The reduction of CaWO_{4} by Si could be expressed as following equation:
$3\mathrm{S}\mathrm{i}+2\mathrm{C}\mathrm{a}\mathrm{W}{\mathrm{O}}_{4}=2\mathrm{C}\mathrm{a}\mathrm{O}+2\mathrm{W}+3\mathrm{S}\mathrm{i}{\mathrm{O}}_{2}$(1)Figures 2–4 showed XRD patterns for samples treated for various periods at 1,423 K, 1,473 K and 1,523 K respectively. It could be seen that diffraction peaks of CaWO_{4}, Si and W could be distinguished in the patterns. However, there is not any product phase except W which could be distinguished in term of XRD patterns. This could be due to the formation of some non-crystalline phases. Nevertheless, the fraction conversion could be determined by only employing diffraction peaks of CaWO_{4} and W.

Figure 2 XRD patterns for samples heated at 1,523 K for various periods

Figure 3 XRD patterns for samples heated at 1,473 K for various periods

Figure 4 XRD patterns for samples heated at 1,423 K for various periods

The fractional conversions were determined using an X-ray quantitative analysis method based on RIR (Relative Intensity Ratio) values [11]. The method of X-ray quantitative analysis adopted in the present work was described briefly as follows.

The ratio of mass percentage of CaWO_{4} to W could be calculated from intensity of the strongest peaks for CaWO_{4} (peak(1,1,2)) and W(peak (1,1,0)) according to following equation.
$\frac{\mathrm{\omega}\left(\mathrm{C}\mathrm{a}\mathrm{W}{\mathrm{O}}_{4}\right)}{\mathrm{\omega}\left(\mathrm{W}\right)}=\frac{\mathrm{R}\mathrm{I}\mathrm{R}(\mathrm{W})}{\mathrm{R}\mathrm{I}\mathrm{R}(\mathrm{C}\mathrm{a}\mathrm{W}{\mathrm{O}}_{4})}\times \frac{\mathrm{I}\left(\mathrm{C}\mathrm{a}\mathrm{W}{\mathrm{O}}_{4}\right)}{\mathrm{I}\left(\mathrm{W}\right)}$(2)where *ω*(CaWO_{4}) and *ω*(W) are mass percentages of CaWO_{4} and W respectively. RIR(W) and RIR(CaWO_{4}) are relative intensity ratio values for W and CaWO_{4} respectively. I(CaWO_{4}) and I(W) are intensities of the strongest peaks for CaWO_{4} (peak(1,1,2)) and W(peak (1,1,0)) respectively.

The fractional conversion could be further calculated from *ω*(CaWO_{4})/*ω*(W) according to the following equations.
$\alpha =\frac{1}{1+\frac{\omega \left({\text{CaWO}}_{\text{4}}\right)}{\omega \left(\text{W}\right)}\times \frac{183.84}{287.84}}$(3)

RIR(W)/RIR(CaWO_{4}) could be calculated from relative intensity ratios of W and CaWO_{4} obtained from ICDD cards. Alternatively, it could be also determined from XRD measurements on mixture of W and CaWO_{4} with ω(CaWO_{4})/ω(W) = 1. Figure 5 showed XRD patterns for CaWO_{4}+W mixture (*ω*(CaWO_{4})/*ω*(W) = 1). The XRD measurements were repeated twice. Mean value for RIR(W)/RIR(CaWO_{4}) is 1.467.

Figure 5 XRD patterns for CaWO_{4}+W mixture (*ω*(CaWO_{4})/*ω*(W)=1)

The fractional conversions for various periods at 1,423 K, 1,473 K and 1,523 K were determined from XRD patterns in Figures 2–4 and shown in Figure 6. According to phase equilibrium data for the present system [12], all reactants and products are in solid state at 1,423–1,523 K. Reaction between solid Si and solid CaWO_{4} particles takes places during reduction process. After contact between Si and CaWO_{4}, a product layers could be generated around reactant. With progress of reaction, this layer could be gradually accumulated. This product layer isolates the reactant from each other, therefore hinders the overall reaction. Continuous reaction could only persist through (1) diffusion of reactant (in the form of atoms or ions) through the product layer to the reaction interface; (2) chemical reactions among reactants at the reaction interface. In most cases, the diffusion of reactant through the product layer is slower than interfacial chemical reaction and often becomes the rate-controlling step of the overall reaction [13, 14]. Jander model [15] and Ginstling–Brounshtein model [16] are most frequently used to describe kinetics of three-dimensional diffusion controlling reaction. The expressions of Jander model and Ginstling–Brounshtein model are as follows:
$\text{Jander}:{(1-{(1-\alpha )}^{1/3})}^{2}=kt$(4)
$\text{Ginstling}-\text{Brounshtein}:\text{\hspace{0.17em}}1-\frac{2}{3\alpha}-{(1-\alpha )}^{2/3}=kt$(5)

Figure 6 Fractional conversions as functions of reaction time at various temperatures

where *α* is fractional conversions. *t* is time, *k* is rate constant and follows Arrhenius equation $k=A\mathrm{exp}(-E/RT)$ where *E* and *A* designate the apparent activation energy and pre-exponential factor respectively.

In some cases, the interfacial chemical reaction is slower than diffusion through the product layer and should be considered as rate-controlling step. The equation for three-dimensional chemical reaction could be as follow.
$1-(1-\mathrm{\alpha}{)}^{1/3}=kt$(6)Experimental kinetics data were fitted using Jander, Ginstling–Brounshtein and 3D chemical reaction model. Fitting results are shown in Figures 7–9. It could be seen from Figures 7 and 8 that both Jander and Ginstling–Brounshtein model fit the kinetic data with fairly accuracy. The correlation coefficients attained by the least square fitting are as large as 0.9945 and 0.9925 for Jander and Ginstling–Brounshtein model respectively. The good agreements between fitted and experimental kinetic data using Jander and Ginstling–Brounstein indicate that 3D diffusion could be rate-determining step of overall reaction. In comparison, as shown in Figure 9, there are large differences between fitted kinetic data using 3D interfacial reaction model and experimental kinetic data, and the correlation coefficient is only. The bad fitting with 3D interfacial reaction model indicates that interfacial chemical reaction could not be rate-determining step of overall reaction. Accordingly, it could be inferred from above fittings that the reduction of CaWO_{4} by Si from 1,423 K to 1,523 K could be controlled by three-dimensional diffusion through the product layer.

Figure 7 Result of fitting kinetic data using Ginstling–Brounshtein model

Figure 8 Result of fitting kinetic data using Jander model

Figure 9 Result of fitting kinetic data using 3D interfacial chemical reaction model

The slopes of straight line in Figures 7 and 8 directly provided rate constants *k* at different temperatures. As shown in Figures 10 and 11, the natural logarithms of rate constants *k* were plotted versus reciprocal of absolute temperature to yield the apparent activation energy and pre-exponential values. The temperature dependences of rate constant obtained from the two fitting lines are expressed as follows:
$\begin{array}{rl}k=\phantom{\rule{thickmathspace}{0ex}}& 3.80\times {10}^{7}({\mathrm{s}}^{-1})exp\left(\frac{-379.93(\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l})}{RT}\right)\\ & \text{\hspace{0.17em}}\left(\mathrm{J}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\text{\hspace{0.17em}}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}\right)\end{array}$(7)
$\begin{array}{rl}k=\phantom{\rule{thickmathspace}{0ex}}& 2.72\times {10}^{2}({\mathrm{s}}^{-1})exp\left(\frac{-387.16(\mathrm{k}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l})}{RT}\right)\\ & \text{\hspace{0.17em}}\left(\mathrm{G}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}-\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{B}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{h}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}\text{\hspace{0.17em}}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}\right)\end{array}$(8)Only few reports could be found in literature on kinetics of silicothermic reduction. These studies mainly focused on kinetics of silicothermic reduction of calcined dolomite with regard to Pidgeon process [17, 18]. These studies all suggested that the reduction of dolomite by silicon was controlled by solid-state diffusion of reactants, which is similar to the reaction mechanism revealed in the present work. Wulandari et al. [17] reported an activation energy value of 299~322 kJ/mol for silicothermic reduction of calcined dolomite in flowing argon, which is close to our activation energy value.

Figure 10 Natural logarithm of rate constant k obtained by fitting Ginstling–Brounshtein model as a function of reciprocal of temperature

Figure 11 Natural logarithm of rate constant k obtained by fitting Jander model as a function of reciprocal of temperature

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