Selectivity in liquid-liquid chromatography

Selectivity in Liquid-Liquid Chromatography D. E, Martire Department of Chemistry, Georgetown University, Washington, D. C . 20007

D. C . Locke Department o j Chemistry, Queens College of the City University of New York, Flushing, N. Y. 11367

The thermodynamic basis for selectivity in liquidliquid chromatography is considered in terms of a new approach to solution free energies. The treatment applies to closely-related and difficultly separable molecules, such as isomers. Both athermal (size) and thermal (energy) factors contribute to the relative retention. The results are expressed quite simply in terms of the molar volumes of the solvents and the pair of solutes to be separated, the solvent solubility parameters, and the fractional difference between the solute critical temperatures. LLC solvent systems designed to separate polar molecules inherently offer the greatest selectivity. Reasonably good agreement with experiment i s found for a selective LLC system used to separate aromatic hydrocarbons. The fundamental approach taken i s shown to be superior to one using conventional regular solution theory.

THEABILITY of any chromatographic technique to perform separations depends upon both the selectivity and efficiency of the chromatographic system. The former is expressed in terms of the relative retention, a, of the pair of components to be separated, while the efficiency is measured by the number of theoretical plates, N (or plate height H ) generated in the chromatographic column by the solutes. While both are important, achievement of large relative retention values is fundamentally more critical. The very nature of chromatographic systems generally ensures respectable column efficiencies, if reasonable care is taken in constructing and operating the column. Further, any resolution equation shows that resolution, the degree of separation of two peaks, increases directly with ( a - 1 ) but only with N.1’2 Finally, in practical systems, optimum column efficiency is usually sacrificed for shorter analysis times. Broadly, plate height depends o n operating conditions such as eluent flow rate, packing particle size, stationary phase loading, and so forth-in short, o n the dynamics of the system. Relative retention, o n the other hand, is a n equilibrium property of a specific system, independent of all its extensive properties. Basically, to increase a, it is necessary to change a t least one of the chromatographic phases so as to improve the selectivity of the system for one of the solutes of interest. To this end, we consider the selection and design of the stationary and mobile phases in liquid-liquid chromatography (LLC), in particular as applied to the separation of closelyrelated molecules such as isomers. Existing solution theories provide an adequate basis for formulating certain principles of selectivity. The results developed apply equally to LLC and to solvent extraction. Selectivity in G L C has been the subject of recent work (2-4) whereas this is the first formulation for LLC as well (1) P. T. Funke, E. R. Malinowski, D. E. Martire, and L. 2. Pollara, Separ. Sci., 1, 661 (1966). (2) S. H. Langer, AKAL.CHEM., 39, 524 (1967). (3) L. Rohrschneider, J . Gas Chronzatogr., 6 , 5 (1968). (4) S. H. Langer and R. J. Sheehan, in “Progress in Gas Chromatography,” J. H. Purnell, Ed., Interscience, New York, N. Y . , 1968, p 289. 68

as the first application of a very recent theoretical development ( 5 ) to chromatographic separations. THEORY

Previously, we showed (6) that a t moderate column pressures, the LLC solute specific retention volume, V g ,is

where y z m and y2* are, respectively, the infinite dilution solute activity coefficients in the mobile and stationary phases which have molecular weights Mm and M,; p m is the eluent density a t the column temperature. I n this paper, all activity coefficients refer to their values at infinite dilution. The relative retention a of two solutes a and b is

Understanding and prediction of selectivity graphic solvent pairs is thus a problem in modynamics, Le., the prediction of relative coefficient values from pure solute and solvent We are o n good grounds for assuming additivity (7-9): In yz = In

+y2th

+ In

728th

of chromatosolution thersolute activity properties. the following (3)

where y p t h is the configurational or athermal contribution to the activity coefficient arising from size differences between solute and solvent molecules, and y , t h is the thermal component which accounts for intermolecular forces between solute and solvent molecules. Whereas the former is strictly an entropic term, In y 2 t h comprises both enthalpic and entropic contributions to solution non-ideality (9). Proceeding, we use the usual Flory-Huggins expression (8-11) for the athermal part of the activity coefficient, where In

y ~ a t h=

In ( l l r )

+ [l - (l/r)]

(4)

we approximate r as the ratio of molar volumes of solvent (component 1 ) to solute (component 2), r = cl0/c2O. The recent theoretical treatment of Luckhurst and Martire (5) provides a n expression for the thermal contribution. These authors relate y2th for some solute 2 of interest to that of a reference solute 2’ in the same solvent under the (5) G. R. Luckhurst and D. E. Martire, Trans. Faraday SOC.,65, 1248 (1968). (6) D. C. Locke and D. E. Martire, ANAL.CHEM., 39,921 (1967). (7) A. J. Ashworth and D. H. Everett, Trans. Faraday Soc., 56, 1609 (1960). (8) D. E. Martire, in “Gas Chromatography,” L. Fowler, Ed., Academic Press, New York, N. Y., 1963. p 33. (9) D. E. Martire, in “Gas Chromatography 1966,” A. B. Littlewood, Ed., Elsevier, Amsterdam, 1967, p 21. (10) P. J. Flory, J. Chem. Phys., 10,51 (1942). (11) M. L. Huggins, Ann. N . Y.Acad. Sci., 43,l (1942).

ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971

-1 0

Figure 1. Athermal contribution to relative retention, with ( u b o - o b o ) as a parameter. a = (vo)a/(Vp)b

- 3.0

-20

-1

0

00

loo(:

1

-

10

-1

where (EJuIO)and ( E Z ~ / Lare . Z ~the ~ ) cohesive energy densities of the solvent and reference solute, respectively, uio and u2(O are their molar volumes, and Tz* and Tz,* are the critical temperatures (K) of the solute and reference solute, respectively. The reference solute is selected to be closely related in solution behavior to the solute of interest. Now Ei -

.-

CiO

AEt" -

( A a U- R T ) Vi0

Vi0

(6)

where AE," and AHiO are the molar energy and enthalpy of vaporization of component i. Hildebrand (12, 13), of ) * ' ~ the solubility parameter course, identified ( i l E i " / ~ . t ~ with 6, (see Appendix 1). Clearly, on this basis we can write Equation 2 as In a

=

(In

a)ath

+ (In

a)th

(7)

where the activity coefficient contributions to a have been divided into athermal and thermal contributions. Athermal Contribution to Relative Retention. From Equations 2-4, (In

a)afh

=

In

(-yum/ybm)ath

- In (-)'us/ybs)ath =

30

"*

"rn

same conditions, by

20

1

of its larger positive excess entropy of solution (7-9). For example, if two isomers a and b of M = 200 have respective densities of 1.025 and 1.000, then (ob0 - uuo) = 5 cc/mole; if urno = 50 cc/mole and uB0 = 500 cc/mole, then (In a ) a t h = 0.09, or a = 1.1, which corresponds to a straightforward separation requiring roughly only 4300 effective plates for a base-line separation. In Figure 1 , the dependence of (In a!)atIi on (l/unLo- l/uso)is illustrated for various (ubo - cue). Large separation factors are attainable on the basis of this size effect alone. The athermal contribution to a is readily estimated from the densities and molecular weights of the solutes and solvents. Thermal Contribution to Relative Retention. From Equation 5 , substituting Hildebrand solubility parameters for cohesive energy densities, (In

a)th

=

(In yum/ybm)th - (In

~ ~ ' / ~ ~ ' )= t h

Since we are restricting attention to energetically closelyrelated solutes (a and b) which are also both closely related to the reference solute (2 '), we can write the following approximate equations, u2,0

= uao

Tz#*= Ta*

where we could as easily have equated 2' and b. Thus Thus the more divergent the values are for the pairs of solvents (omo and uso) and solutes (uno and obo), the greater a! will be, especially for small umO or uso. Experimentally, the eluent in LLC is commonly composed of smaller molecules than is the stationary phase. Energetically identical isomeric solutes are clearly separable by LLC if the molar volumes (densities) are sufficiently different; no selective solute-solvent interactions are required. In this case, the smaller solute is preferentially dissolved in the phase with the larger uo because (12) J. H. Hildebrand and R. L. Scott, "Regular Solutions," Prentice-Hall, Englewood Cliffs, N. J., 1962. (13) J. H. Hildebrand and R. L. Scott, "Solubility of Nonelectrolytes," 3rd ed., Reinhold, New York, N. Y . , 1950.

where we define (Ta* - Tb*)/Ta* J' ( T * ) as the fractional difference between the critical temperatures of pure solutes a and 6 . Separation is clearly favored by large differences in the cohesive energy densities of the chromatographic solvents, which must in any event be the case if the two phases are to be immiscible (12, 13). For any given solvent pair, this contribution to resolution of two solutes depends only on the product uuof(T*); one criterion of separability in LLC is, on this basis, unequal critical temperatures. Sources of data on critical temperatures and solubility parameters are given in Appendix 1.

ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971

69

I

-0.4 -80

I

-60

-40

I

-20

0

I

20

40

60

80

Figure 2. Thermal contribution to relative retention, with f ( T * )as a parameter. CY = (VQ),J(Vg)b The behavior predicted by Equation 1i is entirely reasonable. Luckhurst and Martire (5) selected Ti* as a macroscopic measure of the magnitude of the pairwise potential energy of interaction between two molecules of type i. Thus the solute molecules of the larger interaction profile (as measured by T * ) will tend t o interact more strongly with the solvent of the large interaction profile (measured by Jt2); as they say in the old country, “Like dissolves like.” If the solvent is the chromatographic stationary phase, this solute will be preferentially retained, and if it is the eluent, the solute will be preferentially eluted. The absence of a n explicit solute-solvent interaction term results from the use of a reference solute and the geometric mean assumption for 1-2 interactions. Equation 11 might be illustrated with a numerical example. Iff(T*) = 0.01, v,O/RT = 0.2 cc/cal, and (6,2 - am2) = 50 cal/ cc, then (In a ) t h = 0.10, and a t b = 1 . 1 , which again is entirely sufficient for the base-line separation of the two equallysized solute molecules a and b. The sensitivity of (In a)th to changes inf(T*) is shown in Figure 2. The ranges of values given in the plot are realistic for practical LLC systems for difficult separations. DISCUSSION

The validity of the foregoing depends upon the strict applicability of the basic theory (5) to these LLC systems. We will consider first the assumptions involved and then compare relative retentions calculated from the theory with experimental values. First, the reference solute must be chosen such as to have nearly the same strength and nature of interaction with the solvent as the solute of interest. I n LLC, of course, this condition must apply to both solvent phases. I n other words, our theory applies only to very similar-and difficult t o separate-pairs of solutes. A simple means of checking the validity of this assumption is t o compare values of vb0/Tb* and va0/TQ*. This is so because we have arbitrarily identified the reference solute with one of the pair of interest. The second, and more important, assumption, is that the form of the pairwise intermolecular potential function 70

is the same for both solute-solute and solute-solvent interactions, for identical configurations. If the potential functions above have the same forms, it is reasonable to expect adherence to the geometric mean combining rule for the intermolecular potential functions (5). It is this assumption which fundamentally allows the use of the reference solute. I n addition, the proportionality between the depth of the potential well and the solute critical temperature can in this case justifiably be used. These two considerations lead to the remarkably simple form of Equation 5 and thus of Equation 11. I n LLC, because the two phase solvents must be more o r less immiscible, there will generally be different degrees of interaction of the solutes with the two phases. The LLC stationary phase is usually selected to be a “better” solvent for the “average” solute in a mixture than is the mobile phase; this tends in practice to allow the solvent system to differentiate between solutes and produce separations. Consequently, the above assumption may ordinarily hold for one of the phases, but a t best less rigorously for both. We can thus expect agreement of theory with experiment, and also the power of reasonable prediction, for systems in which the distribution coefficients are close to unity. We can also expect the theory to hold where moderately specific interactions exist (see below). I n these systems, which are of interest because of their selectivity, one phase interacts o r reacts selectively with a t least some of the solutes. Here, the second phase is usually a “neutral” solvent for the solutes, possessing the proper solvency to provide reasonable retention volume values. I n the example given below, the solutes are aromatic hydrocarbons, the interacting phase is acetonitrile, and the neutral stationary phase is squalane. Since we are concerned here with selectivity, those systems in which solute-solvent interactions are much weaker than solvent-solvent interactions, or vice versa, need not be considered. An example is given below of the inappropriateness of our theory to such systems. Here, the actual interaction energy would be less than that predicted by the geometric mean rule. At both extremes, then, the theory breaks down. The geometric mean rule does not apply where the solutesolvent interactions are either rather weak or rather strong. I n Appendix 2, we consider a modified geometric mean rule. Finally, to arrive a t Equation 8, we have assumed that it is permissible to replace the solute infinite dilution partial molar volume in solution with its molar volume in the pure liquid state. This is justified both because the volume changes o n mixing will probably be quite similar for both solutes of interest, and thus introduce only a small error, and additionally because few excess volumes of mixing data exist nor can they be reliably calculated. Let us now illustrate these considerations with a n example. One of the authors (14) has published LLC retention data for hydrocarbon solutes using acetonitrile eluent and squalane stationary phase. Tables I and I1 give the data, and calculated and experimental values of CY for the xylenes and diethylbenzenes. While the quantitative agreement of the two sets of CY values is hardly perfect, there is uncertainty in the experimental data, and more important, the solute selected as the reference compound may not in all cases be suitable. If values of (vQo/Ta*)(T~*/ooo) are calculated and ranked in order of magnitude (the values range from 1.004 for p/m-xylene to 1.046 for p/o-xylene), the order is nearly the same as the order of increasing percentage deviation of (14) D. C . Locke, J . Chromatogr., 35, 24 (1968).

ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971

Table I. Physical Properties and Chromatographic Data for the System Aromatic Hydrocarbons/Acetonitrile/Squalane vo, ccl T* V,, Compound Sa, cal/cc molc (K)d cc/g’ Kf Acetonitrile 140“ 52.8 ... ... ... Squalane 66b 524 ... 0-Xylene 9 . W 121.2 630.3 1.00 01806 m-Xylene 8 . 8 0 ~ 123.7 617.0 1.08 0.871 p-Xylene 8.8W 124.0 616.2 1.10 0.887 o-Diethylbenzene ... 153.2 662.8 1.71 1.380 m-Diethylbenzene ... 156.2 657.1 1.81 1.460 p-Diethylbenzene ... 156.4 660.5 1.78 1.435 a Reference (15). b Reference (16). c Calculated from data in references (14) (acetonitrile), (16) (squalane), and (17) (hydrocarbons). d References (18) (xylenes) and (17) (diethylbenzenes). e Reference (17). Reference (14).

Table 111. Physical Properties, Chromatographic Data, Calculated and Experimental Relative Retentions for Paraffins in the Acetonitrile/Squalane System vo, ccl T* Vg, Solute mala (K)b cc/gc a c x p t l c a c a l c d n-Pentane 116.1 469.6 4.32 1.05 0.201 2-Methvlbutane 117.5 460.4 4.12 2,3-Dimethylbutane 131.1 499.9 5.15 1.15 0.198 2,ZDimethylbutane 133.8 488.7 4.48 ZMethylpentane 133.0 504.4 5.54 .~ 1.02 0.208 3-Methylpentane 130.8 497.5 5.44 a Calculated from data in reference (18). Reference (18). Reference (14). See text.

Table 11. Calculated and Experimental Relative Retentions in the Acetonitrile/Squalane System Solute pair“ In , p t h In a t h Lycalcdb Lyexptlc

these results, which are clearly meaningless. F o r paraffins, the order of elution is no longer the order of decreasing critical temperature, which reflects the fact that paraffins d o not enter into specific interactions with acetonitrile and indeed cannot compete effectively with other acetonitrile molecules; they are “rejected” from solution. This is reflected in the magnitude of the activity coefficients in acetonitrile (14). There is no satisfactory theory for such systems. For paraffins of higher molecular weight, interfacial adsorption effects ( 1 4 ) preclude even calculation of meaningful and reproducible CY values from the experimental retention volumes. Obviously, of course, the actual selectivity of the systems-the observed magnitude of the relative retention-is what is ultimately important, and may in fact be enhanced in some particular systems where adsorption effects, low solubilities, etc., exist. As shown by the above calculations, the thermal contribution to CY is usually more important than the athermal part. The latter can assist or impede separation. Previously (6) we showed that, with all else equal, maximum analysis speed is realized in systems with high molecular weight stationary phases and low molecular weight eluents. As we indicated there and have demonstrated here from the point of view of relative retention, the more important consideration is the effect of the athermal part of a o n the separation achievable. In any case, according t o Equations 8 and 11, the LLC

plo-Xylene plm-Xylene mlo-Xylene p/o-Diethylbenzene plm-Diethylbenzene m/o-Diethylbenzene a First-named solute

-0.049 -0.009

0.354 1.36 1.10 0.021 1.01 1.01 1.34 1.08 -0.040 0.333 1.01 1.04 0.068 - 0.055 0.902 0.983 -0.003 -0.100 1.13 1.06 -0.051 0.169 corresponds to solute a, second to solute

b. = (vv)a/(vc)b. Reference (14). Ly

the experimental and calculated CY values (which range from 0.1 forp/m-xylene to 21 % for p/o-xylene). The discrepancy is thus mainly attributable to the dissimilarity between the solute of interest and the “reference” solute. Nonetheless, in all cases the correct order of elution is predicted; as expected for specific solute-solvent interactions, the solute interacting more strongly with the acetonitrile has the higher critical temperature. I t is interesting that the athermal contribution counteracts the thermal separation. I n the case of p/o-diethylbenzene, the athermal contribution nearly reverses the elution order expected o n the basis of molecular interactions between the solutes and acetonitrile. For comparison, we have calculated CY values from the expression for a t h based solely on solubility parameter theory (12, 13) as discussed in Appendix 1. The values obtained are plo-xylene, 1.44; plm-xylene, 1.07; and mlo-xylene, 1.35; these are all in much poorer agreement with the experimental CY values than those based on our a t h . I t was suggested above that the theory is inapplicable to systems in which stronger solvent-solvent than solutesolvent interactions occur. Substantiation of this conclusion is readily made using data from the same system ( 1 4 ) for saturated hydrocarbon solutes. Table I11 gives a few of

x

(15) R. F. Blanks and J. M. Prausnitz, Ind. Eng. Chem., Fundam., 3, l(1964). (16) D. E. Martire and J. H. Purnell, Trans. Faraday Soc., 62, 710 ( 1966). (17) R. R. Dreisbach, Physical Properties of Chemical Compounds, Adcan. Chem. Ser., 15 (1955), 22 (1959), and 29 (1961). (18) A. P. Kudchadker, G. H. Alani, and B. J. Zwolinski, Chem. Rev., 68, 659 (1968).

solvent pair can be characterized by

--

-

and by (6,2 - ~ 5 ~ ~ Where ) . the stationary phase is less “polar” than the eluent, as would be used for the separation of non-polar or slightly polar solutes, the elution order is that of increasing molecular weight, i.e., uao > Eao. Thus when the eluent is smaller than the stationary phase, as is usually the case, ~ ~ 8 ist h a negative contribution to the overall relative retention. Furthermore, am2 > 6s2, and since T* decreases with molecular weight, a t h will be a positive contribution, and larger than ~ ~ 8 t h .F o r the more favorable case of more polar solutes, a polar fixed phase and a less polar eluent would be in order. For monofunctional compounds, the elution order is that of decreasing molecular weight. If the stationary phase is of higher molecular weight than the eluent, then in this system Uao > Uao, 6s’ > 6,,’, Ta*> Tb+, and both a t h and ~ ~ 8 > t h 1. Higher LLC selectivities are intrinsically possible for polar molecules. The two other conceivable cases need not be considered, because as demonstrated above, our theory is

ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971

71

inapplicable and unnecessary because one would not ordinarily select such systems for difficult separations. Our results should in principle apply to mixed and/or partially miscible solvent systems. These are realistically to be encountered more frequently than two effectively immiscible solvents. F o r predictive purposes, however, we are limited by a lack of information about both molar volumes and cohesive energy densities of mixed solvents. While these may often be simply calculated (12), the assumptions underlying our treatment are now clearly suspect, and to expect realistic quantitative predictions is probably presumptuous. Pursuing our previous assertion (6) that LLC has a n inherently greater capability for selectivity than GLC, if we apply the Luckhurst-Martire (5) treatment t o ~ G L C ,Equation 12 results:

used as GC substrates (21,22). The organo-bentonite clays are also in this category. The organic moiety is chemically bonded to the surface of the support, which is usually siliceous. The product is chemically and physically stable. While they act effectively as solvents, to our knowledge their ‘‘solution” properties have not been studied in detail. I n principle, it would appear feasible to synthesize any desired property into this LLC substrate. I n conclusion, the results reported here (in particular, Equations 8 and 11) can in principle be used to select solvents for difficult LLC separations, and even to design new solvents. By indicating the dependence of a upon both molecular size and energetic factors, we have shown that separations can be achieved even when the solutes interact identically with the phase solvents. Furthermore, it is now clear that separations entirely feasible o n the basis of differential interactions may be confounded if the molar volumes of the two phases are improperly disposed. APPENDIX 1

where p o is the solute vapor pressure, s refers to the stationary phase [which for comparison purposes we take t o be the same compound as is used for the LLC stationary phase ( 6 ) ] , and a and b refer t o the solutes. Clearly, for very similar solutes, the first term will be very small because it is the logarithm of a number close to unity. The analogous CYGLCR~’, the second term, will clearly be smaller than w,,,cath, if the eluent is a smaller molecule than the stationary phase. If for example urno = 100 cc/mol and uso = 500 cc/mol, then (urn0)-’ (vsO)--l = 0.008 mol/cc whereas l / u s o = 0.002 mol/cc. , be smaller than a d h . Finally, the last term, C Y G L C ~ ~will I n general, the pair of LLC solvents will have solubility parameters which bracket those of the solutes; this is so because the two must be mutally less soluble than the solubilities of the solutes in either phase. Consequently, [ Ss2 - am2[ > 16,2 - ?iQ21 and C Y L L C ~>~ C Y G L C ~ which ~, further confirms our original assertion (6). While in principle a n important consequence of our results should be the semi-quantitative prediction of a values, screening of solvents for possible use, and designing new solvents, for real analytical problems the required datamolar volumes, critical temperatures, solubility parametersare scarce. It is, furthermore, a rather ambitious undertaking for a n analyst to calculate these parameters from correlations, etc., even when possible, and consequently it would be specious to exhort him to do so. The present state of knowledge concerning solutions often makes it more feasible to experiment than calculate. I n any case, solvent selection in LLC is inherently limited by the dual requirements of low phase miscibility, yet reasonable and differential solvency for the sample components. I n our terms, the solubility parameters of the two phases must differ (by approximately 4) and they must bracket the 6 values of the solutes. However, they may not diverge excessively or the problem of interfacial adsorption will begin to appear. This limitation can perhaps be considerably ameliorated by the synthesis of insoluble organic-like stationary phases, the so-called brush (19) or fur (20) packings which are also (19) I. Halasz, Fifth International Symposium on Advances in Chromatography, Las Vegas, Nev., January 1969. (20) H. N. M. Stewart and S. G. Perry, J . Chromatogr., 37, 97

(1968). 72

Regular Solution Formulation. We have chosen to represent the thermal contribution to the solute activity coefficient by the Luckhurst-Martire formulation (5) rather than by the regular solution (solubility parameter) formulation (12, 13) because the former is more general, allowing both negative and positive deviations from Raoult’s law; it is directly applicable to the problem of separation of very similar molecules; and it is theoretically more sound. I n order to test these assertions by direct comparison, we have derived a n expression analogous to Equation 1 1 . Starting with the expression (12, 13) for the limiting activity coefficient of solute a or b in solvent 1, e.g.

substitution as above leads to vaQ In ath = - [L2 26,(6, RT

+

Assuming vuo

- ),6 - &2] -

bbo,

which is analogous to Equation 1 1 (see also Equation 31 in reference 23). The most extensive published lists of solubility parameters are those of Burrell (2#), Lieberman (25), Gardon (2@, Crowley et al. ( 2 3 , and Morrison and Freiser (28), in addi(21) W. A. Aueand C . R. Hastings, J. Chromatogr., 42, 319 (1969). (22) E. W. Abel, F. H. Pollard, P. C . Uden, and G. Nickless,

ibid., 22, 23 (1966). (23) D. C. Locke, in “Advances in Chromatography,” Vol. 8, J. C. Giddings and R. A. Keller, Ed., Dekker, New York, N. Y., 1969, p 47. (24) H. Burrell, Znterchem. Reo., 14 (l), 3 (1955); (2), 31. (25) E. P. Lieberman, Uf. Dig., Fed. SOC.Paint Technol., 34, 30 (1962). (26) J. L. Gardon, J. Paint Technol., 38,43 (1966). (27) J. D. Crowley, G. S. Teague, Jr., and J. W. Lowe, Jr., ibid., p

269. ( 2 8 ) G. H. Morrison and H. Freiser, “Solvent Extraction in Analytical Chemistry,” Wiley, New York, N. y.,1957.

ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971

where $ and $’ are both small. Continuing our procedure as above, we find

tion to the list given by Hildebrand and Scott (12, 13) and the cohesive energy densities presented by Polak (29). Small (30) has deduced a group-contribution type correlation which allows estimation to be made of the solubility parameters of compounds with certain functional groups. Hildebrand and Scott (12, 13) also discuss methods by which 6’s may be calculated from physical properties, especially the heat of vaporization (Equation 6). These data are compiled in the usual handbooks and references (17,31-33). The most comprehensive source of critical properties is that of Kudchadker, Alani, and Zwolinski (18). In addition to a critical evaluation of experimental data, these authors give means by which critical temperatures may be estimated from pure component properties.

where $,ba = - fibs), and similarly for phase m,and where the subscripts and superscripts are, respectively, solute and phase indexes. Because of the requirement that the two chromatographic phases be immiscible, and because the stationary phase is generally selected for its greater solvency for the solutes to be separated, very frequently and h“LZ0. In this case, again selecting solute a as the reference solute,

APPENDIX 2

Modified Geometric Mean Combining Rule. In the derivation (5) of Equation 5, it is assumed that the pairwise intermolecular potential function for solute-solute and solutesolvent interactions is, for identical configurations, proportional to the depth of the potential well. It is further assumed that the geometric combining rule can be used,

The last term on the right hand side emphasizes the dependence of a t h on rather subtle differences between solute molecules. An alternate combining rule to Equation B2, uiz.,

In cases where there is a small contribution to the overall interaction energy from, for example, a dipole-induced dipole interaction, we use instead fij

= (Eiitdl’2

(1

+

$iJ

(B2)

where $ . I j is a small correction factor,