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.

1. Introduction

The migration of pollutant(s) in aqueous media and subsequent development of containment measures have resulted in the use of adsorption among other techniques [1, 2]. Adsorption equilibrium information is the most important piece of information needed for a proper understanding of an adsorption process.

A proper understanding and interpretation of adsorption isotherms is critical for the overall improvement of adsorption mechanism pathways and effective design of adsorption system [3].

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 [4].

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.

2. One-Parameter Isotherm

2.1. Henry’s Isotherms

This is the simplest adsorption isotherm in which the amount of surface adsorbate is proportional to the partial pressure of the adsorptive gas [4]. This isotherm model describes an appropriate fit to the adsorption of adsorbate at relatively low concentrations such that all adsorbate molecules are secluded from their nearest neighbours [5].

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

()

where q_e is amount of the adsorbate at equilibrium (mg/g), K_HE is Henry’s adsorption constant, and C_e is equilibrium concentration of the adsorbate on the adsorbent.

3. Two-Parameter Isotherm

3.1. Hill-Deboer Model

The Hill-Deboer isotherm model describes a case where there is mobile adsorption as well as lateral interaction among adsorbed molecules [6, 7].

The linearized form of this isotherm equation is as follows [8]:

()

where K₁ is Hill-Deboer constant (Lmg⁻¹) and K₂ is the energetic constant of the interaction between adsorbed molecules (KJmol⁻¹). Equilibrium data from adsorption experiments can be analyzed by plotting ln⁡[C_e(1 − θ)/θ] − θ/(1 − θ) versus θ [8–10].

3.2. Fowler-Guggenheim Model

Fowler-Guggenheim proposed this isotherm equation which takes into consideration the lateral interaction of the adsorbed molecules [11]. The linear form of this isotherm model is as follows [8]:

()

where K_FG is Fowler-Guggenheim equilibrium constant (Lmg⁻¹), θ is fractional coverage, R is universal gas constant (KJmol⁻¹ K⁻¹), T is temperature (k), and w is interaction energy between adsorbed molecules (KJmol⁻¹).

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., w = positive). However, if the interaction among adsorbed molecules is repulsive, then the heat of adsorption decreases with loading (i.e., w = negative). But when w = 0 then there is no interaction between adsorbed molecules, and the Fowler-Guggenheim isotherm reduces to the Langmuir equation.

A plot of ln⁡[C_e(1 − θ)/θ] versus θ is used to obtain the values for K_FG and w.

It is important to note that this model is only applicable when surface coverage is less than 0.6 (θ < 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 (w) was positive which indicates that there is attraction between the adsorbed molecules [8].

3.3. Langmuir Isotherm

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 [12]. Langmuir isotherm accounts for the surface coverage by balancing the relative rates of adsorption and desorption (dynamic equilibrium). Adsorption is proportional to the fraction of the surface of the adsorbent that is open while desorption is proportional to the fraction of the adsorbent surface that is covered [13].

The Langmuir equation can be written in the following linear form [14]:

()

where C_e is concentration of adsorbate at equilibrium (mg g⁻¹).

K_L is Langmuir constant related to adsorption capacity (mg g⁻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 R_L [15].

()

where K_L is Langmuir constant (mg g⁻¹) and C_o is initial concentration of adsorbate (mg g⁻¹).

R_L values indicate the adsorption to be unfavourable when R_L > 1, linear when R_L = 1, favourable when 0 < R_L < 1, and irreversible when R_L = 0.

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 [14].

3.4. Freundlich Isotherm

Freundlich isotherm is applicable to adsorption processes that occur on heterogonous surfaces [15]. This isotherm gives an expression which defines the surface heterogeneity and the exponential distribution of active sites and their energies [16].

The linear form of the Freundlich isotherm is as follows [17]:

()

where K_F is adsorption capacity (L/mg) and 1/n is adsorption intensity; it also indicates the relative distribution of the energy and the heterogeneity of the adsorbate sites.

Boparai et al. investigate the adsorption of lead (II) ions [17] from aqueous solutions using coir dust and its modified extract resins. Although several isotherm models were applied, the equilibrium data was best represented by Freundlich and Flory-Huggins isotherms due to high correlation coefficients [18].

3.5. Dubinin-Radushkevich Isotherm

Dubinin-Radushkevich isotherm model [19] is an empirical adsorption model that is generally applied to express adsorption mechanism with Gaussian energy distribution onto heterogeneous surfaces [20].

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 [21].

The model is a semiempirical equation in which adsorption follows a pore filling mechanism [22]. It presumes a multilayer character involving Van Der Waal’s forces, applicable for physical adsorption processes, and is a fundamental equation that qualitatively describes the adsorption of gases and vapours on microporous sorbents [23].

It is usually applied to differentiate between physical and chemical adsorption of metal ions [22]. A distinguishing feature of the Dubinin-Radushkevich isotherm is the fact that it is temperature dependent; hence when adsorption data at different temperatures are plotted as a function of logarithm of amount adsorbed versus the square of potential energy, all suitable data can be obtained [13].

Dubinin-Radushkevich isotherm is expressed as follows [16]:

()

where ϵ is Polanyi potential, β is Dubinin-Radushkevich constant, R is gas constant (8.31 Jmol⁻¹ k⁻¹), T is absolute temperature, and E is mean adsorption energy.

Ayawei et al. and Vijayaraghavan et al. applied the Dubinin-Radushkevich isotherm in their investigation of Congo red adsorption behavior on Ni/Al-CO₃ and sorption behavior of cadmium on nanozero-valent iron particles, respectively [16, 22].

3.6. Temkin Isotherm

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 (ΔH_ads) of all molecules in the layer decreases linearly as a result of increase surface coverage [24]. The Temkin isotherm is valid only for an intermediate range of ion concentrations [25]. The linear form of Temkin isotherm model is given by the following [22]:

()

where b is Temkin constant which is related to the heat of sorption (Jmol⁻¹) and K_T is Temkin isotherm constant (Lg⁻¹) [26].

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 [18].

3.7. Flory-Huggins Isotherm

Flory-Huggins isotherm describes the degree of surface coverage characteristics of the adsorbate on the adsorbent [27].

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

()

where θ is degree of surface coverage, n is number of adsorbates occupying adsorption sites, and K_FH is Flory-Huggins equilibrium constant (Lmol⁻¹).

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

The equilibrium constant K_FH is used to calculate spontaneity Gibbs free energy as shown in the following expression [28]:

()

where ΔG^o is standard free energy change, R is universal gas constant 8.314 Jmol⁻¹ K⁻¹, and T is absolute temperature.

Hamdaoui and Naffrechoux used the Flory-Huggins isotherm model in their study of the biosorption of Zinc from aqueous solution using coconut coir dust [29].

3.8. Hill Isotherm

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 [30].

The linear form of this isotherm is expressed as follows [29]:

()

where K_D, n_H, and q_H are constants.

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 R² = 0.99 for all adsorbates [29].

3.9. Halsey Isotherm

The Halsey isotherm is used to evaluate multilayer adsorption at a relatively large distance from the surface [16]. The adsorption isotherm can be given as follows [31]:

()

where K_H and n are Halsey isotherm constant and they can be obtained from the slope and intercept of the plot of ln⁡q_e versus ln⁡C_e.

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 [31]. Similarly, Song et al. applied the Halsey isotherm for the study of coconut shell carbon prepared by KOH activation for the removal of ions from aqueous solutions. The Halsey isotherm fits the experimental data well due to high correlation coefficient (R²), which may be attributed to the heterogeneous distribution of activate sites and multilayer adsorption on coconut shell carbons [16].

3.10. Harkin-Jura Isotherm

Harkin-Jura isotherm model assumes the possibility of multilayer adsorption on the surface of absorbents having heterogeneous pore distribution [32]. This model is expressed as follows:

()

where B and A are Harkin-Jura constants that can be obtained from plotting

versus log⁡C_e.

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 [32].

3.11. Jovanovic Isotherm

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 [33].

The linear form of the Jovanovic isotherm is expressed as follows [34]:

()

where q_e is amount of adsorbate in the adsorbent at equilibrium (mg g⁻¹), q_max is maximum uptake of adsorbate obtained from the plot of ln⁡q_e versus C_e, and K_J is Jovanovic constant.

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 [33].

3.12. Elovich Isotherm

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 [35]. The equation was first developed to describe the kinetics of chemisorption of gas onto solids [36].

The linear forms of the Elovich model are expressed as follows [37]:

()

but the linear form is expressed as follows [8]:

()

Elovich maximum adsorption capacity and Elovich constant can be calculated from the slope and intercept of the plot of ln⁡(q_e/C_e) versus q_e.

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 (R²) for the Elovich model was 0.808 which is higher than that of Langmuir; therefore the adsorption of copper (II) onto Chitin was best described by the Elovich isotherm.

3.13. Kiselev Isotherm

The Kiselev adsorption isotherm equation also known as localized monomolecular layer model [38] is only valid for surface coverage θ > 0.68 and its linearized expression is as follows:

()

where K_i is Kiselev equilibrium constant (Lmg⁻¹) and K_n is equilibrium constant of the formation of complex between adsorbed molecules.

Equilibrium data from adsorption processes can be modelled by plotting 1/C_e(1 − θ) versus 1/θ [8, 38–40].

4. Three-Parameter Isotherms

4.1. Redlich-Peterson Isotherm

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 [41].

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 [42].

This model is defined by the following expression:

()

where A is Redlich-Peterson isotherm constant (Lg−1), B is constant (Lmg⁻¹), β is exponent that lies between 0 and 1, C_e is equilibrium liquid-phase concentration of the adsorbent (mgl⁻¹), and q_e is equilibrium adsorbate loading on the adsorbent (mg g⁻¹).

At high liquid-phase concentrations of the adsorbate, (16) reduces to the Freundlich equation:

()

where A/B = K_F and (1 − β) = 1/n of the Freundlich isotherm model.

When β = 1, (18) reduces to Langmuir equation with b = B (Langmuir adsorption constant (Lmg⁻¹ which is related to the energy of adsorption.

A = bq_ml where q_ml is Langmuir maximum adsorption capacity of the adsorbent (mg g⁻¹); when β = 0, (18) reduces to Henry’s equation with 1/(1 + b) representing Henry’s constant.

The linear form of the Redlich-Peterson isotherm can be expressed as follows [34]:

()

A plot of ln⁡(C_e/q_e) versus ln⁡C_e enables the determination of Redlich-Peterson constants, where β is slope and A is intercept [30, 42–45].

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 [46, 47].

4.2. Sips Isotherm

Sips isotherm is a combination of the Langmuir and Freundlich isotherms and it is given the following general expression [48]:

()

where K_s is Sips isotherm model constant (Lg⁻¹), β_s is Sips isotherm exponent, and a_s is Sips isotherm model constant (Lg⁻¹). The linearized form is given as follows [12]:

()

This model is suitable for predicting adsorption on heterogeneous surfaces, thereby avoiding the limitation of increased adsorbate concentration normally associated with the Freundlich model [19]. Therefore at low adsorbate concentration this model reduces to the Freundlich model, but at high concentration of adsorbate, it predicts the Langmuir model (monolayer adsorption). The parameters of the Sips isotherm model are p^H, temperature, and concentration dependent [12, 49] and isotherm constants differ by linearization and nonlinear regression [50].

4.3. Toth Isotherm

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 [51]. This model is most useful in describing heterogeneous adsorption systems which satisfy both low and high end boundary of adsorbate concentration [52]. The Toth isotherm model is expressed as follows [52]:

()

where K_L is Toth isotherm constant (mg g⁻¹) and n is Toth isotherm constant (mg g⁻¹).

It is clear that when n = 1, this equation reduces to Langmuir isotherm equation. Therefore the parameter n characterizes the heterogeneity of the adsorption system [51] and if it deviates further away from unity (1), then the system is said to be heterogeneous. The Toth isotherm may be rearranged to give a linear form as follows:

()

The values of parameters of the Toth model can be evaluated by nonlinear curve fitting method using sigma plot software [53].

This isotherm model has been applied for the modelling of several multilayer and heterogeneous adsorption systems [53, 54].

4.4. Koble-Carrigan Isotherm

Koble-Carrigan isotherm model is a three-parameter equation which incorporates both Langmuir and Freundlich isotherms for representing equilibrium adsorption data [55]. The linear form of this module is represented by the following equation [56]:

()

where A_k is Koble-Carrigan’s isotherm constant, B_k is Koble-Carrigan’s isotherm constant, and p is Koble-Carrigan’s isotherm constant.

All three Koble-Carrigan isotherm constants can be evaluated with the use of a solver add-in function of the Microsoft Excel [56]. At high adsorbate concentrations, this model reduces to Freundlich isotherm. It is only valid when the constant “p” is greater than or equal to 1. When “p” is less than unity (1), it signifies that the model is incapable of defining the experimental data despite high concentration coefficient or low error value [54].

4.5. Kahn Isotherm

The Kahn isotherm model is a general model for adsorption of biadsorbate from pure dilute equations solutions [57].

This isotherm model is expressed as follows [58]:

()

where a_k is Kahn isotherm model exponent, b_k is Khan isotherm model constant, and Q_max is Khan isotherm maximum adsorption capacity (mg g⁻¹).

Nonlinear methods have been applied by several researchers to obtain the Khan isotherm model parameters [59, 60].

4.6. Radke-Prausniiz Isotherm

The Radke-Prausnitz isotherm model has several important properties which makes it more preferred in most adsorption systems at low adsorbate concentration [61].

The isotherm is given by the following expression:

()

where q_MRP is Radke-Prausnitz maximum adsorption capacity (mg g⁻¹), K_RP is Radke-Prausnitz equilibrium constant, and MRP is Radke-Prausnitz model exponent.

At low adsorbate concentration, this isotherm model reduces to a linear isotherm, while at high adsorbate concentration it becomes the Freundlich isotherm and when M_RP = 0, it becomes the Langmuir isotherm. Another important characteristic of this isotherm is that it gives a good fit over a wide range of adsorbate concentration. The Radke-Prausnitz model parameters are obtained by nonlinear statistical fit of experimental data [61, 62].

4.7. Langmuir-Freundlich Isotherm

Langmuir-Freundlich isotherm includes the knowledge of adsorption heterogeneous surfaces. It describes the distribution of adsorption energy onto heterogeneous surface of the adsorbent [54]. At low adsorbate concentration this model becomes the Freundlich isotherm model, while at high adsorbate concentration it becomes the Langmuir isotherm. Langmuir-Freundlich isotherm can be expressed as follows:

()

where q_MLF is Langmuir-Freundlich maximum adsorption capacity (mg g⁻¹), K_LF is equilibrium constant for heterogeneous solid, and M_LF is heterogeneous parameter and it lies between 0 and 1. These parameters can be obtained by using the nonlinear regression techniques [63].

4.8. Jossens Isotherm

The Jossens isotherm model predicts a simple equation based on the energy distribution of adsorbate-adsorbent interactions at adsorption sites [64]. This model assumes that the adsorbent has heterogeneous surface with respect to the interactions it has with the adsorbate. The Jossen isotherm can be represented as follows:

()

where H is Jossens isotherm constant (it corresponds to Henry’s constant), p is Jossens isotherm constant and it is characteristic of the adsorbent irrespective of temperature and the nature of adsorbents, and F is Jossens isotherm constant.

The equation reduces to Henry’s law at low capacities. However, upon rearranging (29) [65],

()

The values of H and F can be obtained from either a plot of ln⁡(C_e/Q_e) versus q_e or using a least square fitting procedure.

A good representation of equilibrium data using this equation was reported for phenolic compounds on activated carbon [66] and on amberlite XAD-4 and XAD-7 macroreticular resins [67].

5. Four-Parameter Isotherms

5.1. Fritz-Schlunder Isotherm

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 [68].

This isotherm model has the following equation:

()

where q_mFS is Fritz-Schlunder maximum adsorption capacity (mg g⁻¹), K_FS is Fritz-Schlunder equilibrium constant (mg g⁻¹), and MFS is Fritz-Schlunder model exponent.

If M_FS = 1, then the Fritz-Schlunder model becomes the Langmuir model, but, for high adsorbate concentrations, the model reduces to Freundlich model.

Fritz-Schlunder isotherm parameters can be determined by nonlinear regression analysis [69, 70].

5.2. Baudu Isotherm

Bauder observed that the estimation of the Langmuir coefficients, b and q_ml, by measurement of tangents at different equilibrium concentrations shows that they are not constants in a broad range [71]; therefore the Langmuir isotherm has been reduced to the Bauder isotherm [62]:

()

where q_m id Bauder maximum adsorption capacity (mg g⁻¹), bo is equilibrium constant, x is Baudu parameter, and Y is Baudu parameter.

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 [72].

5.3. Weber-Van Vliet Isotherm

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 [73].

The isotherm developed by weber and Van Vliet has the following form:

()

where C_e is equilibrium concentration of the adsorbate (mg g⁻¹), q_e is adsorption capacity mg g⁻¹, p₁, p₂, p₃, and p₄ are Weber-Van Vliet isotherm parameters

The isotherm parameters (p₁, p₂, p₃, and p₄) can be defined by multiple nonlinear curve fitting techniques which is predicated on the minimization of sum of square of residual [72–74].

5.4. Marczewski-Jaroniec Isotherm

The Marczewski-Jaroniec isotherm is also known as the four-parameter general Langmuir equation [75]. It is recommended on the basis of the supposition of local Langmuir isotherm and adsorption energies distribution in the active sites on adsorbent [76].

The isotherm equation is expressed as follows:

()

where n_MJ and M_MJ are parameters that characterize the heterogeneity of the adsorbent surface, M_MJ describes the spreading of distribution in the path of higher adsorption energy, and n_MJ describes the spreading in the path of lesser adsorption energies.

The isotherm reduces to Langmuir isotherm when n_MJ and M_MJ = 1, when n_MJ = M_MJ; it reduces to Langmuir-Freundlich model.

6. Five-Parameter Isotherms

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 [74].

The isotherm equation is

()

where q_mFS5 is Fritz-Schlunder maximum adsorption capacity (mg g⁻¹) and K₁, K₂, α_FS, and β_FS are Fritz-Schlunder parameters.

This isotherm is valid only in the range of L_FS value less than or equal to 1.

This model approaches Langmuir model while the value of both exponents α_FS and β_FS equals 1 and for higher adsorbate concentrations it reduces to Freundlich model.

7. Error Analysis

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 [77, 78].

Studies have shown that the error structure of experimental data is usually changed during the transformation of adsorption isotherms into their linearized forms [79]. It is against this backdrop that nonlinearized regression analysis became inevitable, since it provides a mathematically rigorous method for determining adsorption parameters using original form of isotherm equations [80, 81].

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 [82]. This operation is no longer computationally difficult because of availability of computer algorithms [21].

7.1. The Sum Square of Errors (ERRSQ)

The sum of square of errors (ERRSQ) is said to be the most widely used error function [83]. This method can be represented by the following expression [45]:

()

where q_e,i,calc is the theoretical concentration of adsorbate on the adsorbent, which have been calculated from one of the isotherm models.

q_e,i,meas is the experimentally measured adsorbed solid phase concentration of the adsorbate adsorbed on the adsorbent.

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 [45, 84].

7.2. Hybrid Fractional Error Function (HYBRID)

The hybrid fractional error function (HYBRID) was developed by Kapoor and Yang, to improve the fit of the sum square of errors (ERRSQ) [85] at low concentrations by dividing it by the measured value. This function includes the number of data points (n), minus the number of parameters (p) or isotherm equation as a divisor [85].

The equation for this error function is

()

7.3. Average Relative Error (ARE)

The average relative error was developed by Marquardt [86] with the aim of minimizing the fractional error distribution across the entire concentration range. It is given by the following expression:

()

7.4. Marquardt’s Percent Standard Deviation (MPSD)

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 [87]. It is given by the following expression:

()

7.5. Sum of Absolute Errors (EABS)

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 [88]. It is represented by the following equation:

()

7.6. Sum of Normalized Errors (SNE)

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 [88–91].

Calculation procedure is as follows:

(i)
Selection of an isotherm model and error function and determination of the adjustable parameters which minimized the error function
(ii)
Determination of all other error functions by referring to the parameters set
(iii)
Computation of other parameter sets associated with their error function values
(iv)
Normalization and selection of maximum parameters sets with respect to the largest error measurement
(v)
Summation of all these normalized errors for each parameter set

7.7. Coefficient of Determination (R²) Spearman’s Correlation Coefficient (R_s) and Standard Deviation of Relative Errors (S_RE)

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 [12, 92]. The coefficient of determination (R²) is defined by the following equation [93]:

()

where q_exp is amount of adsorbate adsorbed by adsorbent during the experiment (mg g⁻¹), q_cal is amount of adsorbate obtained by kinetic isotherm models (mg g⁻¹), and q_mexp is average of q_exp (mg g⁻¹).

7.8. Nonlinear Chi-Square Test (X²)

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 [90]. The value of this function can be obtained from the following equation:

()

7.9. Coefficient of Nondetermination

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 [93].

()

where R² is coefficient of determination.

8. Conclusion

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.

Conflicts of Interest

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

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Modelling and Interpretation of Adsorption Isotherms

Abstract

1. Introduction

2. One-Parameter Isotherm

2.1. Henry’s Isotherms

3. Two-Parameter Isotherm

3.1. Hill-Deboer Model

3.2. Fowler-Guggenheim Model

3.3. Langmuir Isotherm

3.4. Freundlich Isotherm

3.5. Dubinin-Radushkevich Isotherm

3.6. Temkin Isotherm

3.7. Flory-Huggins Isotherm

3.8. Hill Isotherm

3.9. Halsey Isotherm

3.10. Harkin-Jura Isotherm

3.11. Jovanovic Isotherm

3.12. Elovich Isotherm

3.13. Kiselev Isotherm

4. Three-Parameter Isotherms

4.1. Redlich-Peterson Isotherm

4.2. Sips Isotherm

4.3. Toth Isotherm

4.4. Koble-Carrigan Isotherm

4.5. Kahn Isotherm

4.6. Radke-Prausniiz Isotherm

4.7. Langmuir-Freundlich Isotherm

4.8. Jossens Isotherm

5. Four-Parameter Isotherms

5.1. Fritz-Schlunder Isotherm

5.2. Baudu Isotherm

5.3. Weber-Van Vliet Isotherm

5.4. Marczewski-Jaroniec Isotherm

6. Five-Parameter Isotherms

7. Error Analysis

7.1. The Sum Square of Errors (ERRSQ)

7.2. Hybrid Fractional Error Function (HYBRID)

7.3. Average Relative Error (ARE)

7.4. Marquardt’s Percent Standard Deviation (MPSD)

7.5. Sum of Absolute Errors (EABS)

7.6. Sum of Normalized Errors (SNE)

7.7. Coefficient of Determination (R2) Spearman’s Correlation Coefficient (Rs) and Standard Deviation of Relative Errors (SRE)

7.8. Nonlinear Chi-Square Test (X2)

7.9. Coefficient of Nondetermination

8. Conclusion

Conflicts of Interest

References

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References

Related

Information

7.7. Coefficient of Determination (R²) Spearman’s Correlation Coefficient (R_s) and Standard Deviation of Relative Errors (S_RE)

7.8. Nonlinear Chi-Square Test (X²)