The influence of analyte concentration when compared with the concentration of a charged ligand in background electrolyte (BGE) on the measured values of electrophoretic mobilities and stability constants (association, binding or formation constants) is studied using capillary electrophoresis (CE) and a dynamic mathematical simulator of CE. The study is performed using labile complexes (with fast kinetics) of iron (III) and 5-sulfosalicylate ions (ISC) as an example. It is shown that because the ligand concentration in the analyte zone is not equal to that in BGE, considerable changes in the migration times and electrophoretic mobilities are observed, resulting in systematic errors in the stability constant values. Of crucial significance is the slope of the dependence of the electrophoretic mobility decrease on the ligand equilibrium concentration. Without prior information on this dependence to accurately evaluate the stability constants for similar systems, the total ligand concentration must be at least >50–100 times higher than the total concentration of analyte. Experimental ISC peak fronting and the difference between the direction of the experimental pH dependence of the electrophoretic mobility decrease and the mathematical simulation allow assuming the presence of capillary wall interaction.

## Introduction

The evaluation of the stability, association, formation or binding constants of metals, solutes or substrates with different ligands is an actual task in different fields of chemistry, medicine, ecology and industry. Recently, capillary electrophoresis (CE) and related electroseparation techniques have been widely used for this purpose because of its rapidity, low cost of reagents and high performance (1–5). The analytes such as metals, pharmaceuticals, nucleotides, proteins, oligosaccharides, poliovirus and nanobodies (6) are studied. The complex formation with different ligands is also often employed to improve the selectivity of inorganic and organic analyte separation by CE (7, 8). It is important to select the conditions in which the technique used would not result in significant systematic errors in the measured values of the stability constants (2, 5, 9). One of these conditions is the concentration ratio of the charged ligand to analyte; however, this problem is paid little attention in the literature. It is noted in the literature that the ligand concentration should be 10–100 times higher than the substrate concentration for a strong interaction between the ligand and substrate. However, this statement was deduced from other methods where the ligand and substrate interacted in the vessel (2, 5) rather than from the CE theory. In (10), the peak top is shifted with increasing the analyte concentration and an approach is proposed to correct the migration time for the high concentration level using the Haarhoff–Van der Linde (HVL) function for a triangular peak obtained by affinity CE. However, in this case, the study was concerned with the neutral ligand (β-cyclodextrin).

To study the interaction between analyte and ligand, several electropherograms with varying ligand content in background electrolyte (BGE) and analyte injected as a sample are recorded. On the basis of the calculated values of the electrophoretic mobility and ligand concentration the stability, formation, association or binding constants are obtained. Sometimes, the analyte is added to the BGE, and the ligand is injected as a sample. The statement “In the electrophoretic experiment, the concentration of the ligand is an a priori known value” (11) is right under the assumption of a neutral ligand or when the ligand concentration is much higher than the analyte one. Only under these conditions, the equilibrium concentration of the ligand may be calculated on the basis of its total concentration and pH.

In CE, the concentrations of all BGE constituents in the analyte zone differ from the ones in the initial BGE because of the partial displacement of the BGE ions governed by the charge and mass conservation laws (12). This phenomenon is used for indirect detection in CE. For non-UV absorbing analytes, BGE is composed of UV absorbing co-ions and its decrease in the analyte zone is recorded. A mathematical expression for calculating the displacement amount of BGE in the analyte zone is derived only for systems with strong or weak univalent constituents. For the system containing a single cation K^{z+} and a single anion An^{z−} constituents in the pH region where both H^{+} and OH^{−} can be neglected, the change in the BGE anion concentration in the analyte zone can be calculated using the constancy of the Kohlrausch regulating function as follows:

*С*

_{An}is the BGE anion concentration in the analyte zone, $CAn0$ is the initial BGE anion concentration,

*µ*is the electrophoretic mobility,

*z*is the charge,

*C*is the analyte concentration and TR

_{X}_{An}is the transfer ratio. TR

_{An}shows how many times the change in the BGE anions concentration is higher or lower than the analyte concentration. If TR

_{An}< 0, the BGE anion concentration in the analyte zone decreases. For the cationic analyte, the change in the BGE anion concentration can be calculated by Equation (2):

*μ*=

_{K}*μ*. If

_{X}*μ*<

_{K}*μ*

_{X}, the BGE anion concentration in the analyte zone increases.

The decrease of the ligand concentration in the analyte zone results in shifting the effective electrophoretic mobilities and errors in the stability constant values because of their calculation being based on the ligand concentration in BGE. It is clear that, with decreasing the relative analyte concentration, the errors would decrease as well. The lower limit of the analyte concentration is restricted by the CE detector sensitivity. The relative analyte concentration can be decreased by increasing the ligand concentration in BGE, but the upper limit of BGE is restricted by Joule heating. Excessively high analyte concentration relative to the ligand concentration is sometimes used, which is likely to result in the difference between the results obtained and data available in the literature (11, 13).

For complicated system, amount of the displaced BGE in the analyte zone can be calculated only using the dynamic computer simulation of electrophoresis (14–22). It is based on the numerical solution of partial differential equations for charge and mass transfer using a set of assumptions. Though the dynamic computer simulation of electrophoresis has been used for >30 years (15), it is only recently that the interest in taking into account the complex formation equilibria during the electrophoresis process has increased (14, 19–22). However, since the calculation is complicated, neutral ligands are used in most dynamic computer electrophoresis simulators, though there are several publications where the ligand is charged while the analyte is neutral (14, 21, 23). For this case, the BGE ions and ligand concentrations in the analyte zone are likely to change to a very slight degree because the charge of the complex between ligand and analyte is equal to the charge of ligand.

The aim of this paper is to study the influence of the negatively charged ligand to analyte concentration ratio on the measured values of the electrophoretic mobility and stability constant values obtained by experiments and dynamic mathematical simulator of CE. The study is performed using kinetically labile complexes of iron (III) and 5-sulfosalicylate ions (ISC, H_{3}L) as an example. This is a case in which one analyte molecule relates to one ion of charged ligand, i.e. S^{0}+ L^{z} = SL^{z} (S^{0}−FeL^{0}, *z* = −3). The system where the ligand and its protonated forms are sole anions of BGE is considered. In such systems, BGE co-ions are replaced by analyte ions, i.e. concentration of the co-ions in the analyte zone is always lower than that in the BGE. For BGE with two anionic constituents, it is possible that one constituent in the analyte zone is lower than that in the BGE, whereas the second constituent concentration increases (24). In addition, the systems with one anionic constituent are convenient due to a small number of the system peaks appearing because of the presence of several constituents in BGE (18, 25). Hydrodynamic pressure is used instead of the electroosmotic flow (EOF) modifier, this having been widely employed for physicochemical measurements (26) and analysis (27, 28) recently.

## Experimental

### Chemicals

The chemicals used, namely, iron (III) chloride, ethylenediaminetetraacetic acid, 5-sulfosalicylic acid dihydrate (from Sigma-Aldrich, Moscow, Russia), and 0.1 M hydrochloric acid solution and potassium hydroxide (from Himreaktivsnab, Ufa, Russia), were of analytical-reagent grade. The other mentioned solutions were purchased from Agilent Technologies (Waldbronn, Germany). Deionized water from a water purification system Direct-Q3 (Millipore, France) was used for the solution preparation.

A 0.1 M stock solution of iron (III) chloride was prepared by dissolving a weighed amount of FeCl_{3} in a 0.1 M solution of HCl. The precise concentration of iron (III) was obtained by titrating an aliquot of this solution with ethylenediaminetetraacetic acid as a titrant and 5-sulfosalicylic acid as an indicator in a triplicate. Samples were daily prepared by diluting the above stock solution with BGE to avoid stacking effects during the sample injection. A 0.1% (v/v) acetone was used as an EOF marker.

### Instrumentation

The study was carried out using a CE system with a diode-array detector Agilent ^{3D}CE G1600A (Agilent Technologies) of the Krasnoyarsk Regional Center of Research Equipment, Siberian Branch of the Russian Academy of Sciences. Untreated fused silica capillaries with 50 μm id and the total and effective lengths of 48.5 and 40 cm were used (Agilent Technologies). The capillary temperature was kept constant at 25.00 ± 0.04°C. The data acquisition and processing were performed with the computer program ChemStation Rev.A.10.02. The separation was achieved by applying a voltage of –19.0 to –20.7 kV so that for each BGE the product of voltage and current was kept constant (Table I). The negative voltage was applied to the capillary inlet. The detection wavelength, 260 nm, was chosen on the basis of the highest possible value of signal-to-noise ratio of ISC peak. The samples were injected hydrodynamically for 2 s at a pressure of 50 mbar. To change EOF velocity, the hydrodynamic pressure of 15–50 mbar is used.

pH | $CL0$ (mM) | BGE composition (mM) | [L^{3−}]_{bge} (10^{−11} M) | U (kV) | Current (μA) | U·current (V·A) | ||
---|---|---|---|---|---|---|---|---|

[H_{2}L^{−}] | [HL^{2−}] | [K^{+}] | ||||||

2.400 | 22.2 | 8.3 | 13.9 | 32.1 | 0.874 | −19.0 | 29.7 | 0.564 |

2.493 | 21.3 | 6.9 | 14.4 | 32.5 | 1.12 | −19.4 | 29.1 | 0.565 |

2.627 | 20.2 | 5.3 | 14.9 | 32.7 | 1.58 | −19.8 | 28.5 | 0.564 |

2.700 | 19.7 | 4.5 | 15.2 | 32.9 | 1.90 | −19.7 | 28.5 | 0.561 |

2.803 | 19.1 | 3.7 | 15.4 | 32.9 | 2.46 | −20.0 | 28.0 | 0.560 |

2.904 | 18.6 | 2.9 | 15.7 | 33.1 | 3.15 | −20.4 | 27.6 | 0.563 |

3.112 | 17.9 | 1.9 | 16.0 | 33.1 | 5.21 | −20.7 | 27.2 | 0.563 |

pH | $CL0$ (mM) | BGE composition (mM) | [L^{3−}]_{bge} (10^{−11} M) | U (kV) | Current (μA) | U·current (V·A) | ||
---|---|---|---|---|---|---|---|---|

[H_{2}L^{−}] | [HL^{2−}] | [K^{+}] | ||||||

2.400 | 22.2 | 8.3 | 13.9 | 32.1 | 0.874 | −19.0 | 29.7 | 0.564 |

2.493 | 21.3 | 6.9 | 14.4 | 32.5 | 1.12 | −19.4 | 29.1 | 0.565 |

2.627 | 20.2 | 5.3 | 14.9 | 32.7 | 1.58 | −19.8 | 28.5 | 0.564 |

2.700 | 19.7 | 4.5 | 15.2 | 32.9 | 1.90 | −19.7 | 28.5 | 0.561 |

2.803 | 19.1 | 3.7 | 15.4 | 32.9 | 2.46 | −20.0 | 28.0 | 0.560 |

2.904 | 18.6 | 2.9 | 15.7 | 33.1 | 3.15 | −20.4 | 27.6 | 0.563 |

3.112 | 17.9 | 1.9 | 16.0 | 33.1 | 5.21 | −20.7 | 27.2 | 0.563 |

А new capillary was first flushed with 1 M NaOH for 10 min and then with ultra-pure water for 10 min. To avoid baseline fluctuation and drift, at the beginning of each day, the capillary was first flushed with 0.1 М NaOH for 5 min, twice with ultra-pure water for 10 min, with 20 mM 5-sulfosalicylic acid for 5 min and then with running BGE for 15 min. Between the runs, the capillary was flushed with BGE for 5 min.

All pH measurements were made using a calibrated precise pH instrument “Expert-001-1” (Econix-Expert, Moscow, Russia) with a precision of 0.005 pH units.

### Measurements and data processing

The BGE used consisted of 5-sulfosalicylic acid H_{3}L and its potassium salts (Table I). The equilibrium ligand concentrations in BGEs ([L^{3−}]_{bge}) were varied by changing BGE pH from 2.4 to 3.1 because the 5-sulfosalicylic acid can be used as single BGE anion constituent with acceptable buffering capacity of the BGE in this pH region $(pKH2L\u2212=2.18)$:

^{3−}]

_{bge}was calculated as follows:

*I*= 0.05 (30). The total ligand concentrations $CL0$ in the BGEs (Table I) were taken such as the ionic strength

*I*should be 0.0500 (see Supplementary data, S1 for detail):

*β*is the stability constant of FeL

_{2}_{2}

^{3−}and

*β*is the stability constant of FeL

_{1}^{0}. Table II presents p

*K*and log

*β*for all the species used in the calculation. The values of the stability constants were taken from “NIST Critically selected stability constants of metal complexes” (Database 46 freeware downloaded from www.nist.gov/srd/nist46.cfm). The stability and dissociation constants were recalculated by the extended Debye–Huckel expression for the ionic strength

_{i}*I*= 0.05 (30). The complexes between iron (III) and OH

^{−}ions are neglected for these conditions (Supplementary data, Table S1). The ISC are kinetically labile and resulted in a sole peak in the electropherograms. The ISC zone charge is weighted average of molar forms, FeL

^{0}and FeL

_{2}

^{3−}(Supplementary data, Table S1). Thus, the ISC zone moves as anion. Since the charge and ionic mobility of FeL are equal to zero, the ISC effective electrophoretic mobility is calculated as follows:

_{2}

^{3−}. The ratio

*β*

_{2}/

*β*

_{1}and confidence interval were determined from Equation (10) using non-linear regression fitting of the program Origin 8.1 (OriginLab Corporation, Northampton, MA, USA). The effective electrophoretic mobility from the experimental data was calculated as follows:

*l*and

*l*

_{eff}are the total and effective capillary lengths, respectively;

*U*is the voltage;

*t*

_{ISC}is the ISC migration time measured at the top of the electrophoretic peak; and

*t*

_{eof}is the migration time of the EOF marker.

pK | log β | |
---|---|---|

Hydrosulfosalicylate HL^{2−} | 11.60 | |

Dihydrosulfosalicylate H_{2}L^{−} | 2.18 | |

[FeL]^{0} | 15.07 | |

[FeL_{2}]^{3−} | 25.52 | |

[FeL_{3}]^{6−} | 32.15 |

pK | log β | |
---|---|---|

Hydrosulfosalicylate HL^{2−} | 11.60 | |

Dihydrosulfosalicylate H_{2}L^{−} | 2.18 | |

[FeL]^{0} | 15.07 | |

[FeL_{2}]^{3−} | 25.52 | |

[FeL_{3}]^{6−} | 32.15 |

*I* = 0.05 at 25°C.

It is worth noting that the accuracy of the рН measurements is of great importance. The accuracy of the рН measurement of ∼0.1 resulted in an error of 40% in the calculated (L^{3−}) value (see Supplementary data, Table S2 in Section S4) for рН ranging from 2.4 to 3.1:

_{pH}, (L)

_{pН+ΔpH}and [L]

_{pН−ΔpH}are the [L

^{3−}] values calculated using Equation (4) with рН, рН+ΔpH or рН−ΔpH. Therefore, the accuracy of the рН measurements should be no worse than 0.005.

The HVL function in the program PeakFit v4.11 (SPSS, Chicago, IL, USA) was used to fit experimental peaks.

### Mathematical simulator of CE

The dynamic mathematical simulator of CE (18, 31) using the slice method of calculation was employed. In the simulator, the general equation governing the ion flux along the length of the capillary tube (*x*-axis) was solved numerically

*C*(

_{i}*x, t*) is the total concentration of the

*-*th constituent,

*t*is the time,

*j*is the current density,

*z*is the charge of

_{i}*i*th constituent,

*µ*

_{eff,i}is the effective electrophoretic mobility of the

*i*th constituent,

*κ*is the electrical conductivity and

*D*is the diffusion coefficient of

_{i}*i*th constituent. As a model system, the capillary was taken to consist of several thousands of slices. The rate of the concentration change of the

*i*th constituent in the

*n*th slice at the time point

*t*was calculated by Equation (14)

*n-*th slice in the time point

*t*, including the analyte ions, with

*μ*

_{eof}being the electroosmotic mobility.

Based on Equation (14), the changes in the concentration of the *i*th constituent in the *n*th slice $\Delta Ci,n\tau $ and their concentrations $Ci,nt+\tau $ with the time *τ* were calculated as follows:

*i*th constituent in the

*n*th slice at the time point

*t*,

*l*

_{slice}is the slice length which is equal to the ratio of the effective capillary length to the slice number

*l*

_{eff}

*/N*,

*R*is the gas constant,

*T*is the temperature and

*F*is the Faraday constant. When calculating the hydrogen concentration, the neighboring slices were taken into account because of the motion redirection being possible under certain conditions:

*t*reached a desired value. The calculations were performed by the program based on Visual Basic for Applications using PC on the basis of Intel Core i5-3550 processor with a frequency of 3.3 GHz. The initial shape of the analyte zone is a trapezoid occupying 1/250 of the capillary length. The composition of injected samples is the same as described in the

*Chemicals*section. To decrease the calculation time use was made of a moving window. The calculation was performed for the zone where the total analyte concentration was >1×10

^{−4}×$CM0$ or the total hydrogen concentration was 0.0001% higher or lower than that in BGE. Each calculation for certain conditions (BGE concentration and pH, ligand to analyte concentration ratio and slice number) lasted for several hours or even tens of hours.

In the analyte zone, the equilibrium concentrations of the constituents were found using iteration since the electroneutrality equation was a fourth-degree polynomial in [L^{3−}]. The free ligand concentration in BGE calculated using Equation (4) was used as the initial value. Furthermore, the equilibrium concentrations were calculated as follows:

*C*is the total iron concentration in the considered slice of the mathematical simulator.

_{M}The concentration of H^{+} ions was calculated by Equations (22) and (23):

^{3−}] were found at which the

*y*function was equal to zero. Concentrations of hydrosulfosalicylate and dihydrosulfosalicylate ions were calculated by

The effective electrophoretic mobilities of the *i*th constituent in each slice were calculated as the weighted average of the mole fractions of all *i*th constituent ions (for simplicity, the ion charges were omitted):

The negative sign before the ionic mobility of H^{+} ions was used because their direction was opposite to the motion direction of the analyte. Table III presents the ionic mobilities for all the species used in the calculation. The ionic mobility of [FeL_{2}]^{3−} was measured by experiments (see Equation 10). The other ionic mobilities were taken from (33) and recalculated for *I* = 0.05 as described in (34).

Ionic species | μ (10^{−9} m^{2} V^{−1} s^{−1}) |
---|---|

K^{+} | 64 |

H^{+} | 307 |

Hydrosulfosalicylate HL^{2−} | 44^{a} |

15 | |

5 | |

Dihydrosulfosalicylate H_{2}L^{−} | 30^{a} |

10 | |

3 | |

Cl^{−} | 71 |

[FeL]^{0} | 0 |

[FeL_{2}]^{3−} | 42 |

[FeL_{3}]^{6−} |

Ionic species | μ (10^{−9} m^{2} V^{−1} s^{−1}) |
---|---|

K^{+} | 64 |

H^{+} | 307 |

Hydrosulfosalicylate HL^{2−} | 44^{a} |

15 | |

5 | |

Dihydrosulfosalicylate H_{2}L^{−} | 30^{a} |

10 | |

3 | |

Cl^{−} | 71 |

[FeL]^{0} | 0 |

[FeL_{2}]^{3−} | 42 |

[FeL_{3}]^{6−} |

*I* = 0.05 at 25°C.

^{a}Calculated using equation for sulfonic acids (35).

The simulated electropherograms were plotted as the total analyte concentration, ligand equilibrium concentrations, total and free ligand concentrations, H^{+} ion concentration and ligand effective electrophoretic mobilities in the detector zone versus time. Using the metal peak, the total electrophoretic mobility of the zone was calculated by Equation (30)

*N*

_{slice}is the number of slices in the simulator and

*N*

_{slice}/500 is the correction of the effective capillary length taking into account the sample length (

*N*

_{slice}/250).

The peak efficiency (theoretical plate numbers) *N* was calculated using Equation (31)

*w*

_{1/2}is the peak width at half height.

## Results

### Experiments

For the whole рН range studied, the shift of the ISC peak top was found toward greater migration times with increasing the analyte concentration in the injected sample $(CM0)$ and, correspondingly, toward smaller electrophoretic mobilities. Figure 2 shows the example of the electropherograms of ISC, with the $CL0/CM0$ ratios being 10 and 100. The differences between pH of injected samples and BGE pH did not exceed 0.5 (pH of such samples is shown in Supplementary data, Tables S3 in S5). For all pH range studied, the differences between ISC electrophoretic mobilities for samples simply diluted by BGE and the samples diluted by BGE and adjusted to BGE pH were negligible.

It is worth noting that in spite of peak fronting, for a high concentration, the peak is not of a triangular shape and, as shown in Figure 2, the HVL function (10) is poorly fitted ISC peak asymmetry, particularly for $CL0/CM0$ is 10. Besides, the HVL function does not remove the shift of ISC peak with increasing analyte concentration (see vertical lines corresponded to the parameters *a*_{1} of the HVL function). Such peak shape is likely to relate with partial displacement of the BGE ions governed by the charge and mass conservation laws.

Figure 3 shows the pH dependence of the relative difference between the electrophoretic mobilities measured at $CL0/CM0=10(\mu eff,10)$ and $CL0/CM0=100(\mu eff,100)\Delta \mu eff(exp)$ calculated as follows:

^{0}and [FeL

_{2}]

^{3−}(Figure 1). And the lower is pH, the more considerable change is observed. On the other hand, for the lower pH, the average charge of ISC is smaller, while the quantity of the displaced BGE is proportional to the charge. The рН dependence of Δ

*μ*

_{eff}(exp) can be accounted for by these opposite tendencies.

Because the difference in the electrophoretic mobilities is dependent on the analyte concentration, it would also depend on the conditions under which the electropherograms are recorded. It was experimentally found that, with varying the values of the applied hydrodynamic pressure from 15 to 50 mbar, the values of electrophoretic mobilities and Δ*μ*_{eff} (exp) were not changed. With varying the voltage and capillary length, these values changed. This is likely to be due to the effect of Joule heating. Therefore, all the measurements should be performed with the product of voltage and current being constant.

The difference in the electrophoretic mobilities causes the systematic errors in the evaluated values of the stability constants. For $CL0/CM0=100$, the stability constant ratio *β*_{2}/*β*_{1} calculated using Equation (10) is (2.7 ± 0.2) × 10^{10} M^{−1} (*R*^{2}> 0.999), that is in agreement with the literature data, namely, 2.82 × 10^{10} M^{−1} (*I* = 0.05). At the same time, for $CL0/CM0=10$, the stability constant calculated using Equation (10) is (2.2 ± 0.2) × 10^{10} M^{−1} (*R*^{2}>0.999). Thus, the systematic deviation of −19% is observed relative to *β*_{2}/*β*_{1} calculated for $CL0/CM0=100$. But it is not possible to find experimentally the concentration at which the systematic errors Δ*μ*_{eff} (exp) in the electrophoretic mobilities are negligible because of the low detection sensitivity for this system. Taking the above mentioned into account, this phenomenon is investigated using the mathematical simulation.

### Mathematical simulation

The examples of the simulated electropherograms of ISC with $CL0/CM0=10$ for three cases of different electrophoretic mobilities of the BGE anions are presented in Figure 4. For all the cases, the shift of the ISC peak top from the theoretical position is observed. The shift is larger when the electrophoretic mobilities of the BGE anions are smaller. But for the higher electrophoretic mobilities of the BGE anions, the migration time decreases (Curve 1), while for the smaller electrophoretic mobilities of the BGE anions, the migration time increases (Curves 2 and 3).

This can easily be explained, when examining the simulated electropherograms in the coordinates of [L^{3−}] and time. For higher electrophoretic mobilities of the BGE anions, in the ISC zone, smaller decrease in the equilibrium ligand concentration [L^{3−}] as related to the initial concentration in the BGE is observed. This regularity is contrasted to the regularity observed for strong and weak univalent electrolytes, with TR increasing when increasing the electrophoretic mobilities of BGE anions (see Equation (1)). For higher electrophoretic mobilities of the BGE anions, higher decrease in the free total ligand concentration $CL\u2032$ in the ISC zone is observed along with higher decrease in the H^{+} ion concentration, resulting in an enhanced dissociation of the hydrosulfosalicylate ions to form ISC. The decrease in the migration time for higher electrophoretic mobilities of the BGE anions can be explained by changing the ISC peak asymmetry.

It is noteworthy that for lower electrophoretic mobilities of the BGE anions (*μ*(HL^{2−}), *μ*(H_{2}L^{−}): 5 and 3 10^{−9}m^{2} V^{−1} s^{−1}), the free total ligand concentration $CL\u2032$ increases in the ISC zone. And, since the H^{+} ion concentration increases in the ISC zone, the equilibrium ligand concentration decreases relative to the initial BGE concentration. Besides, in the ISC zone, the effective electrophoretic mobilities of the BGE anions change (Figure 4 and Equation 28).

To compare the experimental and simulated data, the efficiency of the simulated electrophoretic peaks must be in agreement with the efficiency of the experimental electrophoretic peaks. It is found that the simulated peak efficiency depends on the effective capillary length and time τ (see Equations 15, 16 and 18) as follows:

*k*is the proportionality coefficient which is constant for the given values of pH, BGE concentration, electrophoretic mobilities for all species and $CL0/CM0$ ratio. It is found that the minimal time (in seconds) so that the oscillation is absent is the following:

*k′*is equal to

*k*/20. The dependence of

*N*on

*l*

_{eff}

*·N*

_{slice}is linear for the entire рН region studied and for different electrophoretic mobilities of the BGE anions with the correlation coefficient of

*R*

^{2}> 0.97 (Figures 5A and B). The coefficient

*k′*ranges from 3.6 to 5.1 × 10

^{−3}cm

^{−1}. A small deviation of the points from the straight line is likely to be due to the peak asymmetry influence or difference between the sample and BGE conductivity.

The decrease in the effective electrophoretic mobility is calculated as a percent deviation of the values calculated by Equation (36) with respect to the initial value of the effective electrophoretic mobility introduced to the simulation *μ*_{eff,init}:

*k″*is the proportionality coefficient which is constant for the given values of pH, BGE concentration, electrophoretic mobilities for all species and partly for $CL0/CM0$. Actually, the $N/leff$ ratio is a value that is inversely proportional to the peak width. For the higher electrophoretic mobilities of the BGE anions, Δ

*μ*

_{eff}(sim) is almost independent of

*$N/leff$*(Figure 5D). The asymmetry of ISC peak is likely to compensate the migration time increase due to [L

^{3−}] decrease in the analyte zone.

For $CL0/CM0=10$, the pH dependences of Δ*μ*_{eff}(sim) (satisfying the experimental conditions: *N* = 16,400, *l*_{eff}= 40 cm) calculated by Equation (37) and those obtained by the mathematical simulation (for the highest electrophoretic mobility of the BGE anions) are shown in Figure 5E. Δ*μ*_{eff}(sim) increases with increasing pH and decreases with increasing the electrophoretic mobilities of the BGE anions. The dependences have the same slope, but in the pH region of 2.7–2.8 the third curve as well as the experimental dependence has a local minimum (Figure 3).

As expected, |Δ*μ*_{eff}(sim)*|* increases with enhancing the total metal concentration (Figure 5F). The dependence of Δ*μ*_{eff}(sim) on $CM0/CL0$ is almost linear with *R*^{2}> 0.999 for all the cases of different electrophoretic mobilities of the BGE anions with a small point deviation for $CM0/CL0$ being 0.1. Table IV shows the values of $CL0/CM0$ when |Δ*μ*_{eff}(sim)| is equal to 0.6% (the experimental accuracy of the electrophoretic mobility measurement).

μ(HL^{2−}), μ(H_{2}L^{−})(10 ^{−9} m^{2} V^{−1} s^{−1}) | рН | |
---|---|---|

2.4 | 3.1 | |

44, 30 | 30 | 8 |

15, 10 | 47 | 75 |

5, 3 | 68 | 103 |

μ(HL^{2−}), μ(H_{2}L^{−})(10 ^{−9} m^{2} V^{−1} s^{−1}) | рН | |
---|---|---|

2.4 | 3.1 | |

44, 30 | 30 | 8 |

15, 10 | 47 | 75 |

5, 3 | 68 | 103 |

It is worth noting that the 10% change in the total BGE concentration in the ISC zone at zero time results in the 0.2% change in the electrophoretic mobilities, which is lower than the random error of experimental measurement of the electrophoretic mobilities (0.1–0.6%).

The decrease of the ISC effective electrophoretic mobilities can be treated as the increase in the apparent value of *β*_{2}/*β*_{1} for the given conditions or decrease in the equilibrium ligand concentration. The relative error of the apparent value of the stability constants Δ(*β*_{2}/*β*_{1}) relative to the initial value (*β*_{2}/*β*_{1})^{0} is equal to the relative error of the equilibrium ligand concentration:

*β*

_{2}/

*β*

_{1}) (Equation 38) and the systematic errors of the relative stability constants calculated by Equation (10). As shown in Table V, the resulting errors are more dependent on the slope of the рН dependence of the electrophoretic mobility decrease than on the errors of every point. And, the sign of the errors is specified by this slope. In addition, the non-monotonic behavior of the dependence of the electrophoretic mobility decrease (the existence of plateau) reduces the measurement precision of the stability constant values obtained by CE. Under the conditions studied, the systematic errors of the electrophoretic mobilities result in the systematic errors of the relative stability constants of <7%. This is comparable with the experimental random error of the stability constant measurements. However, it is worth noting that such small systematic errors are due to a slight slope of the pH dependence of the effective electrophoretic mobility decrease. The systematic errors of the relative stability constants are almost proportional to the $CM0/CL0$ (Table V).

μ(HL^{2−}), μ(H_{2}L^{−})(10 ^{−9} m^{2} V^{−1} s^{−1}) | $CL0/CM0$ | Apparent β_{2}/β_{1} values, Equation (38) (M^{−1}) | Equation (10) | ||
---|---|---|---|---|---|

рН | β_{2}/β_{1}×10^{−10} (M^{−1}) | Δ (β_{2}/β_{1}) (%) | |||

2.4 | 3.1 | ||||

44, 30 | 10 | 2.2 | 1.2 | 2.88 ± 0.02 | 2.2 |

15, 10 | 10 | −3.5 | −10 | 2.93 ± 0.02 | 4.0 |

5, 3 | 10 | −5.1 | −14 | 2.9 ± 0.1 | 3.4 |

50 | −1.1 | −3.0 | 2.84 ± 0.02 | 0.7 | |

100 | −0.5 | −1.6 | 2.825 ± 0.009 | 0.2 |

μ(HL^{2−}), μ(H_{2}L^{−})(10 ^{−9} m^{2} V^{−1} s^{−1}) | $CL0/CM0$ | Apparent β_{2}/β_{1} values, Equation (38) (M^{−1}) | Equation (10) | ||
---|---|---|---|---|---|

рН | β_{2}/β_{1}×10^{−10} (M^{−1}) | Δ (β_{2}/β_{1}) (%) | |||

2.4 | 3.1 | ||||

44, 30 | 10 | 2.2 | 1.2 | 2.88 ± 0.02 | 2.2 |

15, 10 | 10 | −3.5 | −10 | 2.93 ± 0.02 | 4.0 |

5, 3 | 10 | −5.1 | −14 | 2.9 ± 0.1 | 3.4 |

50 | −1.1 | −3.0 | 2.84 ± 0.02 | 0.7 | |

100 | −0.5 | −1.6 | 2.825 ± 0.009 | 0.2 |

(*β*_{2}/*β*_{1})^{0}=2.82·10^{10} M^{−1}.

## Discussion

The influence of analyte concentration when compared with the concentration of a charged ligand in BGE on the measured values of electrophoretic mobilities and stability constants is studied using CE and a dynamic mathematical simulator of CE. As shown in Figures 3 and 5E, the differences in the electrophoretic mobilities obtained by the experiments and simulation do not coincide as to the direction of the pH dependence. This can be explained as follows. It is known that the silanol groups (SiOH) on the interior capillary wall are partially dissociated. The extent of their dissociation depends on the concentration, рН and temperature of BGE. As shown in the simulated electropherograms in Figure 4, the concentration of all species and pH in the ISC zone are different from the initial BGE concentration and pH. This would result in changing the dissociation of the silanol groups and, correspondingly, changing the equilibrium concentrations of all ions in the ISC zone when moving across the capillary. It causes the decrease in the ligand concentration and higher retardation of the ISC zone motion. This can explain the experimental peak fronting and appearance of sp2 (Figure 2), which are not observed in the simulated electropherograms.

It can be assumed that the analyte concentration dependence of the experimental electrophoretic mobilities decrease is linear. Thus, for a similar system in which an analyte relates to a charged ligand, to eliminate the systematic errors, the total ligand concentration should be >30–160 times higher than the analyte concentration. But since the experimental error of the stability constants are ∼7%, the ratio of 50–100 would probably be sufficient.

## Conclusion

Thus, using the complexes between iron (III) and ISC as an example for a system where ligands and their protonated forms are single BGE anions, the influence of the charged ligand to analyte concentration ratio on the measured values of the electrophoretic mobility and stability constant values was studied by ACE and the dynamic mathematical simulator of CE. It was found that since the equilibrium ligand concentration in the analyte zone was not equal to that in the BGE, a considerable change in the migration times and electrophoretic mobilities was observed, resulting in the systematic errors of the stability constant values. It is shown that of crucial importance is the slope of the pH dependence of the electrophoretic mobility decrease on the ligand equilibrium concentration. For such systems, without prior knowledge of this dependence, to accurately evaluate the stability constants, the total ligand concentration must be at least >50–100 times higher than the total concentration of analyte.

It is worth noting that a considerable difference between the BGE concentration in the sample and that in the running buffer results in the enrichment effect (stacking). This is quite advantageous for the analysis, but when determining the stability constants this phenomenon is to be eliminated. Therefore, the electrokinetic injection is not particularly recommended. To avoid systematic errors occurrence, the total BGE concentration in the sample must be close to its concentration in the running buffer.