Mixed-solvent Theory for Liquid Chromatography MCCANN,HOWARD AND C. ANTHONY BY MERANEY PURNELL WELLINGTON Department of Chemistry University College of Swansea Singleton Park Swansea SA2 8PP Received 28th July 1980 The competitive- and solution-interaction models are further developed. Certain assumptions previously made in the theory are shown to be either unacceptable or unnecessary. It is shown that both the Langmuir and diachoric-solution models lead to an equation relating inverse retention volume in a simple manner with volume-fraction composition of the mixed solvent. Results are presented for elution of several solutes from carbon tetrachloride + diethyl ether mixtures over the whole range q= 0-1. This provides the first data covering the whole concentration range.Although the same general equation describes both theoretical approaches it is shown how they may be distinguished in the light of experiment. It is concluded that all data so far published fall into a group in which over the approximate range q = 0.1-1,there is no competitive adsorption of the solvent components and the diachoric model applies. The importance of extending studies over the range q = 0-1 is eniphasised as also is the need to work with solvent pairs capable of competitive adsorption over a wider range of q. The overwhelming majority of liquid-chromatographic (1.c.) analyses are conducted with a multi-component mobile phase. In view of the explosive growth in the use of 1.c. techniques it has become a matter of importance to analysts that some system should be developed whereby the choice of mixed solvents is simplified.The only current system available is that developed by Snyder' and in the absence of any alter- native its use at least as a qualitative guide is expanding. Snyder adopts a highly approximated theoretical approach ; the following represent the assumptions under- lying the basis others being necessary to the full elaboration homogeneity of the adsorbent; infinite dilution of the solute; ideality of the mixed liquid phase; equality of the molar volume of any species in the mobile and in (on) the adsorbed layer ; equality of all molar volumes of all species in the system; full coverage of the surface (monolayer) at all times; retention of solute when eluted by a mixed solvent is the sum of individual contributions due to competition for adsorbent by solute with the individual solvent components.Since separation is seen as resulting from the contest for surface sites with the solvent playing the role of an inert mobile phase as in principle does the gas in gas chromato- graphy this approach has come to be called the competitive model. Snyder then writes as the basic equation for partition of solute S between solvent (A + B) and adsorbent K(AB)s = xgK(A)~+ x&B)s (1) where xa represents a mole fraction in the surface layer and K's represent stoichio- MIXED-SOLVENT THEORY FOR LIQUID CHROMATOGRAPHY metric partition coefficients for elution by solvents A B and A + B respectively. Here K is defined by e.g.C representing mol/adsorbent weight in the surface layer and Cs representing mol/ volume in the liquid phase. Thus K(A)s has the dimensions of volume/weight so that according to convention the retention volume of S eluted by pure A would be v(A)s = &A)s wa (3) where W is the weight of adsorbent in the column. Several theoretical groups 2-7 have sought either to provide a rationalisation of Snyder’s method or to modify it or even to supplant it. With only one exception’ however they have retained most of the foregoing assumptions but for reasons of their own have chosen to work not with stoichiometric partition coefficients but with alternatives defined in terms of mole fraction e.g. with pure A K‘(A)s = (x?/x~)A* (4) It is a simple matter to show that a corresponding pair of K and K’ for a pure eluent are related via e.g.where V is the monolayer volume of adsorbed A. Since all molar volumes are assumed to be identical the monolayer volume is constant for all compositions of (A + B) mixtures hence it is legitimate subject to this substantial restriction to replace all K by K’ in eqn (1). Thus K‘(AB)s = xOAK’(A)~ + ~S’(ms-(6) It is immediately apparent that this equation is of no direct practical value since the several xu cannot be determined. However by introducing the equilibria setting rearranging and substituting via KBA = &A/.& yields the basic equation from which most recent theoretical development 43 has sprung. It is in fact unnecessary to assume equality of molar volumes to justify use of eqn (7).As has been pointed out V(AB)sis the sum of the individual retentions due to A and B. Thus V(AB)~= KA)SWA + &B)sWB (8) where WAand WBare respectively the weights of adsorbent covered by A and by B. Evidently WA+ WB= Wa and = K’(A)sVX + K’(B)~V~ V~AB)~ = K‘(A)snZrA + K‘(B),n;pB (9) M. MCCANN H. PURNELL AND C. A. WELLINGTON where Vz and V are the volumes of ng and ng moles of A and B in the surface layer and FAand vBare their molar volumes. Now V(AB)s = &AB)sWa = Kf(AB)s(VA + YB) (n + ng)l(nA+ nB)* Thus and The introduction of the molar volume of the eluting mixture VAB depends upon the assumption that any excess volume of mixing is trivial a situation so common as to be almost general.We thus see that eqn (1) and (6) are correct only when rA= vB and are subject to very considerable arithmetic error since even quite common mixed solvents may have molar volumes differing by factors of two or more. Further we see that a more general expression is available although VAB is of course composi- tion-dependent. Eqn (10) may now be manipulated to eliminate x terms exactly as previously des- cribed and leads without assumption to eqn (7). Thus whilst eqn (1) and (6) are totally limited to the case of FA= rB,eqn (7) appears to have a wider validity. This rather surprising result prompts us to look for an explanation. One is found by substituting back via their definitions for the various K'. This yields whereas by definition In the foregoing "ng is the full monolayer number of moles of A deposited from pure A containing nA@ moles and similarly for €3 while (n + n:) is the total number of moles in the monolayer deposited from A + B.These two equations become equivalent when (n mn;lnAo)= ng and (nB ,ng/nBo> = n:. Since the surface layers of A and B are assumed to be independent i.e. non-interacting clearly in each region n = ,,,ng and ng = ,,,ng. Thus implicit in the model is the requirement nA = nAo and nB = nBo which in turn means that the solvent components are to be regarded as effectively immiscible. Such behaviour has been reported by us8 for a number of gas-liquid chromatography systems and termed diachoric. The model discussed above is thus equivalent to that of elution from a column containing W of adsorbent by solvent A of volume VAfollowed by a subsequent elution from a column containing WB of adsorbent by volume VBof B.THE LANGMUIR MODEL It is commonly stated that the foregoing model is equivalent to Langmuir adsorp- tion of A B and solute S. Indeed in the special case that rA= vB= Psit is possible to derive eqn (7) as we show below; however the result is not general. MIXED-SOLVENT THEORY FOR LIQUID CHROMATOGRAPHY The standard equation for competitive Langmuir adsorption yields for the frac- tional surface coverage (OS) by S Full monolayer coverage can only occur when ZK C + 1 and since in addition KsCs is small we have Each K in eqn (12) represents a ratio of the forward and backward rate constants in the relevant adsorptive equilibrium and should not be confused with partition coeffi- cients and equilibrium constants such as were defined earlier.Now 6s =nga,f WaAa where a is the molar area of S and A is the area per unit weight of adsorbent. But n;/Wa= Cg and we may set u,/Aa = a. Then (acf/cS) = aK(AB)~= [(KA/KS)cA +(KB/KS)cBl-l-We may set CA= qAvA and CB= vBvBwhere p represents a volume fraction pro- vided that we again assume that any excess volume of mixing is trivial. Thus It is a standard result of the Langmuir approach that e.g. thus (aKA/ FA&) = (as/Aa) (cSlcAFA) (eA/oS)' Since S is at infinite dilution CAcorresponds to the pure liquid value uiz.CAo= FA-' hence (Cs/CAVA)= Cs. Further (a8AlAaes) =asniaAiAaasn and at a full monolayer nga =AA and AAIA = Wa.Thus (aKA/rAK,)= CsWafn = C,/C; = K(*)s-l. We thus find VA &4B)s K(A>s This result is more attractive than is eqn (7) for a variety of reasons e.g. (a) the interaction on the surface is competitive; (b) there is no restriction with regard to relative molecular sizes; (c) there is no explicit assumption of additivity of retention; (d) it is couched in terms of the practically measurable quantity K rather than the indeterminable quantity K' and so can be tested. M. MCCANN H. PURWELL AND C. A. WELLINGTON It shares with the earlier model only the assumptions of surface homogeneity and infinite dilution of S both reasonable as a starting point.From the viewpoint of the practising chromatographer the latter need never prove a restriction whilst clearly the former can be modified in the light of practical experience. Although no assump- tion of sdvent ideality is made the solute activity is by implication independent of solvent composition. THE DIACHORIC MODEL In this model which is a development of our g.1.c. findings we make only the initial assuniptions of infinite dilution of S homogeneity of the adsorbent and estab- lishment of equilibrium between phases. We set up first the three-phase equilibrium for the A B and S system. I We now introduce one further assumption based upon our observation that with many mixed liquid solvents the liquid/gas partition coefficients for solute Sare given by the diachoric equation8 so a result identical with that derived viathe Langmuir approach.It is thus not obvious how one would distinguish between the two models but it is at least possible to test them not only generally but in particular since both KIg and KSgcan be determined experimentally for any system. The diachoric model has affinities with that proposed by Scott and K~cera,~ MIXED-SOLVENT THEORY FOR LIQUID CHROMATOGRAPHY which has been labelled the solvent-interaction m~del~-~ and seen as an alternative to the competitive model although Scott and Kucera did not take this view. Since the dead volume corrected retention volume is given by V = KW, i.e. Provided the excess volume of mixing is trivial qAcan be replaced by C and eqn (21) then becomes Scott and Kucera’s empirical relation.Despite the fact that the general form of the various equations presented is reason- ably widely known the literature contains no single example of a study carried out over the whole range p = 0-1. Scott and Kucera7 have provided by far the most comprehensive data yet their work relates only to variations of q~of 0 to ca. 0.3 with the single exception of elution of desoxycorticosterone alcohol by isopropanol + n-heptane mixtures covering 80 volume percent. Furthermore all published data relate to the high-retention-volume region where the greatest probability of failure of the relatively primitive models discussed here is to be expected. Even so it is clearly established that in virtually every instance there is some range of linearity of a plot of inverse retention volume (or capacity factor) against 9.Not surprisingly in the light of the foregoing discussion few examples of any linearity of inverse retention with mole fraction have been identified and log/log plots have been more widely used in this context. It appears to us that detailed theoretical development prior to acquisition of com-prehensive data is likely to be counter-productive. In consequence we present here data for a few typical systems for which data covering the whole solvent composition range have been acquired. EXPERIMENTAL The experimental unit comprised an Altex model 1OOA pump/model 20 microprocessor coupled with a 20 mm3 Rheodyne (7010) injector a Pye-Unicam 1.c.-U.V.detector and Hewlett-Packard 3380A integrator/timer. The columns used were commercial 25 cm x 4.6 mm i.d containing Hypersil(5p). The solvents diethyl ether and carbon tetrachloride werz of the best quality available and were stored over sodium. Both the columns and solvent reservoir were immersed in a water thermostat held at 25 “C. Solutes for injection were highly diluted in the solvent used for the particular elution since this was found to be essential for accurate and reproducible measurement of retention volume. Flow rates were continuously measured during elutions; this again was found to be essential since the short-term stability of the pump was no better than &0.5%. On this account also solute samples were injected repetitively at fixed time intervals usually in groups of ten.Flows measured at room tem- perature were corrected to 25 “C using tabulated values of the coefficient of cubical expan- sion. Each final data point represents an average of between thirty and fifty elutions per solute at each solvent composition. RESULTS Fig. 1 illustrates a plot of the data according to eqn (21) for phenol elution. Shown on this plot too is a series of error bars that define the range of values of inverse retention volume that would result from a spread of -&0.5% in the total retention volume i.e. that inclusive of dead volume at various values of inverse retention MCCANN H. PURNELL AND C. A. WELLINGTON I 'I I I I 0.5 1.1 volume fraction diethyl ether FIG.1.-Plot of inverse retention volume of phenol against volume fraction of diethyi ether in carbon tetrachloride (35 "C).Column 25 cm x 4.5 mm i.d. packed with Hypersil (5,~). volume. It is clear in the light of the associated error bars that there is no alternative to drawing a continuous straight line from 0 = 0 to 1. Fig. 2 shows corresponding plots for elution of nitroethane (A) and nitromethane (B). The error bars of fig. 1 are again relevant and it is again certain that between 8 = 0.1 and 1 the data are linear but curve sharply downward between 8 =0.1 and 0. Finally in fig. 3 are shown the results for elution of 3-phenyl-propan-1-01. In volume fraction diethyl ether FIG.2.-Plot of inverse retention volume of nitroethane (A) and nitromethane (B) against volume fraction of diethyl ether in carbon tetrachloride (25 "C).Column as in fig. 1. MIXED-SOLVENT THEORY FOR LIQUID CHROMATOGRAPHY this case the data are linear from 0 = 0 to ca. 0.8 and then curve so sharply that the retention volume is virtually constant to 8 = 1. The partition coefficient of nitromethane between the solvent pairs diethylether + water and carbon tetrachloride + water was approximately determined. The ratio which gives the value of the partition coefficient between diethyl ether and carbon tetrachloride was 6.8. 0 0.5 1.o volume fraction diethyl ether FIG.3.-Plot of inverse retention volume of 3-phenyl-propan-1-01 against volume fraction of diethyl ether in carbon tetrachloride (25 "C). Column as in fig. 1. DISCUSSION The zero intercepts at 0 = 0 shown in fig.1 and 3 are consistent with our total inability to elute these solutes with pure carbon tetrachloride in any reasonable time although they could subsequently be eluted by adding some ether to the eluent stream. They are however very soluble in carbon tetrachloride and so the effectively infinite retention must indicate essentially irreversible adsorption in the presence of carbon tetrachloride. Since the retention of phenol with pure diethyl ether is ex- tremely small while the solubility is considerable phenol clearly competes very weakly with the ether for adsorption sites. We can thus conclude that over the greater part of the solvent composition range the adsorbent surface is entirely ether coated and that in consequence the variation of retention is dominated by the change of solvent composition since the adsorbent surface is essentially a constant.According to the diachoric model [eqn (18)] the ratio of the intercepts at 9 = 0 and 1 of the extra- polated straight lines should correspond to the partition coefficient ratio for the sol- vent system diethylether + carbon tetrachloride. The value derivable from fig. 2 for the intercept ratio for nitromethane is ca. 6.0. In view of the substantial uncer- tainty in the extrapolation this is not unreasonable agreement with the value deter- mined by us. The data are therefore consistent with a model in which the ether provides the adsorbed monolayer over the greater part of the solvent composition range thus pro- viding a fixed fraction of surface for solute adsorption the extent of which is then determined by its activity in solution which in the present instance if no others is described by eqn (17).Only over a very short range in the region of 100% carbon tetrachloride is there competitive adsorption between the solvent components. Even here though there will be solvent effects and so the simple Langmuir (competitive) model would not necessarily apply. It seems clear that for such systems as these the M. MCCANN H. PURNELL AND C. A. WELLINGTON results for which are very similar to others so far p~blished,~*~-' the competitive model so far as it is currently developed is unlikely to be an adequate basis for general theoretical development.The foregoing view is entirely consistent with that expressed by Scott and Kucera,' who have found no exception to limited linearity of inverse retention with molarity of solvent. Further they have shown that in the low-q region the curvature is des-cribable in terms of Langmuir-like behaviour in a number of instances. However the situation is not entirely clear since in many instances the linear part of their data is correlated with a negative intercept at q = 0. This is clearly impossible and im- plies an upward curvature in this region. If this is true the solute is displaying anti- Langmuir behaviour which implies solute niulti-layer formation as distinct from the solvent multi-layering invoked by Scott and Kucera" in the case of certain other systems.The data of Slaats et al can all be interpreted in the same way since for each of their six systems the data provide very lengthy linear regions of the plots of inverse retention against q with in one effective linearity to q = 0 and in the other five a very rapid fall downward below q = 0.1. Interestingly for all six the value of inverse retention at q -=0 is essentially zero. We thus see that to all intents and purposes all data for normal-phase elution from silica published to date point in the same direction and indicate with some certainty that for these systems at least the competitive model is of limited validity. The semi- quantitative success of the Snyder system is presumably a consequence of the fact that the form of governing equation is the same as that for the solvent model.It must be noted that the view expressed above relates to solvent systems in which the relative adsorption of the solvent components is very disparate. For instance it is a simple matter to show via eqn (12) that for competitive adsorption to become not- able only below q = 0.1 there must be a factor of 100 difference in the Langmuir constants KA and KB. All systems so far studied seem more or less to fulfil this requirement as the adsorption isotherms of Scott and KuceralO show. It would clearly be of great interest now to study more widely solvent systems of comparable KA and KB,when more complex behaviour might be anticipated. Finally it is worth- while to point out that future studies must extend over the whole composition range since only then can all the information necessary to theoretical testing be determined.We thank the S.R.C. for a maintenance grant to M. M. L. R. Snyder PrincQles of Adsorption Chromatography (Marcel Dekker New York 1968); Anal. Chem. 1974 46 1384. J. Oscik Przem. Chem. 1965 44,129; J. Oscik and J. K. Rozylo Chroniatographia 1971 4 516; J. Oscik and G. Chojnacka J. Chromatogr. 1973 93 167. P. Jandera and J. Churacek. J. Chromatogr. 1974 9 207. E. Soczewinski and W. Golkiewicz Chromatographia 1971,4 501 ; 1973,6 269; E. Soczewin-ski J. Chromatogr. 1977 130 23. M. Jaroniec J. K. Rozylo and B. Oscik-Mendyk J. Chromatogr. 1979 179,237; M. Jaroniec B. Klepacka and J. Narkiewicz J. Chromatogr. 1979,170 299; M. Jaroniec J. K. Rozylo and W.Golkiewicz J. Chromatogr. 1979 178 27. E. H. Slaats J. C. Kraak W. J. T. Brugman and H. Poppe J. Chromatogr. 1978 149 255. R. P. W. Scott and P. Kucera J. Chrornatogr. 1978 149 93; 1979 171 37. J. H. Purnell and J. M. Vargas de Andrade J. Am. Chem. SOC.,1975,97,3585; R. J. Laub and J. H. Purnell J. Am. Chem. SOC.,1975. 98 30. R. P. W. Scott J. Chromatogr. 1976 122,35. lo R. P. W. Scott and P. Kucera J. Chromatogr. 1979 171 37.