Anal. Chem. 2004, 76, 953-958
Experimental Proof of a Chromatographic Paradox: Are the Injected Molecules in the Peak? Jo 1 rgen Samuelsson,† Patrik Forsse´n,‡ Morgan Stefansson,§ and Torgny Fornstedt*,†
Center for Surface Biotechnology, Uppsala University, BMC Box 577, SE-751 23 Uppsala, Sweden, Department of IT/Scientific Computing, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden, and AstraZeneca R&D, Mo¨lndal, SE-431 83 Mo¨lndal, Sweden
Forty years ago Helfferich and Peterson published an article in Science regarding a “paradoxical” behavior in nonlinear chromatography (Helfferich, F.; Peterson, D. L. Science 1963, 142, 661-662). They theoretically predicted that when an excess of sample molecules is injected into a chromatographic column that is equilibrated with a constant stream of identical molecules, the observed peak will not contain the injected molecules. Instead the observed peak will only contain molecules from the stream whereas the injected molecules will exit the column in a slower moving, “invisible” peak. They considered it paradoxical that a single injection in a singlecomponent system could cause the successive elution of two peaks (Helfferich, F. J. Chem. Educ. 1964, 41, 410413). In this study, the paradox is experimentally proven for the first time. Two different strategies were employed: (i) a radiochemical approach and (ii) a method based on the use of two enantiomers in a nonchiral separation system. The experiments were compared with computer simulations. When a chromatographic column is equilibrated with a constant stream of molecules in the mobile phase, a so-called concentration plateau is established. Injecting a sample, with an excess of identical molecules as in the concentration plateau, a single peak will be detected. This peak is a well-known phenomenon in chromatography and is referred to as “the system peak” by the analytical community3,4 and “the perturbation peak” by the engineering community.5 The peak is the result of the propagation of a positive concentration pulse of the stream, and its mean retention time is governed by the tangential slope of its isotherm at the concentration plateau (see Figure 1). The “perturbation peak method” utilizes this relation to estimate parameters in an isotherm equation.3-6 * Corresponding author. E-mail:
[email protected]. Fax: +46-18-55 50 16. † Center for Surface Biotechnology, Uppsala University. ‡ Department of IT/Scientific Computing, Uppsala University. § AstraZeneca R&D. (1) Helfferich, F.; Peterson, D. L. Science 1963, 142, 661-662. (2) Helfferich, F. J. Chem. Educ. 1964, 41, 410-413. (3) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography, 1th ed.; Academic Press: Boston, MA, 1994. (4) Sokolowski, A.; Fornstedt, T.; Westerlund, D. J. Liq. Chromatogr. 1987, 10, 1629-1662. (5) Blu ¨ mel, C.; Hugo, P.; Seidel-Morgenstern, A. J. Chromatogr., A 1999, 865, 51-71. 10.1021/ac030268j CCC: $27.50 Published on Web 01/20/2004
© 2004 American Chemical Society
Figure 1. Schematic Langmuir adsorption isotherm showing how the mean retention time of the perturbation peak, mass peak, and plateau perturbation peaks are related to the tangential slope and the slope of the chord (see eqs 2a and 2b) during strongly nonlinear conditions. Here c (mM) is the mobile-phase concentration and q (mM) is the stationary-phase concentration of the component. A single Langmuir model was assumed with a ) 2 and b ) 0.100 mM-1.
According to Helfferich and Peterson, the injected molecules are not in the observed perturbation peak but will instead be in a slower moving peak.1 This peak, here called the mass peak, cannot generally be detected because it has a combined elution with a deficiency peak of the molecules originating from the stream, here called a plateau perturbation peak (see Figure 2). The mean retention time of the mass peak and this plateau perturbation peak is determined by the slope of the chord of the isotherm at the concentration plateau (see Figure 1). The so-called “tracer-pulse” technique utilizes this relation to estimate points on an isotherm curve.7-9 The “Helfferich paradox”, i.e., the successive elution of two peaks caused by one single injection in a single-component system, has fascinated chromatographers for 40 years. It has not been proven until today; probably because it is a difficult task to detect only a selected amount of molecules from a larger population of identical ones.3 The aim of this study was (i) to prove the paradox and (ii) to investigate the effect more systematically. For this purpose, two completely different experimental approaches were employed. The experiments were also compared with computer simulations. (6) Forsse´n, P.; Lindholm, J.; Fornstedt, T. J. Chromatogr., A 2003, 991, 3145. (7) Stalkup, F. I.; Kobayashi R. AIChE J. 1963, 9, 121-128. (8) Stalkup, F. I.; Deans, H. A. AIChE J. 1963, 9, 106-108. (9) Peterson, D. L.; Helfferich, F.; Carr, R. J. AIChE J. 1966, 12, 903-905.
Analytical Chemistry, Vol. 76, No. 4, February 15, 2004 953
Figure 2. Illustration of the different types of peaks possible, (i) the perturbation peak (ii) the mass peak, and (iii) the plateau perturbation peaks, on three concentration plateaus. For the Langmuir parameters, see Figure 1. (a) A linear plateau, c ) 0.05 mM; θ ) 0.5%. (b) A weakly nonlinear plateau, c ) 0.5 mM; θ ) 4.8%. (c) A clearly nonlinear plateau,, c ) 5 mM; θ ) 33%. The chromatogram shows the result of an analytical injection of a mixture of labeled and unlabeled molecules on a concentration plateau of unlabeled molecules. The solid line shows the perturbation peak (left scale), the dashed-dotted line shows the plateau perturbation peaks (left scale), and the dotted line shows the mass peak (right scale). Here cµL (mM) is the concentration of unlabeled molecules, cL is the concentration of labeled molecules, and the x axis is time. The mean retention times (tR,1 and tR,2) are given by eq 2.
THEORETICAL SECTION Consider a chromatographic system with a single component where the solid-phase concentration q is given by a Langmuir isotherm,
q ) ac/(1 + bc)
(1)
where c is the mobile-phase concentration, a is the initial slope of the isotherm, and b is the equilibrium constant of the phase system. Now take some component molecules and “label” them in some way; note that the labeled and unlabeled molecules should have identical chromatographic properties. Consider an experiment with a concentration plateau c0 of unlabeled molecules where we inject a small sample with a mixture of labeled and unlabeled molecules. Assuming a detector capable of distinguishing between the labeled and unlabeled molecules, it is possible to show that the individual response for the unlabeled molecules will be two plateau perturbation peaks with mean retention times tR,1 and tR,2:
(
tR,1 ) t0 1 + F
(
(2a)
q(c0) c0
(2b)
tR,2 ) t0 1 + F 954
| )
dq dc c)c0
)
Analytical Chemistry, Vol. 76, No. 4, February 15, 2004
where t0 is the column hold-up time and F is the phase ratio.2,3 It is also possible to show that that the individual response for the labeled molecules will be a single mass peak with mean retention time tR,2. Although the individual response of the unlabeled molecules is two peaks, only a perturbation peak at tR,1 will be detected when a detector with equal response for the labeled and unlabeled molecules is used. The reason is that the last plateau perturbation peak and the mass peak, both with mean retention time tR,2, will have the same area but with opposite signs (see Figure 2c). It should be noted that the above holds for all concentration plateaus and injections. According to eq 2, the first plateau perturbation peak will have a mean retention time proportional to the tangential slope of the isotherm at the concentration plateau c0. Similarly the last plateau perturbation peak and the mass peak will have a mean retention time proportional to the slope of the isotherm chord at the concentration plateau c0 (see Figure 1). Here it should be noted that, under linear, analytic conditions (c0 “small”) tR,1 ≈ tR,2; i.e., the two peaks will coincide (see Figure 2a). Linear conditions can be quantified by calculating the fractional surface coverage θ, i.e., the fraction of monolayer built on the stationary phase surface when it is at equilibrium with a certain concentration plateau c0. Using the equation θ) bc0/(1 + bc0), where b is the equilibrium constant of the phase system, θ can easily be calculated. At θ% < 1, linear conditions prevail, whereas at 1 e θ% < 10, the conditions are weakly nonlinear and, at θ% g 10, the conditions are clearly nonlinear.3 The separation factor (R) between two peaks is quantified by using the equation R ) k′2/k′1 (k′ ) (tR - t0)/t0). Thus, at R ) 1.00, the peaks have a combined elution; i.e., they exit at the same time. EXPERIMENTAL SECTION System 1. Apparatus. An HP 1100 liquid chromatographic system was used including a gradient pump, autosampler, and DAD detector from Hewlett-Packard (Palo Alto, CA). The radioactivity was measured continuously with a Radiomatic on-line radioactivity detector model C525TRX from Packard Instrument Co. (Meriden, CT) using a 500-µL flow cell. The radiodetector was coupled in series with the HP DAD detector. The mobilephase flow was 1.00 mL/min, and the scintillation back-up flow rate was 3.0 mL/min. The software used for instrument control, collection, and evaluation of data was FLO-ONE for the radiodetection and HP Chemstation for the UV detection. Chemicals. The acetonitrile used was from Merck (Darmstadt, Germany), and the scintillation fluid Ultima-Flo AP used was from Packard. The unlabeled substance was isosorbide mononitrate (ISMN), and the labeled one was tritium isosorbide mononitrate (T-ISMN) with a specific radioactivity of 25.5 kRq/nmol (synthesized at AstraZeneca R&D, Mo¨lndal, Sweden). The tritium-labeled substance utilized in this study was kept as a liquid solution in glass bottles and safely handled according to the Swedish Radiation Protection Agency.10 Column and Solutions. The column was a PLRP-S (150 × 4.6 mM, 5 µm) from Polymer Laboratories Ltd. (Shropshire, U.K.) (10) The Swedish Radiation Authority, www.ssi.se/english/lank_symbol_Eng.html, (29 Sep 2003), Radiation Protection Act; 1998; p 220.
used at ambient temperature (21 °C). The mobile phase was 10% (v/v) acetonitrile in Milli-Q gradient water to which ISMN or T-ISMN were added. The labeled solutions were prepared by adding T-ISMN from a stock solution dissolved in methanol. A given volume was withdrawn, and the solvent was evaporated under a gentle flow of nitrogen. Thereafter, the solute residue was reconstituted in mobile phase to the correct concentration. Procedures. The perturbation experiments were made by either 25-µL injections of labeled molecules (T-ISMN) on different levels of stable unlabeled (ISMN) concentration plateaus or by injection of unlabeled (ISMN) on stable labeled (T-ISMN) concentration plateaus. The adsorption parameters were derived from the simultaneous processing of the perturbation peaks and the mass peaks using eqs 2a and 2b (see Theoretical Section). The nonlinear regression method used was the Gauss-Newton algorithm with the Levenberg modification as implemented in the software PCNONLIN 4.2 from Scientific Consulting (Apex, NC). System Properties. The phase ratio (F) of the system was 1.71, and the Langmuir parameters were a ) 5.91 and b ) 11.11 M-1. The lag time between the two detectors was determined to be 0.08 min by injecting 100 µL of 25.6 µM T-ISMN and subtracting the retention time of the peak measured by the radiodetector from that measured by the UV detector. It is of essential importance that the phase system does not distinguish between unlabeled and labeled molecules. The absence of such an isotope effect was checked by analytical injections (25 µL of 25.6 µM solutions) of ISMN and T-ISMN on a column equilibrated with a mobile phase lacking both components, and no selectivity was detected. System 2. Apparatus. The chromatographic system used consisted of two ESA 580 pumps from ESA (Chelmsford, MA), one of them mastering the other in the gradient programming mode. An automatic injector MIDAS from Spark Holland (AJ Emmen, The Netherlands) and a UV detector Lambda 1010 from Bischoff (Leonberg, Germany) were used. A MN6 Lauda circulating water bath from Lauda (Ko¨ningshofen, Germany) was used for temperature control of the column. A computer data acquisition system using the software CSW 1.7 from DataApex Ltd. (Praha, Czech Republic) recorded the chromatograms. The pH was measured with a Mettler Toledo MP 125 pH meter from MettlerToledo (Schwerzernbach, Switzerland). An Advantec SF-2100W (Toyo Kaisha, Ltd.) fractionator was used. Chemicals. (+)-Methyl L-mandelate (LM), (-)-methyl D-mandelate (DM; purity > 99%), and methanol (purity >99%) were from Fluka Chemika (Buchs, Switzerland). The acetate buffers were prepared from acetic acid (purity >99.8%) from Riedel-de Hae¨n (Seelze, Germany), anhydrous sodium acetate (purity >99%), and 2-propanol LiChrosolv from Merck (Darmstadt, Germany). The water used was from Millipore, MilliQ grade. The buffer solutions were filtered through 0.45-µm filters (Kebo, Spånga, Sweden). Columns and Solutions. The nonchiral system was a Kromasil KR100-5C18 (150 × 4.6 mm) column from Eka Chemicals (Bohus, Sweden), and 35% (v/v) methanol in Milli-Q water was used as mobile phase. The chiral system was a CHIRAL-AGP (100 × 4.0 mm) column from ChromTech AB, (Ha¨gersten, Sweden), and acetate buffer at pH 6.0 (I ) 0.050) mixed with 0.25% (v/v) 2-propanol was used as mobile phase.
Procedures. The perturbation experiments were made on the nonchiral system. Before the perturbations, the system was equilibrated with a certain concentration of the LM enantiomer of methyl mandelate in the mobile phase. Perturbations were made by 50-µL injections of DM dissolved in the actual LM plateau. The column was placed in an in-house-made water jacket keeping its temperature constant at 27.0 °C. The UV detector was set at 260 nm for
