## 3 Results and Discussion
### Maximum sorption Capacity of MES and MMS  
Maximum sorption capacity data for MES and MMS are modeled using three isotherms: Langmuir 1, Langmuir 2, Freundlich, and Dubinin-Radushkevich (Table 3, Figure 6). Judging by the R\({}^{2}\) coefficient of determination and the Chi-squared test (\(\chi^{2}\)), the Langmuir 2 model shows the greatest fit to data in all experimental groups. This indicates that at least two types of sorption centers play an important role in RB5/RY84 sorption on MES/MMS. Presumably, for both of the tested sorbents, the active sites in question are protonated amine (-NH\({}_{3}\)+), acetamide (-NH\({}_{2}\)COCH\({}_{3}\)+), and hydroxyl groups (-OH\({}_{2}\)+) of chitin and protein. Figure 5: Intraparticle diffusion model for sorption of: (**a**) RB5 on MES, (**b**) RB5 on MMS, (**c**) RY84 on MES, and (**d**) RY84 on MMS. The K\({}_{1}\)/K\({}_{2}\) constants derived from Langmuir 2, which measure the affinity of sorbate functional groups to those of the sorbent, have similar ranges for both RB5 and RY84 sorption (0.091-0.104 for K\({}_{1}\) and 0.002-0.06 for K\({}_{2}\)). This suggests that RB5 and RY84 follow similar mechanisms of sorption on MES and MMS, presumably based on ionic interaction between the ionized groups of the dye (e.g., -SO\({}_{3}\)-) and the protonated function groups of the sorbent (-NH\({}_{3}\)+, -NH\({}_{2}\)COCH\({}_{3}\)+ -OH\({}_{2}\)+). The maximum RB5 and RY84 sorption capacity of the tested sorbents is 78.70 mg/g and 60.49 mg/g, respectively, for MES, as well as 55.72 mg/g and 44.25 mg/g, respectively, for MMS. The inferior sorption capacity of MMS compared with MES may be due to the lower density of sorption centers on the sorbent's surface. The slightly lower sorption capacity of RY84 compared with RB5 may stem from the higher molar mass of the former (RY84--1628 g/mol; RB5--991 g/mol). The heavier dye may, to some extent, restrict access to the active centers in the deeper layers of the sorbent. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & & \multicolumn{3}{c}{**Langmuir 1 Model**} & \multicolumn{3}{c}{**Freundlich Model**} & \multicolumn{3}{c}{**Dubinin–Radushkevich Model**} \\ \cline{3-13}
**Sorbent** & **Dye** & **Qmax** & **K\({}_{\mathbf{c}}\)** & **R\({}^{2}\)** & **x\({}^{2}\)** & **k** & **\(n\)** & **R\({}^{2}\)** & **x\({}^{2}\)** & **Qmax** & **B** & **R\({}^{2}\)** & **x\({}^{2}\)** \\ \cline{2-13}  & & **[mg/g]** & **[L/mg]** & **-** & **-** & **-** & **-** & **-** & **-** & **[mg/g]** & **[mol\({}^{2}\)/\({}^{2}\)]** & **-** & **-** \\ \hline \multirow{2}{*}{**MES**} & **RB5** & 71.65 & 0.084 & 0.986 & 2.822 & 16.3 & 0.27 & 0.899 & 21.18 & 63.68 & 0.0168 & 0.923 & 35.56 \\  & **RY84** & 57.75 & 0.076 & 0.998 & 0.297 & 13.5 & 0.26 & 0.902 & 717.89 & 51.39 & 0.0211 & 0.910 & 24.95 \\ \hline \multirow{2}{*}{**MMS**} & **RB5** & 49.38 & 0.063 & 0.998 & 0.101 & 11.1 & 0.26 & 0.895 & 11.56 & 43.88 & 0.0346 & 0.914 & 28.44 \\  & **RY84** & 41.82 & 0.031 & 0.993 & 0.979 & 6.5 & 0.31 & 0.946 & 4.88 & 34.88 & 0.1104 & 0.871 & 641.5 \\ \hline \hline \multicolumn{13}{c}{**Langmuir 2 Model**} \\ \cline{3-13}
**Sorbent** & **Dye** & **Qmax** & **b\({}_{1}\)** & **K\({}_{1}\)** & **b\({}_{2}\)** & **K\({}_{2}\)** & **R\({}^{2}\)** & **x\({}^{2}\)** & & & & & \\ \cline{2-13}  & & **[mg/g]** & **[mg/g]** & **[L/mg]** & **[mg/g]** & **[L/mg]** & **-** & **-** & & & & & \\ \hline \multirow{2}{*}{**MES**} & **RB5** & 78.70 & 45.84 & 0.099 & 32.86 & 0.002 & 0.990 & 2.433 & & & & & \\ \cline{2-13}  & **RY84** & 60.49 & 51.95 & 0.091 & 8.54 & 0.004 & 0.999 & 0.279 & & & & & \\ \hline \multirow{2}{*}{**MMS**} & **RB5** & 55.72 & 48.86 & 0.095 & 6.86 & 0.002 & 0.999 & 0.098 & & & & & \\ \cline{2-13}  & **RY84** & 44.25 & 30.08 & 0.104 & 14.17 & 0.006 & 0.997 & 0.204 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Constants determined from Langmuir 1, Langmuir 2, Freundlich, and Dubinin–Radushkevich models. Figure 6: Isotherms of sorption of (**a**) RB5 on MES and MMS and (**b**) RY84 on MES and MMS.