Activity Coefficients at Infinite Dilution from Gas-Liquid Chromatography BY TREVOR M. LETCHER Department of Chemistry Rhodes University Grahamstown 6140 South Africa Received 22nd July 1980 Activity coefficients can be determined at infinite dilution with great precision using gas-liquid chromatography. The development of this technique is reviewed and recent results and treatment of results discussed. Gas-liquid chromatography (g.1.c.) is undoubtedly one of the most useful scientific inventions of the century. Its rapid development from James and Martin’s first experiment of 1952 bears testimony to its importance and usefulness.’ Since the basic process is an equilibration of a solute between two immiscible phases the chromato- graphic technique may be used to measure such physical properties as activity co- efficients second virial coefficients of gas mixtures partition coefficients adsorption and partition isotherms and complex formation constants.Other properties which can be measured with less accuracy from secondary measurements or from tempera- ture-variation studies include surface areas heats of adsorption excess enthalpies and excess entropies of solution. A number of reviews and discussions on these measurements have appeared in the literat~re.~’~~ The present work is restricted to a review of activity coefficient measurements. Activity coefficients are perhaps the most important and fundamental property in the thermodynamic study of liquid mixtures. They can be used to obtain all the other thermodynamic solution properties such as excess enthalpies.Gas-liquid elu- tion chromatography offers a rapid method of determining activity coefficients at infinite dilution. Conder and Purnell have developed a method of determining acti- vity coefficients at finite concentration^^^-^^ and it has recently been used by other worker~.l~-~~ To do this the elution technique must be supplemented by controlling the solute concentration in the carrier gas. In this type of chromatography the recor- der no longer records a peak but an integral form of the peak. This paper will be concerned with activity coefficients at infinite dilution obtained by the elution method. The nature of gas-liquid chromatography unfortunately limits the choice of liquid mixture.The solute must be rather volatile if retention times are to be reasonable and the solvent or stationary phase must be a liquid at the temperature of the experi- ment with a sufficiently low vapour pressure so as not to “ bleed ” off the column during the course of an experiment. Further limitations restrict the choice of carrier gas to those which are insoluble or nearly insoluble in the stationary phase and the range of mixtures to those in which adsorption effects of any kind are negligible. For systems which fall within these limitations accurate activity coefficients at infinite dilu- tion can be obtained which would be very difficult to determine in other ways. The restrictions on some of these properties can be relaxed but it does involve added experimentation.For example solvent bleeding can be overcome by either weighing the column before and after or by inserting a pre-column containing the 104 ACTIVITY COEFFICIENTS FROM G. L.C. volatile solvent. in any event regular checks must be made on the column specifica- tions. Adsorption effects have been dealt with by measuring retentions using columns of different solvent loadings.22 Activity coefficients at infinite dilution are important and useful properties to chemical engineers solution chemists and theoreticians. To the chemical engineer they are of interest in the design of plants that involve liquid-vapour equilibrium. To the solution chemist they are important in understanding the mixing process. Perhaps their most interesting and successful application has been in the testing of various solution theories.This is possibly due to the large number of systems that have been analysed (because the results can be obtained so rapidly using g.1.c.) and to the infinitely-dilute condition which in many cases presents the theorist with simpler equations and fewer complications. THEORETICAL THE UNREFINED THEORY The idea of gas-liquid chromatography goes back to 1941 when Martin and Synge mentioned in a paper on general ~hromatography~~ that it should be possible to use a gas as a mobile phase and a liquid as a stationary phase. They related the equilibrium partition coefficient K to retardation properties using a plate theory. Their general equation when considered in relation to gas-liquid chromatography (for zero pressure difference across column) relates the retention volume of the solute VR,to the gas hold-up volume V, and the solvent volume V3according to VR = vc + KV3.(1) (In this work the solute will be referred to as component 1 the carrier gas as com- ponent 2 and the solvent as component 3.) The idea was not taken up and it was left to Martin this time in conjunction with James,' to describe a separation using g.1.c. (They separated some volatile fatty acids by partitioning between nitrogen and a mixed liquid phase of silicone oil and stearic acid.) More important they presented the first theory specifically applicable to gas-liquid chromatography by taking into account the compressibility of the mobile phase. This involved applying a correction factor to the gas volumes of eqn (1).In terms of Everett's notation24 this correction term J; can be generalised as n (Pi/P0)"-1 Jr = -m (PJP,)"-1 where Piand Porefer to inlet and outlet pressures respectively. In 1956 Martin25 again hinted at future developments in this field and speculated that gas-liquid chromatography might be useful in studying the solution thermo- dynamics of gas and liquid phases. In the same year Porter et aZ.26related the net retention volume V, to the activity coefficient of the solute at infinite dilution YE according to where n3is the amount of liquid solvent on the column and Pothe vapour pressure of the solute at the temperature T of the experiment. The retention volume V is determined from the column outlet flow-rate U, by T.M. LETCHER where tRand tG are the retention times for the solute and an unretained gas and V is the dead-space volume or gas hold-up volume at mean column pressure PoJ;. This theory assumes that (a) idealized chromatographic conditions exist ; (b)the car- rier gas and solute vapour behave as ideal gases; (c) adsorption effects of any kind are absent and (d) the carrier gas is insoluble in the solvent. The measurements based on the above theory give activity coefficients for n-alkane systems which are within a few percent of the results obtained from the more refined theory discussed below. REFINEMENTS TO THE THEORY The next major step in the evolution of determinations of accurate activity co-efficients came in 1961 when Everett and Stoddard2' took into account the solute vapour and solute $-carrier-gas imperfections.An important outcome of this work was the possibility of obtaining the mixed virial coefficient B12. DestyZ8 applied these ideas to the determination of BI2values and used an extrapolation procedure based on the equation In VN= In VNo+ ppoJ3 (5) where V," is the extrapolated retention volume at zer9 mean column pressure where V;is the molar volume of the solute and Bllthe second virial coefficient of pure solute. p is given by where Vy is the partial molar volume of solute at infinite dilution in the stationary phase. Both of the above treatments assume that the partition function does not change significantly along the column but remains constant at the mean column pres- sure value.Everett24 attempted to avoid this assumption and developed a detailed theory of the pressure-drop effect which led to a different extrapolation procedure. The Bristol group29-31 reformulated the differential equation describing the local elu- tion rate in the column and suggested a third extrapolation procedure In VN = In VN" + ppnJ:. (8) This was tested using a numerical integration procedure and shown to be superior to the previous extrapolation techniques.28*24 Moreover this theory takes into account small imperfections in the carrier gas and is thus suitable for carrier gases such as hydrogen helium nitrogen oxygen and argon. For carrier gases which are appreciably non-ideal they proposed where b = B2,/RTand is the second virial coefficient of the carrier gas.A further refinement done by the Bristol group32 involved the solubility of the carrier gas in the stationary liquid. They showed neglecting second-order effects that the retention volume (for a pressure drop across the column of (200 kPa) is related to pressure po according to In Vk = In F'; + /3'poJ$ (10) where /3' = /?-/-3,[1 -a ;2y"1 I06 ACTIVITY COEFFICIENTS FROM G.L.C. and A is defined by the expansion of x2 (mole fraction of carrier gas in the solvent) as a series in the local carrier-gas pressure x2 = AP2 + pP2”+ . . . . (12) where p is the coefficient of the second-order pressure term. Eqn (1 1) includes the two effects resulting from the solubility of the carrier gas namely the increase in the number of moles of stationary liquid and the change in the activity coefficient due to the change in the nature of stationary liquid.The importance of deriving eqn (10) and (1 1) lies in the fact that it shows that because the quantity (2 1n yz/2x2)is virtually inaccessible the property BI2 cannot be obtained unambiguously from g.1.c. measure- ments with a solvent in which the carrier gas is appreciably soluble. The significance of the carrier-gas solubility and eqn (1 1) in particular has been discussed in terms of experimental results by the Bristol gr~up~’’~~ and by Pecsok and Wind~or.~~ THE TRUE RETENTION TIME Existing theories of g.1.c. predict a unique retention time. Experimentally how- ever a peak spread is observed so it is therefore necessary to speculate where on this peak the true retention time may be found.I have included some of the peak pro- perties that have been used to define this “ true ” retention time. The peak-initial time tI and peak-final time tF are determined from the intersection of the base-line time with the tangents to the leading edge and trailing edge respectively. The peak- tangent time tT,on the other hand is determined from the point of intersection of the tangents to the leading edge and trailing edge. The peak-half-area time is defined as the net time that divides the area under the peak into two equal parts. The peak- maximum time tMis obtained from the time of peak maximum and the peak-average retention time tIFis obtained from the mean of tI and tF.Because of the lack of knowledge concerning the detailed processes taking place in a g.1.c. column the present-day theories cannot hope to unravel the problems associ- ated with peak asymmetry. Possible causes that have been given by various wor- kers 7*35*36including eddy diffusion in packed columns non-equilibria between phases sample size surface heterogeneity and non-zero response time of the detecting system. A somewhat surprising answer to the question of true retention time was given in some of the earlier attempts to derive thermodynamic properties from g.1.c. measure- ment~.~~-~~*~~ The peak initial time tl was used the justification being that this gave the best agreement with data from static measurements.Later it was that if sufficient care was taken to achieve effective infinite dilution of the solute and if the measurements were done over a range of pressure and analysed by a theory which culminates in eqn (8) then the peak tangent gave good agreement with static measure- ments. This seems far more reasonable as it is somewhere near the top of the peak although for skew peaks this may not be true. Many worker~~~q~’ have discussed the “ first time moment ” or “ centre of gravity ” of a chromatographic peak undergoing elution. In the absence of longitudinal or eddy diffusion this property has been shown to be equal to the ideal thermodynamic retention time for zero-pressure-drop columns. More recently theories regarding the first time moment has been extended by and by Buffham4’ to include pressure- drop columns and the first time moment has been related to thermodynamic proper- ties.Buffham used the “ mean residence time ” f which is equivalent to the “ first time moment ” f = /om t ci(t)dt //om ci(t)dt T. M. LETCHER 107 where cl(t)is the response of the solute molecules i at time t after injection. He related ito thermodynamic properties and showed that it can be used to obtain results similar to those of Cruickshank et aZ.,30Stalkup and Kobaya~hi~~ and Koonce et ~l.,~~ without the restrictions previously considered necessary.45 has recently arrived at a solution to the g.1.c. situation for which the diffu- sion of solute in the liquid phase is the rate-determining step for equilibration.In this work Hicks shows that for this situation the peak-average retention volume VIF can be used to obtain thermodynamic properties. To support this he showed that the flow-rate dependence of retention volume as observed in the benzene + glycerol system22 disappears if peak-average retention volumes are used instead of peak-tangent retention volumes. Experimentally tIFcan be obtained directly from the peak tan- gents without elaborate equipment or with the problems associated with the “ first time m~rnent.”~~*~’ The flow independence of V, does seem to point to tlFbeing an excellent estimate of the ideal retention time although much more work is required on systems with asymmetric peaks to show its range of validity. The first moment is possibly better at least in theory because it does not require assumptions about the nature of the kinetic processes governing the equilibration between phases but it is experimentally very inconvenient.The problem of locating the true retention volume is however usually only important for solutes which have short residence times and have very asymmetric peaks. It is only then that the peak-average retention times differ significantly from the peak-maximum peak-tangent peak-half-area or mean-residence retention times. SURFACE ADSORPTION EFFECTS Surface adsorption is perhaps the most important limitation of the g.1.c. for deter- mining activity coefficients. Martin48 suggested that this adsorption at the gas-liquid interface can be related to the retention volume and proposed VN = kV3 + k A3 where k is the adsorption coefficient and A3 the area of the liquid surface.Unfor-tunately there is no way of separately determining these two terms by chromatographic experiments alone and an extrapolation procedure must be used to obtain VN.22 RESULTS AND EQUIPMENT GENERAL If the activity coefficients are not required to any great accuracy (&5% uncer-tainty) then eqn (3) will suffice provided that adsorption effects are insignificant. For such measurements a simple gas chromatograph with the column in a well-controlled water bath is suitable. The type of detection is not important so long as sample detection of 0.5 pmol is possible. The advantage of katharometer detection is that the flow meter can be placed downstream of the column.For more accurate determinations of activity coefficients it is necessary to take into account carrier-gas and solute imperfections. Assuming no adsorption of the solute on the solvent or solid support and assuming the carrier gas is not appreciably soluble in the solvent then eqn (10) is suitable The uncertainty in the activity coefficients determined in this way has been estimated ACTIVITY COEFFICIENTS FROM G.L.C. to be <0.4%. Application of eqn (10) usually infers the determination of the reten- tion volume as a function of pressure (p J;) from which the mixed second virial co- efficient BI2,and the activity coefficient yg can be obtained. Medium-high-pressure g.1.c. does require more sophisticated apparatus.It must include a high-pressure injector pressure control values special metal-glass seals and a flow meter capable of operating at pressures of 1200 kPa. A detailed breakdown of equipment design will not be given here. This has been well covered by recent review^.^*'^-^^ The activity coefficients of many hundreds of systems have been determined using this method. I will discuss some of the results especially those that have a bearing on solution thermodynamics. The results before I967 have been discussed in reviews by Young’ and Kobayashi et aL5 Many of these results were done on ill-defined sub- stances such as silicon oils and apiezon greases and will not be discussed here. Most of the systems discussed here have been reviewed by the author.I2 Condor and Young13 have also recently reviewed g.1.c.-determined activity coefficients.n-ALKANE MIXTURES AND RELATED SYSTEMS The activity coefficients for these systems measured by g.1.c. have been the most useful and successful in testing solution theories. The repetitive nature of the carbon chains make these systems ideal in this respect. Lattice theories in particular have proved very successful. Alkane systems are fortunately convenient to study because the properties such as molar volumes vapour pressures and virial coefficients have been well documented. The pioneering work for these systems was done by Kwantes and Rijnder~~~ who studied normal and branched-chain alkanes in n-octane n-decane n-hexadecane n- tetracosane and n-pentatriacontane.Measurements have also The results are consistent with static mea~urement~.~~~~~ been reported by Little~ood,’~ Martire and P01lara,~* and by Pease and Thorburns3 but these do not agree well with static measurements. The most reliable and comprehensive g.1.c. activity-coefficient measurements for n-alkane systems have been done by the Bristol group31~33~39~s4-56 using medium- high- pressure g.1.c. and taking all carrier-gas and solute imperfections into account. They have examined the C4-Cs n-a1 kane solutes in c,&2 n-alkane solvents. Generally the results indicate that the smaller the disparity in carbon number between solute and solvent the closer is the activity coefficient to unity. The measured activity co- efficients range from 0.930 for the heptane + hexadecane system at 303 K to 0.695 for heptane + dotriacontane at 348 K.Activity coefficients for many aik-1-ene + alkane systems have also been measured by this Tewari et aLS*have also measured the activity coefficients of n-alkane and n-alk- 1-ene solutes in long-chain n-hydrocarbon solvents (in this case C24 C30 and C36). Their results substantiate the general trends and results obtained by the Bristol group. Letcher and Marsicano 59 have measured the activity coefficients of other unsatur- ated Cs and c6 straight-chain hydrocarbon solutes in n-octadecane n-octadec- 1-ene n-hexadecane and n-hexadec- 1-ene solvents. The activity coefficients for these systems do not form simple trends because the various types of double bonds (ter- minal internal and conjugated) influence intermolecular interactions in different ways.Definite trends can however be seen in the interactional contribution T‘. Branched-chain alkane systems are more difficult to fit into a lattice-theory picture. Nevertheless activity coefficients for such systems have been rep~rted.~~-~~ Cycloalkane + n-alkane systems have been extensively investigated by Letche~.~O-~~ T. M. LETCHER 109 Benzene in n-alkane solvents has been studied by the Brist ol and by Let~her.~~ Recent work by Letcher on hydrocarbon + siloxane systems66 and Group IVA tetramethyl compounds + hydrocarbon^^^ have been done with the expressed pur- pose of testing theories of liquid mixtures. Many other systems have been investi- gated and been reviewed12-14 and will not be discussed here.Perhaps one of the most useful applications of this technique is in predicting finite-concentration activity co- efficients from the infinitely-dilute result. Work in this field has been done by Let- cher and Netherton,68 Bogeatzes and Tassios6’ and Hussey and Parcher.” THEORETICAL TREATMENT OF ACTIVITY COEFFICIENTS The activity coefficients at infinite dilution have been analysed in many different ways in attempts to understand the interactions of solute with solvent. The earliest efforts based on empirical relationships for homologous series of solutes interpreted the results in terms of group interactions structural effects polarity and electron donor and acceptor capacitie~.~-I~ These ideas have served as foundations for the more sophisticated theoretical interpretations.The fundamental approximation that has been used in most of the theoretical treat- ments considers the activity coefficient to be separable into two parts In yl (configuration) + In y1 (interaction). (15) The configurational and interactional contributions can be considered independent so long as the interactions are small. The configurational contribution due to the mix-ing of long-chain and short-chain molecules can be related to simple lattice proper- ties.7 A simplified version of the theory originally given independently by Flory 70 and Huggins,71 gives -In Yi (config.) = [(I -93)/Xil -k (1 -k I/r)vl3 (16) where 93 = rx3/(xl + rx3)* (17) The symbol q3refers to the volume fraction of the long-chain molecules and r to the ratio of sites occupied by the long- and short-chain molecules.For the infinitely dilute condition that applies in g.l.c. eqn (16) becomes In y (config.) = ln(1jr) + (1 -l/r). (18) The ratio r is often taken as the ratio of molar volumes of the long- and short-chain compounds. For the infinitely-dilute condition the interactional contribution is given by72 In y1 (interaction) =x (19) where x is the interaction parameter. It is in the interpretation of x that the various solution theories differ. Many theories such as Hildebrand-Scatchard solubility parameter theory pertur- bation methods congruence principle Flory Orwoll and Vrij theory and the Prigo- gine cell theory have been tried.These attempts have been reviewed by Condor and Young13 and will not be discussed here. The segment and contact-point treatment has been very successful and will be discussed briefly. The original theory was presented by Tompa73 and was later developed by McGlashan et aZ.74 Various forms of it have been very successful in treating the interactional parameter x,obtained from experimental activity coefficients at infinite ACTIVITY COEFFICIENTS FROM G.L.C. dilution and the calculated configurational contribution obtained from eqn (1 8). Applying this theory to mixtures involving only two types of segments or contact points (A and B) and to data obtained at infinite dilution the interactional contribu- tion is given by W x = rl(Ol -0,)‘ kT -where 0 and O3 are the A type segment fractions of molecules of type 1 and 3 and W is the interchange energy of the two types of segments defined in terms of hypothetical molecules containing only one kind of segment.This interchange energy should be constant at a given temperature for all mixtures containing only the two types of seg- ments. and Tewari et aLs8have been most successful in applying The Bristol gro~p~~-’~ this theory to n-alkane mixtures. Young29 has extended this theory to include three types of segments. In this case the interactional parameter x,becomes where ai Piand d1 are the fractions of segments of type A B or C of a molecule i (i = 1 or 3) and WAB,WBcand WACare the interchange energies of A and B B and C and A and C respectively and n is the number of segments on the smaller molecule.This theory has been applied by the Bristol group,57 by Tewari et aLs8and by Let- &er63,65 -68 to n-alkene + n-alkane and to benzene + n-alkane systems. Letcher and Marsicano 59 have extended this to include n-alkene + n-alkane systems. Tewari et aL7’ have also applied it to halogenoalkane solutes in alkane solvents. This theory becomes unwieldy when more than three segments are used. The slopes of the plots of x against carbon number for many systems have been obtained by LetcheP7 and give an insight into the predictive ability of this technique. 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