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Anal. Chem. 1998, 70, 1692-1700

Interpretation of Ion-Exchange Chromatographic Retention Based on an Electrical Double-Layer Model Tetsuo Okada

Department of Chemistry, Tokyo Institute of Technology, Meguro-ku, Tokyo 152, Japan

An ion-exchange chromatographic model based on the Stern-Gouy-Chapman electrical double-layer theory is presented. We assume several equilibria occurring at the surface of an ion-exchange resin, such as the ion-pair formation of counterions with an ion-exchange site, the adsorption of ions, and the ion-pair formation of an adsorbed ions with their counterions. These equilibria are affected by the potential at the planes (the surface and the Stern layer potential) where the reactions occur. In addition, the nonselective accumulation of ions in the diffuse layer is also taken into account. Based on the developed model and derived equations, the nature of log k′-log X (X is the concentration of an eluent ion) plots is investigated for various sets of parameters and is compared with that of possible experimental results. Three extreme cases can be distinguished: (1) ion-pair-controlled, (2) adsorption-controlled, and (3) accumulation in the diffuse layer-controlled retention. Though log k′log X plots, when they are studied over wide eluent concentration ranges (more than 2 orders), do not necessarily show precise linearity especially in the presence of eluent adsorption and extremely high eluent ion-pair formation, the linear regression analyses give negative unit slopes (within 10% deviations) for monovalent soluteeluent combinations. Since the deviation from linearity relation is mostly found in very low k′ ranges (e.g., k′ < 1), it is possible only in very limited cases to confirm nonlinearity experimentally. This may have mistakenly led to the idea that selectivity coefficients can be regarded as constants and thus the log k′-log X plots should be linear with the slope equal to (charge ratios) × (-1) in ion-exchange chromatographic experiments. The efficiency of the developed model is verified by its ability to predict experimental results, e.g., nonlinear log k′-log X plots, changes in separation factors with varying ionexchange affinity of an eluent ion, adsorption isotherms at different salt concentration, etc. Ion-exchange chromatography is a useful tool for the separation of not only simple ions but also macromolecules such as proteins and has been extensively used for the analysis and purification of ionic compounds. A large number of partition (alternatively chromatographic retention) data of ions have been accumulated for various ion-exchange resins differing in resin * (e-mail) [email protected].

1692 Analytical Chemistry, Vol. 70, No. 9, May 1, 1998

matrixes and/or chemical structures of ion-exchange sites as well as for various mobile phases since ion chromatography was introduced as an effective means for ionic analysis.1-3 A number of studies are still devoted to enhancing separation selectivity and performance by varying resin structures and mobile-phase compositions4-9 in order to exploit novel applications and to elucidate proper ion-exchange mechanisms. It is not only of practical importance but also of fundamental importance to understand ion-exchange mechanisms properly. A stoichiometric (phase separation) model based on an ion-exchange equilibrium shown by

X h +YhX+Y h

(1)

is most extensively applied to the interpretation of results in ionexchange chromatography, where X h and Y h denote ions X and Y in an ion-exchange resin phase. Selectivity coefficients instead of thermodynamic ion-exchange equilibrium constants have been used for the analyses of ion-exchange chromatographic data; i.e., activity coefficients are neglected in this treatment. Although corrections for activity coefficients are possible in batch experiments for both solution and an ion-exchange resin phase, those for an ion-exchange resin phase are difficult in chromatographic experiments because only linear portions in adsorption isotherms are usually used for chromatographic measurements (very low solute concentration under dominant eluent concentration). This assumption where activity coefficients are negligible claims the validity of a famous and very simple linear relation between log k′ and log X with a -zY/zX slope (k′ and X refer to a capacity factor and the concentration of an eluent ion in a mobile phase, and zY and zX are charges of X and Y). It is obvious that this relation has no thermodynamic meaning; this has already been repeatedly criticized by, for example, Leaderer10 and Ståhlberg.11 Some models based on electrical double-layer theories have been developed to explain ion exchange in more sophisticated (1) Small, H.; Stevens, T. S.; Bauman, W.C. Anal. Chem. 1975, 47, 1801. (2) Okada, T. Bunseki Kagaku 1995, 44, 579. (3) Barron, R. E.; Fritz, J. S. J. Chromatogr. 1984, 284, 13. (4) Nishimura, M.; Hayashi, M.; Yamamoto, A.; Horikawa, T.; Hayakawa, K.; Miyazaki, M. J. Chromatogr., A 1995, 708, 195. (5) Shamsi, S. A.; Danielson, N. D. J. Chromatogr. 1993, 653, 153. (6) McNeff, C.; Zhao, Q.; Carr, P. W. J. Chromatogr., A 1994, 684, 201. (7) Krokhin, O. V.; Smolenkov, A. D.; Obrezkov, O. N.; Shpigun, O. A. J. Chromatogr., A 1995, 706, 93. (8) Okada, T. Anal. Chem. 1996, 68, 1158. (9) Okada, T. J. Chromatogr., A 1997, 758, 29. (10) Lederer, M. J. Chromatogr. 1988, 452, 265. (11) Ståhlberg, J. Anal. Chem. 1994, 66, 440. S0003-2700(97)00655-0 CCC: $15.00

© 1998 American Chemical Society Published on Web 03/25/1998

ways. Cantwell and co-workers12-14 have derived some equations involving the contributions from the ion exchange, the ion adsorption, and the ion exclusion terms to the overall ion-exchange capacity factor of a given ionic solute by starting from SternGouy-Chapman (SGC) theory. Though Cantwell’s work12-14 includes a very important contribution from the change in solvation and has successfully explained ion-exchange chromatographic behaviors of some solutes, derived equations include ambiguities coming from some approximation. The ion-exchange term, which was defined by a difference in the standard chemical potential of transfer between an eluent and a solute ion, was evaluated from the assumption of complete occupation of ionexchange sites by eluent ions. This treatment includes the same problem as seen in the usual work based on stoichiometric ionexchange equilibria mentioned above. Also, if the standard chemical potentials of transfer of ions are included in an ionexchange equilibrium constant, changes in the chemical potential of ion-exchange sites should also be involved because the solvation states of ion-exchange sites also changes during ion-exchange processes. In addition, the potential at the outer Helmholtz plane (OHP) was calculated on the basis of constant surface charges and the Debye-Hu¨ckel approximation, which holds only for very small surface potential. Ståhlberg’s model11 also very excellently explained ionexchange chromatographic capacity factor on the basis of a nonstoichiometric (successive interface) model. Ståhlberg’s main criticism against stoichiometric (separated-phase) models, including the Cantwell model, is that stoichiometric consideration is invalid for very strong and long-range interaction such as electrostatics. From this viewpoint, he derived useful equations capable of predicting the log k′-log X relationship. His model is based on the Gouy-Chapman (GC) model rather than the SGC model. As is well-known in electrochemistry, the SGC model has predicted behaviors of electrolytes at the vicinity of electrode surfaces much better than the GC model.15-16 Thus, the physical significance of Ståhlberg’s model should be reevaluated with the SGC model. It will be one of the most important points of physicochemical models as to how ion-exchange selectivity is involved therein. Cantwell12-14 took the ion-exchange, ion-exclusion, and adsorption terms into account as stated above. Ståhlberg11 considered that the adsorption of ions on the matrix surfaces of ion-exchange resins is the origin of the ion-exchange selectivity; ions are accumulated in the diffuse layer in the identical manner (dependent only on charges) if no adsorption occurs. However, there are a number of data indicating that the ion-exchange selectivity can be varied or sometimes reversed by changing the structure of ion-exchange sites and/or the solvents used in mobile phases,2,3,17 suggesting that the chemical interaction between solutes and ionexchange sites should be taken into account to explain ionexchange selectivity. Ho¨ll’s surface complex formation theory18-21 (12) Afrashtehfer, S.; Cantwell, F. F. Anal. Chem. 1982, 54, 2422 (13) Hux, R. A.; Cantwell, F. F. Anal. Chem. 1984, 56, 1258. (14) Cantwell, F. F. In Ion-Exchange and Solvent Extraction, Marinsky, J. A., Marcus, Y., Eds.; Marcel Dekker: New York; 1985; Vol. 9, Chapter 6. (15) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1985 (Japanese version; McGraw Hill: Tokyo, 1991). (16) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; John Wiley & Sons: New York, 1980. (17) Okada, T. J. Chromatogr., A 1997, 758, 19

Figure 1. Schematic representations of the electrical double layer at the interface between an anion-exchange resin and solution: (A) change in electrostatic potential with the distance from the resin surface; (B) schematic representation of surface equilibria; (C) equivalent circuit.

is instructive in this concern. In the present paper, equations are derived by assuming the surface ion-pair formation and the validity of the SGC theory to explain ion-exchange chromatographic behaviors of monovalent ions and numerically solved for various sets of parameters. THEORETICAL ASPECTS A schematic diagram is shown in Figure 1. We assumed that the mechanisms in ion-exchange chromatography are separated into three major contributions, e.g., the specific adsorption of solute ions on the resin, the ion-pair formation of solute ions with ion-exchange sites, and the electrostatic accumulation of solute ions in the diffuse layer. The specific adsorption has been recognized as one of the most important mechanisms in ion exchange, and, as mentioned above, has been involved in various ion-exchange models. According to our previous study of nonaqueous ion-exchange chromatography,9 this mechanism is one of the main factors governing anion-exchange selectivity in hydroxylic solvents such as water and methanol but plays minor (18) Horst, J.; Ho ¨ll, W. H.; Eberle, S. H. React. Polym. 1990, 13, 209. (19) Ho ¨ll, W. H.; Horst, J.; Wernet, M. React. Polym. 1991, 14, 251. (20) Ho ¨ll, W. H.; Horst, J.; Franzreb, M. In New Developments in Ion-Exchange, Abe, M., Kataoka, T., Suzuki, T., Eds.; Kodansha: Tokyo, 1991; p 277. (21) Ho ¨ll, W. H.; Franzreb, M.; Horst, J. In Ion-Exchange and Solvent Extraction; Marinsky, J. A., Marcus, Y., Eds.; Marcel Dekker: New York, 1993; Vol. 11, Chapter 3.

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roles in aprotic solvents. Instead of specific adsorption, the second mechanism, ion-pair formation between counterions and ionexchange sites more dominantly determines overall selectivity in aprotic media, where anions are less solvated and form ion pairs much more easily than in hydroxylic solvents. Though, in water, the ion-pair formation must be less important because of the strong solvation ability of this solvent, ion-exchange data compiled so far strongly imply that this mechanism should be involved in the separation mechanisms. In addition to the ion-pair formation between ion-exchange sites and counterions, the ion pair between the adsorbed species and their counterions is also taken into account. As shown later, this interaction is necessary to explain salt effects on the adsorption isotherms. Double-layer accumulation must also be an essential mechanism characteristic of a model derived on the basis of electrical double-layer theories. The specific surface adsorption occurs at the inner Helmholtz layer, and thus is affected by the surface potential (ψs). The ionpair formation occurs at the Stern layer and thus should be affected by the Stern layer potential (ψst). The double-layer accumulation is also influenced by the Stern layer potential. Although Ho¨ll et al. assumed different Stern layers for different ions,18-21 a single Stern layer (Figure 1) is assumed in the present study for the following reasons: (1) the equation derivation is simple with a single Stern layer; (2) in ion-exchange chromatography, an eluent ion is dominant even in a sample zone; (3) there are very small differences in ionic (or solvated ionic) radii between usual ions. The Stern layer potential ψst is obtained from the following equations (see Appendix I for equation derivation).

x8RT0Z° sinh

(

-

)

zxFψst ) RT -zie(Γtot ie

- Γad-X - Γad-Y + Γad-Z)F (2)

cd(ψst - ψd) - zie(Γtot ie - Γad-X - Γad-Y + Γad-Z)F )

Γtot ad ,

( )

x

2F2Z°0 Fψst cosh RT 2RT

(

ΓadX° exp -

)

zXFψs RT

1694 Analytical Chemistry, Vol. 70, No. 9, May 1, 1998

Γad-Z

)

(7)

(

zZFψs RT

ΓadZ° exp -

Γad-XZ

(

)

(8)

)

(9)

zZFψst Γad-XZ° exp RT Γad-YZ

KYZ ad )

(

Γad-YZ° exp Γie-X

KXip )

(

ΓieX° exp KYip )

zZFψst RT

)

(10)

)

(11)

zXFψst RT

Γie-Y

(

zYFψst ΓieY° exp RT

YZ X Y where KXad, KYad, KZad, KXZ ad , Kad , Kip, and Kip are the adsorption constants of X, Y, and Z, the ion-pair formation constants of adsorbed X or adsorbed Y with Z, and the ion-pair formation constants of X and Y with ion-exchange sites, respectively. Γad and Γie are the surface concentration of free adsorption sites and free ion-exchange sites. The surface potential, ψs, can be calculated based on the assumption of the constant closest approach (d),

(12)

cs(ψs - ψst) ) zie(Γtot ie - Γad-X - Γad-Y + Γad-Z - Γad-XZ Γad-YZ + Γad-ZX + Γad-ZY) (13)

where Γie-X, and Γie-Y are the surface concentrations of total adsorption sites, adsorbed eluent ions (X), adsorbed solute ions (Y), adsorbed co-ion (Z), total ion-exchange sites, ion pairs between X and the ion-exchange site, and ion pairs between Y and the ion-exchange site, Z° is the bulk concentration of Z (a single co-ion is assumed, and Z° must be equal to the sum of the bulk concentrations of X and Y, X° + Y°), zie and zX are the charge of the ion-exchange site and X, cd is the capacitance of the diffuse layer, and , 0, F, R, and T are relative permittivity of a medium, the dielectric constant of vacuum, the Faraday constant, the gas constant, and the absolute temperature, respectively. The surface concentrations are calculated on the basis of the following equilibrium constants.

Γad-X

(

KZad )

KXZ ad )

)

(6)

zYFψs ΓadY° exp RT

(4)

Γad-X, Γad-Y, Γad-Z, Γtot ie ,

KXad )

Γad-Y

cs ) d/0

-zie(Γie-X + Γie-Y)F (3)

cd )

KYad )

where cs is the capacitance of the Stern layer. Assuming an appropriate value for d, we can solve simultaneous equations to obtain necessary potential values for a given set of equilibrium constants. The calculated potential can be used for the prediction of experimental results. The prediction of the adsorption isotherms of charged solutes on electrically neutral surfaces is very simple; e.g., the adsorption isotherm of Y (in the absence of ion exchange) is given by a change in (Γad-Y + Γad-YZ) with the equilibrium concentration of Y. In chromatographic experiments, it is convenient and informative to study the nature of k′ (or corrected retention volumes). The contributions from ion-pair formation (kip) and adsorption stoichiometric interactions (kad) to k′ for Y can be calculated from the infinite slope of an individual adsorption isotherm at the limiting dilution as described by

(5) kip )

( )

A ∂Γie-Y V0 ∂Y°

Y0)0

(14)

kad )

(

)

A ∂Γad-Y V0 ∂Y°

(15)

Y0)0

where A and V0 are the total surface area of the stationary phase and the void volume of a column, respectively. Thus, these stoichiometric contributions from the surface adsorption and the surface ion-pair formation can be evaluated. In addition, the Stern layer potential obtained from the above calculation allows us to evaluate the contribution of the accumulation of ions in the diffuse layer. According to Ståhlberg’s method,11 this contribution to a capacity factor (kDL) can be written as

kDL ) ψ(x) )

A V0

∫ {exp(a

d

) }

zXFψ(x) - 1 dx RT

{

}

1 + tanh(Fψst/4RT) exp{-κ(x - d)} 2RT ln (16) F 1 - tanh(Fψst/4RT) exp{-κ(x - d)}

where 1/κ is the Debye length. Ståhlberg calculated a from the volume of the stagnant mobile phase.11 In the present study, we evaluated a in the manner shown in the Experimental Section. Hence, the overall capacity factor is represented by

k′ ) kip + kad + kDL

(17)

Thus, we can predict experimental results from the present calculation. However, unfortunately, we have no direct methods to determine equilibrium constants. In the present study, though the extraction of equilibrium constants from experiments has been, in some cases, attempted by curve fitting, the values obtained are not very reliable because there are possibly many equilibrium constants capable of explaining experimental results. EXPERIMENTAL SECTION The chromatographic system was composed of a Shimadzu HPLC pump model LC 10AT, a Rheodyne injection valve equipped with a 100-µL sample loop, a Tosoh UV-visible detector model UV-8020, and a thermostated water bath. A PTFE column (4.6 mm i.d. 50 mm) was used after packing an appropriate stationary phase. MCIGEL 5HP 5C (polystyrene-divinylbenzene copolymer gel, particle size 9-11 µm, specific surface area 500 m2 g-1) was used for adsorption experiments. The same resin was used for anion-exchange chromatographic studies after chloromethylation followed by the reaction with trimethylamine (NEt3+) or with ammonia (NH3+). The ion-exchange capacity of these was determined from the concentration of NO3- eluted from the NO3- form resin by passing a sufficient amount of ClO4- solution. The capacities of the resins were 0.262 mmol g-1 for the NEt3+ resin and 0.59 mmol g-1 for the NH3+ resin. The former column was mainly used for the following experiments, and thus the parameters necessary for calculations were set for this stationary phase. Reagents were basically of analytical grade. Sodium 2-naphthalenesulfonate (2NS) was recrystallized from methanol. Aqueous solutions were prepared in distilled deionized water. Adsorption amounts of solutes were determined from breakthrough curves.

The integral parameter for eq 16 was determined by the following method. The weight of water imbibed by the resin was determined from the weights of a dry stationary phase packed in a column and of a mobile phase in a column (from the void volume). This water weight was converted into volume and then into the thickness of imbibed water layer. For the NEt3+ resin, a was determined to be 1.66 nm. Numerical calculation was carried out with a mathematical program package Mathcad Plus ver.6 on a DEC personal computer. RESULTS AND DISCUSSION Adsorption Sites. The adsorption of ionic compounds on ionexchange resin matrixes often plays an important role to determine overall selectivity. The adsorption of some compounds on the resin matrix was studied to determine the adsorption capacity of the resin surface and to evaluate adsorption equilibrium constants. The adsorption of neutral molecules is usually explained by a Langmuir adsorption isotherm. However, this equation is not applicable to the adsorption of ionic compounds; the surface potential is a function of the surface concentration of ionic compounds, and the surface concentration is also affected by the surface potential. The adsorption isotherm can be derived in a manner similar to the above equation derivation (see Appendix II). The total adsorbed mole of X (Xad) on the surface of the total -2 adsorption capacity (Γtot ad /mol m ) is given by

(

(

)

-zXFψs + KXadKXZ ip X°Z° × RT -zXF(ψs - ψst) -zXFψs exp / 1 + KXadX° exp + RT RT -zXF(ψs - ψst) KXadKXZ (18) ip X°Z° exp RT

X Xad ) Γtot ad A KadX° exp

{

}) (

{

(

) })

where the first and the second terms in the numerator represent the contribution from adsorbed species X and XZ, respectively. A similar equation has been reported but involves neither the ionpair formation term nor the Stern layer potential.22-23 The ionpair formation of adsorbed molecules with a counterion is necessary to explain coexistent salt effects, as shown below. In the present paper, anion-exchange resin was synthesized from styrene-divinylbenzene copolymer. Since it was difficult to obtain adsorption isotherms for ionic compounds on ion-exchange resin because ion-exchange and adsorption equilibria takes place simultaneously, the adsorption on the base resin with no ionexchange sites was studied. Figure 2 shows adsorption isotherms of Na+ 2NS with and without added NaCl. As well-known as salting out in many instances, adsorption is enhanced by adding salts.24,25 This can be explained at least by two factors involved in the present system. One is the enhanced ion-pair formation; this is caused by increasing concentration of counterions (Na+ in the present case). The other is a change in the potential. Even if KXZ ip is not involved in the calculation, the adsorbed amount (22) Ståhlberg, J. J. Chromatogr. 1986, 356, 231. (23) Bartha, AÄ .; Ståhlberg, J. J. Chromatogr., A 1994, 668, 255. (24) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1980. (25) Rosen, M. Surfactants and Interfacial Phenomena; Wiley: New York, 1978.

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1695

Figure 2. Effects of added NaCl on adsorption isotherms of sodium naphthalene-2-sulfonate. Adsorption on unmodified MCIGEL CHP 5C from aqueous solution. Curves are calculated based on eq 18 with parameters, KXad ) 1.9 m-3 mol, Kad-XZ ) 0.2 m-3 mol, and Γtot ad ) 1.0 × 10-7 mol m-2.

should increase with increasing concentration of salts as shown by the dashed curves in Figure 3B. This effect is caused by less extended electrical double layers or numerically lowered surface potential, as shown by the solid curves in Figure 3. However, the increases in the adsorption due to the potential effect are not large enough to fully explain the actual adsorption isotherms. Solid curves in Figure 2, as well as dashed curves in Figure 3A, were obtained by the calculation based on eq 18 with a set of parameters, KXad ) 1.9 m-3 mol, Kad-XZ ) 0.2 m-3 mol, and Γtot ad ) 1.0 × 10-7 mol m-2. Adsorption isotherms are approximately described by this equation, suggesting that the ion pairs are formed between adsorbed species and counterions. The comparison of part A of Figure 3 with part B indicates that the surface potential is numerically lowered by the ion-pair formation and that the adsorption is enhanced as a result in the former case. Figure 3C indicates that ion pairs are major adsorbed species when the salt concentration is rather high (10 mM). Thus, it can be concluded that there are two factors responsible for the adsorption enhancement by adding salts: i.e., (1) the shift of the adsorption equilibrium and (2) reduced Stern layer potentials resulting from the neutralization of the charges of adsorbed species and reduced diffuse layer thickness. In the Absence of Adsorption. The following calculations and experiments are basically for anion-exchange chromatography of monovalent solute anions with monovalent eluent anions. The discussion for multivalent ions was omitted because the above theory becomes unreliable as the valence of the ions increases. The application to cation exchange is feasible simply by reversing surface potential. The adsorption of usual inorganic ions on an ion-exchange resin is not very strong. In particular, this will be true for ion exchange between small and well-hydrated ions as well as for the equilibria in most nonaqueous solvents. Though we tested the adsorption of simple inorganic ions (Cl-, I-, ClO4-, etc), it was so weak that we could not obtain quantitative adsorption isotherms even for a so-called “hydrophobic” anion, ClO4-. Though such ions must be adsorbed to some extent, an adsorption term(kad) is negligible in many cases for the separation of simple inorganic 1696 Analytical Chemistry, Vol. 70, No. 9, May 1, 1998

Figure 3. Changes in adsorption amounts of ionic compounds (2NS) and in surface potential with increasing salt concentration. Parameters were set for 2NS as shown in Figure 2: (A) with Kad-XZ term; (B) without Kad-XZ term; (C) adsorption amount from Figure 2 and calculated adsorbed species with 10 mM NaCl. (a) with 0 mM NaCl; (b) with 2 mM NaCl; (c) with 10 mM NaCl.

ions that are not very adsorptive. Using the equations derived above, we can simulate log k′-log X relations for some sets of equilibrium constants. Selected results are shown in Figure 4, and the slopes and correlation coefficients obtained with the assumption of linear log k′-log X relations are summarized in Table 1. In the absence of adsorption, k′ values are composed of a ion-pair term (kip) and a diffusion-layer accumulation term (kDL). When KXip ) KYip (Figure 4A), the retention is dominantly governed by kip; log k′-log X is linear with negative unit slope lines. When KXip increases keeping KYip constant (Figure 4B-D), kDL becomes dominant. As shown in Table 1, the log kip-log X plots give a negative unit slope within 1% deviation, while log kDL-log X plots show slightly different slopes. Thus, log k′-log X plots also show negative unit slopes if overall retention is determined dominantly by the contribution from ion-pair formation of a solute ion, while the slope of log k′-log X plots will depart from -1 to some extent if the diffuse layer accumulation of a solute ion governs total retention. The dominant contribution from kDL is seen when KXip is much larger than KYip, because Y is retained only through diffuse layer accumulation but not through ion-pair formation. When the ratio, KXip/KYip, is very large, log k′-log X plots obviously become nonlinear. Such an example is seen in

Figure 5. Experimental example of kDL-dominant log (tr - t0) - log X plot: stationary phase, NH3+; mobile phase, tetraethylammonium bromide in acetonitrile; sample, tetraethylammonium iodide. Broken curve, calculated with parameters KXip ) 0.18 m3 mol-1 and KYip ) 0.01 m3 mol-1.

Figure 4. Calculated log k′-log X plots and the contribution from components of k′. σs and d were set to 0.048 C m-2 and 2.6 Å. Equilibrium constants: (A) KXip ) 0.1 m3 mol-1, KYip ) 0.1 m3 mol-1; (B) KXip ) 0.1 m3 mol-1, KYip ) 0.01 m3 mol-1; (C) KXip ) 1 m3 mol-1, KYip ) 0.01 m3 mol-1; (D) KXip ) 10 m3 mol-1, KYip ) 0.01 m3 mol-1. Table 1. Slope of Calculated log k′-log X Plots in the Absence of Adsorption -1 × slope parameter set KXip KYip KXip KYip KXip KYip KXip KYip KXip KYip KXip KYip KXip KYip KXip KYip

mol-1

) 0.001 ) 0.01 m3 mol-1 ) 0.01 m3 mol-1 ) 0.01 m3 mol-1 ) 0.1 m3 mol-1 ) 0.1 m3 mol-1 ) 0.1 m3 mol-1 ) 0.01 m3 mol-1 ) 0.1 m3 mol-1 ) 0.001 m3 mol-1 ) 1 m3 mol-1 ) 0.01 m3 mol-1 ) 1 m3 mol-1 ) 0.001 m3 mol-1 ) 10 m3 mol-1 ) 0.01 m3 mol-1 m3

k′

(ra)

kip

kDL

1.004 (1.000)

1.003

1.007

1.007 (1.000)

1.003

1.016

1.005 (1.000)

1.001

1.034

1.019 (1.000)

1.001

1.034

1.031 (1.000)

1.001

1.034

1.052 (0.9997)

0.9982

1.071

1.068 (0.9995)

0.9982

1.071

1.123 (0.9988)

0.9971

1.147

a Correlation coefficient for log k′-log X plots at 17 different concentrations ranging from 1 × 10-5 M to 18.4 mM (every 0.204 log scale). σs and d were set to 0.048 C m-2 and 2.6 Å.

Figure 4D, where the plot over relatively higher eluent concentration range lies below the negative unit slope line. The ion-pair formation between ion-exchange sites and counterions is weaker in water than in most organic solvents having weaker donor and acceptor abilities. In nonaqueous media, a smaller ion usually forms stronger ion pairs; several order differences in ion-pair formation constants are often seen for the

usual ions.26,27 This suggests that either the kDL-dominant or kipdominant condition can be more easily realized in organic solvents only if appropriate ions are selected as an eluent and a solute; a nonlinear log k′-log X plot is expectable in the kDL-dominant case. We found such an example in the retention of I- with Br- eluent in acetonitrile with the NH3+ resin. The result is shown in Figure 5, where the deviation from the line of a slope of -1 to the lower direction can be seen as predicted from kDL-dominant separation as shown, for example, in Figure 4D. This is caused by the much larger ion-pair formation ability of Br- than I-. Though larger differences in ion-pair formation ability are expectable for other pairs of anions, e.g., I--Cl-, too large a difference in ion-pair formation ability prevented the measurements of retention times; the elution of Cl- from a I--form anion-exchange resin was not confirmed, but I- was so weakly retained by a Cl--form anionexchange resin that its retention time was not measured. Counterion Adsorption. The adsorption of eluent ions numerically lowers potentials and decreases the retention of a solute; thus, ions having strong adsorption ability can be regarded as strong eluents irrespective of their ion-pair formation ability. The nature of log k′-log X plots is slightly different in the presence of the adsorption of counterions (including solute ions) from that in the absence of adsorption. Figure 6 shows calculated log k′log X plots. When parameters of an eluent ion are assumed equal to those for 2NS, log k′-log X plots seem linear (Figure 6A-D). However, detailed studies indicated that the plot in Figure 6A is not linear but bends upward. This upward bending is so small that those who believe the constant selectivity coefficient assumption must regard it as linear. When a larger Γtot ad value (1.0 × 10-6 mol m-2) is assumed for calculation, the deviation from linearity becomes more obvious; characteristic behaviors can be seen in the plots in Figure 6E and F. Figure 7 shows the log(tr - t0) - log(X/M) plot for ClO4solute and naphthalene-2-sulfonate eluent. As predicted in the above calculation, the plot is not linear over low eluent concentration ranges. A broken curve shows calculated results with parameters, KXip ) 0.085 m3 mol-1, KYip ) 0.05 m3 mol-1, and those listed in Figure 2 for 2NS adsorption. The reversed situation (26) Hojo, M.; Takiguch, T.; Hagiwara, M.; Nagai, H.; Imai, Y. J. Phys. Chem. 1989, 93, 955. (27) Okada, T.; Usui, T. J. Chem. Soc., Faraday Trans. 1996, 92, 4977.

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1697

Figure 7. Experimental example of log (tr - t0) - log X plot in the presence of strong adsorption of an eluent ion: stationary phase, NEt3+; mobile phase, sodium 2NS aqueous solution; sample, ClO4-. Broken curve, calculated with parameters KXip ) 0.085 m3 mol-1, KYip ) 0.05 m3 mol-1, and adsorption parameters for sodium 2NS listed in Figure 2.

dynamic values, the following conclusion can be drawn for the separation between monovalent ions

KBAKCB ) KCA

Figure 6. Calculated log k′-log X plots and the contribution from components of k′. σs, and d were set to 0.048 C m-2, 2.6 Å, and 1.0 -7 mol m-2, × 10-6 mol m-2. Equilibrium constants: (A) Γtot ad ) 1 × 10 X X XZ 3 -1 3 -1 Kip ) 0.1 m mol , Kad ) 1.9 m mol , and Kad ) 0.2 m3 mol-1; (B) -7 mol m-2, KX ) 0.1 m3 mol-1, KY ) 0.01 m3 mol-1, Γtot ad ) 1 × 10 ip ip X tot 3 3 -1 -7 mol Kad ) 1.9 m mol-1, and KXZ ad ) 0.2 m mol ; (C) Γad ) 1 × 10 X Y X Y -2 3 -1 m , Kip ) Kip ) 0.1 m mol , Kad ) Kad ) 1.9 m3 mol-1, and KXZ ad ) tot 3 -1 -7 mol m-2, KX ) 0.1 m3 mol-1, KYZ ad ) 0.2 m mol ; (D) Γad ) 1 × 10 ip 3 -1 KYip ) 0.01 m3 mol-1, KXad ) KYad ) 1.9 m3 mol-1, KXZ ad ) 0.2 m mol , YZ tot X 3 -1 -6 -2 and Kad ) 0.02 m mol ; (E) Γad ) 1 × 10 mol m , Kip ) 0.1 m3 3 mol-1, KYip ) 0.01 m3 mol-1, KXad ) 1.9 m3 mol-1, and KXZ ad ) 0.2 m tot X Y -1 -6 -2 3 -1 3 mol ; (F) Γad ) 1 × 10 mol m , Kip ) 0.1 m mol , Kip ) 0.01 m Y 3 -1 3 mol-1, KXad ) 1.9 m3 mol-1, KXZ ad ) 0.2 m mol , Kad ) 0.19 m YZ -1 3 -1 mol , and Kad ) 0.02 m mol .

(ClO4- as an eluent and 2NS as a sample) could not be tested because of the very strong adsorption ability of 2NS. The surface potential of an anion-exchange resin is basically tot positive, but can be reversed when Γtot ad > Γie . In such cases ion exclusion takes place; i.e., anions are excluded from a diffuse layer due to a negative Stern layer potential. This contribution was taken into consideration as a different term by Cantwell et al.12-14 However, in the present model, ion-exclusion contribution is automatically included in kDL terms as negative values. Ion exclusion is seen in Figure 6E, where kip becomes larger than k′ at higher eluent concentration because of the negative kDL values. Separation of Ions. One of serious problems in the constant selectivity coefficient assumption is the failure in predicting the dependence of separation ratios (R) on the nature of eluents. When selectivity coefficients are treated as if they were thermo1698 Analytical Chemistry, Vol. 70, No. 9, May 1, 1998

(19)

where KBA is the selectivity coefficient between A and B ion. Suppose that the selectivity coefficients are determined by ionexchange chromatographic experiments with an eluent ion B. Equation 19 indicates that the selectivity coefficient between A and C should be constant irrespective of the nature of B and that a separation ratio is also constant because RCA ) k′C/k′A ) KCA according to usual ion-exchange theory. This must be against most ion-exchange chromatographic experience that better separation (larger separation ratio) is usually obtained with weaker eluents. Anion-exchange separation of NO2- and NO3- was studied with Cl-, Br-, and ClO4- as an eluent, where the concentrations of eluents were adjusted to obtain an identical k′ for NO2- (k′ ) ∼4). Two anion-exchange resins differing in resin matrixes were used: (1) the NEt3+ column (polymer resin) and (2) TSKgel IC-Anion-SW (silica gel, NMeEt2+ as active sites). The adsorption ability for these columns should be different; silica gel is a much weaker anion adsorbent. Regardless of the adsorption ability of the resins, separation ratios decrease with the affinity of eluent anions to the resins; R ) 3.42 (Cl- eluent), 2.46 (Breluent), and 1.65 (ClO4- eluent) for the NEt3+ column and R ) 1.23(Cl- eluent), 1.08 (Br- eluent), and 1.03 (ClO4- eluent) for TSKgel IC-Anion-SW. The difference between two columns is apparently due to the adsorption ability of the resins. This trend can also be followed by the present model. Figure 8A shows a plot of calculated separation ratios with the ion-pair formation ability of an eluent ion, where the adsorption was ignored. Similarly, Figure 8B shows the dependence of R on the adsorption constant of an eluent ion on the basis of the G-C model (not the SGC model). Though the above experimental tendency is qualitatively explained by either model, R values are less sensitive to changes in the adsorption constants of eluents. Since it must be reasonable to assume no anion adsorption on the silica gel resin, changes in R with the TSKgel IC-Anion-SW column can be

sites nor the ion-exchange separation in the absence of adsorption, e.g., in nonaqueous solvents or with resins of low adsorbing ability, we can now treat these phenomena by the present model. APPENDIX I An electrical double layer at the surface of an ion-exchange resin, divided into three layers, is electrically equivalent to the connection of two condensers. Total differential capacitance of the electrical double layer (ct) is represented by

1/ct ) 1/cs + 1/cd

(A-1)

According to the Gouy-Chapman model, cd is given by

cR ) ∂σd/∂ψ ) cosh

( )x

2F20X° RT

Fψst 2RT

(A-2)

where σd is the charge density at the diffuse layer. For electroneutrality,

σs + σst + σd ) 0

(A-3)

where σst is the charge density at the Stern layer. These charge densities can be represented by

σs ) zieF(Γie + Γie-X + Γie-Y - Γad-X - Γad-Y + Γad-Z - Γad-XZ - Γad-YZ + Γad-ZX + Γad-ZY) (A-4) Figure 8. Change in calculated separation ratio with eluent strength: (A) dependence of R on KXip in the absence of adsorption. Separation ratios between a solute of KYip ) 0.005 m3 mol-1 and of KYip ) 0.006 m3 mol-1. (B) Dependence of R on KXad. Calculation based on the G-C model (no ion-pair formation). Separation ratios between a solute of KYad ) 0.9 m3 mol-1 and of KYad ) 1 m3 mol-1. -7 mol m-2. (C) Dependence of R on KX with Γtot ad is assumed 5 × 10 ip X constant Kad ()0.01 m3 mol-1). Separation ratios between a solute Y Y of Kip ) 0.005 m3 mol-1 and Kad ) 0.03 m3 mol-1 and a solute of KYip ) 0.025 m3 mol-1 and KYad ) 0.01 m3 mol-1. Γtot ad is assumed 1 × 10-7 mol m-2.

followed by the calculation not involving adsorption. However, very large changes in R with the NEt3+ column cannot be explained if adsorption is ignored. If two solutes to be separated are retained by a different mechanisms, e.g., one mainly by ionpair formation but the other by adsorption, much larger decreases in R can be expected. Figure 8C is an example, where the separation of two solutes, retained by different mechanisms, is estimated. As KXip increases, the separation mechanism changes from ion pair to adsorption, and the separation drastically becomes worse. These strongly imply that ion exchange should be explained by several mechanisms rather than a single mechanism. In conclusion, a retention model based on the SGC theory for ion-exchange chromatography is presented. This model is basically an extension of several studies in this discipline but describes various experimental facts better than previous models. Though models developed by Ståhlberg11 or Cantwell12-14 explain neither variation of ion-exchange selectivity with varied ion-exchange resin

σst ) -zieF(Γie-X + Γie-Y - Γad-XZ - Γad-YZ + Γad-ZX + Γad-ZY) (A-5) σd ) -zieF(Γie - Γad-X - Γad-Y + Γad-Z)

(A-6)

σs ) cs(ψs - ψst)

(A-7)

σst ) cs(ψst - ψs) + cd(ψst - ψd)

(A-8)

or

σd ) cd(ψd - ψst)

(

) sinh -

)x

Fψst 2RT

8RT0X° (A-9)

On the other hand, we can calculate surface concentration of all species from eqs 5-11 for a given set of equilibrium constants. Thus, eqs A-4-A-6 can be written by functions of ψst. From eqs A-6 and A-9, we obtain

-zieF(Γie - Γad-X - Γad-Y + Γad-Z - Γad-XZ + Γad-ZX +

(

Γad-ZY) ) sinh -

)x

Fψst 2RT

8RT0X° (A-10)

Analytical Chemistry, Vol. 70, No. 9, May 1, 1998

1699

Thus, we can calculate ψs and ψst by solving simultaneous eqs 12, 13, and A-10. APPENDIX II The adsorption of ionic molecules (X) can be treated in a manner similar to Appendix I. If the adsorption of counterion Z is negligible, eqs A-11-A-13 can be rewritten as

σs ) zXF(Γad-X + Γad-XZ)

(A-11)

σst ) -zXFΓad-XZ

(A-12)

σd ) -zXFΓad-X

(A-13)

where XZ refers to the ion pair between X and Z. In this case, the adsorbed molecule X exists at the surface, while the counterion Z exists at the Stern layer. Finally, we can calculate necessary potentials and capacitance for a given set of the adsorption equilibrium constant of X and the ion-pair formation constant between X and Z. The total adsorbed amount of X (Xad) is given by

(

(

)

-zXFψs + KXadKXZ ip X°Z° × RT -zXF(ψs - ψst) -zXFψs exp / 1 + KXadX° exp + RT RT -zXF(ψs - ψst) KXadKXZ (A-14) ip X°Z° exp RT

X Xad ) Γtot ad A KadX° exp

{

}) (

{

(

})

)

This equation is applicable to the case where Z is added as a foreign salt accompanied by a counterion not adsorbed on the stationary phase. GLOSSARY

KYad

adsorption constant of a solute ion (Y) (mol-1 m3)

KZad

adsorption constant of a coion (Z) (mol-1 m3)

KXZ ad

ion-pair formation constant of adsorbed X with Z (mol-1 m3)

KYZ ad

ion-pair formation constant of adsorbed Y with Z (mol-1 m3)

KXip

ion-pair formation constant of X (mol-1 m3)

KYip

ion-pair formation constant of Y (mol-1 m3)

σs

surface charge density (C m-2)

σst

charge density at the Stern layer (C m-2)

σd

charge density at the diffuse layer (C m-2)

cst

capacitance of the Stern layer (F m-2)

cd

capacitance of the diffuse layer (F m-2)



relative permittivity

0

dielectric constant of vacuum, 8.854 × 10-12 C2 N-1 m-2

d

the closest approach (m)

Γtot ad

total surface concentration of adsorption sites (mol m-2)

Γad

surface concentration of free adsorption sites (mol m-2)

Γad-A

surface concentration of adsorbed A (A ) X, Y, or Z) (mol m-2)

Γad-AB

surface concentration of adsorbed ion pair AB (AB ) XZ or YZ) (mol m-2)

Γtot ie

total surface concentration of ion-exchange sites (mol m-2)

Γie-A

surface concentration of ion pair between A (A ) X or Y) and ion-exchange sites (mol m-2)

Γie

surface concentration of free ion-exchange sites (mol m-2)

zX

charge of X

zie

charge of an ion-exchange site

X

eluent ion

Y

solute ion

Z

coion

A°

bulk concentration of ion A (A ) X, Y, or Z) (mol m-3)

ψs

surface potential (V)

ψst

Stern layer potential (V)

ψd

diffuse layer potential (V)

ψ(x)

potential at a point, distance x from the charged surface (V)

Received for review June 24, 1997. Accepted January 30, 1998.

KXad

adsorption constant of an eluent ion (X) (mol-1 m3)

AC970655R

1700 Analytical Chemistry, Vol. 70, No. 9, May 1, 1998

ACKNOWLEDGMENT This work was partly supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Culture, and Sports, Japan.