An Analysis based on Established Theory of Mixed-solvent Behaviour in Gas-Liquid Chromatography BYPETERF. TILEY School of Chemistry University of Bath Bath BA2 7AY Received 27th June 1980 Seven published sets of results on gas-liquid chromatographic mixed-solvent systems yield data on nearly two hundred ternary liquid systems and these were analysed for conformity with the Scatchard-Hildebrand-Flory-Huggins model of liquid mixtures. Whilst the expected quadratic (In KR,q) relation applies in most cases consistent values of solvent-solvent interaction parameters are not obtained when specific solute-solvent attractions exist. The use of alkane solutes appears to give constant and meaningful values of these parameters even when specific solvent-solvent interac- tion occurs.A computed simulation of solute-additive complexing in a solvent medium shows a linear (KR,c) relation at relatively low additive concentration but indicates that any calculation of formation constants by this method must involve very considerable quantitative uncertainty. Two recent books lv2 have reviewed the application of gas-liquid chromatography (g.1.c.) to the determination of thermodynamic data including the precautions neces- sary to ensure that the experimental measurements of retention volumes and partition coefficients may be directly related to the equilibrium between liquid and gas phases. Under such circumstances the measurement of partition coefficients in a binary mixed-solvent phase yields information about the ternary liquid system solute (1) + solvent (2) + solvent (3).The fact that the g.1.c. measurement is made with the solute approximating very closely to a state of infinite dilution gives a considerable simplification of any theoretical equations and consequently the solute can be used as a probe to explore the solvent-solvent interactions. The advantage of this approach is that once a given set of mixed-solvent columns have been prepared results can be rapidly obtained for a variety of probe molecules thus providing a severe test of any model of ternary systems. That ternary and multicomponent-fluid phase equilibria are of vital interest in chemical engineering is self-evident and the stimulus from this source has probably provoked as much theoretical work in this field as any other.Recent aspects are reported in ref. (3) wherein a plea can be found4 for greater academic attention to multicomponent mixtures of polar compounds. A relatively new theoretical model the Uniquac eq~ation,~ which effectively comprehends many previous well-known equations has met with considerable success in modelling the excess free energy and hence activity coefficients in liquid mixtures. The same approach has been extended to the calculation of activity coefficients from group contributions the Unifac method,6 although it is still controversial whether this approach is satisfactory for specific molecular association. Probably the most rigorous theories of liquid mixtures involving large non-polar molecules are those of Flory and co-workers and of Janini and Martire [e.g.see ref. (I) chap. 51. The latter theory has been applied quantitatively* and with some suc- cess to g.1.c. systems comprising n-alkane solutes in mixed n-alkane solvents. Exten-sion of the treatment to systems involving polar components with or without molecular MIXED-SOLVENT BEHAVIOUR complexing is not at present possible. However current thermodynamic theories of specific interactions in non-electrolytes have been recently re~iewed.~ In short workers in the g.1.c. field have no lack of theory for correlation and inter- pretation of their results. It is not the intention of this paper to investigate the precise application of the more sophisticated models to g.1.c. mixed-solvent systems but rather to explore the extent to which one of the long-established models may provide a useful approximation to such systems.The model is that of the " regular solution " and in considering it the remarks of one of its co-originators'O should be borne in mind namely " The best advice . . . is to use any model in so far as it helps but not to believe that any moderately simple model corresponds very closely to any real mixture ". In view of the varying definitions extant at the time Hildebrandl' at a Faraday Discussion in 1953 defined a regular solution as " one in which thermal agitation is sufficient to give practically complete randomness ". He pointed out that the concept of random mixing was embodied in the original Flory-Huggins expression for the entropy of mixing of molecules of very unequal size but would not be applicable to solutions in which there are specific " chemical " hydrogen-bonding or orientational effects.It is noteworthy that although the solubility-parameter interpretation of molecular interactions has often been associated with the regular-solution model the solubility-parameter concept is in no way an essential feature of the hypothesis of random mixing. Using the regular-solution model with the Flory-Huggins combinatorial term included results in the following eq~ation'~,'~ for g.1.c. mixed-solvent partition co- efficients KRC2,) KR(3)are the partition coefficients in the pure solvents KR(2.3) is that in the mixed solvent of volume fraction q2of component (2). V is the molal volume of the solute and ~23 is a parameter supposedly characteristic of the solvent-solvent interaction.Provided all partition-coefficient values have been measured at precisely the same temperature the application of eqn (1) requires no solute data other than molal volume. The testing of eqn (1) against experimental results involves three criteria which are in increasing order of severity (a) a quadratic dependency of In KR(2.3) on q2 (b) an independence of the value of ~23 on the solute and (c) the significance of the g.1.c. value of ~23 in relation to any non-g.1.c. phenomenon. Various published data sets of KR values in mixed-solvent systems are examined below in order to test the applicability of eqn (1). The complete antithesis of the regular solution is the micropartitioning model of a liquid mixture proposed by Purnell and LaubI4 and elaborated at length by Laub and Pecsok.2 On this model a homogeneous liquid mixture even at temperatures far removed from critical solution departs so far from random mixing that it may be treated as an assembly of microscopically immiscible groups of like molecules.The evidence for this theory is that a number of mixed-solvent g.1.c. systems show an approximately linear relation between KR(2,3) and q2as in eqn (2) Both MartireI5 and TileyI3 have shown that more conventional models can lead to such behaviour under given circumstances and until further theoretical developments have been explored it seems better to treat this as a purely empirical phenomenon " the mixed-solvent linear approximation ".It should be noted that there have been many P. F. TILEY serious theoretical attempts to treat the problem of non-random mixing two relatively recent ones being the Wilson16 and Uniquac' equations. COMPUTED RESULTS Eqn (1) was fitted to the systems show below. With the exception of data set (0) under each data set are listed the mixed solvents the temperature and the number of compositions studied and for each solute is given the computed value of the solvent- solvent interaction parameter ~~~/mol The curve-fitting procedure was based dm-3. on minimising the square of the residuals of In KR which is equivalent to assuming an equal percentage error on all KRvalues for a given solute. The percentage standard deviation of KRis reported as an average for all solutes in a given solvent system.For any given solute this was calculated as 0 100{~[(KR exptl. -~t calc.)/~Rexptl.l2/fi) 'a n DATA SET (0) In order to assess the constancy or otherwise of the computed ~23 values for a given solvent system an attempt was made to examine the effect of ran- dom error in the KR measurements. Sets of (KR p) values were generated with pre-set values of the parameters of eqn (1) using seven p values equally spaced. Using a statistical algorithm pseudo-random errors at a pre-set level of standard deviation were built into the KR values. Ten such data sets were generated for each level of standard deviation and all sets were then fitted to eqn (1) to recover the ~23 values.The stan- dard deviation of the ~23 values was calculated which was found to be independent of the values of the equation parameters and dependent only on the percentage error in KRand in particular on the range of compositions chosen. It would of course be dependent on the number of composition points but this effect was not investigated since about seven composition points i.e. five in addition to the pure solvents have been used by many workers. TABLEEFFECT OF RANDOM ERROR IN KRON THE ~23 VALUES OBTAINED FROM EQN (1) composition range dx23) p values pre-set (%) calc./mol dm-3 0.0-1.o 2 0.73 0.0-1.o 1 0.51 0.0-1 .o 0.5 0.21 0.0-0.5 0.5 0.95 0.0-0.25 0.5 3.9 The results are shown in table 1 where it is immediately apparent that significant values of ~23 can only be obtained by studies over the full composition range of p = 0.0-1.O.This point has already been noted by Parcher and Westlake,17 and therefore published g.1.c. results which only cover a restricted composition range have been omitted from this study. DATA SET (1)18 Squalane + dinonylphthalate at 30 "C using 12 composition points* for the following solutes pentane 2.29; hexane 2.36; heptane 2.39; octane 2.47; * The KRvalues are not published but the authors themselves report the fitting of eqn (1) to their results and the x23 values above were calculated from their reported values of x(=)~[=VZxz3with com-ponent (2) being dinonylphthalate]. MIXED-SOLVENT BEHAVIOUR cyclohexane 2.43 ; methylcyclohexane 2.46; benzene 3.70; toluene 3.41.Average a = 0.4". DATA SET (2)12 Dinonylphthalate + trinitrotoluene at 82 "C using 7 composition points for the following solutes octane 8.3; nonane 8.3; decane 8.3 ; methylcyclo-hexane 8.3; benzene 9.0; toluene 7.8 ; ethylbenzene 6.4; isopropylbenezene 6.6; o-xylene 8.2; m-xylene 8.5; p-xylene 7.5. Average a = 1.2:4. DATA SET (3)19 Squalane + dodecyl laurate at 60 "cusing 7 composition points for the following solutes cyclohexane 0.10; heptane 0.15; methylcyclohexane 0.21 ; ethylcyclohexane 0.17 ; cis-2-heptene 0.19 ; trans-2-heptene 0.19 ; trichlorethene 1.02; tetrachlorethene 0.42 ; benzene 0.52 ; toluene 0.48 ; xylene 0.55; fluoro-benzene 0.64; chlorobenzene 0.94; methoxybenzene 0.78 ; benzotrifluoride 1.06; octaflurotoluene 2.32.Average o = 0.3%. DATA SET (4)20 Squalane + dibutyl tetrachlorophthalate at 80.3 "C using 8 composi-tion points for the following solutes (T = thiophene) cyclopentene 0.1 ; cyclopen-tadiene 2.1 ; benzene 3.4; diethyl ether 0.03; divinyl ether 2.1 ; furan 4.4; methyl- furan 2.8; T 4.6; 2-methylT 2.5; 3-methylT 3.3; 2,5-dimethylT 3.0; 2-ethylT 25; 2-chloroT 4.3; 2,5-dichloroT 2.9; 2-bromoT 4.9; pyrrole 16.2; l-methyl- pyrrole 6.6. Average a (omitting last two) = 0.50//0. DATA SET (5)21 Tetradecane + tributyl phosphate at 25 "C using 10 composition points for the following solutes pentane 7.3; hexane 6.5; cyclohexane 7.5; ben-zene 10.3 ; carbon tetrachloride 8.6; chloroform 50.3; dichloromethane 42.9 ; dichloroethane 3 1.9 ; trichloroethane 13.1.Average o (first five only) = 3.0%. DATA SET (6)28 Two mixed-solvent systems were studied namely squalane + dode-canol and squalane + lauronitrile at temperatures 40-80 "Cusing 6 composition points for 27 aliphatic solutes. Apart from pentene hexane and heptene all the soluest were polar comprising alcohols ketones acetates halides cyanides and nitroalkanes. The fit of eqn (1) was generally poor (a = 3-10%) and the ~23 values varied widely frcm 2 to 40 mol dm-3. The lowest ~23 values for both systems were obtained with the non-polar solutes. DATA SET (7)23 Using 5-9 composition points the systems shown below were studied at 25 "C (except System 1 at 43 OC) System 1 phenol -/-aniline. System 2 m-cresol + benzylamine. System 3 m-cresol -+ benzyl alcohol.System 4 m-cresol -k acetophenone. System 5 m-cresol + benzonitrile. System 6 benzylamine + aniline. System 7 nitrobenzene + aniline. System 8 decalin + benzyl alcohol. System 9 acetophenone + tetrabromethane. System 10 acetophenone + dichlor-acetic acid. System 11 N-methyl pyrollidone + dimethyl sulphoxide. System 12 benzonitrile + dimethyl sulphoxide. Seven solutes were used as shown in table 2. Note that these published data are of y" values of the solute which were converted to equivalent KR data for use in eqn (1) on the basis that KR(2,3) = a/y" V(2.3) where a is a constant for a given solute which disappears in the derivation of eqn (1). DISCUSSION QUADRATIC (In K,,Cp) RELATION The fit of eqn (1) appears to be within experimental error for most results data set (6) being the exception.The linear (KR,p) relation is certainly not applicable to data sets (l) (2) (5) and (7) but gives a considerably better fit for data set (4) (with a remarkably low standard deviation of 0.1 %) a marginally worse fit for data set (3) and there are a few instances in data set (6) which seem to show a linear relation. It is interesting that the results of Eon et al. in data set (4) were on their model of com- P. F. TILEY 97 plexing shown to give a linear relation between (V;2,3) &(2,3)) and x2,where V(2,3) is the molar volume of the mixed phase and x2 the mole fraction of additive. Noting that for zero excess volume V(2,3)= x2V2+ (1 -x2)V3and that q2 = X?V~/V(~,~) it can readily be shown that the linear relation of Eon et al.is mathematically identical with eqn (2)of Purnell and Laub. In other words a linear (KR,y)relation can equally well be claimed to support a theory of complexing as to support a micro-partitioning theory. Two further points must be made. Firstly a g.1.c. column prepared from a TABLE 2.-cOMPUTED x23 VALUES FROM DATA SET (7) system 1 2 3 4 5 6 pentane -4.2 -5.6 -3.0 -1.8 hexane -4.5 -18.5 -4.9 -2.2 -1.4 1.1 -heptane -3.8 -4.0 -2.0 -1.4 cyclohexane -3.9 -18.3 -4.6 -2.5 -1.9 0.8 methyl butene -5.2 -25.8 --2.2 0.0 isoprene -4.9 -23.3 --2.0 -1.1 benzene -2.9 -18.6 -1.6 -2.4 -0.6 -2.7 o(%) 2.2 2.0 3.5 2.5 3.1 1.7 system 7 8 9 10 11 12 pentane 3.1 I 0.2 -8.6 hexane 3.1 13.7 0.0 -3.9 10.4 7.4 hep tane 2.5 -0.0 --6.8 cyclohexane 2.3 12.6 0.1 -4.7 11.1 9.2 methyl butene 3.2 11.3 -0.07 -6.3 9.6 5.3 isoprene 2.0 10.4 -1.7 -6.9 9.8 4.5 benzene 0.6 9.4 -2.8 -6.6 11.8 5.2 o(%) 1.5 3.7 1.8 2.6 2.6 2.2 mechanical mixture of two separately coated packings (" mixed-bed '' column) would certainly give a linear (KR q) relation so that the partioning behaviour of such a column should be predictable from eqn (2).Provided both phases exist as liquids at the column working temperature then such a column could well be preferable for any analytical purposes. The only minor disadvantage is that a mixed-bed column would be thermodynamically unstable in that gas-phase diffusion would eventually lead to a homogeneous mixed-solvent phase with depending on the system changing partition- ing characteristics.The second point is that if certain solutes particularly the alkanes (see below) give approximately equal KRvalues in the two pure solvents then eqn (1) should approxi- mate very closely to eqn (2) since a ~23 value close to zero would be expected.I3 This is clearly the case with data set (3). CONSTANCY OF X23 VALUES The second test of the model used in eqn (1) is that the ~23 values for a given solvent system should be independent of the solute. Even allowing for the effect of random experimental error as simulated in the results shown in table 1 it is obvious that the MIXED-SOLVENT BEHAVIOUR criterion is not met for any of the data sets. In view of the fact that eqn (1) is based on the hypothesis of random mixing this is not surprising since all the solvent systems include at least one polar constituent and many of the solutes used are themselves polar.Specific attractions are to be expected whether these be dipole association or charge-transfer complexing and indeed much of the work reported is claimed to be studies of complex formation. However when no specific solute-solvent interaction would be expected as with the alkane solutes a study of the results shows a reasonable constancy of ~23 (it is regretted that some workers report no results for any alkanes). The very careful work of data set (l) involving twelve composition points shows remarkably con- sistent ~23 values for the alkanes as also do the results in data sets (2) and (3).In the case of data set (7) where a high accuracy was not claimed (1-3 %) and possible experi- mental error in ~23 of around $11 mol dm-3 would be anticipated the results for the alkanes show reasonable constancy. The only other non-polar solutes studied have been either the alkenes or the aromatic hydrocarbons. Data set (3) seems to show the alkenes yielding the same ~23 values as the alkanes but data sets (l) (2) (3) and to some extent (7) show that benzene and its homologues show a divergence. This is readily explicable on the basis of specific interaction between the delocalised electrons of the benzene ring and the polar group on the solvent molecule. A further point arises as to whether for a given system ~23 is independent of the composition of the mixed-solvent phase.Perry and Tiley l2concluded that even with the alkanes ~23 showed a linear dependence on composition which effectively implied a cubic (In KR,9)relation. In quantifying the excess free energy of binary liquid systems many workers have proposed that two parameters (at least) are required in addition to the combinatorial term or its equivalent. For exact theoretical descrip- tion this may well be true but it must be recognised that once additional adjustable parameters are included in any fitting of experimental results the molecular signifi- cance of the values so obtained may in fact become much more nebulous. In parti- cular there arises in the curve-fitting process a possible correlation between any pair of parameters so that no unique set of values exists.z4 For that reason this paper does not present any investigation of the composition-dependence (if any) of ~23.SIGNIFICANCE OF G.L.C. X23 VALUES The careful and painstaking vacuum microbalance studies of vapour-liquid equilibrium carried out by Ashworth and Hookerz5 effectively validate the g.1.c. determination of ~23 for the squalane + dinonylphthalate system. For the alkanes the former give a mean value of 2.69 mol dm-3 compared with 2.41 mol dm-3 for data set (1) and for benzene values of 3.80 and 3.70 mol dm-3 respectively. Since the microbalance studies were made at finite solute concentration in the ternary systems this seems a very encouraging result. The conclusion that g.1.c. studies of alkane solutes yield a consistent ~23 value for a given solvent system leads to the question of the thermodynamic significance of ~23 in describing the excess free energy of the binary mixture of solvents in particular with respect to binary phase equilibria.The use of probe solutes to explore interactions in polymer systemsz6 is based on the anticipation that g.1.c. studies could be legitimately interpreted for such purposes. The very extensive studies shown in data set (7) should provide a test of this hypothesis since the volatility of the solvents used allows a com-parison with orthodox studies of binary vapour-liquid equilibrium. Unfortunately only one of the solvent systems appears to have been studied at a P. F. TILEY temperature anywhere near 25 "C.Holtzlander and Riggle2' report isothermal vapour-liquid studies for the binary aniline + nitrobenzene system at 57 "C from which the binary ym values may be extrapolated as 1.24 and 1.19 respectively. These yield a mean value of ~23 = 2.0 mol dm-3 at 57 "Ccompared with the g.1.c. mean value for the alkanes of 2.7 mol dm-3 at 25 "C as shown for System 7 in table 2. On simple theory,12~23 would be expected to be inversely proportional to temperature so this shows reasonable agreement between the two values. The highest positive mean value for the alkanes in table 2 is 13.2 mol dm-3 which occurs for System 8 decalin + benzyl alcohol. Theory2* shows that a critical solu- tion temperature should occur when Taking V2 (decalin) = 0.157 dm3 and V3 (benzyl alcohol) = 0.104 dm3 this gives ~23 = 15.8 mol dm3 and the two liquids should be completely miscible at 25 "C,which they are.However some simple laboratory experiments with laboratory-grade re- agents showed critical unmixing at around 2 "C. Again allowing for an inverse temperature dependence of ~23 and for considerable uncertainty in the g.1.c. value there seems some measure of agreement here. For the first five systems of table 2 negative ~23 values are computed indicative of specific attractions existing in these solvent pairs. This would be expected from the chemical nature of the components where acid-base (or acceptor-donor) interaction must occur in each case. Systems 2-5 show the interaction of m-cresol with benzyl- amine benzyl alcohol acetophenone and benzonitrile respectively with ~23 values of -18.4 -4.8 -2.4 and -1.6 mol dm-3.The relative strength of these acceptor- donor interactions as measured by g.1.c. show good qualitative agreement with the '' solvent donicities " (donor power) evaluated by G~tmann~~ which give benzyl- amine 9 benzyl alcohol > acetophenone > benzonitrile. The fact that ~23for phenol + aniline (-4.1) is much less negative than for m-cresol + benzylamine (-18.4) is also explicable on this basis. It seems possible that even for these systems where solvent-solvent association must certainly predominate the use of eqn (1) to interpret the g.1.c. results for alkane solutes yields ~23 values which are at least qualita- tively meaningful with respect to the solvent-solvent interactions.Whilst the results of data set (7) provide a useful test of the proposed model it is not in fact a complete test because all components solutes and solvents are of similar size. Hence the Flory-Huggins combinatorial term plays a very small part and the omission of this term giving the original Scatchard-Hildebrand theory would in most cases have little effect on the calculated ~23 values. MOLECULAR COMPLEXING A model postulating the existence of a 1 :1 complex between solute (1) and additive (2) in solvent (3) has given rise to the well-known eauation where Kc/dm3 mol-1 is the formation constant of the complex and C2/mol dm-3 the concentration of additive. Leaving aside the question of the thermodynamic formu- lation of the equilibrium constant Kc,eqn (3) still remains an approximation.The non-ideal behaviour of the binary solute (1)-solvent (3) interaction is describable in terms of non-specific interactions using an activity coefficient y&. However the derivation of eqn (3) assumes that any perturbation of this non-ideality produced by additive (2) can be ascribed entirely to specific interactions conforming to the law of MIXED-SOLVENT BEHAVIOUR chemical equilibrium. The use of alkanes as solutes where complexing is highly improbable has shown that this assumption is not tenable in many cases. Martire30 has published theoretical studies of the approximate nature of eqn (3) and has proposed a better formulation thus where a is a parameter reflecting the " physical " non-ideality perturbation of the ternary system and the variation in molal volume produced by the additive.At low concentrations Martire suggested that a should be roughly constant so that a linear (KR c) relation results but for a given solvent system a will be solute-dependent. Hence Kc values derived from eqn (3) may be in error by uncertain amounts. Using the model and the system parameters of Perry and Tiley,12 a simulated test of eqn (4) was carried out and an estimate made of the magnitude of a for the tri- nitroluene (TNT) + dinonylphthalate (DNP) system wj th benzene and its homo- logues as solutes. In this case hydrocarbon-TNT complexes were postulated on the basis of spectroscopic evidence. Values of KR were computed at TNT concentrations 0-0.5 mol dmV3 using the thermodynamically exact equation l2 K is the thermodynamic formation constant based on pure-component standard states and subscript 4 refers to the complex.Singly-subscripted y values refer to the mixed-solvent phase and were calculated from regular solution theory as describedl2 and the necessary parameters were assigned values taken from ref. (12). The com- puted KRvalues were then fitted to eqn (3) to obtain values of the " apparent '' K in each case. These were compared with the " true " K values calculated from the assigned values of K and from infinite dilution y values for the components in the solvent medium thus,12 Values of cx were obtained as the differences between apparent and true K values and are shown in table 3.The use of this theoretical model with the parameters from ref. (12) produces good TABLE 3.-vALUES OF o! COMPUTED USING THE PARAMETERS OF THE TNT $-DNP SYSTEM12 apparent K true K a 0.088 0.035 0.043 0.081 0.050 0.037 0.041 0.023 0.018 0.001 0.026 -0.025 0.097 0.051 0.046 0.072 0.048 0.024 0.076 0.062 0.014 linear (KR c) relations in accordance with eqn (4) with correlation coefficients all in excess of 0.995 but as can be seen with these low K values the " correcting term " a is of similar magnitude to K itself and varies from solute to solute. As a result even the relative order of the apparent K values is not the same as the true order. It is P. F. TILEY 101 noteworthy that the error resulting from the use of eqn (3) is much greater than that produced solely by the use of an equilibrium constant in terms of concentrations rather than activities.If eqn (6) is used to calculate K at finite concentrations of additive the variation of K is only ca. 10% at C2 = 0.5 mol dm-3. Being based on the parameters found for the TNT + DNP system the results of table 3 cannot obviously be generalised. Owing to the complexity of the expressions for activity coefficients it is not possible mathematically to reduce eqn (5) to any simple variant of eqn (3) such as eqn (4) and indeed Ma~tire~~ derived cqn (4) from a purely empirical polynomial expansion. Further computational studies would be necessary to examine the effect of different values of the parameters on the value of a.Indications are that a can be either smaller or very much larger than the values of table 3. The Martire-RiedlS1 approach to g.1.c. studies of complexing seems less prone to inherent error provided one accepts the basis on which it is founded. This method depends on using a reference solvent as nearly as possible analogous to the complexing solvent with respect to molal volume and polarisability thereby equalising the corn- binatorial term and the dispersion forces contribution to molecular interactions. Any difference in partitioning behaviour after discounting these effects is then ascribed to specific complex formation describable by the law of chemical equilibrium. On the other hand Perry and Tiley12 attempted a quantitative estimate of the non-specific interactions by use of a semi-empirical solubility parameter treatment and any residual effect was then ascribed to complexing.Both approaches have some .logical justifica- tion but it is only fair to indicate that other workers in the g.1.c. field have not used any model of chemical equilibrium in order to quantify the effects of specific inter- actions. For example Karger et aZ.32used an expanded solubility parameter treat- ment which was claimed to include terms for orientation and induction forces as well as for donor-acceptor interaction. Meyer and Bai~cchi~~ used g.1.c. measurements to determine the enthalpy and entropy changes of specific interactions without pro- ceeding via calculation of association constants.Provided the limitations of each approach are appreciated then the choice of model may well be determined by the object of the investigation. CONCLUSION As stated previously,12 g.1.c. studies in the context of this paper are thermodynamic studies of phase equilibrium and the seeking of a unique interpretation in terms of molecular interactions is akin to debating how many angels can dance on the point of a pin. The pragmatic approach of Scatchard,lo quoted above has much to commend it bearing in mind that pragmatism is not to be equated with pure empiricism. In framing models of mixed-solvent g.1.c. systems it would seem wise to take account of the considerable body of established work on liquid mixtures; and today it is a truism to say that the use of the computer facilitates studies of the quantitative impli- cations of theoretical models.The analysis presented in this paper makes only trivial demands on computer resources but reveals the limitations both of regular solution (Scatchard-Hildebrand-Flory-Huggins) theory and of g.1.c. studies of molecular complexes. I am indebted to Dr. A. J. Ashworth and Mr. D. M. Hooker for helpful discussions and to Mr. N. P. Morgan for some assistance with the computing. J. R. Conder and C. L. Young Physicocheniical Measurement by Gas Chromatography (Wiley Chichester 1979) chap. 5 and 6. 102 MIXED-SOLVENT BEHAVIOUR R. J. Laub and R. L. Pecsok Physicochemical Applications of Gas Chromatography (Wiley Chichester 1977) chap. 5 and 6. Phase Equilibria and Fluid Properties in the Chemical Industry ed.T. S. 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