## 3 Results and Discussions
as well as with increase the concentration of the washing solution in chitosan from 2% to 5%. The graphical representation of the dependence of lead removal efficiency on the two variables of the washing process (stirring time and S/L ratios) obtained from the determined mathematical relations is illustrated in Figure 2a,b in the cases of using 2% and 5% chitosan washing solution, respectively. The lead removal efficiency gradually increases as the stirring time of the contaminated soil with the chitosan-containing washing solution increases. Also, the highest lead removal efficiency (73.82%) was obtained at a S/L ratio of 0.125 dec.fr. in the case of using 2% chitosan washing solution. Increasing the chitosan content in the washing solution from 2% to 5% and increasing the S/L ratio from 0.0625 to 0.2 (dec.fr) did not significantly improve the lead removal efficiency during the stirring time (Figure 2). Analyzing the link between the performance of the washing process (the removal efficiency after the removal of lead and copper from the soil by washing with chitosan solution) and the process parameters of S/L ratio (\(X_{I}\)) and stirring time (\(X_{2}\))--as shown in the experimental values obtained--an exponential dependency is revealed in which the removal efficiency rapidly increases in the first time intervals, after which the removal phenomena is considerably subdued. This exponential fluctuation means that the widely used linear regression method [47; 48] cannot be used in the case of this paper. For a better approximation of the real phenomena, it was necessary to use a non-linear equation of the mathematical model. In order to determine the coefficients, this equation was corrected to a linear one, first through variable changes [45] and afterward using the method of linear regression. The developed mathematical model is in agreement with the results obtained  
Figure 2: Pb removal efficiency using chitosan for (**a**) C = 2% and (**b**) C = 5%. by [PERSON] et al. (2022) [32] and [PERSON] et al. (2019) [33] that highlight the ability of mathematical models to significantly reduce experiment workload by predicting the removal efficiency according to several parameters [32], such as pH, the initial metal concentration [33], solid-liquid ratio, and washing time [23]. Therefore, the removal efficiency of Cu reaches 55.09% after 6 h of stirring the contaminated soil with the washing solution containing chitosan in a concentration of 5% and at a S/L ratio of 0.0625 dec. r. As can be seen from Figure 1b, extending the stirring time from 6 to 40 h did not affect the removal efficiency significantly, and the removal efficiency of Cu after 40 h of stirring only increased by 18.4% as compared to the value obtained after 6 h of stirring (S/L ratio of 0.0625 dec.fr. and a concentration of 5% chitosan). Considering the economy of the washing process, the optimal stirring time was 6 h in the case of Cu. Higher Cu removal efficiencies at shorter stirring times were also reported by other authors [38]. It was suggested that the process of extracting heavy metal from the soil using washing with chelating agents included two step adsorptions, in which a rapid desorption within the first hours was followed by a subsequent gradual release that occurred over the following hours [38]. In order to verify the integrity and conformity of the mathematical model obtained (Table 3), a comparison was performed between the data calculated using the proposed mathematical model and real experimental data obtained in the laboratory. The comparison is illustrated in Figures 3 and 4. The accuracy of the obtained mathematical models was tested by calculating the concordance dispersion \(S_{conc}^{2}\) between the experimentally obtained values and those calculated based on the models. The calculated values of the concordance dispersion indicate a good agreement between the developed models and the real process they describe.