A critical review of the model fitting quality and parameter stability of equilibrium adsorption models

Abstract

We reviewed eight commonly used equilibrium adsorption models and examined their underlying assumptions, fitting qualities, and parameter stabilities. We compared several objective functions that have been applied to curve fitting analysis and a few statistics tests that have been performed to evaluate regression quality. The iteratively reweighted least squares algorithm was selected as the most suitable regression method for adsorption models in the presence of heteroscedasticity. The fraction of unexplained variance was selected to indicate the model fitting quality. Two sources of parameter instability were identified: residue instability and function instability. While the definition of the instability caused by residue is well established, we are the first to consider the instability caused by an adsorption model. The models discussed in this article can be applied to many surfactants, such as normal alcohols, polyglycol ethers, and sodium dodecyl sulfate at different salt concentrations. Our results show that both the model fitting quality and parameter instability increase with the number of parameters subject to curve fitting. For the Frumkin-type of reorientation model, the parameter instability can be as high as 25%. The high degree of instability in some complicated adsorption models may invalidate the estimated parameters.Therefore, additional measurements or simulations are required for complicated models to extract reliable model parameters.

Systematic evaluation, verification and comparison of tuberculosis‐related non‐coding RNA diagnostic panels

“…In this paper, we developed models as close as possible to original papers and thought that the impact of such alterations on the model can be negligible, which is also supported by other scholars 17 . Now, numerous models have been proliferated, whereas most of them do not provide detailed parameters of modelling which is of utmost importance for the performance of models 45 . Missing parameters impede model reproduction, external validation and further improvement.…”

Adsorption for copper(II) ion with chitosan-SP/PET composite adsorbent enhanced by electric field

“…The nonlinear equation for Langmuir isotherm is represented by the following equation (Foo and Hameed, 2010; Peng et al., 2018) where q e and C e are the amount adsorbed (mg g −1 ) and Cu(II) concentration in solution (mg L −1 ) at equilibrium, respectively. Q 0 and b are the Langmuir constants for the maximum monolayer adsorption capacity and the adsorption equilibrium constant, respectively.…”

“…The nonlinear equation for Freudlich isotherm is an empirical equation and given below (Foo and Hameed, 2010; Peng et al., 2018) where K F and n are Freundlich constants related to adsorption capacity (mg g −1 ) and intensity, respectively.…”

“…In this section, Langmuir and Freundlich models are used to analyze the Cu(II) adsorption isotherms onto the CS-SP/PET composite adsorbent. The nonlinear equation for Langmuir isotherm is represented by the following equation (Foo and Hameed, 2010;Peng et al, 2018) q e ¼ Q 0 bC e 1 þ bC e…”

“…Q 0 and b are the Langmuir constants for the maximum monolayer adsorption capacity and the adsorption equilibrium constant, respectively. The nonlinear equation for Freudlich isotherm is an empirical equation and given below (Foo and Hameed, 2010;Peng et al, 2018) q e ¼ K F C 1=n e (6) where K F and n are Freundlich constants related to adsorption capacity (mg g À1 ) and intensity, respectively.…”