## 3 Results and Discussion
### The Adsorption Properties of the Mesoporous Materials
For the remaining systems (with polymeric materials), the values of parameters m = 1 and n = 1 suggest that these adsorbents are characterized by energetic homogeneity. To obtain comprehensive adsorption studies of methylene blue, malachite green, and crystal violet on carbon and polymer materials, the scope of the experimental work was supplemented by kinetic measurements. It is widely known that the applicability of porous solids in water and wastewater treatment technologies depends on both their adsorption capacity and the process rate. Sometimes, slow adsorption kinetics is a critical factor that makes a particular adsorbent unusable in real applications. Due to the significant difference in the adsorption capacities of both groups of adsorbents, the same experimental parameters were used for the kinetic studies of the dye-activated carbon systems and the dye-polymer systems, such as adsorbate concentration, solution volume, mixing rate, and temperature, while the adsorbent mass was treated as a variable parameter. For adsorption systems with polymers, a solid weight was exactly 7.5 times larger than this carbon (m\(\sim\)0.15 g and 0.02 g, respectively). Figure 16A-F show the kinetic curves of a given dye adsorption on the materials C3, C4, P1, and P2 in the form of adsorbate concentration as a function of time and adsorbate concentration as a function of the square root of time. The use of the c-t\({}_{1/2}\) scale allows for more precise observation of the initial course of the kinetic curve. It is well known that the adsorption process in a liquid-solid system comprises several stages: (i) mass transfer of adsorbate molecules by diffusion from a bulk solution to the external surface of a solid, (ii) intramolecular diffusion (diffusion in adsorbent pores into its internal surface), (iii) adsorption according to physical or chemical mechanisms at the active sites of the solid surface [75]. The slowest stage defines the course of the entire process. Therefore, the kinetics of dye adsorption is closely correlated with the structural properties of the adsorbents and the molecular size of adsorbates. In the group of carbons, the adsorption process on the carbon C4 is the fastest. This is typical for mesoporous materials, which are additionally characterized by a high affinity for the adsorbate. One can observe the extended linearity of the concentration profiles as a function of the square root of time, which proves the high dynamics of dye loss in the bulk solution caused by its efficient diffusion into the pore space and the occupation of adsorption sites on the solid surface. In the case of the mixed type (micro- and mesoporous) carbon C3, the stage of intramolecular diffusion of the adsorbate is a critical one that slows down the adsorption process. With the increase in the adsorbate molecule size and the development of its shape, the linearity region of kinetic profiles is shortened. For example, the adsorption of methylene blue, a \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**adsorption System** & **a\({}_{\text{m}}\)** & **m** & **n** & **log K** & **R\({}^{2}\)** \\ \hline MB (C3) & 1.01 & 0.25 & 1 & 1.33 & 0.98 \\ MB (C4) & 0.88 & 0.26 & 1 & 1.62 & 0.97 \\ MB (P1) & 0.038 & 1 & 1 & 1.12 & 0.84 \\ MB(P2) & 0.044 & 1 & 1 & 1.26 & 0.89 \\ MG (C3) & 0.60 & 0.32 & 1 & 0.96 & 0.97 \\ MG (C4) & 0.65 & 0.26 & 1 & 1.19 & 0.98 \\ CV (C3) & 0.33 & 0.22 & 1 & 1.24 & 0.96 \\ CV (C4) & 0.44 & 0.30 & 1 & 1.35 & 0.99 \\ CV (P1) & 0.050 & 1 & 1 & 2.41 & 0.86 \\ CV (P2) & 0.038 & 1 & 1 & 2.26 & 0.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of the Generalized Langmuir equation for the dye adsorption on selected activated carbons and polymer materials. molecule that is characterized by the smallest size and linear shape, gives a profile with the largest range of linearity, while the crystal violet adsorption with the largest molecular size and spatially extended shape is significantly shortened.