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The electrosorption valency and partial charge transfer
Journal of
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 562 (2004) 273–276 www.elsevier.com/locate/jelechem
Short Communication
The electrosorption valency and partial charge transfer Robert de Levie
*
Department of Chemistry, Bowdoin College, Brunswick, ME 04011, USA Received 11 February 2003; received in revised form 8 August 2003; accepted 27 August 2003
Abstract The electrosorption valency as usually defined is an extra-thermodynamic and self-contradictory concept. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Thermodynamics; Electrosorption valency; Charge; Charge transfer
When ions adsorb at the electrode solution interface, partial charge transfer is possible, and sometimes even plausible. Grahame [1] had already indicated the close parallel between halide adsorption on mercury and (covalent) chemical bonding. The related concepts of formal partial charge transfer and electrosorption valency were subsequently taken up by Lorenz and Salie [2] and by Schultze and Koppe [3], and their present status was reviewed recently by Schultze and Rolle [4]. These are concepts that properly belong in the domain of quantum mechanics [5]. However, the electrosorption valency c is often suggested to be a thermodynamic property. In this short communication we will consider its thermodynamic validity. In order to do so we will first recall two well-studied cases, viz. those of the ideally polarizable and the nonpolarizable interface. In order to keep the mathematics simple we will in all cases consider the interface between mercury (identified as phase I) and an aqueous solution of a single strong z, z-electrolyte (phase II), with a reference electrode assembly (containing the metal phase III plus, possibly, its own solution phase II0 together with a barrier membrane selectively permeable to the necessary ion) that responds thermodynamically to either cation or anion. We will not take into account the metal wires connecting the cell with the measuring equipment, because the contact potentials between the various metals are constant at constant temperature, *
Tel.: +1-207-725-3028; fax: +1-207-725-3017. E-mail address: [email protected] (R. de Levie).
0022-0728/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2003.08.027
and we will not be concerned here with the absolute values of the cell potential E ¼ /I /III , but only with its changes dE ¼ dð/I /III Þ ¼ d/I d/III , where / is the inner potential. Such changes dE are unaffected by constant contact potentials. Under these conditions, the Gibbs adsorption equation will take the form dr ¼ Cþ d~ lþ þ C d~ l þ Ce d~ le;I þ Cw dlw ;
ð1Þ
where r is the interfacial tension (as distinct from the electrosorption valency c), l denotes the chemical potential, and l~ the electrochemical potential, while the subscripts +, ), e, and w indicate cations, anions, metal electrons, and water molecules, respectively. Where necessary to avoid possible ambiguity, roman numerals are used to identify the phase involved.
1. The ideally polarizable electrode In the absence of charge transfer we convert Eq. (1) into dr ¼ Cþ dls þ qe dE þ Cw dlw
ð2aÞ
or dr ¼ C dls þ qe dEþ þ Cw dlw
ð2bÞ
depending on whether we use a reference electrode responding to the solution anions or cations, respectively. In order to go from Eq. (1) to (2a), (2b) or (3) we use the following relations:
274
R. de Levie / Journal of Electroanalytical Chemistry 562 (2004) 273–276
(a) the electrochemical potential of metal electrons l~e;I ¼ loe;I F /I and the definition qe ¼ F Ce so that Ce d~ le;I ¼ F Ce d/I ¼ qe d/I where qe is the electronic charge density; (b) the equilibrium condition ls ¼ l~þ þ l~ describing the electrolyte dissociation, so that d~ l ¼ dls d~ lþ or d~ lþ ¼ dls d~ l , respectively; (c) the electroneutrality condition Cþ C ¼ Ce =z ¼ qe =zF ; and (d) the Nernst equation for the reference electrode, which for a cation-responsive electrode based on o ~þ;II þ l~þ;II ¼ l~þ;II0 and Mzþ þ ze III MIII is l II0 z~ le;III ¼ l~þ;II0 þ z~ le;III ¼ lM;III or d~ lþ d~ lþ;II ¼ zd~ le;III ¼ zF d/III , while an anion-selective electrode based on, say, l~;II ¼ l~;II0 and MXz;II0 þ o ~e;III ¼ lM;III þ z~ ze l III MIII þ zXII0 yields ls þ l hence d~ l ¼ zF d/III , where the subscript s denotes the poorly soluble salt MXz , so that dls ¼ 0, and likewise dlM;III ¼ 0 for a pure metal phase MIII . It is customary to combine Eqs. (2a) and (2b) into X dr ¼ C dls þ qe dE þ Cu dlu ; ð3Þ u
where either the top or the bottom signs are used. Consequently we find the Lippmann equation or ¼ qe ð4Þ oE P ;T ;l and or x ¼ C Cw ¼ CðwÞ ; ols P ;T ;E ;lu6¼j xw
ð5Þ
where we have used the Gibbs–Duhem–Margules rule ðolj =oli ÞP ;T ;E ;lk6¼i;j ¼ xi =xj where x denotes the mole fraction. Furthermore, since dr in Eq. (3) is a complete differential, we can obtain cross-relations such as oqe ols ¼ : ð6Þ oCðwÞ E oE CðwÞ
where we use the following sequence of substitutions: le;I ¼ l~R hence d~ lR ¼ d~ lO þ nd~ le;I ; the definition l~O þ n~ Ce ¼ qe =F plus d~ le;I ¼ F d/I so that ðnCR þ Ce Þd~ le;I ¼ ðqe nF CR Þd/I ; the salt dissociation equilibrium lOXz ¼ l~O þ z~ l which yields d~ lO ¼ dlOXz zd~ l ; and the electroneutrality condition zCO þ ðz nÞCR C ¼ qe =F .Elimination of d~ lO rather than d~ lR yields the alternative form dr ¼ ðCO þ CR ÞdlRXzn þ ðqe þ nF CO ÞdE þ Cw dlw ð8Þ which shows that now only the linear combinations CO þ CR and qe þ nF CO are thermodynamically defined. One of these can be replaced by qe nF CR ¼ ðqe þ nF CO Þ nF ðCO þ CR Þ. In aqueous solution, the typical experimental results will, therefore, be COðwÞ þ CRðwÞ and either qe þ nF COðwÞ or qe nF CRðwÞ . The electronic charge density qe per se is no longer thermodynamically defined. We can also find cross-correlations at constant pressure and temperature, such as oðqe þ nF COðwÞ Þ ols ¼ ; ð9Þ oE ðCOðwÞ þCRðwÞ Þ oðCOðwÞ þ CRðwÞ Þ E where s denotes the salt RXzn , or equivalent results based on the salt OXz . We can consider the redox couple Mnþ þ ne Mo a special case of (7) or (8) for which the reduced form is M, so that dlRzn ¼ dlM ¼ 0, while CO ¼ Cþ , in which case one obtains dr ¼ ðqe þ nF Cþ ÞdE þ Cw dlw ¼ ðqe þ nF CþðwÞ ÞdE
ð10Þ
from which no cross relations can be derived. The above results are readily obtained and well known, and are summarized here only to re-emphasize that many results for the ideally polarized electrode are not valid in the presence of interfacial electron transfer, but must be modified to reflect such transfer.
3. Partial charge transfer
2. The non-polarizable electrode In the case of a non-polarizable electrode the metaljsolution interface IjII contains an equilibrium redox couple. We will here consider the redox equilibrium Ozþ þ ne RðznÞþ where both redox components OXz and RXzn are strong electrolytes with a common anion, X . The Gibbs adsorption equation for the IjII interface then reads lO þ CR d~ lR þ C d~ l þ Ce d~ le;I þ Cw dlw dr ¼ CO d~
We now consider partial charge transfer, in which one surmises that, on adsorption, an anion such as I may lose part of its negative charge. Say that we consider mercury in contact with an aqueous solution of LiI, and assume that the iodide in contact with the electrode has a partial charge d, where 0 6 d 6 1, so that the Gibbs adsorption equation reads dr ¼ CLiþ d~ lLiþ þ ðCI þ CId Þd~ lI þ Ce d~ le þ Cw dlw ; ð11Þ
¼ ðCO þ CR ÞdlOXz þ ðqe nF CR ÞdE þ Cw dlw ; ð7Þ
where, compared with Eq. (1), we have one additional interfacial excess, and are, therefore, one relation short,
R. de Levie / Journal of Electroanalytical Chemistry 562 (2004) 273–276
so that the problem is thermodynamically undefined. We can of course finesse the situation by introducing some additional relation, e.g., by assuming that Eq. (6) still applies, even though that equation was derived specifically for the absence of any charge transfer, and certainly does not apply to, e.g., the case of complete charge transfer, see Eq. (9). Postulating that Eq. (6) is nonetheless valid is clearly an arbitrary assumption, and any results so obtained are similarly arbitrary quantities [6]. This does not mean that partial charge transfer does or does not exist, merely that we do not have thermodynamic means to define and measure it. One might assume that the above is merely a matter of selecting some plausible relation between CI and CId , but the problem appears to be more fundamental than that. Gibbsian thermodynamics recognizes only bulk (i.e., three-dimensional) phases. Even solvent-free, ÔcompactÕ monolayers with a crystal-like organization are fully described in terms of the corresponding bulk components in the aqueous solution [7]. Without such a bulk phase, the ÔlayerÕ of partially discharged anions cannot be described thermodynamically, because there is no corresponding, thermodynamically recognizable component of the system. Thermodynamically, the interface cannot exist without its adjoining bulk phases. Similar restrictions apply to chemisorption and underpotential deposition, even with an integer value of d. This is clearly illustrated with partial charge transfer upon adsorption of a neutral solution species. In that case one need not consider non-contact adsorption, such as that in the Gouy space charge layer. Consider, e.g., mercury in contact with an aqueous solution of a single uncharged species U plus a single, strong 1,1-electrolyte, with a reference electrode responding thermodynamically to, say, the electrolyte anions. In that case we have the Gibbs adsorption equation dr ¼ Cþ d~ lþ þ C d~ l þ Ce d~ le þ Cu dlu þ Cw dlw ; ð12Þ d while partial charge transfer UII þ de I Uads yields the interfacial electroneutrality condition
Cþ C dCd ¼ qe =F ;
ð13Þ
where C refers to the adsorbed electrolyte anion, and Cd to the adsorbed species Ud ads . However, the absence of an appropriate equilibrium expression for the partial d ~U þ charge transfer process UII þ de I Uads (i.e., l d~ le;I ¼ l~??? ) leaves us with terms such as dCd d~ l or dF Cd d/III that cannot be expressed in directly measurable parameters. Finally, any relations used in defining the electrosorption valency c must, of course, include its two limiting cases, d ¼ 0 and d ¼ 1. Using Eq. (6) clearly violates that requirement for d ¼ 0. In the presence of integral (as distinct from partial) interfacial electron transfer, i.e., for d ¼ 0, there is no thermodynamically
275
definable electronic charge density qe , but only the socalled free charge density such as qe þ nF COðwÞ or qe nF CRðwÞ . For all the above reasons, the usual definition of the electrosorption valency oqe ols cF ¼ ¼ ð14Þ oCðwÞ E oE CðwÞ cannot be considered to be a valid part of interfacial thermodynamics, since the term cF implies partial charge transfer, while the relation ðoqe =oCðwÞ ÞE ¼ ðols =oE ÞCðwÞ is derivable only for the explicit absence of charge transfer. A similar conclusion, using somewhat different arguments, was reached much earlier by Damaskin [6] and Frumkin et al. [8,9].
4. The surface layer approach In the above we have used the Gibbs method of treating the interface as a single, sharp boundary. There is the alternative of considering the interface as a distinct phase [10–13], and we will now briefly consider that approach. In that case, Eq. (11) must be replaced by dr ¼ Cþ d~ lþ þ C d~ l þ Cd d~ ld þ Ce d~ le;I þ Cw dlw ;
ð15Þ
where the subscript d denotes the partially discharged iodide. Using similar reasoning as before, i.e., I Id þ ð1 dÞe or d~ l ¼ d~ ld þ ð1 dÞd~ le;I and Cþ C dCd ¼ qe =F we find dr ¼ Cþ dls þ fqe þ ð1 dÞF Cd gdE þ Cw dlw ; where s denotes the salt LiI, so that ofqe þ ð1 dÞF Cd g ols ¼ oCþðwÞ oE CþðwÞ E
ð16Þ
ð17Þ
or dr ¼ ðC þ Cd Þdls fqe þ ð1 dÞF Cd gdEþ þ Cw dlw ð18Þ so that ofqe þ ð1 dÞF Cd ðwÞ g ols ¼ ; oEþ fCðwÞ þCd ðwÞ g ofCðwÞ þ Cd ðwÞ g Eþ ð19Þ where neither (17) nor (19) yields Eq. (14). Instead, both (17) and (19) suggest that, in the case of partial charge transfer, neither qe nor d is experimentally accessible by thermodynamic means, but at best only the composite quantity qe þ ð1 dÞF Cd . This is not surprising, because it fits with its two extremes, d ¼ 0 and d ¼ 1, respectively. For the ideally polarizable electrode, d ¼ 0
276
R. de Levie / Journal of Electroanalytical Chemistry 562 (2004) 273–276
and qe is thermodynamically defined, whereas for the non-polarizable electrode, d ¼ 1 and the thermodynamically defined quantity is qe þ F C . Similarly, the case of the uncharged species U yields oðqe dF Cd ðwÞ Þ ols ¼ ð20Þ oCþðwÞ oE CþðwÞ E which does not lead to Eq. (14) either.
5. Conclusion The electrosorption valency is a mixed metaphor, applying to the presumed presence of (partial) charge transfer the electronic charge density qe that is thermodynamically definable (and, therefore, experimentally accessible through the common electrical measurements of current, voltage and/or time) only in the absence of (partial) charge transfer. This, then, is a Catch 22: with the usual electrical methods one can only determine qe , and specify its derivative ðoqe =oCðwÞ ÞE , in the explicit absence of partial charge transfer, so that these data cannot be used subsequently to specify the amount of any such transfer. There are, of course, many valid concepts and methods that fall outside the range of thermodynamics. Partial charge transfer most likely is such a concept. But the attempt to define it in terms of measured data that are interpreted thermodynamically (e.g., by assigning a numerical value to qe ) is inherently self-contradictory. Consequently the concept will remain qualitative as long as no extra-thermodynamic means for its quantification have been found.
In short, the electrosorption valency ðoqe =oCðwÞ ÞE has no valid scientific interpretation in terms of partial charge transfer, and is, therefore, best left alone.
Acknowledgements The author thanks Prof. Panos Nikitas of the Aristotle University in Thessaloniki, Greece, for providing the material for Eqs. (15)–(20) and Prof. Roger Parsons for helpful comments.
References [1] D.C. Grahame, J. Am. Chem. Soc. 80 (1958) 4201. [2] W. Lorenz, G. Salie, Z. Phys. Chem. 218 (1961) 259; Z. Phys. Chem. NF. 29 (1961) 390, 408. [3] J.W. Schultze, F.D. Koppe, Electrochim. Acta 21 (1976) 327, 337. [4] J.W. Schultze, D. Rolle, Can. J. Chem. 75 (1997) 1750. [5] D. D€ ohnert, J. Koutecky, J.W. Schultze, J. Electroanal. Chem. 82 (1977) 81. [6] B.B. Damaskin, Elektrokhim. 5 (1969) 771. [7] M.H. Saffarian, R. Sridharan, R. de Levie, J. Electroanal. Chem. 218 (1987) 273. [8] A. Frumkin, O. Petrii, B. Damaskin, J. Electroanal. Chem. 27 (1970) 81. [9] A. Frumkin, B. Damaskin, O. Petrii, J. Electroanal. Chem. 53 (1974) 57. [10] J.D. van der Waals, Verh. Koninkl. Akad. Wetensch. Amsterdam Sect. 1 (1893). [11] G. Bakker, Kapillarit€at und Oberfl€achenspannung (Handbook Experimental Physics, vol. 6), Akad. Verlagsges., Leipzig, 1928. [12] J.E. Verschaffelt, Bull. Acad. Belg. Cl. Sci. 22 (1936) 373, 390, 402. [13] E.A. Guggenheim, Trans. Faraday Soc. 36 (1940) 398.
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 562 (2004) 273–276 www.elsevier.com/locate/jelechem
Short Communication
The electrosorption valency and partial charge transfer Robert de Levie
*
Department of Chemistry, Bowdoin College, Brunswick, ME 04011, USA Received 11 February 2003; received in revised form 8 August 2003; accepted 27 August 2003
Abstract The electrosorption valency as usually defined is an extra-thermodynamic and self-contradictory concept. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Thermodynamics; Electrosorption valency; Charge; Charge transfer
When ions adsorb at the electrode solution interface, partial charge transfer is possible, and sometimes even plausible. Grahame [1] had already indicated the close parallel between halide adsorption on mercury and (covalent) chemical bonding. The related concepts of formal partial charge transfer and electrosorption valency were subsequently taken up by Lorenz and Salie [2] and by Schultze and Koppe [3], and their present status was reviewed recently by Schultze and Rolle [4]. These are concepts that properly belong in the domain of quantum mechanics [5]. However, the electrosorption valency c is often suggested to be a thermodynamic property. In this short communication we will consider its thermodynamic validity. In order to do so we will first recall two well-studied cases, viz. those of the ideally polarizable and the nonpolarizable interface. In order to keep the mathematics simple we will in all cases consider the interface between mercury (identified as phase I) and an aqueous solution of a single strong z, z-electrolyte (phase II), with a reference electrode assembly (containing the metal phase III plus, possibly, its own solution phase II0 together with a barrier membrane selectively permeable to the necessary ion) that responds thermodynamically to either cation or anion. We will not take into account the metal wires connecting the cell with the measuring equipment, because the contact potentials between the various metals are constant at constant temperature, *
Tel.: +1-207-725-3028; fax: +1-207-725-3017. E-mail address: [email protected] (R. de Levie).
0022-0728/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2003.08.027
and we will not be concerned here with the absolute values of the cell potential E ¼ /I /III , but only with its changes dE ¼ dð/I /III Þ ¼ d/I d/III , where / is the inner potential. Such changes dE are unaffected by constant contact potentials. Under these conditions, the Gibbs adsorption equation will take the form dr ¼ Cþ d~ lþ þ C d~ l þ Ce d~ le;I þ Cw dlw ;
ð1Þ
where r is the interfacial tension (as distinct from the electrosorption valency c), l denotes the chemical potential, and l~ the electrochemical potential, while the subscripts +, ), e, and w indicate cations, anions, metal electrons, and water molecules, respectively. Where necessary to avoid possible ambiguity, roman numerals are used to identify the phase involved.
1. The ideally polarizable electrode In the absence of charge transfer we convert Eq. (1) into dr ¼ Cþ dls þ qe dE þ Cw dlw
ð2aÞ
or dr ¼ C dls þ qe dEþ þ Cw dlw
ð2bÞ
depending on whether we use a reference electrode responding to the solution anions or cations, respectively. In order to go from Eq. (1) to (2a), (2b) or (3) we use the following relations:
274
R. de Levie / Journal of Electroanalytical Chemistry 562 (2004) 273–276
(a) the electrochemical potential of metal electrons l~e;I ¼ loe;I F /I and the definition qe ¼ F Ce so that Ce d~ le;I ¼ F Ce d/I ¼ qe d/I where qe is the electronic charge density; (b) the equilibrium condition ls ¼ l~þ þ l~ describing the electrolyte dissociation, so that d~ l ¼ dls d~ lþ or d~ lþ ¼ dls d~ l , respectively; (c) the electroneutrality condition Cþ C ¼ Ce =z ¼ qe =zF ; and (d) the Nernst equation for the reference electrode, which for a cation-responsive electrode based on o ~þ;II þ l~þ;II ¼ l~þ;II0 and Mzþ þ ze III MIII is l II0 z~ le;III ¼ l~þ;II0 þ z~ le;III ¼ lM;III or d~ lþ d~ lþ;II ¼ zd~ le;III ¼ zF d/III , while an anion-selective electrode based on, say, l~;II ¼ l~;II0 and MXz;II0 þ o ~e;III ¼ lM;III þ z~ ze l III MIII þ zXII0 yields ls þ l hence d~ l ¼ zF d/III , where the subscript s denotes the poorly soluble salt MXz , so that dls ¼ 0, and likewise dlM;III ¼ 0 for a pure metal phase MIII . It is customary to combine Eqs. (2a) and (2b) into X dr ¼ C dls þ qe dE þ Cu dlu ; ð3Þ u
where either the top or the bottom signs are used. Consequently we find the Lippmann equation or ¼ qe ð4Þ oE P ;T ;l and or x ¼ C Cw ¼ CðwÞ ; ols P ;T ;E ;lu6¼j xw
ð5Þ
where we have used the Gibbs–Duhem–Margules rule ðolj =oli ÞP ;T ;E ;lk6¼i;j ¼ xi =xj where x denotes the mole fraction. Furthermore, since dr in Eq. (3) is a complete differential, we can obtain cross-relations such as oqe ols ¼ : ð6Þ oCðwÞ E oE CðwÞ
where we use the following sequence of substitutions: le;I ¼ l~R hence d~ lR ¼ d~ lO þ nd~ le;I ; the definition l~O þ n~ Ce ¼ qe =F plus d~ le;I ¼ F d/I so that ðnCR þ Ce Þd~ le;I ¼ ðqe nF CR Þd/I ; the salt dissociation equilibrium lOXz ¼ l~O þ z~ l which yields d~ lO ¼ dlOXz zd~ l ; and the electroneutrality condition zCO þ ðz nÞCR C ¼ qe =F .Elimination of d~ lO rather than d~ lR yields the alternative form dr ¼ ðCO þ CR ÞdlRXzn þ ðqe þ nF CO ÞdE þ Cw dlw ð8Þ which shows that now only the linear combinations CO þ CR and qe þ nF CO are thermodynamically defined. One of these can be replaced by qe nF CR ¼ ðqe þ nF CO Þ nF ðCO þ CR Þ. In aqueous solution, the typical experimental results will, therefore, be COðwÞ þ CRðwÞ and either qe þ nF COðwÞ or qe nF CRðwÞ . The electronic charge density qe per se is no longer thermodynamically defined. We can also find cross-correlations at constant pressure and temperature, such as oðqe þ nF COðwÞ Þ ols ¼ ; ð9Þ oE ðCOðwÞ þCRðwÞ Þ oðCOðwÞ þ CRðwÞ Þ E where s denotes the salt RXzn , or equivalent results based on the salt OXz . We can consider the redox couple Mnþ þ ne Mo a special case of (7) or (8) for which the reduced form is M, so that dlRzn ¼ dlM ¼ 0, while CO ¼ Cþ , in which case one obtains dr ¼ ðqe þ nF Cþ ÞdE þ Cw dlw ¼ ðqe þ nF CþðwÞ ÞdE
ð10Þ
from which no cross relations can be derived. The above results are readily obtained and well known, and are summarized here only to re-emphasize that many results for the ideally polarized electrode are not valid in the presence of interfacial electron transfer, but must be modified to reflect such transfer.
3. Partial charge transfer
2. The non-polarizable electrode In the case of a non-polarizable electrode the metaljsolution interface IjII contains an equilibrium redox couple. We will here consider the redox equilibrium Ozþ þ ne RðznÞþ where both redox components OXz and RXzn are strong electrolytes with a common anion, X . The Gibbs adsorption equation for the IjII interface then reads lO þ CR d~ lR þ C d~ l þ Ce d~ le;I þ Cw dlw dr ¼ CO d~
We now consider partial charge transfer, in which one surmises that, on adsorption, an anion such as I may lose part of its negative charge. Say that we consider mercury in contact with an aqueous solution of LiI, and assume that the iodide in contact with the electrode has a partial charge d, where 0 6 d 6 1, so that the Gibbs adsorption equation reads dr ¼ CLiþ d~ lLiþ þ ðCI þ CId Þd~ lI þ Ce d~ le þ Cw dlw ; ð11Þ
¼ ðCO þ CR ÞdlOXz þ ðqe nF CR ÞdE þ Cw dlw ; ð7Þ
where, compared with Eq. (1), we have one additional interfacial excess, and are, therefore, one relation short,
R. de Levie / Journal of Electroanalytical Chemistry 562 (2004) 273–276
so that the problem is thermodynamically undefined. We can of course finesse the situation by introducing some additional relation, e.g., by assuming that Eq. (6) still applies, even though that equation was derived specifically for the absence of any charge transfer, and certainly does not apply to, e.g., the case of complete charge transfer, see Eq. (9). Postulating that Eq. (6) is nonetheless valid is clearly an arbitrary assumption, and any results so obtained are similarly arbitrary quantities [6]. This does not mean that partial charge transfer does or does not exist, merely that we do not have thermodynamic means to define and measure it. One might assume that the above is merely a matter of selecting some plausible relation between CI and CId , but the problem appears to be more fundamental than that. Gibbsian thermodynamics recognizes only bulk (i.e., three-dimensional) phases. Even solvent-free, ÔcompactÕ monolayers with a crystal-like organization are fully described in terms of the corresponding bulk components in the aqueous solution [7]. Without such a bulk phase, the ÔlayerÕ of partially discharged anions cannot be described thermodynamically, because there is no corresponding, thermodynamically recognizable component of the system. Thermodynamically, the interface cannot exist without its adjoining bulk phases. Similar restrictions apply to chemisorption and underpotential deposition, even with an integer value of d. This is clearly illustrated with partial charge transfer upon adsorption of a neutral solution species. In that case one need not consider non-contact adsorption, such as that in the Gouy space charge layer. Consider, e.g., mercury in contact with an aqueous solution of a single uncharged species U plus a single, strong 1,1-electrolyte, with a reference electrode responding thermodynamically to, say, the electrolyte anions. In that case we have the Gibbs adsorption equation dr ¼ Cþ d~ lþ þ C d~ l þ Ce d~ le þ Cu dlu þ Cw dlw ; ð12Þ d while partial charge transfer UII þ de I Uads yields the interfacial electroneutrality condition
Cþ C dCd ¼ qe =F ;
ð13Þ
where C refers to the adsorbed electrolyte anion, and Cd to the adsorbed species Ud ads . However, the absence of an appropriate equilibrium expression for the partial d ~U þ charge transfer process UII þ de I Uads (i.e., l d~ le;I ¼ l~??? ) leaves us with terms such as dCd d~ l or dF Cd d/III that cannot be expressed in directly measurable parameters. Finally, any relations used in defining the electrosorption valency c must, of course, include its two limiting cases, d ¼ 0 and d ¼ 1. Using Eq. (6) clearly violates that requirement for d ¼ 0. In the presence of integral (as distinct from partial) interfacial electron transfer, i.e., for d ¼ 0, there is no thermodynamically
275
definable electronic charge density qe , but only the socalled free charge density such as qe þ nF COðwÞ or qe nF CRðwÞ . For all the above reasons, the usual definition of the electrosorption valency oqe ols cF ¼ ¼ ð14Þ oCðwÞ E oE CðwÞ cannot be considered to be a valid part of interfacial thermodynamics, since the term cF implies partial charge transfer, while the relation ðoqe =oCðwÞ ÞE ¼ ðols =oE ÞCðwÞ is derivable only for the explicit absence of charge transfer. A similar conclusion, using somewhat different arguments, was reached much earlier by Damaskin [6] and Frumkin et al. [8,9].
4. The surface layer approach In the above we have used the Gibbs method of treating the interface as a single, sharp boundary. There is the alternative of considering the interface as a distinct phase [10–13], and we will now briefly consider that approach. In that case, Eq. (11) must be replaced by dr ¼ Cþ d~ lþ þ C d~ l þ Cd d~ ld þ Ce d~ le;I þ Cw dlw ;
ð15Þ
where the subscript d denotes the partially discharged iodide. Using similar reasoning as before, i.e., I Id þ ð1 dÞe or d~ l ¼ d~ ld þ ð1 dÞd~ le;I and Cþ C dCd ¼ qe =F we find dr ¼ Cþ dls þ fqe þ ð1 dÞF Cd gdE þ Cw dlw ; where s denotes the salt LiI, so that ofqe þ ð1 dÞF Cd g ols ¼ oCþðwÞ oE CþðwÞ E
ð16Þ
ð17Þ
or dr ¼ ðC þ Cd Þdls fqe þ ð1 dÞF Cd gdEþ þ Cw dlw ð18Þ so that ofqe þ ð1 dÞF Cd ðwÞ g ols ¼ ; oEþ fCðwÞ þCd ðwÞ g ofCðwÞ þ Cd ðwÞ g Eþ ð19Þ where neither (17) nor (19) yields Eq. (14). Instead, both (17) and (19) suggest that, in the case of partial charge transfer, neither qe nor d is experimentally accessible by thermodynamic means, but at best only the composite quantity qe þ ð1 dÞF Cd . This is not surprising, because it fits with its two extremes, d ¼ 0 and d ¼ 1, respectively. For the ideally polarizable electrode, d ¼ 0
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and qe is thermodynamically defined, whereas for the non-polarizable electrode, d ¼ 1 and the thermodynamically defined quantity is qe þ F C . Similarly, the case of the uncharged species U yields oðqe dF Cd ðwÞ Þ ols ¼ ð20Þ oCþðwÞ oE CþðwÞ E which does not lead to Eq. (14) either.
5. Conclusion The electrosorption valency is a mixed metaphor, applying to the presumed presence of (partial) charge transfer the electronic charge density qe that is thermodynamically definable (and, therefore, experimentally accessible through the common electrical measurements of current, voltage and/or time) only in the absence of (partial) charge transfer. This, then, is a Catch 22: with the usual electrical methods one can only determine qe , and specify its derivative ðoqe =oCðwÞ ÞE , in the explicit absence of partial charge transfer, so that these data cannot be used subsequently to specify the amount of any such transfer. There are, of course, many valid concepts and methods that fall outside the range of thermodynamics. Partial charge transfer most likely is such a concept. But the attempt to define it in terms of measured data that are interpreted thermodynamically (e.g., by assigning a numerical value to qe ) is inherently self-contradictory. Consequently the concept will remain qualitative as long as no extra-thermodynamic means for its quantification have been found.
In short, the electrosorption valency ðoqe =oCðwÞ ÞE has no valid scientific interpretation in terms of partial charge transfer, and is, therefore, best left alone.
Acknowledgements The author thanks Prof. Panos Nikitas of the Aristotle University in Thessaloniki, Greece, for providing the material for Eqs. (15)–(20) and Prof. Roger Parsons for helpful comments.
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