Advances in Interfacial Adsorption Thermodynamics: Metastable-Equilibrium Adsorption (MEA) Theory 529 Fig. 8. Comparison between calculated and measured isotherms under different C p conditions in Cd–goethite system. Lines are calculated from the C p effect isotherm equation 0.435 1.778 eq C  . 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. 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 the MEA principle. Macroscopic thermodynamic results 19, 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-MnO 2 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 Thermodynamics – Interaction Studies – Solids, Liquids and Gases 530 between δ-MnO 2 and manganite, as well as the linkage modes of adsorbed Zn(II) on δ- MnO 2 and manganite. 19 Fig. 9. Adsorption (closed symbols) and desorption (open symbols) isotherms of Zn(II) on manganite. EXAFS samples were indicated by arrows. Fig. 10. Adsorption (■) and desorption (□) isotherms of Zn(II) on δ-MnO 2 . EXAFS samples were symboled with blank triangles (Δ). Manganite had a structure with rows of edge-sharing Mn(II)O 6 octahedra linked to adjacent rows through corners. Due to the Jahn–Teller effect of Mn(II) ions and to the presence of both O and OH groups, the MnO 6 octahedra were highly distorted: each Mn is bound to four equatorial oxygen and two axial oxygen atoms. 21, 22 This distortion gave rise to a mild layered structure. Hydrolyzable Zn could be bonded on MnO 6 octahedra of manganite surface via edge and corner-sharing coordination modes. 21, 22 The basic structure of δ-MnO 2 consisted of layers of edge-sharing MnO 6 octahedra alternating with a layer of water molecules. One-sixth of Mn 4+ positions were empty, which gave a layer charge that was compensated by two Zn atoms located above and below the vacancy. 23, 24 Hydrolyzable Zn could be taken up in the interlayer to form tridentate corner-sharing complexes. 25, 26 These differences in crystallographic structure resulted in different linkage modes for the adsorption of Zn on manganite and δ-MnO 2 . Advances in Interfacial Adsorption Thermodynamics: Metastable-Equilibrium Adsorption (MEA) Theory 531 Fig. 11. Corner-sharing linkage (a) and interlayer structures of Zn(II) adsorbed on δ-MnO 2 (b). (a) R Zn–O = 2.07 Å, R Mn–O = 1.92 Å, R Zn–Mn = 3.52 Å. (b) Squares were vacant sites, illustration diagram adapted from Wadsley, 27 Post and Appleman, 28 and Manceau et al 25 Fig. 12. Two types of linkage between adsorbed Zn(II) (octahedron and tetrahedron) and MnO 6 octahedra on the γ-MnOOH surfaces. (a) Double-corner linkage mode; (b) edge- linkage mode. Extended X-ray absorption fine structure (EXAFS) analysis showed that Zn(II) was adsorbed onto δ-MnO 2 in a mode of corner-sharing linkage, which corresponded to only one Zn–Mn distance of 3.52 Å (Figure 11). However, there were two linkage modes for adsorbed Zn(II) on manganite surface as inner-sphere complexes, edge-sharing linkage and corner-sharing linkage, which corresponded to two Zn–Mn distances of 3.07 and 3.52 Å (Figure 12). The Thermodynamics – Interaction Studies – Solids, Liquids and Gases 532 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-MnO 2 . This implied that the adsorption reversibility was influenced by the proportion of different bonding modes between adsorbate and adsorbent in nature. Due to the contrast adsorption linkage mode, Zn(II) adsorbed on δ-MnO 2 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-MnO 2 and manganite may therefore be used as a pair of model systems for comparative studies of metastable- equilibrium adsorption. 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 results 29 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 eq eq eq eq ln ln TII ln ln D SD CC CC     (23) where eq S C is the solution concentration of the adsorption state S ( eq S C , eq q S ) from which desorption is initiated; eq D C is the solution concentration of the desorption state D ( eq D C , eq q D ); eq C  is the solution concentration of hypothetical reversible desorption state γ ( eq C  , eq q  ). eq S C and eq D 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 eq q D . 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 TiO 2 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 Advances in Interfacial Adsorption Thermodynamics: Metastable-Equilibrium Adsorption (MEA) Theory 533 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. S 5 , S 20 , and S 40 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 (edge- sharing 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 corner- sharing 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 metastable- equilibrium adsorption of Zn(II) on anatase. Thermodynamics – Interaction Studies – Solids, Liquids and Gases 534 Fig. 14. Calculated Zn(II)–TiO 2 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. Advances in Interfacial Adsorption Thermodynamics: Metastable-Equilibrium Adsorption (MEA) Theory 535 9660 9680 9700 9720 9740 0.6 1.2 1.8 2.4 5-coord. BM 4-coord. BB exp. pH=6.3 exp. pH=6.8 Photon Energy (eV) Normalized Relative Absorption Fig. 15. Calculated XANES spectra of 4-oxygen coordinated BB and 5-oxygen coordinated BM complex and experimental XANES spectra. 6. pH dependence of metastable-equilibrium adsorption 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 spectral and computational methods. 14, 31, 32 The thermodynamic results 14 showed that, when the total mass of arsenate was added to the TiO 2 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 influence varied with pH. 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 concentration when they alter the final MEA states. 11, 12 Thermodynamics – Interaction Studies – Solids, Liquids and Gases 536 Fig. 16. Adsorption isotherms of As (V) on TiO 2 in 0.01mol/L NaNO 3 solution at 25 °C under different pH. TiO 2 particle concentration is 1g/L. 1-batch stands for a series of total arsenate being added to TiO 2 suspension in one time, and 3-batch stands for the total arsenate being added averagely to TiO 2 suspension in 3 times every 4 hours. EXAFS samples were marked by ellipse, in which the initial total As (V) concentration is 0.80 mmol/L. Sample As-O As-Ti Res. CN 1 /CN 2 BB MM CN R(Å) σ 2 CN 1 R 1 (Å) σ 2 CN 2 R 2 (Å) σ 2 1-batch pH5.5 3.9 1.68 0.002 1.9 3.17 0.008 1.1 3.60 0.01 8.6 1.8 3-batch pH5.5 4.0 1.68 0.002 2.2 3.26 0.01 0.9 3.61 0.008 14.2 2.4 1-batch pH6.2 4.0 1.68 0.002 1.8 3.16 0.007 1.0 3.59 0.006 11.0 1.7 3-batch pH6.2 3.9 1.68 0.002 2.1 3.19 0.008 0.8 3.59 0.01 9.0 2.5 1-batch pH7.0 4.1 1.69 0.002 1.8 3.17 0.007 1.1 3.59 0.001 13.2 1.6 3-batch pH7.0 4.1 1.68 0.002 2.2 3.22 0.004 1.0 3.60 0.001 10.9 2.2 As(V)-pH5.5 4.1 1.68 0.004 6.7 As(V)-pH7.0 4.1 1.69 0.003 5.3 Calculated values 4.0 1.70 2.0 3.25 1.0 3.52 Table 1. Summary of As(V) K-edge EXAFS results for 1-batch and 3-batch adsorption samples at pH 5.5, 6.2 and 7.0. The comparison of EXAFS measured and DFT calculated results indicated that arsenate mainly formed inner-sphere bidentate binuclear (BB) and monodentate mononuclear (MM) surface complexes on TiO 2 , where EXAFS measured two As-Ti distances of 3.20±0.05 and 3.60 ±0.02 Å (Table 1) corresponded to the DFT calculated values of BB (3.25 Å) and MM (3.52 Å) complexes (Figure 17), respectively. 14 Advances in Interfacial Adsorption Thermodynamics: Metastable-Equilibrium Adsorption (MEA) Theory 537 Fig. 17. DFT calculated structure of inner-sphere and H-bond adsorption products of arsenate on TiO 2 : (a) monodentate mononuclear arsenate H-bonded to a H 2 O surface functional group occupying the adjacent surface site (MM 1 ); (b) monodentate mononuclear arsenate H-bonded to a -OH surface functional group occupying the adjacent surface site (MM 2 ); (c) bidentate binuclear (BB) complex; (d) H-bonded complex. Red, big gray, small gray, purple circles denote O, Ti, H, As atoms, respectively. Distances are shown in angstroms. The EXAFS coordination number of CN 1 and CN 2 represented statistically the average number of nearest Ti atoms around the As atom corresponding to a specific interatomic distance. We used the coordination number ratio of CN 1 /CN 2 to describe the relative proportion of BB mode to MM mode in adsorption samples. The CN 1 /CN 2 was 1.6 and 2.2 for 1-batch and 3-batch adsorption samples at pH 7.0, respectively (Table 1), 14 indicating that 3-batch adsorption samples contained more BB adsorbed arsenate than that of 1-batch adsorption samples. This result was cross-confirmed by measuring the spectral shift of X-ray absorption near edge structure (XANES) and Fourier transform infrared spectroscopy (FTIR). Thermodynamics – Interaction Studies – Solids, Liquids and Gases 538 DFT calculation showed that the theoretical XANES transition energy of BB complex was 0.62eV higher than that of MM complex. Therefore, the blue-shift of As (V) K-absorption edge observed from 1-batch to 3-batch adsorption samples suggested a structural evolution from MM to BB adsorption as the multi-batch increased (Figure 18). 31 Fig. 18. The first derivative K-edge XANES spectra of As (V) adsorption on anatase. The DFT calculated frequency analysis showed that the As-OTi asymmetric stretching vibration (υ as ) of MM and BB complexes located at 855 and 835 cm-1, respectively. On the basis of this theoretical analysis, the FTIR measured red-shift of As-OTi υ as vibration from 1- batch sample (849 cm-1) to 3-batch sample (835 cm -1 ) suggested that the ratio of BB/MM in 3-batch sample was higher than that in 1-batch sample (Figure 19). 32 The good agreement of EXAFS results of CN 1 /CN 2 with XANES and FTIR analysis also validated the reliability of the CN ratio used as an index to approximate the proportion change of surface complexation modes. BB complex occupies two active sites on adsorbent surface whereas MM occupies only one. For monolayer chemiadsorption, a unit surface area of a given adsorbent can contain more arsenate molecules adsorbed in MM mode than that in BB mode. Therefore, the increase of the proportion of BB complex from 1-batch to 3-batch addition mode was shown as the decrease of adsorption density in 3-batch isotherm (Figure 16). Table 1 showed that the relative proportion of BB and MM complex was rarely affected by pH change from 5.5 to 7.0, indicating that the pH dependence for the influence of adsorption kinetics (1-batch/multi-batch) on adsorption isotherm was not due to inner-sphere chemiadsorption. 14 The influence of pH on adsorption was simulated by DFT theory through changing the number of H + in model clusters. Calculation of adsorption energy showed that the thermodynamic favorability of inner-sphere and outer-sphere adsorption was directly related to pH (Table 2). 14 As pH decreased, the thermodynamic favorability of inner-sphere and outer-sphere arsenate adsorption on Ti-(hydr)oxides increased. This DFT result explained why the adsorption densities of arsenate (Figure 16) and equilibrium adsorption constant (Table 2) increased with the decrease of pH. [...]... the particle itself will change during both the processes Dyson (1949a;b) discussed that the changing the particle energy itself by the perturbative treatment of 562 Thermodynamics – Interaction Studies – Solids, Liquids and Gases Will-be-set-by-IN-TECH 20 the fermion interaction when we consider single particle propagation from 1 to 2 The energy of the particle differs from the noninteracting particle... can create the particle-hole pair from the vacuum fluctuations Now consider a system of two particles a and b propagate from 1 to 3 interacting at 5 for the particle a and from 2 to 4 interacting at 6 for the particle b In the case of free particles, the kernel K0 is a simple multiple of two free particle kernels K0a and K0b as K0 (3, 4; 1, 2) = K0a (3, 1) K0b (4, 2) (102 ) ˆ When two particles are interacting... interpretation of the antiparticle, which has the reversed charge of the particle; for example, a positron is the antiparticle of an electron The Dirac equation (Dirac, 1928a;b) has negative energy states of an electron Dirac interpreted himself that the negative energy states are fully occupied in vacuum, and an 560 Thermodynamics – Interaction Studies – Solids, Liquids and Gases Will-be-set-by-IN-TECH... two facts: (i) the elementary exciations are not necessarily to be a Goldstone boson and (ii) they are not 552 Thermodynamics – Interaction Studies – Solids, Liquids and Gases Will-be-set-by-IN-TECH 10 necessarily limited to the ionic subsystem, but also electronic one If the elementary excitations are fermionic, thermodynamics are basically calculable as we did for the non-interacting electrons gas... and Klein (1927), known as the Klein-Gordon equation The Klein-Gordon equation is valid for the Bose-Einstein particles, while the Dirac equation is valid for the Fermi-Dirac particles 558 Thermodynamics – Interaction Studies – Solids, Liquids and Gases Will-be-set-by-IN-TECH 16 electrons and the radiations as quantized objects in such a way of a canonical transformation to the normal modes of their... 3.13 103 9 32.1 2.37 10- 6 -135.6 5.72 102 3 27.5 1.54 10- 5 H-bond complexes 0 H2AsO4- ( H2O)12+ [Ti2(OH)4(H2O)6]4+ → [Ti2(OH)4(H2O)6AsO2(OH)2]3+ + 12H2O 1 H2AsO4- ( H2O)12+ [Ti2(OH)5(H2O)5]3+ → [Ti2(OH)4(H2O)6AsO2(OH)2]3+ + OH-( H2O)11 54.4 2 H2AsO4- ( H2O)12+ [Ti2(OH)6(H2O)4]2+ → [Ti2(OH)4(H2O)6AsO2(OH)2]3+ + 2OH-(H2O )10 252.9 5.01 10- 45 -203.1 3.91 103 5 2.96 10- 10 Table 2 Calculated ΔGads (kJ/mol) and. .. Y.; Zhao, D Y.; Yang, Y H.; Chen, H.; He, G Z., J Colloid Interface Sci 2008, 319 (2), 385-391 [30] Sander, M.; Lu, Y.; Pignatello, J J A thermodynamically based method to quantify true sorption hysteresis; Am Soc Agronom: 2005; pp 106 3 -107 2 542 Thermodynamics – Interaction Studies – Solids, Liquids and Gases [31] He, G Z.; Pan, G.; Zhang, M Y.; Wu, Z Y., J Phys Chem C 2009, 113 (39), 17076-17081 [32]... In order to obtain the Gibbs free energy from first principles, it is convenient to use the equilibrium statistical mechanics for grand canonical ensemble by introducing the grand 544 Thermodynamics – Interaction Studies – Solids, Liquids and Gases Will-be-set-by-IN-TECH 2 partition function Ξ ( T, V, {μ i }) = ∑ ∑ exp Ni ζ − β Eζ (V ) − ∑ μ i Ni , (2) i where β is the inverse temperature (kB T )−1 with... the macroscopic relationship between equilibrium concentrations in solution and on solid surfaces The reasoning behind the adsorbent and adsorbate concentration effects is that the conventional adsorption thermodynamic methods such as adsorption isotherms, which are 540 Thermodynamics – Interaction Studies – Solids, Liquids and Gases defined by the macroscopic parameter of adsorption density (mol/m2),... discussed here (79) 556 Thermodynamics – Interaction Studies – Solids, Liquids and Gases Will-be-set-by-IN-TECH 14 The procedure for the photonic subsystem is quite useful in computing thermodynamics for many kinds of elementary exciations, which are usually massless (Goldstone) bosons in a condensed matter system It is predicted that any crystal must be completely ordered, and the atoms of each kind . edge-sharing linkage and corner-sharing linkage, which corresponded to two Zn–Mn distances of 3.07 and 3.52 Å (Figure 12). The Thermodynamics – Interaction Studies – Solids, Liquids and Gases 532. to develop methods to calculate the grand potential Ω. 544 Thermodynamics – Interaction Studies – Solids, Liquids and Gases Towards the Authentic Ab Intio Thermodynamics 3 In principle, we can. and Goldenfeld (1992). 546 Thermodynamics – Interaction Studies – Solids, Liquids and Gases Towards the Authentic Ab Intio Thermodynamics 5 for the minima points A and B. For the case of the