## 3 Results and Discussion  
### Authentication of Isotherm Model Data Using Microsoft Excel Solver Spreadsheet-Based Program  
The experimental data of methylene blue adsorbed onto magnetic nanosorbent were evaluated and analyzed via the Microsoft Excel Solver spreadsheet-based program for the six different nonlinearized forms of [PERSON], Langmuir, Temkin, Liu, Toth and Khan models described above. The error functions (the goodness of fit) were also examined with the same time period to achieve the statistical values from the isotherm models, such as _ERRSQ/SSE_, \(\chi^{2}\), \(R^{2}\), \(HYBRID\), _ARE_, \(MPSD\) and _RMSE_. The values of the isotherm parameters and the regression statistics with formulas (factors for the model and statistical results) are shown in Tables 2-7. The nonlinear regression of isotherm models and the goodness of fit were built-in functions using the Microsoft Excel Solver spreadsheet-based program available in Microsoft Office. They were appropriate to describe the experimental data with accuracy to predict the best fit for the methylene blue adsorbed onto the magnetic nanosorbent material. These parameter values were evaluated with MATLAB and OriginPro and described under Section 3.2. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \(C_{c}\) & \(q_{c\_exp}\) & \(q_{c\_cal}\) & **Residual** & **Residual’2** & \begin{tabular}{c} **Upper,** \\ **CI** \\ \end{tabular}  & \begin{tabular}{c} **Lower,** \\ **CI** \\ \end{tabular}  & \ \begin{tabular}{c} **Factors for the Model** \\ \end{tabular}  & \begin{tabular}{c} **Statistical Results** \\ **Statistical Results** \\ \end{tabular}  \\ \hline
**Row/Column** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** & **I** & **J** & **K** \\ \hline
1 & 0 & 0 & 0 & 0 & 0 & 6.9196 & \(-\)6.9196 & _SSR_ & 72.0332 & _ERRSQ/SSE_ & 72.0332 \\
2 & 0.5693 & 4.7154 & 4.3210 & 0.3943 & 0.1555 & 11.2406 & -2.5986 & _bt_ & 6.0492 & _Chi-square_ & 4.0547 \\
3 & 1.3183 & 6.8409 & 9.4006 & \(-\)2.5598 & 6.5254 & 16.3022 & 2.5810 & _bt_ & 3.5884 & _ARE_ & \(-\)4.0175 \\
4 & 2.0673 & 8.9664 & 12.1222 & \(-\)3.1558 & 9.9593 & 19.0418 & 5.2026 & Mean of \(q_{c\_exp}\) & 20.0481 & _RMSE_ & 2.8291 \\
5 & 4.3142 & 17.8429 & 16.5726 & 1.2703 & 1.6137 & 23.4922 & 9.6530 & _dt_ & 8.0000 & _HYBRID_ & \(-\)4.4639 \\
6 & 9.2576 & 25.3712 & 21.1913 & 4.1799 & 17.4718 & 28.1109 & 14.2277 & SE of \(q_{c\_exp}\) & 3.0007 & _HYBRID_ & 28.4083 \\
7 & 19.5937 & 30.2031 & 25.7267 & 4.4764 & 20.0382 & 32.6463 & 18.8071 & R-square & 0.9553 & \\
8 & 38.0190 & 30.9905 & 29.7366 & 1.2539 & 1.5723 & 36.6562 & 22.8170 & Critical t & 2.3060 & & \\
9 & 57.9422 & 31.0289 & 32.2885 & \(-\)12.5866 & 1.5790 & 39.2051 & 25.3659 & CI & 6.9196 & & \\
10 & 74.7197 & 32.6401 & 33.8238 & \(-\)1.1836 & 1.4010 & 40.7434 & 26.9042 & Adjustst R-square & 0.9503 & & \\
11 & 96.1409 & 31.9295 & 35.3486 & \(-\)3.4191 & 11.6900 & 42.2682 & 28.4290 & & & \\ \hline \multicolumn{10}{c}{Formula or Excel code} \\ \hline \(q_{c\_cal}=\) & \(\$152\) * LN(\$153\$ ^{*}\) A) Array, Note: Values in column C (\(q_{c\_cal}\)) were depended on the values in column A (C), * = Multiplication sign (\(\times\)). & & & & \\ \(\$SR=\$\)UM(\$E:\$11) Array & \(\$\) & \(\$\) & \(\$\) & \(\$\) & \(\$\) & \(\$\) & \(\$\) \\ Mean  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \(C_{r}\) & \(q_{e,exp}\) & \(q_{e,cal}\) & **Residual** & **ResidualT2** & \begin{tabular}{c} **Upper,** \\ **CI** \\ \end{tabular}  & \ \begin{tabular}{c} **Lower,** \\ **CI** \\ \end{tabular}  & \begin{tabular}{c} **Factors for the Model** \\ \end{tabular}