Anal. Chem. 1999, 65, 2643-2849
2649
New Parameters for the Characterization of Relationship between Gas Chromatographic Retention and Temperature M. de Frutos, J. Sanz,' I. Martinez-Castro,and M. I. Jim6nez Instituto de Qulmica Orgbnica General (CSIC), Juan de la Cierva, 3,28006 Madrid, Spain
Estimation of the gas chromatographic retention at different temperatures can be carried out by calculation of the parameters relating capacity ratio (k)and temperature; these parameters can also be used in the optimization of separations. However, their use in the characterization of gas chromatographic retention requires precise measures, since they are very sensitive to experimental errors. We propose in this study the use of the parameters Tik (defined as the isothermal column temperature at which a compound i has a capacity ratio k)and of other parameters derived from them by principal component analysis, in order to characterize, in several approximation levels, the relationship between k and Tfor any compound/ column pair, and as a starting point for the calculation of gas chromatographicretention and the optimization of separations.
dynamicmagnitudes of interest. A first objective in the second approach is the calculation of retention times or retention indexes under different conditions, including temperature programming. In both cases, the parameters that express the relationshipbetweenk and T could be used to characterize, in a quantitative way, the retention of a compound in a gas chromatographic system. However, their values are very sensitive to experimental errors, and very precise measures are necessary for their calculation. The objective of the present study is the development of new parameters for the empiric characterization of the dependence between k and T . We propose the parameters Tik, defined as the isothermal column temperature at which a compound i has a capacity ratio k in a given column, and other parameters derived from them by principal component analysis. These parameters allow the calculation of gas chromatographic retentions and also the optimization of separations.
THEORY INTRODUCT10N The dependence between capacity ratio (k)and temperature has been used by several authors in order to determine the thermodynamic properties of a gas chromatographic ~ystem.l-~Also, a number of workers interested in the practical aspects of gas chromatography have used this dependence in the calculation of gas chromatographic retention behavioP13 and in the optimization of gas chromatographic separations.14-20 The first approach requires of an accurate calculation method to compute, from experimental data, the thermo(1) Podmaniczky, L.; Szepesy, L.; Lakszner, K.; Schomburg, G.
Chrornatographia 1986,21,91-94.
(2) Bincheng, L.; Binchang, L.; Koppenhoefer, B. Anal. Chem. 1988, 60,2135-2137. (3) Guan, Y.; Kiraly, J.;Rijks, J. A. J.Chrornatogr. 1989,472,129-143. (4) Dose,E. V. Anal. Chern. 1987,59,2414-2419.
(5)Podmaniaky, L.; Szepesy, L.; Lakezner, K.; Schomburg, G.
The basic equation for the retention time ( t r ) is
tr = to(l + k)
(1) In open tubular columns, dead time (to) can be calculated from column geometry (length L and radius r ) and inlet (Pi) and outlet (Po)pressures by using
where j, the mobile-phase compressibilityfactor, depends on Po and Pi. Carrier gas viscosity (7) is related to temperature T and must be taken from tabulated values. For a given column, if Pi and Po are constant, eq 2 can be rewritten as t o = (c0nstant)q. Ettre21proposedrelating carrier gas viscosity and temperature by q = A(T1273)B. Equation 2 then becomes
Chrornatographia 1985,20,591-595.
(6) Castells, R. C.; Arancibia, E. L.; Nardillo, A. M. J. Chromatogr. 1990,504,4543. (7) Comor, J. J.; Kopecni, M. M. Anal. Chem. 1990,62,991-994. (8) Shrotri, P. Y.; Mckashi, A.; Mukesh, D.J. Chromatogr. 1987,387, 39-03, (9) Bautz, D.E.; Dolan,J. W.; Snyder, L. R. J. Chrornatogr. 1991,541, 1-19. (10) D o h , J. W.; Snyder,L. R.; Bautz, D. E. J. Chromatogr. 1991,541, 21-34. (11) Snyder,L.R.;Bautz,D.E.;Dolan, J. W. J.Chromatogr. 1991,541, 35-58. (12) Wright, L. H.; Walling, J. F. J. Chromatogr. 1991,540,311-322. (13) Bautz, D.E.; D o h , J. W.; Raddatz, W. D.; Snyder, L. R. Anal. Chem. 1990,62,1560-1567. (14) BBrtu, V. J. Chromatogr. 1983,260,255-264. (15) BBrtu, V.; Wicar, S. Anal. Chim. Acta 1983, 150, 245-252. (16) Sam, J.; Martlnez-Castro, I.; Reglero, G.; Cabezudo,M. D. Anal. Chtm. Acta 1987,194,91-98. (17) Lanin, S. N.; Lyako, 0. V.; Nikitin, Y. S. Chrornatographia 1988, 26, 63-69.
(18)Peichang, L.; Bingcheng,L.; Xinhua, C.; Chunrong, L.; Guangda, L.; Haochun, L. HRC & CC, J. High Resolut. Chromatogr. Chromatogr. Commun. 1986,9,702-707.
(19) Akporhonor, E. E.; Le Vent, S.; Taylor, D. R. J. Chrornatogr. 1987,405,67-76. 0003-2700/93/0385-2643$04.00/0
From experimental to values measured at a given pressure and different temperatures, At and Bt can be calculated by regression and then used to calculate dead time at any temperature. The influence of stationary phase on the retention is expressed in eq 1 by the capacity ratio (k). The partition coefficient K is related to k through the phase ratio 8:
k =KIP
(4)
For a given solute in a givenstationary phase K only depends on the temperature according to eq 5, where R is the gas
In K = -AGo/RT
(5)
constant and AGO the change in the Gibbs free energy for the (20) Akporhonor, E. E.; Le Vent, S.; Taylor, D.R. J. Chromatogr. 1990,504,269-278. (21) Ettre, L. S. Chrornatographia 1984,19, 243.
0 1883 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 65, NO. 19, OCTOBER 1, 1993
Table I. Capillary Columns Used in the Experimental Determinations stationary length i.d. df supplier column phase (m) (mm) (pm) A B C D
ov-1 ov-1 SP-loo0 ov-1
20 22 25 22
0.2 0.22 0.22 0.22
0.5 0.14 0.25 0.14
Table 111. Equations Used for Correlating k and T Values. mean square residuals* (E(kerp - k d 2 / n code equation A B C D
homemade homemade Perkin-Elmer homemade
El E2 E3 E4 E5 E6
Table 11. Compounds Used in the Determination of k Values at Several Temperatures in Columns A-D and Temperature Ranges for Each Compound Class. column
compounds
carbon no.
A (OV-1) B (OV-1)
n-alkanes methyl esters n-acids 2-ketones ethyl esters n-alkanes C (SP-1000) methyl esters n-acids 2-ketones ethyl esters n-alkanes D (OV-1) n-tetradecane
(9-19) (5-12)
(6-12,14,16,18) (8-13,15,17,19) (6-12,14,16) (10-20,22,24) (5-11) (5-8) (8-13) (4-12)
(14-20)
temp
range ("C) 10C-240 60-200 75-250 80-225 75-250 75-270 80-180 120-180 80-180 80-180 80-180 80-250
" Numbers in parentheses indicate carbon chain length. evaporation of the solute from the stationary phase. Then, for a given compound and phase, capacity ratio depends only on the phase ratio (eq 4) and on the temperature (eq 5, where A G O also depends on the temperature). The temperature of gas chromatographic separations is usually selected in order to obtain suitable k values; frequently a temperature program is used when the volatility of sample components covers a wide range. In the temperature-programmed mode, the speed of the chromatographic band changes continuously, since both gas velocity and capacity ratio depend on the temperature.** Retention time can be calculated16 by a numerical procedure, supposing that band movement is decomposed in steps corresponding to small time intervals:
2 t=O
t,(l
t
+ k,)
=1
where kt and are the values of capacity ratio and dead time in the isothermal mode at the mean temperature in the time interval t.
EXPERIMENTAL SECTION Retention times were determined, using the capillary columns listed in Table I, in a Hewlett-Packard 5890 gas chromatograph equipped with a flame ionization detector and a split/splitless injector. Nitrogen was used as carrier gas. Inlet pressure was usually selected in order to obtain a flow rate near the Van Deemter optimum for each column; different flow rates were selected in some determinations. Chromatograms were recorded in a Spectra Physics SP-4270 integrator. Table I1 lists the compounds whose k values were determined at four to six different temperatures in the range 80-240 "C in columns A-C. Column D was used in the determination of k values for n-tetradecane at 11 different temperatures between 80 and 250 "C. The compounds are members of several homologous series; their carbon chain lengths are listed in parentheses. Oven temperatures were chosen for each compound in order t o obtain k values in the usual range for gas chroma(22) Harris, W. E.; Habgood, H. W. Programmed Temperature Gas Chromatography;John Wiley & Sons: New York, 1966.
k = piTeP*/T k k k k k
=
= p1Pn = p1@2
+ T)Pa
= p l e P ~+/ ~p3 =p1TP2eP31T
0.026 0.028 0.046 0.022 0.031 0.033
0.039 0.041 0.082 0.019 0.023 0.022
0.048 0.054 0.136 0.013 0.013 0.029
0.104 0.112 0.229 0,019 0.090
0.034
Other Equations Showing Less Quality of Fit k k k k
k = p&'sT k = @ I + T)/@2T+ P S ) = p p z T + p3 k = p i p + P Z T + p3 k = @ l + T)/@2P+ P S ) = l / @ i T + p2) k = 1/@1 log T + p2) k = p i + p z / T = l/@iT+ p2) k = p i + pzT
= p l / r P l + p3
"Quality of fit (mean square residual) is given for the best equations. nt, total number of k pairs in each data set. 6 For data sets A-D. tography (between 1and 20): k was calculated as the mean value from several measures. Dead time was always determined by the injection of methane; dependence of dead time with T was characterized for each column by the At and Bt values obtained from experimental data at different temperatures using eq 3. Calculations were carried out in an M-20 Olivetti and in an Amstrad PC 1640 microcomputers. Programs were written in Basic and QuickBasic.
RESULTS AND DISCUSSION k-T Dependence through pr Parameters. The dependence of k on the temperature for a given compound i and a column c can be written (7) k,& = f@1, ..*,P,, **', T ) where the value of the parameters p , depends on compound and stationary phase, but also on the column phase ratio. The different retention behavior of the compounds should be reflected in their different p , values. In a first approximation, it could be supposed that only one parameter p , would be enough to describe the variation of k with T. This would mean that compounds having the same k a t a given T would also coelute a t other temperatures. Although this can be true for compounds having a similar chemical structure, many cases are known of different retention behavior and even of elution order inversion in the elution of pairs of components when the column temperature is ~ h a n g e d Then, . ~ ~ ~a t ~least ~ ~two parameters seem to be necessary for characterizing the capacity ratio of a compound in a column. In order to describe the variation of k with T, we have empirically selected 17 different equations based in eq 7, using two or three parametersp, (see Table 111). Equations E l , E2, and E5 had been previously used.4JJ2JP16J9 Since eq E2 can be deduced from eqs 4 and 5 (see Theory section), its p r pararneters5J2Jg have a physical significance. The k-T data pairs for columns A-D (data seta A-D in Table 11)were submitted to nonlinear regression calculations% in order to determine the p , parameters and the quality of fit for the 17 equations. Table I11 lists the average quality (23) Saha, N. C.; Mitra, G. D. J. Chromatogr. Sci. 1970, 8, 84-90. (24) Saxton, W. L. J . Chromatogr. 1986, 357, 1-10, (25) Pell, R. J.; Gearhart, H. L. HRC & CC, J . High Resolut. Chromatogr. Chromatogr. Commun. 1987, 10, 388-391. (26) Draper, N.R.;Smith,H. AppliedRegression Anulysis;John Wiley & Sons: New York, 1980; Chapter 10.
ANALYTICAL CHEMISTRY, VOL. 65, NO. 19, OCTOBER 1, 1993
2646
1- -5
I
I-
-0.22
-I
-0.24-
I/
t
22
I
-0.26-
1164.1
-101.4
I ... ...I
I
.
11
I
i4
115.0-
-1
(-100.0 ,
156
,
,
157
,
,
150
I
159
1GO
B TIz Flgure 2. (A) p, values In eq E4 calculated for 60 k-T data pairs obtained by random modtficationof data set D. p2 (0)and ps (0)are plotted against pl. (8) TRvalues calculated from the same k-Tdata pairs. Tw(0)and TB (0)are plotted aglnst TQ. I
0
7--
10
20
30 rnin
Flgure 1. Chromatograms of a standard mixtureof compounds usually present In cheese volatlle fractions. Upper plot: calculated chre matogram. Lower plot: real chromatogram. Column characteristics: 21 m X 0.25 mm Ld., stationary phase 43% FFAP/57% OV-1. Peak idenmication: 1, methyl hexanoate; 2, ethyl hexanoate; 3, methyl heptanoate; 4, ethylheptanoate; S, 2-nonenone; 6, methyl octanoate; 7, ethyl octanoate; 6, 2decanone; 0, methyl nonanoate; 10, ethyl nonanoate; 11, 2-undecanone; 12, methyl decanoate; 13, hexanolc add; 14, ethyldecanoate; 15,2-dodecenone; 16, methyl undecanoate; 17, heptanoic acid; 16, ethyl undecanoate; 19, 2-tridecanone; 20, methyl dodecanoate; 21, octanoic acid; 22, ethyl dodecanoate; 23, nonanolc a c e 24,2-pentadecanone; 25, decanoic acM. See ref 27 for details.
of fit for the three "best" two-parameterand three-parameter equations, expressed as the mean square residual: where k,, is the experimental k value, k d c is calculated by nonlinear least squares regression, and the mean is extended to the total number ( n t )of k-T determinations in each data set. A study of the residuals ( k , - k d c ) shows the presence of systematicdeviationsin small and high k values. The presence of this kind of errors indicates that great care should be taken when these equations are used in extrapolation.
When a data set covering a wider k range (n-tetradecane at 11 different temperatures in column D) was used, the same kind of error appeared. The use of the best three-parameter equation (eq E4 in Table 111) reduces the error. However, the quality of fit of the two-parameter eqs El and E2 is enough for several practical purposes; for instance, they can be used in the optimization of chromatographicseparations. Optimization of the Resolution. Determination of the resolution requires the calculation of both t, and peak width (w)values. According to eq 1 (isotherm operation) or eq 6 (programmed temperature operation), t, can be calculated from to (related to T through eq 3) and k . Peak width mainly dependson flow rate and k ; the influence of column efficiency in w presents an additional problem. Data sets Band C were used to design a mixed-phase column optimized for the separation of 24 volatile constituents of chee8e.n Equation El was chosen for the determination of p , parameters. Optimizationwas carried out by calculation of retention time and peak width for different column composition (OV-l/FTAF' ratio and film thickness df) and operating conditions (initial temperature and programming rate); the worst resolution between pairs of components was taken as the resolution of the mixture, and the variables maximizing this value were selected. The highest resolution was obtained for a column (2'7) Sam,J.;deFrutm,M.;Martlnez-Caetro, I. Chromutographiu1992,
33,213-217.
2646
ANALYTICAL CHEMISTRY, VOL. 65,
l
NO. 19, OCTOBER 1, 1993
I-
o @
@
8-/
1-0.2G 7-
O
t
Gi
1-0.28 0 .
0
0
1-0.3
0 0 O
31
0
10
12
Q
O
4
0
__,
-,
.,-I
-- 1-t-
16
14
A
10
t
0
20
22
-0.32
24
IlC
I 0
1-225
0
220-
-200
Y 200-
0
-175
6)
180-
0
Ti4
-150
0 Ti2,GO{
0
:I- 1 125
100
100-
75
10
12 # 1 a4
1G
10- -20 ,-_,- 22
24
B riC Fb@m3. (A) A v a k s in eq E4 calculated from the n-akanesIn data set B. p1(0)and h ( 0 )are plotted agalnst carbon number. (B) Tk values calculated from the 881118 data. Tn (0)and TM(0)are plotted
against carbon number.
having 43% FFAP and 57% OV-1,using 80 O C as initial temperature and a program rate of 4 O C I m i n . More details are given elsewhere." Figure 1 shows (lower plot) the real chromatogram of a mixture of standards in the mixed-phase column, preparedaccording the above results and (upperplot) the calculated chromatogram for the optimal separation. Characterization of the Retention Behavior. The use of both two- and three-parameter equations presents a problem, which appears in nonlinear estimation in some illconditioned cases:% the parameters are highly correlated and the error of prvalues can become very high, especially when there are few data points or the experimental error in their determination is high. The error also depends on the least squares criterion used in the fit (i.e., fitting by nonlinear regression or by linear regression after a logarithmic transformation, in eq E2)or on the weights that can be associated with data in the calculations. Aa an example, 60 sets of pr coefficients were calculated after adding a random error to the experimental k values in column D data set in order to obtain a 1 % variation coefficient in the simulated k values. Variation coefficients obtained for p1 and pa values in eq El were 17.8% and 0.996, respectively. The problem is worse in the three-parameter equations. p1, p2, and pa parameters in eq E4 were calculated from the
Table IV. Tu Parametere for the Homologous Sedea Compounde in Data Set B compd T* ("C) fork codesa 0.5 1 2 4 6 8 M6 M6 M7 M8 M10 M11 M12 A6 A7 A8 A9 A10 All A12 A14 A16 A18 K8 K9 K10 K11 K12 K13 K15 K17 K19 E6 E7 E8 E9 E10 Ell E12 E14 E16 c10
c11 c12 C13 C14 C15 C16 C17 C18 c19 c20 c22 C24
82.78 103.66 122.74 140.09 172.81 187.06 200.13 111.47 129.44 146.06 161.88 176.31 190.73 204.20 228.58 251.18 271.16 116.96 135.11 152.10 167.77 183.31 196.92 222.61 245.81 266.80 118.09 135.86 162.13 167.64 181.93 196.34 209.30 234.00 255.77 121.96 140.10 156.92 170.60 184.96 199.41 211.87 224.30 236.64 247.13 257.77 276.31 294.82
63.87 83.62 102.12 119.27 150.39 164.64 177.67 93.66 110.29 125.82 141.46 166.49 168.63 181.50 205.48 227.17 247.04 96.05 113.68 130.34 145.84 160.64 174.17 199.29 221.94 242.60 97.43 114.83 130.92 146.98 160.09 173.66 186.35 210.15 231.50 100.74 118.26 134.70 149.01 163.08 176.67 189.09 200.93 212.31 223.12 233.63 252.43 270.21
46.02 66.09 83.98 100.73 130.67 144.54 157.66 78.87 94.03 108.42 123.42 136.99 149.33 161.68 186.00 206.01 226.56 77.63 94.93 111.12 126.30 140.39 163.87 178.44 200.68 220.73 79.66 96.35 112.08 126.75 140.58 153.49 165.97 189.03 209.95 82.48 99.27 115.17 129.73 143.50 156.49 168.73 180.22 191.49 201.86 212.0 230.90 248.17
29.17 51.01 68.02 84.24 113.33 126.79 139.67 66.60 80.23 93.46 107.61 120.65 132.71 144.37 166.84 187.36 206.41 61.40 78.60 94.12 108.87 122.67 136.74 169.78 181.47 201.24 64-10 80.11 95.35 109.67 123.17 135.81 147.89 170.32 190.81 66.77 82.77 98.01 112.61 125.96 138.58 150.52 161.86 172.84 183.01 192.88 211.50 228.42
23.94 46.60 63.30 79.32 108.20 121.61 134.19 63.12 76.26 89.11 102.79 115.66 127.86 139.28 161.44 181.82 200.71 66.60 73.67 89.08 103.68 117.28 130.33 154.20 175.76 196.41 59.58 75.31 90.37 104.69 117.97 130.68 142.51 164.76 186.11 62.20 77.93 92.94 107.37 120.70 133.26 146.08 156.41 167.26 177.42 187.19 206.67 222.62
13.26 37.96 63.97 69.66 98.08 111.03 123.46 66.43 68.62 80.80 93.47 105.94 118.34 129.26 150.73 170.89 189.36 47.10 64.13 79.10 93.34 106.81 119.56 143.09 164.36 183.78 60.73 66.84 80.49 94.61 107.61 120.23 131.83 153.76 173.81 63.26 68.42 82.93 97.14 110.24 122.70 134.26 146.59 166.12 166.33 176.89 194.01 210.74
10 8.31 34.14 49.82 66.19 93.67 106.34 118.64 63.64 66.14 76.86 89.31 101.60 114.13 124.81 146.94 166.02 184.28 42.87 69.89 74.64 88.71 102.13 114.72 138.10 169.26 178.66 46.82 61.62 76.07 89.99 102.97 116.62 127.05 148.83 168.76 49.33 64.21 78.47 92.66 106.64 117.97 129.38 140.76 161.12 161.37 170.84 188.76 206.44
I, M, methyl esters; A, n-acids, K, 2-ketones; E, ethyl estem; C, mallranee. Numbers in codes indicate carbon chain lerigth.
same simulated data; their variation coefficients are 59.4% , 5.4% ,and 9.9%, respectively. Figure 2A plots the PI,pa, and p3calculated values; it can be seen that the changes in their values caused by the added error are related. These pr coefficients can be used without problem to calculate k values at different temperatures. However, their use in the characterization of chromatographicbehavior or the interpretation of their possible physical significance is very dangerous unless special care in the experimental determinations and calculations is taken. ABan example, p1 and p2 values in eq E4 calculated for the n-alkanesin column C are plotted against carbon number in Figure 3A. The expected regular behavior ismasked by the high relative error. Parameters T*. An ideal set of parameters for the characterization of chromatographicretention behavior should be easily calculated from experimentaldata for a compound and column, should show small variation with experimental error, and should also easily allow the calculation of k at any temperature.
ANALYTICAL CHEMISTRY, VOL. 65, NO. 19, OCTOBER 1, 1993
We propose to characterize the retention of a compound
i in a given column from ita parameters Tjk, defined as the isothermal column temperature at which the compound has a capacity ratio k. When an equation from those listed in Table I11 can be written as T = f ( k , p r )Tik , can be easily calculated from k and pr parameters. In the other cases, they can be obtained iteratively by the Newton-Raphson method.% Since many values of k can be chosen, any number of Tik parameters can be used in the characterization; it seems reasonable to choose k values in the usual chromatographic range. Table IV shows Tik values for the compounds in data set B for k = 0.5,1,2,4,5,8,and 10;values for other k can be easily calculated. When the simulated k values with random error are used as experimental data, the variation coefficient for calculated Tjk values range only from 0.9% to 0.1% ( k between 0.5 and 10). Figure 2B shows a plot of TUand Ti6 against Tjz. While in Figure 2A the errors in parameters p1 and p2 are systematicallyrelatad, no trends are observed in the simulated errors in Figure 2B. In Figure 3B,Ti2 and Ti4 calculated for the n-alkanes in data set B are plotted against carbon number. The regular behavior (compare with Figure 3A) is expected for a homologous series. k values at any temperature can be calculated from Tik parameters, which are used as starting data for a nonlinear regression. In a first step the prcoefficients of a given k-T equation are obtained, then the equation can be used to calculate k values at any temperature. Although in the example of Table IV each compound is characterized by seven values, it can be seen that there is a high correlation among the values corresponding to the different columns of the table. For instance, if we take Ti2 ( k = 2) as the independent variable, the other Tjk in Table IV can be calculated using a single linear regression (Tik = aTi2 b) with a good quality of fit (r between 0.9991 and 0.9999). That means that most of the information contained in them is redundant: the number of Tjk values used in the characterization could then be reduced. A way to do it is principal component analysis. New Parameters Calculated by Principal Component Analysis. Principal component analysis29 can be used to study the structure of a data matrix and the correlations among its variables. The principal component model decomposesa data matrix into the product of a row matrix and a column matrix. Row (rill and column (clj) matrix values are calculatedin order to reproduce the data (dij)with the highest possible accuracy for each value of n (components):
+
n
dij = FriFlj
(9)
2647
Table V. Results of Principal Component Analysis of Data Set B. clj coeff (k 03-10) 1
0.5
1
2
4
5
8
10
1
0.49 -0.61 -0.52
0.44 -0.30 0.27
0.39 -0.03 0.51
0.35 0.21 0.29
0.33 0.28 0.16
0.31 0.42 -0.24
0.30
2 3
1
coeff ( 1 = 1-3) 2
111.29 168.39 213.94 256.84 335.15 370.66 404.04 204.01 242.31 278.97 316.53 351.37 384.81 416.31 475.95 530.83 581.00 197.26 242.57 284.07 323.03 360.02 394.51 458.21 515.70 568.00 203.39 246.14 286.42 324.52 360.15 394.40 426.37 486.34 540.53 211.21 254.22 294.88 331.94 367.53 401.75 433.12 463.47 492.11 519.69 545.87 594.15 639.36
-48.24 -33.43 -28.42 -22.58 -12.53 -7.56 -2.30 -16.40 -15.41 -13.35 -9.77 -5.51 -1.21 1.85 9.52 17.03 24.41 -32.47 -25.40 -20.04 -14.56 -9.87 -4.65 4.61 13.02 20.83 -28.77 -23.87 -18.35 -12.99 -7.83 -3.31 1.00 9.06 17.05 -28.61 -23.91 -19.00 -12.04 -6.93 -2.65 1.98 6.46 10.76 14.48 18.09 26.12 32.55
ril
M5 M6 M7 M8 M10 M11 M12 A6 A7 A8 A9 A10 A11 A12 A14 A16 A18 K8 K9 K10 K11 K12 K13 K15 K17 K19 E6 E7 E8 E9 E10 Ell E12 E14 E16 c10
c11 c12 C13 C14 C15 C16 C17 C18 c19 c20 c22 C24
0.48
-0.48
3 2.98 -0.57 -0.03 0.37 -0.22 0.14 0.45 -1.88 -1.33 -1.00 -0.09
0.15 -0.97 -0.64 -0.06 -0.39 0.01 0.06 -0.26 0.07 0.38 0.01 0.36 0.37 0.32 0.30 -0.65 -0.11 0.25 0.20 0.39 -0.22 0.06 -0.11 -0.07 -0.89 -0.57 -0.18 0.42 0.51 0.07 0.37 -0.06 0.40 -0.16 -0.07 0.55 0.21
=1
A set of values (eigenvalues)which measure the amount of variance in the original data explained by each component is also calcu1ated.m A study by principal component analysis of the Tik values calculated from data set B (Table IV) shows the results summarized in Table V. A 99.80% of total variance is associated with the Fist component, while the second component explains 0.18%. ril and clj coefficients are also shown in the table. The mean error in the recalculation of (28) Press, W. H.; Flannery, B.9.;Teykoleky,S. D.; Vetterling,W. T. NumencalRecrpes;CambridgeUniversityPress: Cambridge,UK, 1986; Chapter 9. (29) Malinoweki, E. R.; Howery, D. G. Factor Analysis in Chemistry; John Wdey & Sons: New York, 1980. (30) Wolff,D. D.;Pareom, M. L. Pattern Recognition Approach to Data Identification; Plenum Press: New York, 1983.
a I , number of components (1-3); i, compounds (code names as in Table IV); j , k values.
Tik values is 4.68,0.15,and 0.004 "C when, respectively, one, two, or three componentsare used; the rest of the components do not have practical significance. When the same calculations are carried out using the experimental data set from column C, 99.77 % and 0.22% of variance are associated respectively with first and second components and 5.15,0.15,and 0.003"C are the mean errors in the recalculationof Tik with one,two, and three components. Figure 4 plots c ~ (Ij = 1,2,3)against k for data sets B and C; clj values are surprisingly similar for the two data sets. Since in both cases components higher than 3 have no relevance, three approximation levels using one, two, and
ANALYTICAL CHEMISTRY, VOL. 65, NO. 19, OCTOBER 1, 1993
2848
Scheme I
0.Gy
R E A L COLUMN
STANDARD C O L U M N
SELECTION OF I( -T EQUATION
0.40.2 -
-0.2-
-0.41
I/
\
'0
-0.Gl
-0
8
0
1
2
~
1
4
~
G
I
~
U
I
~I I
10
12
K
B Flgurr 4. Column coefficients q obtained by principal component analysis of the Tk matrix: (A) data set B; (B) data set C. (0)first component: (0)second component; (0)third component. q values are plotted against k.
three components can be defined in order to describe the data (n= 1,2,or 3 in eq 9). Column coefficientscljare related to the k values used to calculate the matrix, while row coefficients ril are characteristic of the compounds. In the first level, Tik can be calculated as the product rilclj. Row coefficient ril is highly correlated with Tik (r = 0.99940.9999)and has the meaning of an "average" retention. Since each compound is characterized by a single number, the elution order for a series of compounds would be the same regardless of temperature. The second approximation level uses the second component to take into account the different variation of retention with temperature for different compounds. Since c~ coefficients are negative for low k values and positive for high k values, compounds with ri2 positive will be more retained that an average compound at low temperatures and less at high temperatures. For instance, r;2 for ethyl hexadecanoate in column C is 17.05. While the first approximation (first component) indicates that it would need a temperature of 210.43 "C to elute with k = 2, and 160.49 "C for k = 10,the corrected temperatures using the second component coefficients are 210.00 and 168.71 "C, respectively. The third level introduces a further approximation, described by the values csj. Calculated Ti2 and Tilo for ethyl hexadecanoate, corrected using the third component, are 209.96 and 168.74 "C (Ti2 and Tilo values in Table IV are 209.95 and 168.76 "C). The existence of three levels agrees with the results of Hawkes31 about the relation between retention index I and (31) Hawkes, S.J. AMI. Chem. 1989,61, 88-90,
T. The use of our first approximationlevel supposes that I is constant; the second level is approximatelyrelated to the linear dependencebetween Iand T while the third level could explain the small deviations from linearity. Characterization Scheme. In the proposed scheme, the characterization of retention includes parameters related to column and compound behavior. The calculation of these parameters from experimental k-T data for several compounds (including a set of "standard" compounds,in our case, n-alkanes) is summarized in the left side of Scheme I and can be carried out in the following steps: (1)A suitable equation based on eq 7 relating k and T is selected (eq E4 seems to be advisable when data at four temperatures at least are available). (2)The prparameters in the equation are calculated for each compound using nonlinear regression. (3)Several k values, in the same range that those experimentally obtained, are selected. T&are calculated for each compound and k value. (4)Principal componentanalysis is performed over the T* data matrix. Row and column matrices as in Table V are obtained. While the retention behavior of the chromatographic column is characterized by the matrix column values Clj, compounds are characterized by their row values ril. If T& values of new compounds are available, their ro values can be calculated by regression from T;k and Clj data. Calculation of k Values. Capacity ratio values at any temperature for a given column and compound can be easily calculated from rg and Clj values (Scheme I). In a f i i t step, the number of parameters (1 = 1,2,or 3)is selected according to the necessary precision level. T&matrix is then calculated by the Newton-Raphson method. A suitable equation (i.e., eq El or eq E4 in the Table 111) relating k and T is then selected and its parameters (pr)are calculated by regression from Tik data. The equation can be used in the calculation of k at any T . Extension to Stationary-Phase Characterization. When the equations based on eq 7 are written ki,c= p J ( ...,pr, ...,r )
(10)
from eq 4 we have where @ is the phase ratio. In this way parameters independent from column geometry can be calculated, according to the equation selected, they can have thermodynamic significancefrom their relationship with partition coefficient K. Although these parameters could be used in the char-
ANALYTICAL CHEMISTRY, VOL. 65, NO. 19, OCTOBER 1, 1993
acterization of the stationary-phase behavior, they would present the same problems previously mentioned. A more practical approach could be the use of a hypothetical “standard” chromatographic column. For instance, such column can be defined as having 0.25-mm i.d. and 0.25-pm df ( A 250) (12) k , , = hi,, ( P i / & ) = P1(8i/25O)f(-., pr, r ) where ki and pi are capacityratio and phase ratio in any column and k, is the capacity ratio in the standard column. pr parameters in any column can be easily referred to the standard column: only thepl parameter needs to be changed. Tik and related parameters ril and clj in the standard column can be calculated and used to characterize the chromatographic interactions of a given compound and a given stationary phase in the steps summarized in the right side of Scheme I. ..a,
CONCLUSIONS
Scheme I shows that it is possible to characterize the variation of k with T using different types of parameters. (a)p,Parameters. They can be easily calculatedfor many expressionsrelating k with T: Table I11 shows the equations that we have found to give the best quality of fit. According to the equation selected, pr parameters can have physical significance, but great care in their determination is needed to avoid errors in their absolute values caused by correlation between parameters. The calculation of p , on the basis of structural considerations (i.e., in a homologous series) seems to be very difficult. However, they can be used in the optimization of chromatographic separations, as shown in Figure 1. (b) Ti&Parameters. Their calculation can be carried out from any set of prparameters. The use of their values in the characterizationof the gas chromatographic relation between retention and temperature presents several advantages: (1) Their values are less sensible to experimental errors than those of pr (see Figure 3A,B). (2) They have an easy chromatographic interpretation since they represent the temperature necessary to elute a compound with a given k value. Their values are then always included in the usual gas chromatographic temperature range. (3) Tik values for a fixed k can be considered as a sort of retention scale and can be used to determine elution order. (4) Their estimationusing interpolationseemsto be possible for compounds in homologous series.
264Q
Although they must be calculated from the prparameters, and their use in optimization requires a previous conversion to them, the calculations involved can be easily carried out with a microcomputer. The characterizationof the elutionorder inversion behavior requires, however, the use of several Tik values at different k for each compound. Their values are highly correlated, making difficult their precise interpretation. ( c ) ril and clj Parameters. The redundancy introduced by choosing many correlated Tik values can be eliminated by using the ril and clj parameters. Three levels of precisioncan be attained (I = 1,2, or 3). ril does not depend on k and ita chromatographic significanceis similar to that of Tik, while ri2 and ri3 describe quantitativelythe difference in the variation of k with T for different compounds. The main disadvantage of these parameters is the possible dependence of clj on the data set used in the calculations.The two data seta analyzed show that the retention behavior of their compounds can be accurately described with three parameters and that the distribution of clj values is surprisingly similar for the two columns. However, the retention of compounds with different chemical structures in columns with other stationary phases must be studied to confirm whether the behavior observed in this work is really representative of the gas chromatographic retention. The selection of the type of parameter depends on the aspect of the chromatographic retention behavior which must be studied. While Tik, ril, and clj parameters are useful for characterization purposes, the use of pr values allows the optimization of gas chromatographic separations in different conditions,includingmixed phases, seriallycoupled columns, and any kind of temperature programming. The possibility, shown in Scheme I, of using any of these parameters in the calculation of the others, even for another chromatographic column of the same phase, allows a flexible way for the characterization of both columns and stationary phases. ACKNOWLEDGMENT
This work was supported by the Comisi6nInterministerial de Investigaci6nCientlficay TBcnica (ProjectsPB88-034and PB91-0077).
RECEIVED for review May 14, 1993. Accepted June 10, 1993.
