Anal. Chem. 2003, 75, 5197-5206
Effect of Weak Electrolytes on Electromigration Dispersion in Capillary Zone Electrophoresis Guillaume L. Erny, Edmund T. Bergstro 1 m, and David M. Goodall*
Department of Chemistry, University of York, York YO10 5DD, U.K. Sally Grieb
Pfizer Global Research and Development, Pfizer Limited, Kent CT13 9NJ, U.K.
In this paper, simple equations are introduced to predict the electromigration dispersion (EMD) in weak background electrolytes (BGEs) at high pH in capillary zone electrophoresis (CZE) separations. Analytical solutions have been developed in the case of a basic BGE constituted of a neutral acid in equilibrium with its anionic conjugate base, which acts as co-ion to the analytes, and the treatment has been extended to other buffering systems. The treatment assumes that the concentration of the neutral form of the buffer species remains constant throughout the separation and that the pair of ionic species, hydroxide ion and co-ion, effectively act as a single co-ion. Hydroxide ion is shown to have a significant effect on EMD for pH values g10. The change in hydroxide ion concentration in the sample zone can compensate for changes in conductivity due to the difference of mobility between the analyte and the co-ion. Under appropriate conditions, the hydroxide species can correct the distortion due to the difference of mobility, and a good peak shape can be obtained, even with a high analyte concentration. This effect has been modeled using numerical treatments and experimentally validated using benzoate and 4-hydroxybenzoate as test analytes in a BGE constituted of a 3-(cyclohexylamino)-1-propanesulfonic acid (CAPS) buffer with sodium as counterion. At constant co-ion concentration, the peak efficiency improved from 5.1 × 104 to 1.3 × 105 by changing the pH from 10.85 to 11.00. At a constant pH of 10.94, the peak distortion reversed in sign on increasing the concentration of CAPS co-ion from 15 to 25 mM. All experimental electropherograms were successfully fitted using the Haarhoff-Van der Linde function and distortion parameters extracted. Experimental and predicted distortion parameters are in good agreement. This work offers the potential to improve peak shapes in CZE by taking into account the effect of an ion involved in the water autoprotolysis equilibrium, and could lead to improvement in formulation of low and high pH buffers for CZE. In a recent paper,1 it has been shown that a mathematical function, called the Haarhoff-Van der Linde (HVL) function, 10.1021/ac030098a CCC: $25.00 Published on Web 09/03/2003
© 2003 American Chemical Society
perfectly describes triangular CZE peaks distorted by EMD. The HVL function is given by2
a0a2 a1a3x2π
f(x) ) exp
1 a 1a 3
( ) 2
a2
[ ( )]
exp +
-1
1 x - a1 2 a2
2
[ ( )]
x - a1 1 1 + erf 2 x2a2
(1)
where a0 is the peak area, a1 and a2 are the peak center and the standard deviation of the Gaussian part respectively, and a3 is the peak distortion; x is either time or distance; and the units of the various a parameters are consistent with this. This function has been successfully used to fit experimental electropherograms over a wide concentration range where the peaks are neither Gaussian nor triangular.3 The relationships of a0, a1, a2, and a3 to electrophoretic parameters have been given in previous papers.1,3 In particular, the peak distortion in distance units is given by
a3 ) - 2zAaAcIAl0
(2)
where cIA and l0 are the analyte concentration in the injection zone and the injection length respectively, zA is the analyte charge number, and aA is a constant. aA is related to the ratio of conductivity in BGE and analyte zones by
E(x, t) BGE
E
)
κBGE 1 ) κ(x, t) 1 - zAcA(x, t)aA
(3)
where cA(x, t) is the analyte concentration as a function of time and distance; EBGE and E(x, t), the electric field in the BGE and sample zone respectively; and κBGE and κ(x, t), the conductivity in the BGE and sample zone, respectively. aA can be negative, positive, or zero; with aA ) 0, the peak shape is symmetrical, as in this case there; is no EMD. (1) Erny, G. L.; Bergstro ¨m, E. T.; Goodall, D. M.; Grieb, S. Anal. Chem. 2001, 73, 4862-4872. (2) Haarhoff, P. H.; Van der Linde, H. J. Anal. Chem. 1966, 38, 3. (3) Erny, G. L.; Bergstro ¨m, E. T.; Goodall, D. M. J. Chromatogr., A 2002, 229239.
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003 5197
For the case of BGEs constituted of fully charged species, Mikkers showed that the aA coefficient is given by4
kA
aA ) -
∑c
(4)
uj
BGE kj j
uA - uj
j
where ki ) 1/ui + 1/ucount; the summation is over all co-ions; uA, uco, and ucount are the magnitude of the ionic mobilities of the analyte, co-ion and counterion, respectively; and cBGE is the co concentration of co-ion in the BGE zone. Thus, with one co-ion,
aA ) -
(uA - uco)(ucount + uA) uAcBGE co (uco + ucount)
(5)
From now on, magnitude of the ionic mobility will be referred as ionic mobility. In a more general case, it can be shown that an HVL peak will be obtained if the changes in concentration of each ionic species in the sample zone is linearly dependent on the analyte concentration5
cji(x, t) ) cBGE + βjicA(x, t) ji
(6)
where cji(x, t) and cBGE are the concentrations of the ith ionic ji form of species j present in the BGE, in the sample zone, and the BGE, respectively, and βji is a constant. In this case, the conductivity in the sample zone can be put in a form equivalent to that of eq 3, and aA is5 j)1-k i)1-nj
∑ ( ∑ (|z |β u ))
uA +
ji
j
aA ) -
ji ji
i
(7)
j)1-k i)1-nj
∑ ( ∑ (|z |c ji
j
BGE uji)) ji
i
The βji coefficients are related to the transfer ratios; the transfer ratio of a species j (co- or counterion), TRj, is the change in its concentration due to the change in analyte concentration,
TRj )
(
dcjj(x, t)
dcjA(x, t)
)
i)1-nj
∑β
ji
i
+
jcj
∑u ) ω j
jcA(x, t)f0
( ) dcj0(x, t) dcA(x, t)
(9)
1
(10)
j
and
∑z jc ) ω j j
2
(11)
j
where zj is the charge of the species (+1 for cations and -1 for
cA(x, t)f0
where the summation is over all nj ionic forms of the species j, and cj0(x, t) is the concentration of the neutral form of the species j in the sample zone. From eqs 7 and 9, it can be seen that, 5198
THEORY Definition of the System. The system is defined to be made up of a co-ion to the analyte of mobility uco, ionic concentration cco, and total concentration jcco; a counterion of mobility ucount, ionic concentration ccount, and total concentration jccount; and a fully monocharged analyte of mobility uA and ionic concentration cA. Charge numbers |zA|, |zco|, and |zcount| are equal to 1. The system may be in either acid or basic conditions, and the BGE can be buffered by either the co- or the counterion in equilibrium with their neutral forms as conjugate acid or base species. All concentrations in the BGE zone, the region where all analyte concentrations are zero, will be labeled with the superscript BGE. This zone is characterized by constant concentrations and, thus, electric field strength and conductivity. Within the sample zone, all concentrations at a general position x and time t, together with the electric field strength and conductivity, are dependent on the concentration of the analyte.9-12 Electromigration Dispersion in Weak Electrolyte when the Effects of H+ or OH- Ions Are Negligible. The Kohlraush regulating functions (KRFs)13 in monovalent weak electrolytes when the effects of H+ and OH- ions can be neglected are
(8)
where jcj(x, t) is the total concentration of the species j (sum over neutral and ionic forms).6 Using eqs 6 and 8, we obtain for the species j, with a fully charged analyte
TRj )
although the transfer ratios involve the changes in concentration of all forms of a given species, the aA coefficient is dependent only on the changes in concentration of all ionic form of each species. The aims of this paper are (i) to show that for fully charged monovalent analytes in simple BGEs made up of a monovalent buffering species in equilibrium with its neutral form with charge balance provided by a fully charged monovalent counterion, the conductivity in the sample zone can be expressed in a form equivalent to eq 3; (ii) to derive a simple equation linking the aA coefficient and the different electrophoretic parameters; and (iii) to validate the model through experimental studies. The model will be developed first, when effects of H+ and OH- ions are negligible, and second, using simple approximations, when effects of H+ and OH- ions are non-negligible. This work builds on previous studies by Gasˇ et al.,7 for the Kohlraush regulating function (KRF) in weak electrolyte and the computer-aided simulation of velocity slope8 and on the work from Gebauer and Bocˇek for the velocity slope approach to parametrize peak distortion.6
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
(4) Mikkers, F. E. P. Anal. Chem. 1999, 71, 522-533 (5) Erny, G. L. Ph. D. Thesis, University of York, U.K., 2002. (6) Gebauer, P.; Bocˇek, P. Anal. Chem. 1997, 69, 1557-1563. (7) Gasˇ, B.; Vacıˇk, J.; Zelenskyˆ, I. J. Chromatogr., A 1991, 545, 225-237. (8) Gasˇ, B.; Coufal, P.; Jaros, M.; Muzikar, J.; Jelinek, I. J. Chromatogr., A 2001, 905, 269-279.
anions), jcj is the total concentration, and the summation is over all species present within the zone.14,15 The ω values are independent of the applied current, and the equality of the Kohlraush regulating function on either side of a migrating boundary has been proven experimentally and by computer modeling.12 Applying these equations to the case of a system constituted of a fully charged anionic analyte, a fully charged counterion, and a co-ion in equilibrium with its neutral form gives
magnitude higher than the mobilities of other ions. Using uco ≈ ucount ≈ 0.1 × uX, where X is either H+ or OH-, it follows that BGE OH- or H+ can be neglected when cBGE e 0.002. For X /cco example, with a co-ion ionic concentration of 50 mM, eqs 16-18 will hold if the pH is between 4 and 10. Electromigration Dispersion in Weak Electrolyte when the Effect of OH- Ions is Non-Negligible. When the effect of OHis nonnegligible, the conductivity in the sample zone is
jcco(x, t) cA(x, t) ccount(x, t) jcBGE cBGE co count + + ) + uco uA ucount uco ucount
κ(x, t) ) F(cco(x, t)uco + cOH-(x, t)uOH- + ccount(x, t)ucount +
(12)
and BGE jcco(x, t) + cA(x, t) - ccount(x, t) ) jcBGE co - ccount
(13)
cA(x, t)uA) (19) The OH- concentration is related to that of the base species Bin equilibrium with its conjugate acid BH via the dissociation equilibrium Kb
where the left-hand side refers to the sample zone, and the righthand side, to the BGE. Substituting jcco in eq 13 by cco + c0co (cco is the ionic concentration of co-ion, and c0co, the neutral concentration) and using the electroneutrality relationship in the sample and BGE zones (∑j zjcj ) 0) gives c0co(x, t) ) constant ) c0,BGE . co Thus, eq 12 reduces to
cco(x, t) cA(x, t) ccount(x, t) cBGE cBGE co count + + ) + uco uA ucount uco ucount
κ(x, t) ) F(cco(x, t)uco + ccount(x, t)ucount + cA(x, t)uA) (15) where F is the Faraday constant. Equations 14 and 15 involve ionic concentrations rather than total concentrations and are equivalent to those which hold in a BGE comprising a monovalent fully charged co- and counterions. Thus, aA will be given by
(uA - uco)(ucount + uA) uAcBGE co (uco + ucount)
TRcount )
uco(uA + ucount) uA(ucount + uco) ucount(uA - uco) uA(ucount + uco)
cBBGE - Kb
BGE cOH - )
cBGE BH
(21)
and in the sample zone,
cOH-(x, t) )
cB-(x, t)Kb cBH(x, t)
(22)
Assuming that the concentration of the neutral form of the buffering species will still be constant, cBGE BH ) cBH(x, t), we obtain BGE cOH -
cBBGE -
)
Kb cBBH
)
cOH-(x, t) cB-(x, t)
(23)
(16)
although in this case, it must be stressed that cBGE is the ionic co concentration of the co-ion which is not equal to the total concentration. The co- and counterion transfer ratios, TRco and TRcount, respectively, will be given by
TRco ) -
(20)
The general equation for a base dissociation constant in terms of concentrations, Kb ) cBHcOH-/cB- (which holds when activity coefficients may be approximated to unity), leads to the following expressions for the concentration of OH- in the BGE
(14)
It can be readily shown that the same equation will be obtained if the buffering species is the counterion. In this system, the conductivity in the sample zone will be given by
aA ) -
B- + H2O y\z BH + OH-
(17)
(18)
It may be assumed that OH- or H+ can be neglected in the treatment of conductivity if their contribution to the total conduc0 0 tivity is 2.0 × 10-2, and the assumption of negligible effects of OH- in the electroneutrality relationship and KRF is not valid anymore. Whereas good agreement is found between experimental and theoretical value of aA and TRcount, this is not the case for TRco; the theory predicts that the values will increase as the concentration ratio increases, while the software showed an opposite trend. The theoretical value of TRco given by eq 30 is the change in the ionic concentration of co-ion, whereas the value calculated by the software is the change in the total concentration of the co-ion. The good agreement between the experimental and theoretical values for TRcount and aA indicate that the ionic concentration distribution of the co-ion is satisfactorily predicted using eq 30. However, contrary to what was postulated, the concentration of the neutral form of the buffered species cannot be assumed to be constant when OH- has an influence on the conductivity. The equations proposed for the co- and counterion transfer ratios were intuitive. A more rigorous treatment may be developed using the form of the KRF proposed by Gasˇ et al. for weak monovalent electrolyte at high or low pH,7 in which eq 10 is modified to
(
1
∑u j
j
-
zj
)
uOH-
jcj ) ω
Applied to the system studied, eq 12 becomes
(35)
cA(x, t) jcco(x, t) ccount(x, t) + + uA uco ucount jcco(x, t) + cA(x, t) - ccount(x, t) ) uOH BGE cBGE jcBGE jcBGE co count co - ccount + (36) uco ucount uOH-
Substituting jcco by cco + c0co and using cj(x, t) - cBGE ) cA(x, t)TRj j and the electroneutrality relationship gives
cA(x, t)
(
)
TRco TRcount 1 + + + uA uco ucount
c0co(x,
t) uco
c0,BGE co
Figure 1. (a) Neutral, (b) ionic, and (c) total co-ion transfer ratios of analyte A1- calculated with eqs 38, 28, and 39 together with values calculated by the software (().
)
- c0co(x, t) c0,BGE co (37) uOH-
where TRco and TRcount are given by eqs 28 and 29. Substituting from these equations (and using eq 26) in eq 37 and rearranging gives
c0co(x, t) - c0,BGE co cA(x, t)
) TR0co ) BGE cOH ucouOH- uOH-(ucount + uA) (38) BGE u (u c A count + u′co) uOH- - uco co
where u′co is given by eq 26 and TR0co is the transfer ratio of the neutral form of the co-ion. The total co-ion transfer ratio, TRco, as calculated by the software, will be given by
TRco ) TRco + TR0co
(39)
The neutral, ionic, and total co-ion transfer ratios for analyte A1- are shown together with the values obtain by the software in Figure 1. There is now very good agreement between the theoretical and simulated values of the co-ion transfer ratio. Similar results will be obtained for analyte A2-. The accuracy in the prediction of the aA coefficient could in principle be improved by taking into account the concentration changes in the neutral form of the co-ion. However, this would lead to very complex equations. The aim of this paper is not to provide exact solutions, but rather to propose approximate simple equations and apply them to design optimize BGE. Results presented in this section have shown that the peak shape of a given analyte can be improved by adjusting the BGE BGE concentration ratio cOH -/cco . This will now be demonstrated in a real separation. Experimental Validation. The test mixture consisted of 4-hydroxybenzoate (A1-) and benzoate (A2-), each at a concentration of 0.8 mM, with a BGE made of CAPS brought to high pH to provide the co-ion buffer species and Na+ as counterion. Figure 2 shows a typical separation at pH 10.85. The mobilities of A1and A2- were measured to be 2.48 and 2.77 × 10-8 m2 V-1 s-1, respectively. Correction to zero ionic strength using the formulas
Figure 2. Separation of 4-hydroxybenzoate (A1-) and benzoate (A2-) ions at a concentration of 0.8 mM in a CAPS buffer made up of 25 mM NaOH and 31.6 mM CAPS (pH 10.85). The separation was performed in a 40/47 cm, 75-µm-i.d. capillary, at a voltage of 20 kV. The sample was injected in 20% BGE for 5 s at a pressure of 0.5 psi. The detection wavelength was set at 214 nm.
of Survay et al.20 gives values for u° of 2.96 and 3.27 × 10-8 m2 V-1 s-1 for A1- and A2-, respectively. These are in satisfactory agreement with values reported previously: 3.14 × 10-8 m2 V-1 s-1 for A1-,21 and 3.4321 and 3.29 × 10-8 m2 V-1 s-1 for A2-.22 In the first series of experiments, the concentration of sodium ions was held constant at 25 mM; the concentration of the anionic conjugate base of CAPS was equal to this, and the pH varied by using different total concentrations of CAPS. Figure 3 shows the change in the peak shape of A1- as function of the pH of the BGE. The total concentration of CAPS ranged from 31.6 (pH 10.85) to 29.1 mM (pH 11.00). The HVL function fits the experimental data points perfectly at pH 10.85 and 10.94. At pH 10.97 and 11.00, because of the improvement of peak shape, the absorbance at peak maximum exceeded the linear range of the detector, leading to a flat-topped peak. As predicted, the peak shape improves as the pH increases. Under the experimental conditions, reversal of sign of the peak distortion was not observed. The optimal condition was found at pH 11.00, with a peak efficiency, N, of 1.2 × 105; this compares with pH 10.85, where N ) 5.1 × 104. The efficiency has been increased by a factor of ∼3 by increasing the pH by only 0.15 unit. A second series of experiments was carried out using a set of BGEs at the same pH, but with different concentrations of co(20) Survay, M. A.; Goodall, D. M.; Wren, S. A. C.; Rowe, R. C. J. Chromatogr., A 1996, 741, 99-113. (21) Li, D. M.; Fu, S. L.; Lucy, C. A. Anal. Chem. 1999, 71, 687-699. (22) Pospical, J.; Gebauer, P.; Bocˇek, P. Chem. Rev. 1989, 71, 419-430.
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
5203
Figure 3. 4-Hydroxybenzoate peak shapes obtained at different pH values. In each buffer, the concentration of Na+ was constant at 25 mM; the concentration of CAPS was 31.6 (pH 10.85), 30.0 (pH 10.94), 29.5 (pH 10.97), and 29.1 mM (pH 11.00). All other conditions are the same as in Figure 2. Dots are the experimental data points, and the line, the fitted HVL function.
Figure 4. 4-Hydroxybenzoate peak shapes at constant pH (10.94) and different ionic concentrations. Concentrations: 15 mM Na+/17.6 mM CAPS, 20 mM Na+/23.8 mM CAPS, and 25 mM Na+/30.0 mM CAPS. All other conditions are the same as in Figure 2. Dots are the experimental data points, and the line, the fitted HVL function.
Figure 5. a3cBGE as a function of the ratio of hydroxide to co-ion co ionic concentration for (a) 4-hydroxybenzoate and (b) benzoate: 9, values from electropherograms at constant Na+ concentration; 2, values from electropherograms at constant pH. The lines give the theoretical values calculated with eq 17.
b for A1- and A2- respectively. Very good agreement is observed between the theoretical and the experimental values of a3cBGE co BGE BGE until cOH ≈ 4.6 × 10-2. This proves without ambiguity that -/cco in the case of a buffered BGE, the effect of hydroxide ion can be satisfactorily approximated using the equilibrium treatment developed in this paper, in which the co-ion plus hydroxide ion pair is treated as a hypothetical single co-ion. Application to Other Buffering Systems. In the general case (see appendix for derivation of the equations), if the ionic buffer species is the co-ion to the analyte, the effects of H+ or OH- (X in eq 40) on distortion can be predicted by postulating a co-ion of concentration cco, and mobility equal to
zX cBGE X u′co ) uco + u zco cBGE X
(40)
co
ion. The three BGEs used consisted of 15.0 mM Na+/17.6 mM CAPS, 20.0 mM Na+/23.8 mM CAPS, and 25.0 mM Na+/30.0 mM CAPS; all had an identical pH of 10.94. Results are shown in Figure 4. The transition from fronting through symmetrical to tailing peak is striking, yet totally in accord with expectations from the theory developed in the previous section. Results are contrary to what would have been expected assuming dependence of distortion on effective mobility. By fitting the electropherograms from the two sets of experiments with an HVL function, the a3 parameters were measured and converted from time to distance units. The products a3cBGE co are plotted in Figure 5 as a function of the ratio of hydroxide to co-ion ionic concentration. According to eqs 2, 26, and 30, this BGE BGE product should be linearly dependent on cOH for any co-/cco ion concentration. Good linearity is observed for both analytes for both series of experiments (constant concentration of Na+ BGE BGE (squares) or constant pH (triangles)) when cOH e 4.5 × -/cco BGE BGE -2 10 . The concentration ratio cOH-/cco where a3 is 0 is estimated at 4.6 × 10-2 for A1-. Because aA is 0 when a3 is 0, it follows from eq 30 that the mobility of the fully charged form of CAPS is u0co ) 2.03 × 10-8 m2 V-1 s-1, using 2.96 and 20.25 × 10-8 m2 V-1 s-1 for the u0 values of the analyte and hydroxide ion, respectively. The theoretical values of a3 were calculated using eqs 2 and 30, 0 knowing the co-ion mobility, the counterion mobility (uNa + ) -8 2 -1 -1 5.19 × 10 m V s ), the concentration in the injection zone (0.8 mM for each analyte) and the injection length (7.35 mm calculated from the injection time of 5 s and pressure of 0.5 psi). The products of a3 and cBGE co are plotted in Figure 5 as lines a and 5204
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
and the aA coefficient will be given by
aA ) -
(uA + ucount)(uA - u′co) cBGE co uA(u′co + ucount)
(41)
If the ionic buffer species is the counterion to the analyte, the effects of H+ or OH- on distortion can be predicted by postulating a counterion of concentration ccount and mobility equal to
u′count ) ucount +
zX cBGE X zcount cBGE
uX
(42)
count
and the aA coefficient will be given by
aA ) -
(uA + u′count)(uA - uco) cBGE co uA(uco + u′count)
(43)
To validate this generic treatment, three analytes were used, A, B, and C, with mobilities of 2.0, 2.5, and 3.0 × 10-8 m2 V-1 s-1, respectively. BGEs were made up of a co-ion and counterion of mobilities 2.5 × 10-8 and 4.0 × 10-8 m2 V-1 s-1, respectively. BGEs were modeled in both acidic and basic conditions, covering all variants in Table 1. In the acidic BGE, the buffer species was given a pKa of 3.0, and in the basic BGE, a pKa of 11.0. The pKa of all
Table 4. Coefficients aA (M-1) from (1) Analytical Theory, and (2) Numerical Treatment for Three Analytes in a Range of BGE Types and pH Valuesa BGE buffered by the co-ion to the analyte analyte A
analyte B
BGE buffered by the counterion to the analyte
analyte C
analyte A
analyte B
analyte C
pH
1
2
1
2
1
2
1
2
1
2
1
2
7.00 10.00 10.79 11.00 11.50
9.2 7.8 0.0 -6.5 -62.8
9.6 7.7 1.5 -1.5 -10.6
0.0 -1.3 -8.7 -14.8 -67.6
0.0 -1.0 -5.3 -7.5 -14.2
-7.2 -8.4 -15.5 -21.5 -72.5
-7.5 -7.9 -11.1 -12.5 -17.4
9.2 9.2 9.3 9.3 9.4
9.2 9.3 9.3 9.3 9.3
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
-7.2 -7.2 -7.1 -7.1 -7.0
-7.2 -7.2 -7.1 -7.1 -7.1
7.00 4.00 3.46 3.00 2.50
9.2 11.6 17.1 29.3 54.9
9.2 11.4 17.0 42.9 -5.3
0.0 2.3 7.4 18.9 42.8
0.0 2.1 7.2 26.5 -20.1
-7.2 -5.0 0.0 11.0 34.2
-7.2 -5.1 -0.2 16.2 -35.8
9.2 9.2 9.2 9.0 7.4
9.2 9.6 10.9 72.8 3.9
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
-7.2 -7.2 -7.2 -7.3 -8.4
-7.2 -7.1 -6.9 -5.7 -1.8
a Mobilities are 2.0, 2.5, and 3.0 × 10-8 m2 V-1 s-1 for A, B, and C, respectively. pH values in bold correspond to the theoretical values for which EMD for one of the analytes should be 0; aA values in italic indicate the presence of a system peak from numerical treatment using the relative velocity slope.
analytes was set at 14. In all BGEs, the ionic concentration of the co-ion was constant and equal to 25 mM. Results are given in Table 4. pH values in bold are the theoretical values where EMD should be 0; these are calculated from eqs 40 and 41. When the numerical treatment predicts a system peak at a mobility sufficiently close to the analyte mobility to significantly modify the peak distortion, the aA values are in italics in the table. BGE BGE BGE BGE and cOH When the concentration ratios cOH -/cco -/ccount are negligible (pH 7) the aA parameters for each analyte are identical in all buffer systems and equal to the theoretical values of 9.2, 0.0, and -7.2 dm3 mol-1 for analytes A, B, and C, respectively. At higher pH, with an electrolyte buffered by its co-ion, the predicted values from eqs 41 and 43 are in satisfactory agreement with the values obtained using the relative velocity slope parameters until pH ∼11 in basic conditions and pH ∼3 in acidic conditions. Equations 40 and 41 successfully predict the pH where no EMD will be observed for A in basic conditions (theoretical value 10.79; value obtained from numerical treatment 10.92) and for C, in acidic conditions (theoretical value 3.46, value obtained from numerical treatment 3.45). When the BGE is buffered by the counterion to the analyte, under conditions for which no system peak is predicted, there is perfect agreement between the predicted and simulated values. In such a case, the peak distortion can be assumed to be constant BGE BGE at all values of the concentration ratio cOH -/ccount. This is because the influence of the counterion mobility on peak distortion is negligible.1 CONCLUSIONS In this paper, it has been shown that for BGEs consisting of single co- and counterions, the effect of OH- on electromigration dispersion can be predicted by assuming the ionic form of the buffer species and OH- acts as a single ion, of concentration equal to the ionic concentration of the buffer species, and mobility equal to the sum of the ionic mobility of the buffer species and the mobility of OH- weighted by the ratio of the OH- and co-ion concentration. This simple relation holds for all buffers involving univalent ions, and the peak distortion can be predicted assuming that the system is equivalent to a simple system in which the effect
of OH- is negligible. The treatment has been validated using experimental CZE studies of two anionic species in a CAPS buffer, and it was demonstrated that peak efficiency could be improved by a factor of 3 by optimizing the OH-/co-ion concentration ratio. Because peak distortion can be minimized by tuning the OH-/ co-ion concentration ratio, this suggests a generic approach to increase in loading of the main component, without deleterious effects on resolution, in situations for which it is necessary to quantify impurities present at low levels. Analogous considerations hold for the H+/co-ion concentration ratio in an acidic buffer. Because the counterion mobility has a very low impact on EMD, effects of varying pH are far smaller in the case of BGEs in which the ionic form of the buffer species is a counterion to the analyte. In such a case, minimization of any peak overload problems requires the mobility of the main component to be matched to that of the BGE co-ion, irrespective of the pH. In future work, it would be helpful to test these conclusions more widely and extend the treatment outlined in the present paper to systems including multicharged BGEs and analytes. ACKNOWLEDGMENT Support from the University of York and Pfizer Global Research and Development for a graduate research studentship for G. L. Erny is gratefully acknowledged. We thank Professor B. Gasˇ and Dr. J. Reijenga for provision of software and helpful discussions. APPENDIX Case of a System Buffered by an Anionic Base or a Cationic Acid. The treatment of a basic BGE buffered by an anionic base in which the ionic form acts as a co-ion has been developed in eqs 21-31, and extended by symmetry to the treatment of an acidic BGE buffered by a cationic acid (Table 1). In the case of separation of cationic analytes in a basic BGE buffered by an anionic base which acts as a counterion, the conductivity will be given by
κ(x, t) ) F(cco(x, t)uco + cOH-(x, t)uOH- + ccount(x, t)ucount + cA(x, t)uA) (1A) Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
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In this case, the relation between OH- ions and the ionic form of the buffer species (cB- ) ccount) is BGE cOH -
cOH-(x, t) ) ccount(x, t) BGE ccount
(
(
cOH-(x, t) )
(2A)
BGE cOH -
(7A)
cco(x, t)/cBGE co
Because cco(x, t)/cBGE is always close to 1, eq 7A can be co approximated using 1/(1 + k) ≈ 1 - k when k is close to 0, giving
Combining eqs 1A and 2A gives
κ(x, t) )
(cBH+ ) cco in Table 1) is
F cco(x, t)uco + ccount(x, t) ucount +
BGE cOH -
)
uOHcBGE count
+ cA(x, t)uA
)
(
BGE cOH-(x, t) ) cOH - 2 -
)
cco(x, t) cBGE co
(8A)
(3A) Combining eqs 6A and 8A gives By postulating that the pair of ionic species hydroxide plus counterion act as a single counterion of concentration ccount(x, t) and mobility
u′count ) ucount +
BGE cOH -
cBGE count
BGE κ(x, t) - 2FcOH -uOH- ) κ′(x, t) BGE κ(x, t) - 2FcOH -uOH- ) κ′(x, t) )
(4A)
uOH-
( (
F cco(x, t) uco -
BGE cOH -
cBGE co
)
(9A)
the conductivity becomes equivalent to
κ(x, t) ) F(cco(x, t)uco + ccount(x, t)u′count + cA(x, t)uA) (5A) The co- and the counterion concentration distributions and the aA coefficient will be given by appropriate modifications of the equations obtained when the BGE was buffered by the co-ion (i.e., ucount rather than uco carries the prime everywhere). Analogous results will be obtained with separation of anionic analytes in an acidic BGE buffered by a cationic acid, with subscript H+ rather than OH-. Case of a System with a Basic Buffer Involving a Cationic Conjugate Acid. For separation of cationic analytes in a basic buffer involving a cationic conjugate acid (e.g., a protonated amine) in which the ionic form of the buffer acts as co-ion to the analyte, the conductivity is
κ(x, t) ) F(cco(x, t)uco + cOH-(x, t)uOH- + ccount(x, t)ucount + cA(x, t)uA) (6A) The relation between OH- and the ionic form of the buffer species
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)
uOH- + ccount(x, t)ucount + cA(x, t)uA
Analytical Chemistry, Vol. 75, No. 19, October 1, 2003
As in previous cases, postulating that
u′co ) uco -
BGE cOH u BGE OHcco
(10A)
and because
κ′ BGE κBGE 1 ≈ ) κ′(x, t) 1 - zAcA(x, t)aA κ(x, t)
(11A)
this case becomes equivalent to the cases discussed previously. All remaining cases can be solved in a similar way. Received for review March 12, 2003. Accepted June 27, 2003. AC030098A
