## 3 Results and Discussion
### Comparison of Parameter Values for the Isotherm Models Obtained by Nonlinear Fitting
\begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Model Parameter**} & \multirow{2}{*}{**Equation**} & \multicolumn{3}{c}{**Parameter Values for the Isotherm Models**} \\ \cline{3-4}  & & **Microsoft Excel Solver Functions** & **MATLAB** & **OriginPro** \\ \hline \hline  & & Freundlich & & \\ \(k_{f}\) & & 10.76 & 10.76 & 10.76 \\ \(n_{f}\) & & 3.7564 & 3.757 & 3.756 \\ \(1/n_{f}\) & & 0.2662 & 0.2662 & 2.2662 \\ \(R\)-square & & 0.8955 & 0.8955 & 0.8955 \\ \(Adj\)\(R\)-square & & 0.8839 & 0.8839 & 0.8839 \\ \(ERRSQ/SSE\) & \(q_{e}=k_{f}\cdot C_{e}\,^{1/n_{f}}\) & 168.2962 & 168.2962 & 168.2963 \\ \(Chi\)-square & & 13.39 & - & 13.39 \\ \(ARE\) & & \(-16.71\) & - & - \\ \(RMSE\) & & 4.3243 & 4.3243 & - \\ \(HYBRID\) & & \(-20.88\) & - & - \\ \(MPSD\) & & 118.1294 & - & - \\ \hline \hline  & & Langmuir & & \\ \(q_{m}\) & & 34.48 & 34.48 & 34.48 \\ \(k_{l}\) & & 0.2346 & 0.2343 & 0.2346 \\ \(R\)-square & & 0.9897 & 0.9897 & 0.9897 \\ \(Adj\)\(R\)-square & & 0.9885 & 0.9885 & 0.9885 \\ \(ERRSQ/SSE\) & \(q_{e}=q_{m}\cdot k_{l}\frac{C}{1+k_{f}\cdot C_{e}}\) & 16.6590 & 16.6590 & 16.6590 \\ \(Chi\)-square & & 1.254 & - & 1.254 \\ \(ARE\) & & \(-2.154\) & - & - \\ \(RMSE\) & & 1.3605 & 1.3605 & - \\ \(HYBRID\) & & \(-2.393\) & - & - \\ \(MPSD\) & & 15.2297 & - & - \\ \hline  & & Temkin & & \\ \(b_{t}\) & & 6.049 & 6.049 & 6.049 \\ \(k_{t}\) & & 3.588 & 3.588 & 3.588 \\ \(R\)-square & & 0.9553 & 0.9384 & 0.9553 \\ \(Adj\)\(R\)-square & & 0.9503 & 0.9306 & 0.9503 \\ \(ERRSQ/SSE\) & & 72.0332 & 72.0332 & 72.0332 \\ \(Chi\)-square & & 4.055 & - & 4.055 \\ \(ARE\) & & \(-4.018\) & - & - \\ \(RMSE\) & & 2.8291 & 3.0001 & - \\ \(HYBRID\) & & \(-4.464\) & - & - \\ \(MPSD\) & & 28.4083 & - & - \\ \hline  & & Liu [45] & & \\ \(q_{m}\) & & 32.78 & 32.78 & 32.78 \\ \(k_{g}\) & & 0.2660 & 0.2660 & 0.2660 \\ \(n_{g}\) & & 0.7735 & 0.7735 & 1.2928 \\ \(R\)-square & & 0.9948 & 0.9948 & 0.9948 \\ \(Adj\)\(R\)-square & & 0.9935 & 0.9935 & 0.9935 \\ \(ERRSQ/SSE\) & \(q_{e}=\frac{q_{m}(k_{e}\,C_{e})\,^{n_{g}}}{1+(k_{g}\,C_{e})\,^{n_{g}}}\) & 8.3791 & 8.3791 & 8.3791 \\ \(Chi\)-square & & 1.209 & - & 1.209 \\ \(ARE\) & & 3.215 & - & - \\ \(RMSE\) & & 1.0234 & 1.0234 & - \\ \(HYBRID\) & & 4.593 & - & - \\ \(MPSD\) & & 18.5609 & - & - \\ \hline \hline  & & Temkin & & \\ \(b_{t}\) & & 6.049 & 6.049 & 6.049 \\ \(k_{t}\) & & 3.588 & 3.588 & 3.588 \\ \(R\)-square & & 0.9553 & 0.9384 & 0.9553 \\ \(Adj\)\(R\)-square & & 0.9503 & 0.9306 & 0.9503 \\ \(ERRSQ/SSE\) & & 72.0332 & 72.0332 & 72.0332 \\ \(Chi\)-square & & 4.055 & - & 4.055 \\ \(ARE\) & & \(-4.018\) & - & - \\ \(RMSE\) & & 2.8291 & 3.0001 & - \\ \(HYBRID\) & & \(-4.464\) & - & - \\ \(MPSD\) & & 28.4083 & - & - \\ \hline  & & Liu [45] & & \\ \(q_{m}\) & & 32.78 & 32.78 & 32.78 \\ \(k_{g}\) & & 0.2660 & 0.2660 & 0.2660 \\ \(n_{g}\) & & 0.7735 & 0.7735 & 1.2928 \\ \(R\)-square & & 0.9948 & 0.9948 & 0.9948 \\ \(Adj\)\(R\)-square & & 0.9935 & 0.9935 & 0.9935 \\ \(ERRSQ/SSE\) & & 8.3791 & 8.3791 & 8.3791 \\ \(Chi\)-square & & 1.209 & - & 1.209 \\ \(ARE\) & & 3.215 & - & - \\ \(RMSE\) & & 1.0234 & 1.0234 & - \\ \(HYBRID\) & & 4.593 & - & - \\ \(MPSD\) & & 18.5609 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of isotherm parameters and error functions from Microsoft Excel Solver function with MATLAB and OriginPro. It can be seen that the result of R-square (0.9384) of the Temkin isotherm model for MATLAB was different with Microsoft Excel (0.9553) and OriginPro (0.9553) due to the real natural logarithm function In (experimental data, \(C_{e}\)) being defined only for \(C_{e}\) as more than zero; the natural logarithm of zero was undefined (log 0, it is not a real number) in MATLAB. For that reason, it might be that the number of degrees of freedom (_df_, maximum number of logically independent values) were reduced to 8 from the data sample set of \(q_{e}\) and \(C_{e}\) (11, Table 1) as compared to Microsoft Excel (9) and OriginPro (9).