Combined measurement of surface properties of particles and equilibrium parameters of cation exchange from a single kinetic experiment

Abstract

For the Coulombic force adsorption in cation exchange, equilibrium parameters were closely related to the surface properties of particles in aqueous solution. Taking into account the difference in the polarization of two cation species involved in the exchange by introducing a modification coefficient for describing the relatively effective charges of the cation species, the new approaches for estimating exchange equilibrium parameters and surface properties of particles have been developed. Our theory indicates that, just relying on two parameters, the slope and intercept of the first-order rate equation of cation adsorption, the following equilibrium parameters and important surface properties of particles can be obtained easily: (1) surface potential, (2) specific surface area, (3) charge density, (4) electrostatic field strength at the surface, (5) selectivity coefficient, (6) activity coefficients of cation species in the adsorption phase, (7) adsorption quantity of each cation species at equilibrium with strong and weak force adsorption. For the first time, by the quantitative relationship between the selectivity coefficient and surface potential of particles together with polarizability and charge, the number of cation species is established. The results show the difference in the polarizability between two cation species involved in the exchange strongly influences the cation exchange equilibrium and surface properties of particles.

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1 Introduction

Surface potential, surface electric field strength, surface charge number, surface charge density and specific surface area are important characteristics of microscopic particles, which have a significant effect on a great number of chemical, physical and biological properties. However, at present, a non-invasive, more reliable and universally applicable method for the determination of surface properties of particles does not exist although there are many methods for their determination. Hou and Li¹ developed a new method for the determination of surface potential by the batch experiment; this new method not only provides a more reliable determination for surface potential, but also can be widely applied to any charged materials under any condition of pH, temperature and electrolyte concentration. However, the main disadvantages are (1) the reaction balance time is so long that the balance time is not easy to determine; (2) the sample is becomes H-saturated, and acid treatment may lead to material changes in surface properties of the sample, and the reaction balance time of the ion is difficult to achieve in an H-saturated sample. If we change the batch experiment into ion exchange, the kinetic method may overcome the shortcoming of the long reaction time, and the samples do not need to be saturated by H⁺.

An ion exchange reaction is an important process in the interfacial reaction process, and many researchers have devoted great effort to study the ion-exchange process in soil.^2–14 Most of the studies, however, only considered ion movement caused by the concentration gradient and used Fick's first and second laws directly to describe ion diffusion in the soil. Actually, the colloid surface with large negative charges will form an electrostatic field, and the field will strongly influence the ion exchange kinetic.¹⁶

Li and Wu,¹⁵ Li et al.^16,17 and Li and Li¹⁸ have demonstrated that, the adsorption process of cations in a diffuse layer is actually a diffusion process of the cations in the electrostatic field of the diffuse layer driven by both potential gradient and concentration gradient. Based on this, at the same time, Li et al.^16,17 and Li and Li¹⁸ obtained new kinetic models, from the definitions of the variables in the new kinetic models, we can find that some important parameters relevant to solid surface properties and cation exchange equilibrium, most of which were difficult to obtain in the classic theory, can be easily estimated from the analysis of kinetic data of cation exchange.

On the other hand, the polarization of ions will strongly influence cation-exchange equilibrium. A perspective paper published in Phys. Chem. Chem. Phys. in 2011 by Parsons et al. indicated that,¹⁹ the main reason for the Hofmeister sequences in cation exchange was not the hydration radius of a cation species but the structure and softness of the electron cloud (especially the outer electron cloud of a cation), that is polarizability effect. Usually, the surface electrostatic strength will be up to 10⁸(V m⁻¹);¹⁸ thus, the strong electrostatic force will influence the polarizability of the cation. It can be explained that, as the strong electrostatic field is absent, although the outer electron cloud of ions oscillates randomly, the polarizability effect of ions remains unchanged; however, as the strong electrostatic field is present, the adsorbed counterions are strongly polarized, and the produced strong induction force would be an important source of Hofmeister effects in particle interactions; thus, the surface properties of particles and the cation-exchange equilibrium will be strongly influenced by the polarizability effect of adsorbed ions.

In this paper, we studied the Mg²⁺/K⁺ exchange equilibrium for three different materials with net negative charges through the miscible displacement experiment.²⁰ The purpose of this research is to develop new approaches for estimating the exchange equilibrium parameters and the surface properties of particles by taking into account the polarization of a cation species.

2 Theory

Considering a cation exchange reaction,

Z_iM_j(s) + Z_jM_i(aq) ⇒ Z_jM_i(s) + Z_iM_j(aq)

where s represents the solid phase; aq represents the aqueous phase; M_i and M_j are the i and jth exchangeable cation species respectively and Z_i and Z_j are the charges of i and jth cation species respectively.

For the adsorption cation species in the experiment of miscible displacement, by considering ionic interaction in bulk solution, we have^16–18

where N_i(t) is the total adsorption quantity (mol g⁻¹) from t = 0 to t; S is the specific surface area of the sample with unit m² g⁻¹; a_i0 is the activity of the ith ion species in bulk solution; l is the average thickness of the fixed liquid adjacent to the soil particle surface; ϕ(x) is the electric potential in the electric field of double layer (DL); Z_i is the charge of the ion species; F is the Faraday constant; R is the gas constant; T is the absolute temperature; D_pi is the apparent diffusion coefficient of the ion ith species in the system; t is the adsorption time; x is the position of the ion in the electric field of DL.

As strong electrostatic force adsorption is present in the initial stage of cation exchange,¹⁷ the activity of the cation around the surface is approximately equal to 0; from eqn (1), we have¹⁷

where k₁ is the rate coefficient in the stage of the strong electrostatic force adsorption, and

After a given experiment time t, the exchange process may transfer to weak electrostatic force adsorption,¹⁷ and the activity of the cation is not equal to 0:

where k₂ is the rate coefficient in the stage of weak electrostatic force adsorption, N_i(∞) is the total adsorption quantity (mmol kg⁻¹) at the equilibrium of cation exchange, andwhere l′ is the diffusion distance of the ith ion species in the presence of weak electrostatic force adsorption, and

Correspondingly, the discrete forms of eqn (2) and (4) are respectively:

andwhere m = 0, 1, 2, 3,…, N_i(t_m+1/2) = N_i(t_m) + 0.5[N_i(t_m+1) − N_i(t_m)], p = k₂ is the intercept and q = −k₂/N_i(∞) is the slope.

Based on eqn (7) and (8), using kinetic data of cation exchange to plot the relationship curves of [N_i(t_m+1) − N_i(t_m)]/(t_m+1 − t_m) vs. N_i(t_m+1/2), two straight lines, shown in Fig. 1, could be obtained. From the second straight line, the p (intercept) and q (slope) values can be obtained from the kinetic data.

Fig. 1 Adsorption rate curves with both strong and weak force adsorption (① strong force adsorption and ② weak force adsorption).

The following describes how to use the p and q values to estimate the surface properties of solid particles and the cation-exchange equilibrium parameters.

(1) The totally adsorbed quantity of i and jth cation species at the equilibrium of cation exchange.

From eqn (8), the total adsorption quantity of the ith cation species could be obtained from the slope and intercept values of the second straight line, and

where N_i(∞) is the total adsorption quantity of ith cation species (mmol kg⁻¹) at equilibrium.

Therefore, the total adsorption quantity of the jth cation species at equilibrium is

where N_j(∞) is the total adsorption quantity of the jth cation species (mmol kg⁻¹) at equilibrium, and CEC is the cation exchange capacity (here the unit is mmol_c kg⁻¹).

At the same time, according to the straight lines in Fig. 1, the quantities with strong and weak force adsorption for the ith cation species also can be calculated.

The N_i(t_m+1/2) value at the crossover point of the two straight lines in Fig. 1 is the strong force adsorption quantity of ith cation species, N_i(s)(∞); thus, from eqn (7) and (8), we have

where N_i(s)(∞) is the strong force adsorption quantity of the ith cation species.

From eqn (11), we further get

Therefore, the quantity of the ith cation species with weak force adsorption at equilibrium is

N_i(w)(∞) = N_i(∞) − N_i(s)(∞) (13)

where N_i(w)(∞) is the weak force adsorption quantity of the ith cation species.

Unfortunately, the adsorption quantities for the jth cation species with strong and weak force adsorption cannot be evaluated currently.

(2) The average activity coefficient of the i and jth cation species in the adsorption phase.

Taking into account the ionic interaction energy in bulk solution, the definition of the average activity coefficient for the ith cation species in the exchanger phase could be expressed as¹⁵

where ã_i is the average activity of the ith cation species in the exchanger phase, a_i0 is the activity of the ith cation species in the bulk phase; [small gamma, Greek, tilde]_i is the average activity coefficient of the ith cation species in the exchanger phase; f̃_i is the average concentration of the ith cation species in the exchanger phase, and f̃_i = N_i(∞)/Sl; l is the average thickness of the layer of the fixed liquid film around particles in the experiment; l′ is the average thickness of the layer of weak force adsorption; m is a constant; m = 2 for 1 : 1 + 1 : 1 mixed electrolytes and for 1 : 1 + 2 : 1 mixed electrolytes, where f⁰₊ and f⁰₂₊ refer to the concentrations in bulk solution for univalent and divalent cations, respectively.²⁴ Eqn (14) implies that the reference state was ã_i = a_i0 = 1.

From eqn (8), we get

and

The combined solution of eqn (14)–(16) gives

where V (mL kg⁻¹) is the average volume of the fixed liquid film around particles in the flow method, and V = Sl, and Vf_i0 is actually the adsorption quantity in the electrostatic field absence.

Since adsorbed ions in the double diffusion layer is strongly influenced by the surface electric field, and the polarization increased significantly, but the polarization of different ions is not the same because of the difference of the outer electron clouds, the differences must be taken into account. Therefore, the activity coefficient in eqn (14) for the i and jth cation species in the adsorption phase at the exchange equilibrium can be expressed as²³

andwhere [small gamma, Greek, tilde]_j is the average activity coefficient for the jth cation species in the adsorption phase; m is a constant; ϕ₀ is the surface potential of soil particles, w_pi, w_pj is the polarization energy of the ith cation and jth cation, respectively.

However, by taking into account the polarization of a cation species, a modification parameter β for the charge number Z should be introduced for describing the relatively effective charge number βZ for a cation species; thus, the average activity coefficient for the i and jth cation species in the adsorption phase at the exchange equilibrium can be expressed as¹⁵

andwhere β_i and β_j were referred as the relatively effective charge coefficient for the i and jth cation species, respectively, and

where Δw_p is the difference of the polarization energy between the ith cation and jth cation.

From eqn (20) and (21), we have

At the same time, the average activity coefficients with strong and weak force adsorption for the ith cation species also can be evaluated.

According to the definition of the average activity coefficient, we have

where [small gamma, Greek, tilde]_i(s) is the average activity coefficient of the strong force adsorption for the ith cation species; f̃_i(s) is the average concentration of the strong force adsorption, f̃_i(s) = N_i(s)(∞)/S(l − l′).

From eqn (3) and (5), we have

Thus, introducing eqn (24) into eqn (23), the average activity coefficient of the strong force adsorption can be calculated from the following equation:

Correspondingly, the average activity coefficient of weak force adsorption is

where [small gamma, Greek, tilde]_i(w) is the average activity coefficient of weak force adsorption; f̃_i(w) is the average concentration of weak force adsorption, f̃_i(w) = N_i(w)(∞)/Sl′.

Thus, introducing eqn (24) into eqn (26), the average activity coefficient of weak force adsorption could be calculated from the following equation:

Unfortunately, at present, the average activity coefficients for the jth cation species with strong and weak force adsorption cannot be evaluated.

(3) The equilibrium constant, surface potential and surface potential energy of the cation species at the exchange equilibrium.

As the reference state to be employed is ã_i = a_i0 = 1, the equilibrium constant could be expressed as

where K_eq is the exchange equilibrium constant.

Considering [small gamma, Greek, tilde]_i = a_i0/f̃_i and [small gamma, Greek, tilde]_j = a_j0/f̃_j, from eqn (28), there is

Supposing Z_i = 2 and Z_j = 1, from eqn (9), (10), (17) and (22), eqn (29) can be changed to

From eqn (30), we have

andwhere w_i/j = 2β_iFϕ₀ − β_jFϕ₀ is the difference of potential energy at the first layer of DL between the i and jth cation species.

The surface charge density can be calculated from the following equation:²⁰

where σ₀ (C dm⁻²) is the surface charge density.

Thus, obviously the specific surface area can also be estimated,

and the electrostatic field strength at the surface can be estimated,where S (dm² g⁻¹) is the specific surface area, E₀(V dm⁻¹) is the electrostatic field strength and ε is the dielectric constant (8.9 × 10⁻¹⁰ C² J⁻¹ dm⁻¹ for water).

Eqn (31)–(35) indicate that, if the β_i and β_j values are known in advance, the surface potential, difference of potential energy between the two cation species at the surface, surface charge density, specific surface area of particles and electrostatic field strength at the surface can be calculated from the p and q values.

(4) The Vanselow selectivity coefficient.

Here we just discuss the exchange equilibrium between the adsorption phase (with both strong and weak force adsorption) and solution phase. The Vanselow selectivity coefficient or the conditional equilibrium constant could be written as

where a_i0 and a_j0 are the activities of the i and jth cation species in solution phase respectively; X_i and X_j are the mole fraction of the i and jth cation species in the exchanger phase respectively, and X_i = N_i(∞)/[N_i(∞) + N_j(∞)], X_j = N_j(∞)/[N_i(∞) + N_j(∞)].

According to eqn (9) and (10), we have

and

Introducing the calculated X_i and X_j values from eqn (35) and (36) into eqn (34), the Vanselow selectivity coefficient can be obtained, based on the kinetic data of cation exchange.

Supposing Z_i = 2 and Z_j = 1, eqn (36) changed to

According to eqn (29) and (36), as Z_i = 2 and Z_j = 1, also, we have

Considering [small gamma, Greek, tilde]_j = a_j0/f̃_j and [small gamma, Greek, tilde]_i = a_i0/f̃_i, eqn (36) can be rewritten as

Introducing eqn (20) and (21) into eqn (41), we get

Eqn (42) indicates that, β_i, β_j and ϕ₀ determine the relative preference between the two cation species for the particle surface. As 2β_i − β_j = 0 or ϕ₀ = 0, the particle surface bears the same preference for the two cation species, which means K_V = a_i0 + a_j0; as 2β_i − β_j > 0, K_V > a_i0 + a_j0, it shows a preference of the ith cation species for the particle surface as compared to the jth cation species; as 2β_i − β_j > 1, quantum fluctuation forces effect will increase the preference of the ith cation species for a particle surface as compared to the jth cation species.

3 Material and methods

3.1 Material and sample preparation

An engineering nanomaterial TiO₂ (20–30 nm in diameter), a widely applied natural nano-colloidal material montmorillonite in chemical engineering, and a purple soil were used as experimental materials. The specific surface area of materials was determined in advance by the ion-exchange equilibrium method, for which the inner surface of swelling materials can be measured.²¹ The specific surface areas of TiO₂, montmorillonite and soil were 80, 725 and 47.9 m² g⁻¹, respectively. The clay mineral composition of the purple soil was mainly illite (43%) and kaolinite (45%).²² All the materials were prepared as K⁺-saturated in experiments.

3.2 Kinetic studies of adsorption

The kinetics of Mg²⁺ (Mg(NO₃)₂) adsorption in the K⁺-saturated purple soil was studied by the miscible displacement technique under a steady flow condition. The experiment was carried out as follows: approximately 0.5 grams of a K-saturated sample was layered on the exchange column (exact weight of the sample was weighted after the experiment finished). The thickness of the sample layer should be very thin in the sample chamber for the purpose of reducing the effect of the longitudinal concentration gradient in the experiment, because our theoretical analysis was based on the one-dimensional case. In our experiment, a thickness of approximately 0.2–0.3 mm for the sample layer was adopted. The sample area was about 15 cm². The experimental temperature was 298 K. The applied concentration of Mg²⁺ in the flowing liquid was 10⁻⁴ mol L⁻¹. For the experiments of soil, the concentration of the background KNO₃ solution in the flowing liquid were 0, 10⁻⁴, 10⁻³ and 10⁻² mol L⁻¹ respectively, while for experiments of TiO₂ and montomorillonite, the concentration of the background KNO₃ solution in the flowing liquid were 0 and 10⁻⁴ mol L⁻¹. The flow velocity of the flowing liquid was 1.0 mL min⁻¹. Effluent was collected at 10 minute intervals, except for TiO₂ in 10⁻⁴ mol L⁻¹ KNO₃, which was collected at 5 minute intervals. The quantity of the ion that flowed through the interface x = l into the fixed liquid film at x = 0 → l within the time interval t was calculated based on the difference of Mg²⁺ concentration in the influent and effluent.

4 Results and discussion

Here we adopt the soil sample to calibrate the β_Mg and β_K values.

The experimental results of [N_i(t_m+1) − N_i(t_m)]/(t_m+1 − t_m) vs. N_i(t_m+1/2) for Mg²⁺ adsorption in the K⁺-saturated soil samples are shown in Fig. 2. Fig. 2 shows that the slope and intercept values of the second straight line can be obtained accurately.

Fig. 2 Experimental results of the relationship between y = [N_i(t_m+1) − N_i(t_m)]/(t_m+1 − t_m) and x = N_i(t_m+1/2) of Mg²⁺ adsorption for the soil sample (a): 10⁻⁴ mol L⁻¹ Mg²⁺; (b): 10⁻⁴ mol L⁻¹ K⁺ + 10⁻⁴ mol L⁻¹ Mg²⁺; (c): 10⁻³ mol L⁻¹ K⁺ + 10⁻⁴ mol L⁻¹ Mg²⁺; (d): 10⁻² mol L⁻¹ K⁺ + 10⁻⁴ mol L⁻¹ Mg²⁺.

Eqn (31)–(35) indicate that, if the β_Mg and β_K values are known in advance, some important surface properties such as ϕ₀, w_Mg/K, σ₀, E₀ and S can be calculated from the p and q values. Unfortunately, we do not know the β_Mg and β_K values currently. In this experiment, we know the specific surface area of the soil material, and the CEC value can be estimated from the p and q values for the case in Fig. 2(a):

The results give CEC = −2 × 31.79/(−0.4355) = 146 mmol_c kg⁻¹.

The values of slope and intercept depend on the experimental data of first-order dynamics; thus, the uncertainties of them are associated with the critical point of zero and first-order dynamics. Because the obvious inflexion point between zero- and first-order dynamics exists, the error is controlled easily.

By considering β_Mg + β_K = 2,²³ the combined solution of eqn (31), (33) and (34) gives the ϕ₀, σ₀, β_Mg and β_K values; then, introducing those values into eqn (32) and (35), w_Mg/K and E₀ can be calculated respectively. The results are shown in Table 1.

Table 1 β_Mg, β_K values and surface properties of the particles at equilibrium

Concentration (activity)	p	q	βMg	βK	ϕ0	E0	wMg/K
fK0 = 0.1 (aK0 = 0.0977)	19.4	−0.322	0.742	1.258	−0.214	−4.23 × 10^7	−4.664
fMg0 = 0.1 (aMg0 = 0.0912)
fK0 = 1 (aK0 = 0.961)	16.6	−0.552	0.787	1.213	−0.196	−4.23 × 10^7	−6.837
fMg0 = 0.1 (aMg0 = 0.0849)
fK0 = 10 (aK0 = 8.976)	15.9	−2.01	0.879	1.121	−0.172	−4.23 × 10^7	−10.56
fMg0 = 0.1 (aMg0 = 0.0649)

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From the data in Table 1, we can see that.

(1) The surface potential decreases with the increase of ionic strength in bulk solution.

(2) β_K was higher than 1 while β_Mg was lower than 1. This result agreed with the theoretical prediction by taking into account the quantum fluctuation force effects of the cation species in the cation exchange as discussed in the introduction section.

(3) With the increase of ionic strength in bulk solution, β_K decreases while β_Mg increases, and theoretically if the ionic strength is strong enough, β_K = β_Mg = 1. Fig. 3 shows the relationship of β vs. I^0.5, from which we estimate: as I^0.5 ≥ 0.5, β_K = β_Mg = 1. Those can be explained that, higher ionic strength could lead to a weaker electric field around particles, and a weaker electric field will result in a weaker attractive force of the end with positive charges, and will result in a smaller difference in the electron cloud configuration change for the two cation species. The w_Mg/K values in Table 1 also show that the polarizability effects of the two cation species strongly decreased the preference of Mg²⁺ over K⁺ in the exchange. Neglecting the difference in the polarizability of cation means β_K = β_Mg = 1; thus, as ϕ₀ = −0.2139 V, −0.1963 V and −0.1718 V, w_Mg/K will be −20.64 kJ mol⁻¹, −18.94 kJ mol⁻¹ and −16.58 kJ mol⁻¹, respectively, instead of −4.664 kJ mol⁻¹, −6.837 kJ mol⁻¹ and −10.56 kJ mol⁻¹, respectively.

Fig. 3 Empirical relationship between β and I^0.5 for K⁺ and Mg²⁺ in the mixture solution of KNO₃ and Mg(NO₃)₂.

The verification of our new approach in surface properties determination of particles.

For demonstrating the applicability of our new approach in the determination of surface properties, we will use the obtained β_K and β_Mg values from the soil sample to estimate the surface properties of TiO₂ and montomorillonite independently. The experimental results of [N_i(t_m+1) − N_i(t_m)]/(t_m+1 − t_m) vs. N_i(t_m+1/2) for Mg²⁺ adsorption in the K⁺-saturated TiO₂ and montomorillonite are shown in Fig. 4 and 5, respectively.

Fig. 4 Experimental results of the relationship between y = [N_i(t_m+1) − N_i(t_m)]/(t_m+1 − t_m) and x = N_i(t_m+1/2) of Mg²⁺ adsorption (a): 10⁻⁴ mol L⁻¹ Mg²⁺ for montomorillonite; (b): 10⁻⁴ mol L⁻¹ K⁺ + 10⁻⁴ mol L⁻¹ Mg²⁺ for montomorillonite.

Fig. 5 Experimental results of the relationship between y = [N_i(t_m+1) − N_i(t_m)]/(t_m+1 − t_m) and x = N_i(t_m+1/2) of Mg²⁺ adsorption (a): 10⁻⁴ mol L⁻¹ Mg²⁺ for TiO₂; (b): 10⁻⁴ mol L⁻¹ K⁺ + 10⁻⁴ mol l⁻¹ Mg²⁺ for TiO₂.

First, from Fig. 4(a), we got p = 39.543 mmol min⁻¹ g⁻¹, q = −0.0923 min⁻¹; thus, the CEC value of montmorillonite was 2 × (39.543/0.0923) = 856.8 mmol_c kg⁻¹ based on eqn (9). Considering β_K = 1.258 and β_Mg = 0.742 for the bulk solution of “10⁻⁴ mol L⁻¹ KNO₃ + 10⁻⁴ mol L⁻¹ Mg(NO₃)₂” (Table 1), the surface potential, surface charge density and specific surface area can be calculated from eqn (31), (33) and 34, respectively, based on the p and q values in Fig. 4(b), and the results are ϕ₀ = −0.1787 V, σ₀ = −0.001090 C dm⁻² and S_mont = 756 m² g⁻¹, respectively. The specific surface area obtained by this method agreed with that obtained by another independent method, which was S_mont = 725 m² g⁻¹ for the same montmorillonite.

For the TiO₂ material, from Fig. 5(a), p = 31.484 mmol min⁻¹ g⁻¹ and q = −1.8154 min⁻¹; the CEC value of TiO₂ was 2 × (31.484/1.8154) = 34.68 mmol_c kg⁻¹ calculated from eqn (9). Taking the above method, the surface potential and surface charge density of the TiO₂ material in solution of “10⁻⁴ mol L⁻¹ KNO₃ + 10⁻⁴ mol L⁻¹ Mg(NO₃)₂” were ϕ₀ = −0.1478 V and σ₀ = −0.0004609 C dm⁻²; thus, the specific surface area of the TiO₂ material was S_TiO2 = 72.6 m² g⁻¹. The specific surface area of TiO₂ obtained by this method agreed with that obtained by another independent method, which was S_TiO2 ≈ 80 m² g⁻¹ for the same TiO₂.

From eqn (31), (33) and 34, we can see that, only a correct ϕ₀ value could lead to a correct σ₀ value, finally leading to the S value. Obviously, those comparisons of the specific surface area values between our new method and other independent methods can give a reasonable verification for our new approach for surface property determination.

However, for Mg²⁺/K⁺ exchange at low electrolyte concentrations, e.g. 10⁻⁴ mol L⁻¹, Z_Mgβ_Mg − Z_Kβ_K = 0.226 ≪ 1, from eqn (31), we can see that the surface potential value will be sensitive to the experimental error. Thus, in actual applications for surface property determination, we do not recommend to use Mg²⁺/K⁺ exchange but recommend to use Ca²⁺/Na⁺ exchange, since Z_Caβ_Ca − Z_Naβ_Na ≈ 1.699 ≫ 1.19.

Although only Mg/K exchange was selected to determine the surface properties in this study, the other ion pairs, e.g. Ca/Na, Ca/K, Mg/Na and Na/K could also be used theoretically.

Estimation of equilibrium parameters of cation exchange based on kinetic data.

Here we just show the equilibrium parameters of Mg²⁺/K⁺ exchange in the soil. Based on the obtained p and q values, from eqn (9), (10) and (39) (or eqn (42)), the equilibrium parameters of Mg²⁺/K⁺ exchange in the K⁺-saturated purple soil can be obtained, and they are shown in Table 2. From the data in Table 2, we can see (1) the activity coefficients of adsorbed cations increased with the increase of ionic strength in bulk solution, because the increase of ionic strength decreased the potential in the diffuse layer; and (2) K_V/K^*_V ≪ 1 implies the polarizability of the two cation species strongly decreased the preference of Mg²⁺ over K⁺ in Mg²⁺/K⁺ exchange, especially as the surface potential was high, because a higher surface potential means a stronger electric field, which leads a stronger polarizability configuration change for K⁺ than that for Mg²⁺.

Table 2 Equilibrium parameters in Mg/K exchange^a

Concentration (activity)	p	q	NMg(∞)	NK(∞)	KV	KV/K^*V
a Note: K^*V values were calculated from eqn (42) by taking βMg = βK = 1 (neglecting the polarizability effect of cations).
fK0 = 0.1 (aK0 = 0.0977)	19.41	−0.3215	60.4	25.0	0.8508	0.002213
fMg0 = 0.1 (aMg0 = 0.0912)
fK0 = 1 (aK0 = 0.961)	16.62	−0.5517	30.1	85.6	5.151	0.02327
fMg0 = 0.1 (aMg0 = 0.0849)
fK0 = 10 (aK0 = 8.976)	15.91	−2.009	7.92	130	80.08	0.2609
fMg0 = 0.1 (aMg0 = 0.0649)

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5 Conclusions

Taking into account the polarization of two cation species involved in the exchange by introducing a modification coefficient for describing the relatively effective charges of the cation species, new approaches for estimating the exchange equilibrium parameters and surface properties of particles have been developed. Our theory indicated that, the equilibrium parameters and surface properties of particles can be easily calculated from the slope and intercept values of the first-order rate equation of cation adsorption. The surface properties of particles to be estimated by the suggested approach include (1) surface potential, (2) specific surface area of particles, (3) charge density on particles surface and (4) electrostatic field strength at the surface; the equilibrium parameters include (1) selectivity coefficient, (2) average activity coefficients of cation species in the adsorption phase, (3) adsorption quantity of each cation species at equilibrium with strong and weak force adsorption, (4) difference of potential energy between two cation species involved in the exchange. For the first time, the quantitative relationship between the selectivity coefficient of exchange and surface potential of particles together with the charge number of cation species has been established. The results quantitatively show that the difference of polarization between two cation species strongly influences the cation-exchange equilibrium and surface properties of particles. The reliability of our new approach has been verified.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (41201223, 41271292), Specialized Research Fund for the Doctoral Program of Higher Education(20110182120002), and the Fundamental Scientific Research Professional Expenses of Southwest University (XDJK2011C008).

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