- Email: [email protected]
Retention in overloaded columns, an experimental approach
Journal of Chromatography A, 1128 (2006) 203–207
Retention in overloaded columns, an experimental approach Francisco Rex Gonz´alez ∗ , Lilian M´onica Romero Qu´ımica Anal´ıtica, Fac. Ciencias Exactas, Universidad Nacional de La Plata, 47 y 115, 1900 La Plata, Argentina Received 11 April 2006; received in revised form 8 June 2006; accepted 9 June 2006 Available online 3 July 2006
Abstract Employing a micro-bore silica capillary coated with Carbowax 20 M, the dependence of chromatographic retention upon operative variables was studied surpassing the sample capacity of the column. Solution thermodynamics in the non-linear range of the absorption isotherm of n-alkanes on poly(ethylene oxide) were analyzed interpreting the experimental data through a retention equation deduced in a preceding theoretical work. At 120 ◦ C, and pressures up to 11 bar abs, deviations from the ideal-gas behavior are found to be negligible, either for the fluid dynamics of the carrier-gas, or the thermodynamics of solution of the n-alkanes. Within the experimental error, for all practical purposes the mobile phase can be treated as an ideal gas. This constraint allows studying a solute molecule placed in an environment ranging from solvent monomers only, to a mixture of varying composition of solvent and solute, avoiding effects from significant interactions in the gas phase. In the experimental conditions explored, the absorption isotherm can be represented by taking only two-terms of its power series development. © 2006 Elsevier B.V. All rights reserved. Keywords: Gas chromatography; Micro-bore capillary columns; Poly(ethylene oxide); n-Alkanes; Solubility; Non-linear isotherm; Thermodynamic parameters
1. Introduction When the column sample capacity is surpassed, the retention time of an analyte, tR , becomes dependent on all variables which affect its average concentration along the chromatographic elution process: the injected quantity, pressure gradients, column’s geometrical dimensions, mobile phase viscosity, etc. Hence, consecutive injections on a given column, in the same operation conditions, yield different tR if the injected quantities are not exactly the same. This phenomenon is due to the fact that, at increasing concentrations, the distribution of analyte molecules between the mobile phase (M) and the stationary phase (S) takes place in the non-linear range of its sorption isotherm [1–6]. In absorption processes, for a power series development of the isotherm, this is described by taking into account terms higher than the first one [7,8]: 2 CSe = K CMe + K CMe + ...
(1)
CSe and CMe are the equilibrium concentrations of solute molecules, respectively, in the stationary and mobile phase. ∗
Corresponding author. E-mail address: [email protected] (F.R. Gonz´alez).
0021-9673/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2006.06.033
K = ∂CSe /∂CMe is the partition coefficient at infinite dilution, which depends exclusively on solute/solvent interactions in the 2 is a thermodynamic constationary phase. K = ∂2 CSe /2∂CMe stant depending also on solute/solute interactions. If the solute has more affinity for itself than for the solvent, namely for a poor solvent, it is expected a positive K [8]. In such a system, before phase segregation occurs when CSe progressively increases, there should be a region that in principle could be described by taking a limited number of terms in Eq. (1). Experimentally determining the number of terms required for this description bears great interest. A physical or molecular meaning could be settled for the respective coefficients, K, K , etc., through statistical thermodynamics. Thus, an insight into the molecular mechanisms of solvation can be provided for this condition. A typical system of a poor solvent is that of the non-polar n-alkanes solvated by the polar poly(oxyethylene). This has been studied in many aspects, but principally at infinite dilution [9–15]. Liquid poly(oxyethylene) presents the rare peculiarity that its density, ρ, decreases by increasing its number-average ¯ n [14]. This fact is explained by the segremolecular weight M gation of molecular domains due to the folded conformations adopted by chain segments. These predominant conformations in the liquid polymer, which are controlled in part by intramolec-
204
F.R. Gonz´alez, L.M. Romero / J. Chromatogr. A 1128 (2006) 203–207
ular potentials, are reminiscent of the low-energy structures encountered in the solid state. That is the 7/2 helix structure with a tgt (trans-gauche-trans) sequence of dihedral angles COCC, OCCO and CCOC [15,16]. Mentioned features, currently cited in the literature as the “gauche effect”, are the genesis of remarkable solvation and physical properties exhibited by this polymer [17–21]. Due to the small stationary-phase loadings, micro-bore capillary columns admit varying the solute concentration regime from the infinitely diluted region to concentrations well above their sample capacity. Concomitantly, pressures can be varied in the whole range covered by commercial chromatographs, without excessively increasing the velocity of the mobile-phase. In this way are avoided great departures from equilibrium that could give rise to uncontrolled effects. A broad spectrum of operation conditions, still controlled, is thus permitted. Another advantage of micro-capillary columns is that the injected quantities required for surpassing their sample capacity is very small. To preserve detection within the linear range while the column is overloaded, and thus ensuring a constant response coefficient Cd of the detector, is essential for carrying out thermodynamic measurement reliably, eluding the complication of additional variable factors. Presently, there are no theoretical approaches yielding analytical expressions for tR in conditions of overloaded columns, that would allow interpret the chromatographic data in thermodynamic terms. The plain geometry of wall-coated, open tubular capillary columns (WCOT) has great advantages in simplifying a theoretical analysis of the chromatographic elution process. In a preceding theoretical work [8], it is shown that an analytical solution to the differential equation of motion for the peak maximum, there where equilibrium attains, is possible making some approximations. This solution, in conjunction with consequences of its theoretical context, allows an interpretation of the experimental data. 2. Experimental A silica capillary tube (MicroQuartz, M¨unchen, Germany), of length L = 3.83 m, and internal diameter dC = 22.0 m, was coated statically employing Carbowax 20 M (Alltech, Deerfield, IL, U.S.A.) as the stationary-phase. According to the manufacturer, the capillary, made of highly pure synthetic silica, has a concentration of −OH groups in the range 100–500 ppm. No treatments to the internal surface were performed prior to the coating procedure. The coating solution was prepared using pro-analysis dichloromethane as solvent (Merck, Darmstadt, Germany). The resultant phase volume-ratio of the column, β ≡ VM /VS , was determined through procedures described elsewhere [7], using the exact expression β(T) = [(ρ/co )eα(T−To) − 1]. This yielded β = 68.0 at T = 120 ◦ C, where the mass-concentration of the coating solution, co = 0.0150 ± 0.0001 g/ml, corresponds to the temperature at which the column was filled with this, To = 23.0 ± 0.5 ◦ C. The density of Carbowax 20 M, ρ, was calculated through an empirical expression ρ(T) reported by Poole et al. [12,22], which was settled by the regression
of experimental data obtained by classical picnometry. This yields ρ (120 ◦ C) = 1.035 g/ml. The volume thermal expansion coefficient for the amorphous silica of the capillary is α = 1.957 × 10−6 K−1 [14]. The average film thickness, df , was calculated through its exact relationship for the WCOT geometry [7]: dc β df = 1− (2) 2 β+1 For β at 120 ◦ C, Eq. (2) renders df = 0.0800 m. Isothermal chromatographic runs at 120 ◦ C were carried out in a HP 6890 chromatograph with flame ionization detection (FID) and manual split injection. Nitrogen was used as carrier gas. This, and the other employed gases, flame-supply and make-up, were chromatography grade (AGA, Buenos Aires, Argentina). The total carrier-gas flow in the injector was the highest admitted by the electronic pressure control (EPC) of the chromatograph, FT = 200 ml/min. The volume of the injector liner was Vil = 250 l. Then, the injection time is estimated to be, approximately, δtMi = Vil /FT = 0.001 min. In order to ensure the same detector response coefficient, Cd , along each chromatogram, and between different runs, care was taken to keep constant the flow-rate of gas make-up and flame supply gases, which were monitored permanently. This constancy was ensured also by all measured peak heights hmax being comprised by the linear range of the detector. The pressure in the injector, that is, at the column inlet, was varied in the whole range covered by the EPC, up to 150 psig. In the first and second columns of Table 1, are reported respectively the manometric and absolute pressures at which chromatograms were ran, arbitrarily injecting different quantities of analytes. The latter were n-alkanes from pentane to tridecane, all of analytical grade (Carlo Erba, Milan, Italy). In the linear range of the isotherm the gas-hold up time tM is currently determined through the method of regression of n-alkanes retention [7]. The procedure is based on a smooth dependence of the n-alkanes logarithmic distribution with the number of carbon atoms n, when n ≥ 5. This is settled as a smooth function ln K = ln K(n), which leads to an exponential expression for tR (n). Since at infinite dilution holds tR = tM [1 + (K/β)], the regression function must obey to the generic form tR (n) = tM + exp[ln (tM /β) + ln K(n)], where tM is a parameter to be determined through the curve fit, and n is the variable. Table 1 Column headings respectively indicate the gauge pressure at column’s inlet, in psi units; the correspondent absolute pressure, in bar units; the retention time of methane at each pressure, and the gas hold-up calculated through Eq. (3) pi (psig) 50.0 80.0 100 120 150
pi (bar abs)
tR CH4 (min)
tM (min)
u¯ =
4.46 6.53 7.91 9.29 11.3
1.463 0.940 0.771 0.650 0.530
1.455 0.935 0.767 0.646 0.527
4.39 6.83 8.32 9.88 12.1
L tM
(cm/s)
In the last column is tabulated the correspondent temporal average velocity of the carrier-gas.
F.R. Gonz´alez, L.M. Romero / J. Chromatogr. A 1128 (2006) 203–207
205
In the non-linear range of the isotherm, the dependence of tR on concentration hinders this determination. The reason is that under this condition n is not the unique variable in the run, being tR = tR (n, CMe ), where CMe is an average equilibrium concentration in the mobile phase along the elution process. This fact leaves the determination of tM through a marker, injected in small quantities such to keep infinite dilution conditions, as the only viable alternative. Methane is the usual marker for FID detectors, and was the one employed here. Knowing its partition coefficient, K(CH4 ) [13], is indispensable, since at infinite dilution verifies: tM = tR (CH4 )
β β + K(CH4 )
(3)
In a commercial column coated with a poly(ethylene oxide) polymer having solvation properties close to Carbowax 20 M [13], K(CH4 ) is known to be insensitive to temperature, lying between 0.3 and 0.4 in the whole applicable range. The third column of Table 1 reports the retention times of methane at each inlet pressure, and the correspondent tM in the fourth column, calculated through Eq. (3), assuming K(CH4 ) = 0.38. The last column of Table 1 reports the temporal average velocity of the carrier gas. Such low average velocities assure that great departures from equilibrium should not be expected for most of the chromatographic band. 3. Results and discussion At 120 ◦ C, contributions of solute interfacial excesses, respect to the concentration in the bulk liquid, become negligible in capillary columns with current film thicknesses of poly(oxyethylene) [7,13]. Therefore, this would constitute the lower limit for studying accurately the solute distribution between the bulk phases. Runs taken at this, or greater temperatures, should reflect the bulk solvation properties. In addition, if the gas phase behaves ideally, the solute does not interact therein with other molecules, so the retention should be governed exclusively by molecular forces in the bulk stationary phase. That carrier gas molecules do not interact significantly between them is a fact known from the negligible deviation from unity of the compressibility factor Z ≡ p/CkB T, where C is the total numeral concentration of molecules and kB Boltzman’s constant. For the employed carrier gas, nitrogen, it is Z = 1.002 at 120 ◦ C and 10 bar abs [23]. Of course, this quasi-ideal behavior should be reflected by the fluid dynamic pattern of the mobile phase, as the chromatographic band cannot perturb the flow significantly. Since tM is a basic fluid dynamic parameter, and the most accurately measurable in a micro-bore capillary column (flow rates are not easily measurable), it can be checked if its functional behavior corresponds to that of an ideal gas. This behavior is represented by the well-known solution for the ideal gas, obtained by integration of the differential form of HagenPoiseulle’s equation: tM =
4 ηL2 (P 3 − 1) 128 η = 3 Bpo (P 2 − 1)2 3 po
2 3 L (P − 1) 2 d (P 2 − 1)
(4)
Fig. 1. The gas hold-up tM , measured at constant T and varying P, is represented as function of the reduced variable (P3 − 1)/(P2 − 1)2 . The mean-square linear regression of the data yields a slope sregr = 4.806 min.
L is the column length, and η the carrier-gas viscosity. The relative pressure P ≡ pi /po is the ratio of column’s inlet and outlet absolute pressures. B = AM /8π is the permeability of the column, being AM the cross-sectional area available for the flow of mobile phase. According to Eq. (4), tM measured at constant T and varying P, represented as function of the reduced variable (P3 − 1)/(P2 − 1)2 , should be a straight line passing through the origin, having a slope given by s = 128ηL2 /3po d2 , where d = dC − 2df . In Fig. 1 is represented the mean-square linear regression for the tM data of Table 1, which yields a slope sregr = 4.806 ± 0.014 min. By applying the value of viscosity for nitrogen reported in Ref. [24], η = 2.191 × 10−4 g/cm s, and the measured L, po and d, the expression for the slope yields scalc = 4.73 min. The difference between sregr and scalc is clearly in the order of the experimental errors of the factors contained in the latter. In Fig. 2 it is shown a typical chromatogram for a series of n-alkanes, with the aim to illustrate the recorded peak shape h(t) in our particular experimental conditions. An abrupt decay observed in the front should be the consequence of the peak dispersion of convective origin being greater than that generated by axial diffusion [8]. It must be remarked that recorded peaks with a fronted pattern, resembling the profile of a triangle rectangle, is a fact theoretically consistent with the observation of a positive K [8]. The quasi-ideal behavior of the mobile-phase could be reflected also by the thermodynamic behavior of the analyte, if not only the gas interactions are negligible, but if the analyte/carrier-gas and analyte/analyte interactions in M can be neglected too. Theoretically, this situation should lead to plots of Kef vs. CMe which must be independent √ of P [8], where Kef ≡ [(tR /tM ) − 1]β, and CMe = Cop hmax tR /tM /(P 2 − 1). The operative coefficient is Cop = Cn ηL/(Cd A2M po ), being Cn a numerical constant which is a multiple of π, in our case equal
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Fig. 2. Chromatogram for a series of n-alkanes illustrating the recorded peak shape h(t) in our experimental conditions. The largely retained peaks have a fronted pattern, resembling the profile of a triangle rectangle. A very abrupt decay is observed at the peak front, which is interpreted theoretically as the kinetics for the dispersion of convective origin being faster than that by axial diffusion [8].
to 32π. The recorded signal of the detector at the peak maximum is hmax . The yield of ions captured by the electrode of the detector cannot be determined with exactitude, so Cd is not directly calculable. Therefore, Cop is going to be treated here as an undetermined constant. We are interested on the qualitative information √ of the absorption isotherm, obtained by plotting Kef vs. hmax tR /tM /(P 2 − 1). In Fig. 3 it is shown these plots for n-undecane, at pressures reported in Table 1. A largely retained peak was selected in order to include those data corresponding to well-developed bands. Within the experimental errors, all data lie on a straight line, whose linear regression yields r2 = 0.971.
This means that the absorption isotherm can be fairly represented by taking only two terms in Eq. (1). The slope of this, obtained through the regression, renders K Cop = 0.001294 ± 0.000054. At increasing P, no systematic deviations from a common line are observed in Fig. 3. As stated in the previous paragraph, this fact is theoretically possible only if, beyond molecular collisions, the analyte does not interact significantly with other molecules in the compressible mobile-phase. The correlation found in Fig. 3 shows that, in principle, if K and eventually higher coefficients could be determined through these types of plots, the prediction of retention in overloaded columns would be possible through simple analytical functions having the generic form tR = tM [1 + (K/β) + (K /β) CMe + . . .] [8]. The practical relevance for handling the tools necessary to easily predict retention in overloaded columns resides in the fact that this condition is the one of preparative chromatography [6]. Thus, the design and optimization for those separation processes of industrial interest would be facilitated. But, for making this a reality, the present study should be first extended to columns of increasing bore diameters (250 m, and bigger), there where the higher mobile phase velocities and thicker films could originate additional or more complex phenomena. Efforts should be devoted also to extend these studies to liquid chromatography, which bears a greater practical impact for preparative purposes. 4. Conclusions At 120 ◦ C, the absorption isotherm of n-alkanes on poly(ethylene oxide) can be represented by taking only two terms of its power series development, with a positive K . This observation is consistent with the detected peak profiles, fact that induces to think that in the studied conditions there are no other mechanisms contributing significantly to the elution process than those considered theoretically. At the applied temperature, and inlet pressures up to 11 bar, within the experimental error the mobile phase can be treated either fluid dynamically or thermodynamically as an ideal gas. References
√ Fig. 3. Plots of Kef vs. hmax tR /tM /(p2 − 1) for n-undecane, at pressures reported in Table 1. Within the experimental errors the data lie on a straight line. Theoretically, the slope represents the product K Cop .
[1] H. Purnell, Gas Chromatography, Wiley, New York, 1962. [2] J.C. Giddings, Dynamics of Chromatography, Part 1. Principles and Theory, Marcel Dekker, New York, 1965. [3] R.J. Laub, R.L. Pecsok, Physicochemical Applications of Gas Chromatography, Wiley, New York, 1978. [4] J.R. Conder, C.L. Young, Physicochemical Measurement by Gas Chromatography, Wiley, New York, 1979. [5] J.C. Giddings, Unified Separation Science, Wiley, New York, 1991. [6] G. Guiochon, S.G. Shirazi, A.M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, London, 1994. [7] F.R. Gonz´alez, J. Chromatogr. A 1037 (2004) 233. [8] F.R. Gonz´alez, J. Chromatogr. A, in press. [9] M.B. Evans, J.F. Smith, J. Chromatogr. 36 (1968) 489. [10] S.A. Taleb-Bendiab, J.M. Vergnaud, J. Chromatogr. 107 (1975) 15. [11] J. Klein, J.E. Jeberien, Makromol. Chem. 181 (1980) 1237. [12] C.F. Poole, Q. Li, W. Kiridena, W.W. Koziol, J. Chromatogr. A 898 (2000) 211. [13] F.R. Gonz´alez, R.C. Castells, A.M. Nardillo, J. Chromatogr. A 927 (2001) 111.
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[20] G.D. Smith, O. Borodin, M. Pekny, B. Annis, D. Londono, R.L. Jaffe, Spectrochim. Acta A 53 (1997) 1273. [21] S. Neyertz, D. Brown, J.O. Thomas, J. Chem. Phys. 101 (1994) 10064. [22] K.G. Furton, C.F. Poole, J. Chromatogr. 399 (1987) 47. [23] R.C. Reid, J.M. Praunitz, B.E. Poling, The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York, 1987. [24] D.R. Lide (Ed.), Handbook of Chemistry and Physics, 72nd ed., CRC Press, Boca Raton, FL, 1991.
Retention in overloaded columns, an experimental approach Francisco Rex Gonz´alez ∗ , Lilian M´onica Romero Qu´ımica Anal´ıtica, Fac. Ciencias Exactas, Universidad Nacional de La Plata, 47 y 115, 1900 La Plata, Argentina Received 11 April 2006; received in revised form 8 June 2006; accepted 9 June 2006 Available online 3 July 2006
Abstract Employing a micro-bore silica capillary coated with Carbowax 20 M, the dependence of chromatographic retention upon operative variables was studied surpassing the sample capacity of the column. Solution thermodynamics in the non-linear range of the absorption isotherm of n-alkanes on poly(ethylene oxide) were analyzed interpreting the experimental data through a retention equation deduced in a preceding theoretical work. At 120 ◦ C, and pressures up to 11 bar abs, deviations from the ideal-gas behavior are found to be negligible, either for the fluid dynamics of the carrier-gas, or the thermodynamics of solution of the n-alkanes. Within the experimental error, for all practical purposes the mobile phase can be treated as an ideal gas. This constraint allows studying a solute molecule placed in an environment ranging from solvent monomers only, to a mixture of varying composition of solvent and solute, avoiding effects from significant interactions in the gas phase. In the experimental conditions explored, the absorption isotherm can be represented by taking only two-terms of its power series development. © 2006 Elsevier B.V. All rights reserved. Keywords: Gas chromatography; Micro-bore capillary columns; Poly(ethylene oxide); n-Alkanes; Solubility; Non-linear isotherm; Thermodynamic parameters
1. Introduction When the column sample capacity is surpassed, the retention time of an analyte, tR , becomes dependent on all variables which affect its average concentration along the chromatographic elution process: the injected quantity, pressure gradients, column’s geometrical dimensions, mobile phase viscosity, etc. Hence, consecutive injections on a given column, in the same operation conditions, yield different tR if the injected quantities are not exactly the same. This phenomenon is due to the fact that, at increasing concentrations, the distribution of analyte molecules between the mobile phase (M) and the stationary phase (S) takes place in the non-linear range of its sorption isotherm [1–6]. In absorption processes, for a power series development of the isotherm, this is described by taking into account terms higher than the first one [7,8]: 2 CSe = K CMe + K CMe + ...
(1)
CSe and CMe are the equilibrium concentrations of solute molecules, respectively, in the stationary and mobile phase. ∗
Corresponding author. E-mail address: [email protected] (F.R. Gonz´alez).
0021-9673/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2006.06.033
K = ∂CSe /∂CMe is the partition coefficient at infinite dilution, which depends exclusively on solute/solvent interactions in the 2 is a thermodynamic constationary phase. K = ∂2 CSe /2∂CMe stant depending also on solute/solute interactions. If the solute has more affinity for itself than for the solvent, namely for a poor solvent, it is expected a positive K [8]. In such a system, before phase segregation occurs when CSe progressively increases, there should be a region that in principle could be described by taking a limited number of terms in Eq. (1). Experimentally determining the number of terms required for this description bears great interest. A physical or molecular meaning could be settled for the respective coefficients, K, K , etc., through statistical thermodynamics. Thus, an insight into the molecular mechanisms of solvation can be provided for this condition. A typical system of a poor solvent is that of the non-polar n-alkanes solvated by the polar poly(oxyethylene). This has been studied in many aspects, but principally at infinite dilution [9–15]. Liquid poly(oxyethylene) presents the rare peculiarity that its density, ρ, decreases by increasing its number-average ¯ n [14]. This fact is explained by the segremolecular weight M gation of molecular domains due to the folded conformations adopted by chain segments. These predominant conformations in the liquid polymer, which are controlled in part by intramolec-
204
F.R. Gonz´alez, L.M. Romero / J. Chromatogr. A 1128 (2006) 203–207
ular potentials, are reminiscent of the low-energy structures encountered in the solid state. That is the 7/2 helix structure with a tgt (trans-gauche-trans) sequence of dihedral angles COCC, OCCO and CCOC [15,16]. Mentioned features, currently cited in the literature as the “gauche effect”, are the genesis of remarkable solvation and physical properties exhibited by this polymer [17–21]. Due to the small stationary-phase loadings, micro-bore capillary columns admit varying the solute concentration regime from the infinitely diluted region to concentrations well above their sample capacity. Concomitantly, pressures can be varied in the whole range covered by commercial chromatographs, without excessively increasing the velocity of the mobile-phase. In this way are avoided great departures from equilibrium that could give rise to uncontrolled effects. A broad spectrum of operation conditions, still controlled, is thus permitted. Another advantage of micro-capillary columns is that the injected quantities required for surpassing their sample capacity is very small. To preserve detection within the linear range while the column is overloaded, and thus ensuring a constant response coefficient Cd of the detector, is essential for carrying out thermodynamic measurement reliably, eluding the complication of additional variable factors. Presently, there are no theoretical approaches yielding analytical expressions for tR in conditions of overloaded columns, that would allow interpret the chromatographic data in thermodynamic terms. The plain geometry of wall-coated, open tubular capillary columns (WCOT) has great advantages in simplifying a theoretical analysis of the chromatographic elution process. In a preceding theoretical work [8], it is shown that an analytical solution to the differential equation of motion for the peak maximum, there where equilibrium attains, is possible making some approximations. This solution, in conjunction with consequences of its theoretical context, allows an interpretation of the experimental data. 2. Experimental A silica capillary tube (MicroQuartz, M¨unchen, Germany), of length L = 3.83 m, and internal diameter dC = 22.0 m, was coated statically employing Carbowax 20 M (Alltech, Deerfield, IL, U.S.A.) as the stationary-phase. According to the manufacturer, the capillary, made of highly pure synthetic silica, has a concentration of −OH groups in the range 100–500 ppm. No treatments to the internal surface were performed prior to the coating procedure. The coating solution was prepared using pro-analysis dichloromethane as solvent (Merck, Darmstadt, Germany). The resultant phase volume-ratio of the column, β ≡ VM /VS , was determined through procedures described elsewhere [7], using the exact expression β(T) = [(ρ/co )eα(T−To) − 1]. This yielded β = 68.0 at T = 120 ◦ C, where the mass-concentration of the coating solution, co = 0.0150 ± 0.0001 g/ml, corresponds to the temperature at which the column was filled with this, To = 23.0 ± 0.5 ◦ C. The density of Carbowax 20 M, ρ, was calculated through an empirical expression ρ(T) reported by Poole et al. [12,22], which was settled by the regression
of experimental data obtained by classical picnometry. This yields ρ (120 ◦ C) = 1.035 g/ml. The volume thermal expansion coefficient for the amorphous silica of the capillary is α = 1.957 × 10−6 K−1 [14]. The average film thickness, df , was calculated through its exact relationship for the WCOT geometry [7]: dc β df = 1− (2) 2 β+1 For β at 120 ◦ C, Eq. (2) renders df = 0.0800 m. Isothermal chromatographic runs at 120 ◦ C were carried out in a HP 6890 chromatograph with flame ionization detection (FID) and manual split injection. Nitrogen was used as carrier gas. This, and the other employed gases, flame-supply and make-up, were chromatography grade (AGA, Buenos Aires, Argentina). The total carrier-gas flow in the injector was the highest admitted by the electronic pressure control (EPC) of the chromatograph, FT = 200 ml/min. The volume of the injector liner was Vil = 250 l. Then, the injection time is estimated to be, approximately, δtMi = Vil /FT = 0.001 min. In order to ensure the same detector response coefficient, Cd , along each chromatogram, and between different runs, care was taken to keep constant the flow-rate of gas make-up and flame supply gases, which were monitored permanently. This constancy was ensured also by all measured peak heights hmax being comprised by the linear range of the detector. The pressure in the injector, that is, at the column inlet, was varied in the whole range covered by the EPC, up to 150 psig. In the first and second columns of Table 1, are reported respectively the manometric and absolute pressures at which chromatograms were ran, arbitrarily injecting different quantities of analytes. The latter were n-alkanes from pentane to tridecane, all of analytical grade (Carlo Erba, Milan, Italy). In the linear range of the isotherm the gas-hold up time tM is currently determined through the method of regression of n-alkanes retention [7]. The procedure is based on a smooth dependence of the n-alkanes logarithmic distribution with the number of carbon atoms n, when n ≥ 5. This is settled as a smooth function ln K = ln K(n), which leads to an exponential expression for tR (n). Since at infinite dilution holds tR = tM [1 + (K/β)], the regression function must obey to the generic form tR (n) = tM + exp[ln (tM /β) + ln K(n)], where tM is a parameter to be determined through the curve fit, and n is the variable. Table 1 Column headings respectively indicate the gauge pressure at column’s inlet, in psi units; the correspondent absolute pressure, in bar units; the retention time of methane at each pressure, and the gas hold-up calculated through Eq. (3) pi (psig) 50.0 80.0 100 120 150
pi (bar abs)
tR CH4 (min)
tM (min)
u¯ =
4.46 6.53 7.91 9.29 11.3
1.463 0.940 0.771 0.650 0.530
1.455 0.935 0.767 0.646 0.527
4.39 6.83 8.32 9.88 12.1
L tM
(cm/s)
In the last column is tabulated the correspondent temporal average velocity of the carrier-gas.
F.R. Gonz´alez, L.M. Romero / J. Chromatogr. A 1128 (2006) 203–207
205
In the non-linear range of the isotherm, the dependence of tR on concentration hinders this determination. The reason is that under this condition n is not the unique variable in the run, being tR = tR (n, CMe ), where CMe is an average equilibrium concentration in the mobile phase along the elution process. This fact leaves the determination of tM through a marker, injected in small quantities such to keep infinite dilution conditions, as the only viable alternative. Methane is the usual marker for FID detectors, and was the one employed here. Knowing its partition coefficient, K(CH4 ) [13], is indispensable, since at infinite dilution verifies: tM = tR (CH4 )
β β + K(CH4 )
(3)
In a commercial column coated with a poly(ethylene oxide) polymer having solvation properties close to Carbowax 20 M [13], K(CH4 ) is known to be insensitive to temperature, lying between 0.3 and 0.4 in the whole applicable range. The third column of Table 1 reports the retention times of methane at each inlet pressure, and the correspondent tM in the fourth column, calculated through Eq. (3), assuming K(CH4 ) = 0.38. The last column of Table 1 reports the temporal average velocity of the carrier gas. Such low average velocities assure that great departures from equilibrium should not be expected for most of the chromatographic band. 3. Results and discussion At 120 ◦ C, contributions of solute interfacial excesses, respect to the concentration in the bulk liquid, become negligible in capillary columns with current film thicknesses of poly(oxyethylene) [7,13]. Therefore, this would constitute the lower limit for studying accurately the solute distribution between the bulk phases. Runs taken at this, or greater temperatures, should reflect the bulk solvation properties. In addition, if the gas phase behaves ideally, the solute does not interact therein with other molecules, so the retention should be governed exclusively by molecular forces in the bulk stationary phase. That carrier gas molecules do not interact significantly between them is a fact known from the negligible deviation from unity of the compressibility factor Z ≡ p/CkB T, where C is the total numeral concentration of molecules and kB Boltzman’s constant. For the employed carrier gas, nitrogen, it is Z = 1.002 at 120 ◦ C and 10 bar abs [23]. Of course, this quasi-ideal behavior should be reflected by the fluid dynamic pattern of the mobile phase, as the chromatographic band cannot perturb the flow significantly. Since tM is a basic fluid dynamic parameter, and the most accurately measurable in a micro-bore capillary column (flow rates are not easily measurable), it can be checked if its functional behavior corresponds to that of an ideal gas. This behavior is represented by the well-known solution for the ideal gas, obtained by integration of the differential form of HagenPoiseulle’s equation: tM =
4 ηL2 (P 3 − 1) 128 η = 3 Bpo (P 2 − 1)2 3 po
2 3 L (P − 1) 2 d (P 2 − 1)
(4)
Fig. 1. The gas hold-up tM , measured at constant T and varying P, is represented as function of the reduced variable (P3 − 1)/(P2 − 1)2 . The mean-square linear regression of the data yields a slope sregr = 4.806 min.
L is the column length, and η the carrier-gas viscosity. The relative pressure P ≡ pi /po is the ratio of column’s inlet and outlet absolute pressures. B = AM /8π is the permeability of the column, being AM the cross-sectional area available for the flow of mobile phase. According to Eq. (4), tM measured at constant T and varying P, represented as function of the reduced variable (P3 − 1)/(P2 − 1)2 , should be a straight line passing through the origin, having a slope given by s = 128ηL2 /3po d2 , where d = dC − 2df . In Fig. 1 is represented the mean-square linear regression for the tM data of Table 1, which yields a slope sregr = 4.806 ± 0.014 min. By applying the value of viscosity for nitrogen reported in Ref. [24], η = 2.191 × 10−4 g/cm s, and the measured L, po and d, the expression for the slope yields scalc = 4.73 min. The difference between sregr and scalc is clearly in the order of the experimental errors of the factors contained in the latter. In Fig. 2 it is shown a typical chromatogram for a series of n-alkanes, with the aim to illustrate the recorded peak shape h(t) in our particular experimental conditions. An abrupt decay observed in the front should be the consequence of the peak dispersion of convective origin being greater than that generated by axial diffusion [8]. It must be remarked that recorded peaks with a fronted pattern, resembling the profile of a triangle rectangle, is a fact theoretically consistent with the observation of a positive K [8]. The quasi-ideal behavior of the mobile-phase could be reflected also by the thermodynamic behavior of the analyte, if not only the gas interactions are negligible, but if the analyte/carrier-gas and analyte/analyte interactions in M can be neglected too. Theoretically, this situation should lead to plots of Kef vs. CMe which must be independent √ of P [8], where Kef ≡ [(tR /tM ) − 1]β, and CMe = Cop hmax tR /tM /(P 2 − 1). The operative coefficient is Cop = Cn ηL/(Cd A2M po ), being Cn a numerical constant which is a multiple of π, in our case equal
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Fig. 2. Chromatogram for a series of n-alkanes illustrating the recorded peak shape h(t) in our experimental conditions. The largely retained peaks have a fronted pattern, resembling the profile of a triangle rectangle. A very abrupt decay is observed at the peak front, which is interpreted theoretically as the kinetics for the dispersion of convective origin being faster than that by axial diffusion [8].
to 32π. The recorded signal of the detector at the peak maximum is hmax . The yield of ions captured by the electrode of the detector cannot be determined with exactitude, so Cd is not directly calculable. Therefore, Cop is going to be treated here as an undetermined constant. We are interested on the qualitative information √ of the absorption isotherm, obtained by plotting Kef vs. hmax tR /tM /(P 2 − 1). In Fig. 3 it is shown these plots for n-undecane, at pressures reported in Table 1. A largely retained peak was selected in order to include those data corresponding to well-developed bands. Within the experimental errors, all data lie on a straight line, whose linear regression yields r2 = 0.971.
This means that the absorption isotherm can be fairly represented by taking only two terms in Eq. (1). The slope of this, obtained through the regression, renders K Cop = 0.001294 ± 0.000054. At increasing P, no systematic deviations from a common line are observed in Fig. 3. As stated in the previous paragraph, this fact is theoretically possible only if, beyond molecular collisions, the analyte does not interact significantly with other molecules in the compressible mobile-phase. The correlation found in Fig. 3 shows that, in principle, if K and eventually higher coefficients could be determined through these types of plots, the prediction of retention in overloaded columns would be possible through simple analytical functions having the generic form tR = tM [1 + (K/β) + (K /β) CMe + . . .] [8]. The practical relevance for handling the tools necessary to easily predict retention in overloaded columns resides in the fact that this condition is the one of preparative chromatography [6]. Thus, the design and optimization for those separation processes of industrial interest would be facilitated. But, for making this a reality, the present study should be first extended to columns of increasing bore diameters (250 m, and bigger), there where the higher mobile phase velocities and thicker films could originate additional or more complex phenomena. Efforts should be devoted also to extend these studies to liquid chromatography, which bears a greater practical impact for preparative purposes. 4. Conclusions At 120 ◦ C, the absorption isotherm of n-alkanes on poly(ethylene oxide) can be represented by taking only two terms of its power series development, with a positive K . This observation is consistent with the detected peak profiles, fact that induces to think that in the studied conditions there are no other mechanisms contributing significantly to the elution process than those considered theoretically. At the applied temperature, and inlet pressures up to 11 bar, within the experimental error the mobile phase can be treated either fluid dynamically or thermodynamically as an ideal gas. References
√ Fig. 3. Plots of Kef vs. hmax tR /tM /(p2 − 1) for n-undecane, at pressures reported in Table 1. Within the experimental errors the data lie on a straight line. Theoretically, the slope represents the product K Cop .
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