Applicability of the equilibrium adsorption isotherms and the statistical tools on to them: a case study for the adsorption of fluoride onto Mg-Fe-CO3 LDH

The demand for a low-cost, easy-to-use, low-maintenance system of adsorbents for detoxifying wastewater effluents is growing. A proper understanding of adsorption mechanisms, their controls, and adsorbate-adsorbent behavior is essential. Thus, the current study deals with the required understanding of linear equilibrium adsorption isotherm models. The adsorption data of different concentrations of fluoride onto Mg-Fe-CO3 LDH is fitted to isotherm models. The two-parameter models discussed are Langmuir, Freundlich, Temkin, and Dubinin-Radushkevich, whereas the three-parameter model cited is Redlich-Peterson. To identify the best-fitting model(s) in an equilibrium isotherm study to quantitatively represent the relevant sorption system, various error functions and statistical tools, such as average relative error deviation (ARED), sum of square errors (ERRSQ or SSE), hybrid relative error function (HYBRID), Marquardt’s percent standard error deviation (MPSD), sum of absolute errors (EABS), sum of normalized errors (SNE), correlation coefficient of Pearson (r), coefficient of determination (r2), chi-square test (χ2), student’s T-test and F-test were applied. It is found that the Temkin model best fits the isotherm data, and the sorption process occurs over multiple layers as per the Freundlich isotherm and was found to be more promising than Langmuir’s monolayer sorption process. The exothermic physisorption course for the adsorbate-absorbent interactions is deduced from the isotherm parameters.


Introduction
Adsorption is a process that is used to remove contaminants from wastewater either by adsorbing or binding to contaminants in water.This process eliminates a variety of pollutants, including heavy metals [1], organic compounds [2], and bacteria [3].Adsorption can occur due to either strong chemical interactions (chemisorption), or weak physical interactions (physisorption).Physisorption is reversible and occurs at low temperatures, while irreversible chemisorption can follow at all temperatures [4].Additionally, in terms of pollution management, adsorption has been acknowledged for its adaptability and broad applicability [5][6].There are a variety of adsorbents that can be employed for adsorption, including alumina and bauxite [7], silica gel [8], zeolites [9], activated carbon [10], anionic clays [11][12][13], and various other materials.The type of adsorbent that is used will depend on the type of contaminant that needs to be removed.This process is safe, environmentally friendly, and relatively inexpensive [14].
Adsorption isotherms are equilibrium relationships involving the quantity of adsorbate on an adsorbent's surface and the adsorbate's equilibrium concentration in the fluid phase.The ability to accurately forecast the adsorption of molecules onto surfaces is critical for many industrial and environmental purposes [15][16].The equilibrium isotherm is crucial to the study and design of adsorption systems when using predictive modeling.The hypothetical assessment and interpretation of thermodynamic characteristics adsorption isotherms act as a vital tool.An isotherm may be accurate in matching experimental data under specific conditions but not under other conditions.In light of the assumptions behind each model's derivations, it has been determined that no one model is universally applicable.For effective predictive modeling of adsorption systems as well as quantified assessment of adsorption behaviour for various adsorbents or different circumstances within the same system, an exact mathematical explanation of equilibrium adsorption limits, even if subjective, is essential [17].
The present research intends to provide a holistic overview of the modelling of experimental data onto five different isotherm models, namely, Langmuir, Freundlich, Temkin, and Dubinin-Radsuhkevich (two-parameter models), and Redlich-Peterson (three-parameter model).These are selected because of their wide range of applicability and assumptions, thus providing umpteen information regarding the adsorption process and adsorbate-adsorbate interaction.Further, the results are evaluated for their goodness and best fit using the above-mentioned error and statistical tools.It is to be noted that the linearized configuration of nonlinear isotherm models is used for analysis.In the present investigation, to probe into the said adsorption isotherms, it is pertinent to use an example for better understanding.Hence, the adsorption of different concentrations of fluoride, the adsorbate, onto Mg-Fe-CO3 layered double hydroxide (LDH), the adsorbent, is used as a reference (Table S1) [11].The preparation, synthesis and related characteristics of the said LDH can be found in the referred article.The results are obtained using Microsoft Excel software only, while the graphics are created using both Excel and CorelDraw software.

Adsorption Isotherm Models
Adsorption isotherm models were summarized in table 1 and illustrating the linearised and nonlinearised forms, among other information.Table 1: Adsorption Isotherms models, their linearised and non-linearised forms, along with their respective plots.
where   is the amount of adsorbate adsorbed at equilibrium (mg g -1 ),   is the maximum adsorption capacity (mg g -1 ),   is the concentration of adsorbate at equilibrium (mg L -1 ),   is the Langmuir isotherm constant (L mg -1 ) which describes the energy of adsorption.
The values of      can be calculated from the slope and intercept from the plot of     ⁄ .  .The favourability of this isotherm can be applied using   (Separation or Equilibrium Factor -a dimensionless parameter) [27] which is described as: ,   = 0 where   is the original adsorbate concentration (mg L -1 ).

Freundlich Adsorption Isotherm
The Freundlich isotherm, as opposed to the Langmuir isotherm, presumes that the sorption possibly will occur over multiple layers as an alternative of a monolayer [28].Consequently, the more the adsorbent concentration, the more will be the sorption and this infinite amount of adsorption can occur theoretically.Equation (4) represents the Freundlich isotherm model which is an exponential equation (4) and its linear form (5); where   is adsorption intensity (dimensionless),   the Freundlich isotherm constant has units mg g -1 (mg L -1 ) -1/n .  is the marker of adsorption capacity (the larger   , the larger will be overall adsorption capacity) whereas 1/  is representative of the potency of adsorption [29] (larger   value indicates higher adsorption affinity).Value of 1/  has following implications [30]: From the plot between     log   , values of   and   can be computed from the slope and intercept, respectively.If the value of   lies between 1-10, or value of 1   ⁄ lies between 0 to 1, this is indicative of favorable adsorption [31].Furthermore, the adsorption is physical or chemical in nature depending on the value of   greater or less than one, respectively [32].

Temkin Adsorption Isotherm
The indirect interactions between adsorbent-adsorbate are the feature of a parameter in the Temkin model.The equation ( 6) represents Temkin model and assumes that the heat of adsorption, which is the function of temperature, would depreciate linearly instead of logarithmically when the surface coverage increases [33][34].
The Temkin model linear form can be derived from Eq. ( 6) as: where   is the Temkin isotherm equilibrium binding constant (L mg -1 ) describing the affinity of sorbent,  is the absolute temperature of 298 K,   is the Temkin isotherm constant and  relates to heat of sorption (J mol -1 ).The value of B in   −1 can be used to check the physisorption or chemisorption nature of adsorption [35].It is described as: ℎ  20 <  < 50   −1 ; ℎ The values of  and   were assessed from the slope and intercept, respectively, in graph of   versus ln   .

Dubinin-Radushkevich Isotherm
This isotherm is often used to represent the mechanism of adsorption onto a heterogeneous surface with a Gaussian energy distribution [33].In contrast to Langmuir or Freundlich isotherm models, this model's adsorption occurs before the process of pore filling, making it a semi-empirical expression [4].The distinction between chemical and physical adsorption is frequently made using this isotherm model [36].Owing to the Dubinin-Radushkevich isotherm's temperature dependency, all necessary information may be acquired by plotting the square of the potential energy adsorption data vs. a function of the quantity adsorbed at various temperatures.[37], DRK model is expressed as follows: = (  ) −   2 (8) The linearised form of the DRK model is: Where   is the Dubinin-Radushkvich isotherm constant (mol 2 kJ -2 ),   relates to the theoretical isotherm saturation capacity (mg g -1 ),  is the Polanyi (adsorption) potential.With its mean free energy, E (kJ mol -1 ), necessary to displace a molecule of adsorbate from its position at the sorption site to infinity, the model is used to distinguish physical and chemical adsorption [36].Value of E less than 8 kJ mol -1 is indicative of physisorption, while the E falling in range of 8 kJ mol -1 and 16 kJ mol -1 indicates chemical adsorption [32,38].E is computed as: Equation 9 was used to plot the DRK model from which a straight line is obtained which is used to determine the constants   and   respectively from the intercept and slope so obtained.

Redlich-Peterson isotherm
The numerator of this isotherm [39] is linear while the denominator is exponential.At elevated concentrations, it resembles the Freundlich model, and it relies with the Langmuir equation's lowconcentration value as well.The Redlich-Peterson equation includes three parameters into isotherm, and is versatile in nature, such that it can be used in homogeneous as well as heterogeneous systems.The equilibrium on a large range of concentrations is represented by its linear dependency on concentration [40].The model is defined by following expression: The linearised form of Eq. 11 is obtained as follows by taking logarithms on both sides: ln where   is dimensionless Redlich-Peterson exponent which falls within 0 and 1 whereas   and   is isotherm constants with units L g -1 and (  −1 ) −  .
At a high concentration of adsorbate (  approaching zero), Eq. 11 reduces to the Freundlich equation [15]: where When   approaches unity, Eq. 11 reduces to Langmuir equation and   becomes   measured as L mg -1 is a Langmuir adsorption isotherm constant which relates to   , Langmuir maximum adsorption capacity (mg g -1 ) as: =     (14) The plot between ln     and ln   gives slope and intercept of the straight line corresponding to   and   values, respectively.

Error functions and Statistical Tools
A best-fit model is an empirical model that closely resembles a dataset.It is used to find the parameters of a model that best describes and makes predictions about the dataset, and subsequently, the isotherms and adsorption process.The error function is a measure of the discrepancy between the best-fit model and the dataset.Statistical functions are used to assess the correlation, the similarity of two or more groups to one another, and the reliability of fit.Table 2 lists all the error functions and statistical tools used.The lower the values of ERRSQ, EABS, ARED, MPSD, HYBRID, SNE, and 2 , and the higher the values of r, r 2 , t-tests, and f-test, the better the fit of the isotherm.

Sum of Square Errors (SSE/ ERRSQ)
It is among the extensively used error function [20].The magnitude of error values, and thus, squared errors has the propensity to increase at higher end of liquid-phase concentration limits.This demonstrates an improved fit for isotherm parameter derivation.

Marquardt's Percent Standard Deviation (MPSD)
It is given by Marquardt [19] and has parallels to an altered form of geometric mean error distribution in regard to degree of freedoms of the system [21].

Average Relative Error Deviation (ARED)
Marquardt devised the ARE with the goal of decreasing the fractional error distribution over the whole a range of concentration [19].This model tends to under or overstates the experimental data [18].

Sum of absolute errors (EABS)
The function is comparable to the EERSQ; as the errors mount, a better fit will be postulated, favouring data with greater concentrations [29].

Hybrid Fractional Error Function (HYBRID)
Developed by Kapoor and Yang [21], it enhances the fit of ERRSQ function at decreased concentration by dividing it with the measured value.

Correlation Coefficient of Pearson (r) and Coefficient of Determination (r 2 )
The correlation coefficients show how much the experimental and modelled data conforms to each other, [22] and thus can be of great use to know the better fit of isotherms.Spearman's correlation coefficient ranges from -1 to +1, whereas Coefficient of Determination has its range between 0 to 1 [20] since it is always positive owing to its squared value.These coefficients represent the variance about the mean, and higher the values, better the fit.As a general rule of thumb [41], one can use following values of correlation coefficients to know the degree of fit:

Sum of Normalised Errors (SNE)
The drawback of using various error functions is that it produces a plethora of parameters of the isotherm to compare.For selecting the significant comparison among the sets of parameters and to establish the best fitting isotherm, one should follow the following standard procedure [21,42]

Student's T-Test
One uses this test to check whether or not the two group of datasets are different from each other.In present study, it's the observed and calculated data, i.e., the modelled and experimental equilibrium solid state adsorption capacity..The paired t-test with two-tailed distribution was evaluated for the applied isotherm models using formula function of Microsoft's Excel, "T.Test(array1, array2, tails, type)", where array 1 & 2 are the two datasets of observed and calculated equilibrium adsorption capacity, tails is either one-tailed or two-tailed distribution and type is one-sample if groups are compared with a standard value while if groups are coming from different populations then these are considered as two-sample.In case, if groups are coming from single population then it is known as paired sample [21,43].

F-Test
The F-Test is included to check the goodness of fit between the observed and calculated data, i.e., the modelled and experimental equilibrium solid state adsorption capacity [24].The function was calculated directly using the Microsoft Excel's formula syntax, "F.Test(array1, array2)", where array1 & 2 are the two sets of data (observed and calculated).

Results and Discussion
The linearised form of different isotherms for their respective plots (X vs. Y) (Table 1) showing adsorption data of  − onto Mg-Fe-CO3 LDH [11] is represented in figure 1.The R 2 value in the plots of figure 1, is for the respective equation from each isotherm's abscissa vs. ordinate, whereas the r 2 value in figure 2 is also the coefficient of determination value, but for the equilibrium data of all isotherm models plotted on the same abscissa and ordinate ( , . , ).The computed parameters for isotherm models, attained from the slopes and intercepts of respective linear plots (Table 3).For Langmuir isotherm, figure 1a, the average monolayer adsorption capacity (  ) for monolayer formation of  − onto Mg-Fe-CO3 came out to be 3.81 mg g -1 and   value is 0.0113 L mg -1 .Despite the low value of  2 = 0.36, for the plot of     ⁄ .  , which recommends non-applicability of Langmuir model in interpreting equilibrium data of Mg-Fe-CO3 for  − adsorption, Langmuir isotherm is still favourable if looked upon   value (0.81-0.95).The error function values for Langmuir isotherm in table 4 indicates lower ERRSQ, EABS, ARED, MPSD, HYBRID values than other models except Temkin model.The low Chi-square value, the high r 2 (close to unity) (Figure 2), t-test and f-test suggests a good fit of data on to isotherm model.Thus, monolayer adsorption process could be happening for a single type of specie over the fixed adsorptive sites of adsorbent's surface.
From the plot between     log   in figure 1b, values of parameters of Freundlich Isotherm,   and   .1/  value lies between 0 and 1, thus verifying favourability of adsorption.Physisorption is confirmed from the value of   = 1.06 which is greater than 1.   value is 0.04 mg g -1 , and  2 for the equation is 0.98 prove that the sorption mechanism fitted very well to Freundlich isotherm.Furthermore, the low error function values, high correlation values for r 2 (Figure 2), t-test and f-test, and the low  2 (Table 4), points to the goodness of fit of isotherm model.This implies that there are heterogeneous sorption sites, and adsorption of fluoride took place in a multi-layered fashion over the adsorbent's reactive sites.
Temkin isotherm is shown in figure 1c and figure 2, and the determined   and   values from the plot,   vs.ln   , are 8920.63and 0.45 L mg -1 , respectively, while B is 0.28 J mol -1 (6.69×10 - 5   −1 ), confirming physical adsorption process.The exothermic adsorption confirmed with positive value of B. Besides  2 (0.997), for the plot between   and ln   , being maximum among all the modelled isotherms, it is clear from table 4 that the error function values are lowest among all models, goodness of fit functions are highest, and the  2 value is lowest.As a result, it can be said that Temkin isotherm model best describes the equilibrium adsorption of  − onto Mg-Fe-CO3.
From the linear plot of Dubinin-Radushkevich (DRK) isotherm model (Figure 1d),   is 3.69 × 10 −6  2  −2 ,   is 0.53 mg g -1 and mean free energy, E, is 0.368 kJ mol -1 , which is indicative of physisorption process, when value of E< 8 kJ mol -1 .As adsorbent regeneration actually depends on adsorbate-adsorbent interaction, and the probability of regeneration increases when physical adsorption dominantes, the values for mean adsorption energy accurately represent the scope of regeneration of the used adsorbent [44].Also, the DRK model can be said to be applicable for interpreting the equilibrium adsorption data onto adsorbent as the  2 and r 2 (Figure 2) value of 0.9654 and 0.9460, respectively, indicates a good fit of adsorption data on the isotherm.In addition to this, table 4 insinuates higher error function values, lower goodness of fit tools (r 2 , t-test and f-test), and higher  2 , compared to previously discussed models.
Figure 1e demonstrates the linearised form of Redlich-Peterson (RP) isotherm model with a plot of      and   .The value of  obtained is 0.059 which is nearly zero, thus, upholding the Freundlich model's process occurring onto heterogenous surface in multilayer fashion.The Redlich-Peterson constants,   and   , obtained are 0.044 L g -1 and 1.108 (mg −1 ) −  .The  2 obtained is 0.17 which is deemed as a very poor fit of equilibrium data onto RP model and is not recommended for explaining adsorption of fluoride by Mg-Fe-CO3.In contrast, the r 2 value (Figure 2) is close to unity (0.9829), but the other error functions (ERRSQ, ARED and EABS) are highest among all isotherms, the f-test and t-test values (Table 4) are least among other applied models, thus, concluding nonapplicability of RP model.The HYBRID and MPSD values couldn't be obtained for this function as number of data points in dataset and number of parameters was same.Hence, the SNE calculated for RP model was sum of other three error functions, which still came out to be highest.The sum of normalised errors (SNE) is presented for each isotherm parameter set of error functions obtained in table 4. It can be seen that the SNE order in the increasing order follows Temkin < Langmuir < Freundlich < Dubinin-Radushkevich < Redlich-Peterson.Thus, with least SNE value, Temkin isotherm best fits as a model for adsorption of  − onto Mg-Fe-CO3.The r 2 , t-test and f-test values further concretes the goodness of fit of Temkin model with their highest values for the said model.

Conclusion
The equilibrium data taken in the present research for  − adsorption on to Mg-Fe-CO3 LDH anionic clay has been analysed by five different isotherm model equations, viz.Langmuir, Freundlich, Temkin, Dubinin-Radushkevich, and Redlich-Peterson through linear regression.The six error functions are successfully used to find the incongruity in the datasets were ERRSQ, EABS, ARED, MPSD, HYBRID and SNE.Additionally, the four statistical tools to reasonably govern the best fit isotherm model were evaluated (r 2 ,  2 , student's t-test, and f-test).The examination concludes that Temkin model tailors the best fit for isotherm data, and has the least dissimilarity between its ---0.17experimental and modelled equilibrium adsorption data.Although r 2 values for both Langmuir and Freundlich isotherm are close to unity, both isotherms could be applicable for adsorption process, thus favouring monolayer as well as multilayer adsorption process, consequently, multiple models are explaining the adsorption process.When looked upon the Redlich-Peterson model, it advocates for the latter model.As a result, it can be said that heterogenous adsorptive sites are present on the Mg-Fe-CO3 LDH, and hence adsorption process for fluoride could be happening in multilayer manner.Furthermore, physical adsorption type of interaction is occurring at adsorbate-absorbent interface as suggested by Freundlich, Temkin, and Dubinin-Radushkevich model, although analytical techniques along with thermodynamical data should be further sought to establish if an adsorption mechanism is a chemical and otherwise a physical process.
called sum of normalised error: (a) Choosing an error function and an isotherm model, and figuring out the changeable parameters that minimised the error function.(b) Determining any additional error functions by reviewing the given parameters.(c) Calculation of additional parameters sets and their related error function values.(d) Addition of each set of parameter's normalised errors.(e) Normalisation and choosing the largest parameter set possible while taking into account the largest error determination.

Table 2 :
List of Error functions and Statistical tools (p: number of parameters in the model; n: number of data points in the dataset; qe,mod: calculated (expected/measured)equilibrium adsorption capacity;

Table 3 :
Isotherm parameters obtained using the linear method for the adsorption of fluoride onto Mg-Fe-CO3 LDH The bold values indicate the highest r, r 2 , t-test, and f-test, and lowest values indicate the ERRSQ, EABS, ARED, MPSD, HYBRID, and 2 . a

Table 4 :
Isotherm error deviation and best fit for Mg-Fe-CO3 HT-LDH a