**3. Macroscopic thermodynamic evidences of metastable-equilibrium adsorption**

After Screening study of many adsorption systems, we found that there are obvious *C*<sup>p</sup> effect in many irreversible adsorption systems (e.g., Zn-goethite, Zn-manganite, Zn-anatase, As(V)-anatase), and no *C*p effect in reversible adsorption systems (e.g., Cd-goethite and Znδ-MnO2). Taking the adsorption of Zn and Cd on goethite as a typical example, the detailed interpretation of the existence and disappearance of the *C*p effect using the metastableequilibrium adsorption (MEA) theory are presented below.11, 17

 

(*K*me = 1) , [16] is reduced to the conventional Freundlich equation. Equation [16] indicates

According to reaction rate theory, adsorption speed should increase as particle concentration (i.e., reactant concentration) increases.15, 16 Since the reversibility for a physical adsorption process on a plain solid surface generally declines as the speed of the process increases, the adsorption reversibility could decline as the particle concentration increases. Here, we assume that changes in *C*p can affect the metastable-equilibrium

> *<sup>n</sup> K C me <sup>p</sup>*

Substituting [17] into [15] , we obtain a semi-empirical Langmuir-type *C*p effect isotherm

*n p eq n p eq*

*n sp p eq kC C*

The *C*p effect isotherm equations [18] and [19] predict that, by affecting the metastableequilibrium adsorption state (or the adsorption reversibility ), particle concentration can

After Screening study of many adsorption systems, we found that there are obvious *C*<sup>p</sup> effect in many irreversible adsorption systems (e.g., Zn-goethite, Zn-manganite, Zn-anatase, As(V)-anatase), and no *C*p effect in reversible adsorption systems (e.g., Cd-goethite and Znδ-MnO2). Taking the adsorption of Zn and Cd on goethite as a typical example, the detailed interpretation of the existence and disappearance of the *C*p effect using the metastable-

. For a given adsorption reaction, *k*sp is an equilibrium adsorption constant

*kC C kC C*

 

'

 

1

Substituting [17] into [16] , we obtain a Freundlich-type *C*p effect isotherm equation,

that the adsorption isotherm is shifted to the lower as *K*me decreases.

is a constant and *n* is an empirical parameter, *n* ≥ 0.

adsorption constants which are independent of the *C*eq and *C*p conditions.

fundamentally influence the equilibrium constants or adsorption isotherms.

equilibrium adsorption (MEA) theory are presented below.11, 17

**3. Macroscopic thermodynamic evidences of metastable-equilibrium** 

**2.3 Particle concentration (***C***p) effect isotherm equations** 

where

adsorption state,

where ' ' k b

*sp k* 

and k b

which is independent of the *C*eq and *C*p conditions.

where 

where

**adsorption** 

equation

*K C me eq* 

is a constant under isothermal conditions. Under ideal equilibrium conditions

(16)

(17)

. For a given adsorption reaction, *k'* and *k* are equilibrium

(19)

(18)

Fig. 1. Adsorption (solid lines, closed symbols) and desorption (dotted lines, open symbols) isotherms under different *C*p conditions in Zn–goethite (a) and Cd–goethite (b) systems. (a) *C*p1=0.38 g/L, *C*p2=1.53 g/L, *C*p3=2.3 g/L, pH=6.4, equilibration time for adsorption and desorption are 12 days and 10 days, respectively. a', a comparison of the sizes of hysteresis in Figure 1a, when the first points of the desorption isotherms are translationally moved to the same point. (b) pH=7.1, equilibration time for adsorption and desorption are 20 days and 14 days, respectively.

Fig. 2. Adsorption (a) and desorption (b) kinetic curves under different *C*p conditions in Zn– goethite system. pH=6.4. The inset chart in (a) shows the initial stage of the adsorption. The final Zn concentrations of the four experiments in (a) are (*C*eq )*C*p1=0.18 ppm, (*C*eq )*C*p2=0.17 ppm, (*C*eq )*C*p3=0.16 ppm, (*C*eq )*C*p4=0.15 ppm. In order to examine the desorption rate effectively, only the initial stage of desorption is presented in (b).

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where log log *Bk C sp*

the straight line, *n* is obtained.

constant *C*eq are used to calculate n.

the *C*p effect is different in these two systems.

Metastable-Equilibrium Adsorption (MEA) Theory 527

line, then the influence of *C*p on the isotherm can be described by Eq. [19]. From the slope of

Based on the adsorption isotherm data of Figure 1, the plots of log vs log*Ceq* and the plots of log vs log*Cp* for Zn–goethite and Cd–goethite systems are presented in Figure 5 and 6, respectively. Good linear relationships were obtained. For the Zn–goethite system, *k*sp=1.381,

Fig. 4. Data of ( , *C*eq) under constant *C*p are used to calculate b. Data of ( , *C*p) under

Fig. 5. Plot of log*Cp* vs log under the condition of *C*eq=0.2 ppm for Zn–goethite and Cd– goethite systems, respectively. The different slopes of the two lines indicate that the size of

From the intercepts of either Eq. [21] or Eq. [22], *k*sp can be calculated.

β=0.4136, n=0.0819. For the Cd–goethite system, *k*sp=1.778, β=0.435, n 0.

*eq* . Under this condition, if the plot of log vs log*Cp* is a straight

Fig. 3. Adsorption (a) and desorption (b) kinetic curves under different *C*p conditions in Cd– goethite system. pH=7.1. The final Cd concentrations of the three experiments in (a) are (*C*eq)*C*p1=0.235 ppm, (*C*eq )*C*p2=0.211 ppm, (*C*eq )*C*p3=0.210 ppm.

In a controlled simple aqueous system containing Zn–goethite, where a clear *C*p effect is observed, an increase in particle concentration causes a simultaneous decrease in adsorption reversibility and in the adsorption isotherm (Figure 1a). At the same time, Zn adsorbed under a lower *C*p condition desorbs faster (indicating more adsorption reversibility ) than that under a higher *C*p condition (Figure 2). In another controlled simple aqueous system of Cd–goethite, where no *C*p effect is observed, changes in *C*p does not cause discernible changes in adsorption hysteresis and in the adsorption isotherm (Figure 1b). Little difference in desorption rate is observed for the Cd adsorbed under different *C*p conditions (Figure 3). Both the *C*p effect and the non-*C*p effect results can be qualitatively explained and quantitatively described by the MEA theory.

In the Freundlich-type *C*p effect isotherm equation [19], we called *k*sp the specific adsorption constant and *n* the *C*p effect index. *k*sp is a measure of adsorption capacity and *n* is a measure of the degree of the *C*p effect. Like b, *k*sp and *n* can be calculated from adsorption isotherm data. The method is described below.

Take the logarithm of both sides of Eq. [19] ,

$$
\log \Gamma = \log k\_{sp} - n \log \mathcal{C}\_p + \beta \log \mathcal{C}\_{eq} \tag{20}
$$

For a given adsorption isotherm ( e.g., isotherm a, b, or c in Figure 4) , *C*p is a constant, and Eq. [20] becomes:

$$
\log \Gamma = A + \beta \log \mathcal{C}\_{eq} \tag{21}
$$

where *A* log log *sp p knC* . It can be seen from [21] that, if the relationship between and *C*eq under a specified *C*p condition can be described by Eq. [19], then the plot of log vs log*Ceq* should be a straight line. From the slope of the straight line, can be obtained. Under a given *C*eq ( e.g., data for ' <sup>1</sup> , *C*p1,' ' <sup>2</sup> , *C*p2,' ' <sup>3</sup> , *C*p3' in Figure 4), Eq. [20] becomes

$$
\log \Gamma = B - n \log \mathbb{C}\_p \tag{22}
$$

Fig. 3. Adsorption (a) and desorption (b) kinetic curves under different *C*p conditions in Cd– goethite system. pH=7.1. The final Cd concentrations of the three experiments in (a) are

In a controlled simple aqueous system containing Zn–goethite, where a clear *C*p effect is observed, an increase in particle concentration causes a simultaneous decrease in adsorption reversibility and in the adsorption isotherm (Figure 1a). At the same time, Zn adsorbed under a lower *C*p condition desorbs faster (indicating more adsorption reversibility ) than that under a higher *C*p condition (Figure 2). In another controlled simple aqueous system of Cd–goethite, where no *C*p effect is observed, changes in *C*p does not cause discernible changes in adsorption hysteresis and in the adsorption isotherm (Figure 1b). Little difference in desorption rate is observed for the Cd adsorbed under different *C*p conditions (Figure 3). Both the *C*p effect and the non-*C*p effect results can be qualitatively explained and

In the Freundlich-type *C*p effect isotherm equation [19], we called *k*sp the specific adsorption constant and *n* the *C*p effect index. *k*sp is a measure of adsorption capacity and *n* is a measure of the degree of the *C*p effect. Like b, *k*sp and *n* can be calculated from adsorption isotherm

log log log log *sp p eq knC C*

For a given adsorption isotherm ( e.g., isotherm a, b, or c in Figure 4) , *C*p is a constant, and

log log *A C* 

where *A* log log *sp p knC* . It can be seen from [21] that, if the relationship between and *C*eq under a specified *C*p condition can be described by Eq. [19], then the plot of log vs log*Ceq* should be a straight line. From the slope of the straight line, can be obtained. Under a given *C*eq ( e.g., data for ' <sup>1</sup> , *C*p1,' ' <sup>2</sup> , *C*p2,' ' <sup>3</sup> , *C*p3' in Figure 4), Eq. [20] becomes

(20)

*eq* (21)

log *Bn C* log *<sup>p</sup>* (22)

(*C*eq)*C*p1=0.235 ppm, (*C*eq )*C*p2=0.211 ppm, (*C*eq )*C*p3=0.210 ppm.

quantitatively described by the MEA theory.

data. The method is described below. Take the logarithm of both sides of Eq. [19] ,

Eq. [20] becomes:

where log log *Bk C sp eq* . Under this condition, if the plot of log vs log*Cp* is a straight line, then the influence of *C*p on the isotherm can be described by Eq. [19]. From the slope of the straight line, *n* is obtained.

From the intercepts of either Eq. [21] or Eq. [22], *k*sp can be calculated.

Based on the adsorption isotherm data of Figure 1, the plots of log vs log*Ceq* and the plots of log vs log*Cp* for Zn–goethite and Cd–goethite systems are presented in Figure 5 and 6, respectively. Good linear relationships were obtained. For the Zn–goethite system, *k*sp=1.381, β=0.4136, n=0.0819. For the Cd–goethite system, *k*sp=1.778, β=0.435, n 0.

Fig. 4. Data of ( , *C*eq) under constant *C*p are used to calculate b. Data of ( , *C*p) under constant *C*eq are used to calculate n.

Fig. 5. Plot of log*Cp* vs log under the condition of *C*eq=0.2 ppm for Zn–goethite and Cd– goethite systems, respectively. The different slopes of the two lines indicate that the size of the *C*p effect is different in these two systems.

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Metastable-Equilibrium Adsorption (MEA) Theory 529

Fig. 8. Comparison between calculated and measured isotherms under different *C*<sup>p</sup>

0.435 1.778 *Ceq* . Points are adsorption data from Figure 1b.

and non-*C*p effect from the fundamental thermodynamic principle.

the MEA principle.

conditions in Cd–goethite system. Lines are calculated from the *C*p effect isotherm equation

According to MEA theory, for the ideal reversible adsorption reactions, changes in *C*p have no influence on the reversibility of MEA states, and it should have no *C*p effect in such systems when experimental artifacts are excluded.11, 18 For partially irreversible adsorption reactions, changes in *C*p may significantly affect the irreversibility and the microscopic MEA structures, and a *C*p effect should fundamentally exist in irreversible adsorption systems.11, 17 Therefore, the MEA theory provided a rational explanation for the phenomena of *C*p effect

**4. Microscopic measurement of metastable-equilibrium adsorption state** 

It should be noted that, when the *C*p effect isotherm equations are used in the modeling of practical adsorption processes, they may be totally empirical and does not imply particular physical mechanism. The macroscopic adsorption behavior is fundamentally controlled by the microscopic reaction mechanism of adsorbed molecules on solid surfaces. Therefore, the direct Measurement on the microstructures at solid-water interfaces is crucial to verifying

Macroscopic thermodynamic results19, 20 showed that Zn(II) adsorbed on manganite was largely irreversible (adsorption and desorption isotherms corresponding to the forward and backward reactions did not coincide, see Figure 9), but the adsorption of Zn (II) on δ-MnO2 was highly reversible (there was no apparent hysteresis between the adsorption and desorption isotherms, see Figure 10). This contrast adsorption behavior between the two forms of manganese oxides could be explained from the different microscopic structures

Fig. 6. Plot of log*Ceq* vs log under the condition of *C*p=1.534 g/L for Zn–goethite and Cd– goethite systems, respectively. The similar slope of the two lines indicates that both the adsorption of Zn and Cd on goethite have similar β values.

After the specific adsorption constant (*k*sp ), the *C*p effect index (n), and β are calculated, the *C*p effect adsorption isotherm equations for Zn–goethite and Cd–goethite systems can be expressed as 0.0819 0.4136 1.381 *C C p eq* and 0.435 1.778 *Ceq* , respectively. Figures 7 and 8 show that the calculated isotherms fit the experimental data well.

Fig. 7. Comparison between calculated and measured isotherms under different *C*<sup>p</sup> conditions in the Zn–goethite system. Lines are calculated from the *C*p effect isotherm equation 0.0819 0.4136 1.381 *C C p eq* . Points are adsorption data from Figure 1a.

Fig. 6. Plot of log*Ceq* vs log under the condition of *C*p=1.534 g/L for Zn–goethite and Cd– goethite systems, respectively. The similar slope of the two lines indicates that both the

After the specific adsorption constant (*k*sp ), the *C*p effect index (n), and β are calculated, the *C*p effect adsorption isotherm equations for Zn–goethite and Cd–goethite systems can be

Fig. 7. Comparison between calculated and measured isotherms under different *C*<sup>p</sup> conditions in the Zn–goethite system. Lines are calculated from the *C*p effect isotherm

. Points are adsorption data from Figure 1a.

and 0.435 1.778 *Ceq* , respectively. Figures 7 and 8

adsorption of Zn and Cd on goethite have similar β values.

show that the calculated isotherms fit the experimental data well.

expressed as 0.0819 0.4136 1.381 *C C p eq*

equation 0.0819 0.4136 1.381 *C C p eq*

Fig. 8. Comparison between calculated and measured isotherms under different *C*<sup>p</sup> conditions in Cd–goethite system. Lines are calculated from the *C*p effect isotherm equation 0.435 1.778 *Ceq* . Points are adsorption data from Figure 1b.

According to MEA theory, for the ideal reversible adsorption reactions, changes in *C*p have no influence on the reversibility of MEA states, and it should have no *C*p effect in such systems when experimental artifacts are excluded.11, 18 For partially irreversible adsorption reactions, changes in *C*p may significantly affect the irreversibility and the microscopic MEA structures, and a *C*p effect should fundamentally exist in irreversible adsorption systems.11, 17 Therefore, the MEA theory provided a rational explanation for the phenomena of *C*p effect and non-*C*p effect from the fundamental thermodynamic principle.
