Ionic mobility is a basic physical-chemical parameter of ion. It has some important use in electro-chemistry ^[1-3], drug research ^[4-6], electrolytic solution boundary ^[7-11], electrophoretic titration ^[12-14], capillary electrophoresis (CE) ^[15-24], computer simulation of electrophoresis ^{[20, 21, 25-27]} and sample stacking in CE ^[27-29]. For example, we need numerous ionic mobilities in different electrolytes under different ionic strengths during the digital computation of sample stacking conditions in CE ^{[27, 29]}.

There are at least four kinds of methods used to obtain ionic mobility. The oldest method is moving boundary method (MBM) developed in 1920s-1940s ^{[7-9, 11]}. With MBM, one can detect the ionic transference number in pure electrolyte and calculate the actual mobility with corresponding formula. This method is very accurate but time-consuming ^[7].

The second one is the method using a controlled direct electric field pulsed gradient spin echo nuclear magnetic resonance (NMR) ^[3]. In the run, the NMR cells are composed of four-terminal electrodes and a direct potential is applied by controlling the detected potential at the center of the sample as a constant during the spin echo detection. The method is considered to reflect the real migration properties of ion.

The third is CE used to detect ionic mobility ^[15-24]. Based on CE, a lot of studies for ionic mobility in water-organic system were performed. Miller et al. ^[6] proved the semi-empirical relationships among mobility, charge and relative molecular mass of drugs. Porras et al. ^[15] revealed the relationship between mobility and ionization constant of basic drugs in mixed binary water-organic systems. Jouyban et al. ^[16] developed an equation of mobility in the mixed ternary water-methanol-ethanol systems. Undoubtedly, these results ^{[6, 15, 16]} cannot be used for ionic mobility in aqueous system. To obtain the ionic mobility in aqueous system, plenty of studies were undertaken concerning the influence of ionic strength ^{[17, 18, 30]}, charge ^{[19-22, 29]}, pH value ^[19] and Joule heating ^[24] on mobility. The studies resulted in some empirical equations ^[20-23]. The simple and convenient empirical equations have been used for the moving chemical reaction boundary and isoelectric focusing ^{[10, 12]}. However, obvious bias of the equations was found. As a whole, CE is simple for a run, but still time-consuming for numerous detections of ionic mobility in different electrolytes under different ionic strengths.

The fourth is the theoretical way used for the study on ionic mobility. The classic one is the Debye-Hückel-Onsager law ^[31-33] that shows the relationship between ionic equivalent conductivity (viz., mobility) and square root of ionic strength. But the law only holds validity under 0.001 mol/L ionic strength. Afterward, Robinson et al. ^[31-33] converted and corrected the law as ionic mobility equation that shows the relationship between ionic mobility and ionic strength as well as ionic radius sum. The Robison-Strokes’ equation cannot be actually used for CE since it is difficult to obtain ionic radius sum under different conditions. Li et al. ^[33] achieved accurate ionic mobility for CE by using Pitt’s equation and proposed a general equation revealing the ionic strength dependence of mobility. The general equation is highly proved by numerous data of ionic mobility cited from references and can be precisely used for prediction of mobility (3.3% deviation under 0.10 mol/L ionic strength). But the proposed equation is somehow limited since the constant has to be empirically determined for individual ion, which poses difficulty for the immediate mobility computation for individual ion under several conditions in CE ^[25-29]. Reports concerning ionic mobility were few in recent years, especially via theoretical approach. There was a model built recently by Jouyban et al. ^[34] to predict electrophoretic mobility of analytes based on Abraham solvation parameters, and good agreements between predicted and experimental mobility data were acquired.

Unlike these methods mentioned above, this work advances a novel, accurate, simple and semi-empirical method to investigate ion mobility by using electrolytic conductivity. Here, some empirical equations were chosen as the studied targets, in which there were serious bias needed to be corrected. This study confirms that, on the one hand, electrolyte conductivity can be well used for the formulation of mobility equation, and on the other hand, the results show the high validity of that corrected equation.

1 Theory and methods

Equation (Eqn.) (1) describes a basic method to obtain ionic actual mobility (m_act) by detecting the equivalent conductivity (Λ) of electrolyte and ionic transference number (T) in the electrolyte.

(1)

F is the Faraday’s constant (96 487 C/mol).

By using Pitt’s equation, Lucy et al. ^[33] advanced the Eqns. (2) and (3):

(2)

(3)

where A is a constant, m₀ is the absolute mobility, λ₀ refers to the ionic equivalent conductivity in electrolytic solution diluted infinitely, and I is the ionic strength. The equation holds validity with the ionic strength less than 0.1 mol/L, unlike the Debye-Hückel-Onsager law holding validity with the ionic strength less than 0.001 mol/L.

By using the numerous ionic mobilities detected with CE, an empirical equation for mono-valent ionic mobility is formulated ^{[20, 21, 23]} and expressed as:

(4)

z is the ionic valence (z=1 here). The empirical equation is convenient for the computation of ionic mobility. But some serious deviations are found. Here, Eqn. (4) is chosen as the studied target that needs further corrections with numerous data.

Thus, Eqn. (4) ought to be re-expressed as Eqn. (5):

(5)

z=1 here. A and B are the coefficients needed to be reset.

Assuming there is a pure mono-mono-valent electrolyte, such as sodium chloride. For the chloride and sodium ions, Eqn. (1) for the actual mobilities of chloride and sodium ions can be, respectively, expressed as:

(6)

(7)

m_Na⁺and m_Cl^- refer to the actual mobilities of sodium and chloride ions,respectively. T_Na⁺and T_Cl^- represent the ionic transference numbers of sodium and chloride ions. Λ_NaCl is the equivalent conductivity of sodium chloride.

The combination of Eqns. (6) and (7) yields:

(8)

In a pure electrolyte, there is always:

(9)

Thus, Eqn. (8) is changed as:

(10)

The insertion of Eqn. (4) into Eqn. (10) produces:

(11)

Λ_{the NaCl} is the theoretical equivalent conductivity of sodium chloride, m_0Na⁺ and m_0Cl^- refer to the absolute mobilities of sodium and chloride ions, respectively. Owing to the existence of Eqn. (3), Eqn. (11) becomes:

(12)

λ_{0 Na⁺} and λ_{0 Cl^-} are the ionic equivalent conductivities of sodium and chloride ions, respectively.

(13)

Thus, if the concrete electrolyte of sodium chloride is omitted, then Eqn. (12) can be re-written as:

(14)

The subscript “the” represents the theoretic equivalent conductivity. The symbol “z” indicates a mono-mono-valent electrolyte as well as mono-valent ion.

The theoretical results above demonstrate that Eqn. (4) can be converted to Eqn. (14) for the conductivity computation of mono-valence electrolyte with limit equivalent conductivity of electrolyte, if the two coefficients of “-0.5” and “0.5” in Eqn. (4) or (14) are right. The results also show that one can investigate the validity of the two coefficients in Eqn. (4) by using data of equivalent conductivity of electrolyte as will be proved hereinafter. With the abundant data on electrolytic conductivities, we can conveniently test and reset the coefficients in Eqn. (14) or (4), as will be shown and discussed in the following text.

The measurements of the equivalent conductivity of a pure electrolyte can be accurately and simply performed with a conductivity meter and calculated with Eqn. (15):

(15)

where κ is the specific conductivity of pure electrolyte, c is the equivalent concentration of the electrolyte. The subscript “exp” indicates the experimental data of equivalent conductivity. With the equations mentioned above, it is indicated that ionic mobility can be investigated just by simple detection of specific conductivity of electrolyte.

The relative difference (RD_c) value of equivalent conductivity between the theoretical calculation with Eqn. (14) and the experimental data in Table 1 ^[35] can be expressed as:

(16)

The subscript “c” in RD indicates the equivalent conductivity.

The insertion of the coefficients of Eqn. (5) into Eqn. (14) produces Eqn. (17):

(17)

The coefficients A and B herein need accurate correction as will be shown below.

2 Data and process

2.1 Basic Data

Note here, the salts are all mono-mono-valence electrolytes. In Table 1, there are 15 smaller inorganic salts like sodium chloride and potassium chloride with high charge intensities, while there are also four larger organic salts, e. g., NaO₂C₄H₇ and Na picrate, with low charge intensities. Thus, the electrolytes in Table 1 have an extensive significance to the study on the validity of Eqn. (4) or (14).

表 1 (Table 1)

Table 1 Equivalent conductivity of 19 mono-valence electrolytes in aqueous solutions at 25 ℃ [35] Electrolyte Equivalent conductivity/(10-4m2s/mol) Infinite dilution 0.0005 mol/L 0.001 mol/L 0.005 mol/L 0.010 mol/L 0.020 mol/L 0.050 mol/L 0.100 mol/L AgNO3 133.29 131.29 130.45 127.14 124.70 121.35 115.18 109.09 KCl 149.79 147.74 146.88 143.48 141.20 138.27 133.30 128.90 KBr 151.90 - - 146.02 143.36 140.41 135.61 131.32 KI 150.31 - - 144.30 142.11 139.38 134.90 131.05 KHCO3 117.94 116.04 115.28 112.18 110.03 107.17 101.63 95.91 KClO4 139.97 138.69 137.80 134.09 131.39 127.86 121.56 115.14 KIO4 127.86 125.74 124.88 121.18 118.45 114.08 106.67 98.20 KNO3 144.89 142.70 141.77 138.41 135.95 132.34 126.25 120.34 KReO4 128.20 126.03 125.12 121.31 118.49 114.49 106.40 97.40 LiCl 114.97 113.09 112.34 109.35 107.27 104.60 100.06 95.81 LiClO4 105.93 104.13 103.39 100.52 98.56 96.13 92.15 88.52 NH4Cl 149.60 147.55 146.70 143.40 141.21 138.25 133.22 128.69 NaCl 126.39 124.44 123.68 120.59 118.45 115.70 111.01 106.69 NaClO4 117.42 115.58 114.82 111.70 109.54 106.91 102.35 98.38 NaI 126.88 125.30 124.19 121.19 119.18 116.64 112.73 108.73 NaO2C2H3 91.00 89.20 88.50 85.68 83.72 81.20 76.88 72.76 NaO2C3H5 85.88 84.20 83.50 80.86 79.01 76.59 72.38 68.30 NaO2C4H7 82.66 81.00 80.27 77.54 75.72 73.35 69.29 65.24 Na picrate 80.45 - 78.60 75.70 73.70 71.20 66.30 61.80 - : not given in Ref. [35].

Table 1 Equivalent conductivity of 19 mono-valence electrolytes in aqueous solutions at 25 ℃ [35]

The data of transference number monitored by the accurate moving boundary method ^[7-9] are cited from reference (Ref.) ^[36]. These data, including the relative equivalent conductivities, are all collected in Table 2. The limited transference numbers are obtained from:

(18)

表 2 (Table 2)

Table 2 Transference numbers of mono-valent ions and equivalent conductivities (λ) of some mono-mono-valent electrolytes at 25 ℃ [36] Concent ration/ (mol/L) KCl NaCl HCl Na2SO4 BaCl2 LaCl3 T+ T- λ T+ T- λ T+ T- λ T+ T- λ T+ T- λ T+ T- λ Infinite 0.491 0.509 1.498 0.822 0.822 1.264 0.822 0.178 4.260 0.386 0.614 1.298 0.455 0.545 1.399 0.487 0.513 1.459 0.0005 0.491 0.509 1.477 0.823 0.823 1.244 0.823 0.177 4.225 0.386 0.614 1.257 0.451 0.549 1.359 0.475 0.525 1.396 0.001 0.491 0.509 1.469 0.824 0.824 1.237 0.824 0.176 4.212 0.385 0.615 1.241 0.449 0.551 1.343 0.473 0.527 1.370 0.005 - - - 0.825 0.825 - 0.825 0.175 4.156 0.385 0.615 1.171 0.444 0.556 1.280 0.467 0.533 1.275 0.010 0.490 0.510 1.412 0.825 0.825 1.185 0.825 0.175 4.118 0.385 0.615 1.124 0.440 0.560 1.239 0.463 0.537 1.218 0.020 0.490 0.510 1.383 0.826 0.826 1.157 0.826 0.174 4.070 0.384 0.616 1.067 0.436 0.564 1.190 - - - 0.050 0.490 0.510 1.333 0.829 0.829 1.110 0.829 0.171 3.989 - - - - - - - - - 0.100 0.490 0.510 1.289 0.831 0.831 1.067 0.831 0.169 3.911 - - - - - - - - - λ: 10-2m2s/mol. T+ and T- indicate the cation and anion transference numbers respectively. - : not given in Ref. [36].

Table 2 Transference numbers of mono-valent ions and equivalent conductivities (λ) of some mono-mono-valent electrolytes at 25 ℃ [36]

Subscripts (+ and -) indicate cation and anion respectively, and the subscript (0) means the transference number in infinitely dilute solution. The anion transference numbers (T_-) and cation transference numbers (T₊) are in accordance with:

(19)

The absolute mobilities of some small inorganic ions quoted from Ref. ^[35] are listed herein (10^-8m²V^-1s^-1): 7.62 (potassium), 7.91 (chloride) and 5.19 (sodium).

The data of absolute mobilities and constant in Eqn. (2) of numerous mono-valent organic ions are given in Table 3, which are cited from Table 1 in Ref. ^[33]. With the basic data in Table 3, we can calculate ionic mobilities by Lucy’s equation and the equations corrected from Eqn. (4). The data of square values of correlation coefficient (r²) in Table 3 also partially show the excellent agreement between the theoretical and detection values.

表 3 (Table 3)

Table 3 Data on the constants in Eqn. (2) herein and absolute mobilities for different mono-valent organic anions [33] Ion r2 Constant in Eqn. (2)/(10-8m2V-1s-1mol0.5L0.5) m0/(10-8m2V-1s-1) Benzoate 0.9959 3.68 3.37 p-Aminobenzoate 0.9988 3.11 3.17 o-Bromobenzoate 0.9985 2.92 3.10 m-Bromobenzoate 0.9979 3.08 3.15 p-Bromobenzoate 0.9977 2.92 3.08 o-Chlorobenzoate 0.9952 2.96 3.16 m-Chlorobenzoate 0.9920 3.10 3.23 p-Chlorobenzoate 0.9976 3.06 3.21 Salicylate 0.9984 2.95 3.54 m-Hydroxybenzoate 0.9988 3.18 3.14 p-Hydroxybenzoate 0.9990 3.16 3.08 o-Methoxybenzoate 0.9966 3.04 3.07 m-Methoxybenzoate 0.9996 2.91 3.11 p-Methoxybenzoate 0.9960 3.02 3.06 o-Nitrobenzoate 0.9936 2.89 3.16 m-Nitrobenzoate 0.9985 3.21 3.25 p-Nitrobenzoate 0.9994 3.28 3.25 Propionate 0.9946 3.36 3.75 2,3-Dihydroxybenzoate 0.9998 3.54 3.23 2,4-Dihydroxybenzoate 0.9999 3.40 3.18 2,6-Dihydroxybenzoate 0.9986 3.33 3.74 3,4-Dihydroxybenzoate 0.9998 3.43 2.86 3,5-Dihydroxybenzoate 0.9998 3.29 2.87 p-Nitrophenolate 0.9993 3.72 3.25 2,4,6-Trichlorophenolate 0.9890 3.30 2.81 Benzenesulfonate 0.9993 4.00 3.54 p-Chlorobenzenesulfonate 0.9981 3.95 3.32 p-Toluenesulfonate 0.9987 3.95 3.27 2-Naphthalenesulfonate 0.9985 3.84 3.09 5-Amino-2-chlorobenzensulfonate 0.9964 3.68 3.09 2-Amino-5-chloro-4- 0.9961 3.76 3.02 methylbenzenesulfonate 2-Anthraquinonesulfonate 0.9120 3.48 2.71

Table 3 Data on the constants in Eqn. (2) herein and absolute mobilities for different mono-valent organic anions [33]

2.2 Treatments

The ionic strength (I) is calculated by Eqn. (20):

(20)

c_i is the concentration of species i, and z_i means the charge of species i.

The RD_c values (see Fig. 1) which show the comparisons between the theoretical and experimental conductivities are evaluated with Eqn. (16). The sum of RD_c values of 19 electrolytes under same ionic strength, such as I=0.01 mol/L, yields:

Fig. 1

Fig. 1 RD values for the comparisons between the equivalent conductivities of 19 mono-valence electrolytes obtained by theoretical calculation (Eqn. (14)) and experimental detection (Table 1) with different ionic strengths

(21)

i is these electrolytes like AgNO₃, KCl and KBr…in Table 1. Clearly, under a given ionic strength, Sigma=0 means that the sum of RD_AgNO3+RD_KCl+RD_KBr+RD_KI+…+RD_{Na picrate} is equal to zero. Namely, the conductivities computed with Eqn. (17) are generally in agreement with these detected values in Table 1 and the coefficients in Eqn. (17) are correct. Conversely, Sigma≠0 means that the conductivities computed with Eqn. (17) are generally in disagreement with these detected in Table 1 and the coefficients are in error and should be corrected. Thus, Sigma=0 can be used for the resets or corrections of the coefficients A and B, as will be shown below. If the criterion of accurate theoretical computation is controlled within ±5% deviation from experimental values (viz., RD_c value in Eqn. (16)), the quantitative predictions show that the RD_c values are out off ±7% when I≥0.10 mol/L, but within ±4% when I≤0.05 mol/L. The RD_c values with I=0.10 mol/L are evidently out off the criterion of ±5% deviation. So Eqns. (21a)-(21c), rather than Eqn. (21d), are used for the resets of coefficients A and B in Eqn. (17).

2.3 Digital computation software

All of the computations including statistical analyses are performed with the Original (ver. 7.0, the Microcal Software Inc., USA). After that, all of the data figures are automatically generalized by the same software.

3 Results and discussion

3.1 Resets of coefficients in Eqns. (5) and (17)

Fig. 1 shows the comparisons between the theoretical and experimental values of equivalent conductivities of pure mono-mono-valent electrolytes. The values of RD_c are greater than zero greatly. This indicates that the theoretical values are larger than the experimental data evidently. The situation clearly manifests the errors existing in the coefficients “-0.5”and “0.5” in Eqns. (4), (11), (12) and (14). This means the coefficients in Eqns. (5) and (17) should be accurately reset. Therefore, if the B value is set at 0.50, then Eqn. (17) becomes:

(22)

A₁ is the first coefficient needed to be reset. Setting A₁=0.50, 0.55, 0.60, …, 0.75 and 0.80 respectively, calculating theoretical conductivities with Eqn. (22), computing the RD_c values with Eqn. (16), and calculating Sigma values with Eqns. (21a)-(21c), we have Fig. 2.

Fig. 2

Fig. 2 Linear fit lines between coefficient A1 and the Sigma value of RDc with I=0.01 mol/L, I=0.02 mol/L and I=0.05 mol/L

Fig. 2 shows that if ionic strength is 0.01 mol/L, the Sigma value is as a function of A₁ value. So are the Sigma values under 0.02 and 0.05 mol/L ionic strength. The three linear regression equations of Sigma and A₁ values under 0.01, 0.02 and 0.05 mol/L ionic strength are given in Table 4. It is clearly shown in Table 4 that r² for the three linear regression equations are from 0.999 9 to 1. Adding the three regression equations together and setting the addition:

表 4 (Table 4)

Table 4 Equations of linear fit between coefficient A1and the Sigma value of RDc Ionic strength/(mol/L) Linear regression equation SD r2 0.01 Sigma0.01=1.2484-1.8073×A1 9.2712×10-4 1.0000 0.02 Sigma0.02=1.7263-2.5548×A1 0.0019 0.9999 0.05 Sigma0.05=2.5799-4.0174×A1 0.0046 0.9999

Table 4 Equations of linear fit between coefficient A1and the Sigma value of RDc

(23)

We get A₁=0.662 9, which indicates under the B=0.50, the sum of the three Sigma values is zero as shown in Fig. 2

Next, inserting A₁=0.662 9 in Eqn. (17), we get:

(24)

B₁ is the other coefficient we attempt to revise. Setting B₁=0.44, 0.46, …, 0.56, and performing Sigma of the RD values for the 19 electrolytes in Table 1 under 0.01, 0.02 or 0.05 mol/L ionic strengths with Eqns. (16) and (21a)-(21c), we obtain the linear fit analyses of Fig. 3 and the three linear regression equations in Table 5. The values of r² for the three regression equations in Table 5 are greater than 0.99. Using Eqn. (23) for the three equations in Table 5, we have B₁=0.503 0 as marked in Fig. 3, which shows the sum of the Sigma values is equal to zero at B₁=0.503 0.

Fig. 3

Fig. 3 Linear fit lines between coefficient B1 and the Sigma value of RDc under I=0.01 mol/L, I=0.02 mol/L and I=0.05 mol/L

表 5 (Table 5)

Table 5 Equations of linear fit between coefficient B1 and the Sigma value of RDc Ionic strength/(mol/L) Linear regression equation SD r2 0.01 Sigma0.01=-2.7465+5.5539×B1 0.0195 0.9945 0.02 Sigma0.02=-3.3112+6.6469×B1 0.0192 0.9963 0.05 Sigma0.05=-4.0903+7.9730×B1 0.0167 0.9981

Table 5 Equations of linear fit between coefficient B1 and the Sigma value of RDc

Inserting A₁=0.662 9 and B₁=0.503 0 into Eqn. (17), we have:

(25)

Using Eqns. (25) and (16) for the calculation of RD_c values of these electrolytes in Table 1, we obtain the comparisons of theoretical and experimental conductivities as given in Fig. 4. The comparisons of Fig. 1 and Fig. 4 evidently show that the original systemic bias in Fig. 1 is greatly corrected. In contrast to the bias distributions in Fig. 1, the RD_c values in Fig. 4 are fair even distributed among the zero horizontal line, and the RD_c values are within ±5% if the ionic strength is less than 0.05 mol/L.

Fig. 4

Fig. 4 RDc values of theoretical (Eqn. (25)) and experimental conductivities of 19 mono-valence electrolytes as a function of ionic strength

Even the evident systemic bias in Fig. 1 is significantly corrected by Eqn. (25) as shown in Fig. 4, but fine correction to Eqn. (25) is still necessary to obtain more accurate theoretical conductivity or mobility.

The finer resets to coefficients A and B in Eqn. (17) are carried out. The results are: A₂=0.670 1 and B₂=0.503 5. Hence, we have the finely corrected equation (26):

(26)

The symbol “z” indicates a mono-mono-valent electrolyte as well as mono-valent ion.

Inserting the values of A₂=0.670 1 and B₂=0.503 5 into Eqn. (5),we get the finely corrected empirical equation of mobility of mono-valence ion, viz.,

(27)

Theoretically, Eqn. (27) possesses validity for the computation of mobility of mono-valence ion within 5% maximum deviation from the experimental values under 0.05 mol/L ionic strength and 7% maximum deviation under 0.10 mol/L ionic strength.

The classic approach for coefficient optimization is the Levenberg-Marquardt nonlinear least algorithm ^[37]. But in this work, we use the simplest “trial-and-error” approach, as shown from Fig. 1 to Fig. 4. The validity of the approach is well proved by the following statistical results. The sum value of Sigma_0.0005+Sigma_0.001+…+Sigma_0.05+Sigma_0.10 at the optimized coefficients in Eqn. (26) is 0.003 4, and it’s very close to zero. The following results further prove the validity of the two coefficients in Eqns. (26) and (27).

3.2 Validity of coefficients in Eqns. (26) and (27)

To demonstrate the validity of Eqn. (27), the theoretical mobilities of some mono-valence ions like chloride, sodium, potassium ions are firstly computed with Eqn. (27), and further compared with actual experimental mobilities of the ions detected with the accurate MBM by using Eqn. (28):

(28)

RD_m is the relative deviation of theoretical mobility (m_the) and experimental mobility. The subscript of “MBM” means that the experimental mobility was calculated by moving boundary methods (MBM). With Eqns. (27) and (28), we firstly compare the theoretical and experimental actual mobilities of chloride ion in the mono-mono-valent electrolytes of KCl, NaCl and HCl. Fig. 5a shows that the RD_m values are less than -3.8% deviation, if the ionic strength is within 0.05 mol/L. Fig. 5b manifests that the RD_m values for potassium in KCl and sodium in NaCl are within -3.5% deviation if the ionic strength less than 0.05 mol/L. Fig. 5c shows the RD_m values for chloride in BaCl₂ and sodium ion in Na₂SO₄ are 5.0% and -2.2% deviations respectively. Fig. 5d shows the high agreements between the experimental values of mobility of chloride in LaCl₃ detected by MBM and the theoretical mobility computed with Eqn. (27). As a whole, the values of RD_m in Fig. 5 range from the maximum value of 5% to the minimum value of -3.5% under the ionic strength of 0.05 mol/L. Thus, the results in Fig. 5 directly prove the validity of coefficient A₂=0.670 1 and B₂=0.503 5 to Eqn. (27) under the ionic strength of 0.05 mol/L.

Fig. 5

Fig. 5 RDm values (Eqn. (28)) of theoretical (Eqn. (27)) and experimental mobilities of mono-valence ions as a function of ionic strength.

In order to further test the validity of Eqn. (27), the computations with Eqn. (27) are compared with these with Eqn. (2) which can accurately predict ionic actual mobility as well as absolute mobility as proved in Ref. ^[32]. The comparisons between ionic actual mobilities computed with Eqn. (2) and (27) are carried out by using the Eqn. (29) under different ionic strengths from 5×10^-5 to 0.1 mol/L:

(29)

The subscript of “Lucy” means the actual mobility computed with Eqn. (2), viz., Lucy’s equation. The results of the comparisons for over 32 kinds of organic anions are given in Fig. 6 in which the ionic strength changes from 5×10^-5 to 0.1 mol/L.

Fig. 6

Fig. 6 RDm values (Eqn. (29)) of actual mobilities computed with Eqn. (27) and Eqn. (2) as a function of ionic strengths.

Fig. 6a shows the results of RD_m of mobilities computed with Eqns. (27) and (2) for benzoate, amino-benzoate and bromobenzoate, the RD_m values change only from the maximum value of 2.6% to the minimum of -3.1% even the ionic strength goes up to 0.10 mol/L. Fig. 6b reveals the values range from the maximum 2.1% to the minimum -4.8% for six organic anions. Fig. 6c unveils that the RD_m values are from 1.5% to -3.6%. Fig. 6d displays the values change from 3.1% to -3.7%. In Fig. 6e, the values change only from 3.3% to 0.4%. And in Fig. 6f, the values are from 5.0% to 0.4%. As a whole, the values of RD_m in Fig. 6 change from the maximum value of 5.0% to the minimum value of -4.8% under the ionic strength of 0.10 mol/L. The results excellently prove the agreements of ionic actual mobilities over 30 organic anions achieved with Eqns. (2) and (27).

Thus, the results in Fig. 5 and Fig. 6 well demonstrate: (1) the validity of the coefficients of A=0.670 1 and B=0.503 5 in Eqn. (26); (2) the validity of correction (or reset) of the coefficients in Eqn. (4) by using the equivalent conductivity of mono-mono-valent electrolyte, viz., the validity of the theoretical treatment from Eqn. (5) to Eqn. (17).

More importantly, the integrated results of Fig. 4, Fig. 5 and Fig. 6 evidently show the good agreements among mono-mono-valent equivalent conductivity, ionic mobilities detected with MBM and theoretical mobilities computed with Eqn. (27) and Lucy’s equation, viz., Eqn. (2).

3.3 More accurate equation

If more accurate equations of mobility for mono-valent ion are desired, this paper further supplies Eqn. (30):

(30)

The RD_m values for both of the equations could decrease to within 2%, but the ionic strength should be controlled within 0.02 mol/L under which the normal electrophoresis is carried out.

4 Conclusions

Obviously, the work developed a novel, simple and accurate method to investigate ionic mobility. With the data of conductivity, we finely corrected a target empirical equation of mobility existing evident systemic bias as the accurate empirical equation. The method holds some advantages. Firstly, there are adequate data of electrolyte conductivity in references that can be well used. Secondly, the method is very simple, since only a detection of electrolyte conductivity is needed. Thus, the method makes the further study on ionic mobility very easy. Thirdly, the corrected equation can be used for the direct computation of ionic strength with different ionic strengths. Furthermore, the proposed method might be modified to study multi-valence ion mobilities as well.

参考文献

[1]	Biswas R, Bagchi B, J Chem Phys,1997, 106 :5587.
[2]	Bagchi B, Biswas R, Acc Chem Res,1998, 31 :181.
[3]	Kataoka H, Saito Y, Sakai T, et al, J Phys Chem B,2001, 105 :2546.
[4]	Furlanetto S, Lanteri S, Orlandini S, et al, J Pharm Biomed Anal,2007, 43 :1388.
[5]	Furlanetto S, Lanteri S, Orlandini S, et al, J Pharm Biomed Anal,2007, 43 :1402.
[6]	Miller J M, Blackburn A C, Shi Y, et al, Electrophoresis,2002, 23 :2833.
[7]	Longsworth L, J Am Chem Soc,1945, 67 :1109.
[8]	Alberty R A, J Am Chem Soc,1950, 72 :2361.
[9]	Nichol J, J Am Chem Soc,1950, 72 :2367.
[10]	Cao C X, J Chromatogr A,1998, 813 :153.
[11]	MacInnes D A, Longsworth L G, Chem Rev,1932, 11 :172.
[12]	Wang H, Shi Y, Yan J, et al, Anal Chem,2014, 86 :2888.
[13]	Zhang L X, Cao Y R, Xiao H, et al, Biosens Bioelectron,2016, 77 :284.
[14]	Dong J, Li S, Wang H, et al, Anal Chem.,2013, 85 :5884.
[15]	Porras S P, Riekkola M L, Kenndler E, J Chromatogr A,2001, 905 :259.
[16]	Jouyban A, Grosse S, Chan H, et al, J Chromatogr A,2003, 994 :191.
[17]	Bekri S, Leclercq L, Cottet H, J Chromatogr A,2016, 1432 :145.
[18]	Ibrahim A, Allison S A, Cottet H, Anal Chem,2012, 84 :9422.
[19]	Castagnola M, Rossetti D V, Corda M, et al, Electrophoresis,1998, 19 :2273.
[20]	Reijenga J, Kenndler E, J Chromatogr A,1994, 659 :403.
[21]	Friedl W, Reijenga J C, Kenndler E, J Chromatogr A,1995, 709 :163.
[22]	Shimizu T, Kenndler E, Electrophoresis,1999, 20 :3364.
[23]	Cao C X, J Chromatogr A,1997, 771 :374.
[24]	Cross R F, Cao J, J Chromatogr A,1998, 809 :159.
[25]	Bier M, Palusinski O, Mosher R, et al, Science,1983, 219 :1281.
[26]	Ermakov S V, Zhukov M Y, Capelli L, et al, Electrophoresis,1995, 16 :2149.
[27]	Cao C X, Zhang W, Qin W H, et al, Anal Chem,2005, 77 :955.
[28]	Chien R L, Anal Chem,1991, 63 :2866.
[29]	Cao C X, He Y Z, Li M, et al, Anal Chem,2002, 74 :4167.
[30]	Allison S A, Pei H, Baek S, et al, Electrophoresis,2010, 31 :920.
[31]	Robinson R A, Strokes R H. Electrolyte Solutions. 2nd ed. London:Butterworths Press, 1965
[32]	Survay M A, Goodall D M, Wren S A, et al, J Chromatogr A,1996, 741 :99.
[33]	Li D, Fu S, Lucy C A, Anal Chem,1999, 71 :687.
[34]	Jouyban A, Fazeli-Bakhtiyari R, Shayanfar A, et al, Anal Methods,2015, 7 :8123.
[35]	David R L. CRC Handbook of Chemistry and Physics. 73rd ed. Boca Raton:CRC Press, 1992
[36]	Adamson A W. A Textbook of Physical Chemistry. New York:Academic Press Inc., 1973
[37]	Moré J J, Lecture Notes in Mathematics,1978, 630 :105.