J. Phys. Chem. 1994,98, 6577-6585
6577
A New Approach to Development of Ionic Isotherms of Specific Adsorption in the Electrical Double Layer P. Nikitas Laboratory of Physical Chemistry, Department of Chemistry, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece Received: December 28, 1993; In Final Form: March 21, 19946
A new approach, which combines classical thermodynamics with basic electrostatic arguments, is proposed for the development of new ionic isotherms of specific adsorption on ideally polarized electrode surfaces. The main features of this approach are the following. The inner layer is considered to consist of two zones, along which the electric field is homogeneous. Then the Gauss theorem of electrostatics is used to correlate the potential drop across the inner layer with the electrode charge density. This correlation allows the derivation of the electrochemical potentials of the constituents of the inner layer based exclusively on the properties of certain thermodynamic functions of this layer. The adsorption isotherms, which arise from the electrochemical potentials, can take into account changes in the dielectric constant, the inner layer thickness, and the orientation of the solvent dipoles during the adsorption process. Comparison with experiment shows that even the simplest isotherm, which takes into account only variations of the dielectric constant, can give an almost quantitative description of experimental isotherms.
I. Introduction The significance of the isotherm in studies of adsorption was pointed out long ago, and for this reason many studies have been made in this field.'-" The main effort concerns adsorption isotherms of neutral organic adsorbates, whereas considerably little is the work on ionic isotherms of specific adsorption on electrodesurfaces. The work on the latter subject starts with the attempts of Levine, Bockris, and their colleagues to model the inner layer in the presence of specifically adsorbed ion^.^^^ A complete review of these models is given in ref 5 . Their basic feature is that the inner layer is treated independently of the rest of the electrical double layer. The various particleparticle and particlefield interactions are taken into account directly by means of the electrostatic theory and/or statistical mechanics, adopting certain approximations. The work of Levine et al. is based on the formal separation of the electrical double layer into an inner and a diffuse part. In contrast, the recent attempts are directed toward the description of the electrical double layer as a single entity, an approach which makes difficult the treatment of the specific adsorption of ions. For this reason, the work which has already appeared on this topic is still at the beginning.l* In the present paper we propose a new approach for the development of new ionic isotherms of specific adsorption on ideally polarized electrode surfaces. This approach is based on a certain but generalmodel of the inner layer. Then a combination of the Gauss theorem of electrostatics with the properties of the thermodynamic functions of this layer allows the derivation of the electrochemical potentials of the constituents of the inner layer without additional assumptions and approximationsin what concerns significant particleparticle and particlefield interactions. Thus, the adsorption isotherms which arise, despite their simplicity, are not subject to crude approximations. 11. Expressions for Chemical Potentials
Basic Electrostatic Equations. The traditional approach in treating the electrical double layer is in fact based on the work of Stern and consists of the formal separation of the double layer into two regions: the inner region, which extends from the Abstract published in Aduance ACS Abstracts, June 1, 1994.
0022-3654/94/2098-6577S04.50/0
electrode surface up to the outer Helmholtz plane (OHP), and the diffuse layer, which is the region from the OHP out to the bulk of the electrolyte s01ution.l~ In the presence of specifically adsorbed ions, the locus of their centers determines the inner Helmholtzplane (IHP). This generalsubdivision of the electrical double layer is adopted in the present treatment. Note that the existence of an inner layer can hardly be disputed, although it has been questioned whether the independent treatment of the inner and diffuse layers can lead to an acceptable theory of the electrical double layer.20-22 In the absence of specifically adsorbed ions the electric field across the entire inner layer can be considered homogeneous. This assumption is no longer valid when ions are present in the inner layer. In this case we can distinguish two subregions in which the electric field may be considered homogeneous: the inner zone extending from the electrode surface to the IHP and the outer zone from the IHP to the OHP.9 Thus the general model we adopt to treat the inner layer is that depicted schematically in Figure 1. Suppose that an inner layer with volume Y and area A is composed of Ni adsorbed ions i and Ns solvent molecules S.If II is the thickness of the inner zone, 12 is that of the outer zone, and the Gauss theorem of electrostatics is applied over two closed surfaces around the electrode surface and the OHP respectively, we obtain
+ Q'
eo(qM - qi)A/Il= QM q,(qd- cp')A/I,=
+ Q"= -Q" - e'- Q'
(1)
(2)
where @, d, and 4 are the inner potentials at the electrode surface, the IHP, and the OHP, respectively, Q M is the total electrode charge, Q is the charge of the adsorbed ions, Q'is the induced charge on the surface of the inner layer which is adjacent to the electrode surface, Q"(= -Q? is the induced charge on the other side of the inner layer, and to is the permittivityof a vacuum. The induced charge Q' may be expressed in terms of the molecular properties of the constituents of the inner layer as follows. If P is the total polarization of the adsorbed layer, we have23
P = P,, - Pa = VQ'/A = IQ' 0 1994 American Chemical Society
(3)
Nikitas
6578 The Journal of Physical Chemistry, Vol. 98, No. 26, 199’4 I
-
l
Then the surface coverage 0 due to the specificadsorption of ions is given by
l
0 = nNi/M
(10)
and substitution of eqs 8-10 into eq 7 yields
+
+ f3zie0(M/nA)lz/eoe
Acp = aMl/eoc (1 - O)Xs(M/A)/eoc
(1 1) UM
Ui
Ud
rpM
‘pi
rpd
Here, aM = QM/A is the electrode charge density and e is the “distortional” dielectric constant of the inner layer defined from
Figurel. Schematicmodeloftheinnerlayer in thepresenceofspecifically adsorbed ions of charge density d.
where P,,is thedipole polarizationdue toorientationofpermanent dipoles and Pa is the induced polarization caused by translation effects, If Acp = (PM - (B is the potential drop across the entire inner layer and 1 its thickness, the induced polarization Pa may be expressed as23
Pa = aiNiAcp/l
+ a,N,Acp/l
(4)
where ai is the polarizability of the ions and as is that of the solvent molecules. We should note that this expression of Pa is approximate, since it assumes a uniform field across the inner layer with strength equal to Aqfl. The dipole polarization P,, can be written as23
P,, = X,N, where AS is the average value of the solvent permanent dipole vector normal to the electrode surface. From the electrostatic theory of dielectrics, as well as from simple molecular models, we know that AS is constant and independent of the electric field when the adsorbed solvent molecules retain a certain orientation on the electrode surface. When the solvent molecules reorientate on the electrode surface, then we can distinguish two cases: The reorientation leads to certain distinct polarization states of the adsorbed species on the electrode surface, or it may proceed continuously. The distinct states of an adsorbed molecule can be treated as different adsorbed species, each one having a constant orientation. In what concernsthe case of a continuous reorientated molecule, AS may be expressed, in general, as a power series of Aqll.23 However, in a recent study we have shown that the assumption of continuously reorientated solvent molecules leads to predictions which are at variance with experiment.24 For this reason this case will not be considered in the present treatment. Therefore, the induced charge Q’ may be calculated from
Q’= hsNs/l - asNSAcp/l2 - aiNiAcp/12
(6)
which in combination with eqs 1 and 2 yields
QM= coAcpA/l - zieoNi12/l- X,N,/l
+ asN,Acp/12 + cxiNiAcp/12 (7)
where zi is the charge number of the ions and P is the proton charge. Using Euler’s theorem,25 the area A may be written as A = AiNi
+ ASN,
(8)
where Ai is the partial surface area of the ions and As is that of the solvent S. We define the total number of lattice sites M a t the inner layer and the size ratio parameter n by means of the following equations
M = A/A,
and n = A i / A S
(9)
e
= eie + cs(i
- e)
(12)
where26 ei = 1
+ ai/c0lAi,
e,
=1
+ as/eolAs
(13)
Equations 7 and 11 allow the interconnection of the two electrical variables, QM (or uM) and Acp, of the inner layer. Thermodynamic Functions of the Inner Layer. If we adopt Guggenheim’smodel for interfa~a,2’-~9then the total differential of the internal energy of the inner layer may be expressed as d U = T d S - p d V + y dA
+ d V ’ + pi dNi + ps dN,
(14)
where the infinitesimal work due to the electric field dCF1is given by30 d V ’ = JA cp d a dA
+ J,cp
d a dA = cpM d p
+ cpd dQd + $ d Q (15)
If this expressionofde is introduced intoeq 14and thedefinition of the electrochemical potential of ions is taken into account
-
pi =
+ zieOcpi
we obtain d U = T d S -p dV+ A c p d p
+ (ps + yA,) dN, + (Gi - zieoqd+ yAi) dNi (17)
Two other thermodynamic functions, which will be found useful in our treatment, arise directly from this equation:
+ + + + + = S d T + Vdp - Q“ dAcp + (Gi - zie0$ + yAi) dNi + (p, + yA,) dN,
dG = d ( U + p V - TS) = S d T + Vdp A c p d p (Gi - zieocpd yAi) dNi (p, yA,) dN, (18) d(G - A&)
(19)
We should point out that, according to the analysis presented in ref 3 1, these expressions of dU, dH, and d(H - A@) are not affected by the interlayer interactions of the inner layer with the diffuse one. That is, the formal separation of the electrical double layer into an inner and diffuse part does not influence the validity Of e q ~ 17-19. Chemical Potentials of the Adsorbed Species. The chemical potentials of the constituents of the inner layer can be expressed in terms of either the electrode charge density U M or the potential drop Acp. These expressions may be found as follows. Applying the usual cross-differentiation rules for a complete differential to eq 19, we obtain
Ionic Isotherms in the Electrical Double Layer
The Journal of Physical Chemistry, Vol. 98, No. 26, 1994 6579
which, after integration along a path of constant temperature, pressure, and composition, results in
The dependence of QMupon Ni is given by eq 7. Thus, assuming a constant position of the inner and outer Helmholtz planes, substitution of eq 7 into eq 21 yields
ri = k* - TA, + zieOqd+ z i e 0 ~ ~ 1 2t/ol t- i ~ ' p 2 ~ i / 2(22) 1 Similarly we find p,
= ps* - y A ,
+ XsA'p/l - tocsA(p2As/21
(23)
Here, pi* and ps* are the chemical potentials of the ions and the solvent in the absence of electric field and changes in the area A. Therefore, we may write
k* = p /
+ kT lnJO
and ps* = p:
+ kT In&(
where
which in combination with eq 7 yields
ji= pi** - TA, + zieopd+ (zieol,- geic,,Ai)uM/coc(cico1Ai/2)(aM/e0c)2 (26) and
= PS** - y A ,
+ (x,- g ~ s e , , A s ) ~ / E o C(csco~A,/2)(~M/c,42 (27)
where g = (1 - O)X,(M/A)/t,t
+ Ozieo(M/nA)l2/c0c (28)
Note that pi** and ps** are the chemical potentials of the ions and the solvent molecules when the electrode charge density is equal to zero. Therefore, they do not coincide with the corresponding quantities of eqs 22 and 23. However, they can still be expressed by eqs 24 but with different activity coefficients. 111. Isotherms for Ionic Adsorption
General Expressions. The electrochemicalpotential of the ions and the chemical potential ps of the solvent molecules is uniform throughout the electrical double layer, leading to the following equilibrium equation
ji- np, = p:
- np;
= pyb- np$b
+
(nX, - zie012)A9/lkT (ei - cs)conAcp2/21(M/A)kT(30)
1 - 6) (24)
where p? and ps0 are the values of pi* and ps* in the standard state, and fi and fs are the activity coefficients of the adsorbed ions and solvent molecules, respectively. We should clarify the difference between pi* in eq 21 and pi in eq 16. pi is the electrochemicalpotential of ions when (B = 0. In contrast, pi* is the electrochemicalpotential when A9 = fl = (B = 0 and in the absence of changes in the area A, i.e, when the adsorbed layer is considered a bulk phase. Expressions of the chemical potentials in terms of uMcan be similarly obtained if we apply cross-differentiation to eq 18. We have
P,
where the superscript (b) denotes the bulk solution. At this point it is worth noting the following. Due to the uniformity of c(i and ps eq 29 is valid for every choice of n. However, only if we chose n = Ai/As, the undesirable terms yAi and yAs in eqs 22 and 23 are deleted from theadsorption isotherm. In contrast, expressions similar to eq 29, like eq 8 in ref 13, which are based on the treatment of the adsorbed layer as a bulk solution, are valid only when n = Ai/As, because in this case the chemical potentials are not uniform throughout the ~ystem.28J2-3~ Now the adsorption isotherm in terms of A9 is obtained by substitution of eqs 22 and 23 into eq 29 and may be expressed as
+ kT ln{aib/(asb)") (29)
Here, we must point out that, within the frames of the approximations adopted in the derivation of eq 30, the only contribution to the activity mfficientsfi andfs arises from shortrange lateral interactions between the adsorbed particles and long-range lateral repulsion between the adsorbed ions. Indeed, let us consider the following three types of long-rangeCoulombic interactions: dipoldipole, dipole-ion, and interactions of ions with their images. The interaction of a solvent dipole with the rest of the solvent dipoles is in fact the interaction energy of this dipole with the electric field created by the rest of the dipoles of the inner layer. Since this field is included in Poll, because it contributes to the value of Ap, the term XsA(p/lof eq 23 includes a contribution from the interactionsof a permanent solvent dipole with the rest of the solvent dipoles. Another contribution to this term arises from the permanent solvent dipole interactions with the field due to the electrode charge, because this field (uM/eo) also contributes to the value of A9. Similarly, it can be proved that the last term of the rhs (right-hand side) of eq 23 includes contributionsfrom the interactions of the induced dipoles among themselves and with the field of the electrode charge. The interactions of a solvent dipole with the adsorbed ions is in fact zer0.23 However, the interactions of this dipole with the images of the adsorbed ions may not be negligible. This means that the dipole experiences the field of the images and therefore this field necessarily affects the field Ap/l across the inner layer. Thus the dipole-image ions interactions are also implicitly included in the term kA(p/l. Finally, the interaction energy of an ion with the images of the rest ions at the IHP is the work required to bring the ion from an infinite distance to its site at the IHP. However, this work is equal to zieD(B, it is included in the definition of the electrochemical potential of ions, and therefore it is taken into account indirectly in the derivation of the adsorption isotherm (eq 30). Thus the only interactions which are not considered in the derivation of this isotherm are the short-range London type interactions and the repulsive long-rangelateral Coulombicones between the adsorbed ions at the IHP. These interactions contribute to the activity coefficients and fs. An additional contribution to these coefficients may arise from the difference in the sizes of the ions and solvent molecules. Analytical expressions for fi and fs can be obtained from statistical mechanical models. Thus, if we assume a monolayer inner layer under a mean field approximation and introduce second-neighbor and higher interactions, we obtainZ7J5
Nikitas
6580 The Journal of Physical Chemistry, Vol. 98, No. 26, 1994
x
In -= ( n - l)ln(l - 0
US)"
If we further assume that XS = 0, which means that the solvent dipoles are directed parallel to the electrode surface, we obtain
+ Oq/n) +
-
where
+ 2)/z A,AS = Z(WM - (wAA+ wss)/2j/kT q = (zn - 2n
(33) (34)
OD
(35)
z is the coordination number of the lattice, z1 is the number of ith neighbor lattice sites which surround a certain lattice site, wk, is the interaction energy between nearest-neighbor species k (=A$) and j (=A$) and wk/ is the corresponding interaction energy between ith neighbor species. In eq 32 the first term of the rhs expresses the contribution due to the difference in the sizes of the ions and solvent molecules. The adsorption isotherm may be expressed in terms of the electrode charge density a M if the potential drop Acp is replaced by uM via the Gauss equation (1 1). We obtain
- cs)con/21(M/A)kT = In(&??) - zieOqd/kT+ ( n ~ -, zieo12+ dei- es)con/(~/~))aM/eoek~ +
?(ei
((€1
- ~ , ) e o n l / 2 ( ~ / ~ ) k T J ( a ~ / e o (36) e)'
Alternatively, an ionic isotherm expressed as a function of uM may be derived by substitution of eqs 26 and 27 into eq 29. It can be written as
A* *
e + In -= In(&") - zieOqd/kT+ In (1 -e)"
cr,**)"
where$** and fs** denote the activity coefficients of ions and solvent molecules in the inner layer when the electrode charge density is equal to zero. It is seen that the activity coefficientswhich appear in isotherms 36 and 31 are not identical. Thecoefficients$** andfs**, apart from contributions arising from short-range lateral interactions between the adsorbed particles, are also affected by long-range particle-particle interactions which contributeto the field strength Poll, like the dipole-dipole interactions. This means that when we use the electrode charge density as an independent electrical variable, the long-range particlsparticle interactionswhich affect the field Acpll are not included in the two last terms of the rhs of eqs 26 and 21 and therefore the use of the potential drop Acp as an independent variable leads to more complete expressions for the chemical potentials of the constituents of the inner layer. Limiting Cases. In a number of previous studies the dielectric constant of the inner layer is considered concentration independent.2.3.9.36-38In this case we have = cs = c and isotherm 36 is reduced to
zieoqd/kT+ (nX, - zieo12)aM/coekT(38)
When lnlfi/(fs)"] 0, eq 39 becomes identical to the LBC adsorption isotherm.9 Therefore, our isotherm (36) may be considered an extension of the LBC isotherm, which takes into account differences in polarizabilities of the adsorbed species and the dipole nature of the solvent molecules. However, we should point out the different approach followed in the present paper to reach the isotherms 36 and 39.
IV. Comparison with Experiment The isotherms developed above can be used to analyze experimentaldata of specific adsorptionof ions only if an analytic expression of 4 is available. That is, the present theory must necessarily combine with a diffuse layer theory. Therefore, the validity of these isotherms depends to a certain degree on the validity of the theory of thediffuselayer adopted for the calculation of 4. In the present paper the diffuse layer contribution was calculated by means of the simple Gouy-Chapman theory using the well-known equation36 qd(in
V) = 0.05141 In
s)2 +
f;;,f + ((
l)li2)
where ui = e(M/A)eo and A = 0.0767(edT)'/'
(41)
Here, uM and ai are in pC/cm2, P is the dielectric constant of thediffuse layer which is approximated to that ofthe bulksolution, and c is the bulk 1:l electrolyte concentration in mol/L. A crucial parameter in adsorption studies is the size ratio parameter n. In older studies n was calculated from the ratio of the surface areas or volumes of the adsorbed ions or molecules. However, recent studies using a thermodynamic method for the determination of the size ratio parameter n when neutral organic compounds are adsorbed on an electrode surface have led to the value n = 1 f 0.2 for all experimental systems This result was explained by assuming the existence at the adsorbed layer of solvent clusters S,l* with dimensions greater than those of the adsorbate molecules and the following mechanism for the adsorption process of an organic compound A:43
where r is the monomer units which form a solvent cluster with dimensions equivalent to the adsorbate molecules and rm/(m 1) is the number of monomer units which form an original solvent cluster St". This mechanism indicates that each adsorbate molecule displaces from the adsorbed layer a cluster of solvent molecules with equivalent dimensions, and therefore it is compatible with n = 1. Since the nature of the adsorbate species does not affect this mechanism, it is reasonable to assume that n = 1 is also valid in the specific adsorption of ions. This choice for n was adopted in the present paper. When we choose n = 1, the number of latticesites per unit area M I A can be calculated from the crystal radii of ions, assuming an hexagonal lattice structure. Finally, the bulk activity of the ions aib was approximated by the mean activity of the salt atb.
The Journal of Physical Chemistry, Vol. 98, No. 26, 1994 6581 Oa4
16
8
8
F 6
I."
0
0.0
0.1
I
I
I
0.2
0.3
0.4
20
0.5
e Figure 2. Test of isotherm 43 using Grahame's data for I- adsorption on Hg taken from ref 37. Electrode charge densities in pC cm-2 are indicated by the lines.
Adopting the above choices for 4, n, MIA, and qb, the adsorption isotherms were tested against the experimental data of I- and B r adsorption on a Hg electrode from aqueous solutions taken from refs 37 and 44. These systems were selected due to the complete tables with data of I- and B r adsorption on Hg contained in refs 37 and 44. The first tests were made by assuming a dielectric constant independent of the surface composition, i.e., by means of eq 38, which when n = 1 can be written as
e + z,e0qd/kT- ln(a,b) = 8(2AAS- ( M / A ) ( & F = In 1-e
+
~ ~ e ~ 1 ~ ) ~ / 1 k{ln(B) T e ~+ e }(nX,
- zieo12)uM/eoekT-AAS+
x,(M/A)(x, - zie01,)/lk~eOe}(43) where
+
A M = AIAS AhAS
(44)
It is seen that the quantity F should be a linear function of 6, resembling the Frumkin isotherm. Plots of F vs 6 at various electrode charge densities due to I- specific adsorption on a Hg electrode are shown in Figure 2. The plots are linear, though their slope varies from -36.43 at UM = 16 pC cm-2 to -17.54 at uM = -8 pC cm-2. Within the frames of the model under consideration AM is constant and independent of the surface composition. In addition, partial charge transfer is ignored. Thus, this variation might be due to the variation of e during the adsorption process. Thus, if, for example, we assume for simplicity that AS = 0 and the value of e is equal to 6 at uM = 16 pC cm-2, then e must be 12.5 at uM = -8 pC cm-2. This is a significant variation of e which cannot be ignored. We should note that Levine et al.9 have also been forced to use a variable dielectric constant in order to achieve an acceptable agreement between the LBC isotherm and experiment. The plots of Figure 2 show that the simple isotherm (38)is inappropriate to describe the specific adsorption of ions, at least in what concerns the adsorption of I- on Hg. For this reason we have proceeded to examine the validity of the more general isotherm (30). This isotherm contains seven adjustable parameters, apart from n and MIA. These parameters are es, ei, I, 12, AS, In(@),and AM and can be selected as follows. The first information about the magnitude of the ratio tsll arises from the minimum value of the inner layer capacitance C in the region of ca. -16 pC cm-2. Since in that region we have 0 = 0, eq 1 1 gives C = eoes/l and for C = 16 F m-1 we obtain esll = 1.8 X 1010 m-1. The thickness of the inner layer I is somewhat uncertain. Thereareauthors whoapproximate lwith thediameter
0
10
-10
-20
oM/pCc"2 Figure 3. Comparison of calculated (-) and experimental(symbols)Aq versus U M curves due to I- adsorption on Hg from aqueous solutions of KI at the following concentrations (from top to bottom): 0.025,O. 1, and 1 N. Experimental data were taken from ref 37. Curves have been calculated from eq 11 using the parameters of Table 1 and 8 values calculated from 8 = d/eO(M/A).
TABLE 1: Molecular Parameters Used in eqs 30 and 35 parameter IBr 6 X 1018 7.5 x 10'8 MIA,site/m2 5 x 10-'0 5 x 10-'0 f, m 2.5 X 10-10 3 x 10-10 11, m 4
-1
-1
CS Ci
9
9
28.5
25
h C m
2.5 X 10-% 3.4 -4.5
0.5 X 10-m
In B
AM
-1.5 -4
of a water m o l e c ~ l e (~1 ~5 *0.3 nm), while other authors add to this value the average radius of the electrolyte ions, either hydrated235349 or n0t,99s52 and in some cases the radius of a Hg ion is further added.53 In the present calculations 1 was taken equal to 5 X 1O-Io m9 and therefore es = 9. Moreover, when B = 0, eq 11 reduces to
A9 = a'l/eoes
+ &(M/A)/eoes
(45)
Therefore, by means of this equation and from the experimental values of A9, contained in the tables of refs 37 and 44,at far negative values of uM where 8 = 0 we can estimate the value of AS. For the systems studied we found AS = 2.5 X C m for I- adsorption and XS = 0.5 X C m for Br adsorption. This difference is quite large to be attributed to differences in the size and therefore in composition of the solvent clusters displaced by the two anions. It might indicate the weakness of the one-state solvent model to reproduce the experimental data by selecting reasonable values for all parameters. Now using the above values of 1, es, and XS, eq 11 can be used to calculate Acp values as a function of uM. In this case there are two adjustable parameters, say ei and 12, the values of which are selected to give the best fits between the values of Acp calculated from eq 11 and those given in the tables of refs 37 and 44 (Figure 3). The e values needed in these calculations were obtained from the ui values presented in the same tables. The parameters ei and II = I - 12 chosen in this way are given in Table 1. Note that the value of I I is higher for B r than for I-, despite the fact that the ionic radius of Br is smaller than that of I-. Such discrepancies have already been observed in literature38and, like the discrep ancies in the AS values, might be due to the simplicity of the model. More advanced models, like those discussed below, are expected to give a more realistic picture of the inner layer properties. Finally, the parameters In @ and AM can be selected by means of eq 30 if it is written as
Nikitas
6582 The Journal of Physical Chemistry, Vol. 98, No. 26, 1994
+
zie0vd/kT- ln(a,b) - 2AB ' = In 6 - AM + 1-e (nXs - zieo12)A9/lkT (ti - cs)aoAq2/21(M/A)kT (46)
F = In
+
It is seen that the quantity of the lhs (left-hand side) of eq 46 denoted by F is a function of A 9 only. All the parameters appearing in F can be calculated from the data presented in the tables of refs 37 and 44, except the interaction parameter A*. However, this parameter should take that value which leads the "experimentally" calculated values of F at various electrolyte concentrations to a common curve when they are plotted versus Aq. Figure 4 shows these plots of the systems studied. We note that with some small deviations, probably due to the experimental errors, the points coincide with a common curve. The value of F at A 9 = 0 gives the value of In fl - A*. The values of all parameters obtained following the above procedure are given in Table 1, and they were used to calculate (a) the charge density due to the specifically adsorbed anions as a functionof either A 9 or uMand (b) the dependenceof Avversus UM by means of eqs 11, 30, and 36. Comparison tests of the model predictions with experimentaldata of the systems adopted are shown in Figures 5-8. It is seen that there is a satisfactory description of the adsorption isotherms expressed as ui vs A 9 and UM throughout the region where specific adsorption occurs and over a wide range of salt concentrations. The predicted dependence of Acpupon uMis also satisfactory. Thus, despite the fact that we have used the very simple and questionable GouyChapman theory to calculate 4, the obtained results show that the proposed isotherm (36) can give an almost quantitative description of the specific adsorption of ions. A more advanced test of the proposed model is the shape of the inner layer capacitance curves. The inner layer capacity was calculated by numerical differentiation of eq 11, and its values are compared with experimental data in Figure 9. The experimental values of the inner layer capacity were obtained from the total capacity of the electrical double layer C,, given in tables of refs 37 and 44, and the diffuse layer capacity c d of the GouyChapman theory by means of the following equation54
10
5
0
-5 0.2
0.0
-0.2
-0.4
-0.8
Adv
0
'
I
-5 0.5
0.0
-0.5
Adv Figure 4. Plots of the lhs of q 46 versus Acp due to (A) I- and (B) B r adsorption on Hg at the following concentrations: (A) 0, 1; 0,O.l; A, 0.025; (B)0, 1; 0,O.l; A, 0.01 N. In the calculation of the lhs of eq 46 Am was taken equal to (A) -4.5 and (B) -4.
...,N - 1, then the equilibrium equations may be expressed as Cci-nCL1=Cli
(47)
where the derivative dui/duM was calculated by numerical differentiation of the ui versus UM plots. It is seen that the model correctly predicts the magnitude of the differential capacity and its increase at the positive region. However, it does not predict the capacitance hump at ca. 2 pC cm-2. In my opinion this hump must be attributed to the properties of the water molecules at the inner layer and in particular to their reorientation on the electrode surface.
-0.6
pk
p,,
b
-Ws
k = 2, 3,
b
..., N - 1
(48) (49)
where we have assumed that Ak = AI. Substitution of eqs 22 and 23 into eqs 48 and 49 results in the following system of adsorption isotherms
e +
A
In - In -= In(8a:) - zie0qd/kT+ (6,)" VJ" (nX, - zie012)A(p/lkT (ti - el)ediA(p2/21kT (50)
+
V. More Generalized Isotherms In the treatment presented above the dipole of the solvent molecules was assumed to possess a constant orientation. However, the solvent dipoles may exhibit a number of distinct orientations on the electrode surface. In addition the thickness of the inner layer was assumed to be constant and independent of the surface composition, although its variation during the adsorption process cannot be ruled out. Thus, the present treatment can be improved in two respects: By considering more than one solvent state simultaneouslypresent at the inner layer and changes in its thickness. Adsorbed Solvent Molecules Exhibit Certain Distinct Polarization States. When the solvent molecules exhibit certain distinct orientations on the electrodesurface, then each solvent state may be treated as a different adsorbed species. Therefore, if the inner layer consists of ions i and N - 1 solvent states denoted by 1,2,
(tk- tl)tJlA(p2/21kT (51) where analytical expressions for the activity coefficients A and
fk can be found in ref 55. It is seen that we have to solve a system
of N equations with unknowns of the surface coverages 0, el, 02, 01, ...,ON. However, it is worth noting the following. Studies of the inner layer properties in the absence of specific adsorption have shown that at least three solvent states are needed for an acceptable description of the inner layer capacitance plots.56-Sg Therefore, in the presence of specifically adsorbed ions we have to solve a system of at least four equations. The model is expected to give a better description of the inner layer properties, but the great number of adjustable parameters, which can hardly be determined independently, limits its value to a theoretical level.
Ionic Isotherms in the Electrical Double Layer
The Journal of Physical Chemistry, Vol. 98, No. 26, 1994 6583
Variation of Inner Layer Thickness. If we now suppose that the thickness of the inner layer I is a function of the surface composition, then eq 7 yields
# 40
'e
N
+
+
+
{toAqA/I2 zieoNill/12- &NS/l2 2asNSAq/l3
*)
30
u
c '-?
20
2aiNiAq/13](aNi N, 10
which in combination with eq 21 results in
0 0.2
ii= &* - y A , + zieOqd+ zieoAq12/l- coeiAq2Ai/21+ AqA{uM- eoAq/21 + zieONi/A)(") /I aNi
-0.2
0.0
-0.6
-0.4
-0.8
N,
50
and similarly p s = &*
40
- yAs + &Aq/I - coesAq2As/21+ AcpA(uM-
+) / I
?
coAq/21 + zieoNi/A](
NS
f
0
Ni
30
V
2
The adsorption isotherm which arises from eqs 53,54, and 29 may be written as
-?
20 10 0
+
zie012)Aq/lkT (ei - cs)conAq2/21(M/A)kT- AqA(uM-
+
coAq/21 z i e o N i / A )aNA fA)
">
Ns - n( aNS
NAI I k T
(55)
and it can be expressed in terms of uM if Aq is replaced by uM via eq 11. It is evident that the application of this isotherm necessarily requires a functional dependenceof I upon Ni and Ns. The simplest expression is the linear one
20
10
0
-20
-10
aM/pC em-* Figure 5. Comparison of calculated (-) and experimental (symbols) d versus (A) Acpand (B) uM curves due to I- adsorptionon Hg from aqueous solutionsof KI at the following concentrations(from top to bottom): 1, 0.1, and 0.025 N. Experimental data were taken from ref 37. Curves have been calculated from eqs 30 and 36 using the parameters of Table 1.
leading to
where li and 1s are the thickness of the inner layer composed exclusively of ions andusolvent molecules, respectively. The generalized isotherm ( 5 5 ) was applied to the experimental systems of I- and B r adsorptionon Hg, using the same molecular parameters as those depicted in Table 1 except the inner layer thickness 1. For this parameter we have used 1s = 5 X 10-l0 m and li in the region between 4 X 10-loand 10 X m examining the improvement of the fits. However, the observed improvement for values of li slightly higher than 1s was not significant and shows that for the two systems studied the inner layer thickness appears to be constant.
VI. Conclusions In the present paper we have adopted a combination of classical thermodynamics with basic electrostatic arguments in order to develop new ionic isotherms of specific adsorption on ideally polarized electrode surfaces. From previous studies we can distinguishtwo main approaches in developing adsorption isotherms: a thermodynamic and a molecular one. The thermodynamic is based on the work of
-0.8
'
20
I
10
I
0
t
-10
0
-20
aM/pCcm.' Figure 6. Comparisonof calculated (-) and experimental (symbols) Acp versus uM curves due to I- adsorption on Hg from aqueous solutions of KI at the followingconcentrations(from top to bottom): 0.025,0.1, and 1 N. Experimental data were taken from ref 37. Curves have bcen calculated from q s 1 1 and 30 using the parameters of Table 1.
Mohilner et al.I3 The adsorption isotherm is derived from the equilibrium in eq 29 with the introduction of surface activities. Its serious weakness is that it cannot give any molecular picture of the adsorbed layer, since, according to Parsons,@ "if no information about structure is contained in the original experimental data, no such informationcan be obtainedby the operation of thermodynamic transformations on these data". In the molecular approach2Js5.9a detailed account of thevarious particle particle and particlefield interactions is carried out within the
Nikitas
6584 The Journal of Physical Chemistry, Vol. 98, No. 26, 1994 50 40
0.8 M
30
'8
5
20
0.4
10
0 0.5
0.0 0.3
0.1
-0.1
-0.3
'
20
-0.5
I
0
-1 0
-20
oM/pccm-'
AWN
501
I
I
10
I
I
40 r(
30
'E h
3
20
0.5
10
0
0.0 30
20
10
0
-10
-20
o'/pc cm" Figure 7. Comparison of calculated (-) and experimental (symbols) ui versus (A) Acp and (B) uM curveg due to B r adsorption on Hg from aqueous solutions of KBr at the following concentrations (from top to bottom): 1,O. 1, and 0.01 N. Experimental data were taken from ref 44. Curves have been calculated from eqs 30 and 36 using the parameters of Table 1.
0.2
-0.2
-0.6
-1 .o
30
20
10
0
-10
-20
oM/vCem-' Figure8. Comparison of calculated (-) and experimental (symbols) AI^ versus U M curves due to B r adsorption on Hg from aqueous solutions of
KBr at the following concentrations(from top to bottom): 0.01,O. 1, and 1 N. Experimental data were taken from ref 44. Curvcs have been calculated from eqs 11 and 30 using the parameters of Table 1.
frames of either the electrostatic theory or statistical mechanics. However, any treatment of the various interactions introduces approximations which in turn limit the validity of the treatment. Here, we have developed a quite different approach, which makes the independent treatment of the most important interactions at the inner layer unnecessary. In fact we have adopted a very general model for the inner layer, which considers it as having two parts where the electric field is homogeneous. Then the Gauss theorem of electrostatics is used to correlate the potential
'
20
I
I
I
10
0
-10
-20
oM/pcem-' Comparison of calculated (-) and experimental (symbols) inner layer capacityversus uM curveg due to (A) I- and (B) B r adsorption on Hg from 0.1 N aqueous solutions of KI and KBr, respectively. Experimentaldata were taken from refs 37 and 44 following the procedure described in the text. Curves have been calculated by numerical differentiation of eqs 11 using the parameters of Table 1. Figure 9.
drop across the inner layer Acp with the electrode charge density uM. The only approximation used is that the induced dipole moment at the constituents of the inner layer is proportional of the field Acpll. No other approximation is used for the derivation of the adsorption isotherms, which can take into account changes in the dielectric constant, the inner layer thickness, and the orientation of the solvent dipoles during the adsorption process. The proposed approach is characterized both by simplicity and rigour. Even the simplest isotherm, which takes into account only variations of the dielectricconstant, has been found to describe satisfactorily the experimental data of I- and B r adsorption on the mercury electrode. An adsorption isotherm is usually used to explain the observed properties of an adsorbed layer and/or to gain information about the structure of this layer and the molecular properties of its constituents. Thesimplest of our models reproducessatisfactorily the experimental data but with molecular parameters which may take values inconsistent in the various systems. Therefore, we can conclude that, whereas the model is too simple to be used to obtain quantitative and unambiguous results for the molecular properties (AS, ti, es, 11, and r) of the adsorbed layer, it can be adopted to obtain a general picture of its properties. Thus, the satisfactory agreement between theory and experiment shows that the adsorption mechanism which involves the displacement of solvent clusters by polarizable ions is quite possible. Moreover, we can explain the different behavior between small inorganic ions and organic ions. Thus, from the magnitude of the various parameters it arises that in the case of inorganic ions and for Acp values lower than 0.5 V the term zieOl~A(Pl1kTin eq 46 predominates over the last term of the rhs of this equation. For
Ionic Isotherms in the Electrical Double Layer this reason an almost linear dependence of the surface coverage 8 on Acpis obtained. In contrast, in an adsorbed layer with organic ions 12 0 and therefore the last term of the rhs of eq 46 contributes more than the linear term. This explains why in the case of organic ions 8 varies almost quadratically upon Acp and an adsorption maximum is observed. Finally, we must point out that the present treatment has given explicit expressions of the electrochemicalpotentials of adsorbed ions. These expressions can be used not only for the derivation of adsorption isotherms for simpleinorganic compounds but also in the description of the equilibriumin a number of other systems, like multicomponent adsorbed layers, contactive polymer films on electrode surfaces, intercalation of ions, insoluble ionised monolayers, etc.
-
References and Notes (1) Parsons, R. Trans. Faraday Soc. 1955,51, 1518. (2) Bockris, J. OM.; Dcvanathan, M.; Miiller, K. Proc. R. Soc. London, Ser. A 1%3, 274, 55. (3) Bockris, J. OM.;Gileadi, E.; Miiller, K. Electrochim.Acta 1%7,12, 1301. (4) Damaskin, B. B.; Petrii, 0.A.; Batrakov, V. V. Adsorption of Organic Compounds on Electrodes; Plenum Prcss: New York, 1971. 15) Habib. M.A,: Bockris. J. O'M. Commehensive Treatise of Elecfrociemistry;k k r i s , J. OM., Conway, B., Ykger, E., Eds.;Plenum Pres: New York, 1980; Vol. 1. (6) Guidelli. R. Trends in Interfacial Electrochemistry; Silva, A. F., Ed.; Reidel: Dordrecht, 1986. (7) Nikitas, P. Electrochim. Acta 1987, 32, 205. (8) Guidelli, R.; Aloisi, G. Electrified Interfaces in Physics, Chemistry
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