J. Chem. Soc., Faraday Trans. I, 1984, 80, 3315-3329 Generalized Flory-Huggins Isotherms for Adsorption from Solution BY PANAGHIOTIS NIKITAS Laboratory of Physical Chemistry, University of Thessaloniki, Thessaloniki. Greece Received 2 1st February, 1984 A new approach to the study of solutions composed of two kinds of molecules differing in size and shape has been developed on the basis of the lattice statistical model of athermal solutions. The results have been used for determination of the adsorption isotherms of organic compounds adsorbed from solution onto homogeneous surfaces. It is shown that the resulting isotherms are generalized forms of the Flory-Huggins isotherm, to which they are reduced by Flory's approximation. The possible ways of introducing the interactions between adsorbed molecules have also been examined.Finally, a critical examination of the validity of the isotherms is made by comparison with experimental data. The selection of the appropriate adsorption isotherm is one of the fundamental problems faced during analysis of experimental adsorption data. This is because the adsorption isotherm does not just describe the equilibrium between the concentrations of the adsorbate in the adsorption layer and in the bulk of the phase from which the adsorption is taking place. It is also the mathematical expression of a model of the adsorption layer. Therefore correct evaluation of the standard free energy of adsorption and obtaining reliable information about the interaction forces between the adsorbed molecules are only possible when the correct isotherm has been selected.For analysis of experimental adsorption data various isotherms have been emp1oyed.l The most widely used isotherms are those of Fr~mkinl-~ and Langmuir.l* However, these isotherms, though simple and effective, have the disadvantage of not accounting for the presence of the solvent molecules in the adsorption layer. The assumption that adsorption from solution is a substitution process of the pre-adsorbed solvent molecules by the adsorbate has led to the development of isotherms which take into account the size of the adsorbed particle~.~-l~ The most interesting of these isotherms are those of the Flory-Huggins-type:6y (1) 0 (1 -O)rexp(r- 1) pc = 0 exp ( - 2aO) pc = (1 - e y where p is the adsorption equilibrium constant, c is the bulk concentration of the adsorbate, 0 is the surface coverage, r is the molecular size ratio or the area ratio parameter and cc is the interaction parameter.Eqn (1) and (2) have received attention in studies of electrochemical adsorption Eqn (1) results from the assumption that the adsorption layer behaves as an athermal lattice interfacial solution of molecules of different size and by the 33153316 ISOTHERMS FOR ADSORPTION FROM SOLUTION approximation that the coordination number z of the adsorption layer tends to infinity (Flory’s appro~imation~l).~ Although this approximation leads to simple equations it is not accurate, especially when the dimensions of the molecules of the adsorbate and the solvent are significantly different.,l In addition, in eqn (1) the lateral interactions and the adsorbate-adsorbent interactions are included in the equilibrium constant a.Thus it is difficult to obtain information about the intermolecular particle-particle interactions. The term exp (- 2aO) is included in eqn (2) to allow for these interactions. However, the isotherm from eqn (2) deviates from the experimental data. In the present work a new, fairly simple and accurate method is developed for the study of athermal solutions. The results of this method are applied to the study of adsorption from solution and specifically to the determination of the adsorption isotherm. It is proved that the resulting isotherms are generalized forms of the basic Flory-Huggins isotherm, eqn (l), to which they are reduced by Flory’s approximation. Possible ways of introducing molecular interactions into these isotherms are also examined.Finally, the isotherms obtained are critically tested against experimental data for adsorption of methyldiphenylphosphine oxide (MDPO), ethyldiphenylphos- phine oxide (EDPO), triphenylphosphine oxide (TPO), triphenylphosphine (TPP), triphenylarsine (TPAs) and triphenylantimony (TPSb) on Hg from methanolic solution. ADSORPTION ISOTHERMS FOR ATHERMAL INTERFACIAL MIXTURES The lattice theories of the liquid state21922 are based on the assumption that the molecules of a liquid system are arranged in space according to a certain lattice. Each molecule can occupy only one lattice site (monomer) or simultaneously r sites (r-mer). In the case of a binary system of monomers with r-mers, N, and N, will denote the corresponding number of monomers and r-mers in the lattice.Obviously the following relations are valid: N,+N2 = N (3) N, -+ rN, = A4 (4) where N is the total number of the molecules and M is the number of lattice sites. When the binary system of monomers and r-mers has zero mixing energy (athermal solution) then the thermodynamic properties are easily determined provided that the factor a of Guggenheim and McGlashan21g23 has been determined. In principle, a is the ratio of the probability that a group of r sites, congruent with the r-mer, is wholly occupied by a single r-mer to the probability that the group is entirely occupied by monomers. The relative probability a can be determined by the methods of Guggenheim and McGlashan2l9 23 or Brzo~towski,~~ and the method of Guggenheim and McGlashan is considered to be the most accurate.However, it has the disadvantage that for each category of r-mers different and rather complex relationships for a are valid. In the following, a new and simple method for the calculation of a is developed. DETERMINATION OF THE RELATIVE PROBABILITY a ATHERMAL BINARY SOLUTIONS CONTAINING RIGID MOLECULES In order to proceed to a determination of a it is necessary to define first the parameters bi as follows. p denotes the number of allowed distinguishable ways inP. NIKITAS 3317 which an r-mer can be placed in the lattice. Of the r sites which can be occupied by a single r-mer, the 1,2, . . ., i (i < r) sites are already occupied by monomers.Now if the i+ 1 site is occupied by an element of an r-mer, then the number of the available sites for the r-mer in the lattice is not equal to p but to p- bi. If P, is the probability for the group of r sites to be occupied only by monomers, then we have where Pr/l, 2 , ..., ( r - l ) is the conditional probability of the r site being occupied by a monomer when all the remaining r - 1 sites are also occupied by monomers. For the conditional probability we have Pr = q r - 1 ) pr/1,2, ..., (r-1) ( 5 ) where 0 = rN2/(Nl i- rN2). (7) Therefore However, since Pi = 0 / p where Pi is the probability of the group of r sites r-mer, for the relative probability a we will have where b; = b i / p . (9) being completely occupied by an - b; 0) In table 1 the results obtained using eqn (10) are compared with those obtained by the methods of Guggenheim and McGlashan and Brzostowski and also by Flory's approximation.Table 1 also shows the values of b; used for these calculations, determined on the basis of their definition from the geometrical characteristics of the lattice. In certain cases, as for example in the case of a mixture of square tetramers with monomers in a plane lattice, more than one set of b; values were determined. In these cases the set of b; values which minimizes the free energy of the system [eqn (16) below] was selected. Note that the results of eqn (10) are in a good agreement with those of Guggenheim and McGlashan, which means that eqn (10) offers a good approximation of a. Moreover the advantages of eqn (10) are that it is valid for any value of r and it is relatively simple.ATHERMAL BINARY SOLUTIONS CONTAINING FLEXIBLE LINEAR r-MERS In this case prior to a determination of a it is necessary to define the parameter q.21 Consider an r-mer occupying a given group of r sites. Each of these sites has z neighbouring sites. We denote by zq the number of pairs of neighbouring sites of which one is a member of the group occupied by the given r-mer and the other is not. Under the condition that the r-mers are linear and flexible molecules it can be approximately assumed that - b i = b, i = 1,2 ,..., r. (12) The parameter 6can be determined as follows. In a group of r sites, which can be occupied by one r-mer, the r - 1 sites are already occupied by monomers while to the3318 ISOTHERMS FOR ADSORPTION FROM SOLUTION Table 1.Values of the relative probability a in athermal solutionsa ~ mixtures of monomers with trimers on a plane lattice: r = 3, z = p = 6, mixtures of monomers with trimers on a spatial close-packed lattice: and bi = 0.29 bi = 0, b’, = 0.33 and bi = 0.50 r = 3, p = 24, bk = 0, b’, = 0.167 6 I IT I11 IV I I1 I11 IV 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.02 0.02 0.02 0.02 0.0 1 0.06 0.06 0.05 0.07 0.02 0.1 1 0.1 1 0.10 0.15 0.03 0.2 1 0.21 0.18 0.31 0.06 0.42 0.42 0.34 0.67 0.13 0.88 0.89 0.67 1.56 0.29 2.15 2.22 1.55 4.32 0.76 7.33 7.7 1 4.91 16.67 2.77 57.75 62.39 35.65 150.00 23.51 0.0 1 0.02 0.03 0.06 0.13 0.29 0.77 2.8 1 24.07 ~ 0.0 1 0.01 0.0 1 0.02 0.03 0.04 0.06 0.08 0.12 0.17 0.26 0.39 0.66 1.08 2.35 4.17 19.41 37.50 mixtures of monomers with squares on a plane lattice : r = 4, z = p = 4, and bi = 0.75 mixtures of monomers with squares on a simple cubic lattice : r = 4, z = 6, and bi = 0.58 bI, = 0, bi = 0.50, bi = 0.50 p = 12, b; = 0, Bi = 0.33, b; = 0.33 e I I1 I11 IV I I1 I11 IV 0.1 0.03 0.03 0.2 0.08 0.08 0.3 0.18 0.18 0.4 0.35 0.35 0.5 0.71 0.73 0.6 1.58 1.71 0.7 4.34 5.02 0.8 18.00 24.00 0.9 221.20 424.81 0.03 0.07 0.13 0.23 0.41 0.77 1.69 5.33 45.64 0.04 0.12 0.3 1 0.77 2.00 5.86 21.61 125.00 2250.00 0.01 0.01 0.01 0.01 0.03 0.03 0.03 0.04 0.07 0.07 0.06 0.10 0.15 0.15 0.11 0.26 0.33 0.34 0.22 0.67 0.8 1 0.86 0.50 1.95 2.5 1 2.75 1.36 7.20 11.95 13.91 5.62 41.67 174.56 225.06 69.42 750.00 a I, Results of eqn (10); 11, Results of Guggenheim and McGlashan; 111, Results of Brzostowski ; IV, Flory’s approximation.rth site enters the first segment of an r-mer. From the z sites which can be occupied by the second segment of the r-mer only the zq/r are available. However, to the z possible sites for the second segment correspond p possible sites for the whole molecule. Therefore to the zq/r sites available to the second segment of the r-mer correspond p q / r available sites for the whole molecule. Therefore and GuggenheimZ5 and Huggins26 arrived at the same relationship though they used different methods.P. NIKITAS 3319 BASIC THERMODYNAMIC RELATIONS The relative probability a is related to the entropy ASm and the free energy AAm of mixing through:23 AAm - Asm - - ( -joo lna d8+8j01 lna do). RT R r-(r-l)8 In the case of a binary system of monomers with rigid r-mers, from eqn (10) and (1 5 ) we obtain: Xr-l (l-b;O)ln(l -b;8) - O(l-b;)ln(l-b; + B z ( i-0 b; b; where X = 8 / [ r - ( r - 1) 81.comes from eqn (14) and (1 5 ) : The corresponding relationship for a system of monomers with flexible linear r-mers ADSORPTION ISOTHERM FOR A MIXTURE OF MONOMERS WITH RIGID r-MERS For the determination of the theoretical isotherm we use the following adsorption model: (a) the adsorption layer has a regular lattice structure, (b) the N , molecules of the solvent and the N , molecules of the adsorbate occupy the lattice sites of the adsorption layer in exactly the same way as they occupy the lattice sites in an athermal mixture of monomers with r-mers and (c) the adsorption takes place according to: organic (in solution) + r solvent (ads) + organic (ads) + r solvent (in solution).(1 8) The molecules of the solvent and the adsorbate form in the adsorption layer a binary athermal mixture. Therefore the total partition function Q coincides with the partition function of an athermal mixture of monomers with r-mers provided that in the latter the effect of the field of the adsorbing surface is introduced through the internal partition functions qi (i = 1 or 2) of the molecules of the solvent and the adsorbate, respectively. Then we have : In Q = N , In q1 + N , In q2 + ASm/k (19) or In Q = N , In q1 + N , In q2 - N , In ( 1 - 8) - N , In 8 M N1+(1-bb;)rN2 ) -3A1- b;)] (20) +?[qf( 2-0 M b; where Ax) = x In x. and p$ds of the solvent and the adsorbate at the interface are obtained: If eqn (20) is differentiated with respect to Nl and N2, the chemical potentials pgds3320 ISOTHERMS FOR ADSORPTION FROM SOLUTION or and similarly where 0 is obtained from eqn (7) and represents the coverage of the adsorbed layer by the molecules of the adsorbate.For the chemical potentials in the bulk solution the following well known relations are valid: (23) (24) where a,, yi and Xp are the activity, the activity coefficient and the molar fraction, respectively, of the ith component and c and cs are the concentrations of the adsorbate and the solvent, respectively. If we assume that the reference state for the chemical potentials is the unsymmetrical ~ y s t e r n ~ ~ ? ~ ~ then in the region where Henry's law is valid we have PA =/L;+kTInaA =p;+kTIn()'AXb,) X p",kTln(y,c/c,) ps =pg+kTlna, =&+kTln(y,X;) x pg+kTlnys YA=1, YS=l (25) and therefore p A = p i + k T In (c/c,) (26) When thermodynamic equilibrium is established between the adsorbed particles and those in the solution then from the assumption (c) we obtain pi" - rpids = pA - rpS.(28) Now if eqn (21), (22), (26) and (27) are substituted into eqn (28), the adsorption isotherm is obtained : Ll r-1 where W n (I-biO) pc = (1 -6)rexp[-(A+1)] i=o r-l (1 -bi) ln(1 -bi) A = x ( bi i-0 ADSORPTION ISOTHERM FOR A MIXTURE OF MONOMERS WITH LINEAR FLEXIBLE T-MERS Because in this case - - r - 4 ) / r (3 3) b! = b' = b/p = ( the isotherm which is obtained after the substitution of eqn (33) into eqn (29) and (30) has the following form: r-I 0 (1 -0).exp[-(B+ l)] pc = (34) where (3 5 )P. NIKITAS 332 1 PROPERTIES OF EQN (29) AND (34) Eqn (29) and (34) are generalized expressions of the Flory-Huggins isotherm, eqn With Flory's approximation, z -+ co, the following relations are valid: (l), and they can be simplified in the following cases. lim b; = lim[(r-q)/r] = 0 and lim A = limB = - r (36) %-+a z+co and eqn (29) and (34) are reduced to the Flory-Huggins isotherm, eqn (1). When the adsorbate and the solvent have the same size then we have: r = l , b ; = O and A = B = - r . (37) If the above relationships are introduced into eqn (29) and (34) then Langmuir's isotherm is obtained : (38) Langmuir's isotherm is also obtained when the solvent is assumed to be a continuous medium.Then the following relations are valid: 6/( 1 - 6) = (c/cs) exp (- AGo/k7). 41= 1, Ps=O (39) which also lead to Langmuir's isotherm. Finally, when 6 6 1 from eqn (29) and (34) we have pc = 6 (40) which means that these isotherms are reduced to Henry's isotherm. ADSORPTION ISOTHERMS FOR NON-ATHERMAL INTERFACIAL MIXTURES Eqn (29) and (34) are valid when the interfacial mixture of the adsorbate and solvent molecules possesses zero energy of mixing. Therefore these isotherms must be extended in order to include all the possible interactions between the adsorbed particles. This can be achieved if appropriate activity coefficients are introduced into the chemical potentials of the adsorbed molecules. If fA and fs are the activity coefficients of the adsorbate and the solvent in the adsorption layer, then the isotherm resulting from eqn (29) is (41) and similarly from eqn (34) we obtain: (42) On the molecular level the ratiofA/f& can be determined either by the Bragg- Williams approximation21 or by the quasi-chemical approximation.21 homogeneous molecules the Bragg-Williams approximation leads to [n the case of (43) FAR 1 1083322 ISOTHERMS FOR ADSORPTION FROM SOLUTION while the quasi-chemical approximation yields : where u = q6/[q6+r(l -131, v = 1 -U (45) K = 2/(b + l), b = [ 1 + 4uv(v2 - I)]; (46) v = exp(w/2kT).(47) In the above relations, w is defined in such a way that the contribution of each contact of elements of the adsorbate and solvent molecules to the configurational potential energy is (WAA -I- wss + W ) / Z where wAA and wss are the interaction energies between respective contacts of molecules of the same kind.By means of Flory’s approximation, eqn (43) is reduced to The adsorption isotherm in this crude approximation, which results from eqn (41) and (48) or eqn (42) and (48), is 6 exp [a( 1 - 26)] (1 -6)+exp(r- 1) pc = (49) where a = rw/kT. Note that this isotherm is of the same form as eqn (2). At a more general although macroscopic level the ratiofA/’k can be determined in the following way. The introduction of the activity coefficients, fA and fs, is necessarily equivalent to the assumption that there exists an excess free energy g: with respect to the free energy of an athermal binary mixture. Therefore by analogy with the excess free energy ge with respect to an ideal mixture we assume that g: can be expanded as a power series of the molar fractions XA and Xs of the adsorbate and the solvent in the interfacial region:29 In this case the activity coefficients are determined by:29 In the above equations XA and Xs can be expressed as functions of coverage 8 by: XA = 6/[6+r(l-6)] and XA+Xs = 1.(54) This way of introducing particle-particle interactions has also been used by Mohilner et aZ.,8 who instead of gz used ge, i.e. the excess free energy with respect to an ideal mixture. Therefore the values of Bi are expected to be determined by theP. NIKITAS 3323 contribution of the particle-particle interactions and to some extent by the size difference of the adsorbed molecules. On the other hand, in the present work the effect of the size difference is included in the configurational terms of eqn (41) and (42).Consequently in the present approach the values of Bi are mainly determined by the contribution of the particle-particle interactions. COMPARISON WITH EXPERIMENT AND CONCLUSIONS I Eqn (41)-(43) and (49) may be expressed as fl0) = AGo'/RT = AGo/RT+ In (fA/fL) ( 5 5 ) where for the case of rigid r-mersfl8) is given by For the case of the adsorption of flexible linear r-mers we have and finally for the case when Flory's approximation is employed we have From eqn (55) the tests of these isotherms can be performed as follows. First the experimental values of the free energy AG::, = RTfl8) (59) are determined directly from 8 against c data and the values are compared with the corresponding theoretical values obtained from AGE;,, = AGO + RTln CfA/fL). The values of AGO and Bi (or F and a) which are included in this equation can be determined by the least-squares method from AGO' against 6 data.In the case where eqn (53) is employed for the determination of ln(fA/fL) it is obvious that the more Bi terms used the better the fit of the experimental values will be. However, an increase in the number of the Bi terms leads to a decrease in the number of degrees of freedom of the least-squares fit and consequently in the amount of smoothing. Therefore a careful examination of the results of the least-squares fit is required. In this case we can take as a criterion of the fit the value of 0, the root-mean-square deviation between calculated and experimental values of AGO' : N 0' = ~ [ ( A G ~ ~ , , - A G ~ ~ p ) / R T ] 2 / N (61) where N is the number of data points.The fit can be considered as generally satisfactory for r7 values < ca. 0.2. In the present work the isotherms were tested against experimental data for adsorption of MDPO, EDPO, TPO, TPP, TPAs and TPSb on Hg from methanolic solutions of LiCl. The adsorption was studied by means of electrocapillary measure- ments. A description of the experimental apparatus and details of the results have been reported 31 108-23324 ISOTHERMS FOR ADSORPTION FROM SOLUTION Table 2. Data for the adsorption isotherms used in this work isotherm r AGo/RT B,(a or B") B, B2 B3 0 Frumkin Bennes eqn (41) and (43) eqn (53) and (58) eqn (49) eqn (41) and (53) Frumkin Bennes eqn (49) eqn (41) and (43) eqn (53) and (58) eqn (41) and (53) Frumkin Bennes eqn (41) and (43) eqn (53) and (58) eqn (49) eqn (41) and (53) Frumkin Bennes eqn (41) and (43) eqn (53) and (58) eqn (49) eqn (41) and (53) MDPO = 2, b', = 0.333, b; = 0.50 1 -6.61 -0.39 - - - 4.35 -6.09 1.00 - - - 3 -5.89 1.68 3 -6.96 1.07 - - - -6.18 1.46 - - - 3 -5.96 0.99 -0.48 -0.34 -0.11 3 -7.11 1.03 - - - -6.96 0.73 -0.35 -0.38 -0.07 - - - EDPO 4 = 2, b', = 0.333, b; = 0.50 1 -7.11 - 4.75 -6.85 3 - 6.72 3 - 7.78 3 - 6.94 - 6.99 3 - 7.90 - 7.99 - 0.1 1 - 1.33 - - 2.61 - 1.71 1.63 - 1.76 0.16 0.42 1.31 - 1.48 0.29 0.37 - - - - - TPO q = 3, b; = 0.167, b; = 0.333, bj = 0.50 1 -7.65 0.06 - - 5.45 -7.56 1.64 - - 4 -7.05 5.78 - - 4 -8.18 4.15 - - 4 -7.78 2.28 - - 4 -8.67 2.06 - - -7.73 2.37 -0.03 0.56 -8.73 2.12 0.1 1 0.43 TPP q = 3, b; = 0.167, b; = 0.333, bi = 0.50 1 - 10.78 5.51 -10.56 4 - 10.77 4 - 11.70 4 - 10.66 - 10.69 4 - 11.63 -11.69 - 0.07 - - 1.84 - - 5.48 4.16 - 2.46 2.32 -0.09 0.67 2.22 2.06 0.05 0.58 - - - - - - - 0.08 0.09 0.22 0.15 0.08 0.07 0.08 0.07 - 0.1 1 - 0.14 - 0.44 - 0.32 - 0.14 0.09 0.10 - 0.14 0.12 0.10 0.10 - 0.34 - 1.12 - 0.94 - 0.27 0.16 0.09 - 0.27 0.21 0.08 - - 0.17 - 0.82 - 1.80 - 1.55 - 0.60 0.44 0.17 - 0.62 0.49 0.15P.NIKITAS Table 2. (cont.) 3325 isotherm r AG"/RT B,(a or B") B, B2 B3 d Frumkin Bennes eqn (49) eqn (41) and (43) eqn (53) and (58) eqn (41) and (53) Frumkin Bennes eqn (41) and (43) eqn (53) and (58) eqn (49) eqn (41) and (53) TPAs q = 3, b; = 0.167, b; = 0.333, bi = 0.50 1 -10.99 - 5.60 -10.83 4 - 11.00 4 - 11.91 4 - 10.93 - 10.94 4 - 11.90 - 11.94 - - 0.03 1.77 5.56 4.20 - - 2.40 2.32 -0.10 0.62 2.17 2.06 0.04 0.53 - - - - - - - - TPSb = 3, b; = 0.167, 6; = 0.333, bi = 0.50 1 -12.60 0.03 - - 5.82 -12.13 1.97 - - 4 -10.72 9.12 - - 4 -12.16 6.28 - - 4 -12.54 2.47 - - - 12.86 2.26 -0.03 0.26 4 -13.37 2.31 - - -13.84 2.01 0.10 0.3 1 0.09 0.55 1.64 1.36 0.41 0.09 0.41 0.09 0.10 0.82 1.64 1.44 0.63 0.13 0.60 0.13 For the analysis of the experimental data and for the test of the theoretical isotherms the following simplifying assumptions were made.It was assumed that the adsorbed phase consists of a unimolecular layer of adsorbed solvent and adsorbate molecules. In this approximation the parameter Y is equal to the integer closest to the area ratio sA/sS I Y = int (S,/S,) (62) where SA and Ss are the areas covered by an adsorbate and solvent molecule, respectively, on the saturated surface.The values of SA were obtained from ref. (30) and (3 l), while for Ss a value of 0.19 nm2 was used.6 The values of r resulting from eqn (62) are given in table 2. For the calculation of the 6; parameters it was assumed that the adsorbed molecules at the interface occupy the sites of a regular hexagonal lattice. The values of b; are also given in table 2. Fig. 1-3 show plots of the free energy AGO' against the surface coverage 6 at various potentials in the potential range where the isotherms are congruent. The points correspond to experimental values of AGO' while the lines represent theoretical values of AGO' determined from eqn (60).The various parameters required for the calculation of were determined by the least-squares method and are given in table 2, which also gives the data of fitting of Frumkin's isotherm:1~30 (63) PC = [8/( 1 - O)] exp (- 2aO) and Bennes isotherm:10* 303326 ISOTHERMS FOR ADSORPTION FROM SOLUTION -6 -7 -7 h w u 4 --- 0 -9 -1 1 0.5 1.0 e Fig. 1. Tests of the isotherm from eqn (41) and (53) for the adsorption of (a) MDPO and (b) EDPO on Hg at: 0, Emax; 0, -0.5 and A, -0.7 V (us SCE). Points are experimental data plotted according to eqn (56) and (59), solid lines are calculated from eqn (53) and (60) using four Bi terms taken from table 2 and broken lines are calculated according to eqn (43) and (60). - 6 -1 0 h 3 0 u 4 - 14 -9 -13 -17 - 1 8 1 , , -21 0.5 1 .o 8 I 1 0.5 1.0 e Fig.2. Tests of the isotherm from eqn (41) and (53) for the adsorption of (a) TPO and (b) TPP on Hg. Symbols as in fig. 1 with 0, - 0.6 and A, - 0.8 V (us SCE). For TPP, 0 indicates - 0.4 V (us SCE).P. NIKITAS 3327 -1 0 h w "u -15 d . - 20 0.5 0 1.0 -12 -1 7 -22 \ \ \ \ 0.5 1.0 8 Fig. 3. Tests of the isotherm from eqn (41) and (53) for the adsorption of (a) TPAs and (b) TPSb on Hg. Symbols as in fig. 1 and 2. 0.6 0.2 0.0 0.0 0 . 5 1.0 Fig. 4. Change in the excess free energy gz of adsorption of (1) MDPO, (2) EDPO, (3) TPSb and (4) TPO, TPP and TPAs as a function of their molar fraction. gz is calculated from eqn (50) using four Bi terms of eqn (41) and (50) taken from table 2.3328 ISOTHERMS FOR ADSORPTION FROM SOLUTION and as well as the values of 0. From fig.1-3 and from the values of 0 the following conclusions can be drawn. The simple Flory-Huggins isotherm, eqn (49), deviates from the experimental data, with increasing deviation as r increases. The isotherm of eqn (41) and (43) is slightly better than the Flory-Huggins isotherm and it follows the experimental data qualitatively. Essentially identical to the results obtained using eqn (41) and (43) are those obtained using eqn (41) and (44), which is based on the quasi-chemical approximation. This means that in this case the quasi-chemical approximation does not improve the Bragg-Williams approximation. The description of the adsorption layer becomes quantitative only when using eqn (53). Note that three or four Bi terms are enough for the attainment of satisfactory results.It is also characteristic that if only the Bo term is used the results are still better than the results obtained using Bennes isotherm . From the values of Bi and using eqn (50) the excess free energy gz can be calculated. The excess free energy g: can be considered as a measure of the strength of the interactions between the adsorbed molecules. Fig. 4 shows a plot of g: against the molar fraction of the adsorbate in the adsorption layer. Note that gz is positive over the complete range of X , values. Its values for TPP, TPAs, TPSb and TPO are almost the same, while for the phosphinoxides g: decreases in the order TPO > EDPO > MDPO. The positive values of gz reveal that the solvent as well as the adsorbate molecules tend to cluster on the electrode, while the decrease from TPO to MDPO reflects an analogous increase in the interfacial solubility of these substances in the same order.This seems to be reasonable as the decrease of the attractive London interactions and the increase of the repulsive interactions due to the P = 0 dipole are ordered in the same way.3o In addition, it seems reasonable that TPP, TPAs, TPSb and TPO have, within experimental error, the same value of g;. These substances have the same dimensions and they interact with London forces between their phenyl groups. For the case of TPO we expect the dipole-dipole interactions to be markedly weak because the dipoles of the adsorbed molecules of this substance are separated from each other.For the isotherm resulting from eqn (41) and (53) using Flory’s approximation, i.e. eqn (53) and (58), it is seen that, regardless of the difference of the values of the relative probability a (table I), results analogous to those of the generalized isotherm of eqn (41) and (53) are obtained. Therefore this isotherm can be used instead of eqn (41) and (53) or eqn (42) and (53) for the adsorption of substances with unsymmetrical molecules. Note that eqn (41) and (53) provide a good quantitative description of the adsorption of organic substances from solution. The same results are expected to be obtained using eqn (42) and (53). 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