###### Abstract  
The adsorption equilibrium of methyl blue (MB) at different temperatures was optimized using activated graphene (AG) as an adsorbent. The experimental data were compared using five linear and nonlinear adsorption isotherms, namely, Langmuir, Freundlich, Redlich-Peterson (R-P), Sips, and Toth, to estimate the best fit of the equilibrium data. Five distinct error functions were utilized to conduct nonlinear regression for the adsorption equilibrium: SSE, ARE, HYBRID, MPSD, and EABS. These functions offered a wide range of residuals for comparison. For a more accurate prediction of the isotherm model, two statistical techniques--SNE and CND--were applied. By using these techniques in conjunction, a more objective analysis of the error and deviation between the observed and predicted data was achieved, ultimately leading to improved accuracy in the error analysis. The sorption results demonstrated the highest MB removal of 691.89 mg g\({}^{-1}\), which amounted to 98.32% within 120 min. The error analysis findings indicated that the SSE and HYBRID functions produced the smallest error residuals. Based on the \"goodness of fit\" criterion, the models in this study were ranked as R-P > Toth > Langmuir > Sips > Freundlich. Among these models, the R-P isotherm demonstrated the best fit for the data, exhibiting the lowest variance in residuals. Its CND value ranged between 0.0025 and 0.0048, which further supports its superior fit compared to the other models. The combination of multiple error functions and statistical methods allowed for a comprehensive and objective assessment of the nonlinear regression models. The results highlight the importance of using various techniques to improve the accuracy of error analysis and identify the best-fitting isotherms for adsorption. a 2  
such as vomiting, diarrhea, and nausea. Given these risks, it is imperative to prioritize the complete removal or degradation of MB from discharge in open streams to promote sustainable ecology [7; 8; 9]. Diverse treatment techniques based on physical, chemical, and/or biological degradation have been developed to reduce the damaging effects of dyes on the environment [10; 11; 12; 13]. Over the years, due to its low cost, straightforward operation, and effective remediation of pollutants, the adsorption process has been established as a well-recognized pollutant remediation technique. It has been widely used in various industries [14; 15; 16]. Activated carbon, zeolites, and agricultural wastes are being studied as adsorbents due to their moderate surface and porosity attributes, while graphene derivatives are emerging as promising alternatives due to their structural eminence, high efficiency, and low-cost feasibility. Moreover, graphene derivatives have exhibited excellent performance in removing heavy metals, organic pollutants, and dyes from wastewater [17; 18; 19; 20]. They can also be easily modified to enhance their adsorption capacity and selectivity. However, to determine the sorption performance and capacity of an adsorption system under various experimental circumstances, the adsorption parameters must be understood and quantitatively compared. Since adsorption is a one-step process involving one or more physical and chemical phenomena of the adsorbate and adsorbent, an isotherm study is a straightforward approach to identifying the equilibrium conditions [21; 22; 23]. The isotherm study involves the determination of the type of isotherm model that best fits the experimental data to ensure an accurate prediction of the adsorption behavior. Additionally, understanding the equilibrium relationship between the adsorbent properties and the adsorption parameters is key for designing better adsorption systems for specific applications [9; 24]. In most cases, the best-fit isotherm is determined using linear regression. However, researchers have concluded that linearizing adsorption isotherms usually changes the error structure of experimental data [24; 25].