Exact Treatment of Diffusion in Gas Chromatography H. A. Hartung and R. W. Dwyer Philip Morris, U.S.A., Richmond, Va. A chromatographic peak which is axially symmetrical in space may appear skewed in the concentration-time framework if diffusion continues as the peak passes through the detector. The median time corresponds to the point at which the symmetrical spatial distribution is centered in the detector. An exponential function utilizing median time and one other parameter designated p, describes concentration-time data when diffusion is the primary spreading’effect. Good correlation with this function was obtained with columns ranging from 4 to 105 cm, and the experimental p values are presented for nitrogen in helium carrier gas. The parameter p is determined by the diffusion coefficient, column length, dead volume, and certain end effects. Hence, diffusion constants and obstructive coefficients can be obtained from chromatographic experiments by fitting all of the concentrationtime data from a single peak. In tests with empty tubes, good agreement was obtained with literature values for the interdiffusion constant of Nz and He. The same value was obtained using columns packed with Chromosorb P, indicating no need to introduce an obstructive coefficient with packed columns.
THIS STUDY is part of a long range project to gather basic information on the thermodynamics and kinetics of gassolid interactions by means of gas chromatography. One of the objectives of this program was to determine the physical processes occurring in short chromatographic columns. Even in the simplest of G C experiments with short columns, the interpretation of results based on theories developed and proved with long columns was unsatisfactory. The fundamental problems which needed to be resolved were the fact that all of the time-concentration curves were skewed, and that the volume of the ends was significant compared to the volume of the column, and therefore end effects could not be neglected. Previously the effect of diffusion on the time-concentration curve observed in chromatography was interpreted in light of a Gaussian distribution equation ( I , 2). This equation represents a symmetrical distribution of diffusing molecules about some central point. This equation should hold true in column chromatography for nonadsorbed gases only if, at some time after injecting the sample, the zone could be frozen-Le., diffusion stopped-and the detector passed over the zone. But, in reality, the diffusion cannot be stopped but continues even as the sample zone passes through the detector. In studies with long columns and relatively large flow rates, the output curve approaches a Gaussian curve since the amount of band broadening is relatively slight as the zone passes through the detector. With short columns, however, this effect gives rise to conspicuously skewed time-concentration curves. If diffusion is the only spreading process occurring in gas chromatography, then the resulting time-concentration distribution is due to diffusion not only in the column, but also to diffusion in the ends. In the past, the diffusion in the (1) J. H. Purnell. “Gas Chromatography,’’ John Wiley & Sons, New York, N.Y., 1963, p 53. ( 2 ) J. Calvin Giddings, “Dynamics of Chromatography,’’ Part I, Marcel Dekker, New York, N.Y., 1965.
ends was usually dealt with by using a column with a very large volume compared to the extracolumn volume and assuming end effects to be negligible. For the present work, the effect of band broadening in the ends was handled by varying the column length and separating the end effects from the column processes. The objectives of this study were to resolve the problems of interpreting G C data acquired with short columns and to prove the feasibility of using short columns in studying diffusion and obstructive phenomena. These objectives were attained, and the results provide new insights into the use of chromatography for physical studies of porous materials. THEORY
Consider an unpacked column which has a steady flow of carrier gas passing through it. A narrow band of a nonadsorbed sample gas is injected at the inlet. Brownian diffusion starts at the moment of injection, and it yields a symmetrical spreading of the sample which is described by Equation 1 ( I ) a
where n(x) is the number of vapor molecules in the region between x and x dx measured from the center of the distribution, D is the diffusion coefficient, t is the time measured from the moment of injection, and a is the total number of injected molecules. In gas chromatography there is a fixed length, L , from the injection point to the detector. The experimental observation is a peak height, h(t), which is proportional to the number of molecules passing the detector in the time t to t dt. The relationship between n(x) and h(t) depends upon the mean velocity, ti, of the sample zone, which in the case of a nonadsorbed species, is the same as the velocity of the carrier stream. Therefore:
+
+
h(t) = tin(x)
(2)
Considering the simple case of a uniform column, the mean velocity is constant in the system and the value of x can be set equal to the distance from peak center to detector: x2
=
(L -
1ti)Z
(3)
Hence, the time distribution observed in the chromatographic test is ati
-(L - t a p
(4)
This is identical to the relationships derived by Aris and Amundson (3) starting with differential mixing equations. This is not a symmetrical distribution and there are three separate measures of its central tendency: (3) R. Aris and N. R. Amundson, AIChE J., 3,280 (1957).
ANALYTICAL CHEMISTRY, VOL. 44, NO. 11, SEPTEMBER 1972
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Table I. Results of Experiments with Nitrogen in Helium Carrier Gas at Room Temperature Lengths Empty, cm, L, Packed, cm, Lj 4.0 0.0 4.0 0.0 4.0 0.0 8.0 0.0 8.0 0.0 8.0 0.0 8.0 0.0 16.0 0.0 16.0 0.0 16.0 0.0 16.0 0.0 16.0 0.0 16.0 0.0 105.5 0.0 105.5 0.0 105.5 0.0 105.5 0.0 105.5 0.0 0.0 4.0 0.0 4.0 0.0 4.0 0.0 8.0 0.0 8.0 0.0 8.0 0.0 8.0 0.0 8.0 0.0 8.0 0.0 16.0 0.0 16.0 0.0 16.0 0.0 16.0 0.0 16.0 0.0 105.5 0.0 105.5 0.0 105.5
1.
Packing density, dcm a 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.43 0.43 0.43 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.41 0.43 0.43 0.43
Open area, cm2, A , 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.19 0.19 0.19 0.19 0.19 0.37 0.37 0.37 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.15 0.15 0.15
The statistical mean:
f
=
a
Lm
rh(r)dr
(5)
2. The mode time, which is the point at which dh(t)/dr is zero, and 3. The median time, designated m, defined such that:
c h ( r ) d r = a12
Void volume, cma, V 6.92 6.92 6.92 8.84 8.84 8.84 8.84 12.68 12.68 12.68 12.68 12.68 12.68 25.05 25.05 25.05 25.05 25.05 6.49 6.49 6.49 8.01 8.01 8.01 8.01 8.01 8.01 11.02 11.02 11.02 11.02 11.02 20.51 20.51 20.51
Med. time sec, m 28.0 42.7 65.6 25.1 36.8 55.4 76.0 25.4 35.8 50.8 64.9 110.2 120.9 84.6 141.6 193.1 221.5 236.4 28.3 41.1 58.6 23.6 27.8 21.8 54.6 67.7 79.5 22.4 29.9 46.2 62.1 94.0 89.0 145.2 211.0
P Exptl, sec, p 5.42 9.03 20.01 3.70 6.77 13.04 19.34 2.52 4.34 8.02 12.91 28.79 32.26 1.74 4.44 7.39 9.56 10.58 4.42 7.74 13.17 3.06 4.00 4.00 9.29 14.02 15.98 2.87 2.50 4.69 7.35 17.02 1.97 4.01 8.80
Calcd, sec, j3 4.89 9.73 21.29 3.64 6.41 12.97 23.32 2.76 4.27 7.35 11.22 30.03 35.89 2.25 4.09 6.55 8.23 9.21 4.04 7.15 13.27 2.86 3.50 3.50 9.96 14.66 19.74 2.19 2.94 5.32 8.61 18.15 2.41 4.36 7.83
applicable to real systems where uniformity of diameter is impossible to maintain from injection port to detector. The interpretation of 0,however, is not as straightforward for real systems since the mean sample velocity now varies from section to section. The relationship between /3 and the system geometry can easily be accounted for by considering the dependence of the time variance, r 2 ,on the individual section contributions. We start with the approximation ( 4 )
(6) L
The median time corresponds to the time when half of the sample gas population has passed through the detector. This is the time when the center of the x distribution is at the detector location. Since the carrier gas and the sample have the same mean velocity, then:
a
=
L/m
(7)
When this is used in Equation 4, the time distribution becomes : a
For the individual sections, Equation 9 is applicable, and using a subscript to designate a uniform section:
Since the total time variance, T ~ for , the diffusing sample gas is equal to the sum of the time variances, T i 2 , of each section, then
-(r - m)z
(9)
For nonadsorbed sample molecules, the mass flux of the sample is equal to that of the carrier gas and is held constant during each elution. The mass flux in each section is piVi/mi.
Although Equation 8 was derived for idealized systems of constant cross-sectional area, it is of general utility and is
(4) J. Tindall, Philip Morris, U S A . , Richmond, Va., private communication, 1971.
where :
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ANALYTICAL CHEMISTRY, VOL. 44, NO. 11, SEPTEMBER 1972
Hence the median time in a section can be derived from knowledge of void volumes and densities :
10
where p i and p are the densities in section i and in the whole system, respectively. Substitution in the foregoing equations gives :
The foregoing treatment uses assumptions which are the same as previously propounded by Giddings (5, 6) and the net result is equivalent to that obtained previously.
5
F
EXPERIMENTAL
A Hewlett-Packard Series 700 Laboratory Chromatograph was used in this investigation. A thermal conductivity detector was connected to an Infotronics Digital Readout System (Model CRS-1lo), and the time-concentration data were automatically punched on paper tape. A circuit board giving two-second intervals for the digitized data was used. These data tapes were processed on an IBM 1100 computer. Helium was used as the carrier gas and was purified by a molecular sieve trap. Flow rates were monitored with a Matheson rotameter. The columns used for this work had lengths ranging from 4.0 cm to 105.5 cm and inside areas (before packing) of either 0.19 cm or 0.48 cm. The end tubing was 0.16 cm in outside diameter. The columns, end tubing, and connections were made of type 316 stainless steel. Chromosorb P (45/60 mesh) was used as the stationary phase for the filled column experiments. This material was purged with purified helium and baked at 120 “C for several hours before use. The column lengths, packing densities, and open areas utilized are summarized in Table I. A density of 1.9 g/cm3 was used for the Chromosorb P solid phase in calculating open areas. High purity, dry nitrogen was the nonadsorbed test gas used in the diffusion studies. The volume of nitrogen injected ranged from 50 to 100 pl. All of the experiments in this study were performed at 34 “C and 758 mm Hg. Base-line corrections were carried out on the data assuming a straight line between the extremes of each peak. Base-line corrections, determinations of the median, regression analyses, and curve fitting were done using computers with FORTRAN and APL codings.
h
+
u c
-
m 0
0
I
I
1
-50
-25
0
- ( t-m)’/t
Figure 1. Log [h(t)/t] us. [-(t-m)?/t] plot of some typical experimental data with the best fit line. (L, = 16.0 cm, m = 50.8sec) 0
(2) empty columns of varying length and area. These were denoted by subscript c. (3) filled columns of varying length and area. These were denoted by subscript f. The flow rates were low in all tests so that the density ratios po/p, pe/p, and pf/p, were essentially unity. Thus an appropriate form of Equation 14 for the system model was 4Dm2
RESULTS
To calculate the values for each column length and flow rate, Equation 8 was reduced to its logarithmic form:
The p values were obtained from the slopes of least square [ - ( t - m)?/t]. Figure 1 is a plot of plots of log [h(t). diu#. some typical experimental data in this form with the best fit line. The values of /3 determined by this method are listed under the experimental column in Table 1. The experiments corresponded to a simplified model having three sections in series :
(1) injection port, detector, and all connecting tubing. These were the same in all tests. The net effect was treated as a single section designated by the subscript 0. (5) J. Calvin Giddings, ANAL.CHEM.,35, 353 (1963). (6) J Calvin Giddings, “Dynamics of Chromatography,” Part I, Marcel Dekker, New York, N.Y., 1965, p 79.
Experiment
- Theory
p=--- V3
[L,A,3
+ 7 L f A f 3+ + a01
Po
The coefficient y was introduced to allow for an obstructive effect which reduces diffusion in the filled columns. The constant term Po allows for the fact that there may be a finite width to the injected band. The term a0 is presumably a function only of diffusion in the end sections and a constant in all tests. Equation 16 is applicable to a system with filled and empty columns in series. In practice either L, or Lf was zero for an individual measurement of P. The observed P values were combined with void volume and median time data to derive the dependent variable y
P~va m2
The first mathematical regression utilized was triple linear. The variables and derived coefficients are summarized in Table 11. The derived y coefficient was not significantly different from unity. Hence, a second regression was carried out using the assumption that y = 1 . This was the
ANALYTICAL CHEMISTRY, VOL. 44, NO. 11, SEPTEMBER 1972
1745
Table 11. Summary of Results from Mathematical Regression of Experimental Data Dependent variable: Y = pV3/rn2 Independent Regression Derived variable coefficient coefficient Triple linear, allow- X I = L,AC3 4 0 = 2.481 D = 0.62 4yD = 2.489 y = 1.GO ing for obstruc- Xz = L f A j 3 tive effect X 3 = V3/rnz Po = 1.232 Constant term 4Da, = 0.445 a. = 0.18 Double linear, ne- XI = L,AC3+ 4 0 = 2.479 D = 0.62 glecting obstrucLjAj3 tiveeffect,~= 1 X2 = V3/rnz Po = 1.233 Constant term 400, = 0.448 a. = 0.18 Single linear, y = X = L,A,3 + 4 0 = 3.097 D = 0.77 1, neglect Po L,A/ 3 Constant term 4Da, = 0.775 a. = 0.26
1500
1000
Y
.c D .F
.c I Y
m
0 0
500
interdiffusion constants and obstructive coefficients. The unique aspects of the present study were:
0
time, sec
Figure 2. A typical elution peak with the distribution curve predicted by Equation 4 superimposed. (L, = 16.0 cm, m = 50.8 sec) X
Experiment
(1) concentration-time data are fitted to a distribution function ( 2 ) the function accounts for skewness due to diffusion as the band passes through the detector (3) the median time is recognized as the point when the spatial distribution is centered in the detector (4)an allowance is made for spreading that occurs in the ends ( 5 ) an allowance is made for a finite band width on injection It is interesting to note that the usual form of the diffusion term in the Van Deemter equation is obtained by applying the following assumptions to Equation 16:
- Theory
double linear regression in the second part of Table 11. Calculated values for P based on the second set of coefficients are listed in the last column of Table I. The physical interpretation of the 0, term is that a variance of 0.6m sec2 was imposed on the peak by the injection process. In other words the variance stemming from injection was inversely proportional to flow rate. This seem's physically reasonable because flow rate controls the rate of mass transfer of the sample molecule from the injection port into the system. The magnitude of 0,seems small relative to the values of 0, and one may wonder if it is an important factor. Therefore, a single linear regression was carried out on the data, neglecting the term arising from Po. The results are given in the third part of Table 11. The effect on the derived value for the diffusion constant was dramatic as a 25% difference was noted between the double and single linear regressions. The path length from injection port to column inlet and from column outlet to the detector corresponded to Lo = 65 cm. The void volume of the ends and connections gave a measured V, = 5 cm3. This gave a rough estimate of a, = VOalLo2 = 0.03
(18)
The higher experimental value was reasonable in view of the fact that short sections with larger areas would contribute disproportionately to this factor. DISCUSSION
The theory and experimental methods used in this study were very different from those used in previous determinations of 1746
a,
=
0, Po = 0 , V, = 0, V
AILI,G
=
=
LI/m,L,
=
0, A, = 0.
Direct substitution then gives
H
=
2yDja
(20)
These assumptions are implicit in all published data on y and D ,and we are unaware of any attempt to test their validity. Also it should be noted that all previous workers have used mode times rather than medians. The diffusion constant for He-N2 as reported by Fuller, Schettler, and Giddings (7) was 0.69. This was about 10% higher than the value of 0.62 obtained in this study. Neglect of end effects and peak skewness should give an overestiniation of D . Hence, the discrepancy is in the expected direction. More work is needed, however, to establish the absolute magnitude of end effects. The value of the obstructive coefficient for Chromsorb P as reported by Hargrove and Sawyer (8) was 0.42. This was the lowest y we found in a survey of the literature and the basis for choosing that material for our initial study. Hawkes and Steed (9) have noted that the y coefficient appears to be highly reproducible from test-to-test in a given laboratory but that large discrepancies arise when comparisons are made between laboratories. Thus, at an opposite extreme Boheman (7) ~, E. N. Fuller. P. D. Schettler, J. C. Giddings, Ind. Eng. Chem., 58 ( 5 ) , 19 (1966). (8) G. L. Hargrove and D. T. Sawyer, ANAL. CHEM.,39, 945 (1967). (9) S. J. Hawkes and S. Paul Steed, J . Chrornatoar. Sci., 8, 256-60 (1970).
ANALYTICAL CHEMISTRY, VOL. 44, NO. 11, SEPTEMBER 1972
and Purnell (10) report obstructive coefficients of essentially unity. This suggests that there are consistent discrepancies between laboratories in the measurement of system parameters. It also suggests that end effects may be significant even though the customary precautions to minimize them are taken. In the present theory the parameters which are directly measured are utilized to fit the data. As seen from Equation 16, the magnitudes of the experimentally determined D and y values are evaluated from the slopes of best fit lines. The slopes depend on the values of m2,Ac3,and V 3 ,and, therefore, smallerrors in measuring these parameters are greatly magnified when the exponents are
applied. Regressions of this type are also sensitive to the mathematical formula, and neglect of seemingly small end effects also may have a profound influence on estimates of D and y. ACKNO WLEDGMEh’T
The authors gratefully acknowledge the assistance of Ruth Hale in performing the tests and the assistance of Denis Plane in developing the theory. The advice and encouragement of D. T. Sawyer and David A. Lowitz are also gratefully acknowledged.
RECEIVED for review January 24, 1972. Accepted May 24, 1972.
(10) J. Boheman and J. H. Purnell, J. Chem. SOC.,1961,360.
Support-Bonded Polyaromatic Copolymer Stationary Phases for Use in Gas Chromatography Edward N. Fuller Applied Automation, Inc.-Systems
Research Department, Bartlesuille, Okla. 74004
The preparation of porous polyaromatic copolymers of divinylbenzene, ethylvinylbenzene, and styrene physically bonded to a solid support is described together with initial results illustrating the utility of these materials as GC column packings. While similar in nature to the widely used porous polymer beads, the support-bonded phases provide more rapid separations and greater column efficiency. Experiments showing the effects of cross-linking and of initial dilution with inert solvent on the resulting copolymer product are also discussed.
RECENTLY Hollis (1-3) reported the development of porous polyaromatic copolymers as stationary phases for gas chromatography. Following this first disclosure, a variety of such products prepared in the form of polymer beads have become commercially available and are now widely employed as column packing materials. Their properties, as related to the separation of various classes of compounds, have been investigated by Supina and Rose ( 4 ) and more extensively by Dave (5). Other publications, particularly those devoted to specific applications, have become too numerous to mention here. The present paper describes the preparation of similar porous polyaromatic copolymers formed directly on and physically bonded to the surface of suitable solid support materials and presents initial results of experiments investigating their utility as GC stationary phases and their surface properties. The emulsion polymerization methods normally used to prepare the polymer beads are not suited to the task of pre(1) 0.L. Hollis, ANAL.CHEM., 38,309 (1966). (2) 0. L. Hollis and W. V . Hayes, J . Gas Chromatogr., 4, 235 (1966). (3) 0. L. Hollis and W. V. Hayes in “Gas Chrornatography1966”, A . B. Littlewood, Ed., The Institute of Petroleum, London, 1967, p 57. (4) Walter R. Supina and Lewis P. Rose, J. Chromatogr. Sci., 7 , 192 (1969). ( 5 ) S. B. Dave, ibid., p 389.
paring a polymer on a support surface; consequently, a different approach aimed at accomplishing the above goal was devised. Basically, the technique consists of f i s t coating a support with a solution containing the divinylbenzene (DVB), ethylvinylbenzene (EVB), and styrene (STY) monomers, inert diluent, and an initiator in predetermined proportions. The monomers are then reacted in place by gentle heating to produce the porous copolymer directly on the surface of the support. Since the extensively cross-linked polymer is formed within the surface pore structure, it becomes permanently fixed or bonded to the support. The final step, solvent removal, is accomplished by evaporation under vacuum or else during the column conditioning process. EXPERIMENTAL
Reagents. Polymerization grade styrene (STY), 99+%, and practical grade divinylbenzene (DVB), 55.5 % DVB, 39.3 % ethylvinylbenzene (EVB), and 3.4 % diethylbenzene (DEB), were obtained from stocks at Phillips Petroleum Company Research Center. The DVB composition was checked by gas chromatographic analysis using the procedure of Hannah, Cook, and Blanchette (6). The results were 55.4% DVB, 38.9% EVB, and 3.0% DEB, all three present as the meta and para isomers. A number of minor components were observed, the largest being 1.2% naphthalene. The labeled composition was assumed to be correct. In any case, minor errors in composition would not greatly alter experimental results presented here. Technical grade lauroyl peroxide (LP), used as initiator, was obtained from Wallace & Tiernan, Buffalo, N.Y. Pure grade n-heptane, 99+% normal isomer, was obtained from Phillips Petroleum Company, Bartlesville, Okla. Method of Preparation. A solid support, which may be one of the Chromosorbs, porous glass beads, etc., is placed in a weighed glass jar. The jar and contents are evacuated to remove water and other possible adsorbed contaminants (6) Ray E. Hannah, Mary L. Cook, and Joseph A. Blanchette, ANAL.CHEM., 39,358 (1967).
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