2838
J. PhyS. Chem. 1982, 86, 2838-2840
Congruence of Electrosorption with respect to the Electrical Variable David M. Mohllner" and Marian Karolcrak* Department of chemlsfry, Colorado State Unlvwsliy, Fort ColNns. Colorado 80523 (Received: March 17, 1981; In Final Form:January 2 1, 1982)
The concept of congruence of an organic electrosorption isotherm with respect to the electrical variable (electrode potential or excess charge density on the metal surface) which has until now been viewed as a kind of ideal behavior which can be approximated by real systems is, strictly speaking, invalid. It is proved that congruence is both the necessary and sufficient condition that the activity coefficients of the adsorbed species in the inner layer are independent of the magnitude of the electric field there. This proof is based on the theory of Mohilner, Nakadomari, and Mohilner (MNM), originally published in 1977.
Introduction The idea of congruence of organic electrosorption isotherms with respect to the electrical variable (electrode potential, E, or excess charge density, u, on the metal surface) is that if only the correct electrical variable is held constant the shape of the isotherms plotted against the logarithm of the activity QA, of the organic compound in the bulk solution will remain invariant for all constant values of that electrical variable. Thus,for example, if the isotherms are congruent with respect to electrode potential, the isotherms for all different constant values of electrode potential may be superimposed by translation along the In aA axis, i.e., the isotherms are congruent geometrical figures. A similar statement would hold if the isotherms were congruent with respect to excess charge density. The concept is actually very old. In fact the idea of congruence with respect to electrode potential appeard implicitly in In Frumkin's 1925 publication of his famous i ~ o t h e r m . ~ 1955 Parsons4 proposed that the excess charge density was a better choice for the electrical variable and thereupon a vigorous debate in the literature"33 began over whether
(1) To whom correspondence and request for reprints should be addressed at the Department of Chemistry. (2)Postdoctoral reaearch aeeociate on leave from Uniwersytet im. Adama Mickiewicza, Zaklad Elektrochemii, Instytut Chemii, PoznaA, Poland. (3)A. N.Frumkin, 2.Phys. Chem., 116,466 (1925). (4)R. Parsons, Trans. Faraday SOC.,51, 1518 (1955). (5)B. B. Damaskin, Electrochim. Acta, 9,231 (1964). (6)B. B.Damaskin, J.Electroanal. Chem., 7,155 (1964). (7)A. N. Frumkin and B. B. Damaskin in "Modern Aspects of Electrochemistry", Vol. 3, J. O'M. Bockris and B. E. Conway, Ed., Butterworths, Washington, DC, 1964, Chapter 3. (8)B. B. Damaskin and G. A. Tedoradze, Electrochim. Acta, 10,529 (1964). (9)A. N. Frumkin, B. B. Damaskin, and A. A. Survila, J.Electroanal. Chem., 16, 493 (1968). (IO) B. B. Damaskin, Elektrokhimiya, 5 , 771 (1969). (11) B. B. Damaskin, Elektrokhimiya, 6,1135 (1970). (12)B.B. Damaskin, A. N. Frumkin, and A. Chizov, J. Electroanal. Chem., 28,93 (1970). (13)B. B. Damaskin, 0. A. Petrii, and V. V. Batrakov, "Adsorption of Organic Compounds on Electrodes", Plenum Press, New York, 1971. (14)B. B. Damaskin, and V. A. Vosekalis, Elektrokhimiya, 11, 253 (1975). (15)B. B. Damaskin and Yu. N. Kuryakov, Elektrokhimiya, 13,98 (1977). (16)B. N.Afanasiev, B. B. Damaskin, G. I. Avilova, and N. A. Borisova, Elektrokhimiya, 11, 593 (1975). (17)R.S.Hansen, D. J. Kelsch, and D. V. Grantham, J.Phys. Chem., 67,2316 (1963). (18)D. E.Broadhead, R. S. Hansen, and G. W. Potter, Jr., J. Colloid Interface Sci., 31,61 (1969). 0022-3654/82/2086-2838$01.25/0
congruence with respect to electrode potential or excess charge density was a better approximation to the real behavior of organic sorbates on electrodes. Various ways to test experimental electrosorption data for congruence were devised in the cited literature, some being more sensitive than others. The results of the numerous experimental testa were that some compounds appeared to more closely approximate to congruence with respect to one of the electrical variables, some compounds appeared to be congruent with respect to neither variable, and some appeared to be congruent with respect to both. The general opinion appears to have been that congruence is at best an approximation to reality but that it is a useful concept in that it represents a kind of ideal behavior. However, in all the literature published on the subject the basic concept of congruence has never been considered from a really fundamental viewpoint. That is the purpose of this paper. We shall prove that in fact the concept implies that the activity coefficientsof the adsorbed species are independent of the electric field in the inner layer. This proof will be based on the theory of noncongruent electrosorption of organic compounds originally proposed in 1977 by Mohilner, Nakadomari, and Mohilnel.34 (MNM) which we have recently shown35includes as a special case the most general theory of Damaskin.'lJ2J5 Congruence with respect to Electrode Potential Based on the MNM theory we have shown35that the (19)K. G.Baikerikar and R. S. Hansen, J. Colloid Interface Sci., 52, 277 (1975). (20)K. G.Baikerikar and R. S. Hansen, Surface Sci., 50,527(1975). (21)D. E. Broadhead, K. G. Baikerikar, and R. S. Hansen, J. Phys. Chem., 80,370 (1976). (22)R.Parsons, Proc. R. SOC.(London),Ser. A , 261, 79 (1961). (23)R.Parsons, J.Electroanal. Chem., 7,136 (1964). (24)R. Parsons, J. Electroanal. Chem., 8, 93 (1964). (25)E. Dutkiewicz and R. Parsons, J. Electroanal. Chem., 11, 196 (1966). (26)R. Parsons and F. G. R. Zobel, Trans. Faraday Soc., 62,3511 (1966). , ~ - I
(27)E. Dutkiewicz, J. D. Garnish, and R. Parsons, J. Electroanal.
Chem., 16,505 (1968).
(28)M. A. V. Devanathan, Proc. R. SOC.(London),Sei-. A, 264, 133 (1961). (29)M. A. V. Devanathan, Proc. R. Soc. (London),Ser. A, 267, 133 (1962). (30)J. O'M. Bockris, M. A. V. Devanathan, and K. Miiller, Proc. R. SOC.(London),Ser. A , 274,55 (1963). (31)K. Doblhofer and D. M. Mohilner, J . Phys. Chem., 75, 1698 (1971). (32)H. Nakadomari, D. M. Mohilner, and P. R. Mohilner, J. Phys. Chem., 80,1761 (1976). (33)R. Bennes, J. Electroanal. Chem., 105,85 (1979). (34)D. M. Mohilner, H. Nakadomari, and P. R. Mohilner, J. Phys. Chem., 81,244(1977). (35)M. Karolczak and D. M. Mohilner, J. Phys. Chem., following article in this issue.
0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 75, 1982 2839
Eiectrosorption/Electrical Variable Congruence
generalized equation for the excess charge density at constant electrode potential is d In p a' - a = -RTI',O 7
+
gruence of the electrosorption isotherm with respect to electrode potential is the necessary and sufficient condition are all independent of electrode that the coefficients (a,) potential. But according to the MNM theory,%the activity coefficient of the adsorbed organic compound yA* (based on the symmetrical choice of standard states32)is given by k
In yAads= C (1- j)aj(3tAads)' In eq 1, ao is the excess charge density at the specified electrode potential in the absence of organic compound and a is the excess charge density at the same electrode potential in the presence of organic compound. R is the gas constant. Tis the absolute temperature. I', is the maximum value of the surface concentration of the organic compound when the electrode is fuUy covered, and 0 is the fractional surface coverage. n is the number of adsorbed water molecules replaced by one adsorbing organic molecule. p is related to the standard electrochemical free energy of adsorption, AGads', based on the symmetrical choice of standard states34by the equation
p = f?Xp(-AGad:/RT)
(2)
AGE is the excess electrochemical free energy of mixing of the inner layer (surface solution34)which according to the MNM theory is a polynomial of degree k in the mole fraction of the adsorbed organic compound in the inner layer, ~ A a d s ,i.e. k
AGE = R T z ~ i j ( 3 t A ~ ~ ~ ) J j= 1
(3)
In general, the set of coefficients (aj)of this polynomial depends on the electrical state of the system, i.e., on E or on u. If we assume that all aj are independent of the electrode potential ( d a j / d E ) T , p = 0 for all j
RTI',O (d In P/dE)
When 0 = 1, a = a', and eq 5 becomes a o - a' = -RTI',(d In p / d E )
and the activity coefficient of the adsorbed water, yWads, is given by k
In ywads =
+ a'O
(5)
(6)
(7) In 1971 Doblhofer and Mohilner31proved thermodynamically that eq 7 is the sufficient condition for congruence with respect to electrode potential, i.e., if eq 7 is true the electrosorption isotherm must be congruent with respect to electrode potential. Thus it follows that the set of coefficients (ajj in eq 1 being independent of electrode potential is a sufficient condition for congruence of the electrosorption isotherm with respect to electrode potential. On the other hand, if we assume that the electrosorption isotherm is congruent with respect to electrode potential then eq 7 must be true because it was proven thermodynamically by Parsons36in 1959 that eq 7 is the necessary condition for congruence with respect to electrode potential. But the only way eq 7 can be true is that the set of coefficients (ai)in eq 1 be independent of electrode potential. Thus the set of coefficients (aj)being independent of electrode potential is the necessary and sufficient condition that the electrosorption isotherm is congruent with respect to electrode potential. Therefore, likewise, con(36) R. Parsons, Trans. Faraday SOC.,55, 999 (1959).
j=Z
(I - j ) a j ( ~ A a d s ) j
(9)
Therefore it follows that at constant composition in the inner layer, i.e., constant xAads,congruence with respect to the electrode potential is a necessary and sufficient condition that the activity coefficients of the adsorbed species are independent of electrode potential. But if the activity coefficients are independent of electrode potential they are independent of the electric field in the inner layer since the electric field depends on the electrode potential. Congruence with respect to Excess Charge Density Based on the MNM theory we have derived35the generalized equation for the surface pressure, a = p - t at constant excess charge density. t = y + aE where y is the interfacial tension of the electrode. is the value of t in the absence of organic compound at the specified value of a. The equation for the surface pressure is In (1- xAad8)+
- nr,AGE
+ nRTI',
In
In eq 10 aw is the activity of water in the bulk solution containing organic compound and aweis the activity of water in the absence of organic compound in the solution containing the electrolyte at the same activity as in the solution with organic compound. From the definition of f' it follows3' that
(at/
Dividing eq 5 by eq 6 and rearranging we obtain a = aO(l- 0)
(8)
(4)
and eq 1 reduces to &'-a=-
j=2
k
+ jCjaj(xAada)j-l =1
T,p,r
=E
(11)
where the subscript p implies constant composition. Therefore if we differentiate eq 10 with respect to a at constant composition and substitute35
we obtain d In /3 Eo - E = R T I ' , O y -
where EO is the value of the electrode potential at the specified value of a in the absence of organic compound and E is the value of the electrode potential in the presence of organic compound at the same value of a. If we assume all aj are independent of excess charge density ( d f f j / d a ) T , p= 0 for all j (14) and eq 1 reduces to (37) D. M. Mohilner in "Electroanalytical Chemistry", Vol. 1, A. J.
Bard,Ed., Marcel Dekker, New York, 1966,p 292.
J. phys. Chem. 1982. 86, 2840-2845
2840
Eo - E = R7Tm6(d In @/da) When t9 = 1, E = E', and
EO - E'
= R7Tm(d In @/dc)
(15) (16)
Dividing eq 15 by eq 16 and rearranging we obtain
E = EO(I - e)
+ EV
(17) Doblhofer and MohilneP' proved that eq 17 is a sufficient condition for congruence with respect to excess charge being indedensity. Therefore the set of coefficients {aj) pendent of excess charge density is a sufficient condition for congruence with respect to excess charge density. If, on the other hand, we assume that the isotherm is congruent with respect to excess charge density eq 17 must be true because Parsons36proved that it is the necessary condition for congruence with respect to excess charge density. But the only way eq 17 can be true is that the in eq 9 is independent of excess set of coefficients (aj) charge density. Thus the set of coefficients (aj) being independent of excess charge density is the necessary and sufficient condition that the electrosorption isotherm is congruent with respect to excess charge density. Therefore, likewise, congruence with respect to excess charge density is the necessary and sufficient condition that the coefficients (aj) are all independent of excess charge density. But according to the MNM theory% this implies that at constant compoeition in the inner layer the activity coefficients of the adsorbed species are independent of excess charge density (cf. eq 8 and 9). Therefore it follows that, at constant composition in the inner layer, congruence with respect to excess charge density is the necessary and sufficient condition that the activity coefficients of the adsorbed species are independent of excesa charge density and therefore independent of the electric field in the inner
layer because the magnitude of the electric field depends on the excess charge density. Conclusions We have given a proof on the basis of the MNM theory35 that congruence of the electrosorption isotherm with respect to either electrode potential or excess charge density is the necessary and sufficient condition that the activity coefficients of the adsorbed molecules in the inner layer are independent of the electric field. Since the activity coefficients depend on the intermolecular interaction, it follows that congruence actually implies that the intermolecular interactions in the inner layer are independent of the electric field. This seems highly unlikely because one component of the intermolecular interaction is the induced dipole-induced dipole interactions and the magnitude of the induced dipole moments of the molecule depends on the magnitude of the electric field. I t is of course possible that the activity coefficients of some molecules might have a slight dependence on the electric field and this might conceivably account for why congruence has appeared to be valid in some cases. Acknowledgment. This work was supported by the National Science Foundation under Grant CHE76-83432. Acknowledgment is also made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. This work constitutes a portion of the research in a collaborative international program on thermodynamic studies of the inner part of the electrical double layer supported in part by the North Atlantic Treaty Organization (NATO). M.K. expresses his gratitude to the Department of Chemistry, Colorado State University for providing facilities for this work.
Theory of Noncongruent Electrosorptlon of Organic Compounds Marlan Karolcrak' and Davld M. Mohllner'2 Department of Chemistry, Colorado State Universlly, Fort ColWns, Colorado 80523 (Received: March 17, 1981; I n Final Form: January 21, 1982)
The theory proposed in 1977 by Mohilner, Nakadomari, and Mohilner (MNM) on noncongruent electrosorption of organic compounds is in fact much more general than had originally been thought. In this paper it is shown that the most general theory of Damaskin results from the MNM theory when three simplifying assumptions are made. Implications of thisfact are discussed. In addition, generalized equations of state at constant electrode potential and at constant excess charge density based on the MNM theory are derived. Also a generalized equation for the excess charge density based on the MNM theory is derived. A generalized expression for the electrosorptionisotherm expressed as a function of the excess electrochemicalfree energy of mixing of the inner layer is derived. Introduction In 1977 Mohilner, Nakadomari, and Mohilner3 (MNM) published a new theory of noncongruent electrosorption of organic compounds in the absence of specific ionic ad(1) Postdoctoral research associate on leave from Uniwersytet im. Adama Mickiewicza, Zaklad Elektrochemii, Instytut Chemii, Pozn&, Poland. (2) To whom correspondence and requests for reprints should be addressed at the Department of Chemistry. (3) D. M. Mohilner, H. Nakadomari and P. R. Mohilner, J. Phys. Chem., 81, 244 (1977). 0022-3654/82/2086-2840$0 1.2510
sorption based on a model of the inner layer as a twocomponent nonelectrolyte solution called the surface solution. This theory makes it possible to calculate from experimental electrosorption-data the excess electrochemical free energy of mixing, AGE, of the surface solution and the activity coefficients and activities of both the organic compound and of the water adsorbed in the inner layer. It was shown in that original paper than the famous Frumkin isotherm results as a special case of the MNM theory. We have now investigated further the implications of the MNM theory and we shall show in this paper that 0 1982 American Chemical Society
