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The need to design low-cost adsorbents for the detoxification of industrial effluents has been a growing concern for most environmental researchers. So modelling of experimental data from adsorption processes is a very important means of predicting the mechanisms of various adsorption systems. Therefore, this paper presents an overall review of the applications of adsorption isotherms, the use of linear regression analysis, nonlinear regression analysis, and error functions for optimum adsorption data analysis.

The migration of pollutant(s) in aqueous media and subsequent development of containment measures have resulted in the use of adsorption among other techniques [

A proper understanding and interpretation of adsorption isotherms is critical for the overall improvement of adsorption mechanism pathways and effective design of adsorption system [

In recent times, linear regression analysis has been one of the most applied tools for defining the best fitting adsorption models because it quantifies the distribution of adsorbates, analyzes the adsorption system, and verifies the consistency of theoretical assumptions of adsorption isotherm model [

Because of the inherent bias created by linearization, several error functions have been used to address this shortfall. Concomitant with the evolution of computer technology, the use of nonlinear isotherm modelling has been extensively used.

This is the simplest adsorption isotherm in which the amount of surface adsorbate is proportional to the partial pressure of the adsorptive gas [

Thus, the equilibrium adsorbate concentrations in the liquid and adsorbed phases are related to the linear expression:

The Hill-Deboer isotherm model describes a case where there is mobile adsorption as well as lateral interaction among adsorbed molecules [

The linearized form of this isotherm equation is as follows [^{−1}) and

Fowler-Guggenheim proposed this isotherm equation which takes into consideration the lateral interaction of the adsorbed molecules [

This isotherm model is predicated on the fact that the heat of adsorption varies linearly with loading. Therefore, if the interaction between adsorbed molecules is attractive, then the heat of adsorption will increase with loading because of increased interaction between adsorbed molecules as loading increases (i.e.,

A plot of

It is important to note that this model is only applicable when surface coverage is less than 0.6 (

Kumara et al. analyzed the adsorption data for the phenolic compounds onto granular activated carbon with the Fowler-Guggenheim isotherm and reported that the interaction energy (

Langmuir adsorption which was primarily designed to describe gas-solid phase adsorption is also used to quantify and contrast the adsorptive capacity of various adsorbents [

The Langmuir equation can be written in the following linear form [^{−1}).

^{−1}), which can be correlated with the variation of the suitable area and porosity of the adsorbent which implies that large surface area and pore volume will result in higher adsorption capacity.

The essential characteristics of the Langmuir isotherm can be expressed by a dimensionless constant called the separation factor ^{−1}) and ^{−1}).

Da̧browski studied the adsorption of direct dye onto a Novel Green Adsorbate developed from Uncaria Gambir extract; their equilibrium data were well described by the Langmuir isotherm model [

Freundlich isotherm is applicable to adsorption processes that occur on heterogonous surfaces [

The linear form of the Freundlich isotherm is as follows [

Boparai et al. investigate the adsorption of lead (II) ions [

Dubinin-Radushkevich isotherm model [

This isotherm is only suitable for intermediate range of adsorbate concentrations because it exhibits unrealistic asymptotic behavior and does not predict Henry’s laws at low pressure [

The model is a semiempirical equation in which adsorption follows a pore filling mechanism [

It is usually applied to differentiate between physical and chemical adsorption of metal ions [

Dubinin-Radushkevich isotherm is expressed as follows [^{−1} k^{−1}),

Ayawei et al. and Vijayaraghavan et al. applied the Dubinin-Radushkevich isotherm in their investigation of Congo red adsorption behavior on Ni/Al-CO_{3} and sorption behavior of cadmium on nanozero-valent iron particles, respectively [

Temkin isotherm model takes into account the effects of indirect adsorbate/adsorbate interactions on the adsorption process; it is also assumed that the heat of adsorption

Hutson and Yang applied Temkin isotherm model to confirm that the adsorption of cadmium ion onto nanozero-valent iron particles follows a chemisorption process. Similarly, Elmorsi et al. used the Temkin isotherm model in their investigation of the adsorption of methylene blue onto miswak leaves [

Flory-Huggins isotherm describes the degree of surface coverage characteristics of the adsorbate on the adsorbent [

The linear form of the Flory-Huggins equation is expressed as

This isotherm model can express the feasibility and spontaneity of an adsorption process.

The equilibrium constant ^{−1} K^{−1}, and

Hamdaoui and Naffrechoux used the Flory-Huggins isotherm model in their study of the biosorption of Zinc from aqueous solution using coconut coir dust [

The Hill isotherm equation describes the binding of different species onto homogeneous substrates. This model assumes that adsorption is a cooperative phenomenon with adsorbates at one site of the adsorbent influencing different binding sites on the same adsorbent [

The linear form of this isotherm is expressed as follows [

Hamdaoui and Naffrechoux investigated the equilibrium adsorption of aniline, benzaldehyde, and benzoic acid on granular activated carbon (GAC) using the Hill isotherm model; according to their report, the Hill model was very good in comparison with previous models with

The Halsey isotherm is used to evaluate multilayer adsorption at a relatively large distance from the surface [

Fowler and Guggenheim reported the use of Halsey isotherm in their equilibrium studies of methyl orange sorption by pinecone derived activated carbon. The fitting of their experimental data to the Halsey isotherm model attests to the heteroporous nature of the adsorbent [

Harkin-Jura isotherm model assumes the possibility of multilayer adsorption on the surface of absorbents having heterogeneous pore distribution [

Foo and Hameed reported that the Harkin-Jura isotherm model showed a better fit to the adsorption data than Freundlich, Halsey, and Temkin isotherm models in their investigation of the adsorptive removal of reactive black 5 from wastewater using Bentonite clay [

The Jovanovic model is predicated on the assumptions contained in the Langmuir model, but in addition the possibility of some mechanical contacts between the adsorbate and adsorbent [

The linear form of the Jovanovic isotherm is expressed as follows [^{−1}),

Kiseler reported the use of Jovanovic isotherm model while determining adsorption isotherms for L-Lysine imprinted polymer. Their report showed that the best prediction of retention capacity was obtained by applying the Jovanovic isotherm model [

The equation that defines this model is based on a kinetic principle which assumes that adsorption sites increase exponentially with adsorption; this implies a multilayer adsorption [

The linear forms of the Elovich model are expressed as follows [

Rania et al. reported the use of Elovich isotherm model in their work titled “Equilibrium and Kinetic Studies of Adsorption of Copper (II) Ions on Natural Sorbent.” Their investigation showed that the value of the regression coefficient

The Kiselev adsorption isotherm equation also known as localized monomolecular layer model [^{−1}) and

Equilibrium data from adsorption processes can be modelled by plotting

The Redlich-Peterson isotherm is a mix of the Langmuir and Freundlich isotherms. The numerator is from the Langmuir isotherm and has the benefit of approaching the Henry region at infinite dilution [

This isotherm model is an empirical isotherm incorporating three parameters. It combines elements from both Langmuir and Freundlich equations; therefore the mechanism of adsorption is a mix and does not follow ideal monolayer adsorption [

This model is defined by the following expression:^{−1}), ^{−1}), and ^{−1}).

At high liquid-phase concentrations of the adsorbate, (

When ^{−1} which is related to the energy of adsorption.

^{−1}); when

The linear form of the Redlich-Peterson isotherm can be expressed as follows [

This isotherm model has a linear dependence on concentration in the numerator and an exponential function in the denomination which altogether represent adsorption equilibrium over a wide range of concentration of adsorbate which is applicable in either homogenous or heterogeneous systems because of its versatility [

Sips isotherm is a combination of the Langmuir and Freundlich isotherms and it is given the following general expression [^{−1}), ^{−1}). The linearized form is given as follows [

The Toth isotherm is another empirical modification of the Langmuir equation with the aim of reducing the error between experimental data and predicted value of equilibrium data [^{−1}) and ^{−1}).

It is clear that when

This isotherm model has been applied for the modelling of several multilayer and heterogeneous adsorption systems [

Koble-Carrigan isotherm model is a three-parameter equation which incorporates both Langmuir and Freundlich isotherms for representing equilibrium adsorption data [

All three Koble-Carrigan isotherm constants can be evaluated with the use of a solver add-in function of the Microsoft Excel [

The Kahn isotherm model is a general model for adsorption of biadsorbate from pure dilute equations solutions [

This isotherm model is expressed as follows [^{−1}).

Nonlinear methods have been applied by several researchers to obtain the Khan isotherm model parameters [

The Radke-Prausnitz isotherm model has several important properties which makes it more preferred in most adsorption systems at low adsorbate concentration [

The isotherm is given by the following expression:^{−1}),

At low adsorbate concentration, this isotherm model reduces to a linear isotherm, while at high adsorbate concentration it becomes the Freundlich isotherm and when

Langmuir-Freundlich isotherm includes the knowledge of adsorption heterogeneous surfaces. It describes the distribution of adsorption energy onto heterogeneous surface of the adsorbent [^{−1}),

The Jossens isotherm model predicts a simple equation based on the energy distribution of adsorbate-adsorbent interactions at adsorption sites [

The equation reduces to Henry’s law at low capacities. However, upon rearranging (

A good representation of equilibrium data using this equation was reported for phenolic compounds on activated carbon [

Fritz and Schlunder derived an empirical equation which can fit a wide range of experimental results because of the large number of coefficients in the isotherm [

This isotherm model has the following equation:^{−1}), ^{−1}), and

If

Fritz-Schlunder isotherm parameters can be determined by nonlinear regression analysis [

Bauder observed that the estimation of the Langmuir coefficients, ^{−1}),

For lower surface coverage the Bauder isotherm model reduces to Freundlich model.

Due to the inherent bias resulting from linearization this isotherm parameters are determined by nonlinear regression analysis [

Weber and Van Vliet postulated an empirical relation with four parameters that provided excellent description of data patterns for a wide range of adsorption systems [

The isotherm developed by weber and Van Vliet has the following form:^{−1}), ^{−1},

The isotherm parameters (

The Marczewski-Jaroniec isotherm is also known as the four-parameter general Langmuir equation [

The isotherm equation is expressed as follows:

The isotherm reduces to Langmuir isotherm when

Fritz and Schlunder developed a five-parameter empirical model that is capable of simulating the model variations more precisely for application over a wide range of equilibrium data [

The isotherm equation is^{−1}) and

This isotherm is valid only in the range of

This model approaches Langmuir model while the value of both exponents

In recent times linear regression analysis has been among the most pronounced and viable tools frequently applied for analysis of experimental data obtained from adsorption process. It has been used to define the best fitting relationship that quantify the distribution of adsorbates and also in the verification of the consistency of adsorption models and the theoretical assumptions of adsorption models [

Studies have shown that the error structure of experimental data is usually changed during the transformation of adsorption isotherms into their linearized forms [

Unlike linear regression, nonlinear regression usually involved the minimization of error distribution between the experimental data and the predicted isotherm based on its convergence criteria [

The sum of square of errors (ERRSQ) is said to be the most widely used error function [

One major disadvantage of this error function is that at higher end of liquid-phase adsorbate concentration ranges the isotherm parameters derived using this error function will provide a better fit as the magnitude of the errors and therefore the square of errors tend to increase illustrating a better fit for experimental data obtained at the high end of concentration range [

The hybrid fractional error function (HYBRID) was developed by Kapoor and Yang, to improve the fit of the sum square of errors (ERRSQ) [

The equation for this error function is

The average relative error was developed by Marquardt [

The Marquardt’s percent standard deviation error function is similar to a geometric mean error distribution modified according to the degree of freedom of the system [

This model is similar to the sum square error (ERRSQ) function. In this case isotherm parameters determined using this error function would provide a better fit as the forward high concentration data [

Since each of the error criteria is likely to produce a different set of parameters of the isotherm, a standard procedure known as sum of the normalized errors is adopted to normalize and to combine the error in order to make a move meaningful comparison between the parameters sets. It has been used by several researchers to determine the best fitting isotherm model [

Calculation procedure is as follows:

Selection of an isotherm model and error function and determination of the adjustable parameters which minimized the error function

Determination of all other error functions by referring to the parameters set

Computation of other parameter sets associated with their error function values

Normalization and selection of maximum parameters sets with respect to the largest error measurement

Summation of all these normalized errors for each parameter set

The coefficient of determination represents the variance about the mean; it is used to analyze the fitting degrees of isotherms and kinetic models with experimental data [^{−1}), ^{−1}), and ^{−1}).

This function is very important in the determination of the best fit of an adsorption system. It can be obtained by judging the sum square difference between experimental and calculated data, with each square difference divided by its corresponding values [

This function is very valuable tool for describing the extent of relationship between the transformed experimental data and the predicted isotherms and minimization of error distribution [

The level of accuracy obtained from adsorption processes is greatly dependent on the successful modelling and interpretation of adsorption isotherms.

While linear regression analysis has been frequently used in accessing the quality of fits and adsorption performance because of its wide applicability in a variety of adsorption data, nonlinear regression analysis has also been widely used by a number of researchers in a bid to close the gap between predicted and experimental data. Therefore, there is the need to identify and clarify the usefulness of both linear and nonlinear regression analysis in various adsorption systems.

The authors (Ayawei Nimibofa, Ebelegi Newton Augustus, and Wankasi Donbebe) declare that there are no conflicts of interest regarding the publication of this paper.

_{3}: equilibrium, thermodynamic and kinetic studies

^{2+}ions on nanoparticles of zero-valent iron