## 3 Results
### Kinetic Analysis  
Figure 5 shows Jatropha oil conversion variation with time at different temperatures in a methanol/oil mole ratio of 12:1, using 3 wt.% CaNa/AC to oil. The excess amount of methanol is generally used in order to increase the rate of the reaction and displace the equilibrium toward the products. Under this condition, researchers often ignored the intermediate reactions or assumed that the transformation of intermediates occurs rapidly [55]. As a result, most kinetic investigations of vegetable oil methanolysis and ethanolysis have been focused on the entire process [56]. Therefore, the reaction rate expression of the transesterification of Jatropha oil and methanol can be written in as Equation (1):  
\[r=-\mathrm{d}C_{\mathrm{A}}/\mathrm{d}t=kC_{\mathrm{A}}^{\mathrm{A}}C_{ \mathrm{B}}^{\beta} \tag{1}\]  
where \(k\) is the rate constant; \(C_{\mathrm{A}}\) and \(C_{\mathrm{B}}\) represent the concentration of Jatropha oil and methanol, respectively. \(\alpha\) and \(\beta\) are the orders of reactants. The kinetic model is based on the overall reaction, and \(C_{\mathrm{A}}\) can be calculated by taking \(x\) as the Jatropha oil conversion and \(C_{\mathrm{A}}\) as the initial concentration of Jatropha oil. Due to a significant surplus of methanol in the reaction mixture, the concentration of methanol in transesterification can be safely  
Figure 4: The basicity of used CaNa/AC at different temperatures.  
considered as constant. Thus, the reaction order of methanol is zero order. Therefore, Equation (1) can be simplified as Equation (2):  
\[r=-\mathrm{d}C_{\mathrm{A}}/\mathrm{d}t=kC_{\mathrm{A}}^{n} \tag{2}\]  
In order to determine the overall reaction orders \(n\), Equation (2) can be expressed as Equation (3):  
\[\mathrm{lg}r=\mathrm{lg}(-\mathrm{d}C_{\mathrm{A}}/\mathrm{d}t)=\mathrm{lg}k+n \mathrm{lg}C_{\mathrm{A}} \tag{3}\]  
Thus, \(n\) and \(k\) are obtained through plotting \(\mathrm{lg}(-\mathrm{d}C_{\mathrm{A}}/\mathrm{d}t)\) vs. \(\mathrm{lg}C_{\mathrm{A}}\).  
Meanwhile, the Arrhenius equation can be used to compute the reaction's activation energy as Equation (4):  
\[\mathrm{lg}k=-\frac{E_{\mathrm{a}}}{2.303\mathrm{R}T}+\mathrm{A} \tag{4}\]  
where \(E_{\mathrm{a}}\) is the activation energy, R is the gas constant, \(T\) is the absolute temperature, and A is a constant.  
In Figure 5(a), the plots of \(\mathrm{lg}(-\mathrm{d}C_{\mathrm{A}}/\mathrm{d}t)\) vs. \(\mathrm{lg}C_{\mathrm{A}}\) were acceptably represented by straight lines having slopes of close to 1 at the three temperatures, which indicated that the transesterification for Jatropha oil and methanol was approximately a first-order reaction in the temperature range of 323-338 K. The fitting results shown in Table 3 are consistent with the previous report [57]. The calculated rate constants were calculated as 0.012, 0.0076, 0.0053, and 0.0034 min\({}^{-1}\) at 323, 328, 333, and 338 K, respectively. These results indicated that higher temperatures sped up the reaction and shortened the reaction time in order to obtain the highest product yield, implying that reaction temperature was an important component in the transesterification and total product yield.  
It has been reported that the activation energies of base-catalyzed heterogeneous transesterification reactions were in the range of 38-83 kJ\(\cdot\)mol\({}^{-1}\)[58]. The apparent activation energy was 75.6 kJ\(\cdot\)mol\({}^{-1}\), which was closed to the activation energies (73.7 kJ\(\cdot\)mol\({}^{-1}\)) for the transesterification of soybean oil catalyzed by CaO [59], as determined from the slope of the plot in Figure 5(b).  
Figure 5: Jatropha oil conversion variation of CaNa/AC at different temperatures.