J.C.S. Faraday I, 1980, 76,2486-2495Theory of Adsorption from Multicomponent Liquid Mixtureson Solid Surfaces and its Application toLiquid Adsorption ChromatographyBY MIECZYSLAW JARONIEC:: AND ANDRZEJ PATRYKIEJEWDepartment of Theoretical Chemistry, Institute of Chemistry,M. Curie-Sklodowska University, 2003 1 Lublin, PolandReceived 24th September, 1979The theory of adsorption from multicomponent liquid mixtures on energetically homogeneousand heterogeneous solid surfaces has been developed using the prevailing approaches to this problemand utilizing the theory of mixed-gas adsorption. The general isotherm equations have been derivedand applied to evaluate the distribution coefficient characterizing the process of liquid adsorptionchromatography with a multicomponent mobile phase.In contrast to the extensive literature concerning the adsorption of binary liquidmixtures on solid adsorption from multicomponent liquid mixtureshas been comparatively little studied.Theoretical studies of adsorption from multi-component liquid mixtures on solids have usually been related to the region of lowconcentrations. 2* Investigations of multicomponent liquid-solid systems, in thewhole concentration region, have been made by Oicik and Minka and Myers8They considered the adsorption on energetically homogeneous surfaces only.Recently, their results have been extended to adsorption on heterogeneous solidsurfaces 9 9 lo by applying the thermodynamic treatment of Jar0niec.l'. l 2In this paper some aspects of adsorption from multicomponent liquid mixtures onhomogeneous and heterogeneous solid surfaces are discussed.Different adsorptionmodels assuming ideality as well as non-ideality of both phases have been considered.The results of the theoretical considerations are analytical equations for the molefractions of components in the surface phase. These equations have been appliedto evaluate the distribution coefficient of a chromatographed substance in a multi-component mobile phase.ADSORPTION FROM MULTICOMPONENT LIQUID MIXTURESHOMOGENEOUS SURFACESLet us consider adsorption from an n-component liquid mixture. The followingassumptions are made : (a) the surface phase is supposed to be a monolayer, (b) bothsurface and bulk phases will be considered as ideal or non-ideal, (c) the molecularsizes of all components are identical, (d) the total number of molecules in the surfacephase is constant and (e) the adsorbent surface is energetically homogeneous.Theabove assumptions are frequently used in the theory of adsorption from solutions onsolids.Let the adsorption energy of the nth component be smallest. Then the adsorptionmechanism may be represented by the series of quasichemical reactions between a248M. JARONIEC AND A . PATRYKIEJEW 2487molecule of the ith component (i = 1,2, . . ., n- 1) and a molecule of the nth com-ponent :(i)' + (n)s + (i)s + (n)' (1)where the symbols i and n denote molecules of components i and n, respectively, andthe superscripts 1 and s refer to the bulk and surface phases.The equilibrium constantfor reaction (1) isa;.:Kin = - for i = 1 , 2 , . . .,n-1anaiwhere a] and a; are the activities of the ith component in the bulk and surface phases,respectively. The activities a: and a: are defined as follows :and(3)(4)wheref: andfi are the activity coefficients of the ith component in the bulk and surfacephases and x] and xi' are the mole fractions of the ith component in the bulk andsurface phases, respectively. Eqn (2) may be rewritten in a slightly different form,i.e.,wherex:/x,S = Kinflinx)x:, for i = 1 , 2 , . . ., n-1 ( 5 )Summing the mole fractions xi, xi, . . ., xi-l, the following equation may beobtained :, for i = 1,2 ,..., n-1 KinPinxVxX1 + C KjnBjnxf/xtn - 1 x; =j = 1and(7)Evidently the parameter P i n is a function of the composition of both phases.Eqn (7)with pin = 1 has been derived by Minka and Myers * for the description of theadsorption from liquid mixtures on homogeneous surfaces, when both phases areideal. Next, eqn (7) with Pin = 1 was derived by Jaroniec et aL4 in terms of thestatistical thermodynamics.CONCEPTION OF SURFACE HETEROGENEITY I N ADSORPTIONFROM LIQUID MIXTURESGeneralizing Jaroniec's treatment of the adsorption from multicomponent gasmixtures on heterogeneous solid surfaces " 9 l 2 to liquid adsorption we obtain :nxtt = x:(x', E ) F(E) dE, for i = I, 2, . . ., nJ A(92488 ADSORPTION FROM MULTICOMPONENT LIQUID MIXTURESwhere E = (El, E2, . . ., En) and x 1 = (xi, x$, . . ., xL- 1) are vectors, El is the adsorp-tion energy of the ith component for a heterogeneous surface, A is an n-dimensionalintegration region of E and F(E) is an n-dimensional distribution function of E,which is normalized to unity, i.e.,l A F ( E ) dE = 1.(10)The symbol $(xl, E ) denotes the mole fraction of the ith component in the surfacephase for adsorption sites characterized by E. In this formulation, each adsorptionsite is characterized by an n-dimensional vector E. The adsorbent surface is ener-getically heterogeneous if it contains adsorption sites of different values of E.In eqn (9) the mole fraction $(XI, E ) may be evaluated according to eqn (7). Theconstant Ki, is equal t o :where A i , is the entropy factor. The mole fraction xf, expressed by eqn (7), is thefunction of differences of adsorption energies Ei, (i = 1, 2, , .., n - 1). This factenables us to introduce a new distribution function characterizing the surface hetero-geneity in adsorption from solutions.Let us characterize each adsorption site by an ( n - 1)-dimensional vector E* =(El,, E2,, . . ., The surface is energetically heterogeneous if it containssites of different values of E*. Similarly as in the case of eqn (9), an (n- 1)-dimen-sional distribution function G(E*) has been introduced to characterize the energeticheterogeneity of the adsorbent surface. Thus, the mole fraction x$ may be expressedas follows :x!,~ = xi(xl, E*) G(E*) dE*, for i = 1 , 2 , . . ., n (12)G(E*) dE* = 1. (13)s. whereIn the above, R is an (n - 1)-dimensional integration region of E*.Eqn (12) maybe obtained from eqn (9) by assuming the following definition of the function G(E*) :G(E*) = JF(E)dE,. (14)These conceptions of surface heterogeneity in adsorption from solutions are notequivalent. For illustrative purposes we consider the adsorption from a binaryliquid mixture. Let us assume a homogeneous surface characterized by the functionwhere 6 is the Dirac function. According to the second conception, this surface isalso characterized by the Dirac function :F(E1, E2) = F&?31)-F&) = 6(E1 -ET)*6(E, - E ; ) (1 5 )W 1 2 ) = 6(&2 -EL). (16)In the case when the functions Fl(El) and F2(E2) have identical shape but areshifted on the energy axis, the difference of adsorption energies, E12, may be identicalfor all adsorption sites.13 It might even happen that the adsorbent surface appearsto be heterogeneous according to the first conception and quite homogeneous accord-ing to the secondM.JARONIEC AND A . PATRYKIEJEW 2489In adsorption of binary liquid mixtures on heterogeneous solid surfaces, eqn (12)has usually been 4 9 ' 9 lo The cited papers deal with the theoretical considera-tions of adsorption models assuming ideality of both phases or ideality of the surfacephase and non-ideality of the bulk phase. Then the topography of adsorption siteson the surface is not important.12 However, in the case of a non-ideal surfacephase the distribution of adsorption sites on the surface should be taken into con-sideration.As with the case of mixed-gas ad~orption,~. l2 with adsorption fromliquid mixtures two models of hererogeneous surface are considered : (a) a modelwith a patchwise distribution of adsorption sites and (b) a model with a randomdistribution of sites on the surface. For patchwise surfaces the activity coefficientf:is a function of the composition of the solution adsorbed on a given patch and thecomposition of the bulk liquid, i.e.,j f = f i ( x s , x') where xs = (x;, x i , . . ., x:-~). (17)However, for surfaces with a random distribution of adsorption sites the activitycoefficientf: is a function of the composition of the solution contained in the wholesurface phase and the composition of the bulk liquid, i.e.,Thus, for random surfaces the parameter Pin, appearing in eqn (7), is not dependenton E*.The integration in eqn (12) and (7) is formally identical for adsorption modelsassuming: (a) ideality of both phases, (b) ideality of the surface phase and non-ideality of the bulk phase and (c) non-ideality of both phases and an adsorbent surfacewith random distribution of adsorption sites. In the case of patchwise surfaces eqn(7) is a complex function of xi and integration of eqn (12) is very difficult.f: = f f ( x i , x') where xi = (xi,,, xi,,, . . ., xi- l,t). (1 8)ANALYTICAL EQUATIONS FOR MOLE FRACTION,Let us consider a liquid mixture in which (n- 1) components show similar inter-actions with the adsorbent surface and the behaviour of the nth component is com-pletely different.Such a model of adsorption has been considered for adsorptionfrom gaseous l4 and liquid mixtures.l* Then, the diffrence of adsorption energies,Eil = &-El, = ui = constant (i = 1, 2, . . ., n - l), is identical for the wholesurface. Taking into account this relationship and assuming the random distributionof adsorption sites on the surface, we can transform eqn (7) into the following form :wheren- 1andwher2490 ADSORPTION FROM MULTICOMPONENT LIQUID MIXTURESEqn (22) is formally identical with the integral equation used in the adsorption ofgases l 5 and binary liquid mixtures 3* on solids. Thus, the solution of eqn (22)is analogous to the solutions of the integral equations used in gas and liquid adsorp-Now, we derive equations for x$ by means of eqn (22) for two quasi-gaussianenergy distributions GI n(Eln). These distributions give equations for ~ f , ~ of thetype of the T6th's l 6 and Sips' l 7 isotherms.In the case of T6th's distribution thefollowing equation is obtained :tion.3. 4 9 15Eqn (24) leads to the following expression for x& :The constant c in the above equations denotes the heterogeneity parameter character-izing the shape of T6th's distribution function and the parameter KTn is connectedwith the average difference of adsorption energies, Eln.For the distribution function obtained by Sips l7 we obtainandwhere d is the heterogeneity parameter. Eqn (24)-(27) have been derived by assuminga random distribution of adsorption sites on the surface and constancy of the differ-ences of adsorption energies, ie.,and for all types of adsorption sites.to adsorp-tion from multicomponent solutions on solid surfaces of random distribution ofadsorption sites gives :Eil = ui = constant, for i = 1,2,.. ., n-1 (28)Application of the approximation of Crickmore and Wojciechowskiwhere w is the heterogeneity parameter. This equation is an extension of the adsorp-tion isotherm discussed in ref. (9). Eqn (25) and (29) will be used to derive thedistribution coefficient for liquid adsorption chromatography with a multicomponentmobile phase.LIQUID ADSORPTION CHROMATOGRAPHYThe chromatographic process for the rth substance in the n-component mobilephase relates to the adsorption from the (n+ 1)-component liquid mixture, when theconcentration of the rth substance is infinitely low.Let us assume that the 1stcomponent of a mobile phase is a most efficient eluting solvent. The chromato-graphic process may be represented by the following exchange reaction :( 1 y + ( r y + (1)' (30M. JARONIEC AND A . PATRYKIEJEW 249 1and the reactions represented by eqn (1). The equilibrium constant Krl for theexchange reaction (30) is expressed bywhere Prl is defined analogously as in eqn (6) ; however,n ni = 2 i = 2andn nSince the mole fractions x: and x: are infinitely low, the factor Prl is a function ofx1 and xs only, i.e., it is independent of x: and x:.According to the theory of adsorption chromatography,l the ratio of the molefractions x; and x: is equal to the distribution coefficient k, :k, = x:/x:.(34)(35)A simple combination of eqn (31) and (34) gives :Eqn (35) is a general expression defining the distribution coefficient of the rth sub-stance in the n-component mobile phase when non-ideality of both mobile and surfacephases is assumed. The mole fraction xs, appearing in eqn (35), may be calculatedaccording to eqn (7).kr = Kr 1 x; /(xi P r 1 )*For Prl = 1, eqn (35) reduces to the following expression :20kr = Kr1x;/x: (36)and is related to the model based on the assumption of ideal surface and mobilephases. Then, assuming that Pln = P2n = . , . = Pn-l,n = 1, the mole fraction x;,evaluated from eqn (7), becomesNow, we shall define the equilibrium constant Kin by means of the distributionFollowing Minka and Myers * we can write the relationship : coefficients kri and krn.Kin = KirIKnrwhereBecause the concentration of the rth substance is infinitely low, the mole fractionsxl and xf are very close to unity.Then,where kri is the distribution coefficient in the ith solvent. Analogously,Ki, = I / & = l/kri (40)Knr = l/Krn = 1 /krn.Eqn (38), (40) and (41) give :Kin = krdkri2492 ADSORPTION FROM MULTICOMPONENT LIQUID MIXTURESSubstituting eqn (42) into eqn (37) we obtainHowever, combining eqn (36), (41) and (43) we have :nIlkr = C x;/kri.i = 1(44)Eqn (44) may be also obtained in terms of Snyder's theory of adsorption chromato-graphy.,ISnyder,l Soczewinski 22 and Jandera and Churacek 23 considered mobilephases in which the 1st solvent was considerably more polar than other solvents.Then, for higher values of xi the adsorbent surface is practically occupied by the 1stsolvent, i.e.,Taking into account condition (49, eqn (26) becomesxi % 1.(45)orIn k, = In krl -In x i .I n kr = In K , , -In u: +In f:.(47)However, applying condition (45) to eqn (35) and assuming ideality of the surfacephase, we get :Eqn (48) has been also derived by Slaats et aZ.24 by using another theoretical approach.Eqn (35) is the most general equation in our formulation of liquid adsorptionchromatography with an n-component mobile phase and with a homogeneous solidsurface. The main relationships used in a liquid adsorption chromatography maybe obtained from eqn (35).Moreover, applying analytical expressions for the activitycoefficientsf: andf: * * 2 5 in eqn (39, new equations for the distribution coefficient krcan be obtained.(48)HETEROGENEOUS SURFACESLet us consider a heterogeneous surface showing rn types of adsorption sites.Let g p be the ratio of the number of adsorption sites of thepth type to the total numberof adsorption sites. The ratios g , , g,, . . ., g , satisfy the conditionf g p = l .p = l(49)First, we shall discuss the distribution coefficient k, for a heterogeneous surfaceand ideal surface phase (the mobile phase may be either ideal or non-ideal). Inthis case, the topography of adsorption sites on the surface is not important.12According to eqn ( 3 9 , the distribution coefficient of the rth substance on the pthtype of adsorption sites is given by :wherM.JARONIEC AND A . PATRYKIEJEW 2493The distribution coefficient of the rth substance on the entire heterogeneous surface,kr,t, is defined as follows :Using the approximation discussed by Jaroniec et ~ 1 . ~ ~ in eqn (52) we obtainwhere K:1 is an averaged distribution coefficient for the rth substances in the 1stsolvent, referring to the entire adsorbing surface, w is the heterogeneity parameter,analogous to that introduced in eqn (29).Now, we consider a special case of eqn (53), related to ideal surface and mobilephases, i.e.,k , , = ccKrolx~,t)"wlix: -In this case eqn (29) may be rewritten in the following form :(54)(x,'/k,",)"i = 1when the relationship [see eqn (42)] :is satisfied.Substituting eqn (55) into eqn (54) and remembering eqn (56), we obtainFor w = I, eqn (57) reduces to eqn (44), which was obtained assuming energetichomogeneity of the solid surface.Now, we consider the adsorption model with a non-ideal surface phase. Accord-ing to eqn (17) and (18), the variable Prl depends on the properties of an adsorbentsurface.For a patchwise distribution of adsorption sites this variable is a function of thecomposition of the bulk solution and molecules adsorbed on the pth surface patch.However, for a random distribution of adsorption sites on the surface, the variableP r l is a function of the composition of the bulk and surface solutions. Thus, thedistribution coefficient kr,t is given by :mkr,t = (l/xi) gpKrl,pxF;,p/Pr,,p for a patchwise surface (58)p = landmp = lkr,t = (l/x:flr1) gpKrl,px(il,p for a random surface.(59)Using the approximation used in eqn (53) with the sum appearing in eqn (59), weobtain :Eqn (60) describes the distribution coefficient of the rth substance in the n-componentnon-ideal mobile phase and for a non-ideal surface phase formed on a heterogeneoussurface with a random distribution of adsorption sites2494 ADSORPTION FROM MULTICOMPONENT LIQUID MIXTURESCONCLUSIONSThe equations used in adsorption chromatography refer to very simple adsorptionmodels. These equations are special cases of more general expressions eqn (35),(52), (58) and (59), which have been derived in terms of the theory of adsorption frommulticomponent liquid mixtures on either energetically homogeneous or heterogeneoussolid surfaces.Equations defining the distribution coefficient of the rth substance in the n-component mobile phase contain the mole fractions of solvents in the surface phase.These mole fractions may be determined : (a) from analytical equations correspondingto a given adsorption model or (6) directly from experimental excess adsorption data,using the following relationship :where ns is a total number of moles in the surface phase and ni is an adsorption excessof the ith solvent.The first procedure is useful for determining the influence of the adsorption modelon the distribution coefficient.For some adsorption models the equations for thedistribution coefficient are very simple.The other procedure, using the excessadsorption data, is more useful for the description of experimental data obtainedfrom liquid adsorption chromatography.x: = nr/ns++! (61)LIST OF PRINCIPAL SYMBOLSactivityentropy factor in eqn (1 1)heterogeneity parameter in eqn (24)heterogeneity parameter in eqn (26)activity coefficientdistribution function of Eadsorption energydifference of adsorption energies Ei and Enfraction of adsorption sites of the pth typedistribution function of E*distribution coefficient of the rth substance in a mixed mobile phase on ahomogeneous surfacedistribution coefficient of the rth substance for a heterogeneous surfacedistribution coefficient of the rth substance in the ith solventequilibrium constant for a quasi-chemical reaction (1)number of types of adsorption sitesexcess adsorption isotherm of the 1st solventtotal number of moles in the surface phasemole fractionheterogeneity parameter in eqn (29)parameter defined by eqn (6)VECTORM.JARONIEC AND A. PATRYKIEJEW 2495SUBSCRIPTSi the ith componentn the nth componentp thepth type of adsorption siter the rth chromatographed substancet refers to a heterogeneous surfaceSUPERSCRIPTS1 mobile (bulk) phases surface phase(a) D. H. Everett, Trans. Faraday Soc., 1964, 60, 1803 ; (b) D. H. Everett, in Colloid Scienceed. D. H. Everett (Specialist Periodical Report, The Chemical Society, London, 1973), vol. 1,chap. 2.C. E. Brown and D.H. Everett, in Colloid Science, ed. D. H. Everett (Specialist PeriodicalReports, The Chemical Society, London, 1975), vol. 2, pp. 52-100.(a) J. OScik, A. Dqbrowski, M. Jaroniev and W. Rudzinski, J. Colloid Interface Sci., 1976,56,403 ; (b) A. Dqbrowski, J. OScik, W. Rudzifiski and M. Jaroniec, J. Colloid Interface Sci.,1979, 69, 287.M. Jaroniec, A. Patrykiejew and M. Borowko, in Progress in Surface and Membrane Science(Academic Press, New York, 1980), vol. 14.C. J. Radke and J. M. Prausnitz, Amer. Inst. Chem. Eng., 1972, 18, 761.L. Jossens, J. M. Prausnitz, W. Fritz, E. U. Schlunder and A. L. Myers, Chem. Eng. Sci.,1978,33, 1097.J. OScik, Bull. Acad. Pol. Sci., Cl. 3, 1961, 9, 23, 29.C. Minka and A. L. Myers, Amer. Inst. Chem. Eng., 1973, 19,453.M. Jaroniec, J. Res. Inst. Catalysis, Hokkaido Univ., 1978, 26, 155.lo M. Borowko, M. Jaroniec, J. OScik and R. Kusak, J. Colloid Interface Sci., 1979, 69, 311.l1 (a) M. Jaroniec, J.C.S. Faraday 11, 1977, 73, 933 ; 1978, 74, 1292 ; J. Colloid Interface Sci.,1975, 53, 422 ; 1977, 59, 230, 371 ; (b) M. Jaroniec and W. Rudzinski, J. Res. Inst. Catalysis,Hokkaido Univ., 1977, 25, 197.l2 M. Jaroniec, Thin Solid Films, 1978, 50, 163.l3 M. Jaroniec and J. Toth, Colloid and Polymer Sci., 1976, 254, 643.l4 M. Jaroniec, J. Narkiewicz and W. Rudzifiski, J. Colloid Interface Sci., 1978, 65, 9.l5 M. Jaroniec, Surface Sci., 1975, 50, 553.l6 J. Tbth, W. Rudzinski, A. Waksmundzki, M. Jaroniec and S. Sokolowski, Acta Chim. Acad.l7 R. Sips, J. Chem. Phys., 1950, 18, 1024.l8 P. J. Crickmore and B. W. Wojciechowski, J.C.S. Faraday I, 1977, 73, 1216.l9 L. R. Snyder, Principles of Adsorption Chromatography (Marcel Dekker, New York, 1968).2o M. Jaroniec, J. K. R6iylo and B. OScik-Mendyk, J. Chromatog., 1979, 179, 237.21 M. Jaroniec, J. Narkiewicz and M. Borowko, Chromatographia, 1978, 11, 581.22 E. Soczewinski, J. Chromatog., 1977, 130, 23.23 P. Jandera and J. Churacek, J. Chromatog., 1974, 91, 207.24 E. H. Slaats, J. C. Kraak, W. J. T. Brugman and H. Pope, J. Chromatog., 1978,149,255.25 A. S. Jordan, J. Electrochem. Soc., 1972, 119, 123.26 M. Jaroniec, J. K. R6zylo and W. Golkiewicz, J. Chromatog., 1979, 178, 27.Sci. Hung., 1974, 82, 11.(PAPER 9/1514