Langmuir-Hinshelwood and Light-Intensity Dependence Analyses of

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Langmuir−Hinshelwood and Light-Intensity Dependence Analyses of Photocatalytic Oxidation Rates by Two-Dimensional-Ladder Kinetic Simulation Yoshio Nosaka* and Atsuko Y. Nosaka Department of Materials Science and Technology, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka 940-2188, Japan

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S Supporting Information *

ABSTRACT: Though photocatalytic reactions gather enormous attention, dependences of light intensity and reactant concentration have not been concurrently expressed clearly. In the previously reported studies, a quadratic formula equation obtained from the conventional analysis using the concentration of electron−hole pairs has been modified. In this report we numerically simulated the reaction with a two-dimensional(2D)ladder kinetics without using electron−hole concentrations. In this kinetics, the rates of four fundamental processes, (i) photoabsorption, (ii) reduction, (iii) oxidation, and (iv) recombination, were treated as the transitions between the states of each powder characterized by the numbers of possessing negative and positive charges. Through the numerical 2D-ladder simulation with various rate constants, the light-intensity (I) dependence of the oxidation rate was found to be fully expressed by involving the square of the intrinsic quantum yield into the square root part of the well-known quadratic formula equation. The square root dependence of the reaction rate, r ∝ I1/2, could be expected only when the rate of the reduction is extremely smaller than the recombination rate at the normal light intensity. Then, the resultant equations obtained with this 2D-ladder simulation were transformed to the equation for Langmuir−Hinshelwood kinetics with two parameters, rL and KL, which correspond to the intrinsic oxidation rate and the adsorption equilibrium constant of the reactant, respectively. Light-intensity dependence of KL was expressed by adding two terms proportional to I2 to both the adsorption and desorption rate constants. The reported experimental data sets of the decomposition rates for phenol and 4-chlorophenol were fitted with the proposed equation, and then from the obtained parameter values the formation rate of •O2− could be estimated and found to be compatible to that which has been experimentally measured. Thus, the present analytical treatment is actually the simple and useful method to understand the dependencies of reactant concentration and light-intensity on the photocatalytic oxidation rates. and many efforts have been devoted to it.5−10 In most analyses, the amounts of photoinduced electron−hole pairs have been treated by the concentration similarly to the molecules in solution. On semiconductor photocatalysis, however, electrons and holes are confined in a particle and hence the electron− hole distance does not increase with the decrease of the concentration. This problem has not been invoked in most kinetic analyses in photocatalysis. Recently, without the concept of concentration, the light intensity dependence of quantum yields has been calculated by using a Monte Carlo random walking model.11 The present authors reported previously the dependencies of electron-transfer quantum yield on the light intensity and the particle size for pulsedlaser irradiated CdS colloidal solution and analyzed them by using a two-dimensional (2D)-ladder kinetic model.12 In the

1. INTRODUCTION TiO2 photocatalytic reaction, which causes the oxidation and reduction of molecules by using the photon energy absorbed in a solid semiconductor, gathers enormous attention owing to the unique process to be applied for the decomposition of polutants.1−4 Therefore, many researchers have tried to clarify the photocatalytic processes aiming to improve the reaction efficiency for specified reactants. The primary process of semiconductor photocatalysis is the usage of the photoinduced electrons and holes for the surface redox reactions, and accordingly, the techniques for the analysis of photocatalysis span the research fields of catalysis, electrochemistry, solid state physics, and photochemistry. Therefore, Langmuir− Hinshelwood (L-H) kinetics in catalysis have been used to investigate the dependence of the photocatalytic reaction rate on the amount of reactant. The adsorption constant in the L-H analysis may be altered by photoirradiation. However, explanation of the light-intensity dependence in photochemistry could not be easily combined with the L-H analysis © XXXX American Chemical Society

Received: September 26, 2018 Revised: November 21, 2018 Published: November 26, 2018 A

DOI: 10.1021/acs.jpcc.8b09421 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

solution are reduced by e−tr (paths 3 + 4). The reduction of molecules is completed by the desorption from the surface to become A−. In another case, Aad− suffers the chemical reaction to convert into the other chemical species. Only these two processes should be counted as the reduction, because the formed Aad− can return to Aad by some oxidation process (path 12). (iii) Similarly to the reduction process, the oxidation process (iii) involves the direct oxidation path 5 of adsorbed electron-donor molecules (Dad) with the valence-band holes and the indirect oxidation (paths 6 + 7) via the trapped holes (h+tr). The formation rate of Dad+ is not simply accounted as the oxidation rate because the Dad+ species could return to Dad by the reduction (path 13), which should be counted as recombination. (iv) Recombination process involves several charge-transfer reactions between e−CB or e−tr and h+VB or h+tr (paths 8− 11). As stated above, the oxidation of reduced surface species (path 12) and the reduction of oxidized surface species (path 13) are classified as the recombination in the 2-D ladder simulation because these reactions do not contribute to the net photocatalytic reaction. The first oxidized species, D+ in Figure 1, are commonly radicals which initiate radical chain reactions with O2 in solution;13,14 however, the chain reactions could not control the net reaction rate because the photoexcitation is slow as compared to the surface electron transfer reaction. Furthermore, the reactant concentration for the chain reactions is too high to limit the net oxidation rate. Therefore, though the rate of photocatalytic oxidation is measured as the decrease of the reactant concentration, the oxidation of the first step would control the whole oxidation process, suggesting that the reaction rate for the first oxidizing molecule should be compatible to that for the reactant molecules. From the viewpoint of each photocatalyst particle, the above four processes could be illustrated in Figure 2. The particle at

2D-ladder kinetic analysis, colloidal particles are classified with the number of electrons and holes induced by the photo excitation, and the reduction rate was calculated as the transition rate of particle states in which the number of electrons is decreased by one. Therefore, the 2D-ladder kinetics does not take the concentrations of the photoinduced electrons and holes into the kinetic analysis and could become good simulation to the real experimental situation. Hence, the present study is devoted to apply the 2D-ladder kinetics to calculate the TiO2 photocatalytic reaction rates under continuous light excitation, and the resultant equations will be expressed by the Langmuir−Hinshelwood equation to compare with the reported experimental data.

2. THEORY AND RESULTS 2.1. Simulation with 2D-Ladder Kinetics. In photocatalytic reactions, the number of processes which can occur at the surface of the semiconductor solid could be counted as 13 as shown in Figure 1.4 In addition they could be categorized

Figure 1. General reaction paths of photocatalysis with the oxidation of molecule D and the reduction of molecule A.

into four net reaction processes: (i) photo excitation to generate negative and positive charges, (ii) reduction of surface electron-acceptor molecules, (iii) oxidation of surface electrondonor molecules, and (iv) recombination of charge pairs. The details of the processes follow. (i) The photocatalytic reaction starts from the photoexcitation process in which conduction-band electrons (e−CB) and valence-band holes (h+VB) are generated. The rate of the generation of electron and hole pairs (g) corresponds to the number of photons absorbed in a semiconductor particle within a unit time (1 s). Usually the size of the particle is smaller than the penetration depth or the reciprocal absorption coefficient (1/α) of the light. Therefore, the generation rate g is approximated as g = VαI, where V and I are the volume of one particle and the intensity of the irradiation light, respectively. (ii) The reduction process contains the direct reduction (path 2 in Figure 1 [paths are represented in text as bold numbers and presented as circled numbers in the figure]) by which the conduction-band electrons transfer to the electron-acceptor molecules adsorbed on the photocatalyst surface (Aad). The process also contains the indirect reduction in which the electrons are first stabilized at the surface to produce trapped electrons (e−tr) and then the molecules A diffused from the

Figure 2. 2D-ladder kinetics represented by the change of the numbers of electrons and holes (negative and positive charges) in a photocatalyst particle on the photoexcitation (with the rate constant of g), oxidation (κh), reduction (κe), and recombination (κr).

the left bottom is in the dark state before the light irradiation. When the particle absorbs one photon, an electron and a hole are generated and then the state of the particle moves to the upper right position at the rate of g. In the same way, the reduction, the oxidation, and the recombination change the state of the particle in this kinetics. The numbers of electrons and holes which characterize the states are decreased by one along with the surface reduction and oxidation at the rates of κe B


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The Journal of Physical Chemistry C and κh, respectively. On the other hand, the numbers of electrons and holes are also decreased simultaneously by one with the recombination at the rate of κr. The particles which contain n electrons and m holes are represented by the fraction Xnm to simulate the 2D-ladder kinetics. As shown in Figure 3, when a particle absorbs one

rA =

∑ ∑ nκeXmn n≥1 m≥0

rD =

(2)

∑ ∑ mκhXmn n≥0 m≥1

(3)

Furthermore, the quantum yield is obtained as eq 4 by dividing by the excitation rate r ϕ= D g (4) In the steady state, the rates of the reduction and the oxidation should be equal with each other as eq 5. rA = rD (5) The average number of electrons and holes, ⟨n⟩ and ⟨m⟩, can be expressed by eqs 6 and 7, respectively. ⟨n⟩ =

∑ ∑ nXmn n≥1 m≥0

Figure 3. Two-dimensional ladder kinetics for the photoinduced reaction at semiconductor particles. Xnm represents the existence probability of a particle having n electrons (negative charges) and m holes (positive charges), which are both induced by photon absorption. Transition rate of Xnm is represented by eq 1.

⟨m⟩ =

∑ ∑ mXmn n≥0 m≥1

(7)

Substitution of eqs 2 and 3 for eq 5 indicates that the ratio of the average number of electrons and holes calculated with eqs 6 and 7 is equal to the inverse of the ratio of rate constants as eq 8.

photon, the state of the particle changes from Xnm to Xn+1 m+1. In the case of oxidation, the number of holes decreases from m to m − 1 or the state changes from Xnm to Xnm−1 at the rate of mκh. For convenience, the reaction rate between states was assumed to be proportional to the number of charges in the particle. Thus, the rate equation for the state having the fraction of Xnm would be given by eq 1.12

κ ⟨n⟩ = h ⟨m⟩ κe

(8)

The simulation program was written with FORTRAN language and compiled with a free software15 worked on the MS Windows 7 operating system. The size of the matrix, n × m, was 500 × 500, and the numerical integration of eq 1 was performed up to the time at which the values of rD and rA have been converged. Since the lifetimes of electrons and holes in TiO2 particles in the experiments are roughly 1 μs16 and 1 ns,17 respectively, the simulation was started at the rate constants of 109 s−1 for κe and κh. Figure 4A,B shows examples for the distribution of particles in the steady state under the light irradiation with the parameters of g = 1011 photons s−1, κr = 109 s−1, κh = 109 s−1, and (A) κe = 109 s−1 or (B) κe = 2 × 108 s−1. The simulation of the particle distribution shown in Figure 4 confirms the relation of eq 8. Namely, ⟨n⟩ = ⟨m⟩ ≈ 10 in Figure 4A where

dX mn = (n + 1)κeX mn+ 1 + (m + 1)κhX mn + 1 dt + (n + 1)(m + 1)κrX mn++11 + gX mn −− 11 − (nκe + mκh + nmκ r + g )X mn

(6)

(1)

By numerically integrating this differential equation, the number of the particle at the fraction Xnm in a steady state can be obtained. Then, the obtained particle distributions enable us to calculate the photocatalytic rates of reduction (rA) and oxidation (rD) as the summation of each transition as eqs 2 and 3, respectively.

Figure 4. Example of the steady state distribution of particles Xnm having distinctive numbers of electrons (n) and holes (m), which are simulated with a set of parameters g = 1011 photons s−1, κr = 109 s−1, κh = 109 s−1, and (A )κe = 109 s−1 or (B) κe = 2 × 108 s−1. C


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The Journal of Physical Chemistry C κh/κe = 1, while ⟨n⟩ ≈ 20 and ⟨m⟩ ≈ 4 in Figure 4B where κh/ κe = 5. Figure 5 shows the results of the 2D-ladder kinetic simulation of the quantum yield ϕ with eqs 1−4 as a function

Figure 6. Dependence of κr on the oxidation rate rD simulated as a function of the light intensity I (in the unit of mW cm−2 at 365 nm) with κe = 102 s−1 and κh = 104 s−1.The particle diameter is 26.7 nm (V = 10−17cm3) and the absorption coefficient of the semiconductor is α = 104 cm−1 at the excitation wavelength. Thin lines are calculated for eqs 9−11 by substituting κh = 109 s−1.

Figure 5. Oxidation quantum yield ϕ simulated as a function of excitation rate for a particle (g = VαI) with various kinetic parameters, κh = 109 s−1, and κe = 107 s−1 (red), 108 s−1 (green) and 109 s−1 (blue). Numbers in the lines show κr dependence for κr = 107−1013 s−1 with log(κh/κr).

corresponding two lines well overlap, indicating that eq 9 is justified. Therefore, hereafter eq 9 is employed as the result of the 2D-ladder simulation for convenience. In Figure 6, κe was set to be extremely smaller than κh. However, similar results could be obtained when κh was set to be extremely smaller than κe. We selected the former case based on the experimental results for TiO2 stated above.16,17 Since it is well-known that O2 is reduced and adsorbed at the TiO2 surface,18 the adsorbed Aad− has maybe a long lifetime without suffering the oxidation path 12 shown in Figure 1. Actually, it was reported that the open-current voltage of a TiO2 electrode shifted to negative by 0.4 V on the irradiation with the oxidation of methanol under air.19 Though κe was fixed to κe = 102 s−1 in Figure 6, Figure 7 shows the results calculated with changing κe from 10−2 to 106

9 −1

of g with κh = 10 s . The parameters κe and κr were changed from 107 to 109 s−1 and 107 to 1013 s−1, respectively. In the whole parameter ranges, the simulated results with eqs 2 and 3 could confirm that the reduction rate rA and the oxidation rate rD are the same as shown in eq 5. This means that, when the parameter values for κe and κh were exchanged, the same result was obtained. After the simulation using the 2D-ladder kinetics with various parameters, it was found that the results such as shown in Figure 5 could be given by the following equation (9): 1 ϕ= ( −1 + 1 + 4ηb ) (9) 2b where b=

ηκ rVαI κeκh

(10)

η=

κe + κh κe + κh + κ r

(11)

Since the extent of parameters which could be actually performed with a personal computer was limited, the usage of these equations enabled us to simulate the 2D-ladder kinetics with a wide range of parameters. The properties of eq 9 are shown in the Supporting Information. The inflection point of eq 9 could be obtained to be κκ I = 2 e h , which is obtained as the intersection of the two η κ Vα r

linear lines of eqs (S2) and (S5) corresponding to the linear parts in Figure 5. The inflection point in Figures 5 and S1 is estimated to be I = 1021 photons cm−2 s−1 as calculated in the Supporting Information. This means that when either κe or κh becomes small, the situation of rD ∝ I1/2 could be achieved at lower light intensities. Since the light intensity of usual photocatalytic experiments with UV light (365 nm) is I = 1 mW cm−2, or 1.84 × 1015 photons cm−2 s−1, the parameter value of κe in Figure 5 was decreased by several orders of magnitude. Figure 6 shows the results of the 2D-ladder simulation for the small reduction rate, κe = 102 s−1. The bold lines in Figure 6 are simulated by the numerical integration of eq 1, while the fine lines are calculated with eq 9. The

Figure 7. Dependence of κe (from 10−2 to 106 s−1) on the oxidation rate rD calculated for eqs 9−11 as a function of the light intensity I for κh/κr = 10−2 (red dotted line), 1 (blue solid line), and 102 (orange broken line) with κh = 109 s−1, α = 104 cm−1, and V = 10−17 cm3 (diameter of 26.7 nm).

s−1, where the calculation was performed for three sets of parameters, κh/κr = 10−2, 1, and 102. When κh/κr becomes large, the quantum yield becomes naturally close to unity, ϕ = 1, or rD = VαI, which is shown by the broken line at a large κe in Figure 7. 2.2. Concentration Expression. Rates of photocatalytic reactions are usually expressed by the time dependence of the D


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The Journal of Physical Chemistry C Table 1. Correlation of Parameters between the 2D-Ladder Kinetics and the Concentration-Based Solution System 2D-ladder kinetics

units

solution system

units

κe κh κr g (=Vαl) rD

s−1 s−1 s−1 photons s−1/particle molecules s−1/particle

ke[Aad] kh[Dad] kr βI r

s−1 s−1 s−1 mol dm−3 s−1 mol dm−3 s−1

reduction oxidation recombination excitation reaction rate

have been reported for a few decades ago.20−26 However, η2 was not contained in the square root term of the previously reported quadratic equations. Three decades ago, the present authors have also reported the mathematically equivalent, and simpler quadratic relationship, (1 − ϕ)/ϕ2 = I/χ, for the electron transfer quantum yield in a colloidal CdS system.27 For the simple quadratic equation derived from the concentration treatment, the quantum yield becomes unity (ϕ → 1) at the extremely low light intensity (I → 0) because the electrons and holes are not confined in the particle. Hence, the present authors had to introduce the 2D-ladder kinetics to account for the distinctive characteristics of particles.12 For the 2D-ladder kinetics, the recombination takes place in each particle and then the recombination rate reaches a certain value even when the light intensity is extremely low. In other words, the quantum yield becomes intrinsic value (ϕ → η) at the extremely low light intensity (I → 0). Furthermore, we also reported that, when the recombination rate is small or the size of the particle becomes large, the results of the 2D-ladder simulation could be approximated by the results calculated with the previous quadratic formula equation.12 However, eq 15 could not be obtained at that time. In conclusion, the present simulation with the 2D-ladder kinetics gave us the following information. The photocatalytic oxidation rate could be calculated by eq 15, which could be obtained from the previously reported quadratic formula equation by simply modifying the square root term by involving the square of the intrinsic quantum yield, η. As far as the authors know, there has been so far no report of the quadratic formula equation in which the square of the intrinsic quantum yield is contained in the square root term. 2.3. Langmuir−Hinshelwood Representation. The relationship between the oxidation rate and the reactant concentration in solution is usually analyzed with the Langmuir−Hinshelwood (L-H) kinetics.26,28,29 That is, the oxidation rates r measured at various reactant concentrations [D] are fit to eq 17 with L-H parameters rL and KL.

reactant concentration. Since the amount of reactant is not involved in the present 2D-ladder kinetics, the kinetic parameters should be described with the concentration of reactants to compare with the experimental data. The rates of the reduction and oxidation, κe and κh, in the 2D-ladder kinetics are probably proportional to the amount of adsorbed reactants. Hence, they should be ke[Aad] and kh[Dad], respectively, where the reduction and oxidation are completed when the products, Aad− and Dad+, are released from the surface to the solution or altered to the other species. Thus, ke and kh are not simple surface electron transfer rate constants but involve the rate constants of back reactions corresponding to the paths 12 and 13 in Figure 1. The photoexcitation rate in the solution system should be given by the unit of concentration. This conversion is attributed to a parameter β. The correlation of the parameters in the 2D-ladder kinetics with those in the solution system is listed in Table 1. Based on the correspondence in Table1, the resultant equation of the 2D-ladder simulation, eq 9, could be rewritten with the parameters in the solution system. That is, the parameters b and η in eq 9, which are given by eqs 10 and 11, could be obtained by eqs 12 and 13, respectively. b=

ηk rβI ke[A ad]k h[Dad ]

(12)

η=

ke[A ad] + k h[Dad ] ke[A ad] + k h[Dad ] + k r

(13)

In the cases of photocatalytic reactions for environmental cleaning, electron acceptor molecules, A, are usually oxygen in air. Therefore, the reduction rate could be described as eq 14. ke[A ad] = ke[O2,ad ]

(14)

On the other hand, electron donating molecules, D, could be assigned to water. Then D+ may be assigned to hydroxyl radical (•OH) as suggested in some literatures. However, the molecules first oxidized depend on the reactants, and accordingly D is not specified in the present study. The present authors have discussed the oxidation process in photocatalysis and the role of •OH in a recent review paper.18 Anyhow, the first oxidizing step would be a ratedetermining step as described above. Thus, by using eqs 9 and 12−14, the reaction rate could be expressed by eq 15 with parameters η and χ given by eqs 13 and 16. χ {−1 + 1 + (4η2 /χ )I } r= 2η (15) χ=

r=

(17)

It is equivalent to the following equation: 1 1 ij 1 yzz = jjj1 + z r rL jk KL[D] zz{

(18)

In order to perform the L-H analysis, the dependence of eq 15 on the reactant concentration [D] will be examined as a next step, because the equation presented in the above sections was merely the relationship of the oxidation rate with the light intensity I. The adsorbed concentration of reactant [Dad] in eq 15 should have a relationship to the solution concentration [D] with the adsorption−desorption equilibrium in a steady state. The desorption rate is tentatively assumed to increase by an

ke[O2,ad ]k h[Dad ] kr β

rLKL[D] 1 + KL[D]

(16)

Based on the conventional concentration treatment of electron−hole pairs, the quadratic formula equations of quantum yield similar to eq 15 could be obtained, which E


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The Journal of Physical Chemistry C amount proportional to the light intensity I.5 Thus, the time dependence of the concentration of the adsorbed reactant, d[Dad]/dt, may be given by eq 19,30 d[Dad ] = ka[D]([SD] − [Dad ]) − kd[Dad ] − k iI[Dad ] dt

cases.26,30,37,38 To describe the I dependence of the observed KL values, we employed the equation in which a term of I2 is added to both adsorption and desorption rates. Thus, we adopted eq 25 with two parameters ca and cd to calculate KL as a function of I.

(19)

where [SD] is the total concentration of the adsorption site for D, and ka and kd are the rate constants for adsorption and desorption of D in the dark, respectively, and ki is that for photoinduced desorption. In a steady state, [Dad] becomes constant and consequently eq 19 becomes zero. Therefore, kh[Dad] may be calculated by eq 20. k h[Dad ] =

k hka[D] [S D ] kd + k iI + ka[D]

KL =

βχ1 2η1

(−1 +

1 + (4η12 /χ1 )I )

(20)

(21)

η1 =

ke[O2,ad ]k h[SD] kr β

(22)

ke[O2,ad ] + k h[SD] ke[O2,ad ] + k h[SD] + k r

(23)

where the unit of χ1 is the same as that of the light intensity I and β which corresponds to αV in the 2D-ladder kinetics plays a role to convert the unit of χ1 to the reaction rate in the experimental system. Another L-H parameter, KL, presents a feature different from the parameter rL. The equation for KL was changed from eq S9 to eq S12 by the factor of 2 with the light intensity. The adsorption equilibrium constant, KD, is given by eq 24 as the ratio of the rate constants for adsorption and desorption of D in the dark.

k KD = a kd

(25)

3. DISCUSSIONS 3.1. Application to the Reported Experimental Data Sets. Emeline and co-workers34 reported the photodecomposition rate of phenol as functions of both light intensity and the reactant concentration. Based on their data sets, several kinetic models have been reported. Salvador and co-workers proposed a direct−indirect (D-I) model39 and applied7 it to fit the data sets.34 Marin and co-workers40 reported a model alternative to the D−I model, in which they considered all of the basic steps involved in heterogeneous photocatalysis with TiO2 including the back reaction. Mills and co-workers26 have provided a brief, historical overview of 10 apparently different, but in some cases closely related, popular proposed reaction mechanisms and their associated rate equations. The five main mechanisms were tested using their own data sets of 4chlorophenol41 besides the data sets of Emeline et al.34 and they adopted the disrupted adsorption kinetics of Ollis5 in terms of simplicity, usefulness, and versatility.26 In the terminology of the present report, the model of Ollis is given by eq 26 with parameters c1, c2, c3, and θ.

with χ1 =

1 + cdI 2

The change of the definition of KL does not affect the derivation of eq 21. If eq 25 were effective, the rates are proportional to the square of the photon intensity. This would mean that the adsorption and desorption rates could not be changed until two photons are absorbed in the particle. Thus, the photocatalytic oxidation rate obtained experimentally could be expressed for the reactant-concentration dependence by L-H kinetics of eq 17, with two L-H parameters, rL and KL, whose light-intensity dependence is given by eqs 21 and 25, respectively.

From the relationship of eq 20, the oxidation rate r can be calculated with eq 15 as a function of [D], by evaluating η and χ with eqs 13 and 16, respectively, and could be represented by the form of L-H equation, eq 17. One of the two L-H parameters, rL, could be given by the following quadratic formula of eq 21 as shown in the Supporting Information rL =

KD + caI 2

r=

c1I θ[D] 1 + c 2I θ + c3[D]

(26) 8

Afterward, Mills et al. reported a revised disrupted Langmuir-adsorption kinetics as eq 27 with four parameters c1, c2, c3, and c4.

(24)

For low light intensity, KL may be smaller than KD on considering eq S9. On the other hand, for high light intensity, KL can become larger than KD by the factor of less than 2 as indicated by eq S12. However, the reported KL as the lightintensity variation significantly depended on the experimental systems.26,31−33 There is the case that KL becomes more than ten times larger than KD.9 Therefore, the effect of light intensity on KL may not be given by adding kiI to kd simply as shown in eq 19. Since the photocatalytic reaction takes place at the surface of the semiconductor, it is reasonable that both desorption and adsorption at the surface could be largely changed by the photoirradiation. In the experimental results of photocatalysis, the dependence of the reciprocal adsorption constant, 1/KL, on I could be described with the curve of downwardly convex in some cases,34−36 while it was upwardly convex in the other

r=

c1[D]( −1 + 1 + c3( −1 +

1 + c 2I )

1 + c 2I ) + c4[D]

(27)

In the present study, we adopted the 2D-ladder kinetics, in which the concentrations of electrons and holes are not involved in the derivation of the equation and showed that the oxidation rate could be expressed by the L-H kinetic eq 17 with L-H parameters of rL and KL given by eqs 21−25. In order to compare with eq 27, eq 17 of the 2D-ladder kinetic model can be rewritten as eq 28 r=

c1[D]( −1 +

1 + c 2I )

c3 + [D]

(28)

where c1 = βχ1/2η1, c2 = 4η12/χ1, and c3 = 1/KL = (1 + cdI2)/ (KD + caI2). Therefore, the dependence of photocatalytic F


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The Journal of Physical Chemistry C reaction rate r on the reactant concentration [D] and the irradiation light intensity I can be calculated by six parameters of β, χ1, η1, KD, ca, and cd. However, among the six parameters, the three, KD, ca, and cd, can be evaluated at first from the data of KL measured at different light intensities. Figure 8 shows the fitting of parameters in eq 25 to the reported data34 of 1/KL for the photocatalytic oxidation of

By substituting eq 29 for eq 22, the reduction rate can be evaluated as eq 30. 1 − η1 ke[O2,ad ] = χβ η1 1 (30) When the obtained values for the parameters β, χ1, and η1 are substituted for eq 30, ke[O2,ad] for phenol oxidation is calculated to be 6 × 10−7 mol min−1. The production rates of reduced O2 radicals, •O2−, in solution could be measured by means of a chemiluminescence technique in alkaline solution.18 For P25 TiO2 powder in aqueous suspension of 3.5 mL, the rate of •O2− formation without organic reactant was 40 μM/100 s.42 This formation rate corresponds to 8 × 10−8 mol min−1. Therefore, the reduction rate 6 × 10−7 mol min−1 estimated from eq 30 in acidic phenol solution is comparable to that observed experimentally in alkaline solution.42 This compatibility verifies the correctness of the present analysis. The present equations were also tested for the data sets of the photocatalytic oxidation of 4-chlorophenol reported by Mills and co-workers.26,30,41 Figure 10 shows the result of

Figure 8. Dependence of an L-H parameter ΚL on the light intensity I calculated for eq 25 by fitting the experimental data reported for the photocatalytic decomposition of phenol.34 I0 = 1.1 × 1017 photons cm−2 s−1, KD = 3 × 105 M−1, ca = 1.1 × 10−29 M−1 photons−2 cm4, and cd = 3 × 10−33 photons−2 cm4.

phenol in TiO2 powder suspension. By using the values for KD, ca, and cd obtained in Figure 8, the experimental data sets of reaction rate for the phenol photodecomposition at various light intensities and reactant concentrations were fit to eq 28 by changing other three parameters β, χ1, and η1. As shown in Figure 9, the calculated reaction rate r fits well to the experimental data through the whole area. Figure 10. Dependence of an L-H parameter ΚL on the light intensity I calculated for eq 23 by fitting the experimental data reported for the photocatalytic decomposition of 4-chlorophenol.26 I0 = 6.4 × 1015 photons cm−2 s−1, KD = 1.4 × 104 M−1, ca = 6.2 × 10−27 M−1 photons−2 cm4 s2, and cd = 2.2 × 10−30 photons−2 cm4 s2.

fitting the parameters in eq 25 to the data of 1/KL reported30 at various light intensities, and Figure 11 shows the fitting of the remaining three parameters in eq 28 to the whole data sets. Again we could simulate the experimental data sets by eq 28. By substituting the obtained three parameters, β, χ1, and η1, for eq 30, ke[O2,ad] is calculated to be 1.1 × 10−5 mol min−1. Though the light intensity I0 is lower, the obtained value for the reduction rate is larger by 2 orders of magnitude than that for the decomposition of phenol stated above. The photocatalyst powder used was the same P25 TiO2, and the concentration and pH of the suspension were not much different from each other. However, in the case of phenol decomposition, HCl was added to adjust the pH to 3, while for the 4-chlorophenol decomposition HClO4 was added to adjust the pH to 2. Therefore, in the latter case, ClO4− ions might be reduced in place of O2, because the standard reduction potential of ClO4− (+1.19 V) is more positive than that of O2.18 If it is not the reason for the large reduction rate, the significant difference in the adsorption properties of each reactant, as shown in Figures 8 and 10, may largely increase the desorption rate of •O2−.

Figure 9. Dependence of the rate on the reactant concentration [D] and the light intensity I calculated for the quadratic formula equation, eq 28, by fitting parameters β, χ1, and η1 to the experimental data reported for the photocatalytic decomposition of phenol (ref 34). I0 = 1.1 × 1017 photons cm−2 s−1, χ1 = 1 × 1010 photons cm−2 s−1, η1 = 1 × 10−4, and the unit of reaction rate (mol min−1) corresponds to 1.65 × 1020 molecules cm−2 s−1 (=1/β). The other parameters are given in Figure 8.

Since the lifetime of electrons is usually longer than that of holes as stated above, the relation kh[SD] ≫ ke[O2,ad] holds. Therefore, from eq 23, the following relationship is obtained: η1 k h[SD] = kr 1 − η1 (29) G


Article

The Journal of Physical Chemistry C

of 1/KL reported for phenol and 4-chlorophenol decomposition were well described by fitting the parameters as shown in Figures 8 and 10, respectively. 5. The reaction rates reported as functions of the light intensity and the reactant concentration, could be simulated by the present kinetic model as shown in Figures 9 and 11, and the remaining three parameters for eq 28, β, χ1, and η1, could be evaluated. 6. By using the values for the three parameters stated above, the reduction rate of O2 was calculated with eq 30, which appeared to be compatible to the experimentally measured formation rate of •O2−.42 These conclusions imply that the present analytical treatment is actually the simple and useful method to understand the dependencies of reactant concentration and light-intensity on the photocatalytic oxidation rates.

Figure 11. Dependence of the rate on the reactant concentration [D] and the light intensity I calculated for the quadratic formula equation, eq 28, by fitting parameters β, χ1, and η1 to the experimental data reported for the photocatalytic decomposition of 4-chlorophenol.8,26 I0 = 6.4 × 1015 photons cm−2 s−1, χ1 = 1.8 × 107 photons cm−2 s−1, η1 = 1.1 × 10−4, and the unit of reaction rate (mol min−1) corresponds to 1.5 × 1016 molecules cm−2 s−1 (=1/β). The other parameters are given in Figure 10.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b09421. Properties of eq 9 and the derivation for eq 21, with former item containing the simplified equations for the cases of high and low light intensities in addition to Figure S1 showing I dependence of rD associated with Figure 5 (PDF)

4. CONCLUSIONS In order to analyze the kinetics of photocatalysis in the TiO2 powder suspension system, we adopted 2D-ladder kinetics in which the rates of four fundamental processes, (i) excitation, (ii) reduction, (iii) oxidation, and (iv) recombination, were treated as the transitions between the states of each powder characterized by the numbers of possessing negative and positive charges. From the numerical simulation with a personal computer for the 2D-ladder kinetics, the oxidation rate and the quantum yield in a steady state under the light irradiation were calculated with various rate constants as a function of light intensity. Thus, following conclusions were obtained. 1. The light-intensity dependence of the oxidation rate (rD) obtained from the numerical simulation could be expressed by the well-known quadratic formula equation, eq 9, which could be obtained from the conventional concentration analysis of electron−hole pairs. The characteristic difference is that a term of square of the intrinsic quantum yield, η, is inserted to the square root part of the quadratic equation. 2. Well known light intensity (I) dependence of the reaction rate rD ∝ I1//2 could be recognized under the normal light intensity (Figures 6 and 7) only when the rate of reduction or oxidation is extremely smaller than the recombination rate, where, in the 2D-ladder kinetics, the reduction and oxidation are not merely the surface charge transfer but the reactions are accomplished when the adsorbed products would not return to the adsorbed reactants. 3. The resultant equations obtained from the 2D-ladder simulation in particle systems were transformed to the concentration expression in solution systems as listed in Table 1. In addition the oxidation rate could be given by the form of Langmuir−Hinshelwood kinetics, eq 17, with two parameters of rL and KL, which correspond to the intrinsic rate, eq 21, and the adsorption equilibrium constant of the reactant, respectively. 4. Light-intensity dependence of KL was expressed by adding the term of I2 to both adsorption and desorption rate constants as eq 25. Then, the experimental data sets

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Yoshio Nosaka: 0000-0003-1305-5724 Notes

The authors declare no competing financial interest.

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REFERENCES

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