Modeling the Retention of Neutral Compounds in Gradient Elution RP

Anal. Chem. 2009, 81, 9135–9145

Modeling the Retention of Neutral Compounds in Gradient Elution RP-HPLC by Means of Polarity Parameter Models Adolfo Te´llez, Martı´ Rose´s, and Elisabeth Bosch* Departament de Quı´mica Analı´tica and Institut de Biomedicina (IBUB), Universitat de Barcelona, Martı´ i Franque`s 1-11, E-08028 Barcelona, Spain A three-parameter expression, which was already employed before for the prediction of the retention time in gradient mode in reversed-phase high-performance liquid chromatography (RP-HPLC) with satisfactory results, has been tested here under a variety of gradient patterns, using methanol and acetonitrile as the organic modifiers. A wide variety of compounds have been employed as test solutes, including some complex ones used as drugs, such as hydrocortisone and estriol. Simplifications of this expression have been made by considering two- and oneparameter expressions based on the p polarity parameter model, which was successfully employed before in isocratic mode to perform predictions of retention time. The advantages that this model gives in isocratic conditions, namely simplicity and less previous experimental work, have been applied with profit in its application to gradient mode. Good correlations between the experimental retention times and the predicted ones have been obtained with both equations in most cases. Gradient elution is a valuable working mode in reversed-phase high-performance liquid chromatography (RP-HPLC). It allows the separation of complex mixtures of solutes in a reasonable time, regardless of their different nature and retention factors.1 The use of models able to predict the retention time in gradient elution RP-HPLC results in important savings of time and resources and simplifiestheoptimizationprocedureswhenplanninganexperiment.1,2 Some different models have been successfully employed to describe the behavior of the analytes in gradient elution HPLC.3-6 One approach is based on the gradient elution expressions proposed by Nikitas and Pappa-Louisi.5 It has been used in this work in order to study the suitability of three different models for predicting retention time in gradient elution RP-HPLC. The studied models can be later implemented into more general ones * Corresponding author. Tel: +34 93 402 12 84. Fax: +34 93 402 12 33. E-mail: [email protected]. (1) Poole, C. F. The Essence of Chromatography; Elsevier Science B.V.: Amsterdam, 2003. (2) Nikitas, P.; Pappa-Louisi, A. J. Chromatogr. A 2009, 1216, 1737–1755. (3) Schoenmakers, P. J.; Billiet, H. A. H.; Tijssen, R.; de Galan, L. J. Chromatogr. 1978, 149, 519–537. (4) Wiczling, P.; Markuszewski, M. J.; Kaliszan, M.; Kaliszan, R. Anal. Chem. 2005, 77, 449–458. (5) Nikitas, P.; Pappa-Louisi, A. Anal. Chem. 2005, 77, 5670–5677. (6) Jandera, P. J. Chromatogr. A 2006, 1126, 195–218. 10.1021/ac901723y CCC: $40.75  2009 American Chemical Society Published on Web 10/05/2009

to forecast retention time and resolution, and thus carrying out optimization procedures.7-9 Expressions of the Fundamental Equation of Gradient Elution. Nikitas and Pappa-Louisi, following the work by Snyder et al.10-14 and Jandera et al.,15 carried out an exhaustive study about several existing approaches to the fundamental equation of gradient elution, which is usually expressed as follows:5

∫

0

tR-t0

dt )1 t0kφ

(1)

In this equation, tR is the elution time of the solute, t0 is the column hold-up time, and kφ is the retention factor of the solute for a given mobile phase, the composition of which is equal to the volume of organic solvent per unit of volume of mobile phase, φ. They developed several approaches in order to solve eq 1.5,16,17 One of them is used in this work. It considers the continuous gradient formed in the mixer as if it was a stepwise gradient formed by n infinitesimal small time steps δt.

∫

0

tR-tD*

tD dtc + )1 t0(1 + kφ) t0kφin

(2)

In eq 2, tR is the retention time, dtc is the infinitesimal period of time in which the analyte is under the influence of an individual step of the mobile phase of composition φ, tD is the dwell time of the system, tD* is a corrected dwell time of the system (eq 3), and kφin is the solute retention factor at the initial conditions of the analysis. (7) Torres-Lapasio´, J. R.; Baeza-Baeza, J. J.; Garcıá-A´lvarez-Coque, M. C. Anal. Chem. 1997, 69, 3822–3831. (8) Carda-Broch, S.; Torres-Lapasio´, J. R.; Garcıá-A´lvarez-Coque, M. C. Anal. Chim. Acta 1999, 396, 61–74. (9) Torres-Lapasio´, J. R.; Rose´s, M.; Bosch, E.; Garcıá-A´lvarez-Coque, M. C. J. Chromatogr. A 2000, 886, 31–46. (10) Snyder, L. R. J. Chromatogr. 1964, 13, 415–434. (11) Snyder, L. R. Chromatogr. Rev. 1965, 7, 1–51. (12) Snyder, L. R.; Saunders, D. L. J. Chromatogr. Sci. 1969, 7, 145. (13) Snyder, L. R. In High Performance Liquid Chromatography; Horvath, C., Ed.; Academic Press: New York, 1980; Vol. 1, p 207. (14) Quarry, M. A.; Grob, R. L.; Snyder, L. R. Anal. Chem. 1986, 58, 907–917. (15) Jandera, P.; Churacek, J. Gradient Elution in Liquid Chromatography. Theory and Practice; Elsevier: Amsterdam, 1985. (16) Nikitas, P.; Pappa-Louisi, A.; Papachristos, K. J. Chromatogr. A 2004, 1033, 283–289. (17) Nikitas, P.; Pappa-Louisi, A. J. Chromatogr. A 2005, 1068, 279–287.

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tD* ) tD +

(1 + kφin) kφin

(3)

The condition for an analyte to be eluted is shown in eq 4, and the solution of eq 2 is given in eq 5.

the modifier and the alkyl chains in the stationary phase, and thus it takes a constant value.2 In this instance, the model becomes equivalent to the two-parameter expression of the polarity parameter model22,23 (eq 7). N log kφ ) q + pPm

∫

0

tR-t0

dt )1 t0kφ

n

tR ) tD* + δt

∑ i)1

1 + kφ kφ

(4)

(5)

In both eqs 4 and 5, n is the least integer that validates the condition shown in eq 4. k vs φ Retention Models. To predict retention times, eq 5 requires the knowledge of the relationships between retention factor (k) and φ. Several expressions that relate those two variables through one, two, or three empirical or semiempirical fitting parameters are proposed in the literature.1,2,18-20 They have been satisfactorily employed for the prediction of the retention factor in isocratic RP-HPLC. Three-Parameters Model. One of these expressions, based on three parameters (eq 6), was employed for performing predictions of the retention time in gradient elution RP-HPLC.5

The polarity parameter model is a good approach to the actual behavior of the analytes in RP-HPLC. It is a simplified form of the linear free energy relationship (LFER), established by Abraham and Carr,24-28 and relates k to φ through a parameter that N stands for the polarity of the mobile phase (Pm ). This parameter was defined from the linear relationships observed between log k and the Dimroth-Reichardt polarity parameter (ET(30)).22,29 It can be calculated through the following equations, depending on the organic modifier employed: acetonitrile (eq 8) or methanol (eq 9).22 A reference value of 1 was assigned to PNm for pure water. N Pm ) 1.00 -

2.13φ 1 + 1.42φ

(8)

N Pm ) 1.00 -

1.33φ 1 + 0.47φ

(9)

Equation 7 is easily reachable from eq 6 by considering a ) q + p. For acetonitrile, c ) 2.13p, and b is a constant value equal to 1.42. This leads to eq 10.

log kφ ) q + p log kφ ) a -

cφ 1 + bφ

(6)

Equation 6 contains three parameters, a, b, and c, which are constants of the system (fixed column, organic modifier, and solute). Though it was first empirically derived,21 it can also be obtained theoretically by considering the thermodynamic processes involved in the chromatographic process.2 The model was successfully employed for the prediction of the retention time of eight solutes of the catechol family in gradient elution RP-HPLC, under three different gradient patterns, and using methanol, acetonitrile, or 2-propanol as the organic modifier.5 The three parameters in eq 6 have to be simultaneously determined under isocratic conditions before employing the model in gradient mode. This implies a certain preliminary experimental work to be done. Simplification of the model would lead to a decrease of this previous work, which is one of the goals of this paper. Two-Parameters Model. A first simplification was considering b as the equilibrium constant of the adsorption process between

(18) Rose´s, M.; Subirats, X.; Bosch, E. J. Chromatogr. A 2009, 1216, 1756– 1775. (19) Lopes Marques, R. M.; Schoenmakers, P. J. J. Chromatogr. 1992, 592, 157–182. (20) Schoenmakers, P. J.; Tijssen, R. J. Chromatogr. A 1993, 656, 577–590. (21) Neue, U. D.; Phoebe, C. H.; Tran, K.; Cheng, Y.-F.; Lu, Z. J. Chromatogr. A 2001, 925, 49–67.

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(7)

2.13pφ 1 + 1.42φ

(10)

For methanol, c ) 1.33p and b ) 0.47. This leads to eq 11.

log kφ ) q + p -

1.33pφ 1 + 0.47φ

(11)

Rearrangement of terms in eqs 10 and 11 and substitution of eq 8 or 9 respectively leads to eq 7. This expression depends only on two parameters, which implies less preliminary work compared to eq 6. One-Parameter Model. Equation 7 can be further simplified by assuming a linear relationship between q and p, which was observed for many compounds22 (eq 12). q ) (log k)0 - pPsN

(12)

Substitution of eq 12 into eq 7 and rearrangement of the terms results in eq 13. (22) Bosch, E.; Bou, P.; Rose´s, M. Anal. Chim. Acta 1994, 299, 219–229. (23) Torres-Lapasio´, J. R.; Ruiz-Ańgel, M. J.; Garcıá-A´lvarez-Coque, M. C. J. Chromatogr. A 2007, 1166, 85–96. (24) Sadek, P. C.; Carr, P. W.; Doherty, R. M.; Kamlet, M. J.; Taft, R. W.; Abraham, M. H. Anal. Chem. 1985, 57, 2971–2978. (25) Abraham, M. H.; Rose´s, M. J. Phys. Org. Chem. 1994, 7, 672–684. (26) Abraham, M. H.; Rose´s, M.; Poole, C. F.; Poole, S. K. J. Phys. Org. Chem. 1997, 10, 358–368. (27) Cheong, W. J.; Carr, P. W. Anal. Chem. 1988, 60, 820–826. (28) Cheong, W. J.; Carr, P. W. Anal. Chem. 1989, 61, 1524–1529. (29) Johnson, B. P.; Khaledi, M. G.; Dorsey, J. G. Anal. Chem. 1986, 58, 2354– 2365.

N log kφ ) (log k)0 + p(Pm - PsN)

(13)

This expression, developed previously by Bosch and Rose´s,22,30 contains three parameters, but two of them, (log k)0 and PsN, are constant for a certain chromatographic system (fixed column and organic modifier), which leaves eq 13 as a one-parameter model. (log k)0 stands for the retention of the analyte in the hypothetical case that the polarity values of the mobile and the stationary phases were equal. PsN represents the polarity of the stationary phase. It must be noted that both PsN and (log k)0 are characteristic of the chromatographic system, i.e., not dependent on the solute injected. This model has been successfully employed before to predict the retention factor of neutral solutes in isocratic RP-HPLC and for the characterization of several different chromatographic systems under isocratic conditions.31-39 The difference between the models in eqs N 7 and 13 lies in the fact that the latter considers all log k vs Pm straight lines to cross at the same point. The proper use of eq 13 requires the determination of PsN and (log k)0, which are constant for a given column and organic modifier and mainly depend on the stationary phase (i.e., the working column). The former has a reference value of 0, which corresponds to a Merck LiChrospher 100 RP-18 (100 × 5 mm) column,22 and its value can be positive or negative for stationary phases more and less polar than the reference. The determination of these two parameters implies the measurement of the retention of a proper set of standards.32,37 The use of previously characterized columns, i.e., columns with PsN and (log k)0 parameters determined in advance, allows the chromatographer to require just the value of p of any analyte to be injected, eq 13 then being in fact a one-parameter equation. Once values of PsN and (log k)0 have been obtained, values of p can be calculated for any compound. This parameter mainly depends on the polarity of the solute (polar solutes have low values of p, while nonpolar solutes have high values of this parameter), but it also depends, to a lesser extent, on the nature of the organic modifier and the stationary phase. The calculation of p can be performed in some different ways. The first one consists of performing an isocratic experiment and substituting the value of log k obtained, along with the value N of Pm corresponding to the percentage and kind of organic modifier employed (eq 8 or 9), in eq 13. A second way comes from the fact that the value of p of a solute in a certain system (column and organic modifier) can be calculated from the data in any other different chromatographic system, by means of the following linear relationship:32,33 (30) Rose´s, M.; Bosch, E. Anal. Chim. Acta 1993, 274, 147–162. (31) Rose´s, M.; Bou, P.; Bosch, E.; Siigur, K. Org. React. 1995, 29, 51–53. (32) Torres-Lapasio´, J. R.; Garcıá-A´lvarez-Coque, M. C.; Rose´s, M.; Bosch, E. J. Chromatogr. A 2002, 955, 19–34. (33) Bosque, R.; Sales, J.; Bosch, E.; Rose´s, M.; Garcıá-A´lvarez-Coque, M. C.; Torres-Lapasio´, J. R. J. Chem. Inf. Comput. Sci. 2003, 43, 1240–1247. (34) Torres-Lapasio´, J. R.; Garcıá-A´lvarez-Coque, M. C.; Rose´s, M.; Bosch, E.; Zissimos, A. M.; Abraham, M. H. Anal. Chim. Acta 2004, 515, 209–227. (35) Torres-Lapasio´, J. R.; Garcıá-A´lvarez-Coque, M. C.; Bosch, E.; Rose´s, M. J. Chromatogr. A 2005, 1089, 170–186. (36) La´zaro, E.; Ra`fols, C.; Rose´s, M. J. Chromatogr. A 2005, 1081, 163–173. (37) Izquierdo, P.; Rose´s, M.; Bosch, E. J. Chromatogr. A 2006, 1107, 96–103. (38) Herrero-Martıńez, J. M.; Izquierdo, P.; Sales, J.; Rose´s, M.; Bosch, E. J. Sep. Sci. 2008, 31, 3170–3181. (39) La´zaro, E.; Izquierdo, P.; Ra`fols, C.; Rose´s, M.; Bosch, E. J. Chromatogr. A 2009, 1216, 5214–5227.

psystem2 ) mpsystem1 + n

(14)

In eq 14, m and n are constants that depend on the related chromatographic systems, while psystem2 and psystem1 are the values of p of the same solute in two different systems, 1 and 2. Thus, values of p can be “transferred” from one column to another. Values of p for several solutes, referred to retentions in a Spherisorb ODS-2 column and acetonitrile as the organic modifier of the mobile phase, have been and are still being tabulated in a database currently under construction.33 Values of p can also be transferred between different organic solvents. By calculating its value in a certain column and organic modifier, it is possible to know its value in the same column but a different organic solvent.32 This p parameter is also related to some well-known physical parameters, such as the logarithm of the octanol/water partition coefficient (log Po/w) and some other molecular descriptors including the LFER solute descriptors of Abraham.33,34 Thus, knowing the value of any of them allows the chromatographer to obtain a value of p without any experimental work. An obvious advantage of this polarity parameter model is the possibility of performing predictions of retention times in gradient elution RP-HPLC with no need for previous isocratic experiments for solutes with known p value in any characterized chromatographic system. The goals of this work are (i) to study the model shown in eq 6 under a wide variety of gradients and injected solutes and (ii) to study the simpler models based in the solute polarity parameter model (eqs 7 and 13) as an alternative to eq 6 that reduces the amount of work to be done. For this purpose, a set of 28 solutes selected in order to cover a wide range of different polarities has been tested, and their retention times have been predicted under a wide variety of different gradient profiles: linear, convex, concave, and mixtures of them, of different time lengths (see Figure 1). Acetonitrile (MeCN) and methanol (MeOH) have been used as organic modifiers. EXPERIMENTAL SECTION Chemicals. Acetonitrile, methanol, and acetone were HPLC grade from Merck (Darmstadt, Germany). Water was purified by employing a Milli-Q deionizer from Millipore (Bedford, MA), and its resistivity was 18.2 MΩ · cm. All test solutes employed were of analytical reagent grade or better and were obtained from Carlo Erba (Milano, Italy), Merck, Aldrich (Steinheim, Germany), Sigma (St. Louis, MO), Fluka (Steinheim, Germany), or Baker (Deventer, Netherlands). Instrumentation and Procedure. All experiments were carried out with a Shimadzu liquid chromatograph (Kyoto, Japan) equipped with two Shimadzu LC-10AD pumps and a Shimadzu SPD-M10A diode-array detector. Isocratic and gradient measurements were performed on an XTerra MS C18 column (4.6 × 50 mm) from Waters (Milford, MA). Temperature was kept at 25.0 ± 0.1 °C with a Shimadzu CTO-10AS column oven. The mobile phases employed in the isocratic measurements were mixtures of purified water and acetonitrile or methanol, in percentages ranging from 20 to 80%. The mobile phases employed in the gradient experiments were mixtures of water and acetonitrile or methanol, in percentages increasing from 20 to 80% following 24 different gradient patterns (four linear, eight concave, eight Analytical Chemistry, Vol. 81, No. 21, November 1, 2009

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Figure 1. Gradient patterns employed in this work. All of them start at φ ) 0.2 and finish at φ ) 0.8. Linear (a), concave (b), and convex (c) gradients had durations of 5, 10, 15, and 20 min. Two different curvatures were tested for concave and convex gradients. The other gradients tested (d) were mixtures of these types of gradient and isocratic steps. The dwell time of the system (td ) 3.1 min) is included in the plots. Time (t) is in minutes.

convex, and four mixtures of them; see Figure 1). After each gradient, an isocratic step with a composition of 80% organic modifier was applied until the complete elution of the analytes. The flow rate employed was 1 mL · min-1. Standards were dissolved in methanol at a concentration of 100 ppm and filtered through a 0.45 µm nylon Cameo syringe filter (Sterlitech, Kent, WA). The injection volume was always 10 µL. UV detection was performed at 254 nm. All measurements were taken at least in triplicate. The column hold-up time was determined by using an aqueous solution of potassium bromide at a concentration of 0.1 mg · mL-1 as an unretained solute.40,41 Its detection was performed at 190 nm. In methanol/water systems, the retention time of KBr was constant at every φ, while in acetonitrile/water systems it was dependent on φ. This was taken into account in order to perform the predictions of retention time in gradient elution. The dwell time of the system was determined through the addition of increasing amounts of acetone, as stated in the (40) Rose´s, M.; Canals, I.; Allemann, H.; Siigur, K.; Bosch, E. Anal. Chem. 1996, 68, 4094–4100. (41) Oumada, F. Z.; Rosés, M.; Bosch, E. Talanta 2000, 53, 667–677.

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literature,42 and was demonstrated to be 3.1 min. This was also taken into account in the predictions by considering a previous 20% organic modifier isocratic step 3.1 min long before each gradient. pH measurements of the mobile phases were performed with a Crison 5014 electrode in a Crison pH METER GLP 22 potentiometer, with a precision of ±0.01 pH units. Calculations. Experimental values of retention factor (k) were determined from the experimental retention times of the analyte (tR) and the hold-up time marker (t0), as usual (eq 15).

k)

tR - t0 t0

(15)

Calculations of the predicted tR in gradient mode were performed by Excel spreadsheets, following the system employed by Nikitas et al.5 δt were approximated to increments (42) Snyder, L.; Kirkland, J. J.; Glajch. J. L. Practical HPLC Method Development; John Wiley and Sons: New York, 1997; p 392.

∆t of 0.01 min. The gradients employed were represented by the equation φ(t) ) φi +(φf - φi)f(t), where f(t) ) t/T for linear gradients and f(t) ) (eat/T - 1)/(ea - 1) for curve gradients. In these equations, φi and φf are the initial and the final volume fractions of organic modifier in the mobile phase, T is the duration of the gradient until it reaches φf, and a is a measure of the curvature of the gradient (when nonlinear). RESULTS AND DISCUSSION Considerations Related to the Equation of Gradient Elution. The dwell time (tD), the time that takes the solvent mixture (organic modifier/water) to go from the mixer to the beginning of the column, has to be taken into account when predicting retention times in gradient elution RP-HPLC. In the case of the equipment employed, this dwell time is equal to 3.1 min (see Instrumentation and Procedure). Equations 2, 4, and 5 take this parameter into account to perform predictions. Nevertheless, the retention predictions performed in this work using those equations showed some errors when the experimental retention time of an analyte was lower than or similar to the value of the dwell time, though working very well for the rest of the compounds (which were most of them). These early eluting substances are in some cases actually not affected by the gradient, and thus, they elute in an isocratic way (at a φ equal to φi). However, it is not possible to anticipate this behavior when a substance is injected, and consequently, the model should be able to predict the retention time of these substances as well. In order to solve this problem, we opted just to eliminate the dwell time of eqs 2, 4, and 5 and to add an isocratic previous step of 3.1 min to every gradient pattern employed. In this way, eqs 2, 4, and 5 change to eqs 16, 17, and 18.

∫

0

tR

dtc )1 t0(1 + kφ)

(16)

n

∑ t δtk

g1

(17)

1 + kφ kφ

(18)

i)1 0 φ n

tR ) δt

∑ i)1

The use of the new eqs 16, 17, and 18 instead of eqs 2, 4, and 5 gives good predictions of the retention time for all compounds, regardless of the magnitude of the parameter, as it will be seen below. Prediction of the Retention Time. Three-Parameter Equation. The application of the three-parameter equation (eq 6) was made as follows, for both acetonitrile and methanol as the organic modifier. First, all the analytes were injected under isocratic conditions, at organic modifier compositions between 20 and 80%. Values of log k were calculated from the retention times obtained (eq 15), and they were plotted versus the composition of the mobile phase φ, in order to obtain the values of the parameters a, b, and c for every compound by curve fitting of eq 6. Table 1 shows the values of these parameters for all compounds tested. The retention time of each compound under each gradient pattern tested was then predicted by substitution of these parameters into eq 6 and

Table 1. Values of a, b, and c of the Compounds Studied, Obtained by Curve Fitting of Eq 6 to the Experimental log k Data acetonitrile/water

methanol/water

solute

a

b

c

a

b

c

caffeine resorcinol benzamide antipyrine pyrocatechol estriol acetanilide hydrocortisone 3-methylphenol methyl benzoate propiophenone benzene 4-nitrotoluene butyrophenone chlorobenzene bromobenzene naphthalene benzyl benzoate biphenyl 1,2,4-trimethylbenzene propylbenzene phenanthrene heptanophenone butylbenzene pyrene chysene hexylbenzene hexachlorobenzene

4.15 0.27 2.20 5.23 0.83 14.1 1.59 7.66 1.96 2.51 2.55 2.04 2.73 3.31 2.80 3.01 4.41 5.97 3.99 4.52 4.73 5.73 4.84 4.59 4.94 5.80 5.94 6.05

33.3 0.36 10.4 19.5 1.85 28.1 3.06 14.0 1.59 1.57 1.45 0.64 1.14 1.68 0.94 1.05 2.23 2.91 1.18 1.89 1.91 2.48 1.51 1.33 1.54 1.79 1.54 1.85

166 1.70 33.0 122 4.56 437 9.33 127 6.87 7.73 7.48 4.20 7.02 9.99 6.26 6.97 15.2 24.7 9.44 13.6 14.3 20.5 12.6 11.0 12.6 16.1 14.8 16.6

2.89 0.89 1.33 2.38 0.90 3.82 1.61 4.63 1.94 2.66 6.26 1.93 2.54 3.37 3.15 3.31 3.80 6.37 6.63 4.47 4.20 6.45 5.77 4.70 5.48 6.77 8.05 5.14

4.27 0.67 0.97 1.95 0.11 0.97 0.66 1.42 0.24 0.36 0.45 -0.24 0.14 0.46 0.22 0.22 0.36 1.10 0.83 0.40 0.22 1.01 0.56 0.15 0.41 0.60 0.83 -0.08

20.6 3.77 4.93 10.4 2.51 10.1 4.36 13.6 3.64 4.68 5.08 2.11 3.81 5.97 4.62 4.77 5.88 14.6 11.1 6.75 5.67 13.6 9.76 5.83 8.02 11.1 15.0 4.71

implementation of this equation into eqs 17 and 18. Experimental retention times were then obtained by injection of the analytes under the gradient patterns tested. The plots that represent the calculated retention times as a function of the experimental ones are given in Figures 2a (acetonitrile/water) and 3a (methanol/ water). In these plots, apart from these data, a straight line with a slope of 1 and an intercept of 0 is represented. This line represents the ideal case in which all predicted tR were equal to the experimental ones. The proximity of the points represented to this line proves the quality of the model. Nevertheless, in Figure 2a (acetonitrile as the organic modifier), hydrocortisone shows a slight deviation from its expected behavior according to the model. One must take into account that the structure of hydrocortisone is quite different from the structure of many of the other compounds employed, like, for instance, benzene or heptanophenone. This molecule is bigger and has many functional groups. Thus, it is plausible that the solvation of hydrocortisone differs from that of a common compound, as shown by the plots of the experimental values of log k versus φ under isocratic conditions. Figures 2b and 3b show this representation for four substances tested and compare the experimental plots (where values of retention factor are calculated from the experimental tR values and represented by symbols) with the calculated ones [where log k values were calculated through substitution of the values of a, b, and c (Table 1) into eq 6 and represented by lines]. The model predicts a hyperbolic behavior, which matches quite well with the actual behavior of the compounds, with the before mentioned exception of hydrocortisone. A deviation is found for this compound at a low value of φ. When a gradient is applied under such a pattern that hydrocorAnalytical Chemistry, Vol. 81, No. 21, November 1, 2009

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Figure 2. (a) Retention time calculated by means of eq 6 (tcalc) versus actual retention time (texp) of the 28 analytes studied (see Table 1), at the 24 different acetonitrile/water gradient patterns employed (see Figure 1). Times are in minutes. (b) log k versus φ of several compounds. Symbols employed in both panels a and b: 0, estriol; 4, hydrocortisone; b, propiophenone; and [, methyl benzoate.

Figure 3. (a) Retention time calculated by means of eq 6 (tcalc) versus actual retention time (texp) of the 28 analytes studied (see Table 1), at the 24 different methanol/water gradient patterns employed (see Figure 1). Times are in minutes. (b) log k versus φ of several compounds. Symbols employed in both panels a and b: ×, caffeine; +, antipyrine; b, propiophenone; and [, methyl benzoate.

tisone elutes at a retention time that corresponds to a composition of mobile phase close to 20% acetonitrile, the retention factor predicted by the model will be slightly lower than the real one, and consequently, the predicted retention time under gradient conditions will also be lower than the experimental one. This effect is more important at “slow” gradients (a concave gradient that reaches 80% MeCN in 20 min is the “slowest” one, and a convex 9140


gradient that reaches 80% MeCN in 5 min is the “fastest” one). Nevertheless, the standard deviation of the predictions, including all tested compounds, is ±0.16 min, which indicates that the model actually works. No deviations of the regular behavior has been obtained using methanol as the organic modifier. In this instance, the behavior of hydrocortisone matches well the retention forecasted by the

model. The standard deviation found was in this case ±0.19 min, of the same order as with acetonitrile. Two-Parameter Model. As explained previously, the simplification of eq 6 to eq 7 implies the assumption that the parameter b is constant and equal to 1.42 when acetonitrile is the organic modifier and to 0.47 when it is methanol. Comparison of these reference values with the ones given in Table 1 leads to the conclusion that this assumption is quite right in most cases. The substances that do not clearly follow this approximation in acetonitrile/water have mainly complex structures (hydrocortisone, estriol, antipyrine, caffeine), though there are some other ones with simpler structures that show a similar behavior, like benzamide. Similarly, caffeine, antipyrine, hydrocortisone, and benzyl benzoate show b values that differ much from 0.47 in methanol/water mobile phases. It is interesting to note that, for the rest of the compounds, the values of b are closer to the expected ones when methanol is the organic modifier. According to that, it is possible to forecast that the predictions of the retention time will be good in general, with the exception of some compounds, both using acetonitrile and methanol as the organic modifier (slightly better in the latter case). Another reason to try this simplification comes when the values of b and c shown in Table 1 are more carefully observed. A certain correlation is noticed between these two parameters, since when b increases, c increases, too (correlation coefficients are 0.85 for acetonitrile/ water and 0.79 for methanol/water). This correlation indicates that eq 6 may be overparameterized and that two parameters may be enough to account for the variability of the data. The experimental values of log k obtained in isocratic mode N were substituted into eq 7, where values of Pm corresponding to the different values of φ (and calculated through eqs 8 and 9), between 20 and 80% organic solvent, were employed. The two parameters present in eq 7, p and q, were determined by linear regression of the experimental data for every compound tested. The values of q and p obtained were then directly employed to perform predictions of retention time under the gradient conditions tested (Figure 1), and they are listed in Table 2. Figures 4a and 5a show the plots of the calculated retention time versus the experimental one when using acetonitrile or methanol as the organic modifier, respectively. Predictions performed from eqs 7 and 6 are similar, as shown by comparison of Figures 4a and 5a with Figures 2a and 3a. When eq 7 is used, some compounds show slight deviations from the behavior predicted by the model. These compounds are estriol (clear deviation) and hydrocortisone, when using acetonitrile, and caffeine and antipyrine, when using methanol. All of them are compounds with complex structures and low relative retention times. Figures 4b and 5b show representations of the experimental N (symbols) compared to the calculated ones log k versus Pm (lines) for the same four examples as in Figures 2b and 3b. N Note that the use of Pm instead of φ means that the plots of log N k versus Pm are linear, in contrast to the hyperbolic behavior of the plots of log k versus φ. It is possible to see that for compounds that actually show a linear behavior, like propiophenone and methyl benzoate in the example, which are most of the compounds tested, the model matches very well, N i.e., the variation of log k versus Pm is actually linear. The

Table 2. Values of p and q of the Compounds Studied, Obtained by Curve Fitting of Eq 7 to the Experimental log k Data acetonitrile/water solute caffeine resorcinol benzamide antipyrine pyrocatechol estriol acetanilide hydrocortisone 3-methylphenol methyl benzoate propiophenone benzene 4-nitrotoluene butyrophenone chlorobenzene bromobenzene naphthalene benzyl benzoate biphenyl 1,2,4-trimethylbenzene propylbenzene phenanthrene heptanophenone butylbenzene pyrene chysene hexylbenzene hexachlorobenzene

q -0.89 -1.07 -0.97 -1.07 -1.00 -1.61 -1.04 -1.52 -1.06 -0.91 -0.92 -0.78 -0.96 -0.94 -0.82 -0.82 -0.93 -1.15 -0.92 -0.82 -0.81 -0.88 -0.85 -0.77 -0.71 -0.74 -0.68 -0.51

p 0.97 1.58 1.45 1.86 1.72 3.50 2.15 3.70 2.95 3.36 3.45 3.17 3.84 4.10 3.88 4.04 4.67 5.65 5.12 4.94 5.04 5.42 5.58 5.47 5.50 6.02 6.43 5.96

methanol/water q -1.58 -1.64 -1.52 -1.68 -1.50 -1.93 -1.34 -1.72 -1.22 -1.10 -1.14 -0.76 -1.00 -1.14 -0.99 -0.98 -0.95 -1.29 -1.07 -0.86 -0.89 -0.97 -1.13 -0.92 -0.83 -0.88 -0.91 -0.54

p 2.78 2.45 2.62 3.27 2.56 5.18 2.86 5.27 3.29 3.83 3.88 3.07 3.76 3.88 4.45 4.61 4.89 6.38 6.02 5.43 5.53 6.20 6.68 6.36 6.46 7.29 7.81 7.33

prediction of retention time for estriol and hydrocortisone, in acetonitrile, and caffeine and antipyrine, in methanol, suffer from the same drawback found when using eq 6. The log k values expected according to the model are lower than the experimental ones, so the predicted retention times are also lower than the experimental ones. In the case of hydrocortisone and estriol with acetonitrile as the organic modifier, it is possible to N see that, despite the latter having a calculated log k versus Pm plot more similar to the one expected than the former (Figure 4b), it shows a larger deviation in Figure 4a. The reason for this fact is that estriol elutes at a lower tR than hydrocortisone, and N consequently, it elutes at a higher value of Pm , where the N deviation in the log k versus Pm plot is higher. The variation of N log k versus Pm for these compounds follows a hyperbolic pattern (Figure 4b), and consequently, a model that explains their behavior in that way would suit them better. Fortunately, these compounds are a minority. Caffeine and antipyrine show deviations with methanol, but not with acetonitrile, despite showing very large values of the parameter b (Table 1). The reason for this is that they elute very early, when time differences are low. The standard deviation of the residuals (difference between the experimental and calculated values), omitting hydrocortisone and estriol (acetonitrile/water) and caffeine and antipyrine (methanol/water), is ±0.24 min (acetonitrile/water) and ±0.23 min (methanol/water). Compare these values with the ones obtained by means of the three-parameter model with the same compounds: ±0.12 min (acetonitrile/water) and ±0.20 min (methanol/water). It is interesting to notice that the standard deviation when using the three-parameter model is slightly better for acetonitrile/water mobile phases, while it is almost the same for methanol/water. Analytical Chemistry, Vol. 81, No. 21, November 1, 2009

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Figure 4. (a) Retention time calculated by means of eq 7 (tcalc) versus actual retention time (texp) of the 28 analytes studied (see Table 2), at N the 24 different acetonitrile/water gradient patterns employed (see Figure 1). Times are in minutes. (b) log k versus Pm of several compounds. Symbols employed in both panels a and b: 0, estriol; 4, hydrocortisone; b, propiophenone; and [, methyl benzoate.

Figure 5. (a) Retention time calculated by means of eq 7 (tcalc) versus actual retention time (texp) of the 28 analytes studied (see Table 2), at N the 24 different methanol/water gradient patterns employed (see Figure 1). Times are in minutes. (b) log k versus Pm of several compounds. Symbols employed in both panels a and b: ×, caffeine; +, antipyrine; b, propiophenone; and [, methyl benzoate.

The reason is that, when the three-parameter model was employed, no deviations were found when methanol was the organic solvent employed, while the omission of hydrocortisone in the acetonitrile/water medium leads to an improvement of this statistic. Apart from this, the comparison between models indicates that the use of the two-parameter model gives a slightly higher 9142


value of the standard deviation when acetonitrile is employed, while giving a similar value of this parameter when using methanol/water mobile phases. One-Parameter Model. The application of the one-parameter expression of the polarity model (eq 13) was carried out as follows, for both acetonitrile and methanol as organic modifiers.

Table 3. Values of p of the Compounds Employed As the Calibration Set for the XTerra MS C18 Column Employed in This Work, Obtained As Explained in the Text parametera (log k)0 PsN

acetonitrile/water

methanol/water

-1.095 -0.044

-1.912 -0.192

p solute

acetonitrile/water

methanol/water

benzamide pyrocatechol 3-methylphenol methyl benzoate propiophenone 4-nitrotoluene butyrophenone naphthalene propylbenzene heptanophenone butylbenzene chrysene

1.55 1.77 2.75 3.42 3.50 3.78 4.04 4.58 5.19 5.58 5.93 6.28

2.47 2.45 3.38 3.94 3.91 4.02 4.37 4.94 5.46 5.68 5.91 6.57

a Values of (log k)0 and PsN obtained after the calibration procedure are also shown.

First, a calibration step was carried out. A set of 12 analytes was chosen as the calibration set (see Table 3).37 Note that the goal of this calibration step is to characterize the column. That means that, when working with an already characterized column, the chromatographer can overcome this first step. Values of log k of the selected compounds were measured under isocratic conditions from 20 to 80% organic solvent, and eq 13 was fitted to N them. The values of the parameters obtaineds(log k)0 and Pm for the system, p for each compoundsare shown in Table 3. The advantage of this previous calibration step is the fact that the values of (log k)0 and PsN obtained are fixed for a certain system (organic modifier and column). The second step was the calculation of the values of p of the rest of the analytes to be studied. Experimental determination of these values is the most accurate system, though other ways (transference from different chromatographic systems or calculation from other physical properties) are good approximations as well, having the additional advantage that this second experimental step could be avoided (see One-Parameter Model in the introduction). Substitution of the experimental values of log k determined under isocratic conditions from 20 to 80% organic solvent into eq 13, along with the values of (log k)0 and PsN obtained in the previous step for each isocratic condition, leads to these values of p, whose average values are listed in Table 4 for both acetonitrile and methanol. The third step of the method is the prediction of the retention times under gradient conditions. The retention times calculated through the combined use of eqs 13, 17, and 18, along with the parameters obtained in the previous steps for the system [PNm and N (log k)0] and for each compound (p) and the values of Pm calculated through eq 8 or 9, are then compared with the ones obtained experimentally. Figures 6a (MeCN) and 7a (MeOH) show these compared results.

Table 4. Values of p of the Compounds Not Employed as the Calibration Set, Obtained by Means of Eq 13 solute

acetonitrile/water

methanol/water

caffeine resorcinol antipyrine estriol acetanilide hydrocortisone benzene chlorobenzene bromobenzene benzyl benzoate biphenyl 1,2,4-trimethylbenzene phenanthrene pyrene hexylbenzene hexachlorobenzene

1.39 1.47 1.74 1.83 2.04 2.08 3.61 4.13 4.28 4.84 4.98 5.09 5.35 5.93 6.85 6.98

2.46 2.13 2.63 3.45 2.90 3.87 3.98 4.57 4.70 5.22 5.41 5.47 5.68 6.14 6.71 7.25

The correspondence between calculated and actual retention times in Figure 6a is good in general, with the exception of a couple of compounds, as expected. The predicted retention times of estriol and hydrocortisone versus their experimental retention times show a quite evident deviation from the behavior predicted by eq 13, as already happened with eq 7. This anomalous behavior is again caused by the deviations of the actual behavior of the log N k versus Pm plots for these compounds from that predicted by eq 13. Figure 6b shows these plots for estriol and hydrocortisone and also for propiophenone and methyl benzoate, whose behavior is well explained by the model (like all the rest of the compounds contained in Figure 6a). Note that, as discussed before, the latter compounds show linear experimental log k versus PNm plots, which make them well-explained by the model. The plots in Figure 6b corresponding to estriol and hydrocortisone show nonlinear behaviors, with the highest deviations of the calculated log k (lines) from the experimental log k (symbols) at high values of N Pm . These two compounds have low values of p (see Tables 3 and 4), which means that they mostly elute at the beginning of the gradient. This fact, which is more accentuated as the value of p is lower, causes the analytes to be affected by a composition of mobile phase corresponding to a low value of φ (that is to say, to N a high value of Pm ; see eqs 8 and 9). As expected, the deviations found are higher than those obtained when using eq 7. This is due to the additional restriction imposed in eq 13 that the log k N versus Pm plots for all analytes must cross at the same point. These deviations make the actual value of p that is affecting the compounds when they elute to be higher than the one calculated through the model. Predicted retention times under gradient conditions are then lower than the experimental ones, and this effect is enhanced at slow gradients. The same behavior is found when methanol is the organic modifier employed. Figure 7a shows a representation of the calculated values of the retention time versus the experimental one. In this case, all the values of p are higher than those obtained in acetonitrile (see Tables 3 and 4). Some deviations are found in caffeine and antipyrine, and the reason is the same as in acetonitrile (see Figure 7b). Again, their low values of p make N these compounds to elute when Pm is high, and the difference between the predicted (line) and actual (symbols) log k versus N Pm plots is higher. Analytical Chemistry, Vol. 81, No. 21, November 1, 2009

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Figure 6. (a) Retention time calculated by means of eq 13 (tcalc) versus actual retention time (texp) of the 28 analytes studied (see Tables 3 and 4), at the 24 different acetonitrile/water gradient patterns employed (see Figure 1). Times are in minutes. (b) log k versus PNm of several compounds. Symbols employed in both panels a and b: 0, estriol; 4, hydrocortisone; b, propiophenone; and [, methyl benzoate.

Figure 7. (a) Retention time calculated by means of eq 13 (tcalc) versus actual retention time (texp) of the 28 analytes studied (see Tables 3 and 4), at the 24 different methanol/water gradient patterns employed (see Figure 1). Times are in minutes. (b) log k versus PNm of several compounds. Symbols employed in both panels a and b: ×, caffeine; +, antipyrine; b, propiophenone; and [, methyl benzoate.

The standard deviation is slightly higher when using eq 13, compared with the ones obtained when using eq 6 or 7 (this can be seen by comparing the dispersion of the points in Figures 2a, 4a, and 6a for acetonitrile and Figures 3a, 5a, and 7a for methanol). Values of ±0.34 min (acetonitrile/water) and ±0.44 min (methanol/water) are obtained after omitting the outliers (compare to the values obtained with the other models). Only a few compounds are beyond the predictive capabilities of eq 13. 9144


CONCLUSIONS Several models have been tested to predict the retention time of 28 different compounds in gradient elution RP-HPLC under 24 different gradient patterns. Acetonitrile and methanol have been the organic modifiers employed. A three-parameter expression, which had been already employed before by Nikitas and PappaLouisi, has been tested under a large amount of gradient patterns to predict the retention time of a wide variety of compounds. The

results obtained are very good, and slight deviations were found just for one compound, hydrocortisone, when acetonitrile was the organic modifier employed. Two simplifications of this threeparameter equation, based in the p polarity parameter model, have been tested. One of them, a two-parameter equation, gives very good results under all the conditions tested. Nevertheless, the model shows some deviations for some concrete compounds, which represent less than 10% of the cases. Plots of log k versus N Pm in isocratic mode can serve to forecast this behavior, and the model represents an important reduction of work compared to the one based on the three-parameter equation. The second simplification, a one-parameter equation, shows good results but larger deviations than the two-parameter equation. The use of this equation requires very little experimental work to characterize the solutes when working with previously characterized systems. A single injection of the studied compound at a certain mobile phase composition is enough to predict its retention under any gradient pattern, or even no injection at all due to the existing methods to calculate the value of p of any solute, whereas three or four injections at different mobile phase compositions are required when using the two-parameter

equation, and four or five injections are required when using the three-parameter equation. On the basis of the results obtained, it seems advisable to employ the two-parameter expression, which gives very good predictions for most of the studied compounds, as a general rule. When dealing with complex analytes, the use of the three-parameter expression may be advisable. On the other hand, the use of the oneparameter equation is strongly advisable when working with previously characterized systems, due to the lower amount of previous experimental work that it implies. ACKNOWLEDGMENT Financial support from the Ministerio de Educacioń y Ciencia of the Spanish Government and the Fondo Europeo de Desarrollo Regional of the European Union (project CTQ2007-61623) is acknowledged.

Received for review July 31, 2009. Accepted September 24, 2009. AC901723Y


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Modeling the Retention of Neutral Compounds in Gradient Elution RP

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