Linear and Nonlinear RegressionMethods for Equilibrium Modelling of pp-Nitrophenol Biosorption by Rhizopus oryzae : Comparison of Error Analysis Criteria

e study assessed the applicability of Rhizopus oryzae dead fungi as a biosorbent medium for p-nitrophenol (p-NP) removal from aqueous phase.e extent of biosorption was measured through �ve equilibrium sorption isotherms represented by the Langmuir, Freundlich, Redlich-Peterson, multilayer and Fritz-Schlunder models. Linear and nonlinear regression methods were compared to determine the best-�tting equilibrium model to the experimental data. A detailed error analysis was undertaken to investigate the effect of applying seven error criteria for the determination of the single-component isotherm parameters. According to the comparison of the error functions and to the estimation of the corrected Akaike information criterion (AICC), the Freundlich equation was ranked as the �rst and the Fritz-Schlunder as the second best-�tting models describing the experimental data. e present investigations proved the high efficiency (94%) of Rhizopus Oryzae as an alternative adsorbent for p-NP removal from aqueous phase and revealed the mechanism of the separation process.


Introduction
Nitrophenols are classi�ed as moderately to highly toxic even at low concentrations when present in wastewaters.us, they are a source of serious social and hygienic problems as important occupational and environmental pollutants [1].e undesirable impact of these recalcitrant organics on organisms, as well as their considerable release through industrial wastes, provoked the researchers to search for innovative effective technologies for the remediation of nitrophenols-rich wastes before their discharge into natural water streams [2].Nitroaromatic compounds are widely used as pesticides, explosives, solvents, and intermediates in the synthesis of dyes and other chemicals.Many of these compounds and their transform products are of signi�cant toxicological concern [3].
e monosubstituted p-nitrophenol (p-NP) could enter the environment during its production and use in hightemperature coal conversion, as a byproduct in the enzymatic hydrolysis of parathion, methyl parathion, and N-acetylp-amino-phenol.It is also found in suspended particulate matter in the atmosphere, originating mostly from secondary photochemical reactions in the air [3].e toxicity of pnitrophenol on biological systems has led to its classi�cation as a priority pollutant by the United States Environmental Protection Agency (EPA) [4].
e walls of R. oryzae consist of chitin and chitosan.e mechanism of biosorption is quite complicated.e capacity of dead cells might be higher, equal or even lower when compared to that of live cells.In most cases, however, the application of dead biomass is more effective [17].Chitin is the second most abundant natural carbohydrate polymer next to cellulose.Many applications of R. oryzae are due to the secondary amino groups of chitosan which show polycationic chelating and �lm-forming properties along with high solubility in dilute acetic acid [18].
Over the past few decades, linear regression has been developed as a major option in designing adsorption systems [19][20][21][22][23].However, recent investigations have indicated the growing discrepancy (between the predictions and experimental data) and disability of the model, propagating towards a different outcome [24].A number of error functions (sum of the squares of errors, ERRSQ; hybrid fractional error function, HYBRID; Marquardt's percent standard deviation, MPSD; average relative error, ARE; Sum of the absolute errors, EABS; chi-square function,  2 ; Akaike information criterion, AIC, etc.) were highlighted and discussed for the comparisons of linear and nonlinear isotherm models [24][25][26].According to recent studies, the expanding of the nonlinear isotherms represents a potentially viable and powerful tool, leading to the superior improvement in the area of adsorption science [27].However, according to Foo and Hameed [24], further explorations in this area are recommended.Moreover, these error estimating functions do not take into account the number of parameters in the models.Hence, they cannot be used for selection of model(s) appropriateness and determination of its/their magnitude.e literature reports studies applying the Akaike information criterion (AIC) for model selection and ranking in the �eld of veterinary medicine [28], genetics [29], aquatic biogeochemical modeling [30], FTIR spectroscopy [31], pharmacy [32], and so forth.e number of studies subjected to the applicability of AIC for statistical modelling of toxic organics sorption on alternative sorbents, however, is limited.
In this context, the scope of the present study was to investigate the applicability of Rhizopus oryzae as an alternative "ecofriendly" adsorbent for p-nitrophenol removal from aqueous phase at equilibrium conditions.Linear and nonlinear regression analysis was performed to determine model parameters.Seven error functions were applied to evaluate, compare, and rank the feasibility of the �ve applied isotherm models (Langmuir, Freundlich, Redlich-Peterson, multilayer, and Fritz-Schlunder).

Adsorbent. Rhizopus oryzae used in the recent study as a sorbent was supplied by the International Mycological
Institute in Surrey, UK, in the form of IMI strain 266680.e microorganisms were isolated from a soil in Sri Lanka.e spores dried at low temperature were reactivated and cultivated in malt extract (17 g dm −3 malt extract and 3 g dm −3 mycological peptone, dissolved in distilled water at pH 5.4 ± 0.2).e malt extract was inoculated by a standard sterile method and incubated at 32 ∘ C for 3 days in a platform shaker at 175 rpm.ree ceramic granules were added to each of the batch reactors with malt extract to limit the microorganisms threadlike growth.e biomass obtained was precisely washed out consecutively with maternal lye and distilled water and then dried in an oven at 50 ∘ C. e dried biomass was ground in a hammer mill and screened.e fraction used in the recent investigations was   0.15−0.50mm.

Equilibrium Studies.
e equilibrium experiments were accomplished using model solutions of p-NP in distilled water.e investigations were carried out at temperature 19± 1 ∘ C and pH 6.1.Solutions were in the concentration range   5-25 mg dm −3 .Known amounts of Rhizopus oryzae, 0.3 g, were added to 100 cm 3 of the model solutions in screw cap jars.e jars were shaken on a platform shaker.e solute from each jar was then �ltered.e residual p-NP concentrations in the liquid phase (  ) were determined spectrophotometrically.All investigations were performed in triplicate.e corresponding values of p-NP solid phase concentrations (  ) were calculated by the mass balance: where   , mg dm −3 , is the initial adsorbate concentration in the liquid phase,    0, and , , is the adsorbent mass.SPECORD UV-VIS, Carl Zeiss Jena, spectrophotometer was used for concentration determinations at maximum absorption wavelength  228 nm.e pH was measured using an LHP 403T TACUSSEL pH-meter.Blanks containing no adsorbate and replicates of each adsorption point were used for each series of experiments.

Isotherm
Modeling.e design and efficient operation of adsorption processes require equilibrium adsorption data for use in kinetic, dynamic, and mass transfer models [34].In order to optimize the design of a speci�c sorbate/sorbent system, it is important to establish the most appropriate correlation for the experimental equilibrium data [35].
e biosorption behavior of p-NP on R. oryzae in the present research was modelled by the Langmuir, Freundlich, Redlich-Peterson, multilayer, and Fritz-Schlunder (fourparameter) isotherm equations (Table 1).

Error Analysis.
In the present study, linear and nonlinear regression analysis was performed to determine the values of the isotherm model parameters.Six different error functions ( 2 , ERRSQ, HYBRID, MPSD, ARE, and EABS) were examined to evaluate the applicability of each model isotherm equation to the experimental data (Table 2) using the solver add-in functions of Microso Excel soware.
T 1: Mathematical equations of the applied single component isotherm models.

Results and Discussion
In order to assess the fate of nitrophenols in wastewater and to control their mobility and reactivity during remediation processes, the sorption behaviour and mechanism of these toxic contaminants must be understood and revealed.e knowledge of sorbate/sorbent adsorption behaviour at equilibrium is essential for environmental engineering and science as the derived isotherms reveal the speci�c relation between the pollutant concentration and its uptake degree by the solid phase at constant temperature.In this section, among the �ve studied isotherms models, the best-�tting one was determined by the use of seven well-known error functions to calculate the error deviation between experimental and predicted equilibrium adsorption data, aer both linear and nonlinear analysis.In all of the error methods it was assumed that both the liquid phase concentration and the solid phase concentration contribute equally to weighting the error criterion for the model solution procedure.e experimental investigations were conducted at biomass concentration  3.00 g dm −3 .
e values of the model parameters and the isotherm error deviation data for the Langmuir, Freundlich, and Redlich-Peterson equations determined by linear regression analysis and for the multilayer model obtained through the second-order polynomial form of (7) (see Table 1), are presented in Table 3.
Alternative isotherm parameters were also determined by nonlinear regression using �ve error functions (ERRS�, HYBRID, MPSD, ARE, and EABS) and the corrected Akaike information criterion (AIC C ). e values of the Langmuir, Freundlich, Redlich-Peterson, multilayer and Fritz-Schlunder model constants and the isotherm error deviation data are presented in Table 4.
e experimental data points of p-NP adsorption on Rhizopus oryzae dead biomass (Figures 1, 2, 4, and 5) outlined a distinct steep vertical section in the low-concentration region, where a sharp leap of nitrophenol solid phase concentration from   0 to 2.63 mg g −1 was distinguished.Consequently, higher extent of adsorption during the initial stages of the process could be expected.Besides, a clearly marked plateau, that is, a horizontal section in the higher concentration range, was not observed.e latter could be interpreted by means of the presence of a number of vacant active sites, as well as a larger continuance of the adsorption process in the later stages.e above-stated assumptions were consistent with the higher values of the equilibrium constant for the �rst layer adsorption ( 1 1.935: linear regression;  1 2.7971−3.2225:nonlinear regression) than that for multilayer sorption ( 2 0.026: linear regression;  2 0.0265−0.0336:nonlinear regression), calculated by the multilayer model (Tables 3 and 4).e experimentally established maximum equilibrium capacity of the biomass was   4.6 mg g −1 , the monolayer capacity according to the linear Langmuir model-  /  5.01 mg g −1 (Table 3), and according to the nonlinear model-  /  4.28−7.91mg g −1 (Table 4).e second polynomial form of the multilayer equation predicted a maximum monolayer adsorption capacity-  3.399 mg g −1 (Table 3), while the values of this parameter calculated by  applying the nonlinear approach ranged between   3.0834 and 3.401 mg g −1 (Table 4).e comparative analysis between the values of the error functions, obtained through the linear approach, outlined the three-parameter Redlich-Peterson model as the one with the highest  2 0.9997 (Table 3).However, the Freundlich equation characterized with the lowest ERRSQ, HYBRID, ARE, MPSD and EABS error values.e data from Table 4 showed analogous tendency.Hence, is seemed that the Freundlich equation was the most suitable model presenting the most satisfactory description of the studied biosorption phenomenon.
Statistically, it is expected that the higher the number of parameters in a model equation, the closer the theoretical estimates should be to the empirical data.Moreover the error functions HYBRID and MPSD could be accepted as the most indicative, adequate and essentially meaningful when determining the best �t isotherm model, as the number of the isotherm parameters is accounted only by them [35].e data in Tables 3 and 4, however, displayed the contrary as the three-parameter (Redlich-Peterson and multilayer) and the four-parameter (Fritz-Schlunder) models characterized with lower extend of suitability to the experimental data points due to the higher error values.
To prove the latter observations, the modes of the experimental and model isotherms obtained on the basis of the linear and nonlinear approach were also compared (Figure 1, 2, 4, 5).Obviously, the Langmuir isotherms presented in Figure 1 did not correlate the experimental equilibrium data satisfactorily.It was observed that the HYBRID and EABS methods yielded the best �t in the low concentration range, while the LTFM isotherm could be applied for modeling the high concentration region.Among the �ve applied equations, the Freundlich (Figure 2) and Fritz-Schlunder (Figure 5) isotherms could be identi�ed as the most suitable for modelling the equilibrium sorption behaviour of p-NP on R. oryzae covering the entire concentration range.Besides, the individual model curves, derived on the basis of the studied error functions, practically coincided in both cases.Considering the theoretical bases of the Freundlich model, the studied separation process could be described either as non-ideal and reversible sorption, not restricted to monolayer formation on heterogeneous surface, or as multilayer sorption with non-uniform distribution of adsorption heat and affinities over the heterogeneous surface [24].Besides, as the value of the parameter   is below unity according to all implemented error analysis procedures, it is indicative of a chemisorption process.As stated earlier (Section Introduction), chitin and chitosan are the main components of R. oryzae cell walls.Due to the better hydrophilic properties of chitosan, it expands in an aqueous medium, the active sites which predetermine hydrogen bonding, become more pronounced and easily accessible.e amino (-NH 2 ) and hydroxyl (-OH) groups serve as the coordination and reaction sites.Besides, the pKa value of amino group (R-NH 2 ) in the structure of chitosan is 6.3, and amino group dissociates partly into R-NH 3 + even at pH 6.1 (Figure 3).e pKa value of p-nitrophenol is around 7.1, whereby at certain pH below the pKa of the dissociating solutes, p-nitrophenol exists as neutral form and above the pKa value, p-nitrophenol exists in ionic form [4]. O-H bond can be broken off easily, and nitro group causes to earn the resonance stability to structure by helping to the delocalization of negative charge [7].Hence, there is a possibility of chemical interaction between this positive charge and negative charge existent and delocalized in the anionic structure of the organic molecule (Figure 3).Chitin is carrying on linear amino group per glucose unit and thus exhibitis much higher uptake capacity.e amino group has an electron pair available for coordination and behaves like a strong Lewis base.e amine nitrogen on each chitin monomer has been suggested as the active site for 4-NP adsorption [51].
Hence, it is supposed that the adsorption of p-nitrophenol molecules was due mainly to the formation of H-bonds between chitin and chitosan functional groups and NO 2 -and OH-groups in the organic molecules (Figure 3).e relatively high value of the regression coefficient  2 obtained through the second polynomial form of the multilayer model ( 2 0.9967) (Table 3) con�rmed its applicability for describing the experimental data.e plots presented in Figure 4, however, demonstrated a deviation of all model isotherms for the middle concentration region.Probably, the biosorption of the mononitrophenol was accomplished predominantly on the �rst biosorbent layers, but the possibility of multilayer sorption, although at a lower extend, could not be neglected.
e comparison between the values of the six error estimating functions and the modes of experimental versus model isotherms outlined the Freundlich (Figure 2) and/or Fritz-Schlunder (Figure 5) equations as the most adequate models describing p-NP biosorption on R. oryzae.In the context of the current work, the corrected Akaike information criterion (AIC C ) was used as to verify the conclusions for the best �tting model withdrawn on bases of error analysis, so to rank the �ve isotherm models.
e AIC developed by Akaike is a methodology for model selection in a situation where more than one model has been �tted to experimental data and screening of the candidate models is crucial to the objectives of the research work.Akaike's general approach not only allows the best model to be identi�ed, but also allows the ranking of the rest of the models under consideration [46].e data in Table 4 show that the AIC C for the Freundlich model has the minimum AIC C value (AIC C −14.44).is implies that the theoretical data obtained from this model �ts the experimental results better than the other four equilibrium isotherm equations.Consequently, this statistical tool con�rmed the results from the error analysis and proved the better correlation of the experimental data with the nonlinear forms of the conventional two-parameter Freundlich equation.e ranking of the isotherm equations was made on the basis of the Relative Akaike Weight (RAW),   , (Table 2).e values of   (Table 4) clearly distinguished the appropriateness of the �ve isotherm equations ranking the two-parameter Freundlich model as the �rst (  1.00) and the four-parameter Fritz-Schlunder model as the second (  5.675) best �tted ones to the experimental data.Similar results had been obtained by Chatterjee et al. [49].ey observed that experimental data gave better �tting to Langmuir than to Redlich-Peterson model using AIC to rank them.e investigations of Mutua [46] outlined that the Freundlich model ranked �rst among seven isotherm equations used to model the adsorption of Cu 2+ and Cd 2+ onto bentonite and modi�ed bentonite clay.
Comparative analyses between the presented experimental results and the literature cited biosorption capacity of various R. oryzae-based biosorbents were accomplished and presented in Table 5.
Considering the values of the individual system parameters (initial sorbate concentration, biosorbent concentration, etc.), it could be concluded that the maximum biosorption capacity of R. oryzae dead biomass towards p-NP ( max 4.6 mg g −1 ) was satisfactory.Besides, the present investigations proved the high efficiency (94%) of Rhizopus oryzae, obtained at   5 mg dm −3 and  3 g dm −3 , as an alternative adsorbent for p-NP removal from aqueous phase and revealed the mechanism of the separation process at laboratory scale.Due to the harmful effects of these organic compounds and the urge for sustainable development for the natural water resources at global scale, however, the wastewaters containing them must be treated before being discharged to receiving natural water bodies.Consequently, the future perspectives of the applicability of the proposed "ecofriendly" biosorbent for industrial purposes include large-scale investigations with real wastewaters containing mononitrophenols and/or contaminated natural waters.

Conclusion
e single-component biosorption of p-NP on Rhizopus oryzae dead biomass from aqueous medium was investigated on the basis of equilibrium studies.e experimental data was modelled and evaluated using �ve isotherm models and seven optimization and error functions, including the corrected Akaike information criterion (AIC C ) as a statistical estimating and ranking tool.e linear transform model provided the highest  2 regression coefficient for the case of Redlich-Peterson isotherm.e analysis of the other error functions for linear/nonlinear optimization and AIC C outlined the conventional two-parameter Freundlich equation as generally the best-�tting model among the other two-, three-, and four-parameter isotherms applied.e laboratory studies proved the high efficiency (94% uptake extend) of Rhizopus oryzae dead biomass as an alternative adsorbent for removal of the toxic and recalcitrant organic contaminant p-NP from aqueous phase and outlined its future applicability in largescale wastewater treatment technologies.

F 3 :F 4 :
e proposed mechanism of p-NP biosorption by R. oryzae dead biomass.Note.I: chemical interactions between protonated chitosan −NH 3 groups and p-NP functional groups; II: chemical interactions between amine nitrogens of chitin.Experimental equilibrium data and multilayer isotherms of p-NP biosorption on R. oryzae dead biomass using (m 3.00 g dm −3 , pH 6.1 ± 0.2,  1 ± 2 ∘ C).
T 2: Error functions used to discriminate between models.Isotherm error deviation data related to the biosorption of p-NP on R. oryzae dead biomass using seven commonly applied error functions-linear regression analysis.T 4: Values of Langmuir, Freundlich, Redlich and Peterson, Multilayer, and Fritz and Schlunder model parameters for the system p-NP-Rhizopus Oryzae obtained through nonlinear regression analysis and error deviation data using six error functions.
Note. e highest  2 and the lowest ERRSQ, HYBRID, MPSD, ARE, and EABS are in bold.Note.e lowest ERRSQ, HYBRID, MPSD, ARE, EABS, and AIC c are in bold.
T 5: Biosorption capacities of R. oryzae-based biosorbents towards varous heavy metals and priority organic pollutants.
Schlunder model HYBRID: e hybrid fractional error function   : Langmuir isotherm constant, dm 3 g −1   : Freundlich isotherm constant, dm 3 g −1   : Redlich-Peterson isotherm constant, dm 3 g −1  1 : equilibrium constant for the �rst layer adsorption in the multilayer isotherm model  2 : : Fritz-Schlunder exponent : Fritz-Schlunder exponent : Maximum absorbance wavelength, nm   : Akaike weight for the th model Δ  : e difference between the AIC C of the best �tting model and that of model .