Anal. Chem. 2001, 73, 1324-1329
Prediction of Electrophoretic Mobilities. 4. Multiply Charged Aromatic Carboxylates in Capillary Zone Electrophoresis Dongmei Li† and Charles A. Lucy*
Department of Chemistry, The University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
The electrophoretic mobilites of aromatic carboxylates and sulfonates at zero ionic strength were correlated with models incorporating both hydrodynamic and dielectric friction. The hydrodynamic friction was predicted using either the Hu1 ckel spherical ion model or the Perrin ellipsoidal model. Dielectric friction is the charge-induced drag caused by the reorientation of the solvent dipoles in response to the analyte charge. Based on the HubbardOnsager and Zwanzig expressions, the dielectric friction is related to z2/V. Expressions incorporating both the hydrodynamic and dielectric frictional terms successfully predicted infinite-dilution mobilities to within 4.4%. The influence of dielectric friction ranged from 3-8% of the overall drag for singly charged analytes to 39% of the total frictional drag for 1,2,4,5-benzenetetracarboxylate.
induced) frictional drag. The effective charge is governed by the valence number of the ion (Z) and by the acid dissociation equilibria.7-10 The attraction of the ion to the applied electric field is further attenuated by ionic screening caused by the electrolyte buffer. This ionic screening affects mobility in much the same manner as it does activity coefficients, as discussed in detail previously.11 The hydrodynamic friction (fh) in eq 1 is the drag that is experienced when a molecule moves through a viscous medium. The classical expression describing electrophoretic mobility is the Hu¨ckel equation12-14 in which the hydrodynamic friction is described by Stokes’ law (eq 2), where a sphere of radius r moves
Electrophoretic separation results from the differential migration of ions under an applied electric field. As such, a clear understanding of electrophoretic mobility is essential. This is reflected in the numerous recent papers probing the fundamental parameters governing mobility.1-6 However, these papers fail to give a clear statement of the fundamental forces that influence mobility. In this paper, the electrophoretic mobilities of aromatic carboxylates and sulfonates are used as a framework to discuss the factors (size, shape, and charge) that govern the absolute mobility.
through a continuum medium of viscosity η. Most papers modeling electrophoretic mobility involve modifications of Stokes’ law. These modifications can be as simple as altering the constant15-17 or power dependence on the ionic radius.3-5 Additionally, the hydrodynamic friction may be modified to account for the nonspherical shape of the molecules. These shape corrections have been both theoretical18,19 and empirical.1-3 Interestingly, the grossest assumption in Stokes’ lawsthat the solvent acts as a continuumsdoes not cause significant error.20
BACKGROUND The mobility of an ion can be expressed by
µ ) q/(fh + fdl)
(1)
where q is the effective charge on the ion and fh and fdl are hydrodynamic (size- and shape-related) and dielectric (charge* To whom correspondence should be addressed. Current address: Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada, T6G 2G2. E-mail:
[email protected]. † Current address: The City of Calgary, Bonnybrook Wastewater Treatment Plant (#37), P.O. Box 2100, Station “M”, Calgary, Alberta, T2P 2M5. E-mail:
[email protected]. (1) Grossman, P. D.; Colburn, J. C.; Lauer, H. H. Anal. Biochem. 1989, 179, 28. (2) Mckillop, A. G.; Smith, R. M.; Rowe, R. C.; Wren, S. A. C. Anal. Chem. 1999, 71, 497-503. (3) Fu, S.; Lucy, C. A. Anal. Chem. 1998, 70, 173. (4) Fu, S.; Li, D.; Lucy, C. A. Analyst 1998, 123, 1487.
1324 Analytical Chemistry, Vol. 73, No. 6, March 15, 2001
fh ) 6πηr
(2)
(5) Cross, R. F.; Cao, J. J. Chromatogr., A 1997, 786, 171. (6) Grossman, P. D. In Capillary Electrophoresis: Theory and Practice; Grossman, P. D., Colburn, J. C., Eds.; Academic Press: San Diego, 1992; Chapter 4. (7) Gluck, S. J.; Cleveland, J. A. J. J. Chromatogr., A 1994, 680, 49-56. (8) Cleveland, J. A., Jr.; Benko, M. H.; Gluck, S. J.; Walbroehl, Y. M. J. Chromatogr., A 1993, 652, 301-308. (9) Beckers, J. L.; Everaerts, F. M.; Ackermans, M. T. J. Chromatogr. 1991, 537, 407-428. (10) Smith, S. C.; Khaledi, M. G. Anal. Chem. 1993, 65, 193-198. (11) Li, D.; Fu, S.; Lucy, C. A. Anal. Chem 1999, 71, 687. (12) Heiger, D. N. In High Performance Capillary Electrophoresis: An Introduction; Hewlett-Packard: France, 1992; p 14. (13) In Capillary Electrophoresis: Theory and Practice; Grossman, P. D., Colburn, J. C., Eds.; Academic Press: San Diego, 1992; pp 112-114. (14) Oda, R. P.; Landers, J. P. In Handbook of Capillary Electrophoresis; Landers, J. P., Ed.; CRC Press: Boca Raton, FL, 1994; p 15. (15) Edward, J. T. In Advances in Chromatography; Giddings, J. C., Keller, R. A., Eds.; Marcel Dekker: New York, 1966; Vol. 2, Chapter 2. (16) Edward, J. T. J. Chem. Educ. 1970, 47, 261. (17) Frank, H. S. In Chemical Physics of Ionic Solutions; Conway, B. E., Barradas, R. G., Eds.; John Wiley & Sons: New York, 1966; p 60. (18) Elworthy, P. H. J. Chem. Soc. 1962, 3718. (19) Perrin, F. J. Phys. Radium 1934, 5, 497. (20) Bhattacharyaa, S.; Bagchi, B. J. Chem. Phys. 1998, 109, 7885-7892. 10.1021/ac0006977 CCC: $20.00
© 2001 American Chemical Society Published on Web 02/17/2001
The third term in eq 1, the dielectric friction (fdl), is well established in the physical chemistry and conductivity literature. Nonetheless, it has been largely overlooked in modeling electrophoretic mobilities in capillary zone electrophoresis. The dielectric friction is essentially a charge-induced frictional drag. That the charge can induce an additional frictional term is intuitively understood in terms of waters of hydration (i.e., the hydrated radius of an ion is greater than its crystallographic radius). However, while waters of hydration are capable of explaining some phenomena of ionic solutions (e.g., K+ migrating more rapidly than Li+), they are difficult to quantify. For instance, waters of hydration for Li+ range from 2.3 to 7.4 depending on the determination method.21,22 Further, quoting waters of hydration is physically unrealistic. Solvation is a dynamic process, not a static one. In 1920, Max Born proposed an alternative model to explain the mobilities of alkali metal ions. He postulated that an additional dielectric friction is caused by the orientation of the solvent dipoles in response to the ionic charge. As the ion migrates, it leaves its solvent shell behind. This solvation shell does not relax instantaneously. As a consequence, the dipole moments of this solvation shell induce an additional electric field that opposes the motion of the ion. This opposing force diminishes as the solvent relaxes to its normal orientation. Born’s idea has been further developed by numerous workers using the assumption of point charges within a continuum medium.23 The most advanced formulations of this approach are the Zwanzig and Hubbard-Onsager expressions. The general form of their expressions is
(
)
0 - ∞ (ze)2
fdl ) H τ
02
ri3
(3)
where τ is the solvent dielectric relaxation time and 0 and ∞ are the low- and high-frequency dielectric constants, respectively. All of these terms are constant for a given solvent. The constant H is 3/ for the Zwanzig model and 1/ 8 16.5 for the Hubbard-Onsager model.23 The right-most terms in eq 3 are those that relate to the ion. z is the valence number of the ion charge, e is the charge on the electron, and ri is the crystallographic radius of the ion. The most striking feature of eq 3 is the dielectric friction’s strong dependence on the charge and strong inverse dependence upon the radius of the ion. Thus, dielectric friction will be greatest for highly localized charges. In this work, the absolute mobilities of aromatic sulfonates and carboxylates are correlated with models that incorporate terms reflecting both the hydrodynamic and dielectric frictions. Studies were performed using infinite dilute mobilities to eliminate mobility variations caused by ionic strength effects.11 EXPERIMENTAL SECTION Apparatus. All measurements of mobility were made using a P/ACE 2100 system (Beckman Instruments, Fullerton, CA) with a UV absorbance detector at 214 nm. Untreated fused-silica (21) Bockris, J. O. M.; Reddy, A. K. N. Modern Electrochemistry, 1 ed.; Plenum Press: New York, 1970. (22) Marcus, Y. Ion Properties; Marcel Dekker: New York, 1997. (23) Kay, R. L. Pure Appl. Chem. 1991, 63, 1393.
capillaries (Polymicro Technologies, Phoenix, AZ) of 50 µm i.d., 365 µm o.d., 37 cm total length, and 30 cm to detector were used in all experiments. Before each run, the capillaries were rinsed for 5 min at high pressure (20 psi) with a 0.1 M base which had the same cation as the buffer (potassium, sodium, or lithium) and then with deionized water for 5 min and finally with the buffer for 5 min. Samples were introduced onto the capillary using a low-pressure (0.5 psi) hydrodynamic injection with an injection time of 1-4 s. The separation voltages, varying from 4 to 6 kV, were experimentally verified to be within the linear portion of the Ohm plots for the highest concentration buffers. The capillary temperature was controlled at 25.0 °C, unless otherwise noted. The data acquisition and control were performed on a 386microcomputer using System Gold Software (Beckman, version 8.10). The detector rise time was set at 1 s, and the data acquisition rate at 5 Hz. Chemicals. All solutions were prepared in distilled deionized water (Nanopure water system, Barnsted). All chemicals employed were of analytical grade. All solutions were filtered through 0.45µm nylon filters (Nalge Co., Rochester, NY) prior to use. Potassium phosphate buffers were prepared from potassium phosphate (MCB) and potassium dihydrogen orthophosphate (BDH). Potassium hydroxide (BDH) and phosphoric acid (BDH) were used to adjust the buffer pH. Sodium borate buffers were prepared from sodium borate (Fisher Scientific). Sodium hydroxide (BDH) and boric acid (BDH) were used to adjust pH. Lithium borate buffers were prepared from boric acid (BDH) and lithium hydroxide (Aldrich). Buffer series were prepared over a range of ionic strengths from 0.001 to 0.1 M by dilution of the stock buffer solutions (0.1 M). Sample analytes were from Aldrich, BDH, MCB, Eastman, Fisher Scientific, CHEM Service, and Supelco and were used without any further treatment. Sample anion solutions were prepared at concentrations from 10-5 to 10-4 M. All samples were prepared in the corresponding buffer solution to avoid stacking of the sample zone. Determination of Absolute Mobilities. The absolute mobility (µ0) was determined as described previously.11 Briefly stated, the effective mobility, µe, of an analyte at a given ionic strength was calculated using
µe )
(
)
Ld 1 1 E tm teof
(4)
where Ld is the length to detector (∼30 cm), E is the applied potential (the voltage applied divided by the total length of the capillary (∼37 cm)), tm is the observed analyte migration time, and teof is the migration time of a neutral marker (mesityl oxide). The effective mobilities observed over a range of ionic strengths (0.001-0.1 M) were plotted versus the term xI/(1 + 2.4xI), and linear least-squares regression was used to extrapolate the effective mobilities to zero ionic strength, to give the mobility at the infinite dilutionsthe absolute mobility, µo, of the analyte.11 Molecular Modeling. Molecular modeling was performed as described previously3,24,25 using the Molecular Modeling Pro (24) Fu, S. M. Sc., University of Calgary, Calgary, Canada, 1998. (25) Li, D. M. Sc., University of Calgary, Calgary, Canada, 2000.
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Table 1. Absolute Mobilities and Physicochemical Parameters of Aromatic Carboxylates and Sulfonates
ion
molecular volume V (Å3)
axial ratio
f/f0
charge, z
benzoate o-methoxybenzoate m-methoxybenzoate p-methoxybenzoate o-chlorobenzoate m-chlorobenzoate p-chlorobenzoate o-bromobenzoate m-bromobenzoate p-bromobenzoate o-nitrobenzoate m-nitrobenzoate p-nitrobenzoate p-aminobenzoate 2-naphthalenecarboxylate phthalate 2,6-naphthalenedicarboxylate 1,2,4-benzenetricarboxylate 1,3,5-benzenetricarboxylate 1,2,4,5-benzenetetracarboxylate 1,4,5,8-naphthalenetetracarboxylate benzenesulfonate p-chlorobenzenesulfonate 5-amino-2-chlorobenzenesulfonate 2-amino-5-chloro-4-methylbenzenesulfonate p-toluenesulfonate 2-naphthalenesulfonate 2-anthraquinonesulfonate m-benzenedisulfonate 1,5-naphthalenedisulfonate 2,6-naphthalenedisulfonate 3,6-naphthalenedisulfonate 1,5-anthraquinonedisulfonate 2,6-anthraquinonedisulfonate 1,3,6-naphthalenetrisulfonate 5,5′,7-indigotrisulfonate 5,7,5′,7-indigotetrasulfonate 3,6,8,3′,6′-azopentasulfonate 3,6,8,3′,6′,8′-azohexasulfonate
107 135 135 135 122 122 122 128 128 128 135 135 135 119 154 137 179 166 170 196 250 127 139 155 175 142 168 243 169 214 207 214 297 282 268 349 390 491 541
2.18 2.72 3.33 2.63 2.62 2.92 2.37 2.56 2.91 2.36 2.65 2.77 2.47 2.50 2.47 2.49 2.76 2.64 3.09 2.93 2.98 2.03 2.25 2.54 2.64 2.36 2.37 3.17 2.58 2.83 2.70 2.68 2.79 3.47 3.69 4.12 4.78 5.79 5.75
1.053 1.087 1.125 1.081 1.080 1.100 1.065 1.077 1.099 1.064 1.082 1.090 1.071 1.073 1.071 1.072 1.089 1.082 1.110 1.100 1.104 1.044 1.057 1.075 1.082 1.064 1.065 1.115 1.078 1.094 1.086 1.084 1.092 1.134 1.147 1.173 1.211 1.266 1.264
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 3 4 4 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 4 5 6
a
observed µo (10-4 cm2 V-1 s-1)
% dielectric friction based on eq 13
3.40 3.07 3.11 3.07 3.16 3.22 3.17 3.10 3.13 3.08 3.16 3.25 3.28 3.17 2.98 5.43 4.72 6.79 7.04 7.92 7.55 3.54a 3.32a 3.09a 3.02a 3.27a 3.14 2.71b 6.21d 5.35d 5.49c 5.32b 4.81c 4.84d 7.41d 5.87b 7.60b 7.90b 8.10b
8 6 6 6 7 7 7 7 7 7 6 6 6 7 5 20 15 31 30 39 31 7 6 5 4 6 5 3 16 12 13 12 8 9 19 14 20 22 27
Data from ref 11. b Data were adapted from ref 27 and reanalyzed using Pitts treatment11. c Data from ref 27. d Data from ref 18.
Program (version 1.44, WindowChem Software, Fairfield, CA). This program determines the molecular volumes using the van der Waals increments of Bondi.26 Molecular Modeling Pro does not explicitly give the semiaxes, a, b, and c (a > b > c) of the approximate ellipsoidal of an ion. Estimates of the length and width were determined by rotating the ion on the screen. The maximum length (2a) and maximum width (2b) of the ion were thus found. The depth of the molecule, c, was calculated from V, a, and b, by
c ) 3V/4πab
(5)
Perrin’s ellipsoidal model approximates aromatic molecules as oblate ellipsoids (i.e., it is assumed that a ≈ b). For this approximation, the geometric average of a and b was used, making the expression for the axial ratio (AR)
AR ) xab/c
(6)
Numerical Fitting of Data. The optimum constants used in the expressions were obtained using both the Solver function of (26) Bondi, A. J. Phys. Chem. 1964, 68, 441.
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Microsoft Excel (version 5.0) and the Curve Fitter function of SlideWrite Plus (version 2.0 for Windows, Advanced Graphics Software, Inc., Carlsbad, CA). Solver iteratively alters the parameters to minimize the sum of the squares of the residuals. Curve Fitter uses the iterative Levenberg-Marquardt algorithm, which yields parameters based on the minimization of the sum of the squared deviations. Both methods yielded the same fit constants. The uncertainties quoted in the equations below are the standard deviations of the fit constants determined using Curve Fitter. RESULTS AND DISCUSSION Table 1 lists absolute mobilities for aromatic carboxylates and sulfonates determined either experimentally herein or calculated on the basis of literature data.11,18,27 Also listed in Table 1 are the molecular volumes and axial ratios calculated as described in the Experimental Section and the charge number. These data will be used to test a number of models for describing electrophoretic mobility. Hydrodynamic Models. The classical expression describing electrophoretic mobility in capillary electrophoresis is the Hu¨ckel (27) Friedl, W.; Kenndler, E. Anal. Chem. 1993, 65, 2003-2009.
equation for a spherical ion moving through a viscous continuum medium:
µ0 )
15 × 10-4z q ≈ 6πηr V1/3
(7)
For aqueous solutions at 25.0 °C, the Hu¨ckel equation (in cgs units) is given by the right-hand equality in eq 4. The charge on the ion (q) is equal to ze, where z is the charge number and e is the charge on the electron, and the radius (r) of a sphere is related to its volume (V) via normal geometric relationships. Absolute mobilities calculated using eq 7 and those observed experimentally (Table 1) showed a marked divergence between the two. The average discrepancy was 8.4%. A plot of the mobilities calculated using eq 7 versus the observed absolute mobilities (not shown) gives a slope (1.31 ( 0.06), intercept (-1.2 ( 0.3), and correlation coefficient (0.967), which are far from the ideal values (1.00, 0.00, and 1.000, respectively). The failure of the Hu¨ckel model to predict the mobility of small organic ions has been reported many times in the literature3-5,15,16,19,28,29 and has been attributed to the untenable assumptions behind the Hu¨ckel model. First, the hydrodynamic friction of the ion is modeled as the drag force experienced by a macroscopic spherical particle moving through a continuum solvent.21,28 However, recent theoretical studies using the microscopic approach pioneered by Wolynes have shown that the theoretical values of the hydrodynamic friction for simple spherical ions are surprisingly close to those given by Stokes’ law.20 Second, the Hu¨ckel model assumes the ion is a rigid sphere. Previous mobility studies in our research group suggested that mobility seems to correlate with volume to a power greater than that predicted by the Hu¨ckel equation (1/3) for aliphatic solutes.3-4 Similarly, Cross and Cao observed an r2 dependence for mobility of sulfonamides and peptides.5 We postulated that the flexible aliphatic chains would gyrate as they migrate through solution. This gyration would result in the ion’s migration being retarded by a radius of gyration. Certainly, diffusion coefficients30 and mobility of polypeptides1,6 have been related to the radius of gyration. Similarly, McKillop et al. tried to accommodate the dynamic motion of organic molecules by incorporating additional rotational and inertial radii terms into their model of the electrophoretic mobility of alkylpyridines.2 To accommodate all such variables, the mobilities of the carboxylates and sulfonates (Table 1) were fit to a Hu¨ckel-type hydrodynamic equation, where both the numerator and power dependence on volume were allowed to vary. The resultant best fit expression was
µ0 )
(29 ( 14) × 10-4z V(0.47(0.10)
(8)
where µ0 is the absolute (infinite dilution) mobility (in cm2 V-1 (28) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2 ed.; Marcel Dekker: New York, 1986. (29) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems, 2 ed.; Cambridge University Press: New York, 1997. (30) Hayduk, W.; Minhas, B. S. Can. J. Chem. Eng. 1982, 60, 295.
s-1) and V is the molecular volume (in Å3) as determined using Molecular Modeling Pro. The correlation between calculated and observed mobilities improved slightly by using eq 8 compared with the Hu¨ckel equation (eq 7). The correlation plot (not shown) has a slope (1.16 ( 0.06) and intercept (-0.9 ( 0.3). However, the correlation coefficient (r2) was still only 0.958, and the average relative error using this expression remained very high (10.8%). Finally, investigations were conducted to determine whether a classical hydrodynamic model that incorporated the effect of shape could be used. The hydrodynamic friction resistance for a nonspherical particle moving through a liquid can be expressed relative to the resistance for a sphere of the same volume. That is, a frictional correction ratio (f/f0) can be incorporated into the Hu¨ckel equation:
µ0 ) ze/(f/f0)6πηr
(9)
where f is the frictional coefficient of the nonspherical particle and f0 is the frictional coefficient of the sphere having the same volume.15,18,19 Unfortunately, no method is available for calculating frictional ratios for particles having excessively irregular shapes. However, ellipsoids give good estimates of the frictional ratios for most molecules.16,18 Such ellipsoids are classified as oblate (disk-shaped) or prolate (cigar-shaped) to determine the appropriate frictional ratio. The aromatics studied herein are treated as oblate ellipsoids, and the frictional ratio is calculated using19
[(AR)2 - 1]1/2 f ) f0 (AR)2/3 tan-1[(AR)2 - 1]1/2
(10)
where AR is the axial ratio, determined as described under Molecular Modeling in the Experimental Section. Direct application of eqs 9 and 10 were unsuccessful. The best fit equation obtained for the aromatic molecules in Table 1 was
µ0 )
16 × 10-4z (f/f0)V1/3
(11)
The average relative correlation error for these 35 aromatic ions was 11.7%, which is actually greater than that of the Hu¨ckel equation (8.4%). Our previous studies of aliphatic amines also found the Perrin model (eq 9) to be ineffective3. Dielectric Friction Models. Error analysis of the fitting of eqs 7, 8, and 11 showed that the prediction error was directly related to the charge. This suggests that the failure of these expressions is due to the neglect of the frictional drag caused by the charge itself. This charge-induced drag is known as the dielectric friction, as was discussed in the Background. This dielectric friction can be described by expressions such as those of Zwanzig and Hubbard-Onsager (eq 3). Thus, the absolute electrophoretic mobility (µ0) of the 35 carboxylates and sulfonates in Table 1 were fit to an equation of the general form
µ0 )
V
1/3
Az + Bz2/V
Analytical Chemistry, Vol. 73, No. 6, March 15, 2001
(12) 1327
Figure 1. Correlation between the absolute mobilities calculated using eq 13 with the observed absolute mobilities for the aromatic carboxylates (b) and sulfonates (0) in Table 1. The dotted line is the ideal correlation between the predicted and observed mobilities (slope 1.00, intercept 0.00)
where the value of A and B were allowed to vary. The best fit equation obtained was
µ0 )
17.3 × 10-4z V + 44.6z2/V 1/3
(13)
Figure 1 shows the correlation between the mobilities predicted using eq 13 and the observed absolute mobilities from Table 1. The dotted line in Figure 1 is the ideal correlation (slope 1.00, intercept 0.00). The linear regression line (not shown) has a slope of 0.99 ( 0.03 (1 σ), an intercept of 0.02 ( 0.17 (1 σ), and a correlation (r2) of 0.959. The average error of the mobilities predicted using eq 13 is 4.4%. This is substantially better than that of the Hu¨ckel model (eq 7, 8.4%) but greater than the estimated uncertainty of the tabulated absolute mobilities (2-3%). Variations on the Dielectric Friction Model. Variants of eq 13 were tested without significant improvement. For instance, simplification by elimination of the volume term from the dielectric friction term in eq 13 yielded
µ0 )
16.7 × 10-4z V1/3 + 0.13z2
(14)
The resultant correlation plot (not shown) was statistically equivalent to the ideal (slope 0.98 ( 0.03, intercept 0.059 ( 0.15 (1σ), and r2 ) 0.967). However, the equation did yield a larger average error of (4.8 vs 4.4% for eq 13). It is suspected that eq 14 was successful only because of the limited range in the volume of our test solutes. Alternatively, allowing the power dependence of the charge in the dielectric term to vary yielded
µ0 )
16.4 × 10-4z V + 7.1z3.2/V 1/3
(15)
(correlation plot not shown, slope 0.99 ( 0.02, intercept 0.06 ( 1328
Analytical Chemistry, Vol. 73, No. 6, March 15, 2001
0.12, and correlation (r2) 0.979). Equation 15 yielded a lower average error between predicted and observed mobilities than eq 13 (3.9 vs 4.4%). However, the lower error associated with eq 15 versus that of eq 13 (4.4%) is due solely to the improved fit of the highly charged (-5 and -6) azoalkylsulfonates. Indeed, plots of percentage prediction error versus charge number showed no other correlation (results not shown). Thus, since the z2 in eq 13 has fundamental meaning,23 eq 13 is preferred. However, it should be noted that eq 3, which describes the dielectric friction, is a simplified expression which ignores higher order terms. As a consequence, eq 3 is not fully valid when dielectric friction is significant.31 Thus, the divergence of the highly charged azoalkylsulfonates may reflect the limitations of our expression for dielectric friction. It was speculated that possibly the scatter in Figure 1 might result from differences in the hydrodynamic friction between the solutes. Allowing the power on the volume in the hydrodynamic friction term (left-hand term in denominator) to vary as in eq 8 resulted in no improvement in the predictive power of the resultant equation. Indeed, the average error between prediction and observed absolute mobilities increased to 5.1% (correlation plot not shown, slope 0.957, intercept 0.20, and r2 ) 0.969). As discussed under Hydrodynamic Models above, additional friction due to the nonspherical shape can be accounted for using a frictional correction term (f/f0). Table 1 includes frictional correction terms (f/f0) calculated by approximating the molecules as oblate ellipsoids, as discussed under Molecular Modeling in the Experimental Section. Incorporation of this frictional correction term yielded
µ0 )
18.9 × 10-4z f/f0V1/3 + 40.8(z2/V)
(16)
The correlation between mobilities predicted using eq 16 and the observed absolute mobilities is shown in Figure 2. The slope (0.96 ( 0.03 (1 σ)) and intercept (0.19 ( 0.12 (1 σ)) of the correlation in Figure 2 are statistically equivalent to the ideal values, within the 95% confidence interval. Further, the prediction error of eq 16 is slightly lower than that of eq 13 (4.0 vs 4.4%). The improvement in the predictions in Figure 2 relative to that seen in Figure 1 (eq 13) suggests that some of the scatter evident in Figure 1 may be due to the molecular shape. However, the overall effect of shape on the predicted mobilities is relatively minor. Singly Charged Analytes. Table 2 shows the mobilities observed for the substituted benzoates within our test set, along with the mobilities predicted using eqs 13 and 16. The average errors for eqs 13 and 16 were reasonable, being 2.7 and 3.8%, respectively. However, eq 13 reflects solely the effects of the volume and charge of the analyte. Thus, it predicts geometric isomers (e.g., ortho, meta, para) would have the same mobility. This is clearly not the case based on the observed mobilities in Table 2. Equation 16, wherein the molecule is modeled as an ellipsoid, does predict different mobilities for the different isomers. However eq 16 is wrong in every instance in its predictions of which isomer should migrate most rapidly. (31) Evans, D. F.; Tominaga, T.; Hubbard, J. B.; Wolynes, P. G. J. Phys. Chem. 1979, 83, 2669-2677.
Table 2. Absolute Mobilities for Monosubstituted Benzoates µ0 × 104 (cm2 V-1 s-1)
Figure 2. Correlation between the absolute mobilities calculated using eq 16 with the observed absolute mobilities for the aromatic carboxylates (b) and sulfonates (0) in Table 1. The dotted line is the ideal correlation between the predicted and observed mobilities (slope 1.00, intercept 0.00)
Finally, we previously proposed that the degree of delocalization of charge would affect the charge-induced friction.4 It was proposed that the acid dissociation constant (pKa) could be used as a measure of this charge delocalization. This approach was found to be successful with singly charged aliphatic carboxylates and amines.4 However, no such correlation is evident between literature pKa values and the mobilities of the geometric isomers in Table 2. This is not surprising since dielectric friction only accounts for 6-8% of the total frictional drag of a -1 analyte, based on eq 13. Thus, the small substituent effect on dielectric friction would have little overall effect on the mobility. This suggests that rotational effects such as proposed by McKillop et al.2 must dominate the mobility difference between geometric isomers. CONCLUSIONS The electrophoretic mobilities of organic anions is best correlated using models that reflect both the hydrodynamic frictional drag and the dielectric friction due to the analyte charge. The dielectric friction was successfully modeled using the fundamental relationship derived by Zwanzig and HubbardOnsager.23
ion
pKa
measured
predicted eq 13
predicted eq 16
benzoate o-methoxybenzoate m-methoxybenzoate p-methoxybenzoate o-chlorobenzoate m-chlorobenzoate p-chlorobenzoate o-bromobenzoate m-bromobenzoate p-bromobenzoate o-nitrobenzoate m-nitrobenzoate p-nitrobenzoate p-aminobenzoate
4.19 4.09 4.09 4.48 2.92 3.82 3.99 2.85 3.81 4.00 2.18 3.45 3.44 4.87
3.40 3.07 3.11 3.07 3.16 3.22 3.17 3.10 3.13 3.08 3.16 3.25 3.28 3.17
3.36 3.18 3.18 3.18 3.26 3.26 3.26 3.22 3.22 3.22 3.18 3.18 3.18 3.28
3.51 3.21 3.11 3.23 3.32 3.26 3.36 3.29 3.23 3.33 3.23 3.21 3.26 3.36
For molecules of low charge density, such as substituted benzoates, the effect of dielectric friction on the overall mobility is minor (6-8%). However, for highly charged species, such as tetracarboxybenzene, the dielectric friction accounts for over a third of the overall frictional drag. Thus, it is essential that the effects of dielectric friction be incorporated within any model for electrophoretic mobility. This is particularly true for nonaqueous solvent systems where the effects of dielectric friction are much more pronounced.32 ACKNOWLEDGMENT This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the University of Calgary. Also, the comments of an anonymous reviewer greatly enhanced the manuscript and are gratefully acknowledged. Received for review June 16, 2000. Revised manuscript received December 12, 2000. Accepted January 10, 2001. AC0006977 (32) Roy, K. I.; Lucy, C. A. Anal. Chem., manuscript in preparation.
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