Theory of Gas-Solid Chromatography Potential for Analytical Use and the Study of Surface Kinetics J. CALVIN GlDDlNGS Department of Chemistry, University of Utah, Salt lake City 72, Utah
b The theory of gas-solid chromatography in packed and capillary columns has been developed and discussed. Particular emphasis has been placed on expressing the important plate height coefficient, ck, in terms of known parameters. The accommodation coefficient, a, a parameter which has often been measured in connection with surface studies, is used to express the rate of adsorptivedesorptive mass transfer. The theory shows that Ck is very small, the order of lo-’ second, for typical packed columns with uniform adsorptive sites. Nonuniform surfaces exhibit a Ck which is larger by a heterogeneity factor, u. The equation for u provides the first quantitative basis for the effects of nonuniformity, and indicates the energy difference between sites required for column deterioration. An examination of the analytical potential of gas-solid chromatography indicates real advantages in terms of column efficiency and high speed analysis. A great potential is shown to exist for enhancing selectivity by making use of the steric nature of the adsorption forces. Finally, gas-solid chromatography is shown to have a real potential for the measurement of surface kinetics.
T
of gas liquid chromatography in its first explosive decade (1952-62) nearly obscured its somewhat less versatile companion technique, gas-solid chromatography. This relative obscurity now shows signs of yielding in the face of new breakthroughs in knowledge and instrument technology. I n some areas gas-solid chromatography now shows promise of exceeding the performance of gas-liquid system. Concrete progress in gas-solid chromatography (GSC) , beyond the present art, will originate with the numerous experimentalists who acquire new techniques and new solids with more versatile chromatographic value. The main object of this paper is to define the goals that can reasonably be expected in such work, and thus give direction to the search for new gas-solid systems. GSC would appear to be particularly HE SPECTACULAR ADVANCE
1170
ANALYTICAL CHEMISTRY
promising in providing high column efficiency, high analysis speed (a facet of the subject which has not yet been explored) and high selectivity. These matters will be discussed in the following sections. It would be unreasonable to expect GSC to compete with GLC in overall simplicity and effectiveness. Yet the practical problems of GSC (and thus the gap between GSC and GLC) have been reduced considerably in recent developments. With sensitive ionization detectors it appears that there is no major difficulty in operating GSC in the linear range of the adsorption isotherm. I n addition the recent work of Scott shows that one is no longer limited solely to low molecular weight compounds; hydrocarbons up to 45 carbon atoms were separated with his specially modified solids ($8). The problem of obtaining reproducible adsorption surfaces is also bound to be solved as more is understood concerning GSC. ZONE PROFILE AND PLATE HEIGHT
to GSC have been discussed elsewhere (11). Hence only a summary will be needed here. Most of the processes in a GSC column are so similar to those in a GLC system that the majority of plate height terms are nearly identical. The major difference is the existence of a Ck ( k = kinetic) mass transfer term for the kinetics of adsorption and desorption (11). This term replaces the Cl term of GLC. The theory of adsorption mass transfer was developed earlier ($3), and is in many ways broader in scope (14), than the theory of liquid mass transfer applicable in GLC. A more detailed discussion of the Ck term will be presented in the following section. The plate-height equations for GSC may be summarized as follows:
* H
=
=
c
B
1
l/A’,
+ l/C,iv + ; + C k U (packed column)
B/V f COV f Ckv (capillary column)
Under ideal conditions the elution peak from GSC will be narrow and symmetric. The profile will have a Gaussian shape. There are certain perturbations which may, however, cause zone distortion, usually leading to the formation of tailed peaks. Tailing in GSC is usually blamed on nonlinear adsorption. Tailing may, however, be found in the presence of complete linearity. It may originate in the column as a kinetic effect related to adsorption and desorption from the so-called “tail-producing sites” (10). Tailing may also represent a distortion due to the instrument; the author and his coworkers have found time lags of over 3 seconds in some commercial instruments, a value adequate to cause tailing and efficiency loss in any reasonably fast analysis. Peaks with symmetric or near-symmetric profiles can be partially characterized by the plat,e height H . Although the theoretical-plate model is itself unrealistic in describing chromatographic columns, the plate height is a valid measure of peak spreading. The plate-height equations applicable
(1)
(2)
where the coupling expression (2%’) has been used for the packed column. The terms have their usual significance (see nomenclature section at end). The Ck term, of major importance here, is given by (14) c k
=
2R(1
- R)ld
(3)
where R is the ratio of air-peak time to component time. It is assumed that the adsorbing surface is heterogeneous with adsorption occurring independently on each of the different kinds of sites. The mean desorption time of an equilibrium population of sorbed molecules, id, is the significant parameter in this equation (14). I n simplified cases adsorption and desorption may be presumed to occur on only one type of site. The foregoing equation then reduces to
cx
=
2R(1
-
R)/kd
(4)
where kd is the first-order desorption rate constant.
NATURE AND MAGNITUDE OF
Ck
The magnitude of ('k (or the corresponding C i of GLC) is extremely important in regard to column efficiency and high speed analysis. I t is therefore necessary to do more than simply relat,e ck to an unknown ra4e constant, k d , or desorption time, t d . I n this section a n attempt will be made to estimate the order of magnitude of Ck. K i t h such a n estimate available, i t is possible to be far more specific in outlining the ultimate plltential of GSC. Experiment,al chromatoyaphy is as yet of little value here since no direcl measurements of Ck have been reported. The rates of adsorption and desorption are connected by thermodynamic equilibrium, s~ that only one rate parameter determines both rates a t a given equilibrium value. The parameter commonly used to characterize the rate of adsorptive kinetics is the accommodation coefficient (or st,icking coefficient), CY. This parameter is equal to the fraction of molecules which sticke to the surface upon impact (2.5). Uniform Surfaces. A surface whose sites are all equal is t h e simplest case to be considered. T h e ck term can be related to the accommodation coefficient LY by considering t h e adsorption rate. We may express t h e rate of adsorption (number of molecules adsorbing per unit time within a fixed, arbitrary voluml?) a s a n adsorption rate constant, k , , times t h e number of molecules in the gas phase within the given vohme. (The rate constant k , is not the true bimolecular constant expressing the rate of combination of gaseous molec.iles and surface sites, but is the simpler first-order rate constant in which the number of adsorption sites is assumed constant and the gaseous concentration, only, varies.) The rate of adsorption may alternately as the be expressed as aJA-i.e., product of the number, J , of molecules striking a unit surface area per unit time and the fraction, CY, of these that stick, times the total area, A , involved in the volume under consideration. The number of gaseous molecules is nVa where n is the number per unit volume of gas and Va the total volume of gas in the fised volume. h'ow since the rate constant, k,, is tht. ratio of the adsorption rate, aJ.4, to the number of gas phase molecules, use of the foregoing expressions gives
The desorption rate constant, kd, appearing in the Ck term of Equation 4, can be related to k , by the equilibrium expression, k,/k, = Rj(1-R). [Since R is the fraction of molecules in the gas phase and 1 - R the fraction adsorbed, the ratio R l ( 1 - R ) is simply the desorbed over sorbed fraction.] With this and Equation 4 the desired expression for ck is 8(1 - R)' Ba a~ A
ck
(7)
We shall presently estimate the value of this term for both packed and capillary GSC. Nonuniform Surfaces. A nonuniform surface may be considered as one containing a random array of unequal sites. Adsorption at a given site may occur through a direct collision of t h e adsorbing molecule with t h a t site, or it may occur through adqorption at some other site followed by surface diffusion to the given site. This dual mechanism will be simplified somewhat by assuming t h a t the kinetics a t each site is governed by a single adsorption rate constant and a single desorption rate eonstant. More specifically, i t will be assumed that a n accommodation coefficient, or effective accommodation coefficient, C Y % , governs the adsorption (and deqorption) rate at sites of type i. With this assumption the adsorption rate constant for sites in the ith category may be written in a form analogous to Equation 6 k,,
=
C'a,A,/4Bc7
(8)
where .4$is the surface area taken u p by the i sites. The desorption rate constants, necessary in evaluating ck,can be obtained by considering the equilibrium relationship, along with the k,,, applying to sites i. Thus we have
k,,R
=
kd,X,*
(9)
where kdi is the desorption rate constant for site i. The fraction of molecules which occupy sites of type i a t equilibrium is X,*;the corresponding fraction of molecules which is free in the gas phase is R. Equation 9 can be solved for ICd,. This, in turn, can be used in the expression derived elsewhere (24) for id
where the summation extends over the entire surface. This procedure yields The ratio J / n is given by simple kinetic theory as E/4, where is the mean molecular velocity ( I t ) . Hence
id -- R(1
-
R)E
where Equation 8 has been used for k,i.
When this espression is substituted for ld in Equation 3 we have
I n the limiting case when all sites are equivalent this expression equals Equation 7 for uniform surfaces. The difference between uniform and nonuniform surfaces can best be seen by expressing the fraction of total surface area occupied b y type i sites as a, = A J A , and by expressing the fraction of adsorbed molecules on type i sites as zl*= X,*/(l - R ) . (The term X,*is the fraction of all molecules, adsorbed and desorbed, on i type sites. This of just the differs from the fraction, zL*, adsorbed molecules, on i sites.) With these fractional quantities Equation 12 becomes
Ck
=
8(1
- R)'VG EA
(z,*)'
(13)
etal
an equation obviously close to Equation 7. If we assume t h a t all (Y, terms may be replaced by a n effective value, a , this equation is of the form
The part preceding the summation, u = 2 (z,*)'/a,, is identical to Equation 7 for uniform surfaces. It is therefore important t o estimate the summation term or "heterogeneity factor," u, in order to establish the difference between uniform and nonuniform surfaces. Magnitude of CY. As a general rule t h e accommodation coefficient for molecules impacting on clean surfaces is within two orders of magnitude of unity. This is apparently true for both liquids and solids. T h e more voluminous evidence concerning liquids (of interest in calculating the interfacial term of GLC) is summarized elsewhere (20). A brief discussion of some recent work on solids is given here. Several authors have measured CY values for gases condensing on tungsten. Ehrlich ( 7 ) has found CY to range from 0.16 to 0.37 for N2 as the temperature is decreased from 373" to 243" K. Kisliuk (24) has obtained values in the range 0.07 to 0.16 for the same system over the temperature range of 1210" to 798" K. Carbon monoxide condenses on tungsten with CY E 0.5 near room temperature ( 8 ) . The bulk of evidence on the magnitude of a is related to the condensation of a vapor on its own solid ( 1 , 2 ) 4,26, 27, 29). Nearly all recent workers have found that CY values lie in the range from 0.1 to unity. VOL. 36, NO. 7, JUNE 1964
e
1171
While neither type of evidence on cy pertains directly to GSC, the nearly universal conclusion of recent theory and experiment that cy values are in a range near unity suggests that this is a conclusion generally valid for the kinetics of GSC. Numerical Estimate of c k on Porous Supports. We shall first consider uniform adsorption and thus use Equation 7 . Of necessity these calculations will yield only the order of magnitude of c k . The mean velocity of the component molecules, t (approximately equal to the speed of sound), may be taken as 5 X 104 cm. per second. Since R is generally closer to zero than it is to unity, the may be approximated product 8(1 by 5 . Thus to an approximation C k
-
i0-4Vo/afi
(15)
The quantity V c / A varies immensely with the type of solid used. It is roughly the reciprocal of the surface area (in sq. cm.) per gram. Those materials with the order of 1 sq. meter per gram will have VG/A lo-'. Thus we may expect values of the order
-
formity is large, however, an increase in fd will occur. The precise significance of nonuniformity can be determined from Equation 14. I n this equation cy is a measure of the overall exchange rate and the summation term, U , is a measure of the increase in ck due to nonuniformity. We shall attempt to establish a rough concept of nonuniformity effects by calculating several examples. First, suppose that the surface of a solid is equally divided between low energy sites (type 1) and high energy sites (type 2 ) . Thus al = a2 = '2. The heterogeneity factor, u, becomes 2x1*2 2x2*2. This term varies from unity when the surface is equally occupied, x,* = x2*,to two when the high energy sites are strong enough to attract all molecules, xl* = 0, x2* = 1 (note: this latter condition is equivalent to reducing the surface area of Equation 14 by a factor of two). Thus such a surface a t its extreme can only increase ck by the rather insignificant factor of two. Next, let us suppose that there are an equal number of molecules adsorbed on type 1 and type 2 sites, xl* = x2* = l / 2 , but that the densitv of adsorption is greater, and thus the area less, for the high energy (type 2) sites. The summation term becomes x1*/2al [l (Q*/ a2)/(x1*/al)]. If the density of adsorbed molecules is considerably greater on type 2 sites, then al 1. Since x,* la, is proportional to the density, p t , of adsorption the above summation becomes
+
+
for porous supports. With a located between and unity, c k would likely range between lop6 and lo-*. The value may be slightly higher for low area solids and lower for the many solids with a high surface area. Numerical Estimate of c k in Capillaries. Equation 15 may be used for capillary column GSC. The volume to area ratio, VG/.il, is equal to rOi2S, where r, is the capillary radius and S is the surface roughness factor (actual microscopic area over true area). For capillary walls which are not coated with some porous solid material, S will generally range from one to ten (11). Taking ro = 2 X em., V ~ l - 4thus ranges from 10-2 to 10-3. Porous coatings, such as the Fibrillar Boehmite (alumina) recommended by Kirkland (22) may reduce this ratio considerably. However as an upper limit. with cy a t the unlikely value of and Vd.4 = lo-*, C k would equal More typically we with porous might expect C k solid coatings the value may, depending on thickness, be several orders of magnitude less. Effect of Surface Uniformity. This author has emphasized t h a t the most desirable feature of an adsorptive solid is its ability to facilitate rapid adsorption and desorption of solute molecules (11, 1 5 ) . Thi. is shown by Equation 3 where the > a2 and thus a1 1, and that the adsorption density on type 2 sites is much greater than on type 1 sites, p 2 >> p1 and p 2 / p A >> xl*/xZ*, then the summation term is
-
This expression is a slight generalization of Equation 17; we have removed the restriction that xl* must equal x2*. For a given density ratio this expression reaches its maximum when xl* does equal x2* (the product, x1*z2* will usually be the order of 10-1). We have indicated that the u term must be -lo4 in a typical packed column to have much effect. With xl*x2*-lO-', this means that the adsorption density must be l o 5 times as great in region 2 as region 1. If such nonuniformity is caused purely by the difference in adsorption energy, A E , of the two kinds of sites, then AE would be
AE = 6lT In
p2/p1
-
23T
where @ is the gas constant and T the temperature in degrees Kelvin. When T = 500" K. (227' C.),the energy difference, AE, must be 11.5 kcals. in order to have much effect. At room temperature AE would have to be 7 kcals. For a typical capillary column the summation term would be important only if it were as large as -103. This means that the density difference, p 2 / p 1 would be -lo4. I n this case the energy difference between adsorption 18T. At 500" K. sites would be AE this would be 9 kcals., and a t room temperature it would be -5 kcals. The second source of efficiency loss related to nonuniformity would be a result of tailing. This may originate solely from kinetic effects and have nothing to do with adsorption nonlinearity. The theory of this phenomenon has recently been discussed (10). Tailing may occur whenever there is a high energy "tail-producing" site (this is similar to the type 2 site just discussed). This site must hold adsorbed
-
molecules for a time equal to the standard deviation, T (in time units), of the eluted peak. Since T is related to the number of plates, ?,; and the retention time, t , by T = t / &?, we may assume that T would rarely be less than 0.01 second (this cor-esponds to obtaining 104 plates in one second). Xow the mean time that a site holds a molecule is l / k d , and thit quantity must therefore equal T . The equations given earlier show that for miform surfaces
- -- - -
Assuming that R / ( l - 63) 1: C 5 X 104 cm./second, a 0.1 and V o / A 10-4 cm., we have k d 10’ second-’. I n order for a high-energy site to become tail producing, it must have kd reduced a t least to IO2, and must therefore desorb -105 slower than a normal site. As before, this would require that the high energy site h a w an adsorption energy (or energy burier) about 10 kcals. higher than thi? normal site to become tail producing. The above discussion applies only to the kinetic phenomena of GSC. As indicated earlier, a no.iuniform surface will encourage nonlinesr adsorption on its most tenacious sites. This may be one of the more serious drawbacks of nonuniformity. The previous literature on the subject has been notably noncommital on the precise reason for avoiding nonuniformit:i. The previous evamples have all been calculated on the assumption that there are two types of sites. In reality there will be a continuous distribution of site energies. This fact has been allowed for in the general theory, Equations 3, 13, and 14. However the main effect can undoubtedly be accounted for, as above, assuming a 2-site surface. ANALYTICAL POTENrlAL OF G S C
GSC has two immediate potential advantages over GLC which help offset the latter’s greater versatility. GSC also has a theoreticit1 potential for handling very difficult analyses and highspeed analyses which, pending the solution of some comples technical problems, might provide up to a rather phenomenal 10- plates ger second. The areas of immediate promise will be discussed first, followed b j speculation concerning the use of GSC in providing the “ultimate” column. One immediate advantage of GSC resides in the fact that a surface with any reasonable degree of uniformity will exhibit a C k value substantially smaller than the C L of GLC. Since C L contributes a substantial amount to the plate height in most GLC columns, the effective elimination of this term should provide sharper resolution in the same
length of c d u m n . I n the case of highspeed analysis, the combined C terms fix the minimum analyqis time. I n both GLC and GSC one has the gas phase term, C,, to contend with. However the replacement of C L by Ck in GSC may substsntially reduce a large part of the total C term. Furthermore, when C, becomes the limiting C term, there are possibilities for reducing its effective value through the use of low average column pressures and also through the use of high gas velocity, v (Equation 1 shows that C, loses its influence and is thus effectively zero a t large values of v). The first of these possibilities will be discussed in detail shortly. The second immediate advantage of GSC resides in the great potential selectivity of the adsorption process. “Surface adqorption is potentially capable of offering the most versatile and selective characteristics of any of the known retentive mechanisms” (16). The rigidly fived forces of a solid surface contrast sharply with the fluid forces of a liquid phase. As more is learned about the molecular-level detail of surfaces, it should be possible to use these rigid forces to enhance selectivity. The steric nature of these forces should be particularly valuable in separating isomerq. I n G L C , of course, the mobile force field interacts nearly the same with any group irrespective of its location on the molecule. The contrast is illustrated dramatically in the isomeric series consisting of the three vylenes and ethylbenzene. It has been very difficult to separate these solutes by ordinary GLC methods. However Spencer ($2) has separated these isomers easily using a liquid phase (diisodecylphthalate) containing Bentone 34. While the method is still technically that of GLC, the essential component to the separation is apparently the solid wrface of the Bentone 34. The liquid may u5efully modify the surface, but it does not alter its basic steric advantage. The future development of GSC will probably result in a large number of discoveries of useful “qteric” selectivities. Both empirical studies and theoretical adsorption work of the type done recently by Snyder (SO, 31) should contribute to this development. Theoretical Potential of GSC. T h e search for column selectivity may well promote t h e separation of difficult groups of compounds. It will not, however, materiallv improve t h e situation in which a very large number of chemically similar components must be separated. Such analyses, typical of those needed for petroleum products, require the appearance of numerous distinct peaks on a chromatogram of limited dimensions [qee the chromatograms of gasoline appearing in a paper by Blundell and Griffiths (S)]. This
problem requires that the relative width of the peaks must be reduced so that more of them can be fit into the space available. This can be done only by increasing the number of theoretical plates (or effective theoretical plates). Enhanced selectivity is of little value here as this merely shifts the peaks back and forth and thus transfers serious overlap to new points. One of the main obstacles to acquiring truly high column efficiencies, > 106 plates, is the inordinate amount of time needed to complete the analysis. Another difficulty is found with the high inlet pressures needed to maintain flow through the exceptionally long columns. These difficulties have been compounded by the fact t h a t resolution increases only with N1/2-i.e., the difficult task of doubling the number of plates leads to only about 40% more resolution. For these combined reasons there has been essentially a ceiling in practical work in which plate numbers greater than 10e105 are rarely achieved. Yet the further development of chromatography as an analytical tool demands that higher efficiencies be made available without the usual increase in analysis time. At the present time, GSC shows the greatest promise for achieving this goal. The reason for this is outlined below. The theoretical limit of analysis speed, based on our present understanding of the chromatographic process, is given by the following equations for analysis time (9) : 1
GLC
GSC
where N is the number of plates generated and R I I is the R value for the last component to be separated. These equations are based on the assumption that the gas phase term, C,, has been rendered negligible. This is theoretically possible as will be discussed below. Equation 20, combined with the earlier results on the magnitude of C k , shows that the minimum time for analysis using a solid with a uniform surface is of the order of t
-
lop7 N seconds
(21) i.e., lo7 or so plates could be obtained in one second. Even if one allows for considerable nonuniformity, in which case t must be increased by the heterogeneity factor u, this is still an esceedingly fast and effective analysis by present standards. I t is 103-104 times more rapid than can now be achieved in GLC columns. I n order to approach the theoretical limit for minimum analysis time, we have indicated that C, must be made negligible. This task promises to be exceedingly difficult, although definite VOL. 36, NO. 7, JUNE 1964
1173
advantages will accrue even if C, is reduced below its normal range. The reduction of C, may hinge on using very low average pressures in the column (C, is proportional to presbure). This would call for a drastic departure from conventional techniques since the high flow rates required have always been obtained by substantial pressure gradients along the column length. I t is conceivable that electromagnetic forces applied to an ionized carrier gas would provide the necessary propulsion. It is also conceivable that a series of column segments connected by pumps, such that vacuum conditions exist a t each segment outlet, would considerably reduce the C, term (9). As indicated earlier, however, the use of very high velocities may by itself eliminate C, in packed columns. This author is not attempting to minimize the difficulties in obtaining the phenomenal column performance discussed above. Even if the goal is only partially achieved, it will represent a significant advancement in chromatography. STUDY OF SURFACE KINETICS
Gas chromatography and closely related methods can be used for the measurement of several important physical quantities ( 5 ) . Partition coefficients and adsorption isotherms, and their concommitant enthalpies and entropies, have been measured extensively by GC. I n addition to these thermodynamic quantities, kinetic parameters have also been measured. although less extensively. Of these, the measurement of gaseous diffusion coefficients has been the most significant (16, 17). Diffusion coefficients in liquids can probably be measured also (16). The first suggestion, to this writer’s knowledge, that GC might be applicable to the study of surface kinetics was made by Habgood and Hanlan in 1959 (19). Their work, using charcoal as an adsorbent, indicated “the usefulness of gas chromatography in the study of kinetic as well as equilibrium aspects of adsorption.” h similar view was expressed independently by Giddings in 1960 ( I 4 ) . The Ci term is essentially a time constant for gas solid equilibration ( I S ) . A measure of this quantity will thus reflect the nature of the underlying surface kinetics. Hence with this approach GSC may have an important place in the study of surface kinetics. However its ure cannot answer all questions regarding rates and mechanisms. It is thus necessary to outline some of the apparent limitations and advantages of this method. Thc most serious limitations are, apparently, as follows: 1. Ck is a single parameter which specifies a n average rather than a
1 174
ANALYTICAL CHEMISTRY
detailed behavior of sdqorbed molecules. 2. K i t h present technique-, this time constant cannot be obtained if it is much less than second 16). K i t h lower values the adsorption-desorption effect is dominated by the C, term. I t is probable that the use of reduced average pressure and small particle size will lead to a detectable C k a t 10-4 second. Xontheless this remains as an important limitation to the method. I n contrast to the limitations juqt mentioned. this method qhould have the following advantages for the study of surface kinetics. 1. Providing a values can be obtained with some confidence, then Ckbecomes a second parameter for the will indicate the degree of heterogeneity of the surface-i.e., the heterogeneity factor, u, will be acquired. 2 . JVhile in a given measurement ck provides but a single average property of the surface kinetic., additional information can be obtained in a number of simple ways. The value of Ck a t different temperatureq, for example, will yield information on enthalpy and entropy, and thus on mechanism. Additional information could be gained by measuring, when significant, the departure of zones from their ideal Gaussian shape. This information, combined with equilibrium data and independent rate measurements (as of a values)
