43.
386 6 7 8 9 10 11
S.L. Carnie and D.Y.C. Chan, J. Chem. Phys., 73 (1980) 2949. L. Blum and D. Henderson, J. Chem. Phys., 74 (1981) 1902. L. Blum, D. Henderson and 12. Parsons, J. Electroanal. Chem., 161 (1984) 389. J.P. Badiali, M.L. Rosinberg, F. Vericat and L. Blum, J. Electroanal. Chem., 158 (1983) 253. V. MareEek and Z. Samec, J. Electroanal. Chem., 149 (1983) 185. M.H. Abraham and J. Liszi, J. Inorg. Nucl. Chem., 43 (1981) 143.
Elsevier
Chem.,
Sequoia
170 (1984) 383-386 S.A., Lausanne - Printed
383 in The Netherlands
Preliminary note THE USE OF THE MEAN SPHERICAL APPROXIMATION IN CALCULATION OF THE DOUBLE-LAYER CAPACITANCE FOR THE INTERFACE BETWEEN TWO IMMISCIBLE ELECTROLYTE SOLUTIONS
Z. SAMEC,
V. MARECEK
(Received
and D. HOMOLKA
Institute of Physical Chemistry and Electrochemistry, Czechoslovak U tova’ren 254, 102 00 Prague 10, Hostiva’: (Czechoslovakia)
J. Heyrovsky of Sciences,
Academy
15th May 1984)
It has been recognized that the double-layer capacitance of a metal/electrolyte interface depends on both the metal and the electrolyte solution. From a theoretical point of view, the effect of the metal cannot be accounted for using a classical model with a sharp boundary between the two contacting phases [I] . Instead, a model was suggested in which the metal electrons spill over into the solution and the electrical field penetrates the metal [l--3] . However, a similar diffuse boundary probably arises at the interface between two immiscible electrolyte solutions (ITIES) [4] due to the interfacial mixing of the ions [5]. Recently, the double-layer capacitance was calculated for an ion and dipole mixture using the mean spherical approximation (MSA) [6,7]. Comparison of the MSA theory with the experimental data for the mercury/electrolyte interface indicated that quantitative agreement is achieved only after the effect of the metal is taken into account [ 81. Thus, at low electrolyte concentrations, the inverse capacitance of the double layer can be written as [8] : C-’
=
(~oEK)-’
+
(2~0~)~’
(Ui
+
Ud(E
-
1)/h) + A
(1)
where the first two terms represent the MSA result and A is the concentrationindependent contribution of the metal. In eqn. (l), E is the relative dielectric permittivity, e. is the permittivity of the vacuum, K-~ is the Debye length, okand Od are the diameters of the charged and dipolar hard spheres, respectively, and h is related to e through X2(1 + h)4 = 16~. The physical background for the parameter A consists in the spill-over of electrons into the solution giving rise to a surface dipole of high polarizability, the potential drop across which compensates in part for the dipolar potential drop and thereby reduces the inverse capacitance [13] . Hence, A is negative and its absolute value increases with the increasing distance over which the electronic density extends into the solution [l].For the mercury/aqueous NaH,P03 interface the empirical value of A = -2.6 m2 F-l [8] is close to that calculated theoretically for a jellium model of the metal [9] . In this communication we have compared the experimental capacitance for the system consisting of aqueous NaBr and a solution of tetrabutylammonium tetraphenylborate (TBATPB) in nitrobenzene with the theoretical capacitance 002%0728/84/$03.00
0 1984 Elsevier
Sequoia
S.A.
384
caclulated using the MSA approach. In particular, we have assumed that the ITIES consists of two ion and dipole mixtures facing each other which are separated by a hypothetical plane of contact. For this model, the inverse capacitance is presumably c-1
= C,’
+ C;’
+ a
(2)
where the inverse capacitance C,’
= (EgEWKW)-l +
of the diffuse
= (2eWt-J1
is
(Ei+°Ko)-l
and the concentration-independent C,’
double-layer
(UT + u;(ew
part of the inverse capacitance,
- 1)/X”) + (2PElJ1
(up + .;(P
C;',is
~ l)/hO)
(41
The superscripts w and o in eqns. (3) and (4) refer to water and the organic solvent, respectively. Here, the parameter a is assumed to have the analogous meaning as above, i.e. it accounts for the reduction of the dipolar potential due to inter-penetration of through the of contact the two solutions. For non-permeable plane sharp boundary), would be to zero. double-layer capacitance the water/nitrobenzene was evalfrom the galvanostatic pulse which have de-
CIFti’
0.4
0.3
0.1
0.1
-0.1
0
0.1
A:y'lV
Fig. 1. Plot of capacitance C of the water/nitrobenzene interface as a function of the potcntial difference LIZ* for 0.02 M NaBr in water and 0.02 M TBATPB in nitrobenzene. Capacitance data were evaluated from the slope of the galvanostatic transient (I = k 50 PA) at f = 5 ms. Temperature 25.0* O.l”C.
305
scribed in detail elsewhere [4,10]. The electrolyte concentrations in water and nitrobenzene were the same and in the range from 0.01 to 0.1 M. In Fig. 1 the capacitance C is plotted as a function of the potential differences across the water/nitrobenzene interface Arp = p(w) - p(o). The capacitance has a minimum at AF(p = 0, which can be reasonably considered as the zero-charge potential difference [4]. Figure 2 shows the plot of the inverse capacitance C-’ at Azp = 0. The MSA inverse capacitance C;’ + C,’ was calculated vs. c,’ from eqns. (3) and (4) by taking ew = 78.4, e” = 34.8, ur = 0.276 nm [S], CJ~= 0.555 nm [ll], UT = 0.425 nm and up = 0.850 nm. The fit of the theoretical capacitance (eqn. 2) for the experimental data is very good if the value of -20.2 m2 F-’ is used for A, cf. the straight line in Fig. 2. I
-
8-
Fig. 2. Inverse capacitance C’ of the interface between aqueous NaBr and a solution of TBATPB in nitrobenzene at zero-charge potential difference as a function of C,’ calculated from eqn. (3). The straight line was obtained from eqn. (2) for A = 20.2 m2 F-‘.
We conclude that the MSA result may also be applicable to the electrical double-layer at the ITIES, though this conclusion should be confirmed by the theoretical estimation of A. As compared with the mercury/solution interface, the absolute value of the parameter A for the waterlnitrobenzene interface is greater by almost one order of magnitude, which could be ascribed to more extensive ionic penetration into the opposite solution. In fact, the restraints imposed on the ionic movement across the ITIES are not as strict as those for a metal/electrolyte interface.
REFERENCES A. Kornyshev, W. Schmickler and M. Vorotyntsev, Phys. Rev. B, 25 (1982) 5244. J.P. Badiali, J. Goodisman and M. Rosinberg, J. Electroanal. Chem., 130 (1981) 31; 143 (1983) 73. W. Schmickler, J. Electroanal. Chem., 150 (1983) 19. Z. Samec, V. MareEek and D. Homolka, Faraday Discuss. Chem. Sot., 77 (1984) paper 10. H.H. Girault and D.J. Schiffrin, J. Electroanal. Chem., 150 (1983) 43.
386 6 7 8 9 10 11
S.L. Carnie and D.Y.C. Chan, J. Chem. Phys., 73 (1980) 2949. L. Blum and D. Henderson, J. Chem. Phys., 74 (1981) 1902. L. Blum, D. Henderson and 12. Parsons, J. Electroanal. Chem., 161 (1984) 389. J.P. Badiali, M.L. Rosinberg, F. Vericat and L. Blum, J. Electroanal. Chem., 158 (1983) 253. V. MareEek and Z. Samec, J. Electroanal. Chem., 149 (1983) 185. M.H. Abraham and J. Liszi, J. Inorg. Nucl. Chem., 43 (1981) 143.