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The effects of ionic size on equilibrium double layer properties and the direct determination of specifically adsorbed charge
474
Electroanalytical Chemistry and Interracial Electrochemistry Elsevier Sequoia S~A., Lausanne
Printed in The Netherlands
SHORT COMMUNICATION
The effects of ionic size on equilibrium double layer properties and the direct determination of specifically adsorbed charge
W. R. FAWCETT
Department of Chemistry, University of Guelph, Guelph, Ontario (Canada) (Received 7th February 1972; in revised form 29th March 1972)
Since its proposal in 1965-66, the Hurwitz-Parsons method 1'2 has often been used in interfacial thermodynamic studies to determine directly the amount of ionic specific adsorption at a polarisable electrode. Damaskin3 emphasised the value of the technique in cases where this adsorption is weak and pointed out that specific adsorption would remain undetected by the usual methods 4 when the charge due to the specifically adsorbed species is less in absolute value than that on the electrode. However, the Hurwitz-Parsons analysis contains a non-thermodynamic assumption which may be all important in deciding whether specific adsorption is present in such systems. The purpose of the present note is to examine the effects of this assumption on the analysis. The discussion is directed to the important question of caesium ion specific adsorption which has been studied by this technique in both aqueous 5,6 and non-aqueous systems7'8. On applying the Hurwitz-Parsons method to measure caesium ion adsorption, one obtains interracial thermodynamic data at the Hg/solution interface for a series of systems of the type Hg / xM CsX + ( a - x) M LiX/AgX,Ag
(1)
where X- is a monovalent anion not adsorbed in the potential range of interest, a the ionic strength, which is maintained constant, and x the molarity of the caesium salt which is varied between 0 and a. From the Gibbs adsorption isotherm for a polarisible interface, one may show by the usual thermodynamic methods 1'2 that - d 7 = qdE_ + Fcs+
a-x
FLi+
R r d In x
(2)
where 7 is the surface tension, q the surface charge density, E_ the potential of mercury with respect to the above reference electrode, and FM+, the relative surface excess of cation M +. At this point, a non-thermodynamic assumption is made, namely, that the ratio of the number of non-specifically adsorbed Cs + ions in the double layer to that of non-specifically adsorbed Li + ions is equal to x/(a-x), that is, to the ratio of their concentrations in the bulk. Then, assuming that Li + is not specifically adsorbed, F~s+ =Fcs+ - x
U--X
J. Electroanal. Chem., 39 (1972)
FLi+
(3)
SHORT COMMUNICATION
475
where F~s+ is the part of the relative surface excess of Cs + due to specific adsorption. Thus, the specifically adsorbed charge due to Cs ÷ may be obtained directly by determining - f ( a T / a In x) at constant E_ where f = F / R T . The assumption made in the above analysis is correct within the context of the commonly accepted model of the diffuse double layer, namely, the Gouy-Chapman model. However, this model ignores, amongst other things, finite ionic size, and thus, thepossibility that the distance of closest approach to the electrode of non-specifically adsorbed ions may depend on the nature of the ion 9'1°. If the distances of closest approach are indeed different, the above assumption is not valid, and the specifically adsorbed charge calculated by the Hurwitz-Parsons method is simply that due to the smaller ion in the region adjacent to the electrode which cannot be entered by the charge centres of the larger ions. The extent to which differences in ionic size, and thus in distances of closest approach, are important in determining specific adsorption by the direct method may be estimated to a first approximation by the Gouy-Chapman theory following thc procedure of Joshi and Parsons 9 : r a is defined to be the distance of closest approach of the smaller cation, rb, that of the larger cation and anion, and tka and ~bb,the average potentials on planes parallel to the electrode through r~ and rb, respectively. If qd is the excess charge density in the region r >/rb, then ~bb = (2/f) sinh-1 ( - qd/20a ~)
(4)
where O=(RTe/2rc) ~, e being the dielectric constant of the solvent. The potential distribution in the region ra ~
(5)
where ~c=4r~fOx ~ and Ar= r - r b . The amount by which the surfaces excess of the smaller cation exceeds that of the larger, Aq + is given by
Aq + = FOx ~ [exp (-fqSa/2)- exp (-- f Ob/ 2) ]
(6)
Finally the charge on the electrode is q = -Aq+ --qd
(7)
Values of Aq+ were calculated using the above relationships for an aqueous system at 25°C in which a equalled 1.0 M, x varied between 0.02 and 1.0 M and rb -- ra equalled 1.0/k. These conditions correspond to those used by Parsons and Stockton 5 in their study of Cs ÷ specific adsorption, the value of r b - r~ being based on Monk's estimate 11 of the difference in hydrated radii for Li ÷ and Cs ÷ . Calculated values of Aq + as a function of electrode charge density for various Cs ÷ ion concentrations are shown in Fig. 1.Although the results differ in detail from those obtained experimentally using the Hurwitz-Parsons analysis5'6, the present calculation predicts apparent specific adsorption of the correct order of magnitude. Variation of r b - r~ changed the total amount of specific adsorption for a given concentration and electrode charge, but did not change the general shape of the plots shown in Fig. 1. In particular, no reasonable value of rb-- r~ could reproduce the sharp rise in adsorbed charge found experimentally at charges cathodic Of - 15 #C cm- 2. Thus, according to the above
J. Electroanal. Chem., 39 (1972)
476
SHORT COMMUNICATION
8.0
3. .q4.C
o_- - "- Z''°'/''/'/I''°" I
zI
."
-10
q/l~ C
/"
cm-2
Ol
-20
Fig. 1. Plots of apparent specifica]|y adsorbed charge, Aq +, vs. electrode charge, q, for an ionic strength of 1.0 M and r b - r a= 1/~. Number beside each carve gives concn, of smaller ("specifically adsorbed") cation. The exptl, results of Parsons and Stockton ( - - - - ) obtained from data for LiC] CsC] system analysed by the Hurwitz-Parsons method are shown for three Cs + concns. (0.04, 0.2 and 1.0 M).
model, Cs + specific adsorption is present in this potential range, although the amount adsorbed is considerably less than that reported previously 5'6. Wroblowa et al. 12 calculated the surface excess of caesium ion at the mercury/ CsC1 solution interface from data for interfacia] tension as a function of electrode potential and electrolyte concentration and concluded that specific adsorption of the cation was significant for concentrations greater than 1 M and electrode charge densities cathodic of - 1 0 pC cm-2. Their results agree qualitatively with the conclusions reached by the above calculation. Unfortunately, not much significance can be given to the actual values of calculated adsorbed charge density. Firstly, as stated above, such calculations are subject to large errors when the absolute value of the electrode charge density is less than that on the electrode 3'5. Secondly, data were obtained only at six concentrations over a wide concentration range so that errors in calculated surface excesses are undoubtedly quite large 13. Gierst et ai.14 have advocated determining the amount of cation specific adsorption from kinetic data which exhibit a large double layer effect at cathodic potentials. By this procedure, the effect of alkali metal cations on the reaction rate is attributed entirely to cation specific adsorption. However, other effects such as ion pairing between a reacting anion and the cation, and defects in the G o u y - C h a p m a n model used to analyse these data, which are ignored in their procedure, are often important 15, SO that very little quantitative significance can be associated with such an analysis. The effects of ionic size and other defects in the G o u y - C h a p m a n theory have been considered recently by Parsons and Trasattil 6. These authors measured cationic surface excesses in a KC1-MgC12 mixture by obtaining data at constant K + activity for varying Mg 2+ activity and vice versa. In this way, an exact thermodynamic analysis of the data could be carried out. These authors found positive deviations of the J. Electroanal. Chem., 39 (1972)
SHORT COMMUNICATION
477
measured cationic surface excesses from those predicted by the Gouy~Chapman theory, the deviation being greater for K ÷ than for Mg 2÷. If the failure to consider ionic size were the most important defect in the theory, one would expect a positive deviation in the case of K + and a negative 9 one for Mg 2+. However, defects in the simple theory other than that of ignoring finite ionic volume are undoubtedly more important in a system containing both mono- and divalent ions. Schiffrin 17has pointed out the importance of image forces and suggested that they could be the major reason for the positive deviations observed in dilute solutions 18. Assuming a distance of closest approach of 4.5 A for K +, its concentration at the outer Helmholtz plane at the point of zero charge would be 26 mM for a bulk concentration of 20 mM because of attractive image forces 17 ; the corresponding concentration in a 5 mM Mg2÷ solution for a distance of closest approach of 6.0 ,~ would be 10 mM. In a solution containing both salts at the same concentrations, the concentrations at the respective outer Helmholtz planes would be somewhat less due to the increase in ionic strength. However, the effect of ionic size would be such as to make the positive deviation from that calculated by the Gouy-Chapman theory larger for K ÷ and smaller for Mg 2÷. These trends appear to be present in the results of Parsons and Trasatti x6. It is suggested that a more effective experimental test of the effects of ionic size should involve ions of the same charge. Finally, it is noted that Harrison et al. 19 estimated the thickness of the inner layer in the absence of specific adsorption by measuring relative surface excesses in concentrated solutions where the effect of the reference solvent excess is important. They found no significant difference in inner layer thicknesses for Li ÷ and Na ÷ systems at cathodic potentials. However, these calculations rely on the Gouy-Chapman theory at very high concentrations where it is most likely to fail. Acknowledgement
The author wishes to thank Drs. R. Parsons and D. Schiffrin for discussing the above problem and for helpful suggestions. REFERENCES 1 H. D. Hurwitz, J. Electroanal. Chem., 10 (1965) 35. 2 E. Dutkiewicz and R. Parsons, J. Electroanal. Chem., 11 (1966) 100. 3 B. B. Damaskin, Elektrokhimiya, 5 (1969) 771. 4 D.C. Grahame, J. Amer. Chem. Soc., 80 (1958) 4201. 5 R. Parsons and A. Stockton, J. Electroanal. Chem., 25 (1970) App. 10. 6 R. V. Ivanov, B. B. Damaskin and S. Mazurek, Elektrokhimiya, 6 (1970) 1041. 7 Ya. Doylido, R. V. Ivanova and B. B. Damaskin, Elektrokhimiya, 6 (1970) 3. 8 Sh. S. Dzhaparidze, D. I. Dzhaparidze and B. B. Damaskin, Elektrokhimiya, 7 (1971) 1535. 9 K. M. Joshi and R. Parsons, Electrochim. Acta, 4 (1961) 129. 10 D. M. Mohilner, Electroanal. Chem., 1 (1966) 241. 11 C. B. Monk, Electrolytic Dissociation, Academic Press, New York, 1961, p. 271. 12 H. Wroblowa, Z. Kovac and J. O.'M. Bockris, Trans. Faraday Soc., 61 (1965) 1523. 13 W.R. Fawcett and J. E. Kent, Can. J. Chem., 48 (1970) 47. 14 L. Gierst, E. Nicolas and L. Tytgat-Vandenberghen, Croat. Chem. Acta, 42 (1970) 117. 15 D. J. Bieman and W. R. Fawcett, J. Electroanal. Chem., 34 (1972) 27. 16 R. Parsons and S. Trasatti, Trans. Faraday Soc., 65 (1969) 3314. 17 D. J. Schiffrin, J. Phys. Chem., 73 (1969) 1632. 18 D. J. Schiffrin, Electrochemistry, Specialist Periodical Reports, Vol. 1, Chemical Society (London), 1970, p. 223. 19 J. A. Harrison, J. E. B. Randles and D. J. Schiffrin, J. Electroanal. Chem., 25 (1970) 197.
J. Electroanal. Chem., 39 (1972)
Electroanalytical Chemistry and Interracial Electrochemistry Elsevier Sequoia S~A., Lausanne
Printed in The Netherlands
SHORT COMMUNICATION
The effects of ionic size on equilibrium double layer properties and the direct determination of specifically adsorbed charge
W. R. FAWCETT
Department of Chemistry, University of Guelph, Guelph, Ontario (Canada) (Received 7th February 1972; in revised form 29th March 1972)
Since its proposal in 1965-66, the Hurwitz-Parsons method 1'2 has often been used in interfacial thermodynamic studies to determine directly the amount of ionic specific adsorption at a polarisable electrode. Damaskin3 emphasised the value of the technique in cases where this adsorption is weak and pointed out that specific adsorption would remain undetected by the usual methods 4 when the charge due to the specifically adsorbed species is less in absolute value than that on the electrode. However, the Hurwitz-Parsons analysis contains a non-thermodynamic assumption which may be all important in deciding whether specific adsorption is present in such systems. The purpose of the present note is to examine the effects of this assumption on the analysis. The discussion is directed to the important question of caesium ion specific adsorption which has been studied by this technique in both aqueous 5,6 and non-aqueous systems7'8. On applying the Hurwitz-Parsons method to measure caesium ion adsorption, one obtains interracial thermodynamic data at the Hg/solution interface for a series of systems of the type Hg / xM CsX + ( a - x) M LiX/AgX,Ag
(1)
where X- is a monovalent anion not adsorbed in the potential range of interest, a the ionic strength, which is maintained constant, and x the molarity of the caesium salt which is varied between 0 and a. From the Gibbs adsorption isotherm for a polarisible interface, one may show by the usual thermodynamic methods 1'2 that - d 7 = qdE_ + Fcs+
a-x
FLi+
R r d In x
(2)
where 7 is the surface tension, q the surface charge density, E_ the potential of mercury with respect to the above reference electrode, and FM+, the relative surface excess of cation M +. At this point, a non-thermodynamic assumption is made, namely, that the ratio of the number of non-specifically adsorbed Cs + ions in the double layer to that of non-specifically adsorbed Li + ions is equal to x/(a-x), that is, to the ratio of their concentrations in the bulk. Then, assuming that Li + is not specifically adsorbed, F~s+ =Fcs+ - x
U--X
J. Electroanal. Chem., 39 (1972)
FLi+
(3)
SHORT COMMUNICATION
475
where F~s+ is the part of the relative surface excess of Cs + due to specific adsorption. Thus, the specifically adsorbed charge due to Cs ÷ may be obtained directly by determining - f ( a T / a In x) at constant E_ where f = F / R T . The assumption made in the above analysis is correct within the context of the commonly accepted model of the diffuse double layer, namely, the Gouy-Chapman model. However, this model ignores, amongst other things, finite ionic size, and thus, thepossibility that the distance of closest approach to the electrode of non-specifically adsorbed ions may depend on the nature of the ion 9'1°. If the distances of closest approach are indeed different, the above assumption is not valid, and the specifically adsorbed charge calculated by the Hurwitz-Parsons method is simply that due to the smaller ion in the region adjacent to the electrode which cannot be entered by the charge centres of the larger ions. The extent to which differences in ionic size, and thus in distances of closest approach, are important in determining specific adsorption by the direct method may be estimated to a first approximation by the Gouy-Chapman theory following thc procedure of Joshi and Parsons 9 : r a is defined to be the distance of closest approach of the smaller cation, rb, that of the larger cation and anion, and tka and ~bb,the average potentials on planes parallel to the electrode through r~ and rb, respectively. If qd is the excess charge density in the region r >/rb, then ~bb = (2/f) sinh-1 ( - qd/20a ~)
(4)
where O=(RTe/2rc) ~, e being the dielectric constant of the solvent. The potential distribution in the region ra ~
(5)
where ~c=4r~fOx ~ and Ar= r - r b . The amount by which the surfaces excess of the smaller cation exceeds that of the larger, Aq + is given by
Aq + = FOx ~ [exp (-fqSa/2)- exp (-- f Ob/ 2) ]
(6)
Finally the charge on the electrode is q = -Aq+ --qd
(7)
Values of Aq+ were calculated using the above relationships for an aqueous system at 25°C in which a equalled 1.0 M, x varied between 0.02 and 1.0 M and rb -- ra equalled 1.0/k. These conditions correspond to those used by Parsons and Stockton 5 in their study of Cs ÷ specific adsorption, the value of r b - r~ being based on Monk's estimate 11 of the difference in hydrated radii for Li ÷ and Cs ÷ . Calculated values of Aq + as a function of electrode charge density for various Cs ÷ ion concentrations are shown in Fig. 1.Although the results differ in detail from those obtained experimentally using the Hurwitz-Parsons analysis5'6, the present calculation predicts apparent specific adsorption of the correct order of magnitude. Variation of r b - r~ changed the total amount of specific adsorption for a given concentration and electrode charge, but did not change the general shape of the plots shown in Fig. 1. In particular, no reasonable value of rb-- r~ could reproduce the sharp rise in adsorbed charge found experimentally at charges cathodic Of - 15 #C cm- 2. Thus, according to the above
J. Electroanal. Chem., 39 (1972)
476
SHORT COMMUNICATION
8.0
3. .q4.C
o_- - "- Z''°'/''/'/I''°" I
zI
."
-10
q/l~ C
/"
cm-2
Ol
-20
Fig. 1. Plots of apparent specifica]|y adsorbed charge, Aq +, vs. electrode charge, q, for an ionic strength of 1.0 M and r b - r a= 1/~. Number beside each carve gives concn, of smaller ("specifically adsorbed") cation. The exptl, results of Parsons and Stockton ( - - - - ) obtained from data for LiC] CsC] system analysed by the Hurwitz-Parsons method are shown for three Cs + concns. (0.04, 0.2 and 1.0 M).
model, Cs + specific adsorption is present in this potential range, although the amount adsorbed is considerably less than that reported previously 5'6. Wroblowa et al. 12 calculated the surface excess of caesium ion at the mercury/ CsC1 solution interface from data for interfacia] tension as a function of electrode potential and electrolyte concentration and concluded that specific adsorption of the cation was significant for concentrations greater than 1 M and electrode charge densities cathodic of - 1 0 pC cm-2. Their results agree qualitatively with the conclusions reached by the above calculation. Unfortunately, not much significance can be given to the actual values of calculated adsorbed charge density. Firstly, as stated above, such calculations are subject to large errors when the absolute value of the electrode charge density is less than that on the electrode 3'5. Secondly, data were obtained only at six concentrations over a wide concentration range so that errors in calculated surface excesses are undoubtedly quite large 13. Gierst et ai.14 have advocated determining the amount of cation specific adsorption from kinetic data which exhibit a large double layer effect at cathodic potentials. By this procedure, the effect of alkali metal cations on the reaction rate is attributed entirely to cation specific adsorption. However, other effects such as ion pairing between a reacting anion and the cation, and defects in the G o u y - C h a p m a n model used to analyse these data, which are ignored in their procedure, are often important 15, SO that very little quantitative significance can be associated with such an analysis. The effects of ionic size and other defects in the G o u y - C h a p m a n theory have been considered recently by Parsons and Trasattil 6. These authors measured cationic surface excesses in a KC1-MgC12 mixture by obtaining data at constant K + activity for varying Mg 2+ activity and vice versa. In this way, an exact thermodynamic analysis of the data could be carried out. These authors found positive deviations of the J. Electroanal. Chem., 39 (1972)
SHORT COMMUNICATION
477
measured cationic surface excesses from those predicted by the Gouy~Chapman theory, the deviation being greater for K ÷ than for Mg 2÷. If the failure to consider ionic size were the most important defect in the theory, one would expect a positive deviation in the case of K + and a negative 9 one for Mg 2+. However, defects in the simple theory other than that of ignoring finite ionic volume are undoubtedly more important in a system containing both mono- and divalent ions. Schiffrin 17has pointed out the importance of image forces and suggested that they could be the major reason for the positive deviations observed in dilute solutions 18. Assuming a distance of closest approach of 4.5 A for K +, its concentration at the outer Helmholtz plane at the point of zero charge would be 26 mM for a bulk concentration of 20 mM because of attractive image forces 17 ; the corresponding concentration in a 5 mM Mg2÷ solution for a distance of closest approach of 6.0 ,~ would be 10 mM. In a solution containing both salts at the same concentrations, the concentrations at the respective outer Helmholtz planes would be somewhat less due to the increase in ionic strength. However, the effect of ionic size would be such as to make the positive deviation from that calculated by the Gouy-Chapman theory larger for K ÷ and smaller for Mg 2÷. These trends appear to be present in the results of Parsons and Trasatti x6. It is suggested that a more effective experimental test of the effects of ionic size should involve ions of the same charge. Finally, it is noted that Harrison et al. 19 estimated the thickness of the inner layer in the absence of specific adsorption by measuring relative surface excesses in concentrated solutions where the effect of the reference solvent excess is important. They found no significant difference in inner layer thicknesses for Li ÷ and Na ÷ systems at cathodic potentials. However, these calculations rely on the Gouy-Chapman theory at very high concentrations where it is most likely to fail. Acknowledgement
The author wishes to thank Drs. R. Parsons and D. Schiffrin for discussing the above problem and for helpful suggestions. REFERENCES 1 H. D. Hurwitz, J. Electroanal. Chem., 10 (1965) 35. 2 E. Dutkiewicz and R. Parsons, J. Electroanal. Chem., 11 (1966) 100. 3 B. B. Damaskin, Elektrokhimiya, 5 (1969) 771. 4 D.C. Grahame, J. Amer. Chem. Soc., 80 (1958) 4201. 5 R. Parsons and A. Stockton, J. Electroanal. Chem., 25 (1970) App. 10. 6 R. V. Ivanov, B. B. Damaskin and S. Mazurek, Elektrokhimiya, 6 (1970) 1041. 7 Ya. Doylido, R. V. Ivanova and B. B. Damaskin, Elektrokhimiya, 6 (1970) 3. 8 Sh. S. Dzhaparidze, D. I. Dzhaparidze and B. B. Damaskin, Elektrokhimiya, 7 (1971) 1535. 9 K. M. Joshi and R. Parsons, Electrochim. Acta, 4 (1961) 129. 10 D. M. Mohilner, Electroanal. Chem., 1 (1966) 241. 11 C. B. Monk, Electrolytic Dissociation, Academic Press, New York, 1961, p. 271. 12 H. Wroblowa, Z. Kovac and J. O.'M. Bockris, Trans. Faraday Soc., 61 (1965) 1523. 13 W.R. Fawcett and J. E. Kent, Can. J. Chem., 48 (1970) 47. 14 L. Gierst, E. Nicolas and L. Tytgat-Vandenberghen, Croat. Chem. Acta, 42 (1970) 117. 15 D. J. Bieman and W. R. Fawcett, J. Electroanal. Chem., 34 (1972) 27. 16 R. Parsons and S. Trasatti, Trans. Faraday Soc., 65 (1969) 3314. 17 D. J. Schiffrin, J. Phys. Chem., 73 (1969) 1632. 18 D. J. Schiffrin, Electrochemistry, Specialist Periodical Reports, Vol. 1, Chemical Society (London), 1970, p. 223. 19 J. A. Harrison, J. E. B. Randles and D. J. Schiffrin, J. Electroanal. Chem., 25 (1970) 197.
J. Electroanal. Chem., 39 (1972)