of support solid in column v, = volume weight of solvent in column
Wl =
of the apparatus and t o Dr. D. T. Sawyer for extremely helpful discussion.
The authors are grateful t o Dr. S. H. Langer for advice on the construction
(1) Baker, W.J., Lee, E. ,H.,Wall, R. F., “Gas Chromatography, H. J. Noebels et al., eds., p. 21, Academic Press, New York. 1961. (2) Bohemen, J., Purnell, J. H., J. Chem. Soc., 1961, p. 360. (3) Ibid., p. 2630. (4) Boyd, C. A., Stein, N., Steingrimsson, V., Rumpel, W. F., J. Chem. Phys. 19, 548 (1951). (.5,) Clarke. J. K. Ubbelohde A. R.. J. Chem. Soc., 1957, p. 2050. (6) Dal Nogare, S., Chiu, J., ANAL. CHEM.34, 890 (1962). (7) Giddings, J. C., Mallik K. L., Eikelberger, hL, Ibid., p. 1026. (8) Giddings, J. C., Robison, R. A., Ibid., p. 885. (9) Giddings, J. C., Seager, S. L., Stucki, L. R., Stewart, G. H., Ibid., 32, 867 (1960). (10) Glueckauf, E., “Gas Chromatography,” D. H. Desty, ed., p. 33, Butterworths, London, 1958. (11) Golay, M., “Gas Chromatography,” D. H. Desty, ed., p. 36, Butterworths, London, 1958.
W , = weight of support solid in column = labyrinth constant Y = porosity of column € 6 = interparticle volume/total free gas volume x = eddy diffusion coefficient = reduced mass M = collision cross section U X = coefficient in gas phase mass transfer equation = coefficient in over-all gas phase W mass transfer equation = coefficient in gas phase mass transWfi fer equation for flow inhomogeneity wa; = coefficient in gas phase mass transfer equation associated with solvent inhomogeneity Superscript or subscript o and i define column outlet and inlet values. Primes denote values a t one atmosphere pressure; superscript bars denote average column pressure values. Absence of superscripts denotes arbitrary conditions. ACKNOWLEDGEMENT
LITERATURE CITED
(12) Golay, M., “Gas Chromatography,” H. J. Noebels et al., eds., p. 5, Academic Press, New York, 1961. (13) “Handbook of Chemistry and Physics,” 30th ed., Chemical Rubber Publishing Co., Cleveland, 1947. (14) Jones, W. L., AXAL. CHEM., 33, 829, (1961). (15) Kieselbach, R., Ibid., p. 23. (16) Norem, S. D., Ibid., 34, 40 (1962). (17) Perrett, R. H., Purnell, J. H., Ibid., 34, 1336 (1962). (18) Pratt, G. L., Purnell, J. H., Zbid., 32, 1213 (1960). (19) Sawyer, D. T., private communication. (20) Titani, T., Bull. I n s t . Phys. Chem. Res. (Tokyo) 8 , 433 (1929). (21) Trevoy, D. J., Drickamer, H. G., J . Chem. Phys. 17, 117 (1949). (22) Van Deemter. J. J.. Zuiderwea, F. . J., Klinkenberg,’ A,, Chem. Eng.-’Sct. 5, 271 (1956); see also Ibid., p. 258. RECEIVED for review November 19, 1962. Accepted January 21, 1963. Presented a t International Symposium on Advances in Gas Chromatography, University of Houston, Houston, Texas, January 21-24, 1963. The Foxboro Co., Foxboro, Mass., provided a Research Fellowship to R. H. P. and much of the apparatus with which this work was conducted.
Advances in the Theory of Plate Height in Gas Chromatography J. CALVIN GlDDlNGS Department o f Chemistry, University of Utah, Salt Lake City 72, Utah
b The performance of a chromatographic column, packed or capillary, depends on the detailed structure of the internal solid surface and of the liquid partitioning phase. These column structural factors, usually complex, are converted into exact plate height expressions by means of the generalized nonequilibrium theory. The recent development of this structurenonequilibrium approach has provided the first instance in which structural details can b e accounted for and translated realistically into plate height performance. Consequently, it has been possible, using independent data, to predict the plate height of simple packed columns (glass beads) within the 20 to 50% range, and it should b e possible to extend such independent calculations to more complex supports, including those yet untried. In this paper recent advances in the structure-nonequilibrium approach have been summarized, and new evidence, equations, and proposals are outlined which should b e useful in further narrowing the g a p between calculated and experimental plate heights.
T
of chromatography has advanced t o a new stage of authenticity and completeness in the last fern years. Plate height calculations can now be made which realistically reflect the complex nature of the solid and liquid phases (15, 17). Theae calculations do not depend on oversimplified models, such as the uniform film model which is often in error by seyeral orders of magnitude. I n addition, advances have been made in obtaining the important structural parameters of the solid support and of the liquid, and theoretical approaches have indicated the nature and distribution of the liquid phase in relation t o the solid surface. These developments have paved the way for predicting absolute plate height values in packed columns where only relative values \%-ereavailable before ( 2 7 ) . N o theoretical treatment can be regarded a3 entirely complete until it shows this capacity for quantitative prediction in terms of independent experiment parameters. Consider, by way of reference, the van Deemter equation (10) used almost exclusively through the end of the last HE THEORY
decade. At no time n-as this equation used succeizfully to predict a numerical plate height value from independent data. It was, in fact, incapable of independent use because it contained no provision for fixing the magnitude of the all-important effectil-e film thickness in real columns; error- and omis.;ions related to eddy diffusion, gas phase mass tranifer, and liquid film transfer; and several adjustable parameters t o be determined by the experimental data. While this equation, as well as the theoretical work by others preceding it, did reveal much about the role played by flow velocity and diffusivities, it failed t o correlate quantitatively plate height with support structure and liquid load. It was thus an incomplete equation with only partial success in correlating data, and without success in predicting data-i.e., predicting numerical plate height values in terms of independent experimental parameters. The large gap between theory and practice partially closed, a t least on one front, when Golay invented the capillary column and derived a n elegant equation t o describe i t (29). Golay’s VOL. 35, NO. 4, APRIL 1963
439
vergence of theory and practice would soon he achieved. A stubborn gap has remained between the two, however, and various interfacial mass transfer terms have been postulated to close i t (55, 46). There is little to justify the importance ascribed to these terms, other than the pressing need to fill the existing gap, and opposing evidence can be found (see later). With these difficulties in mind, it is only natural to return to the fundamental assumptions to see if a divergence does not exist between the hypothetical column and the real. One's initial suspicion focuses quite naturally on the uniform liquid film which is assumed to exist on the inside wall. A careful analysis by this author of wall roughness and diameter nonuniformity, coupled with a detailed theory of the very adsorption forms which are needed for liquid solid adherence, has led to the conclusion that the liquid film is very nonuniform and irregular. While further evidence is needed to completely elucidate the nature of the liquid film, the conclusions of the forementioned study, with the additional evidence compiled here, indicate that capillary coiumns are indeed rather complex, and are beyond the reach of simple capillary column theory. Thus the theory-practice gap was but partially closed, and then only for a particular type of column. With the situation appearing more and more complex, both in packed and capillary columns, success might be found only if the theory were upgraded to meet the challenge. Consider what this entails. First, the liquid phase is obviously distributed in pores and cavities of odd size and shape and over surface areas of immense complexity. The f i s t essential of the task is an analysis of this complex liquid configuration to define the very location of the partitioning process and the dimensions over which liquid diffusion must occur. This is a structural prohlem, involving the interaction and configuration of liquids and solids. The second step of the theory, and this must follow the first or structuralanalysis step to he meaningful, is the chromatographic theory itself. Several years ago i t was difficult enough to predict the plate height in the presence of a perfectly uniform a m . The enormous complexity of a liquid's configuration in real columns must necessarily, then, constitute a major challenge to the success of chromatographic theory. The two-pronged theoretical approach just described for liquid phase mass transfer is also necessary for gas phase mass transfer. Previous structural models involving bundles of simple, equal capillaries, and the cor-
440
ANALYTICAL CHEMISTRY
Liquid is readily visible (It the conloct points. The A thrwgh C sequence involves (I decreasing beod size wifh identical liquid iooding, about 0.25%. The liquid loading in D, wilh beads of the same size os B, has obviously been increased, and is in the neighborhood of 1%
responding plate height equations, are not very realistic. The first basic steps have already been taken toward the goal outlined above (15, 17, 87). Adsorption and condensation theorv (15). the science of porous materials (i7j; microscopic observations ($44).and mercurv uenetration ($) have been applied to dktermine meaningful structures for the solid SUPport and liquid. The generalized nohequilibrium theory has been developed to the point where plate height contributions can be obtained in terms of these structural characteristics despite their considerable complexity (14,15, 17, $3). The gap between theory and practice has been decidedly narrowed. While the achievements of the d u d structure-nonequilibrium approach are a long way from ultimate theoretical objectives, they are a great deal beyond the uniform film and canillarv bundle models of the last decade. The progress is indeed sufficient to merit discussion. for even in the case of some packed columns (glass beads) the structurenonequilihrinm approach bas predicted absolute plate height values within 20 to 50y0 of the experimental data ($7). We are well along on the track of new concepts and refinements which, in yielding independent plate height cal.
I
culations, show immense potential in selecting promising and yet unborn columns, and in bringing chromatography into the realm of a controlled and quantitative science. GENERAL CONSIDERATIONS
This paper outlines the concepts which me leadine to increased convergence between- theory and experiment, with particular emphasis on those theoretical quantities which can he independently calculated. An attempt will also he made to pinpoint the major difficulties still remaining and to indicate avenues of approach to these problems. New evidence has been compiled which contributes to our understanding of the physical basis of the chromatographic process. Previously it was indicated that the indenendent calculation of nlate heicht depends in a quantitative k a y on 'the structural oarameters of the column. It is difficuit to imagine a more complex structure than that in a typical column packing. The successful definition and use of structural parameters in porous materials such as these is a serious problem in many fields of engineering and science (45), and constitutes one of the major theoretical challenges of the
scientific world. It is only natural that approximations have t o be made, and that results do not have the quantitative accuracy of an analytical balance. Sonetheless, we should hope for reasonable quantitative agreement, a t least sufficient to predict plate height values and the effect of structural changes on chromatographic performance. While it is impossible to estimate how closely theory and experiment may eventually be made to agree. it is a reasonable goal of the near future. considering our present state of knowledge, to expect no more than a factor of 2 or 3 between experimental plate height and independent theoretical predictions for the high velocity range of porous supports (firebrick and Celite). It is similarly a reasonable goal to expect no more than 20% divergence between theory and experiment for some typical glass bead columns, although in some cases a larger error is to be expected. It is also reasonahle to expect something between these extremes for capillary columns, although the present state of confusion regarding these columns makes this estimate the least reliable of all. At the present time it appears that glass bead columns are the most predictable of all, and the agreement (-20 to 50’%) between theory and experiment surpasses that found with any other column (27). Prediction within a factor of two for :my column with rapid flow must be considered as highly successful. The slow development of theories which could be used to predict numerical plate height values is undoubtedly a result of the forementioned complexity of real columns. More than two decades ago the theory of chromatography was initiated by Wilson in an excellent paper containing many modern concepts (49). Until the recent work on glass bead columns m-as reported, the only significant attempt to predict packed column performance from independent data was made by Boyd, Myers, and ildamson in 1947 (6). Using ion exchange columns, they found practice and theory to differ by a factor of between 5 and 10. A divergence nearly as large has been indicated in the liquid phase mass transfer term of capillary columns (12, 47). The significance of the latter is quite uncertain, however, because the independent rate data consisted of questionable values obtained from a semiempirical diffusion equation. Both of the packed column comparisons mentioned relied directly on experimental data. While a considerable effort spanning many years has been necessary t o develop theories of sufficient depth to cope with the over-all operation of real columns, one special range of operation exists in which the problem has never been very serious-the low flow velocity range where lateral equilibrium (both
particle-wide and column-wide) is essentially complete. I n this case, ordinary molecular diffusion in the longitudinal direction (the B term) dominates the plate height. The theory of this is uncertain only in a tortuosity factor, y , which never departs by more than a factor of 2 from unity. While this case will be considered later, main attention will be given t o the medium and high flow rates where nonequilibrium (given by the mass transfer or C terms) is significant. The practical range of operation always involves the C terms, and their magnitude determines the efficiency and especially the speed of chromatographic operation. Thus, the true test of a theory lies in its capacity t o predict the magnitude of the C terms. The independent calculation of C values can only be successful if certain rate data are available. For diffusion controlled processes, diffusion coefficients must be known. Gaseous diffusion coefficients can be estimated quite accurately but liquid phase diffusivity, unfortunately, is not very predictable (33). The calculated values are in error by a factor of about 4. With such error, it would be impossible t o determine if a particular prediction of Ct was within the “highly successful” range with a tolerance factor of only 2. Capillary columns, relying on calculated diffusivities, have therefore never been adequately tested. Because of this limitation in calculated diffusivities, a program has been initiated in the author’s laboratory for measuring directly the diffusion coefficients of some liquids involved in gas chromatography. These values have been used in the successful prediction of glass bead performance (271 * THE GENERALIZED NONEQUlllBRlUM THEORY
The generalized nonequilibrium theory is the tool with which structural parameters are translated into the C terms of the plate height equation. The theory, formulated by this author, is sufficiently general to incorporate mass transfer processes (both diffusion and kinetic steps) of extreme complexity (15, 20, IS). I n some advanced cases, a computer may be needed for the structure-to-plate height translation, but this is a computational detail which poses no basic problem. Because of the apparent scope of this theory, the really difficult part of the structure-nonequilibrium approach is now the structural problem (15, 17). This was not true a few years ago when, as mentioned, it was difficult to obtain plate height expressions for such simple structures as a uniform liquid film. The mathematical details of the generalized nonequilibrium theory have been described adequately elsewhere
(20, 23). The principal results of this theory which are applicable to gas chromatography are summarized in Table I. Nearly all of these terms are unique results of this theory. d notable exception is C-1, the first rigorous C term ever obtained. This adsorptiondesorption term was derived from the stochastic approach in 1957 (19). It has since been re-derived and used to describe interfacial mass transfer (36) and reaction processes occurring in a column (36). The other exceptions are the two capillary column terms, A-1 and B-5, first derived by Golay in 1958 (29).
Several important but not generally recognized principles are demonstrated by the C terms of Table I. (1) The coefficient of the C I term, given by the constant 8 / r 2 in the van Deemter equation, actually varies strongly with liquid configuration 8s shown by the important role of the taper factor, n, in A-3. If the van Deemter equation m r e used to predict the C term for contact-point liquid in a glass bead column, n = 3, the error would be a factor of nearly 10 if d f were taken as the radius of the ring of liquid, and nearly 40 if d, were taken as the diameter of the ring. (2) If a film with variable thickness but local uniformity exists, the proper value of d is the root mean square thickness, averaged over liquid content rather than cross-sectional area. This is shown by A-5. The same principle is illustrated by A-8. (3) An approximate equation, A-6, relates C to the experimental pore size distribution of porous supports, thus utilizing independent structural data and accounting for structural variations. (4) Interfacial adsorption processes not only add terms to the plate height, but they also change the form of the Cl term as shown by A-7. (5) The interaction of unequal flow channels, B-2, gives a C term of considerable magnitude. This is probably the major contribution to gas phase mass transfer in packed columns. (6) Adsorption chromatography in no way depends for success upon sites of equal energy. The only important parameter is the mean desorption time, td (shown in C-3), and this can be reduced as well with nonequivalent sdsorption sites as with equivalent ones. It is the kind of sites, and not their homogeneity, that is important. Other significant features of Table I will be discussed later. As the theory of chromatography becomes more exacting, and it is desired to account for the fine structural details of the liquid and solid support, i t will be necessary to use a computer for the solution of the generalized nonequilibrium equations. The principal VOL. 35, NO. 4, APRIL 1963
441
Table I.
Plate height contribution given by H C value A. Liquid diffusion contributions 2 d2 1. C I = 3 R (1 - R ) Di 1 d? 2. C i = - R (1 - R ) 12 Di
=
Meaning of terms d
=
film thickness
d = diptance from contact point to
meniscus
Applicability and restrictions
4. Cl =
4
(n
d?
+ 2 ) ( n + 3) R ( l - R ) D-I
5 . C I = -2 R ( l - R ) di s 3
s"
=
taper factor
d n
= =
pore depth taper factor
Reference and year
Liquid of uniform depth
23 (1961)
Liquid collected around contact points in glass bead columns. Low liquid loading (
