Kinetics of gas-phase photooxidation of p-xylene on nano TiO₂ P25 thin film

The effect of calcination temperature on the physico-chemical properties of ${\rm Ti}{{{\rm O}}_{2}}$ P25 catalyst was investigated in [20]. From the results it follows that calcination has led to changes in physico-chemical properties of the samples that affect the activity of the catalysts. The sample calcined at 450 °C was characterized by biggest specific surface area, as a result, it exhibited relatively high photocatalytic activity in the illumination of ultraviolet light. Inheriting the results of the publication [20] and in order to increase the economics of catalytic membrane preparation, in this study, the ${\rm Ti}{{{\rm O}}_{2}}$ P25 film after dipping, drying, was heated at 450 °C for a shorter time, of 2 h instead of 4 h as indicated in [20].

XRD pattern of P25-450-2 sample (figure 1) exhibits diffraction peaks of the anatase ($2\theta =25.2{}^\circ$, $37.8{}^\circ, 47.98{}^\circ, 53.9{}^\circ, 54.97{}^\circ, 62.65{}^\circ, 68.9{}^\circ$) and rutile phase $(2\theta =27.4{}^\circ, 35.95{}^\circ, 41.2{}^\circ, 56.6{}^\circ)$ [22]. The acuteness of peaks in XRD pattern demonstrates high crystallinity of sample. According to the XRD patterns of the catalyst, the ratio of anatase to rutile in obtained catalyst was found at 80.9/19.1. This phase ratio of P25-450-2 sample causes the synergetic effect (40%  ⩽  [anatase]  ⩽  80%) [23]. Hence, it is predicted high photoactivity to degrade organic pollutants. The average size of the P25-450-2 crystallites was estimated approximately 21 nm by using Scherrer formula at $2\theta =25.3{}^\circ$ [24].

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On the Raman spectrum of P25-450-2 sample (figure 2) the characteristic peaks of anatase ${\rm Ti}{{{\rm O}}_{2}}$ at $142\,{\rm c}{{{\rm m}}^{-1}},394\,{\rm c}{{{\rm m}}^{-1}},513\,{\rm c}{{{\rm m}}^{-1}},633\,{\rm c}{{{\rm m}}^{-1}}$ and of rutile phase at 443 and $609\,{\rm c}{{{\rm m}}^{-1}}$ were observed. Furthermore, intensity of characteristic Raman peak for rutile is much lower than that for anatase. This is accordant with dominance of anatase phase as calculating phase content from the XRD result.

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According to isothermal adsorption results, the BET specific surface area of P25-450-2 sample is $49.6\,{{{\rm m}}^{2}}\times {{{\rm g}}^{-1}}$ approximately (table 1). It is close to the results of the sample P25-450-4 in investigation [20].

Table 1. The physico-chemical characteristics of ${\rm Ti}{{{\rm O}}_{2}}{{\rm \text -} P25}$ samples: Values of surface specific area (S_BET), pore volume (${{V}_{{\rm p}}}_{{\rm ore}}$), average pore size (${{d}_{{\rm p}}}_{{\rm ore}}$), TiO₂ crystal size calculated at 2θ = 25.3° (${{d}_{{\rm crys}}}$), particle size (${{d}_{{\rm part}}}$), phase ratio anatase/rutile (A/R), point of zero charge (PZC), wavelength of light absorption (λ), and band gap energy (E_G).

Characteristics	${{S}_{{\rm BET}}}$ $({{{\rm m}}^{2}}\,{{{\rm g}}^{-1}})$	${{V}_{{\rm p}}}_{{\rm ore}}$ $({\rm c}{{{\rm m}}^{{\rm 3}}}\,{{{\rm g}}^{-1}})$	${{d}_{{\rm p}}}_{{\rm ore}}$ $({{\rm }\mathring{\rm A} })$	${{d}_{{\rm crys}}}$ (nm)	${{d}_{{\rm part}}}$ (nm)	$A/R$	PZC	λ (nm)	${{E}_{G}}$ (eV)
P25-450-2	49.6	0.037	33.5	23	20–30	80.9/19.1	6.30	393	3.155
P25-450-4 [20]	50.0	—	—	33	—	81.6/18.4	—	325–430	3.180

According to the SEM image (figure 3), P25-450-2 catalyst is suggested having pseudo spherical in shape. In addition, its distribution is nearly uniform. TEM image (figure 4) indicates the co-existence of both bright and dark fields, typically for the presence of anatase and rutile phase in structure of P25-450-2 catalyst [25]. From the SEM and TEM images, the average size of the primary particles estimates about 20–30 nm. This is a good agreement with the results calculated from the XRD pattern (21 nm).

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As can be seen from UV spectrum in figure 5, P25-450-2 mainly absorbs UV irradiation. Band gap energy of P25-450-2 sample based on Tauc curve (figure 6) was determined at 3.155 eV. The absorbable wavelength of the sample can be calculated by the equation $\lambda ={hc}/{{{E}_{g}}}\;$ and equal 393 nm. Hence, to initiate photochemical effects in semiconductor P25-450-2, UV light was required.

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As can be seen from figure 7, strong peaks in FTIR spectrum are observed at 3419, 1632, and $650\,{\rm c}{{{\rm m}}^{-1}}$. The broad peak at 3419 and the peak at $1632\,{\rm c}{{{\rm m}}^{-}}^{1}$ are believed corresponding to the hydroxyl groups and surface-adsorbed water respectively, while the main peak at from 400 to $700\,{\rm c}{{{\rm m}}^{-}}^{1}$ is ascribed to Ti-O stretching and Ti-O-Ti bridging stretching modes. The existence of this OH-groups explains the generation of ${\rm O}{{{\rm H}}^{\bullet }}$ on the catalyst surface.

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So, compared to P25-450-4 sample in article [20], P25-450-2 catalyst used in this study exhibited a smaller crystal size and slightly lower band gap energy. This may due to considerably shorter calcination duration was used in this study (2 versus 4 h).

${\rm C}{{{\rm O}}_{2}}{{\rm \text -}TPD}$ pattern in the figure 8 exhibited the ${\rm C}{{{\rm O}}_{2}}$ desorption peaks in temperature range of 100 °C–350 °C. According to Wang et al [26] the ${\rm C}{{{\rm O}}_{2}}$ desorption peaks in temperature-programmed desorption (TPD) pattern of ${\rm Ce}{{{\rm O}}_{2}}$ at temperatures of 150 °C–200 °C and above 400 °C represents the weak, medium and strong basic sites, respectively. Then, the received results showed that on P25-450-2 catalyst exist the weak and medium basic sites, moreover weak basic sites dominate. Duffy [27] reported that the basicity of ${\rm Ti}{{{\rm O}}_{2}}$ is much greater than that of ${\rm Si}{{{\rm O}}_{2}}$ and is nearly the same as the basicity of CaO. Bergman [28] determined the optical basicity of ${\rm Ti}{{{\rm O}}_{2}}$ is 0.55.

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Figure 9 showed a uniform distribution of ${\rm Ti}{{{\rm O}}_{2}}$ particles on the film and the thin film P25-450-2 has an average thickness of $2.75\,\mu {\rm m}$.

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From the mentioned above results, it follows that the ${\rm Ti}{{{\rm O}}_{2}}{{\rm \text -}P25}$ powder after calcination at $450\,{}^\circ {\rm C}$ for 2 h predominantly consist of spherical shape crystals, having an anatase/rutile phase ratio of 80.9/19.1 and band gap energy $({{E}_{G}})$ of 3.155 eV, absorbing UV light, are capable to use UV lamps in photocatalytic reactions. This catalyst exhibited relatively high activity and stability under the illumination of ultraviolet light (figure 10). This meets the requirements for studying the kinetics of the reaction. The enhancing photoactivity of P25-450-2 sample is explained by synergetic effect of the mixture anatase and rutile phase of suitable ratio [23] and the contact of two phases makes slowly electron-hole recombination [4]. In addition, the smaller size and lower band gap energy also contributed to increased activity of P25-450-2 catalyst.

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The high stability of P25-450-2 catalyst is explained by its high basicity (as shown from the ${\rm C}{{{\rm O}}_{2}}{{\rm \text -}TPD}$ result) and the small particle size, yielding less favorable conditions to coke deposit on the catalyst surface. Furthermore, the point of zero charge of P25-450-2 catalyst was determined at $\sim 6.3$, so in the gas-phase reaction medium its surface charge is neutral, the absence of electrostatic force causes the minimum of the interplay between surface particles and contaminants. This also contributes to reducing coke deposit on the catalyst surfaces. Indeed, the by-products formed on surface of P25-450-2 catalyst during photocatalytic oxidation of $p{{\rm \text -}}$xylene were analyzed from the solution extracted in the dark from the catalyst layer in methanol after a long period of operation (several days) was estimated less than 1 mg g⁻¹ catalyst. The results of the analysis (figures 11 and 12) showed that the compounds, depositing on the catalyst surface are products of the partial oxidation such as 2-methylbenzaldehyde and phthalic anhydride. They are oxygenate compounds with a low C: H ratio (CH0.5 for phthalic anhydride and CH for methylbenzaldehyde), easily combusted in reaction conditions, leading to the low amount of deposited products, that increased the stability of catalyst.

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The analysis of the experimental data during the course of reaction showed that under the given conditions the catalytic activity did not change with time. Thus, all the experimental data can be used in kinetic calculations. Furthermore, the advantage of thin film catalyst is included in high efficiency and ease of recovery. The catalyst can be regenerated by treatment in air flow at $450\,{}^\circ {\rm C}$ in 2 h.

According to the results obtained above, the thin films of P25-450-2 catalyst should be considered as typical catalyst for $p{{\rm \text -}}$xylene photooxidation in UV light. Therefore, the kinetics of reaction (6) was studied on P25-450-2 catalyst at conditions of ultraviolet irradiation.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p-{{{\rm C}}_{8}}{{{\rm H}}_{10}}+10.5{{{\rm O}}_{2}}\xrightarrow{\lambda }8{\rm C}{{{\rm O}}_{2}}+5{{{\rm H}}_{2}}{\rm O}{\rm .}\nonumber \end{align} \tag{ 6 }$$

In the kinetic experiments the values of gas hourly space velocity on a volume basis (GHSV) varied from ${{26.10}^{3}}$ to ${{468.10}^{3}}{{{\rm h}}^{-1}}$ and a catalyst film with the thickness of $2.75-2.90\,\,\mu {\rm m}$ was used. Under such conditions the influences of external as well as internal diffusions can be avoided. Indeed, in this range of flow rate the conversion extent decreased with increasing flow velocity, which indicates that there is no effect of external diffusion. The values of initial partial pressure of $p{{\rm \text -}}$xylene $(P_{xyl}^{0})$, oxygen $(P_{{{{\rm O}}_{2}}}^{0})$ and water vapor $(P_{{{{\rm H}}_{2}}{\rm O}}^{0})$ varied in ranges: 2.1–11.3 hPa, 105–473 hPa, and 6–32 hPa, respectively. To study the effect of reaction products, carbon dioxide with the values of initial partial pressure $(P_{{\rm C}{{{\rm O}}_{2}}}^{0})$ from 0 to 24.89 hPa was added to the reaction mixture. The distance between the catalyst surface and the illuminating lamps is 7 cm. The intensity of UV light varied in range 650–1147 lux. In order to change the light intensity the nets of different mesh sizes are placed between the light source and the catalyst film. Under these conditions the conversion of $p{{\rm \text -}}$xylene (X) varied from 0.10 to 0.98.

The reaction rate on the studied catalysts was calculated by Temkin's formula [29]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r=\frac{C_{xyl}^{0}vX}{m}({\rm mmol}\times {{{\rm g}}^{-1}}\times {{{\rm h}}^{-1}}),\nonumber \end{align} \tag{ 7 }$$

or

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r=\frac{P_{xyl}^{0}vX}{24.436\,m}({\rm mmol}\times {{{\rm g}}^{-1}}\times {{{\rm h}}^{-1}}),\nonumber \end{align} \tag{ 8 }$$

where $C_{xyl}^{0}$ and $P_{xyl}^{0}$ are the initial concentration $({\rm mmol }\times {{{\rm l}}^{-1}})$ and initial partial pressure (hPa) of $p{{\rm \text -}}$xylene in the gas mixture, respectively, $X$ is the $p{{\rm \text -}}$xylene conversion, $m$ is the catalyst loading mass (g), and $v$ is the total flow velocity of reaction gas mixture $({\rm l}\times {{{\rm h}}^{-1}})$.

The convex shape of the curves reaction rate $(r)$ versus conversion extent $(X)$ on figure 13 shows that the reaction was inhibited by the substances participating in the reaction.

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The dependence of the reaction rate on the partial pressures of substances involved in reactions is shown on figures 14 and 15. It can be seen from figures 14 and 15 that the graphs describing the dependence reaction rate $(r){{\rm \text -}}$partial pressure of $p{{\rm \text -}}$xylene $({{P}_{xyl}})$ and oxygen $({{P}_{{{{\rm O}}_{2}}}})$ are gently arched in first case and almost linear in second case, that means probably the terms ${{P}_{xyl}}$ and ${{P}_{{{{\rm O}}_{2}}}}$ are present in both the numerator and the denominator of the equation, but the quantity of $k_{1}^{\prime }{{P}_{{{{\rm O}}_{2}}}}$ and ${{k}_{3}}{{P}_{xyl}}$ in denominator is relatively small or can be present only in the numerator of the equation with the reaction order of 1.

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The curves expressing the dependence of the reaction rate on concentration of water vapor (figure 16) have extreme shapes. At low concentrations of water vapor the reaction rate increases with the concentration of the given reactant, but reaches maximum values in the range of ${{P}_{{{{\rm H}}_{2}}{\rm O}}}=11-20{\rm hPa}$, after that gradually decreases when the concentration of the considered reactant continues to grow.

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This allows us to drive a common rule for the reaction that the concentration/pressure of water vapor is involved in both numerator and denominator of the kinetic equation, but the value of its exponent in the denominator must be greater than that in the numerator.

The coincidence of the curves showing the change of reaction rate with the degree of conversion (figure 17) at different ${\rm C}{{{\rm O}}_{2}}$ initial concentrations confirms that ${\rm C}{{{\rm O}}_{2}}$ does not affect the reaction rate. The dependence of the reaction rate on the light intensity is shown in figure 18.

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From the presented above kinetic results the following conclusions should be drawn:

• The reaction is inhibited by the substances of reaction ($p{{\rm \text -}}$xylene/oxygen/water) and the kinetic equation is not exponential;
• The quantities ${{P}_{xyl}}$ and ${{P}_{{{{\rm O}}_{2}}}}$ can be present in both the numerator and the denominator of the equation, but quantity of $k_{1}^{\prime }{{P}_{{{{\rm O}}_{2}}}}$ and ${{k}_{3}}{{P}_{xyl}}$ in denominator is relatively small or present only in the numerator with exponent 1;
• Quantity ${{P}_{{{{\rm H}}_{2}}{\rm O}}}$ is present both in the numerator and the dominator of the kinetic equation with its exponent in the numerator smaller than in the denominator;
• Quantity ${{P}_{{\rm C}{{{\rm O}}_{2}}}}$ does not participate in the kinetic equation.
• Reaction rate is proportional to light intensity.

By-products formed during photocatalytic oxidation of $p{{\rm \text -}}$xylene were analyzed directly from the reaction gas mixture (figure 19) and showed that the by-product is 4-metylbenzadehyde presenting as traces. Thus, when considering the reaction kinetics the by-products could be ignored. In addition, the material equilibrium data for C also shows that only the product is ${\rm C}{{{\rm O}}_{2}}$.

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For the rates of photo-oxidation of chloroform in aqueous suspension of ${\rm Ti}{{{\rm O}}_{2}}$ the following kinetic equation has been proposed by Kormann et al [30]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle -\frac{d[\operatorname{Re}d]}{dt}=-\frac{d[Ox]}{dt}={{k}_{d}}{{\theta }_{\operatorname{Re}d}}{{\theta }_{Ox}}.\nonumber \end{align} \tag{ 9 }$$

In this equation quantity ${{\theta }_{{\rm Re}\,d}}$ and ${{\theta }_{Ox}}$ respectively are the fraction of adsorption of the electron-donating reductant and electron-accepting oxidant to the catalyst surface, and ${{k}_{d}}$ is reaction rate constant.

Because ${{E}_{G}}$ of P25-450-2 catalyst is 3.155 eV, under the irradiation of UV light, its electron was excited from valence band to the conduction band $(e_{{\rm CB}}^{-})$ generating a positive hole in the valence band $(h_{{\rm VB}}^{+})$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm Ti}{{{\rm O}}_{2}}+hv(\geqslant {{E}_{G}})\to h_{{\rm VB}}^{+}+e_{{\rm CB}}^{-}.\nonumber \end{align} \tag{ 10 }$$

Next, hydroxyl radicals ${{(}^{\bullet }}{\rm OH})$, a strong oxidizing agent, are formed by oxidation of water molecules adsorbed on the surface of ${\rm Ti}{{{\rm O}}_{2}}$ and –OH group (as seen in IR spectra) by photo-generated holes [31]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle h_{{\rm VB}}^{+}+{\rm T}{{{\rm i}}^{4+}}-{{{\rm H}}_{2}}{\rm O}\to {\rm T}{{{\rm i}}^{4+}}+{}^{\bullet }{\rm OH}+{{{\rm H}}^{+}},\nonumber \end{align} \tag{ 11 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle h_{{\rm VB}}^{+}+{\rm T}{{{\rm i}}^{4+}}-{\rm OH}\to {\rm T}{{{\rm i}}^{4+}}+{}^{\bullet }{\rm OH}{\rm .}\nonumber \end{align} \tag{ 12 }$$

Electrons in the conduction band can be trapped by molecular oxygen adsorbed on the surface of ${\rm Ti}{{{\rm O}}_{2}}$ P25 to form superoxide radical anion ${\rm (O}_{2}^{\bullet -})$ [31]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm T}{{{\rm i}}^{4+}}+e_{{\rm CB}}^{-}\to {\rm T}{{{\rm i}}^{3+}},\nonumber \end{align} \tag{ 13 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm T}{{{\rm i}}^{3+}}+{{{\rm O}}_{2}}\to {\rm T}{{{\rm i}}^{4+}}+{\rm O}_{2}^{\bullet -}.\nonumber \end{align} \tag{ 14 }$$

Further, these powerful oxidants (hydroxyl radicals and reactive oxygen species) participate in the reaction with organic compounds to form ${\rm C}{{{\rm O}}_{2}}$ and ${{{\rm H}}_{2}}{\rm O}$ [32]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {}^{\bullet }{\rm OH}+{\rm pollutant}\to {\rm C}{{{\rm O}}_{2}}+{{{\rm H}}_{2}}{\rm O,}\nonumber \end{align} \tag{ 15 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm O}_{2}^{\bullet -}+{\rm pollutant}\to {\rm C}{{{\rm O}}_{2}}+{{{\rm H}}_{2}}{\rm O}{\rm .}\nonumber \end{align} \tag{ 16 }$$

So, oxidant in the case of photo-oxidation in gas phase is both $^{\bullet }{\rm OH}$ forming from dissociative adsorbed water and ${\rm O}_{2}^{\bullet -}$ generating from adsorbed molecular oxygen. Then, ${{\theta }_{Ox}}$ in this case equal

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\theta }_{Ox}}=\theta _{{{{\rm H}}_{2}}{\rm O}}^{0.5}+{{\theta }_{{{{\rm O}}_{2}}}}.\nonumber \end{align} \tag{ 17 }$$

When considering the near-linear dependence of the reaction rate on the partial pressure of $p{{\rm \text -}}$xylene, the question is whether the reaction proceeds according to the Langmuir-Hinshelwood or Eley-Rideal mechanisms? Turchi and Ollis [31] studied two reaction schemes in which the authors considered either the attack of an adsorbed hydroxyl radical on an adsorbed organic substrate or the attack of given radical on a free organic substrate as the rate-determining steps. In our case, with a small amount of catalyst (15 mg) and a low concentration of $p{{\rm \text -}}$xylene $(0.051-0.276{\rm mol}\times {{{\rm l}}^{-1}})$, but a high concentration of oxygen $(2.56-8.36{\rm mol}\times {{{\rm l}}^{-1}})$ in the reaction stream, the catalyst surface should be completely covered with adsorbed oxygen and only a limited number of $p{{\rm \text -}}$xylene molecules which scarcely access to the surface, can participate in the reaction. Therefore, the reaction rate should be proportional to the surface concentration of $p{{\rm \text -}}$xylene. In principle, the participation of $p{{\rm \text -}}$xylene in the reaction from the gas phase cannot be ruled out; in this case, the order of $p{{\rm \text -}}$xylene concentration in the kinetic equation is equal 1. But there is a relationship between the quantity of adsorbed $p{{\rm \text -}}$xylene and its conversion extent, so the first hypothesis should be more appropriate. On P25-450-4 sample the quantity of adsorbed $p{{\rm \text -}}$xylene and its conversion extent were reached $133{\rm mg}\times {{{\rm g}}^{-1}}$ and 99% respectively, while on the sample, calcined at 550 °C in 2 h (P25-550-2) these values were $56{\rm mg}\times {{{\rm g}}^{-1}}$ and 87%, respectively. Since the reaction follows the Langmuir-Hinshelwood model, in which the attack of an adsorbed hydroxyl radical on an adsorbed organic substrate [31] and the reactions (15) and (16) are the rate-determining steps, the rate of reaction (6) could be written in a following form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r={{k}^{\prime }}{{\theta }_{xyl}}({{\theta }_{{{{\rm O}}_{2}}}}+\theta _{{{{\rm H}}_{2}}{\rm O}}^{0.5}),\nonumber \end{align} \tag{ 18 }$$

where ${{k}^{\prime }}$ is the rate constant, ${{\theta }_{xyl}}$ and ${{\theta }_{{{{\rm O}}_{2}}}}$ are the fraction of the $p{{\rm \text -}}$xylene and oxygen molecules adsorbed to the catalyst surface, respectively, $\theta _{{{{\rm H}}_{2}}{\rm O}}^{0.5}$ is the fraction of the water dissociative adsorbed to the surface.

Mills et al [33] indicated that the constant of proportionality ${{k}_{S}}$ of the photo-mineralization of organic substrates $S$ by oxygen on ${\rm Ti}{{{\rm O}}_{2}}$ catalyst in following kinetic equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r=-\frac{d{{[S]}_{i}}}{dt}=\frac{{{k}_{S}}{{K}_{S}}{{[S]}_{i}}}{1+{{K}_{S}}{{[S]}_{i}}}\nonumber \end{align} \tag{ 19 }$$

is proportional to the rate of light absorption $I_{a}^{\beta }$ as well as the fraction of ${{{\rm O}}_{2}}$ non-competitive adsorption at ${\rm T}{{{\rm i}}^{{\rm III}}}$ sites. In this equation $r$ is the reaction rate, ${{k}_{S}}$ is the kinetic constant, ${{K}_{S}}$ and ${{[ S ] }_{i}}$ respectively are the adsorption equilibrium constant and concentration of substrate $S$. The exponent $\beta$ was 0.5 when the light intensity was high but equal to 1 at low light intensity values. Therefore, the kinetic equation (18) can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r=k{{\theta }_{xyl}}[{{\theta }_{{{{\rm O}}_{2}}}}+\theta _{{{{\rm H}}_{2}}{\rm O}}^{0.5}]I_{a}^{\beta }\nonumber \end{align} \tag{ 20 }$$

where ${{\theta }_{i}}$ is the degree of coverage of the respective species and ${{I}_{a}}$ is total light intensity.

The fraction of adsorption of surfactants to the surface is determined by the following Langmuir formulae

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\theta }_{xyl}}=\frac{{{K}_{xyl}}{{P}_{xyl}}}{1+{{K}_{xyl}}{{P}_{xyl}}+{{K}_{{{{\rm O}}_{2}}}}{{P}_{{{{\rm O}}_{2}}}}+{{({{K}_{{{{\rm H}}_{2}}{\rm O}}}{{P}_{{{{\rm H}}_{2}}{\rm O}}})}^{1/2}}+{{K}_{{\rm C}{{{\rm O}}_{2}}}}{{P}_{{\rm C}{{{\rm O}}_{2}}}}},\nonumber \end{align} \tag{ 21 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\theta }_{{{{\rm O}}_{2}}}}=\frac{{{K}_{{{{\rm O}}_{2}}}}{{P}_{{{{\rm O}}_{2}}}}}{1+{{K}_{xyl}}{{P}_{xyl}}+{{K}_{{{{\rm O}}_{2}}}}{{P}_{{{{\rm O}}_{2}}}}+{{({{K}_{{{{\rm H}}_{2}}{\rm O}}}{{P}_{{{{\rm H}}_{2}}{\rm O}}})}^{1/2}}+{{K}_{{\rm C}{{{\rm O}}_{2}}}}{{P}_{{\rm C}{{{\rm O}}_{2}}}}},\nonumber \end{align} \tag{ 22 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\theta }_{{{{\rm H}}_{2}}{\rm O}}}=\frac{{{({{K}_{{{{\rm H}}_{2}}{\rm O}}}{{P}_{{{{\rm H}}_{2}}{\rm O}}})}^{1/2}}}{1+{{K}_{xyl}}{{P}_{xyl}}+{{K}_{{{{\rm O}}_{2}}}}{{P}_{{{{\rm O}}_{2}}}}+{{({{K}_{{{{\rm H}}_{2}}{\rm O}}}{{P}_{{{{\rm H}}_{2}}{\rm O}}})}^{1/2}}+{{K}_{{\rm C}{{{\rm O}}_{2}}}}{{P}_{{\rm C}{{{\rm O}}_{2}}}}},\nonumber \end{align} \tag{ 23 }$$

where ${{K}_{i}}$ is the Langmuir adsorption constant of the $i{{\rm \text -}}$species on the surface of the catalyst and ${{P}_{i}}$ is their partial pressure. Then, the final expression of the kinetics for the whole process of $p{{\rm \text -}}$xylene photooxidation could be described as follows

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r={{k}^{\prime }}\frac{{{K}_{xyl}}{{P}_{xyl}}\left[ a{{K}_{{\rm O}{}_{2}}}{{P}_{{{{\rm O}}_{2}}}}+{{(b{{K}_{{{{\rm H}}_{2}}{\rm O}}}{{P}_{{{{\rm H}}_{2}}{\rm O}}})}^{1/2}} \right]I_{a}^{\beta }}{{{[1+c{{K}_{xyl}}{{P}_{xyl}}+d{{K}_{{{{\rm O}}_{2}}}}{{P}_{{{{\rm O}}_{2}}}}+{{(e{{K}_{{{{\rm H}}_{2}}{\rm O}}}{{P}_{{{{\rm H}}_{2}}{\rm O}}})}^{1/2}}+f{{K}_{{\rm C}{{{\rm O}}_{2}}}}{{P}_{{\rm C}{{{\rm O}}_{2}}}}]}^{2}}}\nonumber \end{align} \tag{ 24 }$$

where $a, b, c, d, e, f$ are the constants.

Defining ${{k}^{\prime }}{{K}_{xyl}}=k$, $a{{K}_{{{{\rm O}}_{2}}}}={{k}_{1}}$, ${{b}^{1/2}}K_{{{{\rm H}}_{2}}{\rm O}}^{1/2}={{k}_{2}}$, $c{{K}_{xyl}}={{k}_{3}}$, $d{{K}_{{{{\rm O}}_{2}}}}=k_{{\rm 1}}^{\prime }$; ${{e}^{1/2}}K_{{{{\rm H}}_{2}}{\rm O}}^{1/2}=k_{2}^{\prime }$, $f{{K}_{{\rm C}{{{\rm O}}_{2}}}}={{k}_{4}}$, and equation (24) can be written by follow common equation, in accordance with the above conclusions of the reaction kinetics:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r=\frac{kP_{xyl}^{{{l}_{1}}}[{{k}_{1}}P_{{{{\rm O}}_{2}}}^{{{l}_{2}}}+{{k}_{2}}P_{{{{\rm H}}_{2}}{\rm O}}^{{{l}_{3}}}]I_{a}^{\beta }}{{{\left( 1+k_{1}^{\prime }P_{{{{\rm O}}_{2}}}^{{{m}_{1}}}+k_{2}^{\prime }P_{{{{\rm H}}_{2}}{\rm O}}^{{{m}_{2}}}+{{k}_{3}}P_{xyl}^{{{m}_{3}}}+{{k}_{4}}P_{{\rm C}{{{\rm O}}_{2}}}^{{{m}_{4}}} \right)}^{2\alpha }}}\nonumber \end{align} \tag{ 25 }$$

where $r$ is reaction rate of $p{{\rm \text -}}$xylene deep photooxidation; $k,{{k}_{1}},k_{1}^{\prime },{{k}_{2}},k_{2}^{\prime },{{k}_{3}},{{k}_{4}}$ are constants of the kinetic equation; ${{P}_{i}}$ are partial pressure of the corresponding components; ${{I}_{a}}$ is total light intensity; ${{l}_{1}},{{l}_{2}},{{l}_{3}},{{m}_{1}},{{m}_{2}},{{m}_{3}},{{m}_{4}}$ and $\beta$ are reaction orders of the corresponding reactants and light intensity; and $2\alpha$ is surface coverage.

To determine the reaction order and the values of kinetic coefficients, the solver tool in MS excel was used to calculate the experimental data in combination with the following conditions: ${{l}_{i}}=0-2$ (step 0.25), ${{m}_{i}}=0-2$ (step 0.25), $\alpha =0-1$ (step 0.25), ${{m}_{2}}\times 2\alpha > {{l}_{3}}$.

The calculated results gave the best coincidence with the experimental data when: ${{l}_{1}}={{l}_{2}}=1$, ${{l}_{3}}=0.5$; ${{m}_{1}}={{m}_{3}}={{m}_{4}}=1$; ${{m}_{2}}=0.5$; $\beta =0.537$ and $\alpha =1$. The calculation also showed that $k_{1}^{\prime }={{k}_{4}}=0$. Therefore, the kinetic equation for the reaction can be written as follows

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r=\frac{k{{P}_{xyl}}\left[ {{k}_{1}}{{P}_{{{{\rm O}}_{2}}}}+{{k}_{2}}P_{{{{\rm H}}_{2}}{\rm O}}^{0.5} \right]I_{a}^{0.537}}{{{\left( 1+k_{2}^{\prime }P_{{{{\rm H}}_{2}}{\rm O}}^{0.5}+{{k}_{3}}{{P}_{xyl}} \right)}^{2}}}.\nonumber \end{align} \tag{ 26 }$$

The values of the kinetic constants in equation (26) are presented in table 2.

Table 2. Values of the kinetic constants in equation (26).

Kinetic constants	Unit	Value
$k$	${\rm mmol}\times {{{\rm g}}^{-1}}\times {{{\rm h}}^{-1}}\times {\rm lu}{{{\rm x}}^{-0.537}}\times$${\rm hP}{{{\rm a}}^{-1}}$	$1.197\times {{10}^{-2}}$
${{k}_{1}}$	${\rm hP}{{{\rm a}}^{-1}}$	$0.639\times {{10}^{-3}}$
${{k}_{2}}$	${\rm hP}{{{\rm a}}^{-0.5}}$	$1.677\times {{10}^{-2}}$
$k_{1}^{\prime }$	${\rm hP}{{{\rm a}}^{-1}}$	0
$k_{2}^{\prime }$	${\rm hP}{{{\rm a}}^{-0.5}}$	$1.00\times {{10}^{-2}}$
${{k}_{3}}$	${\rm hP}{{{\rm a}}^{-1}}$	$5.078\times {{10}^{-8}}$
${{k}_{4}}$	${\rm hP}{{{\rm a}}^{-1}}$	0
Variance	%	$20.5$

As it follows from table 2, the value of ${{k}_{3}}$ is very small in comparison with $k_{2}^{\prime }$ and equation (26) becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r=\frac{k{{P}_{xyl}}[{{k}_{1}}{{P}_{{{{\rm O}}_{2}}}}+{{k}_{2}}P_{{{{\rm H}}_{2}}{\rm O}}^{0.5}]I_{a}^{0.537}}{{{\left( 1+k_{2}^{\prime }P_{{{{\rm H}}_{2}}{\rm O}}^{0.5} \right)}^{2}}}.\nonumber \end{align} \tag{ 27 }$$

The form of equation (26) showed that, the reaction takes place in the areas of high coverages. The exponent value $(\beta)$ of light intensity in kinetic equation (26) is found to be 0.537. According to the authors [17–19], the order of photon flux $(\beta)$ for photocatalytic degradation of trichloroethylene in gas-phase over ${\rm Ti}{{{\rm O}}_{2}}$ supported on glass bead depends on the intensity and wavelength of light, varying from 0.42 to 1.0. The square-root dependence of reaction rate of chloroform photodegradation in the presence of ${{{\rm O}}_{2}}$ on light intensity was reported by Kormann et al [30]. This relationship was explained by the role of hydroxyl radical in the capture of holes and the initiator of oxidation of electron-donating substrates. In our investigation, as it follows from table 2, the value of coefficient ${{k}_{2}}$ of equation (26) is higher than ${{k}_{1}}$ about 28 times.

The extreme form of the dependence of the reaction rate on the partial pressure of water vapor, as shown in figure 16, is the result of the squared of the denominator of equation (26). When the concentration of water vapor $({{P}_{{{{\rm H}}_{2}}{\rm O}}})$ is low, i.e. $k_{2}^{\prime }P_{{{{\rm H}}_{2}}{\rm O}}^{0.5}$ in the denominator of equation (26) is negligible compared to 1 and it follows that r is proportional to $P_{{{{\rm H}}_{2}}{\rm O}}^{0.5}$, that means the reaction order of water vapor is  +0.5. At high partial pressures of water vapor, the degree of surface coverage ${{\theta }_{{{{\rm H}}_{2}}{\rm O}}}$ is high, i.e. $k_{2}^{\prime }P_{{{{\rm H}}_{2}}{\rm O}}^{0.5}$ in the denominator of equation (26) is much higher than 1 and the rate equation is proportional to ${1}/{P_{{{{\rm H}}_{2}}{\rm O}}^{0.5}}\;$ , that means the reaction order of water vapor is  −0.5. It was reported [13] that the concentration of water vapor in the gas mixture initially favored the photocatalytic process, but an increase in the water content reduced the rate of removal of acetone for a longer reaction time.

Therefore, the chosen kinetic model should be based on the Langmuir-Hinshelwood mechanism, in which the interaction between adsorbed $p{{\rm \text -}}$xylene with molecular chemisorbed oxygen ${{{\rm O}}_{2}}$ and radical $^{\bullet }{\rm OH}$ is a rate-determining step.

All the points of view discussed above also lead to the conclusion that the nature of the catalytic action for photooxidation of $p{{\rm \text -}}$xylene on ${\rm Ti}{{{\rm O}}_{2}}$ should be considered as a heterogeneous catalytic reaction involving ${\rm O}_{2}^{\bullet -}$ and free radicals $^{\bullet }{\rm OH}$ generated under the light irradiation.