## 3 Results and Discussion
### Authentication of Isotherm Model Data Using Microsoft Excel Solver Spreadsheet-Based Program
 & \multicolumn{2}{c}{**Statistical Results**} \\ \hline
**Row/Column** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** & **I** & **J** & **K** \\ \hline
1 & 0 & 0 & 0 & 0 & 0 & 1.9260 & \(-1.9260\) & \(SSR\) & 5.5806 & _ERRS/SSE_ & 5.5806 \\
2 & 0.5693 & 4.7154 & 2.9784 & 1.7369 & 3.0169 & 4.9044 & 1.0525 & \(q\) & 32.1326 & _Chi-square_ & 0.8143 \\
3 & 1.3183 & 6.8409 & 6.6830 & 0.1579 & 0.0249 & 8.6009 & 4.7520 & \(k\) & 0.1645 & _ARE_ & 2.8665 \\
4 & 20673 & 8.9664 & 10.0211 & \(-1.0547\) & 1.1124 & 11.9471 & 8.0951 & \(t\) & 1.7045 & _RMSE_ & 0.8325 \\
5 & 4.3412 & 17.8429 & 17.5856 & 0.2573 & 0.0662 & 19.5116 & 15.6959 & Mean & \(q\),exp_ & 20.4081 & _HYBRID_ & 4.0951 \\
6 & 9.2576 & 25.3712 & 25.4476 & \(-\)0.0764 & 0.0058 & 27.3736 & 23.5216 & df & 8.8000 & _MPSD_ & 16.5500 \\
7 & 19.5937 & 30.2031 & 29.8163 & 0.3869 & 0.1497 & 31.7422 & 27.8903 & SE of \(q\),exp_ & 0.8352 & \\
8 & 38.0190 & 30.9059 & 31.3320 & \(-\)0.3415 & 0.1166 & 33.2580 & 29.4060 & R-square & 0.9965 & \\
9 & 57.9422 & 31.0289 & 31.7354 & \(-\)0.7065 & 0.4991 & 33.616 & 29.8094 & Critical t & 2.3600 & \\
10 & 74.7197 & 32.6401 & 31.8736 & 0.7666 & 0.5577 & 33.7996 & 29.9476 & CI & 1.9260 & \\
11 & 96.1409 & 31.9295 & 31.9634 & \(-\)0.0038 & 0.0011 & 33.8894 & 30.0074 & Adjust R-square & 0.9957 & \\ \hline \hline \multicolumn{10}{c}{Formula or Excel code} \\ \hline \hline \multicolumn{10}{c}{\(\phi_{e,cal}=\) SIS2* (ISIS3* )/((ISIS3* )A)/((ISIS4))/((IS4)) Array, Note: Values in column C (\(q_{e,cal}\)) were depended on the values in column A (\(C_{r}\))* = Multiplication sign (\(\times\)).} \\ \multicolumn{10}{c}{\(SSR\) = SUM(CE1:E11) Array} \\ \multicolumn{10}{c}{Men of \(\phi_{e,cal}=\) AVERAGE(B1:B11) Array} \\ \multicolumn{10}{c}{of \(\mu\) = COUNT(IB:B11)-COUNT(I23) Array} \\ \multicolumn{10}{c}{Set of \(q_{e,cal}\) = S SGT(\(\alpha_{e,cal}\)) = SGT(\(\alpha_{e,cal}\))((IS1-C11))/2/(J6) Array} \\ \multicolumn{10}{c}{Rsquare = 1-SUM((BI:B11-C11C1)/2)/(SUM((BI:B11-I5)/2)) Array} \\ \multicolumn{10}{c}{Critical t = TIN(V05,16) Array} \\ \multicolumn{10}{c}{\(CI=\) I= \({}^{\  
sorption. \(k_{f}\) is the Freundlich isotherm rate constant (mg g\({}^{-1}\)) (mg L\({}^{-1}\))\({}^{1/n}\), and \(n_{f}\) is the order of reaction or the heterogeneous factor surfaces (dimensionless) for the [PERSON] model. \(b_{t}\) is the Temkin isotherm constant (J mol\({}^{-1}\)), and \(k_{t}\) is the Temkin isotherm rate constant (L mol\({}^{-1}\)). \(q_{e,\ cal}\) in column C was computed as appropriate for the values of \(C_{e}\) data and the empirical equation of each model shown in Equations (2)-(4) and/or the formula array shown in Tables 2-4 ([PERSON] = SI$3 * (SI$2 * (A/(1 + (SI$2* A)))), [PERSON] = SI$2 * A\({}^{\star}\)(1/SI$3) and Temkin = SI$2*LN(SI$3*A)), respectively. It can be observed that the \(q_{e,\ cal}\) values of each model and row were different and were affected by the statistical results, R-square and Adjust R-square together with residual values, upper CI, lower CI and \(SSR\) and parameter models also. From the perspective of _ERRSQ/SSE, Chi-square, ARE, RMSE, HYBRID_ and _MPSD_, the built-in formula is shown in Tables 2-7. It was found that the statistical results were different depending on the isotherm model function. Likewise, a comparison for the best fitting of two nonlinear parameters of the isotherm models demonstrated that the Langmuir model presented an R-square (0.9897) and Adjust R-square (0.9885) close to 1. _ERRSQ/SSE_ (16.6590), _Chi-square_ (1.2540), _ARE_ (\(-\)2.1538), _RMSE_ (1.2907), _HYBRID_ (\(-\)2.3931) and _MPSD_ (15.2297) values also indicated that all small values of significance were evidence for the acceptance of the nonlinear regression modeling. It could be confirmed that the adsorption of methylene blue or cationic dye onto the magnetic nanosorbent surface involves homogenous sites, with all active sites having equivalent energy, which had no more adsorption onto the magnetic nanosorbent and/or with no interaction among the adsorbed molecules [30; 40].