Vertification of the Theoretical Prediction of the Resolution Optimization in Length-Temperature Time Normalization Chromatography Eli Grushkal and Francis 6. Lo Department of Chemisfry. State University of New York at Buffalo, Buffalo, N . Y . 1 4 2 7 4
In Length-Temperature Time Normalization Chromatography (LTTNC), both the column length and its temperature are changed concurrently in such a way as to maximize the resolution while keeping the analysis time constant. Until recently, the approach to this type of optimization was an empirical one. This paper investigates the validity of a theoretically sound predicting equation. This equation required experimentation on a reference column at two different temperatures at the carrier velocity around the van Deemter mirlimum. The equation is, indeed, a valid one and it predicts the maximizing length and temperature with good accuracy. The experiments were designed to check the equation in the case where the column length and temperature had to be increased and in the case where both these parameters had to be decreased. In both cases, the agreement between theory and experiment was good.
In addition to quantitation and identification, a common task in chromatography is the optimization of the resolution. After all, neither quantitation nor identification can be accomplished if the chromatographic peaks are not resolved. Much work has centered on improving the chromatographic resolution and two recent and excellent reviews have been devoted to this problem ( I , 2 ) . In general, however, an increase in the resolution means an increase in the analysis time. On the other hand, some workers have concentrated on the problem of minimum analysis time which still allows the separation of two components (uiz., 3-5 and references therein). Still another method of approach is time normalization chromatography (6-13), TNC. Time Normalization is a technique which allows the maximization of the resolution at a given analysis time; i . e . , constant analysis time. In this method, two opera'To whom ali inquiries should be directed
E. Heine. "Progress in Gas Chromatography," J . H. Purnell, Ed.. lnlersclence Publishers, New York. N.Y., 1968, p 135. R. P. W. Scott, "Advances in Chromatography-Vol. 9," J. C. Giddings and R. A. Keller. Ed., Marcel Dekker. New York, N.Y., 1970, p 193 S. J. Hawkes,J Chromatogr. Sci.. 7, 526 (1969). L . Rohrschneider, Chromatographfa. 3, 431 (1970). T. W. Smuts and V. Pretorius, Separ Sci . 6, 583 ( 1 9 7 1 ) . E. L. Karger and W. D. Cooke, Ana/ Chem.. 36, 985 (1964).
( 1 ) I . Halasz and
(2) (3) (4) (5)
(6) (7) /bid., p 9 9 1 .
(8) G. Guiochon, A n a / . Chem.. 38, 1020 (1966). (9) R. W. McCoy and S . P. Cram, Paper No. 147, Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Cleveland. Ohio. 1970. (101 E. Grushka and W. D. Cooke, d Chromatogr Sci , 9, 310 ( 1 9 7 1 ) . ( 1 1 ) E. Grushka. Anal. Chern.. 43, 766 (1971). (12) E. Grushka. M . Yepes-Baraya, and W. D. Cooke, J . Chromatogr. Sci.. 9, 653 ( 1 9 7 1 ) (13) E. Grushkaand G. Guiochon, J. Chromafogr. So.,10, 649 (1972).
tional parameters are changed simultaneously, one to affect the resolution and the other to keep the analysis time constant; for example, the column length and temperature. A typical experimental procedure in TNC is as follows. Install in the chromatograph a certain column and, a t a given set of conditions, check the resolution of the hardest-to-separate solutes. If the resolution is not sufficient, then chanke two parameters; e.g. decrease (or increase) column length and temperature according to a certain specified set of equations, and check the resolution again. This procedure is repeated until a maximum in the resolution is obtained. This, however, can be a tedious affair of trial and error. It is, perhaps, due to this limitation that the method is not as widely used as it should be. Recently, Grushka (11) proposed an equation which allows the prediction of the maximizing length in Length-Temperature TNC (LTTNC).The purpose of this paper is to check the usefulness and validity of that equation with data obtained, on packed columns, specifically for that purpose.
THEORY The basic equation of time normalization of any sort is given by the relation
where L is column length, U is carrier velocity, 12 is capacity ratio, and subscripts A and B identify two different sets of chromatographic conditions. The left-hand side of Equation 1 is the retention time, ~ R A ,on column A while the right-hand side is the retention time, t R B on column B. In LTTNC, the carrier velocity remains constant in all columns introduced to the chromatograph. Thus, as pointed out many times before, as column B is lengthened with respect to A, the temperature at which column B is operated must be higher than A's in order to decrease the value of kB, and thus keep Equation 1 a true equality. From this point on, we shall asssume that subscript A indicates the initial column and set of conditions at which the initial resolution was obtained. Subscript B will henceforth be dropped for the sake of generality. As indicated previously, increasing the length in the quest for better resolution, requires elevation of the temperature if the analysis time is t o remain constant. The required capacity ratio, at any length L , which will keep that time constant; Le., normalize the system, is simply
The temperature (in "K) required to give that k value can be obtained from the discussion of Karger and Cooke (6) ANALYTICAL CHEMISTRY, VOL. 45, NO. 6, M A Y 1973
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Table I. Butylbenzenes in LTTNC L (cm) t R (sec) U (cm/sec)
kexp
kcal
15.7 ... 10.1 9.94 7.34 7.20 5.68 5.56 4.56 4.51 3.17 3.10 2.34 2.28 a Temperature calculated from normalizing T of previous column. 100 150 200 250 300 400 500
157.7 157.1 157.6 157.7 157.6 157.8 157.7
10.6 10.5 10.6 10.7 10.6 10.5 10.6
Texp ( O K ) 393 404 429 432 441 461 478
rn
PT,,
(OK)
rCaI
...
408 418 429 442 456 475
408 41 6 424 437 452 465
H
cy
(OK)
...
(cm)
0.0778 0.0893 0.0809 0.0972 0.091 2 0.104 0.0996
1.13 1.11 1.11 1.10 1.09 1.09 1.08
RS
0.964 0.969 1.05 1.04 1 .oo 0.989 0.889
I------
R is the gas constant and AH is the heat of solution of the last solute; i . e . , the normalized one. Hence, for each length change (be it an increase or a decrease) the normalizing temperature can be easily obtained via Equations 2 and 3. The resolution equation between the hard to separate components is given by the well known expression
t ,5
I
3
2
4
5
L (m)
H is the plate height, a is the relative volatility, and 2 indicates quantities obtained from the second (normalized) peak. In general, H, a , and k are all temperature dependent. However, to a good approximation, when operating in the vicinity of the minimum of the van Deemter plot, the plate height is fairly insensitive to temperature variations. (This assumption might break down a t very low temperatures where H is limited by the mass transfer in the liquid phase.) Using this approximation, Grushka (11) has shown that the optimum capacity ratio is given by:
kopt is the optimum value of the capacity ratio giving the maximum resolution. The ramification of Equation 5 was discussed elsewhere (11). The constant a can be obtained from the expression suggested by Giddings (14) a is a constant to be defined shortly and
a - 1 -CT
- -a T
b
b is another constant and T is the absolute temperature. It should be noted that in Equation 5 , the quantity on the left-hand side can be obtained from data on the original column. The constants a and AH can be obtained from runs on column A at two different temperatures. Alternatively, AH can be approximated from Trouton's rule. Once the optimizing k value is known, Equations 2 and 3 allow the calculation of the column length and temperature which will yield that kopt and thus maximize the resolution. The trial error part of the normalization can hence be eliminated. The usefulness of Equation 5 was checked with some sketchy data available on capillary columns (6, 10) and the results were encouraging. We thus decided to check the Gtddtngs Dynamics of Chromatography, Part I Principles and Theory Marcel Dekker New York N Y 1965
(14) J C
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ANALYTICAL CHEMISTRY, VOL. 45, NO. 6, MAY 1973
Figure 1. Resolution vs. column length. Butylbenzenes mixture
validity of Equation 5 with two solutes systems in some experiments designed specifically for that purpose. The systems which we used were typical in that their a values tend to decrease with increasing temperatures.
EXPERIMENTAL Instrumentation. A Hewlett-Packard Model 700 gas chromatograph equipped with duel flame ionization detectors was used throughout this study. The columns used were %-in. 0.d. copper tubing of various lengths allowing us to obtain data on 1- t o 5 meter long columns. The columns were packed with 15% W/W Apiezon L on SO/lOO mesh AW-DMCS Chromosorb W and were conditioned overnight. An injector marker similar to that described before ( I O ) was used. Reagents. The test mixtures in these studies were ( a ) 1 cm3 of sec-butylbenzene and 1 cm3 tert-butylbenzene diluted to 25 cm3 with benzene. ( b ) n-amyl alcohol and iso-amyl alcohol diluted with reagent grade acetone. All the reagents were purchased from various vendors and were used without further purification. The carrier gas was He. Procedure. A Glenco 10-pl syringe was used to introduce the various solutes and mixtures to the chromatograph. The columns were installed in the chromatograph and their temperature was monitored with a copper-constantan thermocouple in conjunction with a potentiometer.
RESULTS AND DISCUSSION Butyl-Benzenes Mixture. In this study, our initial column was 1 meter long and the temperature was set (arbitrarily) at 120 " C . As indicated previously, one of our approximations entails working at the minimum of the H-U plot of the normalized component, in this case sec-butylbenzene. The minimum H = 0.0778 cm occurred at a carrier velocity of 10.58 cm/sec. A t that velocity, the retention time of the sec-butylbenzene was 155.7 sec. In order to compute the optimum k , the constants a and AH are needed. To obtain a, the mixture was run at 120 and 130 "C. The a values were 1.13 and 1.12, respec-
Table 1 1 . Amyl Alcohols in LTTNC L (cm)
t R (set)
kexu
kC,l
100 150 200 300 400 450
135.0 132.4 132.0 132.6 132.2 131.6 132.2
9.39 5.79 4.08 2.40 1.54 1.25 1.04
9.19
:
500
rexu
rCaI
333 353
357
5.80 4.10 2.40 1.55 1.271
...
(OK)
362 392 408 41 5 431
(OK)
361 381 399 41 5 423
...
CY
1.35 1.32 1.29 1.24 1.22 1.21 1.17
H (cm)
R,
0.160
1.52 1.61 1.60 1.58 1.52 1.36 1.26
0.159 0.1 71 0.150 0.140 0.0704 0.114
460
45Ot-
T(@
1
440c I
Figure 3. Resolution vs. column length. Amyl alcohols mixture
4 3 0 r I
t
42.01
L-
41
4001
3 90 I
2
3
5
Figure 2. Comparison between t h e predicted normalizing temperature at each column length with the experimental values 0 - experimental 0 - theoretical
tively (incidentally, the resolutions a t these two temperatures were 0.964 and 0.860, respectively). Using Equation 6 a t these temperatures, we then have two equations with two unknowns; namely, a and 6 . The constant a was found to be 134 OK. AH was estimated, using Trouton's rule as -9.82 kcal/mole. Equation 5 then becomes
- 9.82 1.13 (1.987 X 10-3)(134) 1.13
(
'>
-I-
Graphical solution shows that hopt is 6.44. This indicates that the optimum length is 224 cm operated a t 423 OK (150 "C). In obtaining these values, Equations 2 and 3 had been used. Figure 1 shows the resolution R, us. various column lengths under time normalization condition. The maximum resolution occurs a t about 200 cm a t temperatures of about 156 "C. Indeed, a very good agreement with the predicted values. All the data relevant to this system are shown in Table I. Subscript "cal" indicates calculated pa-
rameters while "exp" means experimentally obtained quantities. Table I indicates that the agreement between the calculated and experimental k values which will keep the analysis time constant is rather good. The agreement in the temperature that normalized the analysis was slightly higher than predicted. See also Figure 2. One reason for that behavior might lie in the error made in estimating AH. When we calculated the normalizing T by using the temperature of the previous column as T A rather than 393 "C ( a t 1-meter length), the agreement is much better. Another point of interest is the relatively small deviation in the plate height of the various columns at various temperatures. This supports our contention that when working a t carrier velocities around the minimum of the van Deemter plot, the assumption of constant H is not a serious one. Also, Table I seems to indicate that cy is not very temperature dependent, e.g., the values a t 441 and 461 "K. In actuality, however, we could experimentally tell that the a values a t these two temperatures were different; i.e., 1.090 and 1.087, respectively. We felt on the other hand that the precision involved in this work merits only three significant figures. The relative flatness of the R, us. L plot should be noted. This is again a result of the relatively independence of cy from variation in T . All the above discussion was based on quantities derived from the l-meter column. To check the validity of Equation 5, we utilize the data obtained on the 5- and 4meter columns to calculate the constant a which in return would enable us to get the optimum length. Pursuing these calculations indicated that the optimizing length should have been about 280 cm. The agreement with the experimentation here, although not too bad, is inferior when compared with the prediction based on the 1-meter column. The main reason for that fact is that here we used two different columns to calculate a. In addition, the approximation given in Equation 6 is not expected to hold over a large range of temperatures. None the less, Figure 1 shows that working a t 280 cm gives resolutions which are not much below the maximum (about 5%) and that, a t ANALYTICAL CHEMISTRY, VOL. 45, NO. 6, M A Y 1973
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worst, a "ball park" approximation of the optimizing length is indeed obtained from Equation 5 . Amyl Alcohols. In order to further investigate the utility of Equation 5, we took the system of n-pentanol and 3methyl-butanol. We further decided to start with the 5 meter column as the original system. We chose the 5meter column as the standard since we suspected that the optimum length would be less than 5 meters. This was in contrast to the butylbenzenes system where we suspected the optimum length to be longer than the 1-meter column we used as the standard. In other words, our aim was to see whether the approach we chose; namely, Equation 5 , is valid in both possible cases, one of which predicts a longer optimizing column, and the other a shorter one. Here, the normalized solute was n-pentanol. The He velocity 7.67 cm/sec and the retention time of the amyl alcohol was 132.2 sec a t 158.3 "C. AH was estimated as -9.02 kcal/mole. By running the 5-meter column at 168.3 and 158.3 "C, a was found to be 319 "K. The optimum k is thus 5.47 and the maximizing L is 158 cm operated a t 372 O K (99 "C). The experimental results are shown in Figure 3 and Table 11. Figure 3 demonstrates that the maximum resolution does, indeed, occur around 158 cm. The agreement between Equation 5 and the experimental data is very good. Table I1 shows that here H varied a little more than in the case for butylbenzenes. Still it does not seem to affect the validity of the proposed approach. This is due, of course, to the square root relation between the resolution and plate height. Except for the 1-meter column, the agreement between the calculated values of k and T and the experimental ones was good. The 1-meter column was run a t a slightly lower temperature than that required to keep the analysis time of n-pentanol constant. If run a t the corrected temperature, a and k would have been lower than obtained, thus, reducing the resolution to even a greater extent ( R ,
