Influence of pH on Retention in Linear Organic Modifier Gradient RP

Sep 10, 2008 - We started from the general theory of combined pH/organic modifier gradient and provided an explicit however approximate solution relat...
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Anal. Chem. 2008, 80, 7855–7861

Influence of pH on Retention in Linear Organic Modifier Gradient RP HPLC Paweł Wiczling* and Roman Kaliszan Department of Biopharmaceutics and Pharmacodynamics, Medical University of Gdan´sk, Gen. J. Hallera 107, 80-416 Gdan´sk, Poland The purpose of this work was to examine the influence of pH on retention of analytes during the linear organic modifier gradient in RP HPLC. We started from the general theory of combined pH/organic modifier gradient and provided an explicit however approximate solution relating gradient retention time and the pH of the eluent. The accuracy of the proposed model was tested by its ability to describe an experimental data set that comprised retention times for a series of monoprotic acids and bases obtained at different pH and for different gradient durations. The basic analytical properties of the equation relating retention time and pH were determined, such as the inflection point and range of retention times. The use of the pH at inflection point as a measure of pKa was discussed and compared to the chromatographic pKa obtained by a fitting to the proposed model. In conclusion, the work provides theoretical results that augment the knowledge on the impact of pH on the gradient retention. Retention of ionizable analytes in reversed-phase highperformance liquid chromatography (RP HPLC) depends significantly on the pH and organic modifier content of the mobile phase. Various theoretical aspects of the isocratic and gradient RP HPLC technique have been studied in the past. The influence of organic modifier content on retention at isocratic and gradient conditions is well-known,1-6 and the knowledge has been implemented in commercially available softwares.7,8 Theoretical and experimental studies of pH influence on the isocratic retention have been reported by numerous researchers,9-13 and the rules established provide a reliable way for optimizing isocratic separations with * To whom correspondence should be addressed. Tel.: ++48 58 349 3260. Fax: + +48 58 349 3262. E-mail: [email protected]. (1) Snyder, L. R.; Dolan, J. W.; Gant, J. R. J. Chromatogr. 1979, 165, 3–30. (2) Jandera, P.; Churacek, J. Gradient Elution in Column Liquid Chromatography, Theory and Practice; Elsevier: Amsterdam, 1985. (3) Snyder, L. R.; Dolan, J. W. Adv. Chromatogr. 1998, 38, 115–185. (4) Snyder, L. R.; Kirkland, J. J.; Glajch, J. L. Practical HPLC Method Development, 2nd ed.; Wiley-Interscience: New York, 1997. (5) Schoenmakers, P. J.; Billiet, H. A. H.; De Galan, L. J. Chromatogr. 1979, 185, 179–195. (6) Snyder, L. R.; Dolan, J. W. The Linear-Solvent Strength Model of Gradient Elution. In Advances in Chromtography; Brown, P. R., Grushka, E., Eds; Marcel Dekker: New York, 1998; Vol. 38, pp 115-185. (7) Dolan, J. W.; Snyder, L.R. J. Chromatogr. Sci. 1990, 28, 379–384. (8) Molnar, I. J. Chromatogr., A 2002, 965, 175–194. (9) Jano, I.; Hardcastle, J. E.; Zhao, K.; Vermillion-Salsbury, R. J. Chromatogr., A 1997, 762, 63–72. (10) Horva´th, C.; Melander, W.; Molna´r, I. J. Chromatogr. 1975, 125, 129–156. (11) Horva´th, C.; Melander, W.; Molna´r, I. Anal. Chem. 1977, 49, 142–156. (12) Rittich, B.; Pirochtova, M. J. Chromatogr. 1990, 523, 227–233. 10.1021/ac801093u CCC: $40.75  2008 American Chemical Society Published on Web 09/10/2008

respect to pH.14-22 Recently, the pH gradient RP HPLC concept was introduced and applied to pKa determination.23-26 However, there is a lack of sound theoretical assessment of pH effects on the retention during linear organic modifier gradient. Only an empirical function relating pH and retention time has been proposed:27 ⁄

tR )

tR,1 + tR,210s(pH-pKa) ⁄

1 + 10s(pH-pKa)

(1)

where tR,1 and tR,2 denote retention of an ionized or nonionized form of an analyte, pKa′ is an apparent pKa, and s is an empirical parameter. Equation 1 correctly explains changes of retention for the same gradient duration.27 However, it lacks generality and proper theoretical basis. Besides, the estimates of pKa′ have been shown to be biased.30 The use of organic modifier gradient HPLC as a method of pKa determination appeared very promising.28,30 However, still there is a need for a readily applicable and theoretically meaningful (13) Lopes Marques, R. M.; Schoenmakers, P.J. J. Chromatogr. 1992, 592, 157– 182. (14) Heinisch, S.; Rocca, J.L. J. Chromatogr., A 2004, 1048, 183–193. (15) Berges, R.; Sanz-Nebot, V.; Barbosa, J. J. Chromatogr., A 2000, 869, 27– 39. (16) Barbosa, J.; Toro, I.; Sanz-Nebot, V. J. Chromatogr., A 1998, 823, 497– 509. (17) Foley, J. P.; May, W. E. Anal. Chem. 1987, 59, 102–109. (18) Bourguignon, B.; Marcenac, F.; Keller, H. R.; de Aguiar, P. F.; Massart, D. L. J. Chromatogr. 1993, 628, 171–189. (19) Vanbel, P. F. J. Pharm. Biomed. Anal. 1999, 21, 603–610. ¨ zkan, G. A.; Beltran, J. L. Anal. Chim. (20) Sanli, N.; Fonrodona, G.; Barbosa, J.; O Acta 2005, 537, 53–61. ¨ zkan, G. A.; Barbosa, J. J. Chromatogr. (21) Sanli, N.; Fonrodona, G.; Barro´n, D.; O A 2002, 975, 299–309. (22) Torres-Lapasio´, J. R.; Garcı´a-A´lvarez-Coque, M.C. J. Chromatogr., A 2006, 1120, 308–321. (23) Kaliszan, R.; Haber, P.; Ba¸zek, T.; Siluk, D. Pure Appl. Chem. 2001, 73, 1465–1475. (24) Kaliszan, R.; Haber, P.; Ba¸zek, T.; Siluk, D.; Valko, K. J. Chromatogr., A 2002, 965, 117–127. (25) Kaliszan, R.; Wiczling, P.; Markuszewski, M. J. Anal. Chem. 2004, 76, 749– 760. (26) Wiczling, P.; Markuszewski, M. J.; Kaliszan, R. Anal. Chem. 2004, 76, 3069– 3077. (27) Canals, I.; Valku´, K.; Bosch, E.; Hill, A. P.; Rose´s, M. Anal. Chem. 2001, 73, 4937–4945. (28) Chiang, P.; Foster, K. A.; Whittle, M. C.; Su, Ch.; Pretzer, D. K. J. Liq. Chromatogr. Relat. Technol. 2006, 29, 2291–2301. (29) Wiczling, P.; Markuszewski, M. J.; Kaliszan, M.; Kaliszan, R. Anal. Chem. 2005, 77, 449–458. (30) Wiczling, P.; Kawczak, P.; Nasal, A.; Kaliszan, R. Anal. Chem. 2006, 78, 239–249.

Analytical Chemistry, Vol. 80, No. 20, October 15, 2008

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method. Recently, the theory of combined pH/organic modifier gradient has been elaborated.29,30 It is a unified theory of chromatographic retention for any pH and/or organic modifier content changes during a chromatographic run. The proposed model of combined pH/organic modifier gradient has been successfully applied to optimize separations in various chromatographic modes, i.e. isocratic, gradient or when both pH and organic modifier content change in a nonlinear manner.29,30 In this work the general theory of the combined pH/organic modifier gradient was applied to linear organic modifier gradient at constant pH. An explicit approximate solution was proposed and its validity tested on a series of acids and bases. The basic analytical properties of the function relating pH and organic modifier gradient retention time were studied. A special emphasis was put on a possibly complete assessment of the factors affecting the determination of pKa from the gradient data.

where t0 is the column hold-up time and t′R ) tR - t0 is an adjusted retention time equal to the difference between the observed retention time and system hold-up time. Parameter kw,j (j = 1, 2) denotes the retention factor in a neat water eluent of ionized or nonionized form of analyte. Generally, for bases kw,1 < kw,2; thus kw,1 refers to the ionized form and kw,2 to the nonionized form of the analyte; in the case of acids, kw,1>kw,2; thus kw,1 refers to the nonionized form and kw,2 to the ionized form of the analyte. φ(t) and pH(t) indicate changes of organic modifier and pH at the column inlet and depend on the applied pump program. pKa(φ(t)) denotes the changes of pKa with respect to the organic modifier content, which in turn, is a function of time φ(t). In eq 3, the ionized and the nonionized form of analyte are assumed to behave in agreement with the Snyder-Soczewin´ski equation:4

THEORY Linear Organic Modifier Gradient with Constant pH. In linear organic modifier gradient HPLC, the volume fraction of organic solvent φ changes linearly from an initial value φ0 to a final value φf during time tG. When the gradient development is finished the final mobile-phase composition φf remains sustained. The pH is assumed to be constant. Mathematically, the mobile-phase changes at column inlet are described by the following set of equations:

where Sj is a constant, characteristic for each form of the analyte and the chromatographic system involved. Further, pKa is assumed to be independent of organic modifier content during gradient run:

{

φ0 for t e td φ(t) ) φ0 + β(t - td), for td < t e tG + td φf for t > tG + td pH(t) ) pH

(2)

where β is the steepness of the gradient β ) (φf - φ0)/tG and td denotes dwell time (time that gradient needs to start at column inlet). Obviously, proper pH measurements are essential to study the influence of pH on the chromatographic retention. In this work, the mobile-phase pH was measured after mixing an aqueous buffer and the organic modifier. The electrode system was calibrated with the usual aqueous standards. This led to the absolute pH scale swpH. The pKa obtained from data expressed in the ws pH scale provides a thermodynamically meaningful dissociation constant of the compound expressed in the same scale, ws pKa. The details of proper assessment of pH in mixed organic/water mobile phases have been discussed by Rose´s31 and the relevant procedure recommended by IUPAC.32 For the sake of simplicity, the left side notation will not be used further unless it needs to be emphasized. Hence, the symbols pH and pKa mean below ws pH and ws pKa. General Model. The combined pH/organic modifier gradient model has been proposed to describe the effect of both pH and organic modifier content on analyte retention.29,30 This model has been derived under assumptions of linear, ideal chromatography, where sample is injected in a small volume and the gradient is produced as programmed. The model equation is as follows:

∫

tR

0

1 1 + 10pH(t)-pKa(φ(t)) dt ) 1 (3) t0 10log kw,1-S1φ(t) + 10log kw,2-S2φ(t)10pH(t)-pKa(φ(t))

(31) Rose´s, M. J. Chromatogr., A 2004, 1037, 283–298. (32) IUPAC. Compendium of Analytical Nomenclature: Definitive Rules 1997, 3rd ed.; Blackwell: Oxford, 1998.

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Analytical Chemistry, Vol. 80, No. 20, October 15, 2008

log kj ) log kw,j - Sjφ,

j ) 1, 2

pKa(φ) ) pKa chrom

(4)

(5)

The pKa chrom is a zeroth-order approximation to the true relationship between pKa and φ. It is also a measure of ionization of an analyte and can be viewed as a resultant of all the pKa values occurring during the chromatographic run as a consequence of changes in organic modifier content. Combining eq 3 with eq 5 leads to the following model:30

∫

tR

0

1 1 + 10pH-pKa chrom dt ) 1 (6) log k -S φ(t) t0 10 w,1 1 + 10log kw,2-S2φ(t)10pH-pKa chrom

Simplified Linear Organic Modifier Gradient Model. Equation 6 must be solved numerically. However, under the assumption that both the ionized and the nonionized forms of an analyte have the same value of S1 ) S2 ) S, it is possible to obtain an explicit solution. Typically, for the same chromatographic system, the S value does not differ considerably among analyte forms. For a monoprotic analyte, the median difference between S of the ionized and nonionized form of analytes equals 0.6.30 Let us rewrite eq 6 to a form

∫

tR

0

10S(φ(t)-φ0)

dt ) 1

(7)

t0k0

where ¯k0 denotes retention factor of an analyte at the initial mobilephase composition:

k¯0 )

k0,1 + k0,210pH-pKa chrom 1 + 10pH-pKa chrom

(8)

and k0,j is retention factor of the ionized or the nonionized form of analyte at the initial value of organic modifier content φ0, which equals (from eq 4)

k0,j ) 10log kw,j-Sφ0

(9)

For polyprotic analytes, a different form of k¯0 applies, as presented in Appendix 1. By denoting b as a steepness parameter b ) Sβt0

(10)

The solution of eq 7 has already been proposed for the linear organic modifier gradient specified by eq 2:33

{

tR )

td + td + tG -

(

t0k0

( ) )

td t0 log 2.303bk0 1 b tk

(

0 0

+1

for tR e td for td < tR e tG + td

)

t0 t0  + t0k0 - td + 10-btG⁄t0 for tR > tG + td 2.303b 2.303b (11)

Properties of Eq 11. To analyze properties of eq 11, it is reasonable to focus on the most common case: when the adjusted retention time satisfies condition td < t′R e tG + td, and the retention during the dwell time is high, i.e., t0k¯0 . td. It leads to

tR ) t0 + td +

t0 log(2.303bk0 + 1) ) t0 + td + b

(

)

t0 k0,1 + k0,210pH-pKa chrom log 2.303b + 1 (12) b 1 + 10pH-pKa chrom From eq 12, it follows that for a constant pH during the chromatographic run retention of an analyte follows the classical organic modifier gradient HPLC theory. Thus, one can optimize chromatographic conditions for ionizable analytes using the available approaches1,4,5 as long as pH is approximately constant during the chromatographic run. At the pH providing full ionization or full inhibition of ionization, pH , pKchrom or pH . pKchrom, solution of eq 12 is given by the well-known relationship describing retention during an organic modifier gradient:1,4

tR,j ) t0 + td +

t0 log(2.303bk0,j + 1), b

j ) 1, 2

(13)

By combining eqs 12 and 13, one obtains an equation parametrized by retention of the ionized and nonionized forms of analyte:

tR )

(

t0 10btR,1⁄t0 + 10btR,2⁄t010pH-pKa chrom log b 1 + 10pH-pKa chrom

)

(14)

A graphical explanation of the properties of eqs 12 and 14 is given in Figure 1 for hypothetical basic and acidic analytes. In general, the analyte retention during a linear organic modifier gradient depends on the retention of its ionized and nonionized forms, the ratio of steepness parameter b and hold-up time (steepness of gradient and S), and the pKa chrom. Equation 14 and (33) Schoenmakers, P. J.; Billiet, H. A. H.; Tijssen, R.; de Galan, L. J. Chromatogr. 1978, 149, 519–537.

Figure 1. Theoretically predicted retention times for different pH under linear gradient conditions for a hypothetical acidic analyte characterized by log k0,1 ) 2, log k0,2 ) 1, S1 ) S2 ) 4, and pKa chrom ) 6, and basic analyte characterized by log k0,1 ) 1 log k0,2 ) 2, S1 ) S2 ) 4, and pKa chrom ) 6. The following parameters of the model RP HPLC system were used in the calculations: t0 ) 1 min, td ) 0 min, tG ) 20 min, and φ ranging from 0.05 to 0.8. The meaning of symbols is explained in the text.

eq 12 are sigmoidally shaped curves, which range between the retention of the completely ionized and the completely nonionized form of an analyte (eq 13). For b approaching zero (isocratic conditions), eq 14 reduces to a typically used equation that describes the effect of pH on isocratic data:

tR )

tR,1 + tR,210pH-pKa chrom

(15)

1 + 10pH-pKa chrom

It confirms the validity of the proposed theoretical development. The retention of an analyte at pH equal to pKa chrom (tchrom) represents the condition when concentrations of ionized and nonionized forms of an analyte are equal:

tchrom )

(

t0 10btR,1⁄t0 + 10btR,2⁄t0 log b 2

)

(16)

The inflection point of the function relating retention time and pH is an easily obtainable parameter by a graphical method. Thus, it might serve as an easy measure of pKa. However, it differs from pKa chrom. The pH at inflection point pHinfl can be obtained from eqs 12 and 14: pHinfl ) pKa chrom + 0.5b(tR,1 - tR,2) ⁄ t0 ) pKa chrom +

(

0.5 log

)

2.303bk0,1 + 1 (17) 2.303bk0,2 + 1

with retention time tinfl at that pH: Analytical Chemistry, Vol. 80, No. 20, October 15, 2008

7857

tinfl )

tR,1 + tR,2 2

(18)

For an analyte with high retention, eq 17 can be simplified to pHinfl ) pKa chrom + 0.5(log k0,1 - log k0,2) ) pKa chrom + 0.5(log kw,1 - log kw,2)(19) indicating that the inflection point is shifted from the pKa chrom by a factor that depends on the steepness parameter, b, and the retention factor of ionized and nonionized forms of an analyte at the initial mobile-phase composition. Specifically, when kw,i is large, the inflection point is shifted from the pKa chrom by a factor that is not dependent on the gradient characteristic. It depends only on the difference between log k0 (or log kw) of ionized and nonionized forms of the analyte. Thus, it is only determined by the properties of the column. For bases, pHinfl is smaller than pKa chrom, whereas for acids, pHinfl is larger than pKa chrom. pKa chrom. The variations of the solvent composition during gradient elution cause some changes in the acidity of the compounds. These changes are generally linear if the organic modifier content is less than 80%.34-36 The significant transport of an analyte in the column during wide organic modifier gradient occurs only for some small range of organic modifier content. This effective range of organic modifier content can be calculated as the change in organic modifier content, ∆φ, that causes analyte migration through the terminal 90% of a column. It equals (Appendix 2)

∆φ )

(

)

2.303k0βSt0 + 1 1 1 log ≈ S S 2.303k0βSt00.1 + 1

(20)

In the case of analyte with a high retention at the beginning of the gradient, ∆φ is inversely proportional to S. Since for methanol S equals ∼4-6, the analyte retention is accompanied by a 17-25% change in methanol content. Such a value confirms that using the gradient method it is difficult to obtain reliable estimates of the pKa changes with organic modifier content, because a small range of φ is considered. On the other hand, the pKa chrom corresponds to a certain value of swpKa as would be determined at certain organic modifier content φchrom: pKa chrom ) wspKa(φchrom)

(21)

The φchrom can be roughly estimated as the inlet organic modifier content at the time of analyte elution during gradient at pH equal to pKa chrom. From eqs 24 and 8 it follows

φchrom ) φ0 +

(

)

(

)

kw,1 + kw,2 k0,1 + k0,2 1 1 log 2.303 b + 1 ≈ log S 2 S 2 (22)

(34) Rose´s, M.; Rived, F.; Bosch, E. J. Chromatogr., A 2000, 867, 45–56. (35) Rived, F.; Canals, I.; Bosch, E.; Rose´s, M. Anal. Chim. Acta 2001, 439, 315–333. (36) Sy´kora, D.; Tesaa´ova`, E.; Armstrong, D. W. LCGC North Am. 2002, 20, 974–981.

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Interestingly, for strongly retained analytes (the approximation of eq 22), φchrom does not depend on the parameters of gradient. It depends only on the physicochemical properties of the analyte and on the column and the organic modifier chosen. From the point of view of pKa determination, the pKa chrom is constant as long as the same system is considered. Changing the column or switching to different organic modifier will lead to a different pKa chrom value due to the difference in φchrom. EXPERIMENTAL SECTION Materials and HPLC Equipment. Experiments were done using a Merck-Hitachi LaChrome (Darmstadt, Germany, San Jose, CA) apparatus equipped with a diode array detector, autosampler, and thermostat. Chromatographic data were collected using a D-7000 HPLC System Manager, version 3.1 (Merck-Hitachi). Numerical analysis and data processing were done with Matlab Software version 7.0 (The MathWorks, Inc., Natick, MA). A Prototype BEH C-18 column, 150 × 4.6 mm i.d., particle size 5 µm (Waters Corp., Milford, MA), with a low silanol activity was used. The 1% urea was injected to determine the column dead volume, Vo, which was 1.78 ± 0.02 mL. The dwell volume Vd equals of 1.4 mL. The chromatographic measurements were done at 25 °C with eluent flow rate of 1.0 mL/min. All the reagents and the analytes employed were of a highest commercially available quality. pH Measurements. A universal buffer was used to control the pH of the linear methanol gradient. The base buffer solution was formed using three compounds: citric acid, tris(hydroxymethyl)aminomethane, and glycine, each at a concentration of 0.008 M. Buffer of w w pH ) 10.50 (buffer C) was made by adding the necessary volume of 3 M KOH to the base solution. Buffer of wwpH ) 2.50 (buffer D) was made by adding the necessary volume of 1 M HCl. The mobile phases contained buffers D and C in different proportions and methanol as the organic modifier (solvent B). The pH of the buffers was measured at 25 °C. The measurements were done with an HI 9017 pH-meter (Hanna Instruments, Bedfordshire, UK). The relationship between swpH, methanol content, and various buffer compositions was experimentally determined to ensure constant pH during the organic modifier gradient. A large number of solvent combinations was mixed by the Merck-Hitachi LaChrome system, and subsequently, ws pH was measured. Next, the five-step pump program was found that gave constant pH despite an increase in methanol content. Experimental Design. Retention times for series of acids and bases listed in Table 2 were measured at a wide organic modifier gradient concentration range (from 0.05 to 0.8) with gradient durations of 20, 40, and 60 min for 10 values of pH. The pump (37) Bosch, E.; Bou, P.; Allemann, H.; Rose´s, M. Anal. Chem. 1996, 68, 3651– 3657. (38) Jano, I.; Hardcastle, J. E. J. Chromatogr., B 1998, 717, 39–56. (39) Avdeef, A. Absorption and Drug Development. Solubility, Permeability, and Charge State; John Wiley & Sons: New York, 2003. (40) Howard, P., Meylan, W., Eds. Physical/Chemical Property Database (PHYSPROP), 1999 ed.; Syracuse Research Corp.; Environmental Science Center; North Syracuse, NY, 1999 http://www.syrres.com/esc/physdemo. htm. (41) Florence, A. T.; Attwood, D. Physicochemical Principles of Pharmacy, 4th ed.; Macmillan Press: London, 2005. (42) Evstratova, K. I.; Ivanova, A. I. Farmatsiia 1968, 17, 41–45. (43) Tammilehto, S.; Bu ¨ chi, J. Pharm. Acta Helv. 1968, 43, 726–738. (44) Sammes, P. G.; Drayton, C. J.; Hansch, C.; Taylor, J. B. Comprehensive Medicinal Chemistry; Pergamon Press: Oxford, 1990.

Table 1. Pump Program Ensuring Constant Values of pH during Linear Organic Modifier Gradienta w

s

pH

% MeOH

3.40

4. 15

4.89

5.64

6.39

7.13

7.88

8.63

9.37

10.12

5.0 24.0 43.0 61.0 80.0

77.0 66.0 53.0 37.0 20.0

66.0 57.0 46.0 34.0 18.0

56.0 49.0 41.0 30.0 17.0

48.0 42.0 35.0 26.0 15.0

% Buffer C 40.0 32.0 35.0 29.0 29.0 24.0 23.0 19.0 13.0 11.0

25.0 22.0 19.0 15.0 9.0

18.0 16.0 14.0 11.0 7.0

11.0 10.0 9.0 7.0 4.0

4.0 4.0 4.0 2.0 0.0

a

The % buffer D can be calculated by the following : % buffer D ) 100% - % MeOH - % bufer C.

Table 2. Parameter Estimates for a Series of Basic and Acidic Analytes Obtained by Fitting Eq 11 to the Experimental Dataa analyte

log k01

% CV

log k02

% CV

S

% CV

pKa chrom

% CV

φchrom

pHinfl

chloropropamide diclofenac glipizide indomethacin ketoprofen naproxen propylparaben tolbutamide phenylbutazone

2.60 3.78 3.21 4.21 3.08 3.27 2.62 2.74 3.48

1.90 3.19 3.49 4.62 4.08 3.55 2.63 2.68 3.77

1.97 3.24 2.69 3.68 2.44 2.37 1.84 2.22 2.69

1.60 3.00 3.23 4.44 3.62 3.02 2.65 2.38 3.36

4.67 5.01 5.28 5.74 4.71 4.92 4.09 4.64 4.86

Acids 2.95 4.21 4.83 5.82 5.85 4.92 4.25 4.07 5.14

5.24 5.13 5.71 5.52 5.28 5.37 8.93 6.10 5.32

0.71 1.28 1.27 1.60 1.43 0.93 0.59 0.93 1.08

47 68 55 69 57 58 51 51 63

5.55 5.40 5.97 5.78 5.60 5.82 9.31 6.36 5.71

aminophenazone chlorodiazepoxide diphenhydramine doxepine cocaine morphine N,N-diethylaniline perphenazine pindolol propranolol reserpine verapamil indoramine ketoconazole procainamide

0.69 2.16 2.65 2.70 1.97 2.00 0.77 3.71 1.38 2.65 3.96 3.23 2.47 3.02 0.58

6.19 2.10 3.05 2.84 1.95 2.39 4.53 2.25 2.64 2.90 3.79 2.77 2.42 3.87 2.75

2.06 3.16 3.83 3.90 3.55 3.65 2.80 4.26 3.02 3.70 5.15 4.30 3.83 3.73 2.61

1.54 1.93 3.60 3.34 2.66 3.25 3.28 2.37 4.37 3.41 4.05 3.04 2.90 4.18 5.21

Bases 5.17 4.90 5.09 4.96 5.46 5.59 3.33 5.30 6.66 5.25 6.77 5.96 6.07 4.74 8.02

2.66 2.74 4.66 4.33 3.47 4.17 5.26 2.98 5.41 4.40 4.77 3.78 3.67 5.61 6.52

4.61 4.47 8.75 8.37 8.29 8.42 6.24 7.25 9.56 9.24 6.72 7.78 8.21 5.68 9.18

0.52 0.71 0.56 0.52 0.44 0.55 0.86 0.63 0.75 0.57 0.67 0.47 0.45 1.16 0.94

32 56 67 70 58 59 63 74 42 63 73 67 58 70 31

w

wpKa lit

(ref)

4.9240 4.1540 5.9040 4.5040 3.9839 4.2040 8.4743 5.2740 4.5040

4.01 3.97 8.16 7.77 7.50 7.60 5.33 6.97 8.75 8.72 6.13 7.25 7.53 5.33 8.24

5.0040 4.7640 8.9840 8.0041 8.6140 8.2140 6.5740 7.8040 9.5440 9.5339 6.6040 8.9240 7.7042 6.5144 9.2640

RMSE 1.24 1.80 1.93 2.18 2.58 2.08 2.11 1.79 2.23

0.95 1.19 1.96 1.90 1.40 1.68 3.44 1.33 1.57 1.78 1.62 1.37 1.32 2.61 1.58

a The value of φchrom was calculated by eq 24. pHinfl was calculated by eq 17. For gradient duration 20, 40, and 60 min, the values of pHinfl were identical. The literature pKa values are from refs 39-44. RMSE denotes the root mean square error, and % CV denotes the coefficient of variation.

programs for selected pH values are presented in Table 1. Prior to each measurement, the column was equilibrated by passing six column volumes of the mobile phase. The obtained data were fitted to the proposed model (eq 11) by a least-squares algorithm implemented in the Matlab lsqcurvefit function. RESULTS AND DISCUSSION The experimental and the model predicted retention times versus pH profiles are shown in Figure 2. These plots demonstrate that the proposed model predicts retention times very well. The root-mean-square error (RMSE) ranged from 0.95 to 3.44 min. That goodness-of-fit parameter shows that the average difference between the predicted and observed data is small. The parameter estimates obtained by the nonlinear curve fittings are listed in Table 2. All the parameters were estimated with high precision, as confirmed by small coefficients of variation (% CV). The fittings to the more general model (eq 6 with different S for the ionized and nonionized forms of analyte) were not significantly better than

the currently proposed model. The average RMSE of all analytes changed from 1.71 to 1.82 min. It indirectly justifies the assumption of equal S values for the ionized and nonionized forms of analyte. The obtained pKa chrom values were close to the literature aqueous wwpKa. The observed differences were most likely caused by the presence of organic modifier content. The determined φchrom ranged from 31 to 74%. In general, the presence of organic modifier increases the acidic and decreases the basic value of s 37 wpKa. The same trends were observed in this work. The average difference between pKa chrom and wwpKa was 0.68 ± 0.42 for acids and -0.27 ± 0.45 for bases. As has been reported previously, the pKa chrom serves as a good marker of analytes′ acidity. In addition, when presented together with φchrom, the ws pKa is meaningful thermodynamically. The pKa estimated by means of an inflection point (pHinfl) gave biased estimates as was predicted in Theory (eq 17). The difference between pKa chrom and pHinfl is in contradiction with an Analytical Chemistry, Vol. 80, No. 20, October 15, 2008

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Figure 3. Comparison of the estimated pKa chrom and pHinfl with literature pKa values for bases (b) and acids (1). The dotted line represents line of identity. The literature pKa values are from refs 39-44. Figure 2. Theoretical model predictions and the retention time data obtained for 8 selected basic and acidic analytes. The chromatographic data were collected at 10 different pH of mobile-phase compositions for the duration of organic modifier gradient 20, 40, and 60 min (lower, middle, and upper plots within each subplot). The predictions were obtained using eq 11.

equation proposed by Canals (eq 1), where the inflection occurs at pHinfl ) pKa′ .27 Since the difference between the log kw of the ionized and nonionized forms of an analyte when methanol was a mobile-phase modifier and XTerra MS C-18 (Waters Corp.) was the column equaled 1 ± 0.5 (n ) 93),30 one could expect the difference between pKa determined from inflection point and the pKa chrom of -0.5 for bases and 0.5 for acids. This has been noticed in our previous work30 where the difference equaled -0.6 ± 0.3 for bases and 0.3 ± 0.2 for acids. The difference between pKa chrom and pHinfl is moderate but clearly indicates that inflection point should be avoided as the measure of analyte acidity. The comparison of pKachrom and pHinfl to the literature values is presented in Figure 3. The project here presented aimed mostly at the theoretical analysis. Therefore, a large number of experimental data was collected to validate the model. However, in principle, fewer experiments can be designed to obtain reasonable parameter estimates. It is critical to ensure that at least one measurement is close to the pKa of each analyte. Thus, for an unknown sample, a series of gradients should be performed at pH spanning a wide range of pH. Additionally, at least two wide organic modifier gradients should be used that differ in gradient duration to obtain a reliable estimate of the steepness parameter b. Thus, one can assume the minimal design as a series of 8 experiments with four runs each of 20-min duration and pH equal to 3, 5, 7, 9, and 4, with tG ) 60 min at pH 4, 6, 8, and 10. This simple design can be changed depending on the user’s needs, e.g., shorter gradient duration. 7860

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Using the proposed theoretical model, special emphasis must be put on the column properties. The column should have low silanol activity ensuring partitioning as an operative mechanism of analyte retention. There are available a variety of columns with wide operational pH range that can be used for that purpose. Of course, there are stationary phases in analytical use that are not based on the silica matrix. In conclusion, a comprehensive explicit model of HPLC retention in linear organic modifier gradient mode at constant pH was proposed and successfully applied to describe experimental data. The theoretical analysis was presented, which allows for a better understanding of the mechanism controlling HPLC retention at gradient conditions. Possible applications for practical pKa determinations were discussed. The gradient technique of pKa determination has several valuable advantages. First, all the analytes are washed out of the column and there is not need to search for specific conditions providing measurable retention times, as is the case at isocratic conditions. Second, the method is not sensitive to impurities. Third, the simultaneous analysis of several analytes can be performed especially when the chromatograph is coupled with the mass spectrometer. In general, the procedure proposed can be readily adjusted to the medium/high throughput settings. Appendix 1. For the analyte with r dissociation steps, the following equation describing the isocratic retention factor as a function of pH has been proposed:9,38 n

k0,0 + k¯0 )

∑k

r·pH-pKa,chrom(r) 0,r10

r)1 n

1+

∑ 10

r·pH-pKa,chrom(r)

r)1

(23)

where pKa,chrom(r) ) ∑i r) 1 pKa,i,chrom is the sum of the first r logarithm of dissociation constants, k0,r (r = 0...n) represents isocratic retention factors of a consecutive form of an analyte that appears when the pH gradually increases from a low to high value. The analyte with n dissociation steps has n + 1 forms. Appendix 2. The inlet organic modifier organic content when analytes is eluted from the column φel is calculated by combining eqs 2, 12, and 10: 1 φel ) φ0 + log(2.303k0βSt0 + 1) S

(24)

For the same run, the inlet organic modifier content when analyte was at 10% of the column is equivalent to φel with 10 times smaller hold-up time (0.1t0). By subtracting the φel for normal and smaller t0 the ∆φ is obtained: ∆φ )

(

2.303k0βSt0 + 1 1 log S 2.303k0βSt00.1 + 1

)

pH(t) pHinfl s

wpH

pKa pKa,chrom pKa′ s wpKa w w pKa s Sj t0 tinfl tchrom

(25)

which is coincident with eq 20. GLOSSARY φ the volume fraction of an organic modifier φ(t) the changes of the volume fraction of an organic modifier during chromatographic run φ0 initial volume fraction of an organic modifier for linear organic modifier gradient φel inlet organic modifier content when analyte is eluted from the column. φf final volume fraction of an organic modifier for linear organic modifier gradient b steepness parameter of linear gradient as given by eq 9 k0,j retention factor of the ionized or nonionized form of an analyte at the initial volume fraction of an organic modifier kw,j retention factor of ionized or nonionized form of analyte in a neat water eluent k¯0 retention factor of an ionizable analyte at the initial mobile-phase composition

td tG tR tR,j tR′ β φchrom

∆φ

changes of pH during a chromatographic run pH at the inflection point of the function relating gradient retention times and pH pH of the mobile phase measured after mixing the aqueous buffer and the organic modifier acidity constant the apparent acidity constant obtained from the proposed model (eq 6 and eq 11) apparent pKa of eq 1 pKa obtained from data expressed in the ws pH scale aqueous pKa slope parameter of eq 1 slope parameter of the Snyder-Soczewin´ski equation for the ionized or nonionized form of analyte hold-up time retention time for the mobile-phase pH equal to pHinfl retention time for the mobile-phase pH equal to pKa chrom dwell time gradient duration retention time retention of the ionized or nonionized form of an analyte adjusted retention time steepness of the gradient inlet organic modifier content at the time of analyte elution during gradient at pH equal to pKa chrom effective range of organic modifier content ensuring the significant transport of an analyte in the column

ACKNOWLEDGMENT We thank Weronika Krzyno´wek for her assistance in performing the experiments. The project was financially supported by a grant from the Ministry of Science and Higher Education, Warsaw, Poland (Grant KBN 2PO5F05529).

Received for review May 29, 2008. Accepted July 24, 2008. AC801093U

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