## 3 Results
### Kinetic Modeling
#### 3.3.2 Second-Order Reaction Rate  
The data fitting with the second-order model is shown in Figures 10 and 11, respectively, for microcosms with C/N = 120 and C/N = 180. As in the previous case, the modeling was done both considering the initial diesel oil concentration, equal to 70 g\(\cdot\)kg\({}^{-1}\) of soil, and measuring the initial concentration by analysis.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **u\% (By Weight)** &
\begin{tabular}{c} **First-Order Model** \\ **k (d\({}^{-1}\))** \\ \end{tabular} & \(\mathbf{R^{2}}\) & \(\mathbf{t_{1/2}}\) **(d)** \\ \hline \multirow{3}{*}{Theoretical C\({}_{0}\)} & 8 & 0.0072 & 0.88 & 96 \\  & 12 & 0.0086 & 0.70 & 82 \\  & 15 & 0.0084 & 0.83 & 83 \\ \hline \multirow{3}{*}{Experimental C\({}_{0}\)} & 8 & 0.0066 & 0.90 & 105 \\  & 12 & 0.0074 & 0.84 & 94 \\ \cline{1-1}  & 15 & 0.0070 & 0.84 & 99 \\ \hline \hline \end{tabular}
\end{table}
Table 6: First-order reaction rate constants and half-life times for microcosms with C/N = 180.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **u\% (By Weight)** &
\begin{tabular}{c} **First-Order Model** \\ **k (d\({}^{-1}\))** \\ \end{tabular} & \(\mathbf{R^{2}}\) & \(\mathbf{t_{1/2}}\) **(d)** \\ \hline \multirow{3}{*}{Theoretical C\({}_{0}\)} & 8 & 0.0115 & 0.77 & 60 \\  & 12 & 0.0091 & 0.92 & 76 \\  & 15 & 0.0080 & 0.92 & 89 \\ \hline \multirow{3}{*}{Experimental C\({}_{0}\)} & 8 & 0.0115 & 0.75 & 60 \\  & 12 & 0.0080 & 0.91 & 76 \\ \cline{1-1}  & 15 & 0.0082 & 0.92 & 85 \\ \hline \hline \end{tabular}
\end{table}
Table 5: First-order reaction rate constants and half-life times for microcosms with C/N = 120.  
Figure 9: First-order model for microcosms with C/N = 180: (a) C\({}_{0}\) = 70 g\(\cdot\)kg\({}^{-1}\) of soil; (b) C\({}_{0}\) was measured.  
With this model, the two cases gave very similar results for k value, as shown in Table 7 for C/N = 120, and in Table 8 for C/N = 180.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **u\% (By Weight)** &
\begin{tabular}{c} **Second-Order Model** \\ **k (kg of soil:g\({}^{-1}\)-d\({}^{-1}\))** \\ \end{tabular} & **R\({}^{2}\)** & **t\({}_{1/2}\) (d)** \\ \hline \multirow{3}{*}{Theoretical C\({}_{0}\)} & 8 & 0.0003 & 0.85 & 48 \\  & 12 & 0.0002 & 0.95 & 71 \\  & 15 & 0.0002 & 0.90 & 71 \\ \hline \multirow{3}{*}{Experimental C\({}_{0}\)} & 8 & 0.0003 & 0.85 & 46 \\  & 12 & 0.0002 & 0.93 & 81 \\ \cline{1-1}  & 15 & 0.0002 & 0.90 & 71 \\ \hline \end{tabular}
\end{table}
Table 7: Second-order reaction rate constant and half-life time for microcosms with C/N = 120.  
Figure 11: Second-order model for microcosms with C/N = 180: (a) C\({}_{0}\) = 70 g\(\cdot\)kg\({}^{-1}\) of soil; (b) C\({}_{0}\) was measured.  
Figure 10: Second-order model for microcosms with C/N = 120: (a) C\({}_{0}\) = 70 g\(\cdot\)kg\({}^{-1}\) of soil; (b) C\({}_{0}\) was measured.  
The half-life times were determined using Equation (9) and, as already discussed, this parameter depended just on the initial diesel oil concentration.  
Tables 7 and 8 report that the tested microcosms gave very similar results for reaction rate constant. However, it was noted that the differences heavily influenced the half-life time, which became a sensitive parameter, and its use in real cases should be done with attention to avoiding incorrect or overly optimistic estimates of clean-up times.