Send Orders of Reprints at Reprints@benthamscience.net Evaluation of Equilibrium Sorption Isotherm Equations

The sorption isotherm is important information for evaluating the ability of an adsorption system to treat the industrial waste water. In this study, eleven datasets from literatures were selected and three two-parameter and three-parameter equations were used to evaluate adsorption systems. Quantitative criteria included six error functions, and the quantitative criterion was residual plots. Isotherm constants differed by linearization and nonlinear regression analysis, especially for the three-parameter equations. The results indicated the inconsistent results of the error functions for selecting the adequate equations. Residual plots were useful to observe the fitting ability of isotherm equations. Temperature has a significant effect on isotherm constants.


INTRODUCTION
Environmental protection has become the basic requirement of industry development.Water or industrial wastewater treatment is an emergency technique for saving the natural environment.Adsorption is a useful technique for purification and separation in treating industrial waste water.With adsorption, pollutants or toxic materials can be removed from aqueous or non-aqueous solutions.
Biological materials have been applied widely used for the adsorption of heavy metals, because of their inexpensive and the ability to bind pollutants.With a constant adsorbate concentration, the absorbent concentration can reach equilibrium at a fixed temperature.The distribution of the concentrations of adsorbate and adsorbent is called the sorption isotherm.The sorption isotherm must be established to evaluate the capacity of an adsorption system.Parameters of sorption isotherm can be used to describe the surface properties of adsorbents and other properties.Factors other than temperature that affected sorption properties included particle size, metal ion [1,2] and pH [3][4][5].
Four isotherms were selected to compare predictive accuracy with single isotherm data for sorption of three heavy metal ions by pipe bark and to determine binary sorption isotherms [1].The Langmuir isotherm model was less accurate.cy than other models.Seven sorption models were used to evaluate the sorption of three metal ions from aqueous solution by peat [2].Six error functions were proposed to evaluate the fitting ability of these models.The Sips equation was the best model for the experimental data for Cu, Ni and Pb.Three isotherms models were used to test tree fern as a sorbent system for removing three types of metal ions [6].The Langmuir and Redlich-Peterson isotherms could represent the experimental data well.The Langmuir model was extended to predict multi-component sorption data and the fitting ability of the new model was tested [7].Seven models were tested with the coefficient of determination and the value of the sum of normalized errors for biosorption of ochratoxin A (OA) on yeast biomass as the decontamination agent [8].The Hill, Freundlich and BET equations were appropriate models.
Many sorption isotherm models have been proposed to assess their ability to fit experimental data.Six twoparameter and nine three-parameter models have been described [9].Some models were modified to describe multicomponent adsorption.More parameters were incorporated into the original form of the model and termed extension isotherm models [1,7,10].
For fitting a nonlinear isotherm model as a linear model, the dependent and independent variables are transformed [2,6,8,[11][12][13][14].The transformed dependent variables are then calculated and evaluated by different error equations.The problems in variable transformation and the regression results have been discussed [9,[15][16][17].
The criteria to evaluate the error distribution between the experimental data and the predictive values of isotherm models are called error functions.Nine error functions have been reviewed [9].Some researchers select one or two criteria, such as the coefficient of determination R 2 and χ 2 [7,[11][12][13][14]18].Others adopted several criteria to determine with best performance for all functions [2,8,11].
Common errors in analyzing of sorption isotherm data have been discussed [16].With linearization, the assumption of experimental error is violated.The variance errors lie in the dependent and independent variables for linear and nonlinear isotherm models.Other errors are abuse of R 2 and neglecting to account for the different degrees of freedom for model comparison [17].
The error functions are only the quantitative criteria for model evaluation.The qualitative criterion is residual plots [19][20].However, this criterion has previously been used to evaluate only the assumption of homoskedasity [21].
This study used eleven datasets from the literature to evaluate the appropriate model describing sorption isotherm data.Six isotherm models, including two-parameter and three-parameter equations, were used to evaluate the fitting ability of the models.The quantitative criteria included six error functions and the qualitative criterion was residual plots.The effect of temperature on constants was also exampled.

Literature Survey
The eleven datasets, including different temperatures, metal ions and adsorbate range are listed in Table 1.

Isotherm Equations
Six isotherms were adopted.

Freundlich Isotherm
where q e is the amount of solute adsorbed at equilibrium, C e is the equilibrium concentration, and K F and br are the Freundlich constants.
Equation (1) can be expressed as an exponential equation, where a and b are constants.

Langmuir Isotherm
where k L and a L are the Langmuir isotherm constants.
Equation (3) can be written as follows: 1

Tempkim isotherm
where R is the gas constant, T is the temperature in K, and A T and B T are the Tempkim isotherm constants.
Equation ( 5) can be transformed as follows

Redlich-Paterson Isotherm
1 where K R , a R and B are the Redlich-Peterson isotherm constants.
Equation ( 7) can be written as following: where a=K R /a R , b=B and c=1/a R .

Hill Isotherm
where q SH , nH and K D are the Hill isotherm constants.
Equation ( 9) can be expressed as follows: where K s , B s and α s are the Sips isotherm constants.
Equation ( 13) has the same form as Equation ( 10), where a=K s /α s , b= B s and c=1/α s .
For the Koble-Corrigan model, where A k , B k and n are the Koble-Corrigan isotherm constants.
Equation ( 14) has the same form as Equation (10), where a= A k / B k , b= n and c=1/ A k .
The Sips and Koble-Corrigan models were not be used to evaluate fitting ability.

Standard Error of Estimate, s
where ^i y is the predicted value of the isotherm model, y i is the experimental data, n is the number of data and p is the number of parameters.

Qualitative Criterion
The residual plot is an important criterion to assess the adequateness of a regression model (Draper and Smith, 1981;Myers, 1986).The distribution between the model predictive errors (residuals) versus predictive values is called the residual plots.A residual plot with a clean pattern indicated that the isotherm model has fixed model errors and is not adequate.A residual plot with a uniform distribution represented an adequate model [19][20].

The Linearization Problems
For some nonlinear isotherm models, data are transformed to linearization for regression analysis easily.However, this process changes the error structure of the data was changed.The Freundlich isotherm model is used to illustrate the violation of the error structure with transformation.
According to the assumption of regression analysis, the error in the dependent variable is additive error.
where e i is the additive error of the model.
The regression model for transformed data is as follows: The In (q ei ) transformed back to the original form results in the following, The error structures of Equations.( 20) and ( 22) are different, that is, data transformation affected the error structure and could also violate the assumption of homogeneous variance.
Considering the problems with linearization, the six isotherm equations used in this study were also studied by nonlinear regression analysis technique.

Statistical Analysis
Constants were estimated by nonlinear regression analysis with statistical software Sigma plot v10.0 (SPSS Inc., Chicago.IL).

Comparison of Constants with Linearization and Nonlinear Regression
The isotherm constants with linearization and nonlinear regression analysis of datasets are listed in regression methods produced different constants.For the ochratoxin A (OA) adsorption onto EX16 yeast biomass [8], the br value of the Freundlich equation was 0.9625 for the linearization equation and 0.7189 for the nonlinear equations.The k L and a L values of the Langmuir equation were 0.0163 and 0.109 with linearization, and 0.037 and 0.161 for nonlinear equation.
The difference between constants with linearization and nonlinear analysis is presented for the Langmuir and Hill equations with OA adsorption data onto beta-glucanes.The constants of the Hill equation, q SH , nH and k D, were 0.174, 1.040 and 8.004 with linearization and 0.766, 1.087 and 51.322 for nonlinear analysis.
For the constants k f and br for the Freundlich equation with sorption data of MB at 40℃ [21] and for the constants of Redlich-Peterson equation k R , a R and B with biosorption of solophenyl Brown textile dye onto raw P. Oceanica seagrass fibers [22], isotherm constants of three-parameter equations significantly different by regression method.

Three Literatures Datasets
The estimated isotherm constants and error functions for the six isotherm equations for three datasets are listed in Table 3.For the Ho and Wang data [2], the Toth equation produced the highest R 2 value and the Hill equation had the lowest EABS value.However, the Langmuir equation produced smallest s, ARE, HYBRID and MPSD values.The results for error functions were inconsistent.
The relationship between adsorbate and adsorbent concentrations and the predictive curves of the six equations is presented in Fig. (1).Visually determining the fitting ability    With data from Ho [13] (Table 3), the Toth equation produced the highest R 2 value and lowest s, ARE, EABS and MPSD values.However, the Hill equation produced the lowest HYBRID value.The equations produced inconsistent results for error functions, but residual plots revealed a uniform distribution for the three-parameter equations, Redlich-Peterson, Hill and Toth equations.
For the Wang and Qin data [14] (Table 3), the Toth equation produced the highest R 2 value and the Tempkim had the lowest s value.The Langmuir equation produced the lowest ARE, HYBRID, EABS and MPSD values.Residual plots revealed a clean pattern for only the Freundlich equation and a uniform distribution for the other five equations.

Sorption Data for Three Metal Ions
The isotherm constants and error functions for three metal ions from Mckay and Portor [7] are listed in Table 4.For the Cd ion, the Hill equation produced the highest R 2 value and the lowest s, ARE, HYBRID, EABS and MPSD values.Residual plots showed a clean pattern for the twoparameter equations, Freundlich, Langmuir and Tempkim equations, and a uniform distribution for the three-parameter equations, Redlich-Peterson, Hill and Toth.For experimental datasets, the Hill equation could be the best equation to describe the relationship between adsorbate and adsorbent concentrations.
For the Zn ion, the Toth equation produced the highest R 2 value and the Hill equation had the lowest EABS value.The Langmuir equation produced the lowest s, ARE, HYBRID and MPSD values.Residual plots showed the uniform distribution for four equations except the Freundlich and Tempkim equations.

Effect of Temperature on Isotherm Data
The data for ammonium biosorption by Posidonia oceanica seagrass fibers at five temperatures were from Wahab et al. [4].The estimated constants and error functions are listed in Table 5. Residual plots showed the clean patterns for the Freundlich and Tempkim equation and the uniform distribution for other four equations (Fig. 5).However, the error functions showed inconsistent results.For the 313 K temperature data, the Hill equation could be the best fitting ability, it had the highest R 2 value and lowest value for other criteria.
For the 323 K and 333 K temperature data, results for error functions were inconsistent.
The isotherm constants of Langmuir and Hill equations were used to study the effect of temperature on sorption isotherms (Fig. 6).Temperature was significantly associated with k L and a L of the Langmuir equation and q SH and k D but not nH of the Hill equation.
In the study of the sorption isotherm for metal ions on tree fern, the experiments were tested at four temperatures [6].Temperature was significantly associated with k L and a L of the Langmuir isotherm were listed.The results were similar as this study.
Many sorption isotherm equations are nonlinear equations.Linearization methods had been adopted to transform dependent and independent variables for linear regression analysis.The problems have been reviewed [16].
The Langmuir isotherm equation was used to demonstrate the expanded variance of constants with linearization [23].EI-Khaiary et al. [17] analyzed simulated data by nonlinear regression and four linearized forms of the pseudosecond-order equation.Nonlinear regression had the minimum variance and was considered the most accurate methods.The current study compared three two-parameter and other three three-parameter isotherm equations analyzed with the nonlinear regression analysis.Six error functions were  selected to compare the fitting-agreement of isotherm equations.The inconsistent results were found.Only the Hill equation produced the lowest values for all error functions for Cd ions.For most data, no isotherm equation produced all the low values for errors functions.Similar results can be found in the literatures.Five isotherm equations were selected to study adsorption of two basic dyes by kudzu [11], the linear-transformed Redlich-Peterson equation produced the highest R 2 value, but did not have the lowest values for the other error functions.
Seven isotherm equations were tested for toxin adsorption with yeast biomass [8].For OA adsorption by EX16 yeast biomass, the Hill equation produced the lowest values for the six error functions.For OA adsorption by betaglucanes, the Hill equation had the lowest MPSD, ARE and the sum of normalized errors (SNE).However, the Toth equation produced the lowest HYBRID and EABS values and the BET equation had the lowest sum of the squares of errors (ERRSQ).Inconsistent results were found for OA adsorption by a yeast cell wall fraction (LEC).
A new error function, the small-sample-corrected Akaike information criterion (AICc) was proposed to evaluate seven isotherm equations [15].The Freundlich equation best described the adsorption of Cu 2+ and Cd 2+ data by Bentonite adsorbents.However, the HYBRD, ARE, MPSD and EABS values for the Freundlich equation were higher than for other isotherm equation such as Langmuir, Redlich-Peterson and Sip.
Results were inconsistent in evaluating the predictive ability of isotherm equations for adsorption of textile dye by Posidonia oceanica seagrass fibres [22].The Redlich-Peterson equation had highest R 2 value and the lowest ARE, MPSD, HYBRID, ERRSQ and s values.However, this equation had the highest values by Students t test.
Researchers have plotted the data distribution of the amount of ion sorbed (q e ) versus the isotherm equilibrium concentration (C e ) and the predictive curves of several isotherm equations were used by researchers [2,6,8,[12][13][14]18].However, the fitting ability is not easily justified visually.In this study, residual plots were dfound to be an adequate quantitative evaluation method.As the model is inadequate, the predictive values were under-or over-estimated.The inadequate results are easy to observe with the data distribution of residual plots.
The effect of the temperature on sorption isotherms has been analyzed [4][5][6]24].In this study, with data from Wahab et al. [4] , temperature was associated with constants of the Langmuir equation and two constants of the Hill equation.The effect of the temperature on constants could be expressed as a temperature function and then incorporated into isotherm equations to express the relationship among C e , q e , and temperature.The development and validations of an adequate isotherm equations incorporating a temperature term remains for further studied.

CONCULSION
This study used eleven datasets from the literatures to assess the fitting ability of sorption isotherm equations in determining the capacity of an adsorption system to treat industrial wastewater.Nonlinear regression analysis was used to avoid problems with linearization which alters the error distribution and distorts the variance structure.Results for error functions were inconsistent among sorption isotherms, but residual plots were useful to observe the fitting ability of isotherm equations.Temperature had a significant effect on the constants of isotherms.Incorporating a temperature term into isotherm equations needs further study.
difficult.Residual plots provide a useful tool (Fig.2).Residual plots showed clean patterns for the Freundlich and Tempkim equations and uniform distributions for the Langmuir, Redlich-Peterson, Hill and Toth equations.

For
the Cu ion, the Hill equation produced the highest R 2 value and the lowest EABS value.The Toth equation had the lowest HYBRID and MPSD values.The Redlich-Peterson equation produced lowest ARE value.The experimental data and predictive curves for the six equations are shown in Fig. (3) and the residual plots are presented in Fig. (4).Residual plots showed a clean pattern for the Freundlich and Langmuir equation and a uniform distribution for the Tempkim, Redlich-Peterson, Hill and Toth equation.
t an t s o f t h e L an g mu i r eq u at i o 0.003 0.00305 0.0031 0.00315 0.0032 0.00325 0.0033 0.00335 0.0034 0.00345 Temp.K Constants of the Hill equation qSH nH kD

Table 5
Contd….. K temperature data, the Redlich-Peterson and Toth equation produced the same R 2 values.The Toth equation had the lowest s value and the Redlich-Peterson equation had the lowest ARE and EABS values.The Langmuir equation had the lowest HYBRID and MPSD values.For the 303 K temperature data, the Toth equation had the highest R 2 value and lowest ARE, EABS and MPSD values and the Langmuir equation had the lowest s and HY-BRID values.