Anal. Chem. 1984, 56,1557-1561
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Model for Micellar Effects on Liquid Chromatography Capacity Factors and for Determination of Micelle-Solute Equilibrium Constants Manop Arunyanart and L.J. Cline Love* Seton Hall University, Department of Chemistry, South Orange, New Jersey 07079
A three-phase equlllbrlum model relatlng capaclty factor to mlceliar moblle phase concentration in liquid chromatography Is proposed, and equatlons are developed which allow calculation of the equlllbrlum constant for the solute between the bulk aqueous phase and micellar aggregate. I n addition, if the equlllbrlum constant Is available from Independent methods, the equations can accurately predict the chromatographic capaclty factor at zero or greater moblle phase mlcelle concentrations. The elementary model descrlbed here assumes mlnlmal electrostatic lnteractlons and Is sulted for applicatlon to neutral solutes; however, it Is posslbie to extend It to account for other simultaneous equilibria such as prototropic. The two principal equlllbria for the solute are bulk phase-micelle and bulk phase-statlonary phase. A thlrd component, micelle-statlonary phase, can be neglected. Use of an equlllbrlum expresslon approach permits calculatlon of micelle-solute equlllbrlum constants which are In good agreement wlth literature values for small to moderate sire specles, such as naphthalene, and In falr agreement for larger molecules, such as pyrene.
Reversed-phase high-performance liquid chromatography (HPLC) employing hydrocarbonaceous-bonded stationary phases has become one of the most widely used modes of liquid chromatography, in part, because of the impressive selectivity available via mobile phase participation in the equilibrium distribution of solute molecules between the stationary phase and mobile phase. Retention in reversedphase liquid chromatography is dominated by the solutesolvent interactions, with solute-stationary phase interactions making important but secondary contributions (I). Thus, the key to selective separations is the ability to control solutesolvent interactions by changing the composition of the mobile phase. Charged surfactants at concentrations below the critical micelle concentration (CMC) have been widely used as mobile phase modifiers to enhance separation of oppositely charged solute ions. Because of the amphiphilic nature of. the surfactant, there exists some uncertainty concerning the retention mechanism(s) of this mode of separation. This is reflected in the variety of names given to the technique (2-9), the more common name being ion-pair chromatography. Several fundamental considerations have been described by Horvath et al. (7),and these are used in developing the results presented in this paper. Aqueous solutions of surfactants a t concentrations above their CMC, where micelles exist along with monomers, dimers, etc., constitute a more complex mobile-phase modifier. These micellar solutions are microscopically heterogeneous, being composed of the amphiphilic micellar aggregate and the bulk surrounding solvent which contains surfactants whose concentration is approximately equal to the CMC. The solute can be preferentially solubilized into or onto the micellar assembly, a process which is dynamic and characterized by
various rate constants (10). These unique properties of micellar solutions have been used to advantage in generation of room-temperature phosphorescence (11),in drug adsorption studies (12),and in chromatographic separations. The partitioning of solutes between micellar and aqueous phases in liquid chromatography was first treated theoretically by Herries (13). Since then, other reports have illustrated certain advantages of micellar chromatography (14-1 7), including the enhancement of selectivity by proper choice of surfactant type and mobile phase concentration (18)and by their use in generating room-temperature phosphorescence in fluid solution for use as a selective phosphorescence detector in HPLC (19). One of the problems encountered with micellar chromatography, its rather low chromatographic efficiency, has been overcome recently by Dorsey and co-workers (1). They demonstrated that efficiencies approaching those of hydroorganic mobile phases can be achieved by adding 3% propanol to the micellar mobile phase and working at an elevated temperature (40 O C ) . The large number of possible interactions associated with micellar mobile phase separations, e.g., electrostatic, hydrophobic, and steric, as well as the modification of the stationary phase by adsorption of monomer surfactants, makes these systems more complicated than conventional reversed-phase HPLC. One existing model allows calculation of partition coefficients for un-ionizable solutes but does not appear to explain the elution behavior of ionizable solutes at different pHs (14). By using the equilibrium treatment proposed here, the model can be expanded to explain the elution behavior of ionizable solutes with varying pH. In addition, for the equilibrium model, the volume of the stationary phase, V,, and the partial specific volume of the surfactant, 8, need not be known in order to calculate equilibrium constants or predict capacity factors. For the purpose of demonstrating the validity of the equilibrium approach in this paper, the mechanisms of interactions have been simplified by eliminating electrostatic interactions through the use of neutral solutes, and elution behavior is described by using a straightforward equilibrium treatment. This paper describes a model for the elution behavior of a series of neutral arenes, such as benzene, p-xylene, 1-bromonaphthalene, and perylene. Equations for calculation of solute-micelle equilibrium constants are derived, based on the model, and the results are compared with literature values obtained by different techniques. Very good agreement was found which suggests the model approximates reality for neutral species. The advantages and limitations of this method of calculating equilibrium constants are discussed. Extension of this model for ionizable solutes will be discussed in a separate paper.
THEORY Reversed-phase micellar chromatography can be described by two principal equilibria, one being a reversible equilibrium of solute in the bulk solvent mobile phase, E,, with the stationary phase sites, L,, tQ form a complex, EL,, and the second a reversible equilibrium of solute in the bulk solvent mobile phase, E,, with the surfactant in the micelle in the mobile
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phase, M,, to form the complex, EM,. A third reversible equilibrium involving the direct transfer of solute in the micelle, EM,, to the stationary phase is also possible and may be neglected in the present treatment (discussed below), and it is assumed that the solute binds independently to the stationary phase and to the micelle in the bulk solvent. The quantitative effect of the magnitudes of these equilibrium constants on the HPLC capacity factors may be derived as follows. The equilibrium expressions are presented by the following set of equations in which the species concentrations in the stationary phase and in the mobile phase are denoted by subscripts s and m, respectively. The concentrations of all species are defined in moles/liter, and [M,] is defined as the concentration of surfactant in the micelle in the mobile phase, and is obtained from [M,] = [surfactant] - CMC, where CMC is the critical micelle concentration.
+ L, Kl EL, K2 E, + M, EM, K3 EM, + L, r EL, + M, E,
(1)
(2)
(3)
The equilibrium constants corresponding to eq 1, 2, and 3 are denoted by Kl, Kg, and K3, respectively. Of the three equilibria, only two are independent such that K3is neglected in the following discussion. By multiplying K2times K3,one obtains K1, and the retention mechanism is controlled by equilibria 1 and 2. The capacity factor of the solute, k’, is defined in the usual way as
k‘
~[EL,I
(4) [Ern1 + [EMmI where is the phase ratio, i.e., the ratio of the volume of the stationary phase, V,, to the volume of the mobile phase, V,, in the column. The combination of eq 1-4 yields the following expression for the capacity factor:
This expression predicts a parabolic curve dependence of k’ on [M,] which will have an intercept value of k’ = $[L,]Kl a t [M,] = 0. If K2 is known (calculated from independently measured KBqvalues), then the value of 4[L,]Kl can be estimated from only one measurement of k ’a t any particular concentration of surfactant in the micelle, [M,]. This intercept value can then be used to estimate the value of k‘ at any concentration of M, and should agree with the value of k ’obtained from the experimental data. Equation 5 can be linearized by taking the reciprocal of both sides, viz.
By plotting llk’vs. [M,] in moles/liter, one should obtain a straight line. The value of K2is the equilibrium constant between solute per monomer of surfactant and is given by the ratio of slope/intercept from the plot. To obtain the equilibrium constant per micelle, Keq, one must multiply the Kz value by the surfactant’s aggregation number. For pure sodium dodecyl sulfate aqueous micelles, a value of 62 is used for the aggregation number (20). The values of Keqobtained by this chromatography technique should be the same, within experimental error, as those obtained by other techniques such as luminescence or solubility (10). However, if K z is large, K1 is also large, which is true for large molecules such as pyrene and perylene, and the second term on the right-hand side of eq 6 is approximately zero. This means that for compounds
which have large K1 values, the Kegvalue may not be obtainable with any accuracy from this type of plot. Equation 6 reduces to the classic chromatographic equation, V, = V, + KV,, when the concentration of surfactant in the micelles goes to zero. Although it is not the purpose of this study, it should be noted that for ionizable solutes such as organic acids and bases, the pH-dependent dissociation of these species must be considered. By writing all principal equilibrium expressions, one can obtain expressions similar to eq 5 which can be used to explain the dependence of capacity factor on pH. Therefore, this approach should give much more information needed to explain the elution behavior of solutes in micellar chromatography.
EXPERIMENTAL SECTION Apparatus. Modular components of the Technicon FAST.LC system were connected as required by the various experimental modes. These included a high-pressure liquid chromatographic (LC) pump, a pneumatic injector with a six-port valve and 20-pL loop, and system controller (Micromeritics, Inc.). A Model FS970 LC fluorometer (Kratos Instruments, Ramsey, NJ) detector was connected to a 254-nm Model 1203 UV monitor (Laboratory Data Control Co., Riviera Beach, FL). The fluorometer settings were as follows: the excitation wavelength was 254 nm, the emission cutoff filter was 300 nm, and the photomultiplier tube was maintained at 750 V. The column (30 cm long X 3.9 mm i.d.) was packed with pBondapak C-18 (Waters, Inc.). A precolumn (12.5 cm long X 4.6 mm i.d.) (Whatman, Inc., Clifton, NJ) packed with silica gel (25-40 pm) was located between the pump and sample injector in order to saturate the mobile phase with silica to minimize dissolution of the column packing. A Model 5000 Fisher Recordall strip chart recorder (Fisher Scientific Co., Springfield,NJ) was used to record the chromatograms. Column temperature was controlled by immersing the precolumn and analytical column in a water bath where the temperature was maintained at 25 O C by a Model 73 circulating pump/heater (Fisher Scientific Co.). Reagents. The surfactant, sodium dodecyl sulfate (SDS),was electrophoresis grade obtained from Bio-Rad, Inc., and was used as received. The benzene, toluene, and naphthalene (Fisher Scientific Co.), anthracene and perylene (Aldrich Chemical Co., Inc.), 1-bromonaphthalene and p-xylene (Eastman Kodak Co., Rochester, NY), biphenyl and pyrene (MC/B Co.), and 1methylnaphthalene (Chem Services Inc.) were used as received. Procedure. The micellar mobile phase was prepared by dissolving the appropriate quantity of SDS in distilled water followed by filtration through a 0.5-pm cellulosic membrane filter (Rainin Instrument Co., Inc., Woburn, MA). Stock solutions of the test solutes were prepared in methanol (Fisher Scientific Co.) and then diluted to the appropriate working concentration with 0.05 M SDS. The working concentrations were benzene (380 pg/mL), toluene (950 pg/mL), naphthalene (11pg/mL), p-xylene (650 pg/mL), 1-methylnaphthalene (19 pg/mL), biphenyl (5.5 pg/mL), 1-bromonaphthalene (200 pg/mL), anthracene (0.1 pg/mL), pyrene (0.5 pg/mL), and perylene (3.6 pg/mL). Retention times were measured manually. A flow rate of 2.0 mL/min was monitored by measuring the effluent in a 10-mL graduated cylinder for a sufficient length of time to collect at least 5 mL. The void volume, V,, of the system was measured by using a 0.05 M SDS mobile phase. The time equivalent, t,, of the void volume was determined by injecting 20 pL of 10% methanol in 0.05 M SDS and measuring from the time of injection to the first deviation from the base line. The time was converted to retention volume by using the flow rate. The average volume obtained, 2.34 mL, was used for all k’calculations. It should be noted at this point that with a water mobile phase, the average void volume found by injecting 10% aqueous methanol was 2.94 mL. The smaller void volume found using mobile phases of SDS at concentrations above its CMC is most probably due to the fact that SDS monomers are adsorbed by the stationary phase with the negative head groups in contact with the mobile phase. The increment of surface coverage of the packing material may result in a decreased void volume. In addition, electrostatic effects most probably prevent the micelles from entering the stationary phase
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Table I. Variation of Capacity Factors of a Series of Arenes with Sodium Dodecyl Sulfate Concentration in the Mobile Phase total SDS in mobile phase, mol/L 0.10 0.025 0.04 0.05 0.06 SDS in micelles, [M,], mol/La
compound benzene toluene naphthalene p-xylene 1-methylnaphthalene biphenyl 1-bromonaphthalene anthracene pyrene perylene
0.0169
0.0319
0.0419
0.0519
0.0919
12.85 23.02 29.85 32.68 37.55 37.97 39.43 41.39 45.21 58.26
11.65 19.00 24.47 27.03 30.54 30.97 31.91 33.53 36.61 46.29
8.23 12.42 14.21 16.09 17.29 17.47 18.15 19.00 20.65 25.67
14.04 26.00 37.03 40.20 49.00 49.40 51.14 53.70 59.26 17.35
20.75 37.89 65.67 67.20 89.88 91.59 100.99 104.98 111.82 151.99
a The concentration of surfactant in the micelle is M [ ], = [surfactant] - CMC, where [surfactant] is the total concentraM (ref 20). tion of surfactant in solution and CMC is the critical micelle concentration. The CMC for SDS is 8.1 X
Table 11. Calculated Solute-Micelle Equilibrium Constants with Associated Statistical Analysis for Various Aromatic Solutes K,,,'" L/mol compound slope intercept (x104) exptl benzene toluene naphthalene p-xylene 1-methylnaphthalene biphenyl 1-bromonaphthalene anthracene pyrene perylene
0.94 0.72 0.73 0.63 0.62 0.62 0.60 0.57 0.53 0.43
(1.6 i. 0.3) x (3.1 i. 0.3) x (1.5 k 0.2) x (0.9 k 0.1) x (7.2 t 1.0) x (8.5 1.9) x (2.9 k 2.0) x (3.3 2.2) x (0.5 i. 0.2) x 1.8 X
370
146 31 45 5.3 4.5 1.3 1.1
0.6 0.015'
lo3 lo3 lo4 104 104 104
105 105
lo6
lit. b
1.6 x 103 5.3 x 103 (0.7-1.7) X lo4 1.6 x 104 7.1 x 104 (2.2-7.3) X l o 4 (2.0-3.0) X lo5 4.0 x 105 1.7 X l o 6 1.7 X l o 7
K,, is the solute bulk phase-micelle equilibrium constant per micelle obtained by multiplying the ratio of the slope/ Reference 10. Predicted value of k ' at [M,] = 0; see text for intercept by the aggregation number, 62, of SDS. discussion. Predicted value of K,, ;see text for discussion. pores, resulting in more reduction in the void volume. However, viscosity and hydrophobicity changes likely have some influence on the effective void volume measurement, preventing a straightforward rationalization.
/ 0.10
RESULTS AND DISCUSSION In order to establish the validity of the approach used to obtain eq 6, the solute-micelle equilibrium constants were calculated for a series of arenes using experimentally determined liquid chromatographic terms. The retention volumes of selected test compounds were measured a t five different concentrations of aqueous SDS mobile phases. The capacity factor, k', for each solute was calculated from the retention data by using the ratio [ VR - V,] / V,, where VR is the elution volume of the solute and V, is the mobile phase dead volume. The experimental capacity factors are tabulated in Table I. The reciprocals of the capacity factors were plotted as a function of the concentration of surfactant in the micelles and the resultant graph is shown in Figures 1and 2. It is apparent from the statistical treatment of the plots in Figures 1 and 2 that the relationship between the reciprocal of the capacity factors and the concentration of surfactant in the micelle is linear, with relative standard deviations of the slopes of less than 1.0%, except for benzene (7.3%) and toluene (2.5%). The results of linear regression analysis of these data are given in Table 11. The calculated equilibrium constants are generally in very good agreement in view of the rather large error limits as shown by comparison of the Keg values obtained by using linear regression analysis in this study and those taken from
1 0.075 -
k'
0.05
0.025
0 0
0.025 0.05 0.075 c o n c e n t r a t i o n of SDS i n micelles, M
0.10
Flgure 1. Dependence of the reciprocal of the capacity factor on
concentration of surfactant in the micelle in the mobile phase for five aromatic test molecules: (a)benzene, (0)toluene, (*) naphthalene, (A)p-dimethylbenzene, and (0)biphenyl; column, WEondapak C-18; flow rate, 2.0 mL/min. the literature obtained by using spectroscopic or solubility techniques (Table 11). The results show excellent linearity based on linear regression analysis of the plots for all test compounds, even for the largest solute, perylene. Unfortunately, a perylene-micelle equilibrium constant could not be obtained because the intercept of the plot of llk'vs. [M,] was essentially zero. Small changes in the slope will produce large changes in the intercept and, consequently, large un-
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Table 111. Estimated Values of Capacity Factors of a Series of Arenes with Sodium Dodecyl Sulfate Concentration in the Mobile Phase and the Values of the Intercept, @[L,]K, SDS in micelles, [M,],= mol/L
compound benzene toluene naphthalene p-xylene 1-methylnaphthalene biphenyl 1-bromonaphthalene anthracene pyrene perylene
@[L,]K, (X10-3) 0.0169 0.0319 0.0419 0.0519 0.0919 0.0267 18.62 14.67 12.85 11.43 7.93 0.105 43.14 28.30 23.02 19.40 11.91 0.373 66.17 38.25 29.85 24.48 14.23 0.386 72.01 41.81 32.68 26.82 15.62 1.84 90.31 49.01 37.55 30.43 17.31 1.91 91.45 49.56 37.97 30.77 17.50 8.03 97.05 51.71 39.43 31.86 18.03 11.23 102.06 54.30 41.39 33.44 18.91 51.99 111.94 59.37 45.21 36.51 20.62 669.4 144.42 76.52 58.26 47.04 26.56 a The concentration of surfactant in the micelle is [M,] = [surfactant] - CMC, where [surfactant] is the total concentration of surfactant in the solution and CMC is the critical micelle concentration. The CMC for SDS is 8.1 x l o + M (ref 20). @[L,]K, is the value of the capacity factor at [M,] = 0.
-
0.05 140 110
-k’1
0.0375
t \i
0,025
0 ,0125
0 0
0 025
0 05
0.075
0 10
c o n c e n t r a t i o n of S D S in micelles,M
Figure 2. Dependence of the reciprocal of the capacity factor on concentration of surfactant in the micelle In the mobile phase for five aromatic test molecules: (0) 1-methylnaphthalene, (0) l-bromonaphthalene, (A)anthracene, ( i s )pyrene, and (0)perylene; column, pBondapak C-18; flow rate, 2.0 mL/min.
certainty in the, intercept value (the intercept for perylene was negative in these experiments). However, a k’value of 6.69 X lo5 for perylene a t [M,] = 0 can be predicted with eq 5 by taking the value of k’at [M,] = 0.0419 M (Table I) along with the known value of the perylene-micelle equilibrium constant, Keq,from the literature (IO). (Note that K z = K,,/aggregation number.) In this manner, eq 5 was used to estimate the value of the intercept, +[L,]Kl, which corresponds to k’at [M,] = 0, and these k’ values are given in Table I11 along with predicted k’ values at different concentrations of [M,]. By use of the value of +[L,]K1 calculated above and the slope of the plot of eq 6 (Figure 2 and Table 11),the equilibrium constant, Keq,for perylene was estimated to be 1.8 X lo7L/mol. The literature value for perylene is 1.7 X lo7 L/mol. The good agreement between the predicted behavior plotted as k’vs. [M,] and experimental values of k‘, as shown in Figure 3, is a strong indication that eq 5 fits the experimental data and that the model and equations derived from it describe the system. This approach to determination of Keqhas the advantages that the solute concentration need not be known and that impurities, if present in the sample, are chromatographically separated and will not interfere. In addition, V, and 2 need not be known. Significantly, Keqobtained from independent experiments can be used to predict k ’values thus providing a fruitful link between spectroscopy and chromatography. The analytical procedure is straightforward and the chromatographic parameter, retention volume, can be quite accurately and reproducibly measured. The time required for measurement of retention volumes can be reduced considerably
I
0.12 0.03 0.06 0.09 [ S D S MICELLES] , M
Figure 3. Dependence of k’ on [Mm]for five aromatic species: (0) benzene, (0) toluene, (V) p-xylene, (m) pyrene, and (*) perylene; experimental data points from Table I. Solid lines were obtained by plotting eq 5 using 4 [L,]K, parameter estimates shown in Table 111.
by using a mixture of test solutes in one injection on the chromatograph.
ACKNOWLEDGMENT The authors are grateful to Esther Guerin, Mary Lynn Grayeski, and Rod Woods for their helpful comments on the chromatographic equations. We also thank Paul Yarmchuk and Robert Weinberger for their continuing support in unraveling the mechanisms of micellar chromatography. The generous donations of the FAST.LC system by Technicon and the system controller by Micromeritics and the use of the FS970 LC luminescence detector by Kratos are gratefully acknowledged.
LITERATURE CITED Dorsey, J. G.; DeEchegaray, M. T.; Landy, J. S.Anal. Chem. W83, 5 5 , 924-928. Fransson, B.; Wahlund, K. G.; Johansson, I . M.; Schill, G. J. J. Chromatogr. 1978, 125, 327-344. Kraak, J. C.; Jonker, K. M.; Huber, F. K. J. Chromatogr. 1077, 142. 671-688. Hoffman, N. E.; Liao, J. C. Anal. Chem. 1977, 4 9 , 2231-2234. Tomlinson, E.; Jefferies, T. M.; Riley, C. M. J. Chromatogr. 1978, 159, 315-358. Knox, J. H.; Laird, G. R. J. Chromatogr. 1978, 122, 17-34. Horvath, C.; Melander, W.; Molnar, I.; Molnar, P. Anal. Chem. 1977, 4 9 , 2295-2305. Bidllngmeyer, B. A,; Deming, S. N.;Price, W. P.; Sachok, 8.; Petrusek, M. J. Chromatogr. 1879, 186, 419-434. Horvath, C.; Lin, H.-J. J . Chromatogr. 1978, 149, 43-70. Almciren, M.; Grieser, F.; Thomas, J. K. J. Am. Chem. SOC. 1979, 101; 279-291. Skrllec, M.; Cline Love, L. J. J. Phys. Chem. 1981, 85, 2047-2050. Granneman, G. R.; Sennello, L. T. J. Chromatogr. 1982, 2 2 9 , 149-157. Herries, D. G.; Bishop, W.; Richards, F. M. J. Phys. Chem. 1964, 68, 1842-1052. Armstrong, D. W.; Norne, F. Anal. Chem. 1981, 53, 1662-1666.
Anal. Chem. 1984, 56,1561-1566 (15) Weinberger, R.; Yarmchuk, P.; Cline Love, L. J. In “Surfactants in Solution”; Lindman, Bjorn, Mittal, K. L., Eds.; Plenum Press: New York, 1984;pp 1139-1158. (16) Yarmchuk, P.; Weinberger, R.;Hirsch, R. F.; Cline Love, L. J. J . Chromatogr. 1984, 283, 47-60. (17) Pramauro, E.; Pelizzetti, E. Anal. Chim. Acta 1983, 154,153-158. (18) Yarmchuk, P.; Weinberger, R.; Hirsch, R. F.; Cllne Love, L. J. Anal. Chem. 1982, 54,2233-2238. (19) Weinberger, R.; Yarmchuk, P.; Cline Love, L. J. Anal. Chem. 1982, 54, 1552-1558.
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(20) Mukerjee, P. J . Phys. Chem. 1962, 66, 1733-1735.
RECEIVED for review September 27, 1983. Accepted March, 19,1984. This work was presented in part at the March 1984 Pittsburgh Conference, Abstract No. 47. This work was supported, in part, by the National Science Foundation Grant NO. CHE-8216878.
Model for Chromatographic Separations Based on Renewal Theory Dan M. Scott Statistics Department, Iowa State University, Ames, Iowa 50011 James S. Fritz*
Ames Laboratory, Iowa State University, Ames, Iowa 50011
A simple but reasonably reallstlc model is formulated for describing the behavior of chromatographic peaks. Our approach Is based on statlstlcal concepts and completely avolds the physically nonexistent “theoretical plates” of classical theory. Thls work complements the “rate theory” of chromatography In that we provide a more detailed look at the “reslstance to mass-transfer process” (what we call the “Interphase process”). The model Is a stochastlc one; because molecular level processes are random In nature, we feel that thls is a natural approach. Although a varlety of stochastlc models have been proposed prevlously, they have been damaged by the necesslty of assumlng a particular mechanism. The present theory is largely immune from thls crltlclsm. The paper makes use of results from the theory of renewal processes, but the resun8 should be comprehensible to anyone wlth only a modest acquaintance wlth statlsticai notions.
One of the major conceptual defects of the plate theory of chromatography is the division of the column into so-called “theoretical plates”. As this completely arbitrary division of the chromatographic column bears no relation to any physical realities, the plate theory, for all its descriptive successes, remains unsatisfying. (Giddings (1)gives an excellent discussion of the plate theory and its limitations.) Rate theory has been very useful in explaining the contribution of various dynamic processes to chromatographic peak broadening, but the idea of theoretical plates is still present in that various broadening effects are almost always given in terms of plate height and therefore in terms of theoretical plates. The present work is an attempt to provide a useful theory that does not depend on the use of theoretical plates. There continues to be an active interest in chromatographic theory as evidenced by the selection of recent publications (2-9). The present authors have reviewed some of the past work on chromatographic plate theory and have proposed a simple statistical approach (10). Some of the drawbacks of existing plate theory were discussed and a plate number was introduced that is independent of the capacity factors ( k ) of the various chromatographic peaks.
A statistical approach is really a simple and understandable way to look at chromatography. Each molecule of a sample component alternately enters the stationary phase and returns to the mobile phase many times during its passage through a column. Since all molecules of the chemical do not behave identically, there is a distribution of exit times and the recorded peak is essentially Gaussian in nature. The statistical plate model published recently (10) gives viable results, but the method is not a realistic representation of the physical process that is in progress. The same limitations apply to another statistical approach, the random walk model (11,12). In this model a molecule is assumed to take a step of random length down the column at fiied intervals of time. Or, in what amounts to almost the same thing, it is assumed to take a step of fiied length in a random direction at fixed intervals of time. In either of the above cases the idea of a fixed length of time between steps seems artificial and unrealistic. In this paper we present a model which gives the chromatographic practitioner a nonplate way of thinking of chromatography that can be readily understood. Its basis is a statistical model which is called the “renewal model”. In a sense this is a generalization of the random walk model in that our molecule is assumed to take steps of random length with a random time between these steps. In the theory of stochastic processes, a stochastic process which “renews”or “regenerates” itself is called a renewal or regenerative process. As applied to chromatography, this means that each time a molecule enters the stationary phase (or alternatively each time it enters the mobile phase) the process is “renewed” because from this point the future motion of the molecule is the same, probabilistically speaking, as it was (and will be) at any other such point of revewal. Technically what we are investigating is called an alternating renewal process. It is known that several factors contribute to the broadening of chromatographic peaks; these include extracolumn broadening in the connecting lines and detector, injection broadening resulting from injecting the sample over a finite time period, broadening resulting from axial diffusion in the column, and multipath effects in packed columns. These effects are now well understood and are, in effect, artifacts of the actual chromatographic separation process. Since it is generally assumed that the variances of various contributions to
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