5 % catalyst for oxygen analysis, vapor-phase conversions for these elemental species are highly efficient. Also, the extremely small sample sizes capable of being analyzed lengthen the lifetimes of reactor packings in this system. Modification of the two-stage catalytic reactors with noncatalytic surfaces (quartz or Au-coated stainless-steel foil) permitted functional-group analysis for unknowns by reference to previous Pyrochrom thermolytic-dissociation patterns (6-9). Hydrocarbons resulted in only C and H fragments; ketones, in addition, give CO; alcohols, CO and HzO; esters, CO, Cot,and H 2 0 ;chlorohydrocarbons, HCI. Demonstration of the ability to transfer sequential G C peaks from G C No. 1 to the Pyrochrom and accurately determine their elemental constituents is shown in Figure 5 , a and 6, for benzene and tert-butyl benzene. The calculated and experimental C and H values are indicated for these representative compounds and are within the desired accuracy limits of +0.5 %. In order to perform these above processes of thermolytic dissociation, C, H, N , and 0 determinations on a series of G C peaks emerging from G C No. 1 more rapidly than ca. 8 minutes apart (the time needed for complete C, H, N , or 0 data for any one peak), it was necessary to provide a stop-flow arrangement. A six-port microvalve (Carle Instruments, Inc., Fullerton, Calif.) and two matched analytical columns were installed in G C No. 1 to accomplish this requirement. Several trials determined that stop-flow of more than 10 minutes showed no adverse effects in peak shapes or relative peak relationships. With this arrangement, a 1-microliter mixture of four compounds of differing functionality was injected into GC No. 1 and sequentially dissociated within the Pyrochrom noncatalytic reactor, then directly analyzed on T internal analytical column. The the 11-ft Poropak Q results are shown in Figure 6. It is evident that the stop-flow system has enabled each peak to be separated, transferred,
dissociated, and individually analyzed as a hydrocarbon, ester, ketone, or alcohol species. Should the (catalytic) elemental reactors be utilized, the stop-flow arrangement allows ample time (ca. 8 minutes) for the C, H, N, or 0 data to be obtained for each peak in sequence as it is separated in the GC No. 1 analytical column. Alternatively, if desired, only one or certain peaks may be removed from a multicomponent mixture and analyzed as above. Therefore, for a mixture of five compounds, each species could be analyzed for C, H, and N content in 8 minutes and one would be provided with data within cu. 40 minutes for determining the likely empirical formula (C,H,N,) for each peak. CONCLUSIONS
It is felt that further direct application of this arrangement to complex problems of organic reaction mechanisms, kinetics, catalysis, and trace-product analysis will be possible. Extension of the method will be attempted to detect and quantitatively determine at these trace levels additional elements, such as sulfur and halogens, by selective combustion procedures. Organic compounds with more complex structures and combinations of heteroatoms will be studied. Development of a systems approach to problem-solving in organic analysis will continue. ACKNOWLEDGMENT
We appreciate the helpful discussion and advice provided by S. A. Sarner, CDS, Inc., and the aid in computer programming by C. D. Nauman and D . C. Messersmith, Armstrong Cork Company.
+
RECEIVED for review November 5 , 1971. Accepted March 17,1972.
Predicting Gas-Chromatographic Resolution for Pairs of N-Alkane Homologs Robert S. Swinglel and L. B. Rogers Department of Chemistry, Purdue UniGersity, Lafayette, Ind. 47907 Using values for relative retention, number of theoretical plates, and relative peak widths determined for two members of a homologous series, reasonably good predictions can be made for resolutions at a given value of capacity ratio for other pairs of homologs in a relatively well behaved chromatographic system. The limitations of the assumptions, upon which some of the calculations were based, have been examined using as an example the n-alkanes on SE-30.
ANALYSIS TIME is often critical in process-control applications ( I ) where analysis lag must be minimized and where many samples are to be analyzed. With the increasing use of digital computers as data processing devices for chromatographic Present address, Central Research Department, Experimental Station, E. I. DuPont de Nemours and Co., Wilmington, Dela. 19898. (1) I. G. McWilliam, in “Advances in Chromatography,” Vol. 7,
J. G. Giddings and R. A. Keller, Ed., Dekker, New York, N.Y., 1968, pp 163-220.
instruments ( 2 , 3), the total analysis time should no longer be limited by calculation of the experimental data but by the separation time itself. The problem of separation time in gas chromatography has been considered from the standpoints of minimum time for a prescribed resulution (4-6) and for normalized-time analysis (7-9) in which resolution is maximized for a prescribed analysis time. Hawkes (IO) has used computer (2) A. W. Westerberg, ANAL.CHEM., 41,1595 (1969). (3) P. P. Briggs, ControlEng., 14(9), 75 (1967). (4) B. 0. Ayres, R. J. Loyd, and D. D. DeFord, ANAL.CHEM., 33, 986 (1961). (5) J. C. Giddings, Am/. Clzem., 34, 314 (1962). (6) I. Halasz and E. Heine, in “Advances in Analytical Chemistry,” Vol. 6, J. H. Purnell, Ed., Interscience, New York, N.Y., 1968, pp 153-208. (7) B. L. Karger and W. D. Cooke, ANAL. CHEM., 36,985 (1964). (8) Ibid.,p 991. (9) G. Guiochon, ANAL. CHEM., 38,1020 (1966). (10) S . J. Hawkes, J . Chromatogr. Sci., 7, 526 (1969). ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
1415
Column
Table I. Column Characteristics Length, Loading, Packing density, cm wlw g/cm
z
50.0 99.8 319.8 100.2 100.0
1 2 3 4 5
2.0 2.0 2.0 3.5 6.4
0.01716 0,01752 0.01730 0.01720 0.01796
simulation, combined with descriptive equations and specific retention data, to determine minimum-time conditions. Scott (11) has recently reported a useful experimental method that one can follow in the laboratory to determine conditions for minimum analysis time. Optimal values of liquid loading, temperature, and flow rate were found for a simple case. One possible limitation of his procedure is that the column length was arbitrarily fixed. While that may be convenient in practice, choice of a different length might have led to different optimized conditions and shorter analysis times. One purpose of the present investigation was to point out how a n on-line computer could greatly reduce the time necessary t o complete a n optimization procedure such as that described by Scott. Even if the computer were used only for data acquisition and processing, the savings in time and effort would be large since calculations for any run would be available a few seconds after its completion. Furthermore, because each point on the various graphs illustrated by Scott could be plotted within a few seconds after completion of a n experimental run, trends could be observed which would permit selection of the best operating conditions for the next chromatogram. Hence, time would usually be saved by decreasing the total number of experiments necessary to complete the study as well as in the calculations themselves. This type of optimization procedure presents a strong case for unattended computer control of a gas chromatograph (12). Using complete computer control, the two weeks of data gathering plus the four days of interpretation necessary in Scott’s example could be reduced t o about four 24-hour days. In addition, the analyst would be relatively free during that period to perform other tasks. A second purpose of the present study was t o examine the special case of binary mixtures of a homologous series of nalkanes, in order t o see how chromotographic characteristics such as the number of theoretical plates and, especially, the resolution changed with the carbon numbers of the pair, with temperature, with liquid loading, and with the flow rate of the carrier gas. In principle, relatively few measurements of the retention times a t different temperatures should provide information about changes in retention times, but changes in resolution also depend upon changes in the peak widths. The peak widths reflect diffusion coefficients in the two phases, and predicting their net effect is a more complex problem. Prediction of peak widths may be further complicated by “anomalous” diffusion behavior (13-15). (11) R. p. w.Scott in “Advances in Chromatography,” VOl. 9, J. C. Giddings and R. A. Keller, Ed., Dekker, New York, N.Y., 1970, pp 193-214. (12) R. W. Swingle and L. B. Rogers, ANAL.CHEM., 43,810 (1971). (13) A. K. Moreland and L. B. Rogers. Seoar. Sci.,6. 1 (1971). (14) P. R. Rony and J. E. Funk, ib;., p 383. (15) J. E. Oberholtzer and L. B. Rogers, ANAL.CHEM.,41, 1590 (1969). 1416
0
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
For any given packed column, Guiochon (9) has shown that, in general, the highest resolution in the least time for a solute pair was attained when that column was operated a t a flow rate of 1.4 t o 2 times the flow rate at the van Deemter minimum and at a temperature which gave a capacity ratio of approximately three for the second member of the pair, Therefore, we have examined the resolutions of pairs of adjacent n-alkanes as the column temperature, liquid loading, and flow rate of the carrier gas were changed so as to pass through the region in which the conditions for resolution were presumably optimized. We have shown that, from a minimum of chromatographic information for any two homologs, the temperature that will give a capacity ratio of three can, as expected, be predicted for any member of the series. In addition, the relative retention, CY,for adjacent homologs at that temperature can be estimated from the linear plot of log CY cs. 1jT. Hence, if HETP and resolution (both of which are affected by diffusion phenomena) were relatively independent of carbon number a t a constant value of the capacity ratio, then the optimum flow rate would remain approximately constant for all homologs, and the resolution achieved at this temperature and flow rate could be calculated from the Purnell equation (16). The column length yielding the resolution desired could then be estimated from the proportionality between resolution and the square root of the column length. As a result, one might be able t o speed up optimization studies involving higher homologs by carrying out preliminary studies using lower members of the series which are often more readily available and easier t o handle. The degree of reliability in the estimates of optimal values for temperature, flow, and column length will depend upon the chemical system under investigation and how well it conforms to the approximations used in this approach. This paper will discuss those approximations and illustrate how well they hold up under various experimental conditions for one group of homologs. EXPERIMENTAL
Apparatus. A Hewlett-Packard Model 2115A computer, interfaced to a high-precision gas chromatograph, was used for data gathering, data processing, and control. The system has been described (12). Reagents. One-to-one mixtures of adjacent normal paraffins, C6 to CI1 (Chromatoquality from Matheson, Coleman and Bell) were prepared. Methane (c.P. Grade, Matheson Co.) was used as received. Procedures. Teflon-6 (Varian Aerograph, Walnut Creek, Calif.) was used for the solid support. Column packings of 2.0, 3.5, and 6.4% wjw SE-30 were prepared by heating, and gently swirling from time to time, a toluene solution of SE-30 containing the Teflon support. The packings were sieved to 30/60 mesh before and after coating while cooling both the sieves and packings with liquid nitrogen so as to minimize the buildup of electrical charge on the particles. Gas chromatographic columns were prepared using 0.32-cm i.d. stainless steel tubing which had been rinsed with successive portions of acetone, dichloromethane, and distilled water before being dried with nitrogen. Columns were packed vertically, while tapping, and were then bent into approxi~I lists b thel lengths, ~ loadmately 12-cm diameter coils. ~ ings, and packing densities of the various columns. All columns were conditioned at 180 “cfor several hours using a helium flow of 5 mljmin. (16) J. H. Purnel1,J. Cliem. Soc., 256, 1268 (1960).
Calculations. A FORTRAN program, GOUTZ, was written for processing chromatographic data in real-time. Peak sensing, area determinations, and mean retention times were calculated as discussed previously (12), except that compression of data was used to limit the number of points across a peak to 200. The retention time for the peak mean, T,,, and corrected retention volume, VRo, were determined as discussed in reference 15. HETP was calculated by using the peak width at half height, Wllz. The retention time for the peak maximum, T,,,, was found by least-squares fitting a quadratic equation about the data point closest to the peak-mean retention time. One-sixth of the total points across the peak were used. The fit was accomplished by a method similar to that described by Savitsky and Golay (17). Coefficients were computer-generated in real-time from Gram polynomials (18), and values for the smoothed ordinate, Yo,the first derivative, Y 1 ,and the second derivative, Y l l , were found. T,,, was then calculated from
-
TmaZ= T,,
Y1/Yl1
(1)
and Y,,,,, the ordinate at TmZ,calculated from Y,,,
=
+ YO
-(Y'Y'/2Y")
(2)
W1l2was determined from the two values of the abscissa corresponding to Y,,,/2, which were, in turn, found from linear interpolation between pairs of points whose ordinates were closest to Y,,,/2. The standard deviation of the peak, s, was found from
s = w11212.345
(3)
Resolution, Re, was defined as
Table 11. Slopes, Intercepts, and Goodness-of-Fit Parameters for Plots of Equation 7 Temperature, Range of "C homologs Slope Intercept Std dev" 71.43 87.93 103.42 111.41 120.26 128.07 135.53 148.35 a
2.946 3.269 3.622 3.801 4.012 4.166 4.354 4.745
c6-C~ CC-CS Cs-Cii Ce-Cii
cs-cii C6-Cll Cs-Cli Cg-Cii
5.671 6.289 6.890 7.245 7.564 7.964 8.303 8.903
10.006 10.005 10,023 1 0 .020 f0.029 10.025 10.011 f O 001
Standard deviation of log k values from the calculated line.
Table 111. Relative Retentions of nParaffins on Column 2 Temperature, "C 71.43 87.93 103.42 111.41 120.32 128.07 135.53 148.35
2.03 1.93 1.87 1.84 1.77 1.70
2.02 1.90 1.84 1.77 1.75 1.71
...
...
...
...
...
1.88 1.83 1.74 1.73 1.70
1.88 1.82 1.77 1.72 1.69 1.62
1.86 1.81 1.80 1.72 1.68 1.62
...
Various equations relating resolution to fundamental parameters have been proposed (20). Purnell(16) has shown that for symmetrical Gaussian peaks of equal width, the resolution may be expressed as
(4) (9)
where the numerical subscripts refer to the first and second peaks eluted. All further calculations were performed off-line because of limitations in computer memory. The capacity ratio, k , and the relative retention, a, were calculated in the usual manner. For a homologous series, the well known linear relationship between the logarithm of the capacity ratio and the carbon number (19) can be expressed as log k
=
mC,
where n is the ratio of peak widths defined by
+b
(5)
where C , represents the number of carbons and m and b are constants for a given temperature. The standard molar heats of solution, AH,O, were calculated from a linear least-squares analysis of log k us. 1/T from the relationship log k
= -
AHso 2.3 RT' ~
b2
where 62 is the constant of integration. It can be shown that differential thermodynamic quantities can be calculated from A(AG,O)
=
-RT In
CY
(7)
and lna
=
A(AH,O) --+RT
where N is the number of theoretical plates and the subscript 2 refers to the second peak of the pair. Karger (20) has proposed a modified equation for peaks that are significantly different in width:
A(AS,O) R
(8)
(17) A. Savitsky and M. J. E. Golay, ANAL.CHEM.,36,1627 (1964). (18) H. T. Davis, in "Tables of Mathematical Functions," Principia, San Antonio, Texas, 1963, pp 307-8. (19) A. I. M. Keulemans, "Gas Chromatography," 2nd ed.: Reinhold, New York, N.Y., 1959, p 27.
RESULTS
A homologous series represents a special case in which the linear free-energy relationship allows the experimenter to estimate capacity ratios and relative retentions for an entire series from a minimum number of measurements on two members of that series. If first approximations can be made for the relative peak widths and the number of theoretical plates, then the resolution for any two homologs can be calculated using Equations 9 and 10. Resolution under Isothermal Conditions. Equation 5 expresses the linearity between the logarithm of the capacity ratio and the chain length of a homologous series. Thus, the capacity ratio of any homolog can be calculated at constant temperature and loading provided the capacity ratios of two other members, preferably not the lowest two or three of the series, have been determined. Table I1 illustrates the application of Equation 5 to retention data for a series of n-paraffins (20) B. L. Karger,J. Gas Chromatogr., 5,161 (1967). ANALYTICAL CHEMISTRY, VOL. 44, NO. 8,JULY 1972
1417
0.0
2
6
IO
14
18
22 26 Flow Rate (ml/min)
30
34
38
I
Figure 3. Column efficiency for homologs at 128 "C on column 2 k/(l+ki
Figure 1. Effect of the capacity ratio function on resolution using column 2 0
CS-C? to C8-clO;
0
c6-ci
to ClO-ClL;
A CS-C~ to Cio-Cii; A C7-C8 to Cio-Cii
2.4
I
0.0
2
6
IO
14
18 22 26 F l o w Rate ( m l h i n )
30
34
38
Figure 4. Column efficiency for homologs at 103 "C on column 2
1.0
0
2
6
IO
14
18 22 26 F l o w Rate (ml/min)
30
34
38
Figure 2. Effect of flow rate on resolution for homologs at 128 "C on column 2
on column 2. The slopes, intercepts, and goodness-of-fit parameters that resulted from linear least-squares analyses are shown for several temperatures. In the worst case, a n uncertainty of 10.029 in log k was observed. As seen from Equation 7, the relative retention should be constant for adjacent homologs under isothermal conditions. Table I11 shows a values for n-paraffins on column 2 at several temperatures. The relative retentions a t 120.32 "C show the greatest difference, 6.2 between the C6-C7and C S € ~pairs. Thus, if one assumes that the change in N with carbon number is small or varies in a regular fashion, Equation 9 shows that a plot of resolution us. kn/(l k 2 )should give a straight line for a constant flow rate. This behavior is illustrated in Figure 1 for column 2 at several temperatures. At each temperature, the lowest point corresponds t o the lowest pair of homologs. A rough estimate of the resolution for any pair of adjacent homologs can be determined using a plot of this type.
z,
+
1418
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
This general approach involves several assumptions which will now be examined. First, the linear free-energy relationship must hold for homologs. This linearity has often been found t o break down for lower members of a series (19). The effect of this breakdown may be evident in Table 111. The a values for Cs-Ci are, in general, higher than those for the other pairs a t the same temperature, reflecting a low capacity ratio for n-hexane. Any nonlinearity in the series will have a large effect on the accuracy of predicted resolution, especially if a is close t o one. Small changes in A(AGAo)will be magnified by the ( a - l)/a term of Equation 11, and the predicted resolution may be grossly inaccurate. A second assumption was that N was not a function of carbon number. The effect of plate height upon resolution can be seen in Figures 2 and 3. As evidenced by these data, the assumption that N was independent of carbon number was not reasonable for these conditions, especially when considering higher flow rates. However, for cases such as that shown in Figure 3, the change in N was fairly uniform for the homologs studied. Thus, even though the resolution data in Figure 1 were taken at a flow rate of 15 ml/min, the linearity was still reasonable. Figure 4 illustrates a van Deemter plot taken at 103 O C where the change in N was not uniform with carbon number. This may be a special case where slow diffusion leads to an anomaly so further work is under way t o examine this possibility. In any case, however, resolution depends only upon the square root of N so plots such as Figure 1 can still be useful.
2.6
Table IV. Relative Peak Widths of Adjacent Homologs Tempera- Flow ture, rate, "C ml/min C&7 C7-CS CS-CQ Cs-clo ClO-cIl 103.42
111.41
5.53 12.99 20.66 32.17 46.86
0.70 0.68 0.68 0.67 ,..
...
5.51 10.43 15.19 21.87 27.42 33.30
0.75 0.73 0.72 0.72 0.71 ,..
0.68 0.65 0.66 0.67 0.67 0.67
...
0.61 0.58 0.58 0.59 0.59
0.64 0.63 0.62 0.63 0.64 0.63
0.59 0.61 0.60 0.60 0.60
0.60 0.60 0.59 0.60
0.64 0.62 0.63 0.63
... ... 0.59 0.58 0.58
... 0.60 0.59 0.60 0.59
I
t '
2.4
-
2.2
-
c
'E
5 E 2.0-
n c
-
-
-
1.8
2
1.6-
0
1.4-
1.2I .o
60
70
80
90
110 120 Temperature ('C)
100
140
130
A
IS0
160
Figure 6. Effect of temperature on resolution for column 2
1
I
I
0
1
2
3
4
5
6
7
6
9
k
Figure 7. Effect of capacity ratio on HETP for column 2 IOOO/T
(
A
"C-'l
0
Figure 5. Effect of temperature on relative retention using a value of cy, at each temperature, that was averaged over all pairs of homologs 0 2.0% loading 0
6.4% loading
A third assumption, used in the above development and implicit in Equation 9, was that the peaks were of equal width. For adjacent homologs, this was not a good assumption. In general, Equation 10 was found to give better estimates of resolution under isothermal conditions. Table IV, which lists some relative peak widths for adjacent homologs at representative temperatures and flow rates, shows that n decreased as carbon number increased. From Equation 11, this was expected since the difference in retention became larger as the carbon number increased. However, Table IV also shows that the change in n decreased with higher homologs and that, at any given temperature and flow rate, n seemed to approach a constant value when capacity ratios for both members of the homologous pair were three or above. Hence, relatively good estimates of resolution should be possible. This is further reinforced by Karger (20) who has discussed theoretically the computation of n for various conditions. Note that a relatively large error in estimating n can be tolerated when calculating resolution. For instance, changing n from 0.75 to 0.65 resulted in only about a 5 gain in resolution. Resolution at Constant Capacity Ratio. The temperature that will give a k of three for any homolog can be estimated from Equations 5 and 6 if the A(AH,O) value for a methylene group in the homologous series is known, and if m and b of
CsC7 to Cio-Cn to
clO-cI1
CrC7 to
c,1-c12
c6-c7
Equation 5 have been determined for any given temperature. Values of A(AHso)for n-paraffins were found to be 1.00 =t 0.05 K cal/mole on column 2. Figure 5 showed that, as expected from Equation 8, a plot of log a cs. 1jT was linear, with the slope being equal to A(AHso)/R. Thus, the relative retention at the temperature giving a capacity ratio of three for any hoinolog could be calculated directly once the constant, A(AS,O)/R, was known. The above method for calculating optimal temperatures assumed that AHsowas constant over the temperature range used in the extrapolation of Equation 6. For the system studied here, the log k US. 1/T plots were slightly concave toward the temperature axis which resulted in predicted temperatures that were slightly high. In addition, the calculation of a as a function of temperature assumed that A(AHso>was constant. However, as seen in Figure 5 , A(AH,O) was a function of the amount of liquid loading. Both these effects were attributed to solute adsorption on the Teflon support (21). The effect of temperature upon resolution is shown in Figure 6. In general, both a and kz increased on going to lower temperatures while Nz decreased, thereby leading to a maximum or flattening out of the resolution-temperature curves. While estimations of changes in HETP with temperature and, thus, resolution were difficult, Figure 7 shows that HETP at 15 ml/min was constant, within lo%, for different homologs when the temperature was adjusted to give a ~
~
~~
(21) J. R. Conder, ANAL.CHEM., 43, 367 (1971). ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
1419
Table V. Effect of Column Length on Resolution at 15 ml/min for 2% SE-30 Column length Resolution DifferComL1, Lz, Rezl R,,2 R,A ence. pounds crn crn found found caiid
c7-cs ci-c*
50 50
100 100
1.44 1 27
2.09 1.87
2.03 1.79
3 5
capacity ratio of three. This illustrates the tendency of the effects of increased temperature and increased carbon number to partially compensate one another. Solute diffusion coefficients decreased as the carbon number went up, but the increase in temperature necessary to normalize kp to a value of three compensated for this to a large extent. As indicated earlier, the relative peak width is important in calculating resolution (Equation 10). Table IV shows that the relative peak width increased with temperature for a given flow rate and homologous pair. This was expected term and the for a given value of k z , since both the [N2/N1]1’2 retention-time ratio in Equation 11 should increase with temperature due to the decrease in C Y . Hence, small increases in n were observed as the carbon number of the homolog was increased and the temperature was adjusted to bring k z to a value of three. For example, C9-Closhowed n equal to 0.65 when k was 3.1, and Cs-Cghad n equal to 0.62 when k of C9 was 2.9. Hence, the changes in n were small and varied in a regular fashion. For the liquid loading of column 2, an empirically determined value of 0.04 multiplied by the difference in carbon number between the test and “unknown” homolog was added to n of the test member for prediction purposes. As stated earlier, however, a relatively large error in n exerted only a small effect on the calculated resolution. Using the method outlined above, the resolution for the C9-Clo pair on column 2 was estimated from data on the Ci-Cs pair. The optimal temperature was calculated to be about 130 “C. Values for Nz, C Y , and n were estimated at 364, 1.72, and 0.66, respectively. Substitution of these values into Equation 10 gave a calculated resolution of 1.84 which agreed very well with the value of 1.77 found from Figure 6. Effect of the column length on HETP has been studied by many authors, the latest being Novak (22). Ayres, Loyd, and DeFord ( 4 ) found HETP to be independent of column length when the column-outlet, rather than the average, flow rate of gas was used. Within the experimental error for determining HETP, the same conclusion was also found to be true in all of the present cases. Thus, from Equations 9 and 10, the resolution was proportional to the square root of the column length as illustrated in Table V. It should be pointed out that, as the carbon number increased, the column length required to attain a specific resolution also increased. This was due to the decrease in CY that resulted from the higher required temperatures. At a constant flow rate at the outlet, the higher temperatures and longer columns led to lower values for the gas compressibility factor and longer analysis times. The effect of loading on k was influenced by adsorption as mentioned previously and demonstrated in Figure 5 . In theory, doubling the amount of stationary phase should approximately double the capacity ratios, but, because of the large influence of retention mechanisms other than liquid (22) J. Novak, J. Chrornatogr., 50,385 (1970). 1420
ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
partitioning (2.9, the experimental results obviously would not agree well with the predictions. Thus, in cases where two or more mechanisms both make a large contribution to the separation, the experimenter will have to make measurements at different loadings before conditions for minimum analysis time can be estimated. Finally, the effect of sample size was examined over about a ten-fold concentration range under several sets of conditions, and very little effect upon resolution was noted. For peaks with widths greater than ten seconds, resolution increased slightly with increasing sample size. This was due to the fact that, for a larger sample, GOUT2 detected relatively more of the peak, and the calculated values of Wl)s decreased. Obviously, major effects on resolution would be seen if the sample size were large enough to overload the column.
DISCUSSION
This paper has described a useful technique for estimating chromatographic resolution for members of a homologous series. From a minimum of prior information, experimental conditions which will approximate those for a minimumtime analysis can also be calculated for each member of the series. (This assumes that the member in question is the second one of a pair to be separated.) Those calculated conditions could be used directly in cases where relatively few samples are to be run or as the starting point for a more nearly complete study or optimization. In the above approach, a capacity ratio of three has been assumed to be optimum for the separation of a pair. However, Grushka (24) has recently emphasized that this is only a first-order approximation, and that a more exact value depends upon 01 and A H as shown in his Figure l . If the sample has more than two components, Purnell and Quinn (25) have shown that, when components elute after the most difficult pair to separate, the minimum analysis time may not be the same as that which gives the fastest analysis of that pair alone. Furthermore, for a mixture containing molecules of widely different physical properties, the AHSO values may be such that little or no separation will occur at the temperature which gives a capacity ratio of three. In those cases, the more general procedure outlined by Scott will be required since it is an entirely empirical approach that has no restrictions as to the number of components in the sample or their physical properties. It should be emphasized that it is essential to work at the minimum acceptable resolution in order to attain minimumtime conditions. Guiochon (9) has shown retention data to illustrate the fact that analysis time is approximately proportional t o the 3/2 power of column length. Hence, doubling the resolution between two peaks by increasing the column length will result in an analysis time about eight times greater than the original. Thus, columns should be as short as possible. In addition, if still faster analyses are desired, mathematical peak deconvolutions will permit smaller values of resolution to be selected. Halasz (6) has pointed out that, when performing a complete optimization procedure, the maximum of the curve for resolution DS. temperature defines the optimum operating temperature. Often, however, that temperature will be (23) R. L. Martin, ANAL.CHEM., 35,116(1963). (24) E. Grushka, ibid., 43,766 (1971). (25) J. H. Purnell and C. P. Quinn, “Gas Chromatography, 1960,” R. P. W. Scott, Ed., Butterworths,London, 1960, p 184.
subambient, and inconvenience will force the experimenter to settle for less then optimum conditions. For example, data for resolution us. temperature at a constant flow rate were shown in Figure 6 for several adjacent homologous pairs. It is possible that, at low temperatures, the curves will become nearly parallel if the HETP becomes less dependent upon carbon number (Figure 4). Experiments are now being
run to examine this feature. In any case, if there were less dependence of HETP upon carbon number, it could facilitate prediction of resolution and minimum-time conditions.
RECEIVED for review June 7, 1971. Accepted March 15, 1972. Supported in part by the United States Atomic Energy Commission under Contract AT(l1-1)-1222.
Principal-ComponentAnalysis Applied to Chromatographic Data Donald Macnaughtan, Jr.,l L. B. Rogers, and Grant Wernimont Department of Chemistry, Purdue Uniuersity, Lafayette, Ind. 47907 Principal-component analysis of high-precision gas chromatographic data has been shown to produce meaningful characterization of peak-shape changes. Thus, it provides a means for deconvoluting two overlapped peaks and gives quantitative information on sample composition. The principal-component analysis method is outlined together with the deconvolution procedure.
IN CHROMATOGRAPHY, as in other types of measurements, overlapping peaks can occur quite frequently. The following method is based upon principal-component analysis which, under certain restrictions, can effect a deconvolution of two chromatographic peaks. In chemistry, principal-component analysis has been used, especially in spectrometry, for determining the number of absorbing species (1-5) and their absorption curves (6, 7), and for determination of equilibrium constants (8,9). In chromatography, many methods have been set forth for deconvolution of two or more overlapped peaks. The method studied here requires more than one chromatogram of the mixture and these must be aligned properly on the time axis so that they correlate with one another. Consequently, a chromatographic system capable of high precision is required in order to obtain meaningful results. The combination of principal-component analysis and high-precision data makes possible not only deconvolution of two overlapped peaks, but also characterization of changes in peak shapes for any series of chromatographic curves that arise from a parameter change. It can also allow reliable extrapolation of system behavior beyond the limits that were studied. This mathematical approach is not intended as a replacement for other methods such as moment analysis, but rather as a complementary tool. Its pov :r lies in the fact that it is a self-modeling method and consequently requires no preconceived model. If one wishes to test or apply a model, it can Present address, Chemistry Department, University of Illinois, Urbana, 111. 61801. (1) G. Weber, Nature, 190, 27 (1961). (2) S. Ainsworth, J . Phys. Chem., 65, 1968 (1961). (3) R. M. Wallace, ibid., 64, 899 (1960). (4) Zbid., 68, 3889 (1964). ( 5 ) D. Datakis, ANAL.CHEM., 37,876 (1965). (6) S. Ainsworth, J . Phys. Chem., 67, 1613 (1963). (7) W. H. Lawton and E. A. Sylvestre, Technometrics, 13, 617 (1971). (8) L. P. Varga and F. C. Veatch, ANAL.CHEM., 39, 1101 (1967). (9) J. J. Kankare, ibid.,42,1322 (1970).
be fit to the deconvoluted curves and the appropriate parameters found. The method gives both qualitative and quantitative information about the systematic changes in a set of data. The quantitative results (scalars) can be studied further in the form of a scalar plot. The types of information that can be found in such a plot are discussed in detail in the Appendix. Finally, important information can usually be obtained from the residuals, the data that are left unexplained by the method. For example, one can see if one or two residuals were derived from readings that were “bad,” probably because of errors in recording the data, noise spikes, and so forth. In the Appendix, the complete mathematical procedure is given. Briefly, however, the vector analysis is similar to that used by Simonds (IO) but modified by Wernimont (11) using the second-moment matrix in the calculations to produce characteristic eigenvectors and the corresponding transformed scalar multiples (scalars). This is essentially the same procedure used by Lawton (7), who extended the analysis to determine the pure underlying functions as described in the present discussion. These eigenvectors, V, and scalars, S, can be combined to regenerate the original data matrix, D, of n rows and r columns as follows : n
D?ZJ
=
c I
Sn,i.V2,7
(1)
where i refers to the specific eigenvector and its scalars, up to as many as n eigenvectors. The data are in the form of a matrix of a set or family of response data (vectors). Each data point in a vector must be aligned to the corresponding data point in the other vectors. This is the only condition necessary for principal-component analysis of a series of response data. However, for the deconvolution procedure, other restrictions on the data must be met. This method assumes that the two functions are, at all times, nonnegative and linearly independent. In addition, one of the functions must be zero in some place where the other has a finite value. Although this is not strictly true for overlapped Gaussians, this condition is closely approximated when there is some resolution of the peaks. Finally, the relative amounts of the two functions must be different in each sample. T o resolve a two-component mixture, at least two data vectors are required. However, the results will be statistically better (10) J. L. Simonds,J . Opf. SOC.Amer., 53,968 (1963). (11) G. Werimont, ANAL.CHEM., 39, 554 (1967). ANALYTICAL CHEMISTRY, VOL. 44, NO. 8, JULY 1972
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