Source-normalized error analysis method for accurate prediction of adsorption isotherm: application to Cu(II) adsorption on PVA-blended alginate beads

Surface Modification of NiO‐Al2O3 Nanocomposites using Stearic Acid as a Modifier and their Applications in Self‐cleaning and Removal of Dyes

“…To understand the adsorption process, various models such as Langmuir and Freundlich isotherms have been proposed. Langmuir isotherm assumes that the adsorbate covers the surface of the adsorbent as a monolayer and that each adsorption site has equal energy [98] . The equation used for calculating adsorption capacity, according to Langmuir isotherm model, is as follows: $$$\vcenter{\openup.5em\halign{$\displaystyle{#}$\cr {\rm C}{_{{\rm e}}}/{\rm Q}{_{{\rm e}}}={\rm C}{_{{\rm e}}}/{\rm Q}{_{{\rm m}}}+1/{\rm K}{_{{\rm L}}}{\rm Q}{_{{\rm m}}}\hfill\cr}}$$$ …”

“…Langmuir isotherm assumes that the adsorbate covers the surface of the adsorbent as a monolayer and that each adsorption site has equal energy. [98] The equation used for calculating adsorption capacity, according to Langmuir isotherm model, is as follows:…”

Valorization and Upcycling of Acid Mine Drainage and Plastic Waste via the Preparation of Magnetic Sorbents for Adsorption of Emerging Contaminants

“…57−5859 However, previous studies have reported that linearizing the models is not suitable for the estimation of isotherm parameters, thus leading to erroneous conclusions. 60,61 Furthermore, researchers have recommended the use of coefficient of determination (R 2 ) to decide on the best-fitting isotherm models, which was found to be misleading and insufficient. 62,63 Therefore, nonlinear regression models were used to estimate the isotherm parameters.…”

The Statistical Error Optimization of Dye Sorption Equilibria for the Precise Prediction of Adsorption Isotherms on Activated Graphene

“…However, the nonlinear regression functions yielded different residuals, generated from diverse error mechanisms, making it challenging to determine the so-called optimal or best set of isotherm constants [33]. To overcome this complication and enable a meaningful comparison of the error values, a statistical normalizing process called the sum of the normalized errors (SNE) is used, being a well-established approach among academics [26,27]. The detailed method for SNE calculation is presented in the Supplementary Materials.…”

“…However, researchers have concluded that linearizing adsorption isotherms usually changes the error structure of experimental data [24,25]. Moreover, the linearization of isotherms with three, four, and more parameters is not possible, as two or more unknown components limit simple linearization [26]. Nonlinear regression analysis provides a rigorous method of defining adsorption parameters using isotherm equations in their original form.…”