Optimum isotherms of dyes sorption by activated carbon: Fractional theoretical capacity & error analysis

•Fractional Theoretical Capacity measure was proposed for isotherm model selection.•Using error analysis statistics alone may lead to wrong isotherm model selection.•The sorption system for all dyes was heterogeneous based on the Toth model.•Fractal dimension can be determined based on the theoretical sorption capacity.•The sorbent (GAC) surface was smooth according to the fractal dimension parameter. The applicability of statistical Goodness-of-Fit Measures (GoFMs) and a new measure, Fractional Theoretical Capacity (FTC), to finding the best fitting isotherm model(s) in the adsorption of dyes have been assessed. The experimental data of adsorption of three acid dyes; Acid Blue 80 (AB80), Acid Red 114 (AR114) and Acid Yellow 117 (AY117) onto Granular Activated Carbon (GAC) type F400 were used in the model selection. Three two-parameter, nine three and one four-parameter isotherm models were used to fit the experimental data. In order to determine the best-fit isotherm for each dye/sorbent system, the geometrical structure of the dyes was optimized with a semi-empirical PM3 method. Thus the approximate maximum cross-sectional area of dye molecules and then the minimum FTC were determined by molecular calculations. The model with the highest FTC regarding to each dye was chosen as the best descriptive model. Statistically eleven GoFMs were also applied to evaluate and rank the feasibility of isotherm models. The model with the lowest GoFM was chosen again as the best. The results showed that using GoFMs alone may leads to wrong model selection but FTC can be a better measure for best descriptive model selection. Based on the FTC measure the adsorption isotherm models fitted the experimental data in the orders: Toth>Langmuir–Freundlich>Sips>UniLan, Toth>UniLan>Langmuir–Freundlich>Sips and Toth>UniLan>Sips>Langmuir–Freundlich for the dyes AY117, AR114 and AB80, respectively. The FTC measure application is recommended in the isotherm model selection process.