Comparison of the Theoretical Limit of Separating Ability in Gas and Liquid Chromatography J. CALVIN GlDDlNGS Department o f Chemistry, University o f Utah, Salt lake City, Utah
b It is established that the theoretical limit of separating ability in chromatography is a function of relative selectivity, zone migration rates, and the number of theoretical plates. Only the latter is expected to show a significant difference between gas and liquid chromatography. The main limitation to acquiring an unlimited number of plates is the pressure drop available to the system. For a given pressure drop, equations are derived showing the maximum number of plates which can be possibly achieved in single columns. Liquid chromatography, owing mainly to the difference in diffusivities, shows a 100- or 1000fold advantage in maximum plates. These concepts are related to the critical pressure, where the latter is the minimum pressure leading to a specified separation.
G
chromatography are two powerful and conil)linwntary techniques for chemical analysis. Their theoretical hasis is nearly identical, Desilite this, a wide schism has evolved in the manncr of their scientific growth. Few i.cal efforts havc Iwen inattlc to coniliare the ~icrformaiiceantl potential perforinancnc> of one technique with the other. 'Yhis void makes it more difficult than nwessary for the analyst to choose the propcv alternative when faced with a new separations problem. The present Iiaper is intended to explore one aspectt of the potential of the two methods. This a s l w t is concerned with the theoretical limit of separability. -1 subsequent pa1ier will deal with the comparative speed of analysis. The theoretical limit of separating ability may be difficult to realize fully. Such a limit may require the use of extreme operating conditions and may require an inordinatc amount of time for separation. Yonetlieless the theoretical limit should lie approachable within a factor of two or so, a rather small factor compared to the order-of-magnitude differences involved. The theoretical limit may not be needed for ordinary sellarations. However it does give an indication of the present scope of chromatogi.al)hy and the possible approaches to rstcnding this scope. Once the best possible column inaterial> arid conditions have been AS A N D LIQUII)
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ANALYTICAL CHEMISTRY
established for a given problem, it adequate for inoit purposes. 'I'his value indicates that the zones o v d a p seems obvious that the degree of separation can he increased to any desired only in th(.ii, c i i i t c ~ cdgcx> \vII(~I.v the level by adding to the length of the concentration has tiropped to about 10yo column. One soon reaches a length of of its maximum value. surh mngnitridc, ho\vcvc~,that optimum 1he resolution of nc4ghboring zones flow, c:iii 110 I o i i g ~ t , iiiaintnined within obviou*l>-in(was(yh \vi th caolumti Itsngth, thv ~ ~ o l i i i i i i i . 'I'his 1)roI)leiii c*an I)(! I,, and with the. rvlativc, sclwtivity tcwlior:trily skirtcd Iiy incwasing the exhibited 1)y thcl column toward the piwsiirv tlroli tltroiigh thc cduinn. [)airof c~)nil)onc~iits:.('l'hc latter i i given (-11 iiiiat(,lJ,o t i c i,(~nch(+ :i 1)oint whtw the by A K ; I< w h c i ~I< is t h c i n w n theimoequil)iricirit,will stand no further Iircssure dynamic distribution cwfficirnt and incwu>c,s. Thus the ultimatr limit A K is the difference in I< from one of wlini~ntiilityrimy w l l he ~Ic~t~~rniiiiedcomponent to the othcsi,.) I t also inby the rnaxiniurn Ilressure limitations of creases as the column i)late height, 11, is th(, cquilinicmt. it i3 t r u e that cduinn r e c h d . (Plate h c i ~ h t i.; tho iiwal length can itself lie a trouhlcsonie measure of zone sprcatling 01'u j being barrier if excessive values arc reyuired. i shown given by 11 = g2/1,.) It ~ t i he F€owc:vcr, the use of long caoluinns is that these various tcrms contriht(J to largely a matter of iiieorivenience, and resolution as follows length is therefore probably not such a RS = ( 1 , ~ 1 6 ~ ~ ) " * ( ~ K-~ R~ )< )(2) (l basic limitation as preshiire drop in most systcms. t h e limiting role of pressure The R value, similar but not identical to has also been considered by Knos in R , ( S ) , is the ratio of the zone velocity cmnretiori with the ultimate speed of to that of the mobilc fluid. The rat,io scparation (4). The recent development I,/€€ can be written as thr number of of high-pressure latioratory theoretical plates in the column. 'rhus particularly by Hamilton ( 3 ) (uli to 600 RS = (S, 16)1'2(AK/l in this c ~ s a r yto consitlor the paper, to consider pressure drop as the maximum 1)otential mngnitudr of the most critical limiting factor to ultimat,c various terms in this tqiiat ion. separa1)ility. 1he maximum rclativc selcctivity, K , will vary greatly tlcpcnding on the similarity in niolecdtr htrwtiire FACTORS INVOLVED IN SEPARATION for the pair under conxitlciztion. HowThe success of any separation must ever, for a givrn pair it is logicd to hinge, first, on the tiiffcwntial movement exI)ect that ga. antl licjiiiil c~hi~oiii:ttogof zone centers, and second, on the zones raphy would show about the same remaining sufficiently compact during maximum capachy for AK/'K. migration that they do not spread into Selectivity, after all, reduces to the one another despite the disengagement prolAem of molecular intcractions antl of centers. T h e most commonly used the drgree to whic*h these can he made measure of these requirements for different for t\vo himilar solutes;. The separation is the resolution, Rs. This differences clepends on factors such as quantity can be defined by polarity hytliqgon 1)oiitl ing, ~~olarization, etc., antl pro1)ably can be enRS = Aa/4u (1) hanced to nearly an eqii:tl tlcgree in gas where Az is the distance between zone and liquid chromatogi,a1)lii(< syxteins. centers (indicating the differential move(It is assumed throiiqhout,, of voiirse, ment) and r is the mean standard deviathat the solutes are volatile so that a tion in zone width, or roughly the legitimate comparison a n I)c made.) quarter-width of the zone (indicating The masimum value of (1 - R ) is solute mixing owing to znne spreading). unity. This valiic is al)i)roac.hctl as R For close lying zones the widths are goes to zero. 111 pt'actic~,R must he usually of the same approximate kept above zero to get a finite inig magnitude. A resolution of unity tion rate and a reasonably .;hurt anal> indicates that, the heparation is time. .\s a coinpromihe Iicltween r 7
,.
r .
~
conil)lcs function of flow velocnity, v, whose most irnportant characteristic is dg(z1) Sdff > 0
(7)
for a11 real values of u . This simply incans that g ( v ) , accounting for nonqiiililxiuni antl eddy-tliffusiori 1)hcnomena, is an increasing function of v throughout the f u l l olwrating range. 'I'ho first t m n , 2yD,,,, L', accounts for longitudinal molecular diffusion within the column. (The value of y may cwwd unityI especially in liquid chromatography, due to longitudinal diffusion in the stationary phase. Ordinarily, however, y remains within a narrow range which is about the same for gas and liquid systems.) K h e n the above espression for H is substituted into Equation 5 , we have
This equation shows, for a fixed pressure drop, that A' decreases as v increases due to the vg(v) term in the denominator. The theoretical limit of 9 is found as v alil~,oacahrs zero, viz., (9)
Quite significantly, this equation contains no terms related to the nonequilibrium (or mass-transfer) and eddydiffusion contributions to the plate height. I t is a liniit to be ap1)roached strictly a t very low velocities where the latter cffects are negligible. Ikpation 9 shows that the masimum achievable A' is prol)ortional to the pressure drop available to the alm prol)ortional to the square of particle size and inversely proportional to the carrier's viscosity and diffusivity. In the comparison of gas and liquid chromatography it can be assumed, as a first approsimation, that the maximum pressure drop is the same and that the largest practical particle size is com1)arable. Since p and y are fised structural factors, the ultimate separating potential of the two methods depends mainly on the product of viscoaity and diffusivity, qD,. Thus
where subscripts 1 and g stand for liquid and gas, respectively., The viscosity of liquids is roughly 100 times larger than that of gases. IXffusivity in liquids is nearly lo5 times smaller than that in gases. Hence the theoretical limit to the number of plates is roughly 1000 times largc'r in liquid than in gas chrorriatogral~hy; viz.,
(Since D!, decwases as pressure is increased. this ratio may be increased to 1/100 or more a t high inlet pressures.) Thus for potcntial scpaiating ability without regard for speed, liquid chromatography shows a distinct basic advantagc over gas chromatography. The advantage originates with the slow diffusivity found in liquids. The 1000fold advantagr in h' becomes a 30-fold advantage in actual resolution, Rs, since the dependence in Equation 3 is of the square root type. Cal)illary columns in gas chromatography make a closer approach to classical liquid chromatography because the structural constant -yp equals only 16 for open tubes ( 1 ) (d, must be equated to the tube diameter for this comparison). This brings ;VIin,for vapillary gas chromatogral)hy within a 50- to 100-fold margin of liquid chromatography. This would revert back to the 1000-fold margin if capillaries were used in liquid chromatography. There is no basic reason why this cannot be done although the advantages, escept as one approaches the theoretical limit, may be minor. The order of magnitude of Nli, can be easily calculated by substituting the appropriate quantities, in consistent units, into Equation 9. As an example for liquid chromatography we may assume d p = 0.05 cm. (fairly large 30- to 40-mesh range), Ap = 10' dynes per sq. em. (-10 atm.), 7 = 10-2 poise, 11, = 3 X 10-ssq. cm. per second, y = 0.6, and 'p = 300. This gives a value much larger than any yet reported-i.c., Slim IO'.
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THE CRITICAL PRESSURE
The idea that there is a critical inlet pressure, below which a separation can not be achieved by any manipulation of column length or flow velocity, was introduced by the author in relationship to gas chromatography ( 2 ) . The foregoing arguments show that the concept is a general one. Thus Equation 9 gives the maximum number of plates to be achieved with a pressure drop of A p . The quantity A p may be regarded as the critical prrssure (or pressure drop) for S = .Vilm,since any reduction in A p would mean that this number of plates could no longer be achieved. I f one wishes to make the greatest use of a limited pressure drop, it is desirable to operate the rolumn outlet as close as possible to a v:tcuum (a limit is imposed here by the boiling of liquids under reduced pressures). While an absolute vacuum is not necessary, we will assume here that the outlet pressure is negligible compared to the inlet pressure. Thus we may replace A p in Equation 9 by the critical inlet ~ircssure,p , . This leads to VOL. 36, N O . IO, SEPTEMBER 1964
1891
the following equation for the critical pressure p , = 4pyqDmN/dp2
(12)
where S has replaced S,,, since we are now expressing limiting pressures rather than limiting plate numbers. This equation shows that if N plates are needed for a certain separation on a given kind of column (fixed d,, D,, q ) , an inlet pressure equal to or greater than p , must be available. Once again the demands on liquid chromatography are relatively small due to the slight magnitude of the OD, product. The nature of Equation 12 for gas chromatography is altered by the strong dependence of the gaseous diffusion coefficient, D , = D,, on pressure. This can be eupressed by
D,
=
D,'/p
(13)
where D,' is the value of the diffusion coefficient a t unit pressure. The mean value of D,, for use in Equation 12, can be obtained by using the lengthaverage pressure, p , in place of p . Under vacuum outlet conditions, with p , as the inlet pressure, I, is found to equal 2 p C / 3 . Consequently
D,
3D,'/2pc (14) Cpon substituting this back into Equation 12 we obtain p,' = ( 6 ~ y D,'S/dp2)'/2 q (15) =
;ilthough the gradients existing in gas chromatography have been rather loosely treated here, this equation differs from the earlier rigorous form only by the small numerical constant of d2j3. The development given above indicates the limits associated with single
chromatographic columns. One can imagine column segments joined together by pumps of low dead volume such that each segment experiences the maximum possible pressure drop. Under these circumstances the arguments given above would apply to the individual segments rather than to the column as a whole. LITERATURE CITED
(1) Giddings, J. C., ASAI,. CHEM.34, 314
(1962). (2) Giddings, J. C., Stewart, G. H., Ruoff, A4.L., J . Chromatog. 3 , 239 (1960). (3) Hamilton, P. B., ANAL. CHEM.32, 1779 (r960). ( 4 ) Knox, J. H . , J . Chem. SOC. 1961,~. 433.
RECEIVEDfor review April 1, 1964. Accepted June 1, 1964. JI'ork supported by a research grant from the National Science Foundation.
Rapid Scanning Infrared-Gas Chromatography lnstrument A. M. BARTZ and H. D. RUHL
The Dow Chemical Co., Chemical Physics Research laboratory, Midland, Mich.
b This instrument was designed to utilize the ability of gas chromatography (GC) to physically separate a multicomponent chemical sample into its individual components and the ability of infrared spectroscopy to specifically identify reasonably pure compounds. The effluent from a GC column (vaporized sample plus helium) i s passed through a heated light pipe which serves as an infrared absorption cell with a large optical path length-to-volume ratio. The infrared absorption spectrum of the vapor sample i s obtained by using two singlebeam grating spectrometers in parallel. One spectrometer covers the range 2.5 to 7 microns while the other scans from 6.5 to 16 microns. By using two spectrometers, a high chopping rate, and fast recorders, a complete spectrum may be obtained in 16 seconds comparable in quality to a normal 12-minute scan by commercial spectrometers. The high scanning speed i s necessary if an IR spectrum i s desired of each successive GC peak of a multicomponent sample.
total reflectance. Both methods are tedious, slow, and rather difficult since extremely small quantities of condensed sample are involved. An instrument was needed which would obtain an infrared spectrum of the
PREAMP I
1
GC sample directly, without condensation, and do it quickly so that closely spaced (in time) successive GC peaks could be readily identified. There were two problems to be solved: design a satisfactory sample cell and obtain a
'i COLUMN
P
of combining infrared and gas chromatography have required condensation of the GC sample (effluent) and obtaining an IR spectrum of the condensed sample by use of either a microcell or by attenuated REVIOUS TECHNIQUES
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ANALYTICAL CHEMISTRY
Figure 1.
Optical path of infrared radiation
