Comprehensive Kinetic and Mechanistic Analysis of TiO2

Article pubs.acs.org/JPCC

Comprehensive Kinetic and Mechanistic Analysis of TiO2 Photocatalytic Reactions According to the Direct−Indirect Model: (I) Theoretical Approach Juan Felipe Montoya, José Peral, and Pedro Salvador* Departamento de Química, Universidad Autónoma de Barcelona, 08193, Cerdanyola del Vallés, Spain S Supporting Information *

ABSTRACT: The photocatalytic oxidation kinetics of organic species in semiconductor (sc) gas phase and liquid semiconductor suspensions, strongly depends on the electronic interaction strength of substrate species with the sc surface. According to the Direct− Indirect (D-I) model, developed as an alternative to the Langmuir−Hinshelwood (L-H) model (Salvador, P. et al. Catalysis Today 2007, 129, 247), when chemisorption of dissolved substrate species is not favored and physisorption is the only existing adsorption mechanism, interfacial hole transfer takes place via an indirect transfer (IT) mechanism, the photon oxidation rate exponentially depending on the incident photon flux (Vox = VIT ox ∝ ρ ), with n = 1/2 under high enough photon flux (standard experimental conditions), whatever the dissolved substrate concentration, [(RH2)liq]. In contrast, under simultaneous physisorption and chemisorption of substrate species, hole capture takes place via a combination of an DT DT indirect transfer (IT) and a direct transfer (DT) mechanism (Vox = VIT ox + Vox ), with Vox ∝ n ρ and n = 1 for low enough ρ values, as long as adsorption−desorption equilibrium conditions existing in the dark are not broken under illumination, and monotonically decreasing toward n = 0 as ρ increases and adsorption−desorption equilibrium becomes broken. This behavior invalidates the frequently invoked axiom that the reaction order (exponent n) exclusively depends on the photon flux intensity, being in general n = 1 and n = 1/2 under low and high illumination intensity, respectively, independent of the nature of the sc-substrate electronic interaction. On the basis of a detailed analysis of the parameter defined as a = (Vox)2/2[(RH2)liq]ρ, an experimental test able to determine the influence of both interfacial hole transfer mechanisms, DT and IT, in the photo-oxidation kinetics, is presented. A simple method allowing the estimation of the photon flux critical value where adsorption−desorption equilibrium of chemisorbed substrate species is broken and the reaction order starts to decreases from n = 1 toward n = 0, is described.

1. INTRODUCTION TiO2 photocatalysis has received increasing attention during the last 30 years as a solar light assisted potential technology for environmental decontamination via photo-oxidation of pollutants present both in air1 and water.2 However, and in spite of the fact that considerable effort has been done in order to establish mechanistic and kinetic fundamentals existing behind photocatalytic reactions, this objective is still far from being reached. In this respect, a real handicap is concerning the almost “robotic” use of the classical Langmuir−Hinshelwood (L-H) kinetic equation as a suitable tool for the kinetic analysis of photocatalytic reactions.3−8 Some recognized objections to the use of the L-H model are (1) the a priori assumption that the adsorption−desorption equilibrium of substrate species existing in the dark is not disturbed under illumination, no matter the substrate concentration and photon flux used; (2) the lack of physical meaning of the intervening, L-H kinetic rate constants, something in fact expected for an equation which it’s main merit is its applicability to a great variety of catalytic processes; (3) the nonexplicit intervention of the photon flow as an experimental parameter in kinetic expressions, hindering the photon flux reaction rate dependence prediction; (4) the assumption that © 2014 American Chemical Society

the chemical nature of the sc surface is not modified during photocatalytic reactions; (5) the extended idea that chemisorption of organic species onto the semiconductor surface is as a prerequisite for photocatalytic oxidation to take place; (6) the omission of the degree of electronic interaction of substrate species with the semiconductor surface as a parameter strongly affecting photo-oxidation kinetics, in contradiction with the experimental evidence shown in the second part of this publication that the photo-oxidation of physisorbed organic species weakly interacting with the TiO2 surface, as it is for instance the case of benzene, is far less efficient than the observed for chemisorbed molecules like formic acid.9−12 Summing up, it must not be forgotten that the L-H model was not initially conceived for dealing with photochemical reactions, but to provide a general framework that can be adapted to specific photoinduced processes. LH represents a first approximation, certainly unspecific and incomplete, but useful in some circumstances. Received: December 12, 2013 Revised: May 9, 2014 Published: May 9, 2014 14266

dx.doi.org/10.1021/jp4121645 | J. Phys. Chem. C 2014, 118, 14266−14275

The Journal of Physical Chemistry C

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surface kinks and steps of TiO2 nanoparticles, leading to •− generation of terminal O•− s /Obr radicals (other groups have reported, by ESR experiments, the existence of those radicals on the surface of the illuminated TiO227,28); at a second step, surface trapped holes are isoenergetically transferred via tunnelling to physisorbed substrate species, according to the Marcus-Gerischer model for adiabatic electron transfer at the scelectrolyte interface.29−31 Intervening interfacial charge transfer mechanisms are schematized in Figure 1. IT is the only working

On the one hand, the recent application of physical techniques to the chemical study of surface reactions at atomic scale, as it is for instance the Scanning Tunnelling Microscopy (STM), has provided essential information not only about the role that TiO2 surface defects, like bridging oxygen vacancies, play as chemically active surface centers, but on the chemical changes taking place at the TiO2 surface during photocatalytic reactions.13−15 On the other hand, from UV photoinduced oxygen isotopic exchange (POIE) involving anatase nanoparticles in contact with O2, experimental evidence has been obtained that terminal, 2-fold coordinated bridging oxygen ions (>O2− br ) of the TiO2 surface are exchanged with oxygen atoms of dissolved O2 molecules, POIE reactions being blocked in the presence of organic molecules prone to be photooxidized.16−18 Furthermore, experimental evidence has recently been shown about the anaerobic mineralization of water dissolved benzene using 18O isotopelabeled titania (Ti18O2) as photocatalyst. The generation of C16,18O2 as mineralization product allows to conclude that onefold coordinated bridging oxygen radicals (−18O•− br ), photogenerated via inelastic trapping of valence band (VB) free holes (h+f ) by terminal (>18O2− br ) ions, behave as labile, structural oxygen species able to capture electrons from Ti 18 O 2 physisorbed benzene species via an adiabatic, interfacial charge transfer mechanism.19 A similar photocatalytic oxidation mechanism involving photogenerated −O•− br species has been proposed in the photo-oxidation of 4-chloropenol to catechol and hydroquinone.20 In relation to this subject, a stubborn controversy in TiO2 photocatalysis is that concerning the nature of photoinduced radical species active in the photo-oxidation of organic substrates in aqueous dissolution. From metastable impact electron spectroscopy (MIES) and ultraviolet photoelectron spectroscopy (UPS) studies of hydrated, well-defined TiO2 rutile surfaces, it has been recently shown that the photo-oxidation of adsorbed water molecules appears to be kinetically and thermodynamically hindered,21,22 contradicting the generalized assumption that besides free hydroxyl radicals (•OH) originated from the electroreduction of dissolved oxygen molecules, the photooxidation of chemisorbed water species leads to adsorbed •OH radicals acting as primary oxidizing species in TiO2 photocatalytic reactions.2,23−25 A new, comprehensive approach to TiO2 photocatalysis should necessarily involve the development of an autoconsistent mechanistic model able to take into account the nature of those physicochemical phenomena behind photogenerated photoreactions. With this aim in mind, during the last years we have developed the Direct−Indirect (D-I) model as an alternative to the L-H model.7,26 Depending on the type of electronic interaction of substrate species with the sc surface, the D-I model contemplates two well-defined types of interfacial charge transfer mechanisms. For strong electronic interaction (chemisorption) of dissolved substrate species (Lewis basic species) with terminal TiIV ligands of the TiO2 surface (Lewis acid sites), − and even O2− s /OHs terminal oxygen ions, the D-I model assumes that photo-oxidation is mainly based in an inelastic, interfacial direct transfer (DT) mechanism of photogenerated valence band free holes to TiO2 anchored substrate species. On the other hand, for weak interaction (physisorption) of substrate species with the TiO2 surface, the D-I model assumes an interfacial indirect transfer (IT) mechanism involving two successive steps: at the first step, h+f species are inelastically terminal oxygen ions of the TiO2 surface, trapped by O2− s including >O2− br ions and low coordinated oxygen ions located at

Figure 1. D-I model diagram showing TiO2 energy levels and primary interfacial reactions involving: (i) electro-reduction of dissolved O2 molecules with CB electrons, at a rate v′red, (ii) photo-oxidation of physisorbed RH2 species via an adiabatic IT mechanism of surface •− trapped holes (h+s ), identified with O•− s /Obr terminal oxygen radicals, at IT a rate Vox , and (iii) direct photo-oxidation of chemisorbed (RH2)ads substrate species with VB free holes (h+f ) via an inelastic DT mechanism at a rate VDT ox ; λ represents the reorganization energy according to the Gerischer model for electron transfer at the sc-electrolyte interface.28,29 Since we are dealing with initial photo-oxidation rates, the back reaction of electro-reduction of photogenerated RH• radicals with CB electrons is considered to be negligible and therefore not included in the diagram.32 Reproduced with permission from ref 33. Copyright 2009 Elsevier.

mechanism in the absence of chemisorption, as it is the case for benzene,9 but it always coexists with the DT mechanism when chemisorption takes place. Generally DT prevails on IT, although the contribution of both mechanisms can be comparable in some special cases, as it is for instance the photo-oxidation of phenol in water studied in the second part of this publication.9 Summing up, in contrast with the L-H model, the D-I model allows the development of photo-oxidation kinetic expressions where both the photon flux and the substrate concentration are explicit parameters allowing the photo-oxidation rate dependence on both experimental variables to be predicted. Besides, the D-I model rejects the participation of •OH radicals generated via direct photo-oxidation of water species chemisorbed at terminal •− TiIV ions with h+f species, considering photogenerated O•− s /Obr 18−20 radicals as photocatalytically active oxidant species. Furthermore, TiIVsurface ions are not only considered as chemisorption sites for substrate species but as trapping sites for conduction band (CB) free electrons (e−f ).7 Because of the complexity of real photocatalytic systems, the applicability of the D-I model to the analysis of experimental kinetic data is not always a simple task. A main difficulty concerns the influence of adsorption−desorption equilibrium rupture under high enough illumination intensity conditions on 14267



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the photo-oxidation rate, as first raised by Ollis in 2005,5 and further assumed by the D-I model.7 A second problem of the D-I model is related to the difficulty to discern between DT and IT charge transfer mechanisms from photo-oxidation kinetic measurements under nonequilibrated adsorption−desorption conditions (high enough illumination intensity). Our objective here is 2-fold: on the one hand, we will try to get information about the experimental conditions (photon flux and substrate concentration) at which the adsorption−desorption equilibrium existing at darkness is broken under illumination, leading to a drastic photo-oxidation kinetics change in those cases where DT plays an important role as interfacial charge transfer mechanism; on the other hand, we will try to quantify the participation of DT and IT mechanism in TiO2 photocatalytic reactions. A comparative, photo-oxidation kinetic analysis of model organic molecules, benzene, phenol, and formic acid dissolved in water and/or acetonitrile, will be presented in the second part of this publication9 in order to verify the applicability of the D-I model under both equilibrium and nonequilibrium adsorption− desorption conditions.

1/2 IT Examples of vIT plots according to ox versus ρ and vox versus ρ 14 −2 −1 eq 1, for 0 < ρ < 10 cm s , are shown in Figure 2A and B, 1/2 respectively. As expected, Figure 2B shows that vIT ox versus ρ 12 plot is linear above a critical illumination intensity ρc ≈ 10 cm−2 s−1.

2. RESULTS AND DISCUSSION 2.1. Interfacial Charge Transfer Mechanisms Involved in the D-I Model. (a). IT Mechanism under Nonlimited Diffusion of Dissolved Substrate Species. As defined in previous publications,7,26 when IT is the only possible electron transfer mechanism between dissolved substrate species and the sc surface, that is, in the absence of chemisorption, the theoretical initial photo-oxidation rate dependence on the photon flux and substrate concentration in the liquid (liq) phase, either water or any other organic solvent, is defined by the expression IT = vox

d[(RH 2)liq ] dt

(t → 0)

IT

= [(a [(RH 2)liq ])2 + 2a ITk 0ρ[(RH 2)liq ]]1/2 − a IT[(RH 2)liq ]

(1)

where a IT =

IT 1/2 Figure 2. Plots of vIT (B), according to eqs 1 and ox vs ρ (A) and vox vs ρ eq 1b, respectively, for 0 < ρ < 1014 cm−2 s−1, with k0aIT = 5.0 × 10−10 cm s−1. [(RH2)liq] values are 1 × 1016 (1), 3 × 1016 (2), 1 × 1017 (3), and 5 × 1017 cm−3 (4).

IT ′ [(O2 )liq ] kox k red

′ 4k r,1

(1a)

(b). DT Mechanism under Equilibrated Adsorption− Desorption of Dissolved Substrate Species. For high enough electronic interaction of substrate species with terminal TiIV atoms (chemisorption), DT is the prevailing photo-oxidation mechanism. Under equilibrated adsorption−desorption of substrate species, the initial photo-oxidation rate is defined by the expression:7,26

is a parameter with real physical meaning, that depends on the kinetic rate constants kIT ox and k′red, associated with adiabatic electron transfer, via tunnelling, from physisorbed RH 2 III •− molecules to O•− s /Obr radicals and from terminal Ti ions to electrolyte dissolved O2 molecules, respectively, and on the k′r,1 •− rate constant for recombination of e−f electrons with O•− s /Obr surface trapped holes; [(RH2)liq] represents the concentration of substrate species dissolved in the liquid phase, ρ is the photon flux absorbed by unit surface area of suspended semiconductor aggregates, and k0 represents the relationship between the flow of incident photons and photons absorbed by suspended semiconductor nanoparticles in the photoreactor. Under high enough illumination intensity and nonlimited diffusion conditions, such that ((ρ/[(RH2)liq]) ≫ (aIT/2k0)) and 2aITk0ρ[(RH2)liq] ≫ (aIT[(RH2)liq])2, eq 1 becomes IT vox

IT

1/2 1/2

≈ (2a k 0[(RH 2)liq ])

ρ

DT vox =

DT d[(RH 2)ads ] k 0kox [(RH 2)ads ] = ρ 2− DT dt k1′[Os ] + kox [(RH 2)ads ] (t → 0)

(2)

where [(RH2)ads], considered to have identical values under illumination than at darkness, can be described by a Langmuir type adsorption isotherm: [(RH 2)ads ] =

(1b) 14268

kadsb[RH 2]liq 1 + kads[RH 2]liq

(3)



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(c). Combination of IT and DT Mechanisms under Adsorption−Desorption Equilibrium of Dissolved Substrate IT Species. When mechanisms IT and DT coexist and vDT ox ≈ vox , the global photo-oxidation rate is represented by a combination of eqs 1 and 2:

where kads is the Langmuir adsorption constant, b is the maximum semiconductor surface area available for adsorption of substrate species, and [RH2]liq is their concentration in the liquid phase (water or any other solvent). It must be noted that, although eq 2 provides a chemical interpretation of the intervening kinetic constants, it is mathematically equivalent to the LH model photo-oxidation rate equation. As we will show in the second part of this publication,9 it is DT frequently found that k′1[O2− S ] > k0kox [(RH2)]ads; in this case, eq 2 becomes DT vox

k k DT[(RH 2)ads ] ≈ 0 ox ρ k1′[O2s −]

IT DT + vox vox = vox

≈ [(a IT[(RH 2)liq ])2 + 2a ITk 0ρ[(RH 2)liq ]]1/2 − a IT[(RH 2)liq ] +

DT k 0kox [(RH 2)ads ]ρ DT k1′[O2s −] + kox [(RH 2)ads ]

(4)

A plot of + versus ρ, according to eq 4, is shown in Figure 4A. Specially interesting is the observed decrease of d(vIT ox (vIT ox

(2a)

A plot of vDT ox versus ρ according to eq 2a is shown in Figure 3.

vDT ox )

14 −2 −1 Figure 3. vDT ox vs ρ plot, according to eq 2a, for 0 < ρ < 10 cm s , Kads 2− = 8.91 × 10−18cm3, b = 1.39 × 1015 cm−2, k0kDT /k ′ [O ] = 2 × 10−16 ox 1 s 2 16 cm , under the following [(RH2)liq] values: 1 × 10 (1), 3 × 1016 (2), 1 × 1017 (3), and 5 × 1017 cm−3 (4).

The main difference between eqs 1 and 2 and eq 2a concerns the predicted photon flux reaction order, n, being n = 1/2 for the IT mechanism above the critical photon flux value (ρc ≫ aIT/2k0 [(RH2)liq]), under nonlimited diffusion of dissolved substrate species, and n = 1 for the DT mechanism under adsorption− desorption equilibrium conditions. Therefore, an easy discrimination between IT and DT as prevailing photo-oxidation mechanism is possible from a simple analysis of the photon flux reaction order. As will be shown in the second part of this publication,9 a IT typical example of pure DT photo-oxidation kinetics (vDT ox ≫ vox ) under equilibrated adsorption−desorption conditions, where n = 1, is that corresponding to the photo-oxidation of water dissolved formic acid, a substrate able to become strongly anchored to terminal TiIV ligands of the TiO2 surface in competition with water molecules.33 Literature examples where IT apparently prevails on DT photo-oxidation mechanism, with n = 1/2 under high enough illumination intensity, is that corresponding to the photopolymerization of methyl metacrylate on ZnO34 and the photo-oxidation of organic species like chloroform35 and methyl viologen dissolved in water.36 Another emblematic case of IT mechanism is that corresponding to benzene photo-oxidation in acetonitrile, to be analyzed in detail in the second part of this publication.9

IT DT IT 1/2 Figure 4. Plots of vDT (B), according ox + vox vs ρ (A) and vox + vox vs ρ to eq 4, for 0 < ρ < 1014 cm−2 s−1, k0 = 0.1, aIT = 3.3 × 10−9 cm s−1, kDT ox = 1.42 × 10−8 cm3 s−1, Kads = 1.33 × 10−17 cm3, b = 9.39 × 1013 cm−2, and 8 −1 16 16 k′1 [O2− s ] = 2.36 × 10 cm s . [(RH2)liq] values are 1 × 10 (1), 3 × 10 17 17 −3 (2), 1 × 10 (3), and 5 × 10 cm (4). For comparison, the dashed 1/2 for k0aIT = 3.3 × 10−10 cm s−1. lines represent the linear plot of vIT ox vs ρ

+ vDT ox )/dρ as ρ increases, a behavior similar to the observed in Figure 2A when IT predominates on DT. Accordingly a plot of vox versus ρ is not enough to discriminate between both mechanisms. However, this is not the case when the nonlinear DT 1/2 (vIT plot of Figure 4B is compared with the ox + vox ) versus ρ IT 1/2 linear (vox ) versus ρ plot of Figure 2B, which allows an easy 14269



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2− Figure 5. Influence of parameters kDT ox (A), k1′ [Os ] (B), kads (C), and b (D) on the photon flux dependence of the concentration of adsorbed substrate −8 3 −1 species, according to eq 9, for kads = 5.2 × 10−18 cm3 in (A), (B), and (D); b = 1013 cm−2 in (A), (B), and (C); kDT ox = 10 cm s in (B), (C), and (D); 2− 8 −1 16 −3 and k1′ [Os ] = 2.36 × 10 cm s in (A), (C), (D). Common values of the remaining parameters are [RH2]liq = 10 cm , k−1 = 10−20 cm3 s−1, and k0 = 10−1.

the photon flux and concentration dependence of the photooxidation rate (r):

discrimination of the complex IT + DT mechanism from IT and DT. Against the D-I model prediction that the reaction order strongly depends on the type of electronic sc-substrate interaction, it is generally admitted in photocatalysis that the photo-oxidation reaction order exclusively depends on the photon flux intensity, being in general n = 1 for low enough ρ values and n = 1/2 for high enough illumination intensity.24 As we are going to show in the second part of this publication,9 this assumption contradicts the experimental evidence that within a similar photon flux range (1013 < ρ < 1014 cm−2 s−1), n = 1 for formic acid dissolved in water and phenol dissolved in acetonitrile, while n = 0.5 for benzene dissolved in acetonitrile and 0.5 < n < 1 for phenol dissolved in water. 2.2. DT Photo-Oxidation Kinetics under Nonequilibrated Adsorption−Desorption of Chemisorbed Substrate Species: Photo-Oxidation Rate Dependence on Photon Flux and Substrate Concentration. Usually, photocatalytic kinetic models, including the LH model, have been developed under the assumption that the adsorption− desorption equilibrium existing in the dark is not perturbed under illumination, an assumption only valid under low enough illumination intensity. However, any autoconsistent photocatalytic kinetic model should take into account that the steady state concentration of chemisorbed substrate species strongly depends on the photon flux. According to the L-H model and considering that •OH radicals generated from the photooxidation of water molecules behave as primary photo-oxidizing species, Ollis has developed a pseudo-steady-state kinetic analysis based on the existence of nonequilibrated adsorption of reactants.5 The author arrives to the following expression for

r=

app app kcat kdis [(RH 2)ads ] app kdis + [(RH 2)ads ]

(5)

While in eq 5 the concentration of dissolved substrate species, [(RH2)liq], explicitly appears as an independent experimental variable, the photon flux only appears implicit in the apparent app rate constant, kapp cat , and in the dissociation constant, kdis , as far as both constants are considered to depend on the photon flux. The fact that the photon flux does not appear explicitly in eq 5 represents an important drawback of Ollis kinetic analysis under nonequilibrated adsorption of reactants. This inconvenience disappears when the pseudo-steady-state approach proposed by Ollis is incorporated into the D-I model, and h+f holes, instead of • OH radicals generated from the photo-oxidation of TiO2 adsorbed or free water molecules, are considered as primary photo-oxidizing species of chemisorbed substrate molecules. The following explicit photon flux dependent expression is then obtained for the concentration of adsorbed substrate species:7

{ /{1 + k

}

[(RH 2)ads ] = kadsb[(RH 2)liq ]

ads[(RH 2)liq ]

DT /k −1⎤⎦ + ⎡⎣k ox

DT [(RH 2)ads ]⎤⎦⎤⎦ × ⎡⎣k 0ρ /⎡⎣k1′[O2s −] + kox

} (6)

where k−1 is the diffusion constant of substrate species from the sc surface toward the solution and the remaining parameters 14270



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Finally, Figure 6 shows the photon flux influence on the L-H adsorption isotherm ([(RH2)ads] versus [(RH2)liq] plot) under

have been previously defined. A detailed description of how eq 6 was obtained is shown in the Supporting Information. As pointed out in Figure 1, and against the thesis recently defended by Valencia et al.,37 the back reaction of electroreduction of photogenerated RH• radicals to RH2 molecules is not considered in obtaining eq 6; as much as we are dealing with initial photo-oxidation rates and [(RH•)ads] is negligible versus [(RH2)ads], such a back reaction cannot have a real influence on the adsorption−desorption equilibrium rupture taking place under high enough illumination intensities.32 It must be noted that for low enough ρ values, when DT adsorption−desorption equilibrium exists and (kox )/(k−1) DT ((k0ρ)/(k′1[O2− ] + k [(RH ) ]) ≪ 1 + k [(RH s ox 2 ads ads 2)liq], eq 6 becomes identical to eq 3. Let us reorganize eq 6 in the following way: [(RH 2)ads ] = ⎡⎣kadsb[(RH 2)liq ]k1′[O2s −] + kadsb[(RH 2)liq ]

{

DT [(RH 2)ads ]⎤⎦ kox

/⎡⎣kads[(RH 2)liq ]k1′[O2s −] + k1′[O2s −]

Figure 6. Effect of illumination intensity on the relationship between concentrations of adsorbed and dissolved substrate species, according 8 −1 13 to eq 9. Parameters used are k′1 [O2− s ] = 2.36 × 10 cm s ; b = 9.4 × 10 −2 −20 3 −1 DT −8 3 −1 −1 cm ; k1 = 10 cm s ; kox = 1.42 × 10 cm s ; k0 = 10 ; kads = 1 × 10−17 cm3. ρ values are 0 (1), 5 × 1014 (2), 5 × 1015 (3), and 5 × 1016 cm−2 s−1 (4). Curve (1) corresponds to the L-H isotherm at darkness.

DT [(RH 2)ads ] + kads[(RH 2)liq ]kox DT DT [(RH 2)ads ] + ⎡⎣(kox /k −1) × k 0ρ⎤⎦⎤⎦ + kox

} (7)

Equation 7 can be transformed into the following second order equation: ⎫ ⎧[(RH ) ]2 (k [(RH ) ]k DT + k DT) 2 ads ads 2 liq ox ox ⎪ ⎪ ⎪ ⎪ ⎛ ⎪+ [(RH ) ]⎜k k ′[O2 −][(RH ) ] + k ′[O2 −]⎪ 2 ads ads 1 s 2 liq 1 s ⎬=0 ⎨ ⎝ ⎪ ⎪ DT ⎞ kox ⎪ ⎪ DT ⎟ k 0ρ − kadsb[(RH 2)liq ]kox ⎪ ⎪ + k ⎠ −1 ⎭ ⎩

broken adsorption−desorption equilibrium conditions. In general, a diminution of d[(RH2)ads]/d[(RH2)liq] is observed as the photon flow grows up, this tendency being more evident as [(RH2)liq] decreases. All these changes in tendency of d[(RH2)ads]/d[(RH2)liq] are inherent to the adsorption− desorption equilibrium rupture and will be further discussed in detail. The following photon flux dependence of the DT, initial photo-oxidation rate under nonequilibrated adsorption−desorption conditions is obtained by substituting into eq 2 the expression for [(RH2)ads] in (eq 9):

(8)

The only solution with physical meaning of eq 8 is [(RH 2)ads ] =

−(Xρ − H ) + {(Xρ − H )2 + A}1/2 L

−1 k0kDT ox /k ,

(9)

DT = vox,neq

where X = H = γb − α − β, L = 2(γ + and A = 4 2− 2− αb(γ + kDT ox ), while α = kadsk1′ [Os ][(RH2)liq], β = k1′ [Os ], and γ DT = kadskox [(RH2)liq]. As already pointed out, it must be emphasized that the parameters intervening in eq 9 cannot be considered as adjustable parameters, on the contrary, they have real physical meaning and well-defined values, not always known with exactitude. As expected, under low enough ρ values, such that A ≫ (Xρ − H)2, the adsorption−desorption equilibrium is re-established and [(RH2)ads] becomes photon flux independent. Figure 5 2− illustrates the influence of kDT ox , k′1 [Os ], kads and b, on the photon flux dependence of [(RH2)ads] according to eq 9. In general, the adsorption−desorption equilibrium existing in the dark (i.e., for ρ = 0) becomes broken as ρ reaches high enough values, which leads to a pronounced, monotonous decrease of DT both [(RH2)ads] and d(aDT ap )/dρ = (1/2k0[(RH2)liq]ρ) [2vox DT DT 2 (d(vox )/dρ) − ((vox ) /ρ)]. For ρ = constant, [(RH2)ads] is observed to increase as parameters kads, k1′ [O2− s ], and b increase; in contrast, [(RH2)ads] is observed to decrease as kDT ox increases; DT the decrease of kox and b leads to a clear decrease of (d[(RH2)ads]/dρ), indicating that adsorption is the limiting step in DT photo-oxidation of substrate species. kDT ox ),

DT 2 1/2 kox {⎡⎣[(Xρ − H ) + A] 2− DT 2 k1′[Os ] + kox {⎡⎣[(Xρ − H ) +

− (Xρ − H )⎤⎦/L}ρ A]1/2 − (Xρ − H )⎤⎦/L} (10)

As expected, under low enough ρ values, such that (Xρ − H)2 ≪ A, the equilibrated adsorption−desorption conditions are recovered and a linear dependence of vDT ox on the photon flux is predicted by eq 10. •− In those cases that kDT ox ≤ k′1 and [(RH2)ads] ≪ [Os ], so that DT •− kox [(RH2)ads] ≪ k′1 [Os ], eq 10 leads to the following simplified expression of vDT ox : DT vox ≈

DT kox k1′[O•− s ] 2 ⎡ {X ρ 4 − 2XHρ3 + H2ρ2 + Aρ2 }1/2 − Xρ2 + Hρ ⎤ ⎢ ⎥ L ⎣ ⎦ (10a)

According to eq 10a, under low enough ρ values, such that H2ρ2 ≫ X2ρ4, 2XHρ3, Aρ2 and Aρ2 ≫ X2ρ4, 2XHρ3, and Hρ ≫ Xρ2, eq 10a becomes 14271


The Journal of Physical Chemistry C DT vox ≈

DT kox {H2 + A2 } + H ρ •− k1′[Os ] L

Article DT = aap

(10b)

indicating that a photo-oxidation rate order n = 1 prevails under low enough illumination intensity. In contrast, as ρ increases and the adsorption−desorption equilibrium becomes broken, so that X2ρ4 > 2XHρ3 > H2ρ2 > Aρ2, eq 10b is not anymore valid, the DT DT lineal dependence of vox on ρ disappears and dvox /dρ DT progressively decreases, in such a way that dvox /dρ → 0 (i.e., 0 DT vDT ox approaches a constant value), equivalent to say that vox ∝ ρ , which indicates a tendency to a zero order photo-oxidation rate as the photon flux increases. But considering that n = 1 and 0 under low enough and high ρ values, respectively, a reaction order 0 ≤ n ≤ 1 is to be expected for intermediate photon flux values. This result again contradicts the frequently invoked axiom that, independently of the nature of the sc-substrate electronic interaction, the photo-oxidation rate order exclusively depends on the photon flux intensity, being in general n = 1 and 1/2 for low enough and high illumination intensities, respectively. The behavior predicted by eq 10a is especially evident in the log(vDT ox ) versus log(ρ) plot under variable substrate concentration shown in Figure 7; particularly interesting is the transition from n = 1 to 0 taking place under high enough ρ values, and its dependence on [(RH2)liq].

DT 2 (vox ) 2k 0[(RH 2)liq ]ρ

(12)

According to eq 12, DT d(aap )

dρ

=

DT ⎡ (v DT)2 ⎤ 1 DT d(vox ) ⎢2vox − ox ⎥ 2k 0[(RH 2)liq ]ρ ⎣ dρ ρ ⎦

(13)

On the other hand, on the basis of eq 2, and assuming that DT k1′ [O2− s ] ≅ kox [(RH2)ads], it can be written DT 2 ⎛ k DT[(RH ) ] ⎞2 (vox ) 2 ads ⎟ρ ≈ ⎜ ox 2− ρ ⎝ k1′[Os ] ⎠

(14)

Finally, by combining eqs 12−14, the following expressions DT are obtained for aDT ap and daap /dρ, respectively: DT aap ≈

⎛ k DT[(RH ) ] ⎞2 1 2 ads ⎜ ox ⎟ρ 2[(RH 2)liq ] ⎝ k1′[O2s −] ⎠

(15)

and DT d(aap )

dρ

≈

DT 2 (kox ) 2[(RH 2)liq ](k1′[O2s −])2

⎧ d[(RH 2)ads ] ⎫ ⎬ × ⎨[(RH 2)ads ]2 + 2ρ[(RH 2)ads ] dρ ⎩ ⎭ (16)

Two different cases can be considered: (a). Low Enough Photon Flux, such that (Xρ − H)2 ≪ A in Eq 9, while [(RH2)ads] Does Not Depend on the Photon Flux. In this case, according to eq 15, aDT ap depends linearly on ρ, while d(aDT ap )/dρ > 0 according to eq 16. (b). High Enough Photon Flux, such that (Xρ − H)2 ≫ A in Eq 9. According to Figure 5, we know that [(RH2)ads] decreases as ρ increases, so that the following expressions are obtained from eq 2a: DT DT d(vox ) kox d[(RH 2)ads ] = [(RH 2)ads ] + ρ 2− dρ dρ k1′[Os ]

Figure 7. Logarithmic representation of the VDT ox dependence on the photon flux, according to eq 10a, for 1013 < ρ < 1014 cm−2 s−1, k0 = 0.1, 8 −1 13 Kads = 1.33 × 10−17 cm3, k1′ [O2− s ] = 2.36 × 10 cm s , b = 9.39 × 10 −8 3 −1 15 cm−2, and kDT ox = 1.42 × 10 cm s . [(RH2)liq] values are 1 × 10 (1), 1 × 1016 (2), 5 × 1017 (3), and 5 × 1018 cm−3 (4).

(17)

DT DT ⎞ d(vox ) ⎛ kox d[(RH 2)ads ] ⎟ ([(RH 2)ads ])2 ρ + ρ ≈⎜ 2− ′ dρ dρ ⎝ k1[Os ] ⎠ 2

DT vox

(18)

and DT 2 ⎛ k DT ⎞2 (vox ) ≈ ⎜ ox 2 − ⎟ ([(RH 2)ads ])2 ρ ρ ⎝ k1′[Os ] ⎠

2.3. The (vox)2/2ρ[(RH2)liq] versus ρ Plot under Variable Substrate Concentration: an Experimental Test for Differentiating DT and IT Mechanisms. It was shown in Section 2.1(a) that under high enough illumination intensity values, within the standard experimental range, eq 1a leads to a IT ≡

IT ′ [O2 ]aq kox k red

′ 4k r,1

=

IT 2 (vox ) = const 2k 0[(RH 2)liq ]ρ

(19)

Finally, by substituting eq 17 and eq 19 into eq 13, the following expression for d(aDT ap )/dρ is obtained: DT DT ⎞ d(aox ) [(RH 2)ads ] ⎛ kox ⎜ ⎟ = dρ 2[(RH 2)liq ] ⎝ k1′[O2s −] ⎠

(11)

⎧ d[(RH 2)ads ] ⎫ ⎬ × ⎨[(RH 2)ads ] + 2ρ dρ ⎩ ⎭

indicating that when the photo-oxidation kinetics is governed by 2 an IT mechanism, the term (vIT ox ) /2k0ρ [(RH2)liq] does not depend on the photon flux, whatever the substrate concentration used. By analogy with eq 11, the following “apparent” aDT ap parameter can be defined for the DT mechanism:

(20)

Let us analyze eq 20 in detail. It is known from Figure 5 that under nonequilibrated adsorption−desorption conditions (i.e., for high enough photon flux) d[(RH2)ads]/dρ < 0, so that 14272



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according to eq 20, d(aDT ap )/dρ < 0 for [(RH2)ads] < 2ρ (d[(RH2)ads]/dρ). In contrast, under equilibrated adsorption− desorption conditions (i.e., for low enough photon flux) daDT ap / dρ > 0 and eq 20 predicts the existence of an intermediate photon flux value where d(aDT ap )/dρ = 0, in such a way that aDT (ρ) must get a maximum value for a certain photon flux: ap ρmax

[(RH 2)ads ] = 2[(d[(RH 2)ads ]) /dρ]

IT + DT = aap

=

IT DT 2 + vox (vox ) 2k 0ρ[(RH 2)liq ] IT 2 DT 2 IT DT (vox ) + (vox ) + 2vox vox 2k 0ρ[(RH 2)liq ]

DT = a IT + aap +

(21)

IT DT vox vox k 0ρ[(RH 2)liq ]

(22)

Obviously, eq 11 is a particular case of eq 22 for vDT ox = 0, in the absence of chemisorption of substrate species, with aIT+DT  aIT ap 2 IT = (vDT ) /2k ρ[(RH ) ] = const. On the contrary, for v ≪ vDT ox 0 2 liq ox ox ,

in such a way that [(RH2)ads] = |2ρmax d([(RH2)ads]/dρ)|. Moreover, we know from Figure 6 that [(RH2)ads] increases and d([(RH2)ads]/dρ decreases as [(RH2)liq] increases, so that according to eq 21 ρmax should increase parallel to the substrate concentration. Summing up, for low enough ρ values, under equilibrated n adsorption−desorption conditions, we found that aDT ap ∝ ρ with DT n = 1, which means that aap increases linearly with the photon flux. However, the linear dependence of aDT ap on ρ is lost when high enough photon flux values are reached and adsorption− desorption equilibrium is broken. Then, aDT ap reaches a maximum value for ρ = ρmax and starts to decrease for ρ > ρmax. Moreover, under low enough photon flux, aDT ap is found to decreases as [(RH2)liq] increases, while ρmax is observed to shift toward higher photon flux values. However, this tendency can be inverted under high enough photon flux, in such a way that aDT ap is expected to increase as [(RH2)liq] increases. This is just the behavior observed in Figure 8 for the dependence of aDT ap on ρ, under different values of [(RH2)liq], when eq 9 and eq 15 are combined.

taking into account eqs 2, 2a, and 3, eq 22 becomes IT + DT DT aap ≡ aap

=

=

DT 2 (vox ) 2k 0ρ[(RH 2)liq ] DT 2 k 0(kox ) (kads)2 b2[(RH 2)liq ]

2(1 + kads[(RH 2)liq ])2 (k11[O2s −])2

ρ (23)

indicating that depends linearly on the photon flux. Figure 9 compares the photon flux dependence of aIT+DT and ap aDT ap , respectively. Note that, in spite that the parameters and the photon flux range used in Figures 8 and 9 for obtaining aDT ap versus ρ plots are far different, the aDT ap dependence on photon flux observed in both figures is very similar. Moreover, according to Figure 9, the main difference between aIT+DT and aDT ap ap concern a shift of aIT+DT toward higher values, in such a way that, ap apparently, for ρ = 0, aIT+DT > 0, while aDT ap ap = 0. aIT+DT ap

3. CONCLUSIONS In the absence of chemisorption, when photo-oxidation of substrate species in the liquid phase takes place exclusively via an IT mechanism, the photo-oxidation reaction order under high enough illumination intensity (standard experimental condi1/2 tions) is n = 1/2(vIT ox ∝ ρ ). Contrarily, under chemisorption of substrate species two cases can be considered: (1) under low enough illumination intensity and equilibrated adsorption− 1 desorption conditions, n = 1(vIT ox ∝ ρ ) in those cases were DT DT IT IT prevails on IT (vox ≫ vox ), while for (vDT ox ≈ vox ), it is n < 1 in the whole photon flux range; (2) under high enough illumination intensity, when adsorption−desorption equilibrium rupture is induced, the surface concentration of adsorbed substrate species is found to decrease as ρ increases, while the photo-oxidation reaction order progressively decreases from n = 1 to n = 0, in such a way that unambiguous differentiation between the DT and the IT mechanism is not always evident. However, the analysis of the photon flux and substrate concentration dependence of the parameter defined as aap = (vox,exp)2/2k0ρ[(RH2)liq], where vox,exp represents the initial, IT experimental photo-oxidation rate (vox,exp = vDT ox + vox ), allows to unequivocally determine the nature of the prevailing photooxidation mechanism taking place at the semiconductor−liquid interface, DT, IT, or a combination of both, during the photocatalytic reaction. Three cases can then be considered: (1) in the absence of chemisorption or when physisorption is the operative adsorption IT DT mechanism (vIT ox ≫ vox ), it is found that aap ≡ a = const under high enough ρ values, for any electrolyte concentration; (2) IT when chemisorption and physisorption coexist but vDT ox ≫ vox , it DT n is aap = aap ∝ ρ with n = 1 for any [(RH2)liq] value; (3) when

15 −2 Figure 8. aDT ap photon flux dependence, in the range 0 < ρ < 10 cm −1 s , obtained by combining eq 9 and eq 15, under the following values of the intervening parameters: kads = 8.9 × 10−18 cm3; b = 1.4 × 1015 cm−2; 8 −1 −20 −7 cm3 s−1; kDT cm3 k1′ [O2− s ] = 1 × 10 cm s ; k1 = 1 × 10 ax = 2 × 10 s−1; and k0 = 0.1. [(RH2)liq] values are 1 × 1016 (1), 1 × 1017 (2), 5 × 1017 (3), 1 × 1020 (4), and 3 × 1020 cm−3 (5).

Parameter. Let us now return to the 2.4. Apparent aIT+DT ap general case where both mechanisms IT and DT contribute to the photo-oxidation of substrate species with analogous values, DT so that vox = vIT ox + vox . Emulating eq 11, let us define the apparent parameter

14273



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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 34 93 581 2772. Fax: 34 93 581 2920. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the Spanish “Ministerio de Ciencia e Innovación” through both Project CTQ 2008-00178 and the research fellowship granted to J.F.M. is acknowledged.

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Figure 9. Comparative plot of the photon flux dependence of aIT+DT ap 17 −2 −1 from eq 22 (A) and aDT ap from eq 23 (B), under 0 < ρ < 10 cm s and aIT = 3.3 × 10−9 cm s−1, for the following parameters: kads= 1.33 × 10−17 −8 cm3; b = 9.39 × 1013 cm−2; k1 = 9 × 10−20 cm3 s−1; kDT ox = 1.42 × 10 2− −18 cm3 s−1; k0kDT cm3 s−1 and k0 = 0.1. [(RH2)liq] ox /k1′ [Os ] = 6 × 10 values are 1 × 1017 (1), 5 × 1017 (2), 1 × 1018 (3), 2 × 1020 (4), and 8 × 1020 cm−3 (5).

IT chemisorption and physisorption coexist and vDT ox ≈ vox , it is aap = IT DT DT IT a + aap + vox + vox /k0ρ[(RH2)liq], aap being undetermined for ρ = 0; in contrast, for ρ > 0, aap is observed to increases linearly with ρ under low enough illumination intensity, as long as the adsorption−desorption equilibrium existing in the dark is maintained under illumination, but linearity disappears as the photon flux increases and adsorption−desorption equilibrium is broken, as it was the case for aDT ap , in such a way that a critical photon flux ρ = ρmax is reached where d(aDT ap )/dρ = 0 or, in other DT words, aap reaches a maximum value for ρ = ρmax. Finally, ρmax is found to increase as [(RH2)liq] increases.

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ASSOCIATED CONTENT

S Supporting Information *

Details of the mathematical deduction of eq 6. This material is available free of charge via the Internet at http://pubs.acs.org. 14274

LIST OF ACRONYMS a : indirect transfer parameter defined by eq 1a. aDT ap : apparent, direct charge transfer parameter defined by eq 12. aIT+DT : apparent, (direct + indirect) charge transfer parameter ap defined by eq 22. b: semiconductor surface sites available for chemisorption of dissolved substrate species. DT: electric charge, interfacial direct transfer mechanism. e−f : conduction band free electrons. h+f : valence band free holes. IT: electric charge, interfacial indirect transfer mechanism. kads: substrate species adsorption constant. k1: diffusion constant of dissolved substrate species from the electrolyte toward the sc surface. k−1: diffusion constant of dissolved substrate species from the sc surface toward the electrolyte. kIT ox : indirect transfer photo-oxidation rate constant. kDT ox : direct transfer photo-oxidation rate constant. k′red: recombination rate constant of conduction band free electrons with electrolyte dissolved O2 molecules. k1′ : diffusion constant of dissolved substrate species from the electrolyte toward the semiconductor surface. k0: empirical photon absorption constant. kr′1: interfacial recombination rate constant of conduction band free electrons with O•− s radicals. λ: redox couple reorganization energy. [(O2)liq]: concentration of dissolved O2 molecules in the liquid phase (water or any other solvent). >O2− br : 2-fold coordinated terminal bridging oxygen ions at the TiO2 surface. −O•− br : 1-fold coordinated terminal bridging oxygen radicals at the TiO2 surface, generated via photo-oxidation of >O2− br ions with valence band free holes. >O2− s : in general, low coordinated terminal oxygen ions present at kinks and steps at the surface of suspended TiO2 nanoparticles. O•− s : terminal oxygen radicals at the TiO2 surface, generated via photo-oxidation of >O2− s ions with valence band free holes. [(RH2)liq]: concentration of dissolved substrate species in the liquid phase (water or any other solvent). [(RH2)ads]: semiconductor surface concentration of adsorbed substrate species. ρ: photon flux intensity. vDT ox : initial, direct transfer photo-oxidation rate of semiconductor chemisorbed substrate species, defined by eq 2. vIT ox : initial, indirect transfer photo-oxidation rate of semiconductor physisorbed substrate species, defined by eq 1. vox: global, initial photo-oxidation rate of dissolved substrate DT species (vox = vIT ox + vox ). IT



Article

Θa: fraction of semiconductor surface sites covered with chemisorbed substrate species.

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Comprehensive Kinetic and Mechanistic Analysis of TiO2

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