**5. Temperature dependence of metastable-equilibrium adsorption**

Since temperature (*T*) is expected to affect both adsorption thermodynamics and kinetics, the adsorption–desorption behavior may be *T*-dependent. The adsorption irreversibility of Zn(II) on anatase at various temperatures was studied using a combination of macroscopic thermodynamic methods and microscopic spectral measurement.

Adsorption isotherm results29 showed that, when the temperature increased from 5 to 40 °C, the Zn(II) adsorption capacity increased by 130% (Figure 13). The desorption isotherms significantly deviate from the corresponding adsorption isotherms, indicating that the adsorption of zinc onto anatase was not fully reversible. The thermodynamic index of irreversibility (TII) proposed by Sander et al.30 was used to quantify the adsorption irreversibility. The TII was defined as the ratio of the observed free energy loss to the maximum possible free energy loss due to adsorption hysteresis, which was given by

$$\text{TII} = \frac{\ln \text{C}\_{\text{eq}}^{\text{r}} - \ln \text{C}\_{\text{eq}}^{D}}{\ln \text{C}\_{\text{eq}}^{S} - \ln \text{C}\_{\text{eq}}^{D}} \tag{23}$$

where eq *<sup>S</sup> C* is the solution concentration of the adsorption state S ( eq *<sup>S</sup> C* , qeq *<sup>S</sup>* ) from which desorption is initiated; eq *<sup>D</sup> C* is the solution concentration of the desorption state D ( eq *<sup>D</sup> C* , qeq *<sup>D</sup>* ); *C*eq is the solution concentration of hypothetical reversible desorption state γ (*C*eq , qeq ). eq *<sup>S</sup> C* and eq *<sup>D</sup> C* are determined based on the experimental adsorption and desorption isotherms, and are easily obtained from the adsorption branch where the solid-phase concentration is equal to qeq *<sup>D</sup>* .

Based on the definition, the TII value lies in the range of 0 to 1, with 1 indicating the maximum irreversibility. The TII value (0.63, 0.34, 0.20) decreased by a factor of >3 when the temperature increased from 5 to 40 °C. This result indicated that the adsorption of Zn(II) on the TiO2 surfaces became more reversible with increasing temperature.29

EXAFS spectra results showed that the hydrated Zn(II) was adsorbed on anatase through edge-sharing linkage mode (strong adsorption) and corner-sharing linkage mode (weak adsorption), which corresponded to two average Zn–Ti atomic distances of 3.25±0.02 and 3.69±0.03 Å, respectively.29 According to the DFT results (Figure 14),13 EXAFS measured the

edge-sharing linkage was a stronger adsorption mode than that of the corner-sharing linkage, which would make it more difficult for the edge linkage to be desorbed from the solid surfaces than the corner linkage.20 So adsorption of Zn(II) onto manganite was more irreversible than that on δ-MnO2. This implied that the adsorption reversibility was influenced by the proportion of different bonding modes between adsorbate and adsorbent

Due to the contrast adsorption linkage mode, Zn(II) adsorbed on δ-MnO2 and manganite can be in very different metastable-equilibrium adsorption (MEA) states, which result in the different macroscopic adsorption–desorption behavior. For example, the extents of inconstancy of the equilibrium adsorption constant and the particle concentration effect are very different in the two systems. Adsorption of metals on δ-MnO2 and manganite may therefore be used as a pair of model systems for comparative studies of metastable-

Since temperature (*T*) is expected to affect both adsorption thermodynamics and kinetics, the adsorption–desorption behavior may be *T*-dependent. The adsorption irreversibility of Zn(II) on anatase at various temperatures was studied using a combination of macroscopic

Adsorption isotherm results29 showed that, when the temperature increased from 5 to 40 °C, the Zn(II) adsorption capacity increased by 130% (Figure 13). The desorption isotherms significantly deviate from the corresponding adsorption isotherms, indicating that the adsorption of zinc onto anatase was not fully reversible. The thermodynamic index of irreversibility (TII) proposed by Sander et al.30 was used to quantify the adsorption irreversibility. The TII was defined as the ratio of the observed free energy loss to the

> eq eq eq eq

*S D C C C C*

is the solution concentration of hypothetical reversible desorption state γ (*C*eq

*D*

*<sup>D</sup> C* are determined based on the experimental adsorption and desorption

*<sup>D</sup> C* is the solution concentration of the desorption state D ( eq

(23)

*<sup>S</sup> C* , qeq

*<sup>S</sup>* ) from which

*<sup>D</sup> C* ,

,

ln ln

ln ln

isotherms, and are easily obtained from the adsorption branch where the solid-phase

Based on the definition, the TII value lies in the range of 0 to 1, with 1 indicating the maximum irreversibility. The TII value (0.63, 0.34, 0.20) decreased by a factor of >3 when the temperature increased from 5 to 40 °C. This result indicated that the adsorption of Zn(II) on

EXAFS spectra results showed that the hydrated Zn(II) was adsorbed on anatase through edge-sharing linkage mode (strong adsorption) and corner-sharing linkage mode (weak adsorption), which corresponded to two average Zn–Ti atomic distances of 3.25±0.02 and 3.69±0.03 Å, respectively.29 According to the DFT results (Figure 14),13 EXAFS measured the

maximum possible free energy loss due to adsorption hysteresis, which was given by

TII

the TiO2 surfaces became more reversible with increasing temperature.29

*<sup>D</sup>* .

*<sup>S</sup> C* is the solution concentration of the adsorption state S ( eq

**5. Temperature dependence of metastable-equilibrium adsorption** 

thermodynamic methods and microscopic spectral measurement.

in nature.

where eq

qeq *<sup>D</sup>* ); *C*eq 

qeq ). eq

desorption is initiated; eq

*<sup>S</sup> C* and eq

concentration is equal to qeq

equilibrium adsorption.

Fig. 13. Adsorption and desorption isotherms of Zn(II) on anatase at various temperatures. Symbols, experimental data; solid lines, model-fitted adsorption isotherms; dashed lines, model-fitted desorption isotherms. S5, S20, and S40 indicate where desorption was initiated and samples selected for subsequent EXAFS analysis. Data given as mean of duplicates and errors refer to the difference between the duplicated samples.

corner-sharing linkage mode at the Zn-Ti distance of 3.69 Å may be a mixture of 4 coordinated bidentate binuclear (BB, 3.48 Å) and 6-coordinated monodentate mononuclear (MM, 4.01 Å) MEA states. DFT calculated energies showed that the MM complex was an energetically unstable MEA state compared with the BB (-8.58 kcal/mol) and BM (edgesharing bidentate mononuclear, -15.15 kcal/mol) adsorption modes,13 indicating that the MM linkage mode would be a minor MEA state, compared to the BB and BM MEA state. In the X-ray absorption near-edge structure analysis (XANES), the calculated XANES of BB and BM complexes reproduced all absorption characteristics (absorption edge, post-edge absorption oscillation and shape resonances) from the experimental XANES spectra (Figure 15).13 Therefore, the overall spectral and computational evidence indicated that the cornersharing BB and edge-sharing BM complexation mode coexisted in the adsorption of Zn(II) on anatase.

As the temperature increased from 5 to 40 °C, the number of strong adsorption sites (edge linkage) remained relatively constant while the number of the weak adsorption sites (corner linkage) increased by 31%.29 These results indicate that the net gain in adsorption capacity and the decreased adsorption irreversibility at elevated temperatures were due to the increase in available weak adsorption sites or the decrease in the ratio of edge linkage to corner linkage. Both the macroscopic adsorption/desorption equilibrium data and the molecular level evidence indicated a strong temperature dependence for the metastableequilibrium adsorption of Zn(II) on anatase.

Advances in Interfacial Adsorption Thermodynamics:

**0.6**

BM complex and experimental XANES spectra.

spectral and computational methods.14, 31, 32

concentration when they alter the final MEA states.11, 12

influence varied with pH.

**6. pH dependence of metastable-equilibrium adsorption** 

 **Normalized**

**1.2**

**1.8**

**Relative Absorption**

Metastable-Equilibrium Adsorption (MEA) Theory 535

**2.4** 5-coord. BM

4-coord. BB exp. pH=6.3 exp. pH=6.8

**9660 9680 9700 9720 9740**

**Photon Energy (eV)**

According to MEA theory, both adsorbent/particle concentration (i.e., Cp) and adsorbate concentration could fundamentally affect equilibrium adsorption constants or isotherms when a change in the concentration of reactants (adsorbent or adsorbate) alters the reaction irreversibility or the MEA states of the apparent equilibrium. On the other hand, a general theory should be able to predict and interpret more phenomena. To test new phenomenon predicted by MEA theory can not only cross-confirm the theory itself but also provide new insights/applications in broadly related fields. The influence of adsorbate concentration on adsorption isotherms and equilibrium constants at different pH conditions was therefore studied in As(V)-anatase system using macroscopic thermodynamics and microscopic

The thermodynamic results14 showed that, when the total mass of arsenate was added to the TiO2 suspension by multiple batches, the adsorption isotherms declined as the multi-batch increased, and the extent of the decline decreased gradually as pH decreased from 7.0 to 5.5 (Figure 16). This result provided a direct evidence for the influence of adsorption kinetics (1 batch/multi-batch) on adsorption isotherm and equilibrium constant, and indicated that the

According to MEA theory, for a given batch adsorption reaction under the same thermodynamic conditions, when the reaction is conducted through different kinetic pathways (1-batch/multi-batch), different MEA states (rather than a unique ideal equilibrium state) could be reached when the reaction reaches an apparent equilibrium (within the experimental time such as days).14 Equilibrium constants or adsorption isotherms, which are defined by adsorption density, are inevitably affected by the reactant

Fig. 15. Calculated XANES spectra of 4-oxygen coordinated BB and 5-oxygen coordinated

Fig. 14. Calculated Zn(II)–TiO2 surface complexes using density functional theory: (a) dissolved Zn(II) with six outer-sphere water molecules; (b) monodentate mononuclear (MM); (c) bidentate binuclear (BB); (d) bidentate mononuclear (BM). Purple, red, big gray, small gray circles denote Zn, O, Ti, H atoms, respectively. Distances are shown in angstroms.

Fig. 14. Calculated Zn(II)–TiO2 surface complexes using density functional theory: (a) dissolved Zn(II) with six outer-sphere water molecules; (b) monodentate mononuclear (MM); (c) bidentate binuclear (BB); (d) bidentate mononuclear (BM). Purple, red, big gray,

small gray circles denote Zn, O, Ti, H atoms, respectively. Distances are shown in

angstroms.

Fig. 15. Calculated XANES spectra of 4-oxygen coordinated BB and 5-oxygen coordinated BM complex and experimental XANES spectra.
