Journal of Chemistry

Volume 2013 (2013), Article ID 517631, 10 pages

http://dx.doi.org/10.1155/2013/517631

Research Article

Linear and Nonlinear Regression Methods for Equilibrium Modelling of p-Nitrophenol Biosorption by Rhizopus oryzae: Comparison of Error Analysis Criteria

Zvezdelina Lyubenova Yaneva,1,2 Bogdana Koumanova Koumanova,¹ and Nedyalka Valkanova Georgieva²

¹Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 8 Kliment Ohridski, 1756 Sofia, Bulgaria

²Chemistry Unit, Department of Pharmacology, Animal Physiology and Physiological Chemistry, Faculty of Veterinary Medicine, Trakia University, Students Campus, 6000 Stara Zagora, Bulgaria

Received 13 June 2012; Accepted 9 August 2012

Academic Editor: Javier Hernandez-Borges

Copyright © 2013 Zvezdelina Lyubenova Yaneva et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The study assessed the applicability of Rhizopus oryzae dead fungi as a biosorbent medium for p-nitrophenol (p-NP) removal from aqueous phase. The extent of biosorption was measured through five 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-fitting 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 (), the Freundlich equation was ranked as the first and the Fritz-Schlunder as the second best-fitting models describing the experimental data. The 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.

1. Introduction

Nitrophenols are classified as moderately to highly toxic even at low concentrations when present in wastewaters. Thus, they are a source of serious social and hygienic problems as important occupational and environmental pollutants [1]. The 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 significant toxicological concern [3].

The monosubstituted p-nitrophenol (p-NP) could enter the environment during its production and use in high-temperature coal conversion, as a byproduct in the enzymatic hydrolysis of parathion, methyl parathion, and N-acetyl-p-amino-phenol. It is also found in suspended particulate matter in the atmosphere, originating mostly from secondary photochemical reactions in the air [3]. The toxicity of p-nitrophenol on biological systems has led to its classification as a priority pollutant by the United States Environmental Protection Agency (EPA) [4].

The literature reports a number of studies on the efficiency of various adsorbents for phenols and substituted phenols removal from wastewaters [5]. The high adsorption activity of various biopolymers (unmodified/modified chitosan, lignin, etc) towards phenols, chlorophenols, and dyes was scientifically proven [6–8]. Fungi belonging to the genera Rhizopus and Penicillium have already been studied as a potential biomass for the removal of heavy metals (Cu, Zn, Cr, Cd) [9–11], nonylphenol [12], polycyclic aromatic hydrocarbons (PACs) [13], cyanide [14], and dyes from aqueous solutions [8]. The saprophytic microorganisms Rhizopus oryzae were proved as a good adsorbent for the removal of the organochloropesticide lindane [15], and rhodamine B from water [16].

The walls of R. oryzae consist of chitin and chitosan. The mechanism of biosorption is quite complicated. The 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 film-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 [19–23]. 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, ; Akaike information criterion, AIC, etc.) were highlighted and discussed for the comparisons of linear and nonlinear isotherm models [24–26]. 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. The literature reports studies applying the Akaike information criterion (AIC) for model selection and ranking in the field of veterinary medicine [28], genetics [29], aquatic biogeochemical modeling [30], FTIR spectroscopy [31], pharmacy [32], and so forth. The 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 five applied isotherm models (Langmuir, Freundlich, Redlich-Peterson, multilayer, and Fritz-Schlunder).

2. Materials and Methods

2.1. Adsorbate

p-Nitrophenol (p-NP) (Merck, 99%), without further purification, was used as a sorbate in the investigations. Its physicochemical characteristics are presented elsewhere [33].

2.2. 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. The microorganisms were isolated from a soil in Sri Lanka. The 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 ). The malt extract was inoculated by a standard sterile method and incubated at 32°C for 3 days in a platform shaker at 175 rpm. Three ceramic granules were added to each of the batch reactors with malt extract to limit the microorganisms threadlike growth. The biomass obtained was precisely washed out consecutively with maternal lye and distilled water and then dried in an oven at 50°C. The dried biomass was ground in a hammer mill and screened. The fraction used in the recent investigations was 0.15−0.50 mm.

2.3. Equilibrium Studies

The equilibrium experiments were accomplished using model solutions of p-NP in distilled water. The investigations were carried out at temperature °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 cm³ of the model solutions in screw cap jars. The jars were shaken on a platform shaker. The solute from each jar was then filtered. The residual p-NP concentrations in the liquid phase () were determined spectrophotometrically. All investigations were performed in triplicate. The 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, , and , , is the adsorbent mass.

SPECORD UV-VIS, Carl Zeiss Jena, spectrophotometer was used for concentration determinations at maximum absorption wavelength 228 nm. The 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.

2.4. Isotherm Modeling

The 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 specific sorbate/sorbent system, it is important to establish the most appropriate correlation for the experimental equilibrium data [35].

The biosorption behavior of p-NP on R. oryzae in the present research was modelled by the Langmuir, Freundlich, Redlich-Peterson, multilayer, and Fritz-Schlunder (four-parameter) isotherm equations (Table 1).

Table 1: Mathematical equations of the applied single component isotherm models.

Isotherm model

Nonlinear form

Linear expression
Plot

Langmuir [36]
(1)
,
(2)

versus

Freundlich [37]
(3)

(4)
.
versus .

Redlich and Peterson [38]
(5)

(6)
.
versus

Multilayer [39]
(7)
,

Second-order polynomial form
versus

(8)

Fritz and Schluender [40]
(9)

—
—

2.5. 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 (, 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 Microsoft Excel software.

Table 2: Error functions used to discriminate between models.

Error function

Definition
References

Coefficient of determination ()
(10)

[24, 25, 41]

Sum of the squares of errors (ERRSQ)
(11)

[34]

Hybrid fractional error function (HYBRID)
(12)

[42]

Marquardt’s percent standard deviation (MPSD)
(13)

[43, 44]

Average relative error (ARE)
(14)

[24, 45]

Sum of the absolute errors (EABS)
(15)

[46]

Corrected Akaike information criterion (AICC)
(16)
,
[46–48]

(17)
where

Akaike weight for the ith model ()
(18)
,

(19)
where

3. 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. The knowledge of sorbate/sorbent adsorption behaviour at equilibrium is essential for environmental engineering and science as the derived isotherms reveal the specific relation between the pollutant concentration and its uptake degree by the solid phase at constant temperature.

In this section, among the five studied isotherms models, the best-fitting one was determined by the use of seven well-known error functions to calculate the error deviation between experimental and predicted equilibrium adsorption data, after 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. The experimental investigations were conducted at biomass concentration 3.00 g dm⁻³.

The 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.

Table 3: 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.

Model
Langmuir
Freundlich
Redlich and Peterson
Multilayer (second-order polynomial form)

Parameters
3.535
2.208
1115.864
1.935

0.705
0.303
533.949
0.026

0.666
3.399

Error functions

0.9833
0.9956
0.9997
0.9967

    ERRSQ
0.7090
0.0255
0.5987
0.1827

    HYBRID
12.639
0.3040
19.1130
4.4978

    MPSD
35.5410
5.5138
58.3339
31.9740

    ARE
14.3110
2.0128
10.3833
7.1261

    EABS
0.7156
0.1006
0.5192
0.3563

Note. The highest and the lowest ERRSQ, HYBRID, MPSD, ARE, and EABS are in bold.

Alternative isotherm parameters were also determined by nonlinear regression using five error functions (ERRSQ, HYBRID, MPSD, ARE, and EABS) and the corrected Akaike information criterion (). The values of the Langmuir, Freundlich, Redlich-Peterson, multilayer and Fritz-Schlunder model constants and the isotherm error deviation data are presented in Table 4.

Table 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.

ERRSQ
HYBRID
MPSD
ARE
EABS
AICC

Langmuir model

3.9939
6.1036
6.1037
6.1038
7.1577

0.8428
1.4228
1.4228
1.4228
1.6142

    Error
0.6678
8.8101
29.6806
2.5861
1.6665
2.49

Freundlichmodel

2.1825
2.1945
2.1946
2.1946
2.2393

0.3110
0.3073
0.3073
0.3073
0.3002

    Error
0.0226
0.2928
5.4105
0.1757
0.2210
−14.44
1.00

Redlich and Peterson model

1307.7861
1170.0995
1018.8770
1018.6101
1000.0155

588.4872
531.9036
463.0101
462.8754
572.1642

0.6974
0.6937
0.6938
0.6938
0.5793

    Error
0.0255
0.4483
6.7061
0.1799
1.2184
8.19

Multilayer model

2.8903
3.2225
3.2224
3.2224
2.7971

0.0317
0.0336
0.0336
0.0336
0.0265

3.1784
3.0834
3.0834
3.0834
3.4010

    Error
0.1218
2.0060
14.1635
0.8024
0.5022
16.01

Fritz and Schlunder model

4.4335
4.5110
5.9346
3.5962
2.7281

0.3109
0.3073
0.3073
0.3075
0.2956

1.0314
1.0555
1.7042
0.6388
0.2182

0.0004
−0.0263

    Error
0.0226
0.8783
9.3716
0.1757
0.2213
−10.97
5.67

Note. The lowest ERRSQ, HYBRID, MPSD, ARE, EABS, and AICc are in bold.

The 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. The 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. The above-stated assumptions were consistent with the higher values of the equilibrium constant for the first layer adsorption ( 1.935: linear regression; 2.7971−3.2225: nonlinear regression) than that for multilayer sorption ( 0.026: linear regression; 0.0265−0.0336: nonlinear regression), calculated by the multilayer model (Tables 3 and 4). The 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—4.28−7.91 mg g⁻¹ (Table 4). The 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).

Figure 1: Experimental equilibrium data and Langmuir isotherms of p-NP biosorption on R. oryzae dead biomass ( 3.00 g dm⁻³, pH , °C).

Figure 2: Experimental equilibrium data and Freundlich isotherms of p-NP biosorption on R. oryzae dead biomass ( 3.00 g dm⁻³, pH , °C).

The 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 0.9997 (Table 3). However, the Freundlich equation characterized with the lowest ERRSQ, HYBRID, ARE, MPSD and EABS error values. The 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 fit isotherm model, as the number of the isotherm parameters is accounted only by them [35]. The 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 fit in the low concentration range, while the LTFM isotherm could be applied for modeling the high concentration region.

Among the five applied equations, the Freundlich (Figure 2) and Fritz-Schlunder (Figure 5) isotherms could be identified 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. The amino (–NH₂) and hydroxyl (–OH) groups serve as the coordination and reaction sites. Besides, the pKa value of amino group (R–NH₂) in the structure of chitosan is 6.3, and amino group dissociates partly into even at pH 6.1 (Figure 3). The 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. The amino group has an electron pair available for coordination and behaves like a strong Lewis base. The amine nitrogen on each chitin monomer has been suggested as the active site for 4-NP adsorption [51].

Figure 3: The proposed mechanism of p-NP biosorption by R. oryzae dead biomass. Note. I: chemical interactions between protonated chitosan −NH₃ groups and p-NP functional groups; II: chemical interactions between amine nitrogens of chitin.

Figure 4: Experimental equilibrium data and multilayer isotherms of p-NP biosorption on R. oryzae dead biomass using ( 3.00 g dm⁻³, pH , °C).

Figure 5: Experimental equilibrium data and Fritz-Schlunder (four-parameter model) isotherms of p-NP biosorption on R. oryzae dead biomass ( 3.00 g dm⁻³, pH , °C).

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 NO₂– and OH– groups in the organic molecules (Figure 3).

The relatively high value of the regression coefficient obtained through the second polynomial form of the multilayer model ( 0.9967) (Table 3) confirmed its applicability for describing the experimental data. The 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 first biosorbent layers, but the possibility of multilayer sorption, although at a lower extend, could not be neglected.

The 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 was used as to verify the conclusions for the best fitting model withdrawn on bases of error analysis, so to rank the five isotherm models.

The AIC developed by Akaike is a methodology for model selection in a situation where more than one model has been fitted 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 identified, but also allows the ranking of the rest of the models under consideration [46].

The data in Table 4 show that the for the Freundlich model has the minimum value   (  −14.44). This implies that the theoretical data obtained from this model fits the experimental results better than the other four equilibrium isotherm equations. Consequently, this statistical tool confirmed 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. The ranking of the isotherm equations was made on the basis of the Relative Akaike Weight (RAW), , (Table 2). The values of (Table 4) clearly distinguished the appropriateness of the five isotherm equations ranking the two-parameter Freundlich model as the first ( 1.00) and the four-parameter Fritz-Schlunder model as the second () best fitted ones to the experimental data. Similar results had been obtained by Chatterjee et al. [49]. They observed that experimental data gave better fitting to Langmuir than to Redlich-Peterson model using AIC to rank them. The investigations of Mutua [46] outlined that the Freundlich model ranked first among seven isotherm equations used to model the adsorption of Cu²⁺ and Cd²⁺ onto bentonite and modified 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.

Table 5: Biosorption capacities of R. oryzae-based biosorbents towards varous heavy metals and priority organic pollutants.

Biosorbent
Sorbate
, mg g⁻¹
System parameters
References

, mg dm⁻³
m, g dm⁻³

R. oryzae dried biomass
Cr, As
2.0
50.0
1.0
[9]

R. oryzae dried biomass
Zn
11.0
50.0
1.0
[9]

R. oryzae dried biomass
Cu
17.0
50.0
1.0
[9]

R. oryzae viable biomass
Cu
12.4
50.0
1.0
[11]

R. oryzae cell walls
Rhodamine B
7.6
100.0
4.0
[16]

R. oryzae biomass
Lindane
0.375
1.0
1.7
[15]

R. oryzae biomass
Malathion
0.445
1.5
2.0
[49]

R. oryzae fungi
Reactive Black 5
157.82
250.0
1
[50]

R. oryzae dead biomass
-Nitrophenol
4.5
25.0
3
Present study

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 ( 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.

4. Conclusion

The single-component biosorption of p-NP on Rhizopus oryzae dead biomass from aqueous medium was investigated on the basis of equilibrium studies. The experimental data was modelled and evaluated using five isotherm models and seven optimization and error functions, including the corrected Akaike information criterion () as a statistical estimating and ranking tool. The linear transform model provided the highest regression coefficient for the case of Redlich-Peterson isotherm. The analysis of the other error functions for linear/nonlinear optimization and outlined the conventional two-parameter Freundlich equation as generally the best-fitting model among the other two-, three-, and four-parameter isotherms applied. The 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 large-scale wastewater treatment technologies.

Nomenclature

: | Fritz-Schlunder constant, dm³ g⁻¹
: | Langmuir isotherm constant, dm³ g⁻¹
: | Redlich-Peterson isotherm constant, dm³ g⁻¹
AIC: | Akaike information criterion
: | Corrected Akaike information criterion
ARE: | The average relative error
: | Fritz-Schlunder constant, dm³ g⁻¹
: | Redlich-Peterson isotherm constant
: | Equilibrium adsorbate concentration in the liquid phase, mg dm⁻³
: | Initial adsorbate concentration in the liquid phase, mg dm⁻³
: | Particle diameter, mm
EABS: | The sum of the absolute errors
ERRSQ: | The sum of the squares of errors
FSch: | Fritz-Schlunder model
HYBRID: | The hybrid fractional error function
: | Langmuir isotherm constant, dm³ g⁻¹
: | Freundlich isotherm constant, dm³ g⁻¹
: | Redlich-Peterson isotherm constant, dm³ g⁻¹
: | equilibrium constant for the first layer adsorption in the multilayer isotherm model
: | Equilibrium constant for multilayer adsorption in the multilayer isotherm model
LTFM: | The linear transform model
: | Adsorbent concentration, g dm⁻³
MPSD: | Marquardt’s percent standard deviation
: | number of isotherm parameters
: | the heterogeneity factor in the Freundlich model
: | Number of experimental data points
: | Maximum monolayer adsorption capacity in the multilayer isotherm model, mg g⁻¹
: | The equilibrium sorbate concentration in the solid phase, mg g⁻¹
: | Experimental p-NP equilibrium solid phase concentration, mg g⁻¹
: | Model calculated p-NP equilibrium solid phase concentration, mg g⁻¹
: | The sorption capacity at equilibrium and at time , mg g⁻¹
: | Correlation coefficient
SSR: | The sum of the squares for the residual
: | Temperature, K
: | Time, min
: | Solution volume, dm³
: | Adsorbent mass, g.

Greek  Symbols

: | Fritz-Schlunder exponent
: | Fritz-Schlunder exponent
: | Maximum absorbance wavelength, nm
: | Akaike weight for the th model
: | The difference between the of the best fitting model and that of model .

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