Examination of current adsorption models for Pb(II) and Cu(II) adsorption onto Fe 3 O 4 @Mg 2 Al-NO 3 Layered Double Hydroxide in aqueous solution

Abstract

The adsorption of Pb(II) and Cu(II) onto Fe₃O₄@Mg₂Al-NO₃ Layered Double Hydroxide (LDH) as a function of Fe₃O₄@Mg₂Al-NO₃ LDH concentration was studied. An adsorbent concentration effect (C_s effect), namely adsorption isotherm declines as adsorbent concentration (C_s) increases, was observed. The experimental data were fitted to the adsorption models including the classic Freundlich model, the metastable-equilibrium adsorption theory, the flocculation model, the power function model, and the surface component activity model. The results show that the Freundlich-type metastable-equilibrium adsorption equation, the power function model, and the Freundlich-surface component activity equation can adequately describe the C_s effect observed in the batch adsorption tests as all the correlation coefficients (R²) of the nonlinear plots are higher than 0.96. In other words, their intrinsic parameters simulated from the experimental data are independent of C_s value. It is considered that the Freundlich-surface component activity equation is the best model to describe the C_s effect of the studied adsorption systems by Akaike Information Criterion evaluation criterion.

Keywords

Adsorbent concentration effect Freundlich equation metastable-equilibrium adsorption model power function model surface component activity model

Introduction

In studies of adsorption at the solid–liquid interface, an anomalous phenomenon of “adsorbent concentration effect” (Voice and Weber, 1985) or “solids effect” (O’Connor and Connolly, 1980; Voice et al., 1983) (C_s effect), namely adsorption isotherm declines as adsorbent concentration increases, has been observed since the 1980s. Thermodynamically, the adsorption equilibrium constant (or equilibrium partition coefficient) for a given adsorption reaction under constant temperature, pressure, and medium composition (e.g. pH, ionic strength) should be independent of both adsorbate and adsorbent concentrations (Pan and Liss, 1998). So, the classic adsorption model cannot explain this anomalous phenomenon because it was derived on the assumption that the adsorption equilibrium constant is independent of the adsorbent concentration (C_s).

In fact, there has been controversy about the reasons for the C_s effect over the last 30 years. Some researchers (Cohen-Stuart et al., 1980; Grolimund et al., 1995; Gschwend and Wu, 1985; Nyffeler et al., 1984) have attributed the C_s effect to a variety of experimental artifacts, while other researchers (Chang and Wang, 2002; DiToro et al., 1986; Helmy et al., 2000; Lu et al., 2008; Pan and Liss, 1998; Pan et al., 1999; Utomo and Hunter, 2010; Voice and Weber, 1985; Wu et al., 2006) believed that the C_s effect is a constancy phenomenon based on thermodynamic principles although many artifacts may cause a pseudo C_s effect. Because the substances for which adsorption isotherm apparently declined with increasing C_s include inorganic and organic adsorbates in freshwater and marine sediments, quarts, clays and clays minerals, and digested sewage sludge, it seems unlikely that experimental artifacts could explain widespread agreement of a C_s effect for so diverse a set of adsorbates, adsorbents, and investigators (McKinley and Jenne, 1991). Thus, it can be concluded that the C_s effect is an experimental fact indeed, not an artifact, and there should be a universal reason that causes the C_s effect although it is still not clear now.

In order to explain the reasons for the C_s effect, many models have been proposed, e.g. the solute complexation model (Voice and Weber, 1985), the particle interaction model (DiToro et al., 1986), the metastable-equilibrium adsorption (MEA) theory (Pan and Liss, 1998), the flocculation model (Helmy et al., 2000), the power function (Freundlich-like) model (Chang and Wang, 2002), and the surface component activity (SCA) model (Zhao and Hou, 2012; Zhao et al., 2012, 2013). It needs to fully examine the applicability of the models in describing adsorption equilibriums with the C_s effect. In this study, the C_s effect of the adsorption of Pb(II) and Cu(II) on Fe₃O₄@Mg₂Al-NO₃ LDH was analyzed and the applicability of the current theories was examined, including the MEA theory, the flocculation model, the power function model, and the SCA model. The present work can improve our understanding of the C_s effect adsorption phenomenon as well as the applicability of the current models in describing adsorption equilibriums.

Materials and methods

Materials

All chemicals used in this work were of analytical grade. Ultrapure water was used in all cases.

Magnetic nanoparticles were prepared by dissolving 0.01 mol of FeCl₂·4H₂O and 0.02 mol of FeCl₃·6H₂O in water solution under stirring at 65℃, and 20 wt% of NH₃·H₂O was added dropwise together at a constant pH value of 10–11. The obtained material (Fe₃O₄) was recovered, washed several times with deionized water until the pH was neutral. The obtained Fe₃O₄ was preserved as suspension.

Fe₃O₄@Mg₂Al-NO₃ LDH with an Mg²⁺/Al³⁺ molar ratio of 2:1 was prepared by coprecipitation. An aqueous solution containing 0.1 mol Mg(NO₃)₂·6H₂O and 0.1 mol Al(NO₃)₃·9H₂O was added dropwise to Fe₃O₄ solution with Fe/Mg molar ratio equal to 0.02, under vigorous stirring. During the synthesis, the temperature was maintained at 65℃ and pH at 11–12 by the simultaneous addition of NaOH solution. The reaction mixtures were aged for 45 min in mother solution at room temperature and then filtered, washed with deionized water until the pH was neutral. The filter cakes were further hydrothermally treated at 80℃ for 24 h. The sols were dried, triturated, and sieved to collect the particles of <74 µm in diameter.

Magnetic Fe₃O₄@Mg₂Al-NO₃ LDH was characterized by X-ray diffraction (XRD) and transmission electron microscopy (TEM) (Figures 1 and 2). XRD was performed on a D/maxrA model diffractometer (Bruker, Germany) using Cu Kα radiation (λ = 1.5418 Å) at 40 kV and 40 mA. The morphologies were observed by TEM (JEM-2100, JEOL, Japan). The magnetization curves were obtained at ambient temperature by vibrating sample magnetometry (JDM-13, Jilin University, China) in a magnetic field range of 0–8000 Oe (Figure 3). The characteristic text results show that the Fe₃O₄@Mg₂Al-NO₃ LDH complex still has the characteristic diffraction peaks of the LDH, and there is no obvious diffraction peak of Fe₃O₄ phase. The complex particles are approximately spherical. LDH clad on the surface of Fe₃O₄. These indicate that the magnetic Fe₃O₄ particles are highly dispersed into the LDH phase and Fe₃O₄@Mg₂Al-NO₃ LDH inherits the magnetic property from the Fe₃O₄.

Figure 1.

XRD patterns of Fe₃O₄@Mg₂Al-NO₃ LDH sample. LDH: Layered Double Hydroxide; XRD: X-ray diffraction.

Figure 2.

TEM images of Fe₃O₄@Mg₂Al-NO₃ LDH. LDH: Layered Double Hydroxide; TEM: transmission electron microscopy.

Figure 3.

Magnetization curves of Fe₃O₄@Mg₂Al-NO₃-LDH and the inset is the magnetization curve of Fe₃O₄. LDH: Layered Double Hydroxide.

Adsorption experiments

Adsorption tests of Pb(II) and Cu(II) on Fe₃O₄@Mg₂Al-NO₃ LDH sample were carried out by batch equilibration technique (Ding et al., 2010; Hermosin et al., 1995; Ma et al., 2010). Solutions with various concentrations (0–800 mg/l) of Pb(II) and Cu(II) were prepared in 0.01 mol/l of NaNO₃ with Pb(NO₃)₂ and Cu(NO₃)₂·3H₂O, and the pH values of the solutions were adjusted to 5.0 with 0.1 mol/l HNO₃ and NaOH solutions. Known masses (0.025–0.20 g) of Fe₃O₄@Mg₂Al-NO₃ LDH samples were mixed with 25 ml Pb(II) and Cu(II) solutions of various initial concentrations in polyethylene centrifugal tubes. The centrifugal tubes were put into a thermostatic water bath shaker at 25±0.2℃ for 24 h. Then the suspensions were centrifuged at a speed of 4000 r/min for 5 min. The Pb(II) and Cu(II) equilibrium concentrations in the supernatants were determined by flame atomic absorption spectrometry (TAS-990, Beijing Purkinje General Instrument Co., Ltd). The equilibrium adsorption amounts were calculated using the following equation

Γ_{e} = \frac{(C_{0} - C_{e}) V}{m}

(1)

where C₀ (mg/l) and C_e (mg/l) are the initial and the remaining (equilibrium) concentration, respectively; Γ_e (mg/g) is the equilibrium adsorption amount; V (ml) is the volume of the suspension; and m (g) is the mass of the Fe₃O₄@Mg₂Al-NO₃ LDH sample.

Results and discussion

Figure 4 shows the adsorption isotherms of Pb(II) and Cu(II) on Fe₃O₄@Mg₂Al-NO₃ LDH at different Fe₃O₄@Mg₂Al-NO₃ LDH concentrations. As can be seen, the adsorption isotherms are L-type, and they decline significantly as the adsorbent concentrations increase. The L-type isotherms can be commonly described using the Langmuir and Freundlich isotherms. This phenomenon accords with the law which is described by a C_s effect. The adsorption equilibrium constants should be dependent on the C_s. The applicability of current theories to predict the C_s effect phenomenon was examined as following.

Figure 4.

Adsorption isotherms of Pb(II) (a) and Cu(II) (b) on Fe₃O₄@Mg₂Al-NO₃ LDH at different adsorbent concentrations. LDH: Layered Double Hydroxide.

Classical adsorption equations

The classical Langmuir equation was introduced by Irwin Langmuir in 1916. It was originally formulated for describing the adsorption of gases onto a solid phase. It is now also used to predict the adsorption of the liquid–solid system. The equation is based on following assumptions: (1) the surface of the adsorbent is uniform, that is, all the adsorption sites are equivalent; (2) adsorbed molecules do not interact; (3) all adsorption occurs through the same mechanism; and (4) at the maximum adsorption, only a monolayer is formed, i.e. molecules of adsorbate do not deposit on other, already adsorbed, molecules of adsorbate, only on the free surface of the adsorbent (Langmuir, 1918; Iraolagoitia and Martini, 2010). The Langmuir equation is stated as

Γ_{e} = \frac{Γ_{m} K_{L} C_{e}}{1 + K_{L} C_{e}}

(2)

where Γ_m is maximum adsorption amount and K_L is Langmuir equilibrium constant.

Equation (2) can be expressed by the following linear form

\frac{C_{e}}{Γ_{e}} = \frac{C_{e}}{Γ_{m}} + \frac{1}{K_{L} Γ_{m}}

(3)

As we know, Langmuir model was developed on the basis of the thermodynamic equilibrium theory while Freundlich model was originally an empirical equation. However, it was found that Freundlich equation could be thermodynamically derived based on assuming that the adsorption sites of adsorbent are of different energies and the adsorption of sorbate on the sites having same energy level obeys Langmuir equation (Sheindorf et al., 1981).

Classical Freundlich equation is represented as (Santhi et al., 2010; Sheindorf et al., 1981)

Γ_{e} = K_{F} C_{E}^{n_{F}}

(4)

where Γ_e is equilibrium adsorption amount, C_e is equilibrium concentration, and K_F and n_F are Freundlich constants.

Equation (4) can be expressed by the following linear form

\lg Γ_{e} = \lg K_{F} + n \lg C_{e}

(5)

The adsorption data of Pb(II) and Cu(II) on Fe₃O₄@Mg₂Al-NO₃ LDH (Figure 4) were fitted with the Langmuir equation and the Freundlich equation (see Figures 5 and 6). It was found that Freundlich equation can adequately describe the adsorption equilibriums of Pb(II) and Cu(II) on Fe₃O₄@Mg₂Al-NO₃ LDH for a given C_s value better than Langmuir equation, and all the correlation coefficients (R²) are greater than 0.95. The fitted Freundlich parameters, K_F and n_F, for various C_s values are presented in Table 1. It can be seen from Table 1 that both two Freundlich parameters vary with C_s value and similar results were reported in the literature (Sheindorf et al., 1981). This dependence of the Freundlich parameters on C_s showed that the classical Freundlich model cannot predict the C_s effect. That is to say, using the Freundlich parameters obtained with given C_s values to predict the adsorption behavior of adsorbate at other C_s values will be inaccurate.

Figure 5.

Linear correlation plots for the classical Langmuir equation: (a) Pb–Fe₃O₄@Mg₂Al-NO₃ LDH system and (b) Cu–Fe₃O₄@Mg₂Al-NO₃ LDH system. LDH: Layered Double Hydroxide.

Figure 6.

Linear correlation plots for the classical Freundlich equation: (a) Pb–Fe₃O₄@Mg₂Al-NO₃ LDH system and (b) Cu–Fe₃O₄@Mg₂Al-NO₃ LDH system. LDH: Layered Double Hydroxide.

Table 1.

Parameters of the classical Freundlich equation for Pb(II) and Cu(II) adsorption on Fe₃O₄@Mg₂Al-NO₃ LDH at different adsorbent concentrations.

C_s (g/l)	Pb(II)			Cu(II)
C_s (g/l)	n_F	K_F	R²	n_F	K_F	R²
1.0	0.676(0.017)	5.501(0.029)	0.995	0.402(0.024)	1.080(0.055)	0.975
2.0	0.619(0.023)	4.635(0.038)	0.989	0.466(0.033)	0.805(0.072)	0.966
4.0	0.591(0.019)	3.555(0.029)	0.992	0.541(0.043)	0.511(0.091)	0.957
8.0	0.613(0.030)	2.245(0.042)	0.981	0.527(0.023)	0.453(0.044)	0.986

LDH: Layered Double Hydroxide.

The data in parentheses are the standard deviations.

The fact that Freundlich equation can adequately describe the adsorption behavior for a given C_s value indicates that Freundlich equation is available for the real adsorption systems studied. Therefore, the deviation of the prediction of Freundlich equation from the experimental data for different C_s values should be because the effect of C_s on adsorption was not accounted for in the derivation process of the classical Freundlich equation.

Examining using MEA theory

In order to explain the fundamental mechanism of the C_s effect, Pan and Liss (1998) developed an “MEA theory.” It suggests that, in reality, the experimentally measured “equilibrium” state of solid–liquid adsorption system is generally a metastable-equilibrium state. So, the measured “equilibrium” adsorption density (Γ_e) is no longer a state variable. In this case, the same number of adsorbed molecules on the solid surface could contain different amounts of energy depending on the different metastable-equilibrium states of the adsorbed molecules. A definite value of Γ_e could therefore have different values of chemical potential (μ) and represents different thermodynamic states of the adsorbate on the solid surface. Based on the MEA theory, Pan and Liss (1998) obtained a Langmuir-like and a Freundlich-like MEA isotherm.

The Freundlich-type metastable-equilibrium isotherm equation (Pan and Liss, 1998) is

Γ_{e} = k_{sp} \cdot C_{s}^{- α_{P}} \cdot C_{e}^{n_{P}}

(6)

or in a linear form

\lg Γ_{e} = n_{P} \lg C_{e} + \lg k_{sp} - α_{P} \lg C_{s}

(7)

where k_sp is an equilibrium adsorption constant, and n_P is an empirical parameter. For a given adsorption system, the k_sp, α_P, and n_P are independent of the C_s condition.

The plots of lg Γ_e versus lg C_s at various C_e values were examined for the two adsorption systems studied, respectively. It was found that all plots showed fairly straight lines (Figure 7). The α_P values obtained from the slopes of the plots change in a small amplitude with C_s. Using the average values of the α_P values obtained at various C_e values (see Table 2), the normalized nonlinear plots of $(Γ_{e} \cdot C_{s}^{α_{P}})$ versus C_e for the two systems are shown in Figure 8. As can be seen, the nonlinear plots for various C_s values show a unique curve for each adsorption system, and their correlation coefficients (R²) are higher than 0.96. These results indicate that Freundlich-type MEA equation can adequately describe the C_s effect observed in the two systems. The k_sp and n_P values are listed in Table 2.

Figure 7.

Plots of lg Γ_e versus lg C_s at different equilibrium concentrations: (a) Pb–Fe₃O₄@Mg₂Al-NO₃ LDH system and (b) Cu–Fe₃O₄@Mg₂Al-NO₃ LDH system. LDH: Layered Double Hydroxide.

Table 2.

Parameters of Freundlich-like equations for two adsorption systems.

Adsorbate	Freundlich-like MEA			Power function		Freundlich-SCA
Adsorbate	k_sp	α_P	n_P	K_C	n_C	α	γ	K_S	n_S
Pb(II)	6.509 (0.020)	0.568 (0.027)	0.637 (0.013)	7.272 (0.035)	0.619 (0.012)	0.722 (0.038)	0.256 (0.025)	7.093 (0.199)	0.620 (0.013)
Cu(II)	8.102 (0.037)	0.443 (0.036)	0.483 (0.017)	9.034 (0.056)	0.464 (0.014)	0.446 (0.015)	0.610 (0.032)	1.890 (0.089)	0.443 (0.015)

MEA: metastable-equilibrium adsorption; SCA: surface component activity.

The data in parentheses are the standard deviations.

Figure 8.

The Freundlich-type MEA isotherms: (a) Pb–Fe₃O₄@Mg₂Al-NO₃ LDH system and (b) Cu–Fe₃O₄@Mg₂Al-NO₃ LDH system. LDH: Layered Double Hydroxide; MEA: metastable-equilibrium adsorption.

Examining using the flocculation model

Helmy et al. (2000) proposed a “flocculation model.” It hypothesizes that particles associate through flocculation, thus the surface area available for adsorption decreases. Based on the classic theory of flocculation, a straight-line relation was obtained relating the square root of the adsorption maximum to adsorbent concentration as follows

X_{m}^{1 / 2} = a_{0} + a_{1} C_{s}

(8)

where X_m = Γ_mċC_s is the maximum adsorption concentration and a₀ and a₁ are constants. Helmy et al. (2000) also proposed that the surface area available for adsorption is a function of particle association, which in turn is a function of particle concentration.

Figure 9 shows the plots of $X_{m}^{1 / 2}$ versus C_s for the adsorptions of Pb(II) and Cu(II) on Fe₃O₄@Mg₂Al-NO₃ LDH. The result for Cu(II) is a good linear (R²=0.949), while that for Pb(II) is not quite so linear (R²=0.747). These results indicate that the flocculation model cannot adequately account the C_s effect observed in some experiments.

Figure 9.

Relationships between $X_{m}^{1 / 2}$ and C_s of (a) Pb–Fe₃O₄@Mg₂Al-NO₃ LDH system and (b) Cu–Fe₃O₄@Mg₂Al-NO₃ LDH system. LDH: Layered Double Hydroxide.

Examining using the power function model

A power function (Freundlich-like) model was proposed by Chang and Wang (2002) using batch experimental data with dimensional analysis method. It assumed that the total sorbed amount of sorbate is a function of the equilibrium concentration of sorbate, the volume of sorbate solution, and the adsorbent amount in the system. A relationship among Γ_e, C_e, and C_s was derived as follows

Γ_{e} = K_{C} (\frac{C_{e}}{C_{s}})^{n_{C}}

(9)

where K_C and n_C are constants. Equation (9) may be expressed in a linear form as

\ln Γ_{e} = \ln K_{C} + n_{C} \ln (C_{e} / C_{s})

(10)

If power function model is applicable for a given system, the plot of ln(Γ_e) versus ln(C_e/C_s) will give a unique straight line for various C_s values.

The experimental data of Pb(II)–Fe₃O₄@Mg₂Al-NO₃ LDH and Cu(II)–Fe₃O₄@Mg₂Al-NO₃ LDH systems were fitted by equation (9), and the results are shown in Figure 10. As shown in figures, the nonlinear plots for various C_s values show a unique curve for each adsorption system, and their correlation coefficients (R²) are higher than 0.97. These results indicate that the power function model can adequately describe the C_s effect observed in the two systems. The K_C and n_C values are listed in Table 2.

Figure 10.

Fitting curves of the power function model: (a) Pb–Fe₃O₄@Mg₂Al-NO₃ LDH system and (b) Cu-Fe₃O₄@Mg₂Al-NO₃ LDH system. LDH: Layered Double Hydroxide.

Examining using the Freundlich-SCA equation

Considering the deviation of a real adsorption system from an ideal one, we proposed an alternative model, SCA model (Zhao and Hou, 2012; Zhao et al., 2012, 2013). It suggests that (1) the surface of the adsorbent is uniform, that is, all the adsorption sites are equivalent; (2) all adsorption occurs through the same mechanism; (3) at the maximum adsorption, only a monolayer is formed; (4) the molecular size of solute and solvent are similar; and (5) surface component (adsorption site or adsorbed solute) activity coefficient is not equal to unity because of the deviation of a real adsorption system from an ideal one. A C_s-dependent Freundlich equation (or Freundlich-SCA equation) can be derived as following

Γ_{e} = K_{S} \cdot f_{H_{2} O}^{s} \cdot C_{e}^{n_{S}}

(11)

The linear form of equation (11) is

\lg (\frac{Γ_{e}}{f_{H_{2} O}^{s}}) = n_{S} \lg C_{e} + \lg K_{S}

(12)

The coefficients, n_S and K_S, of a given system are independent of adsorbent concentration.

The C_s-dependent function of $f_{H_{2} O}^{s}$ is an exponential form

f_{H_{2} O}^{s} = \exp (- γ C_{s}^{α})

(13)

where γ and α are empirical constants. Then

K_{F} = K_{S} \exp (- γ C_{s}^{α})

(14)

or in a linear form

\ln K_{F} = \ln K_{S} - γ C_{s}^{α}

(15)

Thus, the C_s-dependent function of $f_{H_{2} O}^{s}$ can be estimated from the relationship between the experimental measured K_F values and C_s values.

The changes of the K_F with C_s for the four adsorption systems were fitted to equation (14) using a nonlinear least-squares method (Excel Solver) (Figure 11), and the best-fit values of the empirical constants, γ and α, are listed in Table 2.

Figure 11.

Relationships between K_F and C_s in two systems: (a) Pb–Fe₃O₄@Mg₂Al-NO₃ LDH system and (b) Cu–Fe₃O₄@Mg₂Al-NO₃ LDH system. LDH: Layered Double Hydroxide.

Using the simulated γ and α value (Table 2), the normalized nonlinear plots of $(Γ_{e} / f_{H_{2} O}^{s})$ versus C_e for the studied systems were obtained (Figure 12). A unique curve independent of C_s was obtained from experimental data for each given adsorption system. The K_S and n_S values are listed in Table 2. All the correlation coefficients (R²) of the nonlinear plots are higher than 0.96, indicating that the Freundlich-SCA equation can adequately describe the C_s effect observed in the two systems.

Figure 12.

The Freundlich-SCA isotherms for (a) Pb–Fe₃O₄@Mg₂Al-NO₃ LDH system and (b) Cu–Fe₃O₄@Mg₂Al-NO₃ LDH system. LDH: Layered Double Hydroxide; SCA: surface component activity.

In addition, the Akaike Information Criterion (AIC) evaluation criterion was applied to select the suitable evaluation model. Generally, it is considered that the fitting degree is higher as the AIC value smaller and the value of the R² closer to 1. The parameters of the three models (including Freundlich-type MEA equation, power function model, and Freundlich-SCA equation) are solved by the nonlinear regression program using SPSS statistical software. The results of the evaluation indicator are listed in Table 3. The AIC value obtained by Freundlich-SCA equation is smaller than other two models. Therefore, it is considered that the Freundlich-SCA equation is the best model to describe the C_s effect of Pb(II)–Fe₃O₄@Mg₂Al-NO₃ LDH and Cu(II)–Fe₃O₄@Mg₂Al-NO₃ LDH adsorption systems.

Table 3.

Evaluation values of the models.

Adsorbate	Freundlich-like MEA		Power function		Freundlich-SCA
Adsorbate	AIC	R ²	AIC	R ²	AIC	R ²
Pb(II)	98.3	0.986	101.1	0.987	97.2	0.981
Cu(II)	151.9	0.962	148.4	0.973	131.6	0.985

AIC: Akaike Information Criterion; MEA: metastable-equilibrium adsorption; SCA: surface component activity.

Conclusions

The adsorption of Pb(II) and Cu(II) onto Fe₃O₄@Mg₂Al-NO₃ LDH in the studied conditions is subject to the C_s effect in which the adsorption isotherm declines as the adsorbent concentration increases. The above fitting results of the adsorption data of the two systems with Freundlich-like equations show that Freundlich-type MEA equation, power function model, and Freundlich-SCA equation can be used to describe the C_s effect observed in the batch adsorption tests. In this case, the values of parameters obtained at given C_s values can be used to predict the adsorption behavior for any C_s values.

The fitting process of the power function model is simplest; however, the model cannot interpret any particular mechanism for the C_s effect as what Chang and Wang (2002) claimed, and it should be regarded as a curve-fitting model or a mathematical tool for describing experimental data. The MEA theory and the SCA model provided the fundamental reason for the C_s effect. The MEA theory implies that the changes in C_s can affect the MEA state, which in turn can fundamentally affect the adsorption isotherm. However, seemingly it cannot explain the C_s effect observed in desorption experiments because in this case the adsorption state of sorbate should be independent of solid concentration utilized in the experimental systems. The SCA model supposes that it is the interaction between the adsorbent particles that induces the deviation of a real adsorption system from an ideal one, just as the deviation of a real solution from an ideal one. With increasing the C_s value, the interaction between the adsorbent particles strengthens, resulting in the decrease of the effective adsorption site density, therefore, the adsorption isotherms decline.

It is considered that the Freundlich-SCA equation is the best model to describe the C_s effect of Pb(II)–Fe₃O₄@Mg₂Al-NO₃ LDH and Cu(II)–Fe₃O₄@Mg₂Al-NO₃ LDH adsorption systems by AIC evaluation criterion. This work improves our understanding of the C_s effect adsorption phenomenon and confirms the applicability of the SCA model in describing adsorption equilibriums with the C_s effect.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Shandong Provincial National Natural Science Foundation of China (No. ZR2015YL014), the Youth Science Funds of Shandong Academy of Sciences (No. 2014QN001), the National Natural Science Foundation of China (No. 21603124), and the Major Science and Technology Program for Water Pollution Control and Treatment (No. 2015ZX07203007 and 2015ZX07203005).

References

Chang

Wang

(2002) Assessment of sorbent/water ratio effect on adsorption using dimensional analysis and batch experiments. Chemosphere 48: 419–426.

Cohen-Stuart

Scheutjens

Fleer

(1980) Polydispersity effects and the interpretation of polymer adsorption isotherms. Journal of Polymer Science: Polymer Physics Edition 18: 559–573.

Ding

Huang

Yang

et al. (2010) Equilibrium and kinetics of adsorption of Ca(II) ions onto KCTS and HKCTS. Journal of Central South University of Technology 17: 277–284.

DiToro

Mahony

Kirchgraber

et al. (1986) Effects of nonreversibility, particle concentration, and ionic strength on heavy metal sorption. Environmental Science and Technology 20: 55–61.

Grolimund

Borkovec

Federer

et al. (1995) Measurement of sorption isotherms with flow-through reactors. Environmental Science and Technology 29: 2317–2321.

Gschwend

(1985) On the constancy of sediment-water partition coefficients of hydrophobic organic pollutants. Environmental Science and Technology 19: 90–96.

Helmy

Ferreiro

Bussetti

(2000) Effect of particle association on 2,2′-bipyridyl adsorption onto kaolinite. Journal of Colloid and Interface Science 225: 398–402.

Hermosin

Pavlovic

Ulibarri

et al. (1995) Hydrotalcite as sorbent for trinitrophenol sorption capacity and mechanism. Water Research 30: 171–177.

Iraolagoitia

Martini

(2010) Ca²⁺ adsorption to lipid membranes and the effect of cholesterol in their Composition. Colloids and Surfaces B 76: 215–220.

10.

Langmuir

(1918) The adsorption of gases on plane surfaces of glass, mica and platinum. American Chemical Society 40: 1361–1403.

11.

Luan

Zhang

et al. (2008) Study on the adsorption of Cr(VI) onto landfill liners containing granular activated carbon or bentonite activated by acid. Journal of China University of Mining and Technology 18: 125–130.

12.

McKinley

Jenne

(1991) Experimental investigation and review of the “solids concentration” effect in adsorption studies. Environmental Science and Technology 25: 2082–2087.

13.

Jiang

(2010) Adsorption and desorption of Cu(II) and Pb(II) in paddy soils cultivated for various years in the subtropical China. Journal of Environmental Science 22: 689–695.

14.

Nyffeler

Santschi

(1984) A kinetic approach to describe trace-element distribution between particles and solution in natural aquatic systems. Geochimica et Cosmochimica Acta 48: 1513–1522.

15.

O’Connor

Connolly

(1980) The effect of concentration of adsorbing solids on the partition coefficient. Water Research 14: 1517–1523.

16.

Pan

Liss

(1998) Metastable-equilibrium adsorption Theory I. Theoretical. Journal of Colloid Interface Science 201: 71–76.

17.

Pan

Liss

Krom

(1999) Particle concentration effect and adsorption reversibility. Colloids and Surfaces A 151: 127–133.

18.

Santhi

Manonmani

Smitha

(2010) Kinetics and isotherm studies on cationic dyes adsorption onto annona squmosa seed activated carbon. International Journal of Engineering, Science and Technology 2: 287–295.

19.

Sheindorf

Rebhun

Sheintuch

(1981) A freundlich-type multicomponent isotherm. Journal of Colloid and Interface Science 79: 136–142.

20.

Utomo

Hunter

(2010) Particle concentration effect: Adsorption of metal ions on coffee grounds. Bioresource Technology 101: 1482–1486.

21.

Voice

Rice

Weber

(1983) Effect of solids concentration on the sorptive partitioning of hydrophobic pollutants in aquatic systems. Environmental Science and Technology 17: 513–518.

22.

Voice

Weber

(1985) Sorbent concentration effects in liquid/solid partitioning. Journal of Environmental Science and Technology 19: 789–796.

23.

Zhao

et al. (2006) Ion adsorption components in liquid/solid systems. Journal of Environmental Science 18: 1167–1175.

24.

Zhao

Hou

(2012) The effect of sorbent concentration on the partition coefficient of pollutants between aqueous and particulate phases. Colloids and Surfaces A 396: 29–34.

25.

Zhao

Song

et al. (2012) A sorbent concentration-dependent Langmuir isotherm. Acta Physico-Chimica Sinica 28: 2905–2910.

26.

Zhao

Song

et al. (2013) A sorbent concentration-dependent Freundlich isotherm. Colloid and Polymer Science 291: 541–550.