Prediction of Electrophoretic Mobilities. 3. Effect of Ionic Strength in

Anal. Chem. 1999, 71, 687-699

Prediction of Electrophoretic Mobilities. 3. Effect of Ionic Strength in Capillary Zone Electrophoresis Dongmei Li, Shilin Fu, and Charles A. Lucy*

Department of Chemistry, The University of Calgary, Calgary, Alberta, Canada T2N 1N4

Plots of mobility versus the square root of ionic strength (I1/2) do not show the linear behavior predicted by Kohlrausch’s law. Classical electrolyte theory states that such deviations are to be expected due to the finite size of the ions. This paper uses the Pitts equation to account for the effect of ionic size on the ionic strength dependence of mobilities in CZE. Experimental mobilities for carboxylates, phenols, and sulfonates of -1 to -6 charge in aqueous buffers ranging from 0.001 to 0.1 M ionic strength were described by µ- ) µ0 - Az (I1/2/(1 + 2.4I1/2)), where the constant in the denominator is empirically determined. Infinite dilution mobilities (µ0) determined by extrapolation of mobility data to zero ionic strength based on this expression yielded excellent agreement (100.3 ( 3.3%) with literature values for 14 compounds in a variety of buffers. The Pitts equation provides a reasonable estimate of the constant A for solutes up to a charge of -5. However, this constant also depends on temperature and the nature of the buffer counterion, presumably due to ion association. Thus it is most appropriate to determine the constant A empirically for a given buffer system. The electrolyte buffer is the most important and the most flexible variable in capillary zone electrophoretic (CZE).1-3 The pH, the concentration, and the type of buffer can all significantly influence the selectivity, the efficiency, and the speed of separation. The effect of pH on the mobility of weakly acidic and basic solutes has been well described in the literature.4-8 The impact of the mobility of the buffer co-ion on electrodispersive peak broadening was recognized early9 and has been extensively discussed.10,11 Similarly, the impact of the buffer concentration and conductivity on Joule heating has been thoroughly reviewed.12 * Corresponding author: (Facsimile) 403-289-9488; (e-mail) Lucy@chem. ucalgary.ca. (1) Landers, J. P.; Oda, R. P.; Spelsberg, T. C.; Nolan, J. A.; Ulfelder, K. J. BioTechniques 1993, 14, 98-111. (2) Altria, K.; Kelly, T.; Clark, B. LC-GC 1996, 14, 398-404. (3) Li, S. F. Y. Capillary Electrophoresis; Elsevier: Amsterdam, 1993; Chapter 5. (4) Beckers, J. L.; Everaerts, F. M.; Ackermans, M. T. J. Chromatogr. 1991, 537, 407-428. (5) Smith, S. C.; Khaledi, M. G. Anal. Chem. 1993, 65, 193-198. (6) Friedl, W.; Kenndler, E. Anal. Chem. 1993, 65, 2003-2009. (7) Cleveland, J. A., Jr.; Benko, M. H.; Gluck, S. J.; Walbroehl, Y. M. J. Chromatogr., A 1993, 652, 301-308. (8) Gluck, S. J.; Cleveland, J. A., Jr. J. Chromatogr., A 1994, 680, 49-56. (9) Mikkers, F. E. P.; Everaerts, F. M.; Verheggen, Th. P. E. M. J. Chromatogr. 1979, 169, 1-10. (10) Hjerte´n, S. Electrophoresis 1990, 11, 665-690. (11) Susta´cek, V.; Foret, F.; Bocek, P. J. Chromatogr. 1991, 545, 239-248. 10.1021/ac980843x CCC: $18.00 Published on Web 01/05/1999

© 1999 American Chemical Society

Buffer concentration and ionic strength have been noted to have significant effects on separation efficiencies and solute mobility. Higher ionic strength buffers have been recommended to reduce electrodispersion broadening,11 and extremely high ionic strength buffers (up to 2 M) have been used to minimize adsorption of cationic proteins on capillary walls.13,14 Further, differences between sample and buffer ionic strengths has been used to concentrate or “stack” sample components on-capillary.15 In comparison, our understanding of the effect of buffer concentration on solute mobility is lacking. Only a few studies reported the effect of buffer concentration on mobilities.16-20 Similarly, only a few experimental means for circumventing its effect have been reported.8,21 Many of the studies purported to discuss this effect report only the effect of buffer concentration on the apparent mobility.16-18 The apparent mobility is a function of both the solute and the electroosmotic mobilities. The ionic strength effect on the larger electroosmotic flow mobility overwhelms and obscures the ionic strength effects on the solute mobility. This is unfortunate since the isotachophoresis literature predicts that the ionic strength of the buffer can contribute significantly to selectivity between monocharged and mulicharged solutes.22,23 Indeed, Friedl et al.24 and Mechref et al.25 have experimentally demonstrated dramatic selectivity changes in CZE due to ionic strength. In this work, the influence of ionic strength on solute electrophoretic mobility is explained on the basis of a simple relationship arising from classical electrolyte theory. (12) Li, S. F. Y. Capillary Electrophoresis; Elsevier: Amsterdam, 1993; Section 1.3.4. (13) Green, J. S.; Jorgenson, J. W. J. Chromatogr. 1989, 478, 63-70. (14) Bushey, M. M.; Jorgenson, J. W. J. Chromatogr. 1989, 480, 301-310. (15) Burgi, D. S.; Chien, R. L. Anal. Chem. 1991, 63, 2042-2047. (16) Bruin, G. J. M.; Chang, J. P.; Kuhlman, R. H.; Zegers, K.; Kraak, J. C.; Poppe, H. J. Chromatogr. 1989, 471, 429-436. (17) Nashabeh, W.; El Rassi, Z. J. Chromatogr. 1990, 514, 57-64. (18) Issaq, H. J.; Atamna, I. Z.; Muschik, G. M.; Janini, G. M. Chromatographia 1991, 32, 155-161. (19) Survay, M. A.; Goodall, D. M.; Wren, S. A. C.; Rowe, R. C. J. Chromatogr., A 1996, 741, 99-113. (20) Reid, R. H. P. J. Chromatogr., A 1994, 669, 151-183. (21) Rawjee, Y. Y.; Vigh, Gy. Anal. Chem. 1994, 66, 619-627. (22) Everaerts, F. M.; Beckers, J. L.; Verheggen, Th. P. E. M. IsotachophoresisTheory, Instrumentation and Applications; Elsevier: Amsterdam, 1976. (23) Bocek, P.; Deml, M.; Gebauer, P.; Dolnik, V. Analytical Isotachophoresis; VCH: Weiheim, 1988. (24) Friedl, W.; Reijenga, J. C.; Kenndler, E. J. Chromatogr., A 1995, 709, 163170. (25) Mechref, Y.; Ostrander, G. K.; El Rassi, Z. J. Chromatogr., A 1997, 792, 75-82.

Analytical Chemistry, Vol. 71, No. 3, February 1, 1999 687

I)

1 2

∑z

2 i

ci

(2)

i

where zi and ci are the charge and concentration of each ion i, including the reference ion, H+ and OH-. In the absence of an applied electric field, the ionic atmosphere is symmetrically distributed around the reference ion. In effect, this ionic atmosphere screens the reference ion from other ions, thereby reducing its activity. If it is assumed that the reference ion can be approximated as a point charge, the resultant Debye-Hu¨ckel limiting law is Figure 1. Schematic representation of the central reference ion and its ionic atmosphere as assumed by the (A) Debye-Hu¨ ckel limiting law for activity coefficients, (B) Onsager-Debye Hu¨ ckel law for electrophoretic mobilities, (C) Debye-Hu¨ ckel extended law for activity coefficients, and (D) Pitts equation for electrophoretic mobility.

THEORY In this discussion, analogies are drawn between the DebyeHu¨ckel equations for estimating activity coefficients and equations describing equivalent conductance. This is not to imply that the processes are similar. Indeed they are not. However, the underlying assumptions used to derive the familiar Debye-Hu¨ckel limiting law and extended law mirror those used to derive the Onsager and Pitts equations for equivalent conductance. Much of the following discussion was derived from the excellent discussion of ion-ion interactions and ion transport in ref 26, which is strongly recommended. Debye-Hu1 ckel Limiting Law. Debye and Hu¨ckel dealt with the complexities of ion-ion interactions by making a number of simplifying assumptions. First they dealt with only a single central reference ion (anion in Figure 1a). Next they assumed that only Coulombic forces were important between ions. This allowed them to ignore the molecularity of the solvent and treat the solvent simply as a continuum medium of dielectric constant . Third, the ions surrounding the reference ion (anion in Figure 1a) have a net charge equal in magnitude but opposite in sign to the reference ion (positive in Figure 1a). These surrounding ions are subject to thermal motion which effectively smoothes them into a charged continuum. This charged continuum is generally referred to as the ionic cloud or ionic atmosphere. The thickness of the ionic atmosphere (in cgs) is given by κ-1, the Debye-Hu¨ ckel reciprocal length:

κ-1 )

(

8πNAe2 1000kT

)

-1/2

I-1/2 (cm)

(1)

where NA is Avogadro’s number, e is the charge on an electron, is the dielectric constant of the medium, k is the Boltzmann constant, and T is the temperature (in kelvin). I is the ionic strength of the solution given by the summation: (26) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum Press: New York, 1970; Vol. 1.

688

Analytical Chemistry, Vol. 71, No. 3, February 1, 1999

2

1 NA|z+z-|e logγ( ) κ ≈ -0.5115|z+z-|xI 2.303 2RT

(3)

where γ( is the mean ionic activity coefficient, z+ and z- are the cation and anion charges, respectively, and R is the gas constant. The right-hand approximation in eq 3 is the commonly quoted equality for aqueous solutions at 25 °C. The Debye-Hu¨ckel limiting law is valid under conditions where the ionic atmosphere (κ-1) is much larger than the reference ion. As can be seen from eq 1, as the ionic strength increases, this assumption will become less valid. As a result, the validity of the Debye-Hu¨ckel limiting law is restricted to dilute solutions, ionic strength e0.001 M. Equivalent Conductance. Precise measurement of the conductivity of strong electrolytes is easily accomplished. Thus excellent experimental values for conductivity of such salts were achieved over a hundred years ago. Typically the conductivities of electrolyte solutions are reported in terms of equivalent conductivity where (in cgs)

Λ ) 103κc/zc (Ω-1 cm2 equiv-1)

(4)

where κc is the specific conductance, z is the ionic charge (assuming a symmetrical electrolyte where z+ ) z- ) z), and c is the electrolyte concentration in molarity. This equivalent conductance is related to the mobilities of the individual ions by

Λ ) λ+ + λ- ) F(µ+ + µ-)

(5)

where λ+ and λ- are the equivalent conductances and µ+ and µare the mobilities of the ions constituting the electrolyte. F is the Faraday constant. The precise nature of the equivalent conductance data facilitated the development of extensive theoretical models of the nature of electrolyte solutions, which will be used in the following discussion. However, it must be borne in mind that these theoretical models refer to the conductivity of the solution (i.e., in terms of a CE experiment, to the conductivity of the electrolyte buffer). Therefore the application of these models to an analyte ion within the electrolyte buffer is somewhat ambiguous, as has been stressed previously by Friedl et al.24 Herein, the theoretical expressions based on classical electrolyte theory are written in terms of equivalent conductance rather than recast into mobility. This is done to emphasis that the expressions were derived for a

different situation than discussed herein. Nonetheless, the expressions provide guidance for the behavior that can be expected in CZE. Debye-Hu1 ckel-Onsager Equation. Onsager used the ion atmosphere model introduced by Debye and Hu¨ckel to derive an expression for equivalent conductance. However, in the case of electrophoretic mobilities, the effect of the ionic atmosphere is more complex because the reference ion is now moving (Figure 1b). If the reference ion moves in response to an applied electric field, the ions making up the ionic atmosphere must move to reestablish themselves around the reference ion. This reestablishment of the symmetrical ionic atmosphere takes a finite amount of time. During that time, the reference ion continues to move in response to the applied electric field. The net result is that, under electrophoretic conditions, the ionic atmosphere becomes eggshaped and lags behind the moving ion. Since the ionic atmosphere possesses an electric charge opposite that of the reference ion, there will now be a Coulombic attraction between the reference ion and the lagging ionic atmosphere. This relaxation effect retards the motion of the reference ion. Additionally, the ionic atmosphere itself possesses a charge, which will respond to the applied electric field. When the ionic atmosphere moves in response to the applied field, it tries to carry with it all of its constituent ionssincluding the reference ion. Thus, this electrophoretic effect on the ionic atmosphere also causes a retardation in the mobility of the reference ion. The Debye-Hu¨ckel-Onsager equation for the equivalent conductance of an anion (λ) is26

λ- ) λ-0 -

(

z-eFκ e2ωκ 0 + λ 1800πη 6kT -

)

(Ω-1 cm2 equiv-1) (6)

λ-0 is the equivalent conductance of the anion at infinite dilution, z- is the magnitude of the charge on the anion, η is the viscosity, and ω is a parameter related to electrolyte type:27

ω ) z+z- (2q/(1 + xq))

(7)

in which:28

q)

z+zλ+0 + λ-0 z+ + z - z λ 0 + z λ 0 + + - -

(8)

Substitution of numerical values into eq 6 yields the DebyeHu¨ckel-Onsager equation for the anion (in cgs units)

λ- ) λ-0 -

(

)

1.40 × 106 41.25 2q z + z+zλ-0 xI 1/2 3/2 η(T) (T) 1 + xq (9)

Commonly, the Debye-Hu¨ckel-Onsager equation is quoted for symmetrical electrolytes (z+ ) z- ) z) wherein q equals 1/2. Equation 9 then simplifies to (in cgs units):28 (27) Reference 26: p 432. (28) Erdey-Gru´z, T. Transport Phenomena in Aqueous Solutions; Wiley: New York, 1974; Chapter 4.

λ ≈ λ0 -

(

)

41.25 8.20 × 105 2 z+ z λ0 xI 1/2 η(T) (T)3/2 (Ω-1 cm2 equiv-1) (10)

However, such a simplification of the Debye-Hu¨ckel-Onsager equation is not appropriate for the mobility of ions in CZE. Therefore, further discussion will refer to the Debye-Hu¨ckelOnsager equation as expressed in eq 9. For aqueous solutions at 25 °C ( ) 78.54; η ) 0.008 904 P), eq 9 simplifies to

λ- ) λ-0 - (30.16z- + 0.391z+z-(2q/(1 + xq))λ-0)xI (11)

Thus, for a given solute, the Debye-Hu¨ckel-Onsager model reduces to

λ ) λ0 - constxI

(12)

Equation 12 is the general form of the empirical Kohlrauch law observed for equivalent conductances in dilute (e0.001 M) solutions. The constant in eq 12 is referred to as the Onsager slope. CZE. Thus, both theoretical and empirical relationships predict that mobility will decrease with the square root of the ionic strength. For this reason, mobilities observed in CZE have generally been plotted versus the square root of the buffer concentration. However, significant nonlinearities are observed for such plots, particularly for highly charged solutes and high ionic strength.24 The failure of the I1/2-dependence has been attributed to three flaws in the Debye-Hu¨ckel-Onsager equation.24,29 First, the equation is only valid up to 0.001 M. Typical CZE buffers range from 0.001 to 0.1 M. Second, this model refers to the equivalent conductivity of electrolytes, dissolved at different concentrations. It is contended that this is a situation different from the mobility of a dilute solute in a background electrolyte. Third, fundamental studies of conductivity are limited to spherical ions of high charge density. Solutes in CZE are generally asymmetrical organic molecules of low charge density. For the above reasons, Reijenga and Kenndler favored empirically derived expressions for the effect of ionic strength on mobility. Their first empirical expression for mobility relative to the infinite dilution mobility (µ0) was29

µ ) µ0 exp(-0.5z1.78xI)

(13)

where z is the charge number of the solute and I is the ionic strength of the buffer. This expression was based on experimental data limited to solute charge numbers from 1 to 3 and ionic strengths up to ∼0.01 M.24 The stated objective of this expression was to achieve a 10% predictability under actual experimental conditions.29 Use of a wider solute and buffer range (sulfonates of charge numbers from 2 to 6 and ionic strengths of 0.001-0.1 M), Friedl et al.24 observed the empirical relationship (29) Reijenga, J. C.; Kenndler, E. J. Chromatogr., A 1994, 659, 403-415.


689

µ ) µ0 exp(-0.85z0.49I0.52) ≈ µ0 exp(-0.77xzI) (14) Given that the power dependences on the charge number and ionic strength were close to 0.5, Friedl et al.24 preferred the inherent simplicity of the square root relationship. This yielded the right-hand expression in eq 14. When this expression was used to calculate mobilities for these same solutes used to generate the expression, a random error of e5% was observed for charge number (z) of 2-6. A systematic deviation (∼5%) was observed for monocharged sulfonates. Despite the success of eq 14, it is nonetheless an empirical expression. It is impossible to gage the reliability of this expression for other solutes or other buffer systems. Reijenga and co-workers abandoned the Debye-Hu¨ckel-Onsager equation since it was limited to solutions e0.001 M, was derived for equivalent conductance, and dealt with spherical ions. In this work, we will take a different tack. We will return to classical electrolyte theory to determine how the equivalent conductivity in higher ionic strength solutions was dealt with and then try to apply such expressions to CZE mobility data. Debye-Hu1 ckel Extended Law. In the Debye-Hu¨ckel limiting law, it was assumed that the ion could be described as a point charge (i.e., rion ) 0). Under dilute conditions, this is a reasonable assumption. For instance, in a 0.001 M solution of a 1:1 electrolyte, the thickness of the ionic atmosphere is 100 times the radius of the ion. However, as the ionic strength of a solution increases, the thickness of the ionic atmosphere decreases, as described by eq 1. Thus, above an ionic strength of 0.001 M, it is necessary to allow for the finite size of the ion (a) (Figure 1c). Incorporation of this factor leads to the Debye-Hu¨ckel extended law for the mean activity coefficient:30

logγ( ) -

2 1 NA|z+z-|e 1 κ ≈ 2.303 2RT 1+κ

where

(

8πNAe2 1000kT

)

1/2

≈ 0.3291 × 108 cm-1 M-1/2 ) 0.3291 Å-1 M-1/2 (16)

Again the right-hand equalities in eqs 15 and 16 are for aqueous solutions at 25 °C. The ion size parameter a in eq 15 is described as the mean distance of closest approach for the ions. As such, it is greater than the crystallographic radius of the ion and slightly less than the hydrated radius of the ion. If the ion size parameter is assumed to be 3 Å and combined with eqs 15 and 16, the resultant expression for the mean activity coefficient is

logγ( ≈ -0.5115|z+z-|(xI/(1 + xI))

(17)

This is the most common form of the Debye-Hu¨ckel extended (30) Reference 26: Section 3.5.2.

690


Λ) Λ0 - Gxc

[

]

H(x2 - 1)xc 1 Hxc T 1 + κa (1 + κa)2(x2 + κa) (1 + κa) 1

[

Λ0

H(x2 - 1)xc

(1 + κa)(x2 + κa)

]

+ 3H2cS1 (18)

where c is the concentration of the symmetrical electrolyte, κ is related to the Debye-Hu¨ckel reciprocal length (eq 1), a is the ion size parameter (see discussion after eq 15), and S1 and T1 are functions of κa.31,33 H and G are given by

H ) z2e2κ/3kTxc

(19)

G ) kTHNA109/πηC2

(20)

where C is the speed of light. Rearrangement of eq 18 using Mathematica (version 3.0, Wolfram Research Inc., Champaign, IL) using a first-order Pade approximation yields (in cgs units)

(

Λ ) Λ 0 - z2

xI -0.5115|z+z-| (15) 1 + BaxI

B)

law. However, it is important to recall that there are temperaturedependent (T), solvent-dependent (), and ionic size-dependent (a) parameters incorporated into the implicit constant of 1 in the I1/2 term in the denominator. Pitts’ Equation. In analogy to the Debye-Hu¨ckel extended law, an electrophoretic expression can be derived based on the assumptions of an ionic atmosphere surrounding a reference ion of finite size (a). The resultant expression derived by Pitts is exceedingly complex,31,32 and so no attempt is made to represent this expression in its entirety. Indeed, even a simplified form restricted to symmetrical electrolytes (z+ ) z- ) z) is daunting:32,33

)

82.5 1.40 × 106 2q Λ0 × + 1/2 η(T) (T)3/2 1 + xq

xI (21) (8.30 × 10 T + 1.41 × 1016Λ0η) axI 1+ (1.65 × 102T + 1.60 × 106Λ0η) (T)1/2 11

For aqueous solutions ( ) 78.54; η ) 0.008 904 P) at 25 °C and equivalent conductances typical of organic ions, the terms in the right-hand denominator including equivalent conductance diminish in significance. This simplifies eq 21 to

Λ ≈ Λ0 -

(

)

| |

82.5 2 1.40 × 106 2q z + z+zΛ0 × 1/2 1/2 η(T) (T) 1 + xq

xI axI 1 + 5.03 × 10 (T)1/2

(22)

9

For fully dissociated 1:1 electrolytes, eq 22 simplifies to the expression reported by Erdey-Gru´z.28 Alternatively, at the limit (31) Pitts, E. Proc. R. Soc. A 1953, 217, 43-70. (32) Pitts, E.; Tabor, B. E.; Daly, J. Trans. Faraday Soc. 1970, 66, 693-707. (33) Falkenhagen, H.; Kelbg, G. In Modern Aspects of Electrochemistry; Bockris, J. O’M., Ed.; Butterworth: London, 1959; Vol. 2, Chapter 1, eq 235.

of low ionic strength, eq 22 should reduce to the Debye-Hu¨ckelOnsager equation (eq 9). At low ionic strength, eq 22 successfully reduces to eq 9 except with respect to the charge dependence on the electrophoretic effect (left-hand term within the brackets of eq 22). The electrophoretic effect is predicted to have a first-order dependence on the ionic charge in the Debye-Hu¨ckel-Onsager equation (eq 9) but displays a second-order dependence in eq 22. While this discrepancy was apparent in Pitts’ original paper,31 it was not discussed. Expressing the Pitts equation in terms of the equivalent conductance of the anion in an aqueous electrolyte solution at 25 °C yields (in cgs units)

(

| |

λ- ≈ λ-0 - 30.16{z}z- + 0.391 z+z-

)

xI 2q λ-0 1 + xq 1 + BaxI (23)

indicated that the Pitts treatment would yield a Debye-Hu¨ckeltype dependence. Thus, the discussion herein uses the Pitts treatment. When it is necessary to discuss ion association, the treatment of Takayanagi et al.40,41 is used because it is simple and was developed for CZE. For a 1:1 ion associate between a cation C+ and an anion An-, the equilibrium and associated constant are

Λ ) Λ0 - SxRc + ERc log(Rc)JRc - KARcγ(2Λ (24) where S is the Onsager limiting slope, R is the degree of dissociation of the electrolyte, c is the electrolyte concentration, and E and J are coefficients of the higher-order terms in the Fuoss-Onsager equation that involve Λ0, solvent properties, and the ionic size parameter (a). KA is the ion association constant, and γ( is the mean activity coefficient. The Fuoss-Onsager equation might have been preferable for the present discussion, as it does directly incorporate the effect of ionic strength. However, it is not readily apparent what ionic strength dependence is reflected by eq 24, whereas the prior work of Erdey-Gru´z28 had (34) Harris, D. C. Quantitative Chemical Analysis, 4th ed.; Freeman: New York, 1995; Table 8.1. (35) Skoog, D. A.; West, D. M.; Holler, F. J. Fundamentals of Analytical Chemistry, 7th ed.; Saunders: Fort Worth, TX, 1996; Table 8-1. (36) Kielland, J. J. Am. Chem. Soc. 1937, 59, 1675-1678. (37) Fuoss, R. M.; Onsager, J. J. Phys. Chem. 1962, 66, 1722-1726. (38) Fuoss, R. M.; Onsager, J. J. Phys. Chem. 1963, 67, 621-628. (39) Fuoss, R. M.; Onsager, J. J. Phys. Chem. 1964, 68, 1-8.

(25)

KA ) [C+‚An-]/[C+][An-]

(26)

The apparent electrophoretic mobility of the anion (µ′An-) in the presence of ion association is given by

µ′An- ) where B is equal to 0.3291 Å-1 M-1/2 as defined in eq 16. The additional charge dependence associated with the electrophoretic effect is in braces to indicate the discrepancy of this term between the Debye-Hu¨ckel-Onsager equation (eq 9) and the Pitts equation (eq 23). Values reported for the ion size parameter (a) used in calculating activity coefficients range between 3 × 10-8 and 6 × 10-8 cm (3-6 Å).34-36 This suggests that the constant within the denominator of the right-hand term in eq 23, hereafter referred to as Ba, should be about 1-2. Finally, it is generally stated that the Pitts treatment accurately reflects equivalent conductance data up to 0.1 M.28,32 However, that range of validity refers solely to 1:1 electrolyte systems. No experimental studies of higher charge electrolytes have been made. Theoretically, the neglect of higher-order terms in the derivation of the Pitts equation is expected to limit the expressions validity to 2:2 electrolyte concentrations below 1.2 × 10-4 M (I ) 0.0005 M).31 Ion Association. One behavior not accounted for within the Pitts treatment is ion association. The Fuoss-Onsager expression incorporates an ion association term:37-39

C+ + An- h C+‚An-

[C+]KA 1 µ + µCA(n-1)nA 1 + [C+]KA 1 + [C+]KA

(27)

where µAn- is the mobility of the unassociated anion and µCA(n-)- is the mobility of the ion associate, respectively. EXPERIMENTAL SECTION Apparatus. All measurements of mobility were made using a P/ACE 2100 system (Beckman Instruments, Fullerton, CA) with a UV absorbance detector. The selected working wavelength was 214 nm. Untreated fused-silica 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), 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 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 0.45-µm 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) was used to adjust pH. Buffer series were prepared over a range of ionic strength from 0.001 to 0.1 M by dilution of the stock buffer solutions (0.1 M). (40) Takayanagi, T.; Wada, E.; Motomizu, S. Analyst (London) 1997, 122, 5762. (41) Takayanagi, T.; Wada, E.; Motomizu, S. Analyst (London) 1997, 122, 13871391.


691

In studies of the counterion effect experiment, potassium, sodium, or lithium borate buffers were prepared from boric acid and potassium hydroxide, sodium hydroxide or lithium hydroxide (Aldrich). In the electrotype effect experiment, 0.001 M ionic strength sodium borate buffer was used to control the pH and varying amounts of sodium chloride (BDH) or sodium sulfate (BDH) were added to the buffer to prepare a series of solutions of differing ionic strength. The pH of the buffer was chosen to ensure all sample anions were fully deprotonated while maintaining high buffer capacity. Chemicals for sample solutions were obtained 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 the Constant Ba. The optimum constant Ba in the term, I1/2/(1 + BaI1/2), was obtained for each analyte/ buffer series using the Curve Fitter function of SlideWrite Plus (version 2.0 for Windows, Advanced Graphics Software, Inc., Carlsbad, CA). Curve Fitter uses the iterative LevenbergMarquardt algorithm, which yields parameters based on the minimization of the sum of the squared deviations. The experimental mobility data was fit to an equation of the general form

(

µ e ) a0 + a1

xI 1 + a2xI

)

(28)

The constant Ba was one of the three constants (a2) obtained from the fitting. The other fit parameters (a0 and a1) were the infinite dilution mobility (µ0) and the Onsager slope, respectively. Determination of Absolute Mobilities. The effective mobility, µe, of an analyte was calculated from the migration time of the analyte (tm) and the migration time of electroosmotic flow (teof) using following equation:

µe )

(

)

Ll 1 1 V tm teof

(29)

where L is the total length of the capillary (∼37 cm), l is the length to detector (∼30 cm), and V is the applied voltage. The effective mobilities of an analyte at different ionic strengths were then plotted versus the term I1/2/(1 + 2.4I1/2), and the linear leastsquares regression was employed to extrapolate the effective mobility to zero ionic strength, which gives the mobility at the infinite dilutionsthe absolute mobility, µo of the analyte. Experimental data for the AZO sulfonates and monoamines were obtained from refs 24 and 42. RESULTS AND DISCUSSION As cautioned by Friedl et al.,24 classical electrolyte theories such as discussed above were developed to explain equivalent conductance and not mobility. In essence, such theories describe the electrophoretic behavior of the buffer rather than the solute. Thus, these expressions were not derived for, and cannot rigorously be extended to, the situation in CZE. However, since (42) Fu, S. M.Sc. Thesis, University of Calgary, Canada, 1998.

692


Figure 2. Dependence of the mobility of monovalent carboxylate ions on (A) the square root of ionic strength (I1/2) and (B) I1/2/(1 + 2.4I1/2). Solutes: [, propionate; 9, salicylate; 2, benzoate; /, p-nitrobenzoate; b, m-bromobenzoate; 0, m-methoxybenzoate. Curves are second-order polynomial fits in plot A and linear regression in plot B. Experimental conditions: detection, direct UV at 214 nm; applied voltage, 6 kV; capillary, 37 cm (30 cm to detector); buffer, potassium phosphate (pH 7.2); temperature, 25 °C; sample concentration, 0.0001 M.

the complexity of electrophoretic mobility as performed in CZE precludes rigorous theoretical treatment, the theories derived for the simpler case of equivalent conductance will be used to provide guidance for the following discussion. To summarize the previous discussion, the Debye-Hu¨ckelOnsager expression predicts that mobilities decrease with the square root of ionic strength of the buffer. This has been the most common means by which the effect of ionic strength in CZE has been expressed. However, the Debye-Hu¨ckel-Onsager model is only valid at 0.99). Table 3 shows the resultant infinite dilution mobilities (intercept) and Onsager slopes at the three temperatures studied. Literature values for the infinite dilution mobilities are only available for 25 °C (for which there is excellent agreement). While no such comparison is possible at 15 and 50 °C, the infinite dilution mobilities in Table 3 do vary with viscosity as expected. Plots of the infinite dilution mobilities in Table 3 versus 1/η show excellent linearity (r2 > 0.996) with intercepts statistically equivalent to zero. Similarly no literature values are available for the Onsager slopes reported in Table 3. However, these Onsager slopes demonstrated the general dependence upon viscosity, dielectric constant and temperature expected. That is, plots of the Onsager slopes in Table 3 versus 1/η(T)1/2 were rectilinear (r2 > 0.95), as predicted by the electrophoretic effect in eq 22. Thus, the general temperature effects observed in Table 3 are consistent with the Pitts treatment. However, much more extensive data sets are required to fully evaluate the dependences depicted in eq 22. Counterion. Table 4 shows the behavior observed for a number of solutes in pH 9.1 borate buffers prepared with a variety of univalent cations. The values for Ba in Table 4 are consistent with the weighted average of 2.4 determined using the data in Analytical Chemistry, Vol. 71, No. 3, February 1, 1999

697

Table 4. Effect of Buffer Counterion on Ionic Strength Effect on Mobilitya µo (10-4 cm2 V-1 s-1)

µo (lit.)b (10-4 cm2 V-1 s-1)

Potassium Borate Buffer -3.44 ( 0.036 -3.46 ( 0.035 -8.97 ( 0.219

3.355 ( 0.0044 3.128 ( 0.0043 5.50 ( 0.027

3.36;h 3.34;i 3.29j 3.13;j 3.05m 5.42;h 5.27j

0.9990 0.9988 0.9994

Sodium Borate Buffer -4.05 ( 0.072 -3.77 ( 0.077 -9.79 ( 0.133

3.405 ( 0.0089 3.163 ( 0.0095 5.56 ( 0.017

3.36;h 3.34;i 3.29j 3.13;j 3.05m 5.42;h 5.27j

0.9994 0.9996 0.9994

Lithium Borate Buffer -4.41 ( 0.063 -3.97 ( 0.048 -10.49 ( 0.146

3.414 ( 0.0078 3.182 ( 0.0060 5.57 ( 0.018

3.36;h 3.34;i 3.29j 3.13;j 3.05m 5.42;h 5.27j

ion

Ba (mol-0.5 L0.5)

r2

benzoate 2-naphthalenesulfonate phthalate

2.33 ( 0.28 2.15 ( 0.19 2.37 ( 0.66

0.9997 0.9997 0.9982


1.99 ( 0.37 1.83 ( 0.34 2.00 ( 0.22


2.31 ( 0.38 2.69 ( 0.29 2.05 ( 0.26

Onsager slope (10-4 cm2 V-1 s-1 mol-0.5 L0.5)

a pH of the buffers was 9.1, and the ionic strength range was 0.001-0.1 M for all three buffer systems. b See Table 1 for the references of the literature values of the mobilities.

Table 5. Effect of Buffer Co-Ion on Ionic Strength Effect on Mobilitya ion

r2


0.9992 0.9992 0.9961


0.9996 0.9996 0.9903

µo (10-4 cm2 V-1 s-1)

µo (lit.)b (10-4 cm2 V-1 s-1)

Buffer with Sodium Chloride -3.78 ( 0.062 -3.47 ( 0.058 -8.92 ( 0.395

3.470 ( 0.0078 3.100 ( 0.0073 5.465 ( 0.0428

3.36;h 3.34;i 3.29j 3.13;j 3.05m 5.42;h 5.27j

Buffer with Sodium Sulfate -2.97 ( 0.034 -2.73 ( 0.031 -6.76 ( 0.474

3.343 ( 0.0043 3.101 ( 0.0039 5.436 ( 0.051

3.36;h 3.34;i 3.29j 3.13;j 3.05m 5.42;h 5.27j

Onsager slope (10-4 cm2 V-1 s-1 mol-0.5 L0.5)

a pH of the buffers was 9.1, and the ionic strength range was 0.001-0.1 M. b See Table 1 for the references of the literature values of the mobilities.

Table 1. Therefore, the type of counterion does not influence this parameter, which is consistent with the term reflecting the compacted hydrated radius of the solute. Infinite dilution mobilities obtained with all of the alkali metal-borate buffers were in excellent agreement with each other, and with the literature values. However, the Onsager slope was significantly different for the three alkali metals. For all solutes studied, Li+ buffers caused a greater decrease in mobility than Na+ than did K+. This trend was observed for both carboxylates and sulfonates and for noncomplexing and complexing solutes. The greater impact of Li+ than Na+ than K+ on mobility observed herein is consistent with the trend noted by Nielen for mobility of aminobenzoate and p-toluenesulfonic acid in 40 mM acetate (pH 5.4) buffer.46 While stability constants for alkali metal complexes are rare, those available (acetate, malate, tartarate, citrate) consistently show association increasing from K+ to Na+ to Li+.47 For a univalent anion associating with a univalent cation, the ion associate (C+‚A-) is neutral and so has a mobility of zero. Thus, the expression derived by Takayanagi et al.40,41 simplifies to

µ′An- ) (1/(1 + [C+]KA))µAn-

(33)

where µAn- is the mobility of the unassociated anion, C+ is the cation within the buffer, and KA is the ion association constant. Thus, if the ion association constant increases from K+ to Na+ to Li+, the effective mobility (µ′An-) would be most affected by Li+. 698


Buffer Co-Ion. It is easy to appreciate that the nature of the buffer counterion can influence the electrophoretic mobility through ion association. It is not so apparent that the buffer coion (i.e., anion) might have an influence. However, the relaxation effect contains an electrolyte-type dependent term (q), defined by eq 8. As eq 31 indicates that the relaxation makes a significant contribution to the Onsager slope, we decided to explore the effect of the buffer co-ion. It was not possible to simply use a 1:1 buffer (e.g., borate) and a 1:2 buffer (e.g., phosphate at pH ∼10). It was feared that ionic strength-induced changes in the buffer pKa could cause systematic errors when different buffers were compared. Therefore, the effect of electrolyte type was explored using a low concentration of sodium borate (I ) 0.001 M) and increasing ionic strength by adding either NaCl or Na2SO4. In this manner, any pKa shifts caused by ionic strength would appear in both data sets. The behavior observed is reported in Table 5. A significantly lower Onsager slope was observed for the 1:2 electrolyte (Na2SO4) than the 1:1 (NaCl). This is opposite to the trend that would be expected for the relaxation effect. The relaxation effect would be expected to be larger for a 1:1 buffer than for a 1:2, since q is 0.5 for NaCl and 0.41 for Na2SO4. It is believed that the variation in the Onsager slope observed in Table 4 is not directly a result of the co-ion. Rather it is a (46) Neilen, M. W. F., J. Chromatogr. 1991, 542, 173-183. (47) Martell, A. E.; Smith, R. M. Critical Stability Constants; Plenum Press: New York, 1977; Vol. 3.

reflection of the counterion concentration. The sodium concentration in a Na2SO4 solution will be two-thirds that of a NaCl solution of the same ionic strength. Thus, the ion retardation due to ion association would be expected to be smaller in the Na2SO4 solution. CONCLUSIONS Traditionally, the mobility of ions in CZE have been assumed to be related to the square root of the ionic strength of the buffer (I1/2). This relationship derives from the Debye-Hu¨ckel-Onsager law in which the ions were assumed to be point charges. The discussion above demonstrates that to accurately reflect the influence of ionic strength upon the mobility of an ion in CZE it is necessary to allow for the finite size of the ion. Use of the Pitts treatment leads to a mobility dependence of the general form

µ- ) µ0 - AzxI/(1 + 2.4xI)

constant of 2.4 in the denominator is an experimentally determined parameter which experimentally was independent of solute charge, temperature, counterion, and co-ion. Equation 34 successfully linearized the ionic strength dependence of all anions under all conditions studied. Based on this relationship, electrophoretic mobilities at only two ionic strengths are necessary to fully elucidate the ionic strength dependence up to I ) 0.1 M. ACKNOWLEDGMENT This work was supported by the Natural Science and Engineering Research Council of Canada and by the University of Calgary, who are gratefully acknowledged. Thanks also to Dr. Reg Paul for his assistance in converting the Pitts equation, and for his helpful suggestions.

(34)

where µ0 is the infinite dilution mobility of the ion, z- is the charge on the solute anion, and I is the ionic strength of the buffer. The

Received for review July 29, 1998. Accepted November 7, 1998. AC980843X


699

Prediction of Electrophoretic Mobilities. 3. Effect of Ionic Strength in

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