Published online 28 September 2007
Published in Soil Sci Soc Am J 71:1694-1698 (2007)
DOI: 10.2136/sssaj2007.0004
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SOIL CHEMISTRY
A New Approach to Estimate Ion Distribution between the Exchanger and Solution Phases
Hang Lia,b,* and
Laosheng Wua
a Dep. of Environmental Science, Univ. of California, Riverside, CA 92521
b College of Resources and Environment, Southwest Univ., Chongqing, 400716, P.R. China
* Corresponding author (hli22002{at}yahoo.com.cn).
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ABSTRACT
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Currently there are two approaches to treat ion exchange equilibrium between the exchanger phase and solution phase: the chemical reaction equilibrium and the Donnan equilibrium. Nevertheless, theoretical calculation of the ion distribution between these two phases remains impossible using these two approaches because of the difficulty in obtaining the activity coefficient of an ion in the exchanger phase for a solution mixture containing two or more types of electrolytes. Our analysis of ion diffusion in an electric field showed that the exchange process of electrostatic adsorption can be treated as a diffusion process. A new equilibrium equation was obtained by treating ion exchange as a diffusion process. The new equation can be used to predict the ion distribution between the exchanger and solution phases without knowing the activity coefficients of the adsorbed ions. The new approach was tested and verified using published experimental data for a Na–Ca-illite system. The experimental data and the theoretical predictions matched very well. The new ion distribution equation can provide another way to calculate ion exchange equilibrium for a well-dispersed suspension of dilute bulk solution containing two types of electrolytes.
Abbreviations: DDL, diffuse double layer
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INTRODUCTION
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Ion exchange is an important process in soil. Currently there are two different approaches to treat the ion exchange equilibrium at the solid–liquid interface: treat ion exchange as a chemical reaction following the concept of thermodynamics, or treat it as a physicochemical process using the theory of Donnan equilibrium.
To treat ion exchange as a chemical reaction (Ogwada and Sparks, 1986a; Morgan et al., 1995; Graul et al.,1999; Hui and Baker, 2001; Berber-Mendoza et al., 2006), ion exchange and diffusion in the diffuse double layer (DDL) are considered as two separate processes, and usually the reaction rate is limited by ion diffusion (Ogwada and Sparks, 1986b; Tang and Sparks, 1993; Suresh et al., 2004). For this approach, the activity coefficients of ions in the exchanger phase are required to calculate the ionic distribution between the exchanger phase and solution phase or to calculate the thermodynamic equilibrium constant of ion exchange. Unfortunately, such a calculation has never been really accomplished (De Bokx and Boots, 1989), because the activity coefficient of the ion in the exchanger phase is difficult to obtain (De Bokx and Boots, 1989; Biesuz et al., 2001; Hui and Baker, 2001; Berber-Mendoza et al., 2006). In the past, the relationship between the selectivity coefficient (defined in terms of thermodynamic activities) and the conventional selectivity coefficient (defined in terms of concentration) (Gaines and Thomas, 1953), the relationship between the thermodynamic equilibrium constant and the Vanselow selectivity coefficient (Argersinger et al., 1950), and a mathematical formula related to the activity coefficient of the ion in the exchanger phase and the Vanselow selectivity coefficient (Sposito, 1994) were established. These relationships allow experimental determination of the selectivity coefficient in terms of activities, as well as the thermodynamic equilibrium constant.
The other approach is to treat ion exchange as a physicochemical process; therefore, the theory of Donnan equilibrium can be applied (Eriksson, 1952; Bolt, 1954; Jacob and Reddy, 1991; Tanioka et al., 1998; Biesuz et al., 2001). Application of the Donnan equilibrium to ion exchange implies that the thermodynamic equilibrium constant of ion exchange is equal to 1 (Eriksson 1952). Even if the Donnan theory is applicable, it still remains difficult to theoretically calculate the distribution of ions between the exchanger phase and the solution phase, because the activity coefficient of ions in the exchanger phase is difficult to obtain, as discussed above. The applicability of the Donnan theory to ion exchange equilibrium may imply that the adsorption of the ion on the solid–liquid interface is not a specific adsorption but an electrostatic adsorption, which explains why the adsorbed ions exhibit a Boltzmann distribution adjacent to the solid surface at the equilibrium of ion exchange.
The aims of this research were to develop: (i) a new approach to treat the ion exchange process in the DDL as a diffusion process driven by the apparent concentration gradient; (ii) a new method to estimate the activity coefficient of ions in the exchanger phase; and (iii) a new distribution equation for describing cation exchange equilibrium.
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THEORY
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Similarity of Ion Exchange and Ion Diffusion in the Diffuse Double Layer
The flux equation of diffusion in an external field can be expressed as (Li and Wu, 2004)
 | [1] |
where J(x,t) is the apparent mass flux, D is Fick's diffusion coefficient, a(x,t) is the apparent concentration or activity, x is a space coordinate, and t is time. The apparent concentration in an electric field can be defined as (Li and Wu, 2004)
 | [2] |
where c(x,t) is the concentration of the ion, Z is the valence of the ion, F is the Faraday constant, 
(x,t) is the potential, R is the gas constant, and T is the absolute temperature. Correspondingly, the apparent mass flux can be expressed as (Li and Wu, 2004)
 | [3] |
where j(x,t) is the actual mass flux.
In the absence of an external electric field, j(x,t) = 0, which implies 
c(x,t) = 0, or an equilibrium state of diffusion under an isothermal condition, as indicated by Fick's law. While in the presence of an external field, J(x,t) = 0 also indicates an equilibrium state if a(x, t) = 0 in Eq. [1], even if c(x, t) 
0. Because the Fick's flux equation is a special case of Eq. [1] with (x,t) = 0, a general criterion for assessing the equilibrium state of diffusion is a(x,t = 
) = 0. Consequently, the criterion for judging the equilibrium of diffusion for a multicomponent system containing i types of ions is
 | [4] |
It is well known that, at the equilibrium state of ion exchange, the ionic distribution in the DDL will obey the Boltzmann equation:
 | [5] |
where ci0 is the concentration of ion i in bulk solution.
Rearranging Eq. [5] and then differentiating it, we have
 | [6] |
Comparing Eq. [4] and [6] shows that the diffusion equilibrium analysis could lead to the same result of exchange equilibrium. This implies that the ion exchange of electrostatic adsorption occurring in the DDL may be essentially considered as a diffusion process. Therefore, the diffusion process and exchange process of electrostatic adsorption in the DDL can be combined and treated as one process by using the apparent concentration, in which the driving force combines the concentration gradient and potential gradient. Based on this concept, the kinetic equation of ion exchange in the DDL can be theoretically derived directly from the diffusion equation of an ion in an electric field. This may further imply the way to evaluate the specific adsorption from experimental data when both electrostatic adsorption and specific adsorption are taking place in a system.
Ion Distribution between Exchanger and Solution Phases
Mean Activity Coefficient of Ions in Exchanger Phase
According to the definition given by Li and Wu (2004), the apparent concentration a(x,t) is actually activity, while c(x,t) is actual concentration. Therefore, according to Eq. [2], the activity coefficient of ion i at a given time t and a given position x in an electric field is
 | [7] |
At an equilibrium state, from Eq. [4], we get
 | [8] |
where ci(x) is the concentration of ion i at position x in the DDL, and 
i(x) is the activity coefficient of ion i at position x in the DDL.
The mean concentration of ion i in the DDL can be defined as
 | [9] |
where 
is the Debye–Hückel parameter, and 1/ is the effective thickness of the DDL. According to the chemical thermodynamic theory, at equilibrium we have
 | [10] |
where µi is the chemical potential of ion i, and µi0 is the reference potential of ion i. At equilibrium, both of them are constant, therefore from Eq. [9] and [10] we have
 | [11] |
If we use the mean concentration to express the chemical potential of ion i, we can write Eq. [10] as
 | [12] |
Here 
i can be considered as the mean activity coefficient of ion i in the DDL. From Eq. [11] and [12], the mean activity coefficient of ion i can be defined as
 | [13] |
Introducing Eq. [5] into Eq. [9], we get
 | [14] |
From Eq. [13] and [14], we get
 | [15] |
According to Eq. [15], for a solution mixture of two types of electrolytes, when the diffusion reaches equilibrium, we have
 | [16] |
where i = 1 or 2 represents Cation 1 or 2, respectively.
Li et al. (2004) established the equations for calculating the values of ci0/ for different monoelectrolyte solutions. For soil particles or clay minerals, those equations can be simplified to the following:
For a symmetric electrolyte:
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For a 1:2 type electrolyte:
 | [18] |
For a 2:1 type electrolyte:
 | [19] |
where Zi is the charge number of the cation in each electrolyte.
In the derivation of Eq. [17–19]
, each ion was considered as a point charge, which implies that different ions with equivalent charges (e.g., Cl– and NO3–) are treated equally. Therefore, those equations are applicable directly for calculating the mean activity coefficient of a cation in the exchanger phase for a solution mixture of two 1:1 electrolytes, two 2:2 electrolytes, two 1:2 electrolytes, or two 2:1 electrolytes. On the other hand, because Eq. [17] is applicable for any symmetric electrolyte solution, we can expect that the mathematical form will be the same as Eq. [17] for calculating the mean activity coefficient of each ion for a solution mixture of two different types of symmetric electrolytes (e.g., a mixture of 1:1 and 2:2 electrolytes).
Calculating the mean activity coefficient of each cation in the exchanger phase for a solution mixture of two different types of electrolytes is more complex, but it can be done. Considering a solution mixture of 1:2 (e.g., K2SO4) and 2:1 (e.g., MgCl2) electrolytes, even though Eq. [18] and [19] are applicable only to the two separate systems of 1:2 and 2:1 electrolytes, respectively, we can use them to define an arithmetic mean of if the values of Zi0 are the same for the two separate systems:
 | [20] |
When the difference of mole number is not very significant for the two types of electrolytes, this solution mixture can also be approximately taken as a solution mixture of 1:1 (KCl) and 2:2 (MgSO4) electrolytes. Therefore Eq. [17] for symmetric electrolytes is also approximately applicable for this solution mixture of 1:2 and 2:1 electrolytes. A comparison between Eq. [17] and [20] shows that the defined arithmetic mean of can be used to calculate the mean activity coefficient of a cation in the exchanger phase for a solution mixture of two different types of electrolytes.
The above discussion indicates that a generalized expression for calculating the mean activity coefficient of a cation in the exchanger phase in any solution mixture can be expressed as
 | [21] |
where Aij is a constant. Based on the above discussion, a list of Aij values for different solution mixtures of two types of electrolytes is presented in Table 1
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Ion Distribution between Exchanger and Solution Phases
Considering the case of cation diffusion equilibrium in a solution mixture of two types of electrolytes, introducing Eq. [21] into Eq. [16], we will have
 | [22] |
Equation [22] can also be written as
 | [23] |
where N1 and N2 are the moles of the total adsorbed cations in the DDL with valence Z1 and Z2 per gram of solid, respectively, S is the specific surface area of solid particles (dm2 g–1), and has units of dm–1.
Given that 
is the surface charge density (molc dm–2), therefore
 | [24] |
Introducing Eq. [24] into Eq. [23], we get
 | [25] |
because
 | [26] |
where the unit for 1/ is decimeter and 
= 8.9 x 10–10 C2 J–1 dm–1 for water.
Introducing Eq. [26] into Eq. [25], one obtains
 | [27] |
Equation [27] is the new distribution equation for describing cation exchange equilibrium on a solid–liquid interface. This equation shows that, if the surface charge density of solid particles is known in advance, the adsorbed quantities of cations or the ionic distribution between the exchanger phase and the solution phase can be theoretically calculated. An important feature of Eq. [27] is that it does not contain the cation activity coefficient of the exchanger phase.
From Eq. [21] and [22], we also have
 | [28] |
Here we consider ai0 = 1ci0 for an ideal bulk solution. Equation [28] is identical to the Donnan equilibrium equation but it was obtained by treating cation exchange occurring in the DDL as a diffusion process. This proves that the cation exchange kinetics occurring in the DDL can indeed be treated as a diffusion process. The advantage of this approach is that the activity coefficients of adsorbed ions are not required in the calculation of the ion distribution between the exchanger phase and the solution phase.
For a different cation exchange system of two types of electrolytes, introducing the values of Z1, Z2, and Aij into Eq. [27] and [28], the corresponding ion distribution equations for cation exchange equilibrium can be obtained.
Application and Verification of the Ion Distribution Equation
We applied the new distribution Eq. [27] to cation exchange between Ca2+ and Na+ in an illite–water suspension. The equilibrium Eq. [27] of cation exchange between 2:1 and 1:1 electrolytes (CaCl2 and NaCl) in a dilute bulk solution and well-dispersed clay system is
 | [29] |
where cCa0 and cK0 are the concentrations of Ca2+ and K+ (mol L–1), respectively, in the bulk solution and NNa and NCa are the respective moles of the total adsorbed Na+ and Ca2+ per gram of illite. At T = 298 K, from Eq. [29] we have
 | [30] |
Equation [30] can be rewritten as
 | [31] |
Equation [31] implies that: (i) if we determine the adsorbed quantity of Na+ and Ca2+ at different concentrations of Na+ and Ca2+ in the bulk solution at equilibrium, a linear relationship between log{NNa2/[NCa(NCa + 2NCa)]} and log{3 x 10–9(cNa0)2/[cCa0
(cNa0+3cCa0]} can be obtained; (ii) the slope of the straight line would be equal to 1; and (iii) the surface charge density of the solid particles can be calculated from the intercept obtained from the linear regression.
Figure 1
shows the experimental data (Bolt, 1954) and the linear regression line obtained by fitting Eq. [31] to the data of cation exchange equilibrium between Na–Ca-illite and in a solution containing NaCl and CaCl2. Considering the requirement of dilute bulk solution for the theory, we deleted two sets of experimental data in which the ionic strength was very high in the bulk solution. Figure 1 shows that: (i) the experimental data indeed exhibit a straight line, the value of r2 is 0.9667; (ii) the slope of the straight line is 1.0616, which is close to the theoretically value of 1.0000; and (iii) the value of the intercept log[(4 x 31/2)/(2 + 31/2)] is 7.816, which means that the surface charge density of the illite is = 2.84 x 10–3 mmolc m–2. This value of surface charge density is close to the value 3.0 x 10–3 mmolc m–2 that was determined independently by a negative adsorption method (Bolt, 1954). The excellent match between the theoretical model and the experimental data indicates the correctness of the new ion distribution Eq. [27].
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Fig. 1. The experimental data points (Bolt, 1954) and the linear regression (line) of the relationship between log{NNa2/[NCa(NNa+2NCa)]} and log{[3 x 10–9(cNa0)2]/[cCa0(cNa0+3cCa0)1/2]} for cation exchange at equilibrium.
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Finally, it is important to remember that the above approach was based on two important assumptions: (i) the exchange process of cations is merely an electrostatic adsorption process occurring in the DDL, and (ii) because the Gouy–Chapman theory was used to describe the concentration and potential distribution in the DDL (the equations for calculating the activity coefficient of an ion in the DDL were obtained based on this theory), it is required that the bulk solution is dilute and that the solid particles are well dispersed so that the effect of DDL overlap can be neglected in the suspension.
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CONCLUSIONS
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Traditionally, ion diffusion and ion exchange occurring in the DDL have been considered as two separate processes. Our research shows that the cation exchange of electrostatic adsorption occurring in the DDL can be treated as a diffusion process driven by the apparent concentration gradient of ions in the external electric field from the solid particle surface. Therefore the exchange process of electrostatic adsorption and the diffusion process in the external electric field are essentially the same process, and can be treated as one. Based on this concept, a new ion equilibrium distribution equation was derived in this study. The new distribution equation has the advantage of calculating the ion distribution between the exchanger phase and the solution phase without knowing the activity coefficients of the adsorbed ions. The correctness and accuracy of the new ion distribution equation were verified by experimental data.
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NOTES
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All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.
Received for publication January 4, 2007.
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ABSTRACT
INTRODUCTION
