1491
J . Phys. Chem. 1984, 88, 1491-1496
The Adsorbed Layer and Adsorption Isotherms Emmanuel Tronel-Peyroz Laboratoire de Physico- Chimie des Systcmes Polyphasds, Centre National de la Recherche Scientifique, 34033 Montpellier-Cedex, France (Received: March 30, 1983; In Final Form: July 15, 1983) We show that, to analyze the various solute-solute-solvent interactions which partly determine the conditions of adsorption, all of the recent isotherms proposed considering the existence of only one adsorption coefficient of interaction B“ do not give a correct picture of experimental results obtained, especially at the polarized mercury-aqueous solution interface. On the other hand, when these isotherms seem satisfactory, as in the case of the adsorption of 2-methyl-2-propanolat an airsolution interface, the adsorbed layer is not a “regular” phase. In most cases, it is necessary to use the isotherm deduced from Mohilner’s theory which includes several interaction coefficients B,“.
I. Introduction The adsorption at an interface of organic molecules from an aqueous solution, with or without electrolyte, is a general phenomenon] which only can implicate the internal layer of the electrical double layer, if the effect of the diffuse layer is totally eliminated or considerably reduced. Adsorption studies carried out under these conditions yield information on the structure of the interfacial double layer and also on the different molecular interactions between the adsorbed molecules (solutesolutesolvent interactions). To achieve these, it is necessary to establish the adsorption isotherm, i.e., the plot of the amount of substance adsorbed, at constant temperature, relative to its concentration in the external bulk phase. Numerous models leading to a mathematical formulation of isotherms have been put forward.2J The earlier ones, as for example Freundlich’s, were purely empirical or then deduced from simple statistics as the one due to Langmuir. In both cases, the adsorbed layer was considered as being constituted of the solute in a solution in which the solvent is taken as a continuum. Solution studies have shown that this procedure of excluding the solvent contribution of the interactions which determine the transport properties of an electrolyte in solution is i n ~ o r r e c t . ~It is hence easy to understand why the isotherms deduced from such simplified models cannot be satisfactory. Frumkin5 was the first to attempt to take into consideration molecular interactions. Since then numerous equations have been put forward which have been recently critically reviewed by Rangarajar~.~.’Ideally, in order to establish a general adsorption theory, one must start from the determination of the molecular distribution at the interphase based on a detailed knowledge of the interaction forces between all the molecules present. It is, of course, impossible to have such information, but attempts have been made to solve the problems by assuming as a first approximation that the molecular interactions in the adsorbed phase are identical with those in an “ideal” or ”regular” mixture.*-” Such an approach has been used r e ~ e n t l y ’ ~ to -’~ interpret the results for the adsorption of neutral substances, at (1) D. Schuhmann, R. Bennes, M. Privat, H. Raous, E. Tronel-Peyroz, and P. Vanel in “Adsorption at the Gas-Solid and Liquid-Solid Interface’, J. Rouquerol and K. S. W. Sing, Eds., Elsevier, Amsterdam, 1982, p 141. (2) D. Schuhmann, J . Chim. Phys. Phys.-Chim. Biol., 64, 1399 (1967). (3) H. P. Dhar, B. E. Conway, and K. M. Joshi, Electrochim. Acta, 18, 789 (1973). (4) V. A. Kir’yanov, V. D. Gruba, and Yu. M. Kessler, Elektrokhimiya, 14, 826 (1978). (5) A. N. Frumkin, Z . Phys. Chem., 116, 466 (1925). (6) M. V. Sangara Narayanan and S. K. Rangarajan, J . Electroanal. Chem., 130, 339 (1981). (7) S . K. Rangarajan in “Electrochemistry”, The Chemical Society, London, 1980, p 203, Specialist Periodical Reports Val. 7. (8) R. Defay, I. Prigogine, A. Bellemans, and E. H. Everett, “Surface Tension and Adsorption”, Longmans, New York, 1966. (9) E. A. Guggenheim, “Mixtures”, Clarendon Press, Oxford, 1952. (IO) E. A. Guggenheim, Trans. Faraday Soc., 41, 150 (1945). (11) D. H. Everett, Trans. Faraday SOC.,60, 1803 (1964); 61, 2478 (1965). (12) R. Bennes, J . Electroanal. Chem., 105, 85 (1979). (13) M. Hamdi, P. Vanel, D. Schuhmann, and R. Bennes, J. Electroanal. Chem., 108, 225 (1980). (14) R. Bennes and E. Boukaram, J . Colloid Interface Sci., 81, 304 (1981).
0022-3654/84/2088-1491$01.50/0
both a polarized mercury-solution interface and an air-solution interface. If the notion of an adsorbed phase of finite thickness is today both generally accepted and acceptable, since the contributions of Verschaffelt,Is Guggenheim,16 and more particularly M ~ h i l n e r , ”nevertheless ~~~ the actual nature of this phase, which is essentially characterized by the activity coefficient of its molecules, is subject to contradictory interpretations. This would appear all the more as use is made of the notion of regular sol u t i o n ~under ~ ~ conditions where it is hardly applicableIg as, for instance, in the case of molecules of different sizes, as is going to be shown here.
11. Background (1) It is generally admitted today, as a result of the work done by Bcckris:O Everett,” and Mohilner,17that adsorption is a process whereby the interfacial water is replaced by the surface-active molecules present in the solution. The adsorption equilibrium is represented by the following classical equation: A(b)
+ nH,O(ads) * A(ads) + nH20(b)
(1)
where b denotes the species in solution and ads those adsorbed in the inner layer, while n is the number of molecules of water replaced by one adsorbed organic molecule. In agreement with Mohilner, n is defined as the ratio of molar volumes of the species A and W (solute and solvent). However, one may well ask to what extent one should take into consideration, in the evaluation of n, the number of replaced water molecules whose thermodynamic energy state is the same as those of the solvent molecules in solution. These should then not be included in the equilibrium represented by eq 1. This point will be discussed s u b s e q ~ e n t l y . ~ ~ By equating the electrochemical potentials of the different species on either side of relation 1, one can deduce the adsorption isotherm:
Here aA and aw are respectively the activities of the organic compound and the water in the bulk phase [or adsorbed layer (15) J. E. Verschaffelt, Acad. R. Belg., Bull. CI.Sci., 22, 373, 390, 402 (19 36). (16) E. A. Guggenheim, Trans. Faraday SOC.,36, 397 (1940). (17) H. Nakadomari, D. M. Mohilner, and P. R. Mohilner, J. Phys. Chem., 80, 1761 (1976). (18) M. Karolczak and D. M. Mohilner, J. Phys. Chem., 86,2840 (1982); 86, 2845 (1982). (19) R. Defay and I. Prigogine, “Chemical Thermodynamics”, 5th ed., Longmans, New York, 1969, p 391. (20) J. O M . Bockris, M. A. V. Devanathan, and K. Muller, Proc. R. SOC. London, Ser. A , 274, 55 (1963). (21) D. M. Mohilner, H. Nakadomari, and P. R. Mohilner, J . Phys. Chem., 81, 244 (1977). (22) J. C. Eriksson, Ark. Kemi, 26, 49 (1966). (23) P. Vanel et al., to be submitted for publication.
0 1984 American Chemical Society
1492 The Journal of Physical Chemistry, Vol. 88, No. 8, 1984
Tronel-Peyroz
(ads)] chosen on the basis of symmetrical standard states, and AGads*is the standard adsorption electrochemical free energy again based on the choice of symmetrical standard states, both in the bulk phase and in the adsorbed phase:
is the maximum superficial excess, n is the ratio of the volumes, and Y = xA/(l
= (FA* - F A * ) - n(bW' - PW') (3) Here pA' is the standard electrochemical potential of the adsorbed organic compound based on Raoult's law standard state, Le., the surface fully covered with organic compound at the specified temperature and electrical state of the system. pA' is the chemical potential of the pure bulk organic compound at the specified temperature. The two corresponding potentials for the water are given by pw' and pw*, the standard state for the internal layer being taken in the presence of only adsorbed water. (2) If one considers the adsorbed layer as a phase, the excess electrochemical free energy of the adsorbed phase is given by the relation AGad;
+
AGE = RT[xAadsIn yAads xWads In yWads]
-
Remarks. (a) When the bulk concentration of the organic compound is small (xA 0), y tends to zero and = rA,W. (b) When xA increases, r A , W also increases and finally tends to a limit. Here is only equal to r A m if y 03, when xA 1, which is the case of the pure substance in solution. Relation 14 thus reveals that one never reaches a total coverage under the most common conditions. (3) This thermodynamic theory of adsorption, essentially due in its extended form developed above to Mohilner,I8 enables one to reach most of the adsorption isotherms found in the literature, these being particular cases of the general expression 8, obtained by a choice, depending on the author, of the particular values of k and n used. One can thus discover, using Mohilner's theory, which simplifying hypotheses are hidden behind the various isotherms, and hence establish their limits and possibilities. For example, Mohilner has shown that, for n = 1 and k = 2, one falls upon the Frumkin isotherm, undoubtedly the one most used in adsorption studies (see Appendix). It is also easy to show that one of the recently published isothermsI2 simply constitutes a particular case of the above general isotherm 8 when k = 2 and n # 1. Under these conditions, eq 5 gives
(4)
(5) with Ejelkaj= 0 for the coefficients aj,which are dependent on the electric state of the system. Applying the Duhem-Margules relations to eq 4 one obtains expressions for In yAads and In yWads: yAads =
AGE
- xe)
( x , = mole fraction of electrolyte).
where R is the gas constant, T i s the absolute temperature, xAads and xWads are the mole fractions of the organic compound (A) and yWadsare and water (W) in the adsorbed phase, and yAads the activity coefficients of the adsorbed species for a system where the standard states are symmetrical. According to Mohilner et al.,1s,21AGE can be expressed as a polynomial of the kth degree of the mole fraction of the adsorbed organic substance xAads:
R T In
- xA
-
-
AGE = R T [ ~ I X A+' ~O2(XAads)'] ~
(15)
dAGE/d~AadS = RT[al
+ 2aZ(XAads)]
(16)
In view of the relation Cj,lka,= 0, the activity coefficients of the adsorbed species W and A can be expressed as
+ (1 - X A ~ ~ ~ ) ( ~ A G ~ / ~ X A ' ~(6) ')T,~,E
RT In yWads= ACE - x ~ ~ ~ ~ ( ( ~ A G / ~ x(7) A ~ ~ ~ ) ~ , ~ , ~ and the isotherm can easily be deduced from eq 8: Utilizing now eq 5, relation 2 leads to the adsorption isotherm:
ads
The surface concentrations of the adsorbed species in the inner layer are related to 8, the fraction of the surface covered by the organic compound by the relations
= 6/( 10-'6Ns~ads)
If the mole fractions xAads and xWads are expressed in terms of n and 6, using relations 11 and 12, and to simplify matters the activity of the water in solution is taken as equal to 1, one obtains
Bc
(9)
=f(@
exP[alg(@I
,
(20)
with B = (1 /55.5) eXp(-AGads'/Rq eXp(p) where SAads is the surface in A2 occupied by the adsorbed organic molecule, and N the Avogadro number. One can thus deduce XAads
= g/ ( e
+ 4 1 - 0))
xWads = n(1 - e ) / ( e + n(1 - e ) }
However, the Gibbs equation, which relates the interfacial tension to the Concentration of the organic substance in solution, enables one to determine only the relative superficial excess r A , W from the following equation: rA,W
=
FA
- (xA/XWIrW
(13)
Assuming that the adsorbed phase is a monolayer, one obtains
rA= where
IIA,W -k nYrA" 1 ny
+
g(0)
(11) (12)
e[e + n(1 - e)ln-I/(nn(l - e).) = n2(1 - e)* - n e z / [ 6 + n(1 -e)]'
fle) =
(21) (22) (23)
In relation 21 the coefficient /3 is such that In
YA
since x, (XA
= Pl(xw) N
+ Pz(xw)' + ...
(Pi
+ P 2 + P3 +
=
P
(24) 1 in the case of dilute solutions of the organic substance
0).
(In the original publication dealing with this matter," P is given as B" and a1 as B"; this notation has been maintained in this present article.) 111. Is the Adsorbed Layer a Regular Phase?
(1) The isotherm given in eq 20 has been used to account for experimental results obtained for the adsorption of organic substances at the mercury-solution interfacei2,13(see section IV). And
The Journal ofPhysical Chemistry, Vol. 88, No. 8, 1984 1493
Adsorbed Layer and Adsorption Isotherms
hence
TABLE I : Values of Ea for 2-Methyl-2-propanolat an
Air-Solution Interface at Different Temperaturesa
a
aBa/aT = 0.76 x 1 0 3 / ~
t, "C
EO
t , "C
BO
15 25
0.55 0.6
3s 45
0.75 0.75
Taking into consideration eq 17 and 18 for the variations of In yladsleads to the following relations:
Prom data in ref 14
insofar as the hypotheses made to derive this isotherm lead to relations 17 and 18 relating the activity coefficients of the adsorbed species to the superficial molar fractions, it has been deduced that this model implied that the adsorbed phase could be considered as a regular solution. In fact, this is not justifiable because the form of eq 17 and 18, if it is a necessary condition, is nevertheless not a sufficient one. Considering nonideal solutions one must indeed consider two limiting cases: l 9 (i) Regular solutions for which >> TlhsEl, where A P and hSEare respectively the molar excess enthalpy ARE and and molar excess entropy of mixing. Here ACE deviations from ideality are due to the heat of mixing. (ii) Athermal solutions for which I f l I > qUEI) leads to ASE 0 and hence ahsE/aq = 0, whatever the values of ni. This in turn means that
and, in fact, R T In yi = cte = k. In the second limiting case, we have AHE = 0 and
aAHE = - pa2AGE/ T = -p-(Ra ani a T
ani
aT
In yi) = 0
which in turn gives R In yi = cte = k'
(26)
Relations 17 and 18, therefore, only correspond to the case of regular solutions if B" varies as k/RT. If, on the contrary, B" does not vary with T, these two relations correspond to those deduced for athermal solutions. This means that it is necessary to study the variation of the adsorption with the temperature in order to be able to characterize the deviations from ideality of the species in the adsorbed phase and to define the nature of the latter.21 (2) The results for the adsorption of 2-methyl-2-propanol from an aqueous solution at an air-solution interface can be used as an e ~ a m p 1 e . lThe ~ equation which describes the adsorption at this interface is the one given by Erikssonz2 which at constant temperature and pressure has the following form: SAadsy
- SAads,o y A o = R T In (aAads/aA)
(27)
where y is the superficial tension of the solution, yAois the one for the pure substance, SAads and SAads,' are the partial molar surfaces of the constituent A in the superficial phase of the solution and the pure substance, and uAads and aAare the activities of the organic compound A in the adsorbed phase and bulk phase. Using eq 17 and 24 and assuming that SAads N SAads,', one obtains for relation 27
Finally, by plotting Y vs. ( x ~ one ~ can ~ evaluate ~ ) ~in particular B".14 The results thus obtained at different temperatures given in Table I reveal that B" increases with T, which is contrary to what one would expect for regular solutions. Indeed, as one finds empirically B" = 3.18 - (0.76
X
103)/T
(30)
(29)
AGE = RTB'(~Wads~Aads)
(31)
AHE = -RpXWadsXAads(dBa/dT)
(32)
which means that, at T = 298.2 K, B"
0.62 and
AGE N 0.37xWadsxAads
ARE N -1.5 lXWadsXAads ThSE N -1
.
~
~
X
W
~
~
~
X
A
(33) ~ ~
ImEl
This confirms that is not >> qUEland that the adsorbed phase is neither regular nor athermal. The behavior of 2methyl-2-propanol in the absorbed phase is hence more complex than the two limiting cases envisaged above which, it must be noted, are not even applicable to the 2-methyl-2-propanol in the bulk phase b. One rediscovers here results found by Hildebrand19 which showed that, for solutions of molecules of different sizes, the regular solution hypothesis leads to different expressions for In yAads and In yWadsin solution to those given by eq 17 and 18. These two expressions are quite general insofar as the nature of the adsorbed phase is concerned and in no way imply a regular solution. They are just the result of a simplifying hypothesis, limiting the variations of In yAads and In yWadsto a first term, and whose only merit is that of making the calculation less complex. If this hypothesis happens to be satisfactory in the case under consideration (the adsorption of 2-methyl-2-propanol at an airsolution interface), it is not applicable in most other cases. IV. Limits of the Isotherm 20 Beyond the fact that this isotherm only corresponds to the case of regular solutions when B" is independent of T, two other reasons seem to justify its rejection in both the general or reduced form given by eq 20. This isotherm can indeed be written in the following form:
AG' =
AGad:
- RTB"
+ RTB"g(8)
(34)
with
(1) B" and AGah* can be easily obtained from eq 34 if one notes that, when 0 = 0, g(0) = 1 (relation 23); thus
(AG'), = AGa,js*- RTB" Furthermore, when 0 = nliz/(l
+ RTB"
(36)
+ n1lz) g(0) = 0 and
(AG'), = AGad: - RTBa
(37)
If one considers the curve in Figure 1 giving the variation of AG'with 0 in the case of the adsorption of hexanol at a mercury-aqueous solution of 0.1 M K F interface,12 one finds (AG'), = -4.7 kcal/mol and (AG'), = -6.3 kcal/mol. Since ff = 6.8, using eq 36 and 37 one gets B" = 2.7 and AG,,' = -2.3 kcal/mol. Evaluating these two parameters graphically by plotting the variation of AG'with g(0) one observes (Figure 2) that one does not obtain a straight line as predicted by eq 34 and that the straight line corresponding to the value of B' = 2.7 is in total disagreement with the experimental results. Repeating this analysis with other experimental results invariably leads to exactly the same type of conclusions. Vane1 et aLZ3have studied systematically these variations for the adsorption of various alcohols and acids in a mercury-aqueous 0.5 M NaCl solution. It has thus been shown that these variations are not due to experimental errors inherent in the determination of superficial excess of concentrations and
~
1494 The Journal of Physical Chemistry, Vol. 88, No. 8, 1984
Tronel-Peyroz a
Lnc
i e
0.5
0
Figure 1. Variations of AG' (kcal/mol) with 0 for the adsorption of hexanol in 0.1 M KF solution at the mercury-aqueous solution interface
(from data in ref 12).
t
AG"' AG'
*
. Figure 3. Schematic dependence of In c and d In c/d0 with 0 .
Now P(0) passes through a minimum when 0 = 0*, Le., at that point
= P* = 0 Deriving eq 40 and applying condition 41 one obtains B" =
[3n(l
- n)19*~+ 2 4 2 n - 1)0* - n2][n + 0*(1 - n)]' 6n2(n - l)[O*(l - 0*)12 (42)
-3
-2
-1
0
9Fl
Figure 2. Variations of AG'and AG'" (kcal/mol) with g(0) (see explanations in text) for the adsorption of hexanol in 0.1 M KF at the mercury-aqueous solution interface (from data in ref 12).
therefore not due to errors in evaluating the degree of coverage. (2) An isotherm is a relation between either In c and F A , the superficial excess of concentration of the adsorbed species A, or In c and the degree of coverage O (0 = r A / r A m ) . The numerous experimental results which have been accumulated by Damaskin et al.24all go to show that the In e 0 curves are always of the type represented in Figure 3a, from which one deduces that the d In cld8-B curve must have the shape of the one represented in Figure 3b. Rewriting eq 20, as has been done by D a m a ~ k i none , ~ ~has
On the other hand, when 0 = 0*, P = P* and with eq 42 and 40 n[n - (n + l)O*] p* = (43) 3(n - 1)[0*(1 - 0 * ) ] 2 [ n + 0*(1 - n)] O* and P* having been determined experimentally, one can hence evaluate n and then B". Now in all cases envisaged the values thus obtained do not correspond to the ones estimated before. n is frequently smaller than the ratio of the volumes by which it is defined, and the value of B" is close to or greater than 2, as, for instance, it was found above for hexanol (B" = 2.7). However, such a value is impossible. A limiting case is in fact interesting; it is the one represented in Figure 3 which corresponds to P* = 0. Here eq 43 leads to
o* = n / ( n
+ 1)
(44)
(40)
Using this value for O* in eq 42 one obtains B" = 2 whatever the value of n. As in practice one never obtains isotherms of the shape given in Figure 3b, B" can never attain a value greater than 2. When n is large, O* 1: the theoretical isotherm for B" = 2 has a correct shape for the usual values of 0 (0.2 < 0 < 0.7) and that part of the isotherm for which P* = 0 is shifted outside the experimental range. Nevertheless, when a study in this particular region is possible, a value of P* = 0 has never been observed and hence B" cannot be greater than 2. These two reasons, Le., values of B' which are greater than the critical value yielded by the isotherm and the systematic deviations between the experimental and theoretical values, have led us to abandon this isotherm and to propose a new one based on Mohilner's theory,"J8 which might prove more general.
(24) B. B. Damaskin, 0. A. Petrii, and V. V. Batrakov, "Adsorption of Organic Compounds on Electrodes", Plenum Press, New York, 1971.
V. A More General Isotherm To obtain the equation for a more general isotherm it is merely necessary to start from relation 5 and to develop this to the kth term, setting down a1= B", a2 = B", etc. Then using eq 5 and
In B
+ In c = 1nflO) + B"g(0) = In F(0)
(38)
This means that the equation for the curve in Figure 3b is
P = d In c/dO = d In F(O)/dO = d In f(O)/dO
+ (dg(O)/dO)B"
(39)
as B" is independent of 0. Finally, this leads to
P(0) =
n O(1
- @ ) [ e+ 4
-
1 - 0)l
2n2Bu
[e + n(1 - 0)]2
-
The Journal of Physical Chemistry, Vol. 88, No. 8, 1984 1495
Adsorbed Layer and Adsorption Isotherms
AG*~~~AG'
~~
-0.5
-1
0
~
0.5
9m
Figure 5. Variations of AG'and AG"/(kcal/mol) with g(0) (see explanations in text) for the adsorption of 2-methyl-2-propanolin 0.1 M KF solution at the mercury-aqueous solution interface (from data in ref 25).
For the case of the hexanol, n = 7; 0' = 0.726 and A l = -6.30 kcal/mol; 82 = 0.896 and A2 = -8.40 kcal/mol (Figure 1). With these values eq 53 becomes
AG" = AG'- 11.96h(0) - 4.35g(6) = (AGad8*- RTB")(l 0.518g(0)
+
Figure 4. Variations of AG"(kcal/mol) with T(0) (see explanations in text) for the adsorption of hexanol in 0.1 M KF solution, at the mercury-aqueous solution interface (from data in ref 12).
going through the same procedure as was done to attain relation 20 one has, this time Bc =f(0) exp(B"g(0)
+ B"'h(0) + B"'"j(0) + ...)
h=g-l+k j = g - l + t k(0) = 02[0 + n(3 - e)]/[e + n(1 - 0)13 t ( 0 ) = 03[0
+ n(4 - @ ] / [ e + n(1 - s)l4
etc. Bringing out the AG' expression 34 becomes AG' = AGads*- RTB" + RTB"g(0) RTB'"h(0) RTB"'"j(0)
+
+
(45) (46) (47) (48)
+ ...
(49) which now replaces eq 34. Returning to the case of hexanol in a 0.1 M K F solution and endeavoring to interpret the experimental results using eq 49 and B / / h = 0, one has AG' = AGads*- RTB"
+ RTB"g(0) + RTB""h(0)
(50)
For
0 = 8' = n1/2/(1 g(0) = 0
+ n1/2)
h(0) = [n1/2(1+ n1l2)l2
(AG'), = A l
(51)
and for
8=02= 4-n g(0) =
+
3n - [n(n 8)]'12 2(n - 1)
+ [n(n+ 8)]'12
2n ( A W , = A2 Rewriting eq 50 in the form
h(0) = 0
(52)
E
(AG,,. - RTB")T(0)
+ 1.899h(0)) (54)
The variation of AG" with T(0) is shown in Figure 4. Here one actually has a straight line whose slope gives AGadl- RTB* = -6.05 kcal/mol. This leads, using relations 50-52, to B" = 2.05 and B"" = 0.80. Using these three values and returning to eq 50 one can evaluate AG'" = AG'- RTB"h(0) and trace curve of AG"'vs. g(0). Figure 2 shows that one obtains now a straight line with a slope giving B" = 2, which is correct and justifies the use of a more general isotherm. Further, for 0 = 0, (AG')O = AG,,' - RTB* RTB" = -6.05 1.21 = -4.84 kcal/mol, which is very close to the extrapolated value: -4.7 (Figure 1). Figure 5 gives the results for the adsorption of 2-methyl-2propanol at a mercurysolution interface.25 These examples reveal that it is simply the use of an incomplete isotherm which gives curves, as opposed to a straight line, for the variation of the energy of adsorption AG'with g(0). The inclusion of a supplementary term is sufficient to establish a correct linear relationship and hence facilitate the graphic evaluation of the coefficients B" and B'". The use of numerical techniques for the determination of all the different parameters would certainly yield the most exact values and would correspond closely to the methods of analysis of the adsorption isotherms which have been developed by Mohilner.21
+
+
VI. Conclusion Our object being to evaluate, from experimentally determined isotherms, the activity coefficients of the species adsorbed, in order to be able to analyze the various solutesolute-solvent interactions which partly determine the conditions of adsorption, it was considered important to test recent equations which describe the adsorption equilibrium. It has been shown that isotherm 20, though easy to handle, considering the existence of only one adsorption coefficient of interaction, does not give a correct picture of the experimental results obtained (curves) and that it leads to incorrect values for B" which are greater than the critical value given by this equation (B" = 2). It is thus necessary in most cases to replace this isotherm by one based on a more general relationship between In c and 0, which has been deduced from Mohilner's theory and which includes several interaction coefficients B,". In a next article we shall show how it is possible to relate these coefficients to parameters characteristic of the adsorbed molecule, such as the surface tension of the pure substance and the size of the molecule, and to what extent they really represent solute-solute-solvent interactions. (25) A. Mazhar, R. Bennes, P. Vanel, and D. Schuhmann, J . Electroanal. Chem., 100, 395 (1979).
1496 The Journal of Physical Chemistry, Vol. 88, No. 8, 1984
Acknowledgment. I express my grateful thanks to Professor E. Verdier, former head of this laboratory, for his help in the preparation of this paper. Appendix It is easy to show, utilizing eq 5, 11, and 12, that in the general case (eq 8) in the condensed form for dilute solutions k
Bc = f(0) exp[ C Bjutj(0)] j=1
(1-1)
Tronel-Peyroz Blomgren and and more recently Afanasiev et al.33 An for example of eq 1-6 is the isotherm of Kasterring and H01leck~~ which K'(0) = -2aO. Now if the ratio O / ( 1 - 0) for relation 1-5 can be deduced from expression 1-3 when n = 1, this condition applied to the term K(0) in the same equations does not lead to the expressions given by the various authors. Similarly, one finds K'(0) = -2aO from eq 1-1 only if one puts n = 1. This being so, one is faced with the problem of how to maintain the coefficient (in n) which precedes the term exp[K'(O)] in isotherm 1-6. Taking as an example the recent isotherm proposed by Afanasiev:
with
From the first relation putting n = 1 and k = 2, and AGOF= AGad,' - RT(Ba - B"), one obtains12 the Frumkin isotherm. This undoubtedly represents the result of a clear simplifying hypothesis. The same can be said of the Langmuir relation for which n = 1 and k = 1, which amounts to considering that the adsorbed phase as an ideal phase. In the case of Bockris and Swinkelsz6 their isotherm corresponds strictly to the case where n # 1 and Bju = 0 v j . This also comes down to considering the adsorbed layer as an ideal phase but takes into account the size of the molecules adsorbed. As for the other isotherms put forward, the hypotheses on which they are based are not always as clear. Apart from the cases mentioned above all the isotherms can be represented by only two general expressions:
e
Bc = 1 - 0 exp[K(O)]
(1-5)
Equs ion 1-5 corresponds to the isotherms of V ~ l m e r , ~ Ama' gat,z8Helfand, Frisch, and L e b o w i t ~Hill-de , ~ ~ Boer,30Parsons,31 (26) J. O M . Bockris and D. A. J. Swinkels, J. Electrochem. Soc., 111, 736 (1964). (27) M. Volmer, 2.Phys. Chem., 115, 253 (1925).
where BA is the potential-dependent adsorption equilibrium constant, a the potential-dependent interaction constant, and nD the double-layer parameter describing the relation between Frumkin and Parson's models of the interface.35 Karolczak and Mohilner'* have shown that nD = l / n where n is the volume ratio and that hence eq 1-7 can be written also as BAc =
0 1-0
-
+ (1 - n)O] [n + (1 - n)0]2
0[2n
I
(1-8)
One then observes that, if 0/(l - 0) is deduced fromf(O), for n = 1, the term between brackets is obtained from Bjtj(0) for n # 1 and for k = 2. It is thus, more often than not, difficult to establish without ambiguity the physical significance of the hypotheses which have led to most of the known isotherms. One must note that Afanasiev's isotherm36is identical with the one which will now be discussed herez0except for the term 0 / ( 1 - 0), which replacesf(0) (eq I-3), and therefore the same limits will also apply to it.
xjzlk
(28) R. Parsons, Trans. Faraday SOC.,51, 1518 (1955); 55, 999 (1959). (29) E. Helfand, H. L. Frisch, and J. L. Lebowitz, J . Chem. Phys., 34, 1037 (1961). (30) J. H. de Boer, "The Dynamical Character of Adsorption", Oxford University Press, London, 1953. (31) R. Parsons, J . Electroanal. Chem., 7, 136 (1964); 8, 93 (1964). (32) E. Blorngren and J. O'M. Bockris, J . Phys. Chem., 63, 1475 (1959). (33) B. N. Afanasiev, B. B. Damaskin, G . J. Avilova, and N. A. Borisova, Elektrokhimiya, 11, 593 (1975). (34) Reference 24, p 95. (35) B. B. Damaskin, Elektrokhimiya, 6, 1135 (1970). (36) M. Karolczak, J . Electroanal. Chem., 122, 373 (1981).
