Chronocoulometric Measurements of Azobenzene Adsorption on Mercury from Water+Ethanol Mixtures of Different Composition BY MARIA LUISA FORESTI, FRANCESCO PERGOLA AND ROLANDO GUIDELLI* Institute of Analytical Chemistry, the University, Via Gin0 Capponi 9, 50121 Florence, Italy Received 24th November, 1977 Chronocoulometric measurements of azobenzene adsorption on mercury at - 0.430 V/s.c.e. from water + ethanol mixtures are reported. Having assumed that, at ethanol concentrations Cx > 4 mol dm-3, water adsorption at the chosen potential is negligible, we interpreted the adsorp- tion data by using the adsorption isotherm KC:/C:n = €I1 e/nae'(l -€Il)., where C: and O1 are the bulk concentration and the surface coverage by azobenzene, n is the number of adsorbed ethanol molecules displaced by one azobenzene molecule, K is an adsorption coefficient and CL is an interaction factor.The possibility of varying both C: and C,* allows the configurational effect, due to a n value equal to 3, to be distinguished from the effect of adsorbate-adsorbate interactions, due to a nonzero cc value. A statistical mechanical derivation of the isotherm employed is provided. In a previous note by Pezzatini and Guidelli the changes in shape and half-wave potential of the polarographic waves for azobenzene reduction and hydrazobenzene oxidation, following an increase in reactant concentration,2 were quantitatively interpreted as due to azobenzene adsorption. In particular, on assuming langmuirian adsorption of azobenzene, an adsorption coefficient Kl = 2.8 x lo3 dm3 mol-1 was derived from polarographic measurements in a pH 9, 90 % (v/v) ethanol+water mixture.The present note deals with a chronocoulometric investigation of azobenzene adsorption on mercury from pH 9 water + ethanol mixtures of different composition. The aim of this investigation is to show how the measurement of the adsorption of a given surfactant from mixtures of two solvents, one of which is only negligibly adsorbed, may provide valuable information as to the competition between the adsorbed solvent and the surfactant for adsorption sites. EXPERIMENTAL APPARATUS Potential-step chronocoulometric measurements were carried out with a computerized system analogous to that described by Lauer et aL3 Thus the current was electronically integrated and the charge-time data acquired and analysed with the aid of a Data General 1220 digital computer.The potentiostat was an Amel Mod. 551 with positive feedback for IR compensation. A special hanging mercury drop electrode which could be submitted to a forced laminar flow of solution lasting for 15 ms under computer control, was used throughout. This electrode was used in connection with the cell described by Cozzi et aL5 In most cases an electrode area of 0.035 cm2 was used. REAGENTS Merck reagent-grade azobenzene was used without further purification. All solutions were prepared from triply distilled water treated with active charcoal. Mercury was 155156 ADSORPTION OF AZOBENZENE purified by a wet process followed by three distillations. The pH 9 buffer contained about 0.2 mol dm-3 NH3 and 0.2 mol dm-3 NH4CI.For further details, see ref. (1). PROCEDURE All solutions were deaerated with a stream of prepurified nitrogen, previously passed through two washing bottles containing the same solution as the electrolysis cell, to avoid a decrease in the ethanol and ammonia content during measurements. All test solutions were thermostatted at 25+ 0.2"C. A platinum wire was used as the counter-electrode. The reference electrode consisted of a saturated calomel electrode, to which all potentials herein reported are referred. At pH 9 azobenzene and hydrazobenzene are both electroinactive over a narrow potential range straddling - 0.430 V.l Azobenzene adsorption was determined by stepping the applied potential from the initial value Ei = - 0.430 V to the final value Ef = - 1.40 V, at which azobenzene is reduced under diffusion limiting conditions from the first milliseconds of electrolysis.The resulting values of the charge Q(t) against time t, averaged over 10 successive drops, were then analysed, via the computerized system, fitting them by least squares to a linear Q(t) against t 3 plot to obtain an intercept Qi+f on the charge axis. In all cases standard deviations for the slopes and the intercepts were no greater than 0.1 %. In a majority of the experimental runs the charge Q(t) was sampled at 1 ms intervals for a period z of 100 ms from t = 0 of the potential jump Ei -+ Ef. To separate the faradaic component of the intercept Qi+f, due to instantaneous reduction of the adsorbed azobenzene molecules, from the corresponding double-layer charging component, the latter was estimated by stepping the applied potential backward from Ef to Ei at time t = z.The charge Qf+i involved in the backward potential jump is exclusively capacitive. To achieve a fast equilibration of the double layer with respect to the bulk solution at Ei, just after the back- ward jump the mercury drop was submitted to a forced laminar flow of solution lasting 15 ms. The quantity Qf+i was estimated by subtracting the charge passed at time t = (z+ 15 ms), namely just after cessation of the forced convection, from the charge Q(z) passed immediately before the backward potential jump Ef-+ Ei. In any case, both with and without forced convection the capacitive charge Qf+i varied by no more than 1 pC cmU2 when varying the azobenzene concentration from 10-5-1.3 x mol dm-3, thus indicating that the charge density on the metal at Ei is only slightly affected by azobenzene adsorption.The surface concentration rl of azobenzene at Ei was calculated through the equation The polarographic behaviour of the azobenzene+ hydrazobenzene system indicates that hydrazobenzene adsorption is entirely negligible. No direct chronocoulometric measure- ment of hydrazobenzene adsorption could be carried out at pH 9, since the jump from the potential Ei = -0.430 V, at which both azo- and hydrazo-benzene are electroinactive, to the most positive potential at which mercury is not yet oxidized, does not allow diffusion- limiting conditions to be attained within a few ms.To achieve the latter result chrono- coulometric measurements of hydrazobenzene adsorption were carried out at higher pH values, taking advantage of the fact that the formal potential of the azobenzene-hydrazo- benzene couple shifts towards more negative values with increasing pH. Thus, the narrow potential range over which azo- and hydrazobenzene are both electroinactive lies at W - 0.63 V in a pH 11 phosphate buffer and at % -0.77 V in a 0.2 mol dm-3 KOH solution. Con- sequently in these media anodic potential jumps of -0.5 V, high enough to attain diffusion limiting conditions, could be performed, Under these conditions saturated solutions of hydrazobenzene in ethanol+ water mixtures of different composition yielded differences between the absolute values of the anodic intercept Qi+f and of the cathodic capacitive charge Qf+i which were no greater than 1 pC cm-2.This confirms the very low adsorptivity of hydrazobenzene. After having ascertained that azobenzene adsorptivity varies only slightly when passing from 0.2 mol dm-3 KOH solutions to pH 9 ammonia buffer solutions of equal ethanol content, we carried out chronocoulometric measurements of at the latter pH. In factM. L. FORESTI, F . PERGOLA AND R . GUIDELLI 157 in the neighbourhood of pH 9 the standard rate constant for azobenzene reduction to hydrazobenzene is a minimum, thus permitting a less critical choice of the initial potential Ei. Measurements of the charge density qM on the metal as a function of the applied potential in the absence of the reactant were carried out at a dropping mercury electrode by the same procedure described by Lauer and Osteryoung.6 RESULTS AND DISCUSSION Fig.1 shows the surface concentration rl of azobenzene, measured at Ei = - 0.430 V as described in the experimental section, against the corresponding bulk concentration Cf for different ethanol concentrations C,* in the ethanol + water mixture. In all these curves rl seems to approach asymptotically a maximum limiting value. This apparent limiting value, however, increases continuously as C,* 40 30 l-i I !z 1 Y k" k *O w 10 I I C: x 103/m01 dm-3 FIG. 1.-Plots of 2Fr1 against C: at -0.430 V in a pH 9 ammonia buffer. Ethanol concentration C,* = 15.65 (a), 8.7 (b), 6.07 mol dm-3 (c). Left-hand scale refers to curves (6) and (c) ; right-hand scale refers to curve (a).is decreased and hence is by no means related to the attainment of full coverage of the electrode surface by azobenzene. Such a conclusion is confirmed by the curves of rl against C,* at constant Cf in fig. 2. Incidentally, if the " apparent " limiting value of rl in curve (a) of fig. 1 is estimated at 6 yC ~ m - ~ / 2 F = 0.31 x 10-l' mol cm-2 and is formally equated to the maximum surface concentration then curve (a) turns out to satisfy a Frumkin isotherm with an apparent adsorption coefficient K1 = 1.8 x lo3 dm3 mol-l ; this value compares favourably with 2.8 x lo3 dm3158 ADSORPTION OF AZOBENZENE mol-l, that estimated from the half-wave potential shift of the azobenzene wave in the same medium,l as produced by an increase in Cf.However, a rl,m value of 0.31 x 10-lo mol cm-2 is decidedly too low to have any physical significance. This emphasises the limitations of the polarographic method for estimating reactant adsorption. In fact this " indirect " method is sensitive to deviations from the Henry-isotherm behaviour,' but not to the magnitude of Tm. I I * 1 1 I I I I 50' ! 40 - CI I E Sj- 30- 2 3 - 26 - 10- L I I I 1 1 I I 2 4 6 0 10 12 14 C:/mol dm-3 C: = (a), 2 x (b), 6 x (c), 7 x loh4 (d), mol dm-3 (e). FIG. 2.-Plots of 2Fr, against C,* at -0.430 V in a pH 9 ammonia buffer. Azobenzene concentration The true value can be regarded as equal to the height, x50 pC cm-2/ 21; = 2.6 x 10-lo mol cm-2, of the step exhibited by the rl against C,* curves of fig. 2 at low C,* values. The preceding Tl,m value matches the maximum surface con- centration, 2.55 x mol cm-2, as calculated from the area, x65 A2, projected by a Leybold space-filling model of the azobenzene molecule, placed with the two benzene rings flat on the electrode surface.The further increase in 2Pr1 beyond 50 pC cm-2, as shown by some of the curves in fig. 2, is probably to be ascribed to a closer packing of the azobenzene molecules, e.g. with the plane of the benzene rings perpendicular to the electrode surface.M. L. FORESTI, F. PERGOLA AND R. GUIDELLI 159 The gradual decrease of I?, as the ethanol content is increased is expiained by competitive adsorption of azobenzene and ethanol for adsorption sites. Experimental data will be interpreted on the basis of the rather general adsorption isotherms * * K,Cr = 8,[nl(l -8, -8,)ni]-1 exp (n, all 8, +nl a,, 0,) K,c,* = 8,[n2(1 -8, -e,)nz]-l exp (n, a,, 8, +n2 a12 0,) (1) (2) which account both for adsorbate-adsorbate interactions and for any differences in the size of the various adsorbed molecules.In eqn (1) and (2) K l , K2 are the adsorp- tion coefficients of azobenzene and ethanol, 8,, 8, are the corresponding surface coverages, and n l , n, are the numbers of water molecules displaced by one molecule to 0 6 pl -10 -20 V1s.c.e. FIG. 3.-Plots of the charge density qm against the applied potential E for different water+ethanol mixtures containing 0.2 mol dm-3 NH3 + 0.2 mol dm-3 NH4Cl. Ethanol concentration C: = 0. (solid curve), 0.87 (C), 1.73 (a), 3.47 (a), 5.21 (H), 8.68 rnol dm-3 (A). of azobenzene and of ethanol, respectively.Moreover, all, and a,, are inter- action factors of azobenzene molecules between themselves, of ethanol molecules between themselves, and of azobenzene with ethanol molecules. In practice, at ethanol concentrations which are not too low, water molecules are almost completely displaced from the adsorbed monolayer in direct contact with the electrode by ethanol and azobenzene molecules. Thus Arevalo et al., lo from polarographic measurements of the capacitive current of water + ethanol mixtures containing 0.1 mol dm-3 KCl, concluded that at C; 2 5.2 mol dm-3 surface saturation by ethanol molecules is attained over a wide potential range. For a further confirmation, we have measured the charge density 4m on the metal in different water+ethanol mixtures containing160 ADSORPTION OF AZOBENZENE 0.2 mol dm-3 NH3 +0.2 mol dm-3 NH,Cl (see fig.3). For C,* >/ 5.2 mol dm-3 the charge density 4m attains a limiting value between x -0.4 and = - 1.5 V, thus indicating that the compact layer does not change appreciably as the ethanol concentration is increased beyond 5 mol dm-3. Over the potential range from = -0.4- = -0.6 V the limiting value of qm is already attained at Cg = 3.47 mol d ~ n - ~ . We may therefore conclude that at the chosen potential Ei = -0.430 V water n * r ( I- L, 8 QPOO 0 I 1 I I I I I I I I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 61 FIG. 4.-Plots of {In (e1/C:)-n In [(l-O1)/C;)} against el = r1/(2.6x 10-lo mol cm-2) for C; = 6.07 (0) and 8.7 mol dm-3 (0) and variable C; ,as well as for C: = (a), 2 x (m), 6 x (A), 7 x (A), mol dm-3 (0) and variable C;.(a) n = 1, (b) n = 2, (c) n = 3, (d) n = 4 and (e) n = 5.M . L . FORESTI, F . PERGOLA AND R . GUIDELLI 161 molecules are almost absent from the adsorbed monolayer in direct contact with the electrode at C,* > 3.47 mol dm-3. Under these conditions, dividing eqn (1) by eqn (2) raised to the nl/n2 power and setting 8, + O2 z 1 yields with a = n1(all+a22-2a12); n = n,/n2. The parameter n is the number of ethanol molecules displaced by one molecule of azobenzene. Eqn (3) is substantially a Frumkin isotherm accounting for the different size of solvent and surfactant molecules, and is also expected to apply to adsorption of a single surfactant from a single solvent.g* l 1 9 l2 Eqn (3) has been derived in a piecemeal fashion from eqn (1) and (2), which in their turn were derived in a similar fashion 8 * by combining the Frumkin isotherm with the Flory-Huggins statistics.We therefore found it useful to derive eqn (3) directly from first principles via a generalization of the statistical derivation of the Frumkin isotherm l 3 (see Appendix). In the present investigation we are in the favourable condition to test eqn (3) by varying not only CT but also C;. This was accomplished by plotting {ln(O,/Cf) - n ln[(l -8,)/C~]) against 8, (see fig. 4) using the data of fig. 1 and 2. To this end Tl,m was equated to 2.6 x 10-lo mol cm-2 and n was alternatively equated to 1,2, 3, 4 and 5. The area projected by a molecular model of ethanol with the hydrocarbon chain normal to the electrode is ~ 2 0 A2 ;14 hence the n value which should comply best with the experimental value, 65A2, of the area covered by an azobenzene molecule at full coverage is 3.The scattering of experimental data in fig. 4 is indeed less for n = 3 than for n = 1 or n = 2. A still lower scattering is observed for n = 4 and 5, but the plots corresponding to these n values show an appreciable curvature at 8, > 0.6. At any rate the n values 4 and 5 are not physically significant. If the data relative to n = 3 are fitted to a straight line by least squares, a In K value of 13.7 and an a value of -3.58 are obtained. This implies that, on average, inter- actions between adsorbed molecules are attractive. The difficulty shown by 8, in approaching unity with increasing C,* must therefore be ascribed not to repulsive adsorbate-adsorbate interactions, but rather to a configurational effect due to the different size of azobenzene and ethanol molecules.The points in fig. 4 corresponding to n = 3 which show greater deviations from the average are those relative to C$ = 6.08 mol dm-3 and to 8, < 0.2. This residual data scattering is probably because at this relatively low ethanol concentration the surface coverage by water is not completely negligible, especially at low 8, values, thus partly invalidating the assumption (8, + 8,) = 1. In conclusion we note that measurements of adsorption of a solute from a single solvent are often insufficient to distinguish and separate the effect of a configurational term involving an n value greater than unity from the effect of a nonzero Frumkin interaction factor a.This result is more convenienty achieved by measuring adsorption of a solute from a mixture of two solvents, one of which is not detectably adsorbed. In fact in this case a given composition of the adsorbed monolayer in direct contact with the electrode at a given applied potential can be attained for different compositions of the bulk solution phase. K = Klnl(K2n2)-" exp [-n1(a12 -a22)] = (C~/C:)O,(l -el)-. exp (a€),) (3) APPENDIX Following Flory,16 assume that the surfactant Al consists of a linear sequence of n segments, any one of which may replace a solvent molecule S2 from the adsorbed monolayer in direct contact with the electrode. Let B be the number of adsorption 1-6162 ADSORPTION OF AZOBENZENE sites on the electrode, as created by the adsorbed solvent molecules prior to the adsorption of Al, and let N , be the number of A , molecules distributed among the B sites (B 2 nN,), one per n contiguous sites.The partition function for the present system is R Q(N,, B, T ) = qylq$B-nN1) exp (-EJRT). (Al) i = l In eqn (Al) q1 and q2 are the single-molecule partition functions without interactions relative to the surfactant A , and to the solvent S,, Ei is the total nearest-neighbour energy of interaction for the ith configuration, and fi is the total number of configura- tions. Assume that the n segments of any A , molecule are perfectly equivalent, as regards both their interchangeability with the solvent molecules and the magnitude of their interaction energies with adjacent solvent molecules or with adjacent segments of other A , molecules. Let ull, u22 and u12 denote the interaction energy between two adjacent segments of two different Al molecules, that between two adjacent solvent molecules, and that of a segment of an A , molecule with an adjacent solvent molecule. Eqn (Al) can then be written in the form U12N12 +u22N22)lkTI.(A21 In eqn (A2) N i l , NZ2, N,, are the numbers of pairs of adjacent adsorption sites occupied by segments of two different A , molecules, by two solvent molecules and by an A , segment and a solvent molecule, respectively. In addition, g(N,, By N i l ) is the number of configurations having Nil pairs of sites occupied by segments of two different A , molecules when there are N1 molecules of Al and B sites.Denoting the number of nearest neighbours for any given adsorption site by r, from elementary statistical considerations 179 where N 1 , is the number of nearest-neighbour pairs of sites occupied by two A l segments, independent of whether they belong to the same or to different A , molecules. Let us assume as a first approximation that all configurations of N, molecules of A l on B sites have the same weight as they would have if u,,, u12 and u22 were null.,'. l 8 With this assumption the canonical partition function of eqn (A2) is written it follows that, for B -+ co, rnN, = 2 N l 1 + N l 2 ; r(B-nNl) = 2N2,+NI2 (A31 where u l l ~ i l , u12N12 and uZ2iV2, are average interaction energies for molecules distributed randomly among the sites.nil is readily estimated by noting that, for a random distribution, an intermediate segment of an A molecule has (r - 2)(nN1 / B ) nearest-neighbour sites occupied by segments of different A , molecules. On the other hand a terminal segment of an A , molecule has (Y - l)(nN, / B ) nearest-neighbour sites occupied by segments of different A , molecules. Noting that the total number of intermediate segments is (n-2)N1 whereas that of terminal segments is 2N1, we have = [(n-2)(r-2)nNf/B+2(r- l)nNf/B]/2 = nqNf/2B (A51 with q -= nr-2n+2M . L . PORESTI, F . PERGOLA AND R . GUIDELLI 163 where the 1 / 2 factor prevents the counting of each pair of contiguous segments twice. Naturally, the value N, , assumed by N , for a random distribution equals PI 1 plus the number, (n- l)N1, of nearest-neighbour pairs occupied by segments of the same A , molecule The expression for the summation Nl = nqNf/2B+ (n + l)N1.(A6) g(N1, B, Nil) is provided by the Flory statistics l6 n; 1 C g(N1, B, iVi 1) = [(r - I)/B]("- ' ~ 1 n""[(B/n)!/(B/n - N1)!]"/2"N1 ! n; 1 (A71 where the 2" factor must be replaced by unity in the trivial case of n = 1. Upon combining eqn (A4), (A5) and (A7) and making use of the Isl2 and N22 values as derived from eqn (A3) and (A6), the following expression for the chemical potential p, of A , in the adsorbed state is readily obtained p l / k T = -(a In Q/i2V,),,, = In [2q",lql(r- 1)"-1]+[qu12-(m-n+ l)u2,]/kT+ where 8, E nNl/B is the fractional surface coverage by A , .In deriving eqn (A8) Stirling's approximation for factorials was adopted. In general the chemical potential bpl of A , in the bulk solution will have an analogous form, with different values of the various interaction energies. Under usual conditions, however, the analogue, = n VVJbB, of 8, for the bulk solution phase is much less then unity, so that the term proportional to in the expression for can be neglected. We may therefore write concisely where CT and Cs are the bulk concentrations of A , and of the solvent S2. At equilibrium p, is equal to bpi. Hence, from eqn (A8) and (A9), the following adsorption isotherm is obtained with and In [el/n(l -~1)"1+!?(u11 +u22 +2u12)(31lkT (A81 ",/kT = bpi/kT+h [b81/n(l -b81)n] = bpy'/kT+ln ( C f / C f . ) (A9) KCT/C$" = 8, exp (a 8,)/n(l-0,)n (fw K = ql(r- l)"-' exp ([bpi'+(rn-n+ l ) ~ ~ ~ - q u ~ 2 1 ) / 2 q ~ a = ~ ( U ~ ~ + U ~ ~ - ~ U ~ ~ ) / ~ T .The preceding adsorption isotherm holds even if A , is adsorbed from a mixture of the solvent S2 previously considered and of a further solvent S3, provided that only S2 and A , are adsorbed at the interphase. In fact, in this case, within the closed system consisting of the interphase and of the solution phase, the desorption of an A , molecule will only occur by replacement of this adsorbed molecule with n molecules of S2 from the solution phase. Assume that the solution phase consists of n bN1 + bN2 +&N3 = bB+ bN, cells containing bN, polymer molecules of A , , bN2 monomer molecules of S, and bN3 monomer molecules of S,.In estimating the total number bg(bN,, bN3, bB) of possible configurations within the solution phase, Flory's process l6 of successive addition of the bN1 polymer molecules of A , to the three- dimensional lattice must now be performed not with the lattice initially empty, but rather with a lattice already containing the bN3 molecules of S3 ; in fact these molecules are not available for replacement by segments of the polymer A l . The number of distinguishable arrangements of the biV, molecules within the initially empty lattice equals (bB+bN3)!/(bB!bN3!). For each of these arrangements there is a number W164 ADSORPTION OF AZOBENZENE of different arrangements of the bN1 molecules of A , in the remaining cells, which can be calculated as follows.After ' N , polymer molecules of A , have been added, the probability that one of the two terminal segments of the next A l molecule will enter the lattice equals (bB -n bN1). With Flory's assumptions l6 the probability that the segment adjacent to the terminal segment just added will find an empty cell amounts to br(bB- n bN,)/(bB+ bN3), whereas the analogous probability for each of the remaining (n -2) segments equals (%- l)(bB- n bNl)/(bB+ bN3). Incidentally, is the coordination number for the solution lattice. Following Flory's procedure,16 Zbg(bN,, bN3, bB) will therefore be given by - (bB+bN,)!( %- 1 ) ( f l - l ) b N 1 ( n" bN1 )[ (bB/n)! 1" bB!bN3 ! bB+ bN3 2bN1bN1! (bB/n-bN,)! It is readily seen that upon differentiating the logarithm of the preceding function with respect to bN1 at constant bB and bN3 an expression of of the form of eqn (A9) is again obtained. Analogous conclusions hold if the S3 molecule consists of more than one segment. G. Pezzatini and R. Guidelli, J.C.S. Faraduy I, 1973, 69, 794. B. NygArd, Arkiv. Kemi, 1962,20, 163 ; 1966, 26,167. G. Lauer, R. Abel and F. C. Anson, Analyt. Chem., 1967,39,765. M. L. Foresti, G. Pezzatini, G. Piccardi and R. Guidelli, unpublished results. D. Cozzi, G. Raspi and L. Nucci, J. Electroanalyt. Chem., 1966, 12, 36. G. Lauer and R. A. Osteryoung, Analyt. Chem., 1967, 39, 1886. R. Guidelli, J. Phys. Chem., 1970, 74, 95. J. M. Parry and R. Parsons, J. Electrochem. SOC., 1966, 113, 992. B. B. Damaskin, 0. A, Petrii and V. V. Batrakov, Adsorption of Organic Compounds at Electrodes (Plenum Press, New York, 1971). lo A. Arevalo, S. Gonzalez and E. Fatas, Anales de Quim., 1975, 71, 273. l 1 R. Parsons, J. Electroanalyt. Chem., 1964, 8,93. l2 B. B. Damaskin, Elektrokhimiya, 1965, 1, 63. l3 R. Fowler and E. A. Guggenheim, Statistical Thermodynamics (University Press, Cambridge, 1939), p. 431. l4 B. B. Damaskin, A. A. Survila and L. E. Rybalka, Elektrokhimiya, 1967, 3, 146. H. P. Dhar, B. E. Conway and K. M. Joshi, Electrochim. Acta, 1973, 18, 789. l6 P. J. Flory, J. Chem. Phys., 1942, 10, 51. l7 A. Clark, Theory of Adsorption and Catalysis (Academic Press, New York, 1970). l8 1. Prigogine, A. Bellemans and V. Mathot, Molecular Theory of Solutions (North-Holland, Amsterdam, 1957). (PAPER 7/2069)