Predicting Peak Symmetry in Capillary Zone Electrophoresis: The

Anal. Chem. 1997, 69, 1557-1563

Predicting Peak Symmetry in Capillary Zone Electrophoresis: The Concept of the Peak Shape Diagram Petr Gebauer* and Petr Bocˇek

Institute of Analytical Chemistry, Academy of Sciences of the Czech Republic, CZ-611 42 Brno, Czech Republic

Electrophoretic peaks generally deviate from the symmetrical Gaussian shape that is expected when diffusion is the dominant dispersion factor. A basic understanding of the electromigrational dispersion processes which give rise to peak asymmetry shows that these effects are related to the difference in mobilities between the sample ions and the buffer electrolyte ions. There is a currently adopted rule saying that if the mobility of the buffer coion is higher than that of the analyte, a tailing peak is obtained, while a lower mobility of the buffer co-ion results in a fronting peak. Unfortunately, the phenomena observed in practice are frequently contradictory to these rules. This paper reveals the principles of these effects and establishes a way to proceed in explaining and predicting the peak shape in electrophoresis. Based on a mathematical model of zone migration in capillary electrophoresis, the key parameter controlling the migration dynamics is found to be the velocity slope, defined as the change in the analyte migration velocity with the analyte molar fraction at infinite analyte dilution. It is shown that the sign of the velocity slope determines the peak asymmetry (fronting or tailing) and that its magnitude relates to peak width due to electromigration dispersion. In experiments where the conversion of a fronting peak of an analyte into a tailing one is induced by changing the pH of the background electrolyte, the proposed theory successfully predicts the conversion point where the peak is symmetric. For a given background electrolyte, the model allows assessment of the symmetry and width of any analyte peak. The concept of the peak shape diagram is introduced where the velocity slope contours are plotted in a ionic mobility-pK coordinate system and its utility for fast and easy predicting of the symmetry and asymmetry of peaks as well as of their migration (detection) order is demonstrated. In zone electrophoresis, the shape of the concentration profile of a sample zone (of a detected sample peak) migrating in a homogeneous background electrolyte (BGE) depends, in addition to its original (starting) shape, only on the dispersion process that each sample zone undergoes during the electrophoresis.1,2 Assuming the original sample concentration profile to be narrow enough to be neglected, dispersion remains the only factor (1) Foret, F.; Krˇiva´nkova´, L.; Bocˇek, P. Capillary Zone Electrophoresis; VCH: Weinheim, Germany, 1993. (2) Mosher, R. A.; Saville, D. A.; Thormann, W. The Dynamics of Electrophoresis; VCH: Weinheim, Germany, 1992. S0003-2700(96)00796-2 CCC: $14.00

© 1997 American Chemical Society

affecting the zone (peak) shape. Besides diffusion and similar dispersion effects that usually apply quite nonselectively, some other types of dispersion may become very selective. Of them, the most common is electromigration dispersion, which applies especially at zones with higher sample concentration;3,4 it also affects the dispersion of the low-concentration sample zones usually met in analytical practice. The effect of electromigration dispersion is long known1 and has been investigated by various authors under different names (electrophoretic dispersion,5 electrokinetic dispersion,6 electrodispersive effects,7 zone broadening induced by conductivity differences,8 concentration overload9). The typical result of electromigrational zone dispersion is assymetric concentration profiles (peak shapes). For the most simple case of a migrating sample zone with both sample and the binary BGE being strong electrolytes, the concentration profile of the sample component can be expressed by explicit equations.3,5,10 The question of special interest in electromigration dispersion is how to predict the peak shape of a given analyte zone in a given electrolyte system, i.e., whether it provides a sharp or a broad zone, a fronting or a tailing detection pattern. The situation is quite easy for the above-mentioned case where only strong electrolytes are involved. The equations describing the (timedependent) concentration profile of a migrating zone allow formulation of simple criteria where the ionic mobilities of the analyte and background co-ion (uX and uBGE, respectively) are the key quantities for the prediction of the zone shape:3,5 it holds simply that an analyte X provides a fronting zone if uX > uBGE and a tailing one if uX < uBGE. The derived expressions3,5 further indicate that zone width is a (nonlinear) function of the mobility ratio, uX/uBGE. Similar rules were shown11 to apply when a BGE with multiple co-ions was used: the shape of a given analyte peak is controlled predominantly by the cation that has its mobility closest to that of the analyte. For weak electrolytes, the situation is unfortunately not as transparent. In some cases, the above rules for strong electrolytes (3) Mikkers, F. E. P.; Everaerts, F. M.; Verheggen, T. P. E. M. J. Chromatogr. 1979, 169, 1-10. (4) Mikkers, F. E. P.; Everaerts, F. M.; Verheggen, T. P. E. M. J. Chromatogr. 1979, 169, 11-20. (5) Thormann, W. Electrophoresis 1983, 4, 383-390. (6) Roberts, G. O.; Rhodes, P. H.; Snyder, R. S. J. Chromatogr. 1989, 480, 3567. (7) Beckers, J. L. Electrophoresis 1995, 16, 1987-1998. (8) Hjerte´n, S. Electrophoresis 1990, 11, 665-690. (9) Poppe, H. Anal. Chem. 1992, 64, 1908-1919. (10) Weber, H. Die Partiellen Differential-Gleichungen der Mathematik und Physik; Friedrich Vieweg u. Sohn: Braunschweig, Germany, 1910; Vol. I, p 503. (11) Bullock, J.; Strasters, J.; Snider, J. Anal. Chem. 1995, 67, 3246-3252.

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can be a good approximation12 but their general application is not advisable. One possible approach to predict correctly the peak shape/width of a given analyte zone in a given BGE is the use of computer simulation, where any of the models presented so far (for an overview, see ref 13 and the literature cited therein) can be used; similar results are obtained by special approaches using an eigenvalue treatment9 or the computerization of a non-steadystate model.14,15 All just mentioned models are even able to predict observed peak shape anomalies,14,15 but they neither explain their origin nor present general rules of predicting peak shape of weak electrolytes. A just published paper by Beckers,16 based on a classification of zone electrophoretic systems into eight groups according to whether the co-ion and counterion of the BGE and sample are weak or strong electrolytes, enabled the author to formulate for some of the groups approximative rules of thumb for prediction of peak shape. An attempt to analyze the factors affecting electromigration dispersion in a general case was made by Roberts et al.6 Understanding electromigrational dispersion to be caused by variations in the analyte migration velocity, the two main factors involved were found to be variations in the mean ionization of the analyte (i.e., in the effective mobility) and in the local electric field strength (i.e., in the local BGE conductivity). The so far most important progress in an exact description of the mentioned effects is the definition of a constant17 (called the electromigration dispersion constant) expressing quantitatively the influence of both conductivity and pH effects on the direction and degree of peak deformation. This paper presents a simple theoretical approach to the criteria that control the peak shape in zone electrophoresis of weak electrolytes. The developed model resulted in introducing the concept of peak shape diagram which, once plotted for a given background electrolyte, allows immediate prediction of detection time, peak shape, and peak width of any analyte. THEORETICAL SECTION The principle of electromigrational dispersion is usually explained by the fact that the sample modifies the conductivity and pH of the BGE, which results in a change in the sample’s effective mobility and thus migration velocity. Because the level of modification depends on the local sample concentration, each point of the sample zone migrates with a different velocity and dispersion may be the result. Therefore, the first aim is to investigate how the local sample migration velocity,

vX,AX ) i(u j X,AX/κAX)

(1)

(where i is the electric current density, uj is the effective mobility, κ is the specific conductivity, and X and AX denote analyte X and its zone AX) changes with the sample concentration in a given BGE. The following theoretical investigations are based on a simple model. It is assumed that a sample zone migrating zone electrophoretically in a background electrolyte modifies its concentration so that all the concentrations are adjusted to keep the Kohlrausch (12) Sˇusta´cˇek, V.; Foret, F.; Bocˇek, P. J. Chromatogr. 1991, 545, 239-248. (13) Reijenga, J. C.; Martens, J. H. P. A.; Everaerts, F. M. Electrophoresis 1995, 16, 2008-2015. (14) Beckers, J. L. J. Chromatogr., A 1995, 693, 347-357. (15) Beckers, J. L. J. Chromatogr., A 1995, 696, 285-294. (16) Beckers, J. L. J. Chromatogr., A 1996, 741, 265-277. (17) Xu, X.; Kok, W. T.; Poppe, H. J. Chromatogr., A 1996, 742, 211-227.

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Figure 1. Scheme of the model system.

Figure 2. Dependence of the migration velocity νX,AX of analyte X on its molar fraction aX in the mixed zone AX calculated for the following: (1) pKHX ) 4, uX ) 40 × 10-9 m2 V-1 s-1; (2) pKHX ) 7, uX ) 20 × 10-9 m2 V-1 s-1; (3) pKHX ) 4, uX ) 20 × 10-9 m2 V-1 s-1 in a BGE consisting of 0.02 M MES and 0.01 M Na+ (pH ) 6.21).

regulating function (ω)1,2,18 at a given point constant (diffusion and other dispersional effects except electromigration are neglected). Any point of a migrating sample zone can be so considered to be a mixture of sample and BGE adjusted to the ω value of the original BGE. For a given concentration ratio of sample and BGE ions, both these concentrations can be calculated from the original BGE concentration using a set of equations expressing the regulating principle. This way offers access to the composition of any point of any sample zone in any BGE, allowing us to investigate the sample zone properties and their dependence on the composition of the BGE. The model assumes a BGE composed of a weak monohydric acid HA (co-ion A-) and a weak monohydric base B (counterion BH+). The sample is a weak monohydric acid HX (sample anion X-). Its zone of a given concentration level of HX is modeled by a moving-boundary mixed zone of HA and HX (zone AX) migrating behind the pure BGE zone (zone A), as shown in Figure 1. The set of equations used for the calculations is given in the Appendix. Using the just described model, we investigated an example system formed by a BGE consisting of 0.02 M morpholinoethanesulfonic acid (MES) and 0.01 M Na+ (pH ) 6.21). Figure 2 shows the calculated dependence of vX,AX on the molar fraction of X in the mixed zone, aX ) jcX,AX/(cjX,AX + jcA,AX) (see Appendix, eq A6), for three model analytes. It can be seen from the figure thatsdepending on the ionic mobility and pKHX of the sample componentsthe resulting dependence vX,AX ) f(aX) is decreasing, increasing, or shows a maximum. This can be explained by the fact that both uj X,AX and κAX in eq 1 are functions of aAX and that vX,AX changes in dependence on the course of both functions uj X,AX ) f(aX) and κAX ) f(aX). It is now interesting to discuss the question of how the course of the dependence vX,AX ) f(aX) relates to zone shape in electromigrational dispersion. If the curve expressing the mentioned relationship is descending for any aX (as is curve 1 in Figure 2), then any segment of the sample zone of lower analyte concentra(18) Kohlrausch, F. Ann. Phys. Chem. 1897, 62, 209-239.

Figure 3. Computer-simulated evolution of the concentration profiles of the analytes from Figure 2 using the same BGE. The simulation was performed using 400 mesh points, a column length of 0.05 m, and a constant current of 150 A m-2. As the initial state, a 2 mm long almost rectangular pulse of a 0.01 M sample solution (as Na+X-) was taken with (a) pKHX ) 4, uX ) 40 × 10-9 m2 V-1 s-1; (b) pKHX ) 7, uX ) 20 × 10-9 m2 V-1 s-1; (c) pKHX ) 4, uX ) 20 × 10-9 m2 V-1 s-1. The numbers at the profiles indicate the minutes of electrophoresis time passed.

tion migrates faster than any other segment of this zone with higher analyte concentration. The result is electromigrational zone fronting where the front zone edge is dispersing while the rear of the zone exhibits sharpening. Vice versa, if the local migration velocity of a lower sample concentration level is slower than that of a higher sample concentration level, the sample zone exhibits tailing at the rear side and sharpening at the front side. This situation corresponds to curve 2 in Figure 2. Note that all curves in Figure 2 end in the same point at aX ) 1; this point corresponds to the isotachophoretic migration velocity of the BGE used as leading electrolyte. From the above discussion it can be concluded that the key parameter that describes the direction and magnitude of electromigrational dispersion of a point of a sample zone is the change in the local sample migration velocity with the local sample fraction, dvX,AX/daX. Obviously, fronting zones are formed when

dvX,AX/daX < 0

(2a)

and the tailing effect applies at all points where

dvX,AX/daX > 0

(2b)

Figure 3 illustrates the problem, showing the computer-simulated evolution of the concentration profiles of the model analytes from Figure 2 during their zone electrophoretic migration. The entire curve 1 in Figure 2 has dvX,AX/daX < 0; as shown in Figure 3a, the zone profile evolves as a strongly fronting one. For any point of curve 2 in Figure 2, it holds that dvX,AX/daX > 0 and the corresponding Figure 3b confirms that the zone profile evolves as a tailing one.

Curve 3 in Figure 2 represents an anomalous case where for low aX, dvX,AX/daX > 0, and for high aX, dvX,AX/daX < 0. From this it can be predicted that a low-concentration zone of this type of sample will be tailing whereas high concentrations in the sample zone will tend to be fronting. The simulation result shown in Figure 3c confirms this conclusion: the top of the pattern after 1 min electrophoresis time exhibits a fronting tendency whereas the rear of the zone already starts tailing. During further migration, the high-concentration part of the zone disappears as a result of dispersion and only the low-concentration part remains where dvX,AX/daX > 0 in the entire zone; a tailing zone is the overall result of electromigrational dispersion of this type of analyte. The above discussion has revealed one generally valid principle: each zone electrophoretically migrating sample zone is sooner or later dispersed toward the low aX region and the slope of the vX,AX ) f(aX) function in this aX region thus determines the final shape of the zone concentration profile irrespective of whether the original sample concentration was high or low. It therefore seems to be purposeful to use the change in local sample migration velocity with local sample fraction at infinite sample dilution, (dvX,AX/daX)aXf0, as the most suitable key quantity for the prediction of the final symmetry/shape of sample zones in CZE. The final key criteria can be then in analogy with conditions 2a,b formulated as

SX ) (dvX,AX/daX)aXf0 < 0

(3a)

for fronting zones and as

SX ) (dvX,AX/daX)aXf0 > 0

(3b)

for tailing zones where the parameter SX called the velocity slope is introduced, the sign of which determines whether the peak has tailing or fronting symmetry. Obviously also the magnitude of SX has a clear and straightforward physical meaning in that it expresses the speed of electromigrational dispersion and thus relates directly to zone width. EXPERIMENTAL SECTION The theoretical calculations were made using a simple program written in QBasic. The velocity slope was calculated (assuming a constant current density of 1000 A m-2) from eq A10 using aX ) 0.001. The computer simulations shown in Figure 3 were performed by using a computer program based on the model by Mosher et al.2 The mobilities and pK values used for the calculations are shown in Table 1. For the experiments, a homemade setup equipped with a 50 cm long coated fused-silica capillary (Polymicro Technologies, Phoenix, AZ) of 75 µm i.d. with a separation length of 38 cm was used. The inner capillary surface was coated with 6% polyacrylamide gel using the procedure described by Hjerte´n20 with the only difference that γ-methacryloxypropylsilane was solved in ethanol instead of water. The capillary hung freely in air at room temperature (25 °C). Sampling was performed using a simple hydrodynamic sampling device with ∆h ) 5 cm and a sampling time of 15 s. The sample zones were detected using an 875 UV (version 5) detector from Jasco (Tokyo, Japan) at a detection wavelength of 210 nm. The detector was (19) Pospı´chal, J.; Gebauer, P.; Bocˇek, P. Chem. Rev. 1989, 89, 419-430. (20) Hjerte´n, S. J. Chromatogr. 1985, 347, 191-198.

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Table 1. Constants of the Substances Used for the Calculationsa

H+ OHNa+ MES propionate picrate a

uXb

pKHX

362.5 202.5 51.9 27.0 36.9 31.7

6.10 4.78 0.71

Taken from ref 19. b In 10-9 m2 V-1 s-1.

Figure 5. Experimental UV absorbance detection records (absorbance range 0.08 AU) of picrate in a BGE formed by 0.02 M propionic acid titrated by NaOH to (a) pH ) 3.86, (b) pH ) 4.24, and (C) pH ) 5.38. The sample solution was 0.002 M picric acid neutralized by 70% with NaOH.

Figure 4. Calculated dependence of the velocity slope SX of picrate on the pH of a BGE formed by 0.02 M propionic acid with variable contents of Na+. The three marked values on the pH axis correspond to the experiments shown in Figure 5.

equipped with a homemade capillary holder where the detection window was sandwiched between two quartz lenses. The detector output was connected to a two-channel 4200 line recorder (Laboratornı´ Prˇ´ıstroje, Praha, Czechoslovakia) (range 20 mV, chart speed 0.5 mm/s). As the power supply, a CZE 1000 R model from Spellmann (Plainview, NY) served with a constant voltage of 15 kV. As the BGE, 0.02 M propionic acid containing 0.05% hydroxypropylcellulose was used; the solution was titrated to various pH’s with NaOH. All chemicals were of analytical-reagent grade. The solutions were prepared in deionized water; their pH was measured by a 240 Model pH meter from Corning (Sudbury, U.K.). RESULTS AND DISCUSSION To get an impression of the practical usefulness of the parameter SX, the dependence of its value on pHBGE was calculated for picrate in a BGE formed by 0.02 M propionic acid with variable contents of Na+. Figure 4 shows the result: when the pHBGE decreases the peak shape changes from tailing to fronting with the point of inversion, SX ) 0, being at pHBGE ) 4.22. From the plotted dependence, it can be concluded that, at pHBGE > 4.22, quite sharp peaks with slight tailing can be expected because the velocity slope SX remains low in the entire pHBGE region above 4.22. When, however, the pHBGE decreases below 4.0, broad fronting can be predicted for picrate because the SX value decreases dramatically here. These theoretical expectations can be confirmed experimentally, as indicated in Figure 5, showing three peak patterns of the same amounts of picrate run at pHBGE of 3.86, 4.24, and 5.38. Whereas the peaks at pHBGE ) 4.24 and 5.38 are quite sharp and only slightly tailing, the peak at pHBGE ) 3.86 is very broad and strongly fronting. Note that the point of inversion where fronting changes into tailing, clearly characterized by SX ) 0, does not have any reliable 1560 Analytical Chemistry, Vol. 69, No. 8, April 15, 1997

Figure 6. PSD for a BGE formed by 0.02 M MES + 0.01 M Na+ (pH ) 6.21). The plotted pKHX vs uX (in 10-9 m2 V-1 s-1) dependence corresponds to anionic analytes whose velocity slope is zero, SX ) 0.

simple relation to effective and/or ionic mobilities of the analyte and BGE co-ion. In the present case of picrate as analyte and propionate as BGE co-ion, their effective mobilities at pHBGE ) 4.22, 31.7 and 9.2 × 10-9 m2 V-1 s-1, respectively, do not show any relation either to each other or to the values of their ionic mobilities, 31.7 and 36.9 × 10-9 m2 V-1 s-1, respectively (picrate is fully dissociated within the pH range investigated here). In the Theoretical Section it was shown that the velocity slope SX, which determines the peak (zone) shape of a weak electrolyte depends, besides quantities related to the BGE, only on the ionic mobility uX and the pKHX of the analyte. It is therefore useful to characterize a given zone electrophoretic system (i.e., a given BGE) by a two-dimensional plot of pKHX vs uX. Figure 6 shows, as an example, such a plot for a BGE formed by 0.02 M MES + 0.01 M Na+ (pH ) 6.21). The points in the plane correspond to zone electrophoretically migrating anionic analytes of given pKHX and uX. If now pairs of uX-pKHX values are found for which SX ) 0 and plotted into the given coordinate system, a curve dividing the plane into two parts according to the sign of SX is obtained. In the present case, for analytes whose points are on the right from the curve, SX < 0, and thus (according to condition 3a) they can be predicted to form fronting peaks. Vice versa, analytes whose points are found on the left from the curve are characterized by SX > 0, and it can be expected to get their peaks as tailing ones (see condition 3b). The curve in Figure 6 corresponds to the transition between fronting and tailing peak shape. Figure 7 shows the same situation but with two more curves included, viz., the straight line indicating

Figure 7. PSD from Figure 6 with included mobility contours (dashed) uX ) uA and u j X,A ) u j A,A. For explanation, see text.

Figure 9. PSD from Figure 6 showing the network of effective mobility contours. For an analyte of known pKHX and uX, its effective mobility in the given BGE can be read out, which allows estimation of its detection time.

Figure 8. PSD from Figure 6 showing the network of velocity slope contours. For an analyte of known pKHX and uX, the value of its velocity slope in the given BGE can be read out which allows estimation of its peak symmetry.

the value of the ionic mobility of the background co-ion (uX ) uA; here A ) MES) and the curve involving points corresponding to substances whose effective mobility in the given BGE is equal to that of the BGE co-ion, uj X,A ) uj A,A. As is seen, the comparison of ionic mobilities (uX higher or lower than uA) is applicable for peak shape testing only for fully dissociated analytes. A comparison of effective mobilities (uj X,A higher or lower than uj A,A) is completely useless for this purpose (compare the dashed curve with the thick one). This clearly demonstrates that only the curve corresponding to SX ) 0 is applicable for a successful prediction of peak symmetry. Let us call such type of pKHX vs uX plot as shown in Figure 6 the peak shape diagram (PSD). The single curve in the PSD is sufficient for the estimation of what symmetry has the peak of any analyte of known pKHX and uX. More information can be included considering that the magnitude of SX expresses the level of electromigrational dispersion. Plotting a set of curves with various values of SX into the PSD also offers the way to estimate the peak width of electromigrationally dispersing peaks. The result (for the above described example BGE) is shown in Figure 8. The plane is now covered by a map that allows an estimation of how strongly the fronting or tailing effect applies. It can be generally postulated that the higher the magnitude of SX, the broader is the corresponding sample zone. Other information that also seems useful to include into the PSD is the effective mobilities of the analytes (uj X,A). Figure 9 shows the same PSD coordinate system as in the previous figures with the plotted set of curves having uj X,A as parameter. These curves allow us to estimate the effective mobility of any analyte

Figure 10. Complete peak shape diagram for the BGE formed by 0.02 M MES and 0.01 M Na+ (pH ) 6.21), as obtained by superimposing Figures 8 and 9. The thick lines show the velocity slope contours (corresponding to the indicated SX values), and the dashed lines show the effective mobility contours (corresponding to the values indicated inside the diagram). All mobility values are in 10-9 m2 V-1 s-1.

and thus its detection time. At multicomponent samples, comparison of the effective mobilities of the sample components allows us to estimate both their migration order (i.e., the detection order of sample peaks) and their selectivity (given by the relative effective mobility difference). Figure 10 shows the complete PSD obtained by superimposing the partial plots from Figures 8 and 9. The PSD involves the following: (i) the set of SX curves for estimation of peak shape and peak width; (ii) the set of uj X,A curves for estimation of migration time, migration order, and selectivity; (iii) a supplementary grid for better reading out the point of a selected substance of given uX and pKHX. The following example illustrates how to use the PSD for prediction of the record pattern of a zone electrophoretic separation. A model sample was selected according to Table 2, which shows five substances A-E and their uX and pKHX values. The BGE was 0.02 M propionic acid with 0.01 M Na+ (pH ) 4.78). The related PSD is shown in Figure 11, including the points corresponding to the five substances. Their uj X,A and SX values can be read out from the PSD, as shown in the last two columns of Table 2. The following prediction can be made from the data obtained: (i) the migration order of the five analytes will be from Analytical Chemistry, Vol. 69, No. 8, April 15, 1997

1561

CONCLUSIONS

Table 2. Properties of the Model Substances from Figures 11 and 12 a,b

pKHX

42.3 35.3 31.7 32.9 24.7

2.49 2.94 0.71 4.17 4.35

uX A B C D E

pyruvate salicylate picrate benzoate p-isopropylbenzoate

a

b,c

uj X

42 35 32 26 18

c,d

SX

-0.15 +0.01 +0.09 -0.02 +0.08

a Tabulated19 values. b In 10-9 m2 V-1 s-1. c Values estimated from Figure 11. d In mm s-1.

Figure 11. PSD for the BGE formed by 0.02 M propionic acid and 0.01 M Na+ (pH ) 4.78). All mobility values are in 10-9 m2 V-1 s-1. The points A-E relate to the substances from Table 2.

Migrating zones in CZE may exhibit tailing or fronting assymetric concentration profiles due to electromigration dispersion. The simple theoretical model developed in this paper allows a general prediction of the shapes of the zone concentration profiles. Assuming a moving boundary system composed of a leading zone formed by the BGE and a trailing mixed zone involving the BGE and the sample with their concentration ratio as parameter, it can be calculated how the regulating principle adjusts the concentrations in the trailing zone to the parameters of the leading one. In this way, the zone properties of a sample zone of any adjusted concentration ratio can be modeled, as, e.g., pH and conductivity, which influence the sample effective mobility and its migration velocity. It is shown that a parameter can be defined whose numerical value unambiguously determines the zone (peak) shape. This parameter expressed as SX ) (dvX,AX/ daX)aXf0 and called the velocity slope shows how the analyte local migration velocity in the BGE at infinite analyte dilution changes with an increase in its concentration. The sign of SX determines the type of peak assymetry (a positive SX indicates a tailing zone and a negative SX a fronting one), and its magnitude indicates the magnitude of electromigration dispersion, i.e., relates to zone width. Calculating the SX values for various analytes with ionic mobility (uX) and pKHX as parameter, and plotting the curves corresponding to same SX values in an uX vs pKHX coordinate system, provides a network that is called the peak shape diagram. The PSD, once plotted for a given BGE, allows estimation of the zone (peak) symmetry/assymetry for any analyte of known uX and pKHX. Based on the PSD, CE experiments may be easily optimized either directly or via computer simulation. The applicability of the PSD is demonstrated on a model example as shown in Figures 11 and 12. ACKNOWLEDGMENT The authors thank Prof. Wolfgang Thormann for kindly providing the electrophoresis simulation software. This work was supported by the grants from the Grant Agency of the Czech Republic (203/94/0998 and 203/96/0124) and of the Grant Agency of the Academy of Sciences of the Czech Republic (431404).

Figure 12. Experimental UV absorbance detection record (absorbance range 0.016 AU) of the separation of the five-component mixture from Table 2 in the BGE formed by 0.02 M propionic acid and 0.01 M Na+ (pH ) 4.78). The sample solution contained 0.008 M pyruvic acid (A), 0.0008 M salicylic acid (B), 0.0024 M picric acid (C), 0.004 M benzoic acid (D), and 0.002 M p-isopropylbenzoic acid (E) with the acid contents neutralized by 70% with NaOH.

A (as the fastest) to E (as the slowest); (ii) the peak symmetry in the given migration order will be from fronting (A) to tailing (E) with the fronting peak of D being an exception; (iii) the peak width will vary from broad (A) via narrow (B) to broad (C-E). The experiment fits well with these predictions, as shown in Figure 12. It can be seen that the peak of substance D represents an anomaly in the peak shape/width vs detection time course. The PSD, however, indicates here clearly that there is nothing anomalous on substance D and that its peak shape and peak width are a natural consequence of the interaction of its uX and pKHX with the given BGE. 1562

Analytical Chemistry, Vol. 69, No. 8, April 15, 1997

APPENDIX: PROCEDURE OF CALCULATION OF THE PARAMETERS OF A SAMPLE ZONE OF A GIVEN CONCENTRATION LEVEL For the model system shown in Figure 1, the following set of moving-boundary equations involving components A, B, and X can be written:

j X,AX jcA,Au j A,A jcA,AXu j A,AX u ) (cj - jcA,AX) κA κAX κAX A,A

(A1)

jcB,Au j X,AX j B,A jcB,AXu j B,AX u ) (cj - jcB,A) κA κAX κAX B,AX

(A2)

where jci,j is the total (analytical) concentration of substance i in zone j, uj i,j is the effective mobility of substance i in zone j, and κj is the specific conductivity of zone j. Removal of κAX results in

jcB,Au j B,A u j A,AX - u j X,AX jcB,AX ) jcA,AX + jcA,Au j A,A u j B,AX + u j X,AX jcB,A

(

and b is a correction term expressing the contribution of BH+ to the transport of H+:

)

u j X,AX u j B,A + 1 (A3) u j B,AX + u j X,AX u j A,A

The electroneutrality condition for zone AX can be expressed as

cA,AX + cX,AX - cBH,AX - cH,AX + (Kw/cH,AX) ) 0 (A4) where ci,j is the concentration of ion i in zone j and Kw is the water constant. Combination of the above equations provides

jcA,AX )

[

[[

cH,AX -

Kw + cH,AX

]

] ]

u j X,AX u j B,A + u j A,A u j B,AX jcB,A (1 - aX) / u j B,AX + u j X,AX u j A,A uBH

[

b)

]

(A6)

The regulating function for zone AX can be expressed as

ωAX )

jcA,AX jcX,AX jcB,AX jcH,AX + + + + b ) ωA uA uX uBH uH

(A7)

(A9)

( ) dvX,AX daX

≈ aXf0

vX,AX - vX,A aX

(A10)

where vX,AX relates to a sufficiently small aX.

(A5) where the concentration ratio of A and X in the mixed zone AX is expressed as the molar fraction of substance HX:

)

The combination of equations 1-9 eliminates all unknown variables except cH,AX; its correct value is found by solving the above set for cH,AX by looking for a value of ωAX that equals the ωA value of the original BGE. As soon as cH,AX is known, all other quantities (in particular uj X,AX, κAX, and vX,AX) can be obtained from the above given equations. The velocity slope SX can be then calculated by approximating the derivative (see conditions 3a and 3b) by a finite differences ratio

SX )

u j X,AX u j A,AX jcB,Au j B,A u j A,AX - u j B,AX u j B,AX aX + (1 - aX) uX uA jcA,Au j A,A u j B,AX + u j X,AX uBH

jcX,AX aX ) jcX,AX + jcA,AX

(

κAX 1 jc u j - jcBH,Au j BH,A u j X,AXuH BH,AX BH,AX κA

GLOSSARY aX

local molar fraction of analyte X in its zone

ci,j

local concentration of ion i in zone j

jci,j

local analytical concentration of substance i in zone j

i

electric current density

Kw

water constant

SX

velocity slope of analyte X

ui

ionic mobility of ion i

uj i,j

local effective mobility of substance i in zone j

vi,j

local migration velocity of substance i in zone j

κj

local specific conductivity in zone j

ωj

Kohlrausch regulating function of zone j

where

uA - u j A,AX jcH,AX ) cHA,AX + cHX,AX + cBH,AX + cH,AX ) jcA,AX + uA uX - u j X,AX u j B,AX jcX,AX + jcB,AX + cH,AX (A8) uX uBH

Received for review August 6, 1996. Accepted January 21, 1997.X AC960796D X

Abstract published in Advance ACS Abstracts, February 15, 1997.

Analytical Chemistry, Vol. 69, No. 8, April 15, 1997

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