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Prediction of optimum adsorption isotherm:comparison of chi-square and Log-likelihood statisticsMahdi Hadi a b , Gordon McKay c , Mohammad Reza Samarghandi d , Afshin Maleki a & MehriSolaimany Aminabad aa Kurdistan Environmental Health Research Center, Faculty of Health, Kurdistan Universityof Medical Sciences, Sanandaj, Iran Phone: Tel. +98 9189061738b Center for Water Quality Research (CWQR), Institute for Environmental Research (IER),Tehran University of Medical Sciences, Tehran, Iranc Department of Chemical and Biomolecular Engineering, Hong Kong University of Scienceand Technology, Clearwater Bay, New Territory, Hong Kong SARd Department of Environmental Health Engineering, School of Public Health, Center forHealth Research, Hamadan University of Medical Sciences, Hamadan, IranPublished online: 21 Nov 2012.

To cite this article: Mahdi Hadi , Gordon McKay , Mohammad Reza Samarghandi , Afshin Maleki & Mehri Solaimany Aminabad(2012) Prediction of optimum adsorption isotherm: comparison of chi-square and Log-likelihood statistics, Desalination andWater Treatment, 49:1-3, 81-94, DOI: 10.1080/19443994.2012.708202

To link to this article: http://dx.doi.org/10.1080/19443994.2012.708202

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Prediction of optimum adsorption isotherm: comparison of chi-squareand Log-likelihood statistics

Mahdi Hadia,b,*, Gordon McKayc, Mohammad Reza Samarghandid, Afshin Malekia,Mehri Solaimany Aminabada

aKurdistan Environmental Health Research Center, Faculty of Health, Kurdistan University of Medical Sciences,Sanandaj, IranTel. +98 9189061738; email: hadi_rfm@yahoo.combCenter for Water Quality Research (CWQR), Institute for Environmental Research (IER), Tehran University ofMedical Sciences, Tehran, IrancDepartment of Chemical and Biomolecular Engineering, Hong Kong University of Science and Technology,Clearwater Bay, New Territory, Hong Kong SARdDepartment of Environmental Health Engineering, School of Public Health, Center for Health Research, HamadanUniversity of Medical Sciences, Hamadan, Iran

Received 8 June 2011; Accepted 9 May 2012

ABSTRACT

A comparison of chi-square (X2) and Log-likelihood (G2) statistics of 19 adsorption isothermmodels––seven two-parameter models (Langmuir, Freundlich, Dubinin–Radushkevich, Tem-kin, Jovanovic, Harkins–Jura and Halsey) and 12 three-parameter models (Koble–Corrigan,Langmuir–Freundlich, Toth, Redlich–Peterson, Radke–Prausnitz (three models), Fritz–Schlun-der, Jossens, Khan, UNILAN, Vieth–Sladek) have been applied to the experiment of two dyes(Acid Blue 113, Acid Black 1) sorption onto Granular PineCone derived Activated Carbon(GPAC) and three dyes (Acid Blue 80, Acid Red 114, Acid Yellow 117) sorption onto Granu-lar Activated Carbon type Filtrasorb 400 (GAC F400). The study has focused on the assess-ment of the adequacy and goodness of the fitted models, using two well-known––X2 andG2––statistics. The results showed that G2 could be better than X2 statistic when the numberof model parameters is three.

Keywords: Adsorption; Isotherm models; Chi-square statistic; Log-likelihood statistic

1. Introduction

Adsorption phenomena have been known to man-kind and increasingly utilized to perform desired bulkseparation or purification purposes. The heart of anadsorption process is usually a porous solid medium.Porous solids provide a very high surface area ormicropore volume thus high adsorptive capacity canbe achieved. Adsorption equilibria information is the

most important piece of information in understandingan adsorption process. The adsorption equilibria ofpure components are essential to the understanding ofhow much those components can be accommodatedby a solid adsorbent [1].

The equilibrium isotherm represents the distribu-tion of the adsorbed material between the adsorbedphase and the solution phase at equilibrium. Thisisotherm is characteristic for a specific system at aparticular thermal condition [2].

So far, several isotherm models with differentassumptions have been developed to examine the

*Corresponding author.

Desalination and Water Treatmentwww.deswater.com

1944-3994/1944-3986 � 2012 Desalination Publications. All rights reserveddoi: 10.1080/19443994.2012.708202

49 (2012) 81–94

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adsorption mechanism. However, many models cannot describe well the experimental data. To findingthe best model among all models statistical methodsare needed to find the best model.

The linear least-squares method with linearlytransformed isotherm equations has been widelyapplied to confirm experimental data and isothermsusing coefficients of determination [3]. However,transformations of nonlinear isotherm equations to lin-ear forms can alter their error structure and may alsoviolate the error variance and normality assumptionsof standard least squares [4].

The linear equation analysis using the R2 calcula-tion associated with the sum of normalized errors(SNE) calculation procedure presented an appropriatemethod to use for the study of Ochratoxin A adsorp-tion onto yeast by-products [5] but the nonlinear chi-square method provided better determination for thethree sets of experimental data for the sorption iso-therms of reactive dye from aqueous solutions bycompost [6].

In recent years, several error analysis methods,such as the coefficient of determination, the sum ofthe errors squared, a hybrid error function, Marqu-ardt’s percent standard deviation, the average relativeerror, the sum of the absolute errors, chi-square, F-testand Students t-test have been used to determine thebest-fitting isotherm equation [3,7–10].

It is not appropriate to use the correlation coeffi-cient (R) or the coefficient of determination (R2) of lin-ear regression analysis for comparing the adsorptionisotherms. Nonlinear chi-square (X2) analysis could bea better method [8].

Nonlinear optimization provides a more complexand mathematically rigorous method for determiningisotherm parameter values [11–13], but still requires ameasure of the goodness-of-fit of an estimated statisti-cal model in order to evaluate the fit of the isothermmodels to the experimental data.

G-square (G2) is another statistic to find the modelthat best explains the data. The G-square test forgoodness-of-fit also known as log-likelihood statisticand is an alternative to the chi-square test of good-ness-of-fit. The distribution of G2 statistic under thenull hypothesis is approximately same as the theoreti-cal chi-square distribution. Thus, the probability ofgetting values of G and X can be calculated using thechi-square distribution [14,15].

In this study, the adsorption equilibrium isothermsof five dyes (Acid Blue 113, Acid Black 1, Acid Blue80, Acid Red 114 and Acid Yellow 117), from aqueoussolution onto activated carbon were studied and mod-eled. A trial-and-error nonlinear method of seven two-parameter isotherms––Langmuir [16], Freundlich [17],

Dubinin–Radushkevich [18], Jovanovic [19], Harkins–Jura [20], Temkin [21], Halsey [22]––12 three-parametermodels––Koble–Corrigan [23], Langmuir–Freundlich[24], Toth [25], Redlich–Peterson [26], Fritz–Schlunder[27], Radke–Prausnitz (three models) [28], Jossens[29], Khan [30], UNILAN [31], Vieth–Sladek [32]––wasused to determine the best fit isotherm. The G2 and X2

statistics were used in assessment of the adequacyand goodness-of-fitted models. The purpose of thisstudy is to compare G2 and X2 statistics to find thebest isotherm model. The values of probability to get-ting G and X for each model were calculated andcompared.

2. Materials and methods

2.1. Facilities

Weighing of materials was performed by using ananalytical balance with precision of ±0.0001 g (modelSartorius ED124S). Drying of materials was carriedout in an electric oven (model PARS TEB) and carbon-ization in a muffle furnace (model Exiton). The pH ofsolutions was measured using a digital pH-meter(model Sartorius Professional Meter PP-50). The dyesolutions were stirred using an inductive stirring sys-tem (Oxitop IS 12) within a WTW-TS 606/2-i incuba-tor. The samples were centrifuged using a 301 SigmaCentrifuge. The dye concentration in the samples wasmeasured spectrophotometrically, using a UV-1,700Pharmaspec Shimadzo spectrophotometer.

2.2. Raw materials

Dried pinecone was used as the raw material toproduce the adsorbent. The pinecones were collectedfrom the Mardom Park in front of Hamadan Univer-sity of Medical Sciences of Iran. The ground pineconederived activated carbon (GPAC) is advantageousover carbons made from other materials because of itshigh density and high purity. This carbon is harderand more resistant to attrition [33,34]. Acid Blue 113(AB113) one disazo type dye and Acid Black 1 (AB1),a sulfonated azo dye were used in this study. AB113and AB1 dyes were obtained from Alvansabet dye-stuff and textile auxiliary manufacturer company inthe west of Iran.

The experiments for three other dyes, namely AcidBlue 80 (AB80), Acid Red 114 (AR114) and AcidYellow 117 (AY117) were conducted by Choy et al.[35]. Their experimental data were provided by Prof.Gordon McKay and used in this study. The dyestuffswere used as the commercial salts. The AB80 andAY117 were supplied by Ciba Speciality Chemicals

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and AR114 was supplied by Sigma–Aldrich chemicalcompany.

Some information regarding the five acid dyes,which were used to measure and prepare standardconcentration dye solutions, is listed in Table 1. Thedata include color index number, molecular mass andthe wavelengths at which maximum absorption oflight occurs, kmax. The structures of the five acid dyesare shown in Fig. 1 and information regarding them islisted in Table 1.

2.3. Adsorbents

2.3.1. Pinecone derived activated carbon

The local granular activated carbon was producedby exposing the raw pinecones to a thermal–chemicalprocess. The pinecones were crushed and washedwith hot water and then dried at 100˚C in an ovenovernight. A 50-g crushed sample was mixed with apre-determined volume of phosphoric acid with con-centration of 95% in the mass ratio of 1:10. This mix-ture was transferred to a stainless steel tube (50mmdiameter and 250mm long). This tube was placed on

a tile as insulator and inserted into a muffle furnacewhich was programmed to gradually reach up to 900˚C within 3 h, this temperature was maintained for 1 h,and then gradually cooled down to the room tempera-ture. The end product was repeatedly washed usinghot distilled water until the washings showed pH>6.9; the washed sample was then again dried at 120˚C in an oven overnight. The final sample was thenground in a household-type blender and passedthrough a series of sieves (20, 30, 40, 50 US standardmesh sizes). A mixture of the residuals on 30, 40, and50 sieves was kept in an air-tight bottle and used asthe adsorbent in this study. The average adsorbentparticle size was 0.5mm.

The specific surface area of local GPAC wasobtained by the determination of the optimal concen-tration of methylene blue dye adsorbed onto theGPAC sorbent at constant temperature 20˚C. Themethylene blue calculated surface area was734m2 g�1.

The potential capacity of an adsorbent for adsorp-tion can be evaluated through iodine adsorptionfrom aqueous solutions using test conditions referredto as the iodine number determination. The iodinenumber was measured according to the standardprocedure [36]. The calculated Iodine Number valuewas 483.5mgg�1. The apparent density was calcu-lated by filling a calibrated cylinder with a givenactivated carbon weight and tapping the cylinderuntil a minimum volume was recorded. This densitywas referred as tapping or bulk density of adsorbent.For the real density a pycnometer method was used,which consisted of filling a pycnometer with the acti-vated carbon, then added a solvent (methanol) to fillthe void, at each step the weight was determined.The apparent and real density values were equal to0.50 and 1.70 g cm�3, respectively. The pore volumeand the porosity were determined by using a volu-metric method which consists of filling a calibratedcylinder with a V1 volume of activated carbon (massm1) and solvent (methanol) until volume V2 (totalmass m2) is reached. Knowing the density of solvent,total porosity volume (1.40 cm3 g�1) and the porosity(70%) of the adsorbent were easily calculated. TheBET nitrogen surface area was determined to be869m2 g�1.

2.3.2. Activated carbon F400

The other adsorbent used in the research (con-ducted by Choy et al. [35]) was a Granular ActivatedCarbon type F400 (GAC F400); it was supplied byChemviron Carbon Ltd.

Table 1Information regarding the acid dyes

Name of dyes AB1 AB113 AB80 AR114 AY117

Color indexnumber

20470 26360 61585 23635 24820

Molecularmass (g)

618 681 676 830 848

kmax (nm) 622 574 626 522 438

Fig. 1. Dye structures of AR114, AB80, AY117, AB1, andAB113.

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GAC F400 was crushed by using a hammer milland washed with distilled water to remove fines. Itwas dried at 110˚C in an oven for 24 h and thensieved. The 500–710lm size range activated carbonwas used for the experiments [35]. The BET nitrogensurface area was reported by the supplier to be1150m2 g�1.

2.4. Adsorption data

Accurately weighed quantities of AB113 and AB1dyes were dissolved in distilled water to preparestock solutions (500mgL�1). The calibration curves forAB113 and AB1 were linear from 0.125 to 100mgL�1

(R2 = 0.999) and 0.062 to 100mgL�1 (R2 = 0.999),respectively. The synthetic dyes solutions were pre-pared by diluting stock solutions to produce solutionsof 150mgL�1 of each dye. The adsorption equilibriumexperiments were carried out in a batch process in250-mL Erlenmeyer flasks housed in an incubator con-tainer. To determine the equilibrium time of AB113and AB1 adsorption onto GPAC, an accurate amountof 0.12 g of GPAC with 250mL of each dye solution(150mgL�1) was added to two Erlenmeyer flasks. Thecontents of flasks were mixed using a magnetic stirrerand 1mL samples were taken at regular times. Thedye concentration in the samples was measured spec-trophotometrically after centrifugation at 3800 rpm for5min. The equilibrium times for AB113 and AB1 weredetermined to be 250 and 167h, respectively. To per-form isotherm experiments, accurately weighedamounts of GPAC adsorbent of 0.01, 0.02, 0.04, 0.06,0.08, 0.1, 0.12, 0.14, 0.16, 0.18, and 0.2 g for AB 113 andAB1 were added to several flasks with 250mL dyesolution (150mgL�1) at pH 7.4 ± 0.2 for AB113 and7.0 ± 0.2 for AB1 dyes. The pH values at the end ofbatch runs were 6.1 ± 0.2 and 5.9 ± 0.2 for AB113 andAB1 dyes, respectively. The content of all Erlenmeyerflasks were mixed thoroughly for 250 and 167h forAB113 and AB1 respectively at 20.3˚C using magneticstirrers at constant revolution. A 5-ml sample wastaken after the equilibrium time and centrifuged at3,800 rpm for 5min. the residual dye concentrationwas measured spectrophotometrically.

The 250mgL�1 concentration dye solutions forAB80, AR114, and AY117 dyes were used to deter-mine the equilibrium contact time. For each acid dyesystem, eight jars of fixed volume (0.05 L) of dye solu-tions were prepared and contacted with 0.05 g acti-vated carbon F400 (GAC F400). Then, the jars wereput into the shaking bath with the same conditions ofthe isotherm adsorption experiment (constant temper-ature 20˚C and 200 rpm shaking rate). At 3-day inter-vals, one of the jars was taken from the shaker and

the dye concentration was measured. By plotting theacid dye adsorption capacity of the activated carbonagainst the time, it was found that the activated car-bon adsorption capacity became constant after a cer-tain period of time. It implied that the dye adsorptionsystem had reached equilibrium at that time. There-fore, the equilibrium contact time can be determinedfrom the graph. The activated carbon adsorptioncapacity of all three acid dyes (AB80, AR114 andAY117) in the samples became constant after 21 days.The equilibrium contact time for the sorption equilib-rium studies has been shown to be 21 days minimum[35].

The amount of dye adsorbed onto the sorbent, wascalculated as follows:

qe ¼ ðC0 � CeÞVm

ð1Þ

Each isotherm study was repeated three times andthe mean values have been reported.

2.5. Assessing goodness-of-fit by G2 and X2

The Pearson chi-square (X2) and log-likelihood(G2) statistics are two well-known statistics for assess-ing the goodness-of-fit of a regression model. Thesestatistics can be used to test that the observed datacame from an experiment in which the fitted model istrue. They are based on observed and expected obser-vations [37].

2.5.1. Chi-square statistic

The advantage of using chi-square test is forcomparing all isotherms on the same abscissa andordinate [8]. The chi-square test statistic is basicallythe sum of the squares of the differences betweenthe experimental data obtained by calculating frommodels with each squared difference divided by thecorresponding data obtained by calculating frommodels. The equivalent mathematical statement is:

X2 ¼XNi¼1

ðqexp;i � qm;iÞqm;i

2

ð2Þ

where qexp,i and qm,i are the equilibrium capacity(mgg�1) obtained from the experiment and model,respectively. Here, the values qexp are called theobserved values and the modeled values qm aresometimes called the predicted values. N is samplesize or number of experimental data points. Thechi-square statistic can be used to calculate a p-value

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by comparing the value of the statistic to a chi-square distribution. The number of degrees of free-dom is equal to the number of experimental datapoints (N), minus the number of model parameters[14].

2.5.2. G-square statistic

The G-square test statistic is calculated by takingan observed number (qexp), dividing it by the pre-dicted number (qm), then taking the natural logarithmof this ratio. The test statistic is usually called G2, andthus this is a G2-test, although it is also sometimescalled a log-likelihood test or a likelihood ratio test.The equation is:

G2 ¼ 2XNi

qexp;i � lnqexp;iqm;i

� �� �ð3Þ

As with most test statistics, the larger the differ-ence between observed and expected, the larger thetest statistic becomes. Once the G-statistic known;the probability of getting that value of G can be cal-culated using the chi-square distribution. The shapeof the chi-square distribution depends on the num-ber of degrees of freedom. The number of degreesof freedom is simply the number of experimentaldata points (N), minus the number of model param-eters [14,37].

2.6. Adsorption models

Nineteen different isotherm models were used fordescribing the single-component experimental iso-therm data. The names and the nonlinear forms ofstudied isotherm models are shown in Table 2. Fittingof the adsorption isotherm models to the experimentaldata is performed using OriginPro (Version 8.0) Soft-ware. In this study, the values of standard error aftermodel convergence by an iteration process were satis-fied because each model converged acceptably untilthe reduced chi-square received the minimum.

To find the optimum model, G2 and X2 statisticswere calculated for all models and the models weresorted by X2 and G2 from the smallest to the largest.

3. Results and discussion

The calculated isotherm parameters of two-param-eter and three-parameter isotherm models and theircorresponding X2 and G2 statistics are shown inTables 3 and 4, respectively.

Several authors have introduced the comparison ofdifferent error functions. The applicability of linear ornonlinear isotherm models has been examined indescribing the adsorption of dyes, heavy metals, andorganic pollutants onto a list of low-cost adsorbents(Table 5). Linear regression analysis has frequentlybeen employed in assessing the quality of fits [38] andadsorption performance. However, during the last few

Table 2The name and nonlinear form of studied isotherm models

Parametersnumber

Isotherm Non-linear form Isotherm Non-linear form

2 Langmuir qe ¼ qmbCe

1þbCe

Harkins-Juraqe ¼ AH

BH�logCe

� �1=2

Freundlich qe ¼ kfC1=ne

Halsey qe ¼ Exp ln kH�lnCe

n

� Dubinin–Radushkevich qe ¼ QsExpð�BDe2Þ,

e ¼ RT ln 1þ 1Ce

� � Temkin qe ¼ RTbTlnðKTe

CeÞ

Jovanovic qe ¼ qmaxð1� eðkjCeÞÞ3 Jossens qe ¼ kjCe

1þjCbe

Koble–Corrigan qe ¼ ACCbe

1þBCCbe

Khan qe ¼ qmkbkCe

ð1þbkCeÞbRedlich-Peterson qe ¼ ARCe

1þBRCbe

Toth qe ¼ qmTCe

1=KTþCmTeð Þ1=mT

Vieth-Sladek qe ¼ SmaxkLCe

1þkLCe

� �þ kdCe

Radke I qe ¼ qmRPKRPCe

ð1þKRPCeÞmRPFritz qe ¼ qmFSKFSCe

1þkFSCmFSe

Radke II qe ¼ qmRPKRPCe

1þKRPCmRPe

UNILAN qe ¼ qmu

2s ln1þKuCees

1þKuCee�s

� �Radke III qe ¼ qmRPKRPC

mRPe

1þKRPCmRP�1e

Langmuir–Freundlich qe ¼ qmLFðKLFCeÞmLF

1þðKLFCeÞmLF

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years, a development interest in the utilization of non-linear optimization modeling has been noted [5].

As Table 5 shows, the G2 is one function that hasbeen rarely used in literatures. Ho [8] examined thatnonlinear chi-square (X2) analysis could be a bettermethod for comparing the best fitting of isothermmodels. The applicability of five statistical tools to sat-isfactorily determine the best-fitting isotherm modelfor both linear and nonlinear analysis was also inves-tigated by Ncibi [6]. He showed that the X2 and Stu-dent’s t-tests could be useful, with R2, in the case oflinear analysis. The R, R2, X2, and F-test seem to beadequate to point out the best-fitting isotherm modelafter a nonlinear regression approach. On the otherhand, nonlinear approaches for avoiding the errorswill impact the final determination. In this study, wecompared G2 and X2 statistics to finding the best-fittedisotherm models using nonlinear regression.

To decide what statistic could be better to comparethe isotherm models; each statistic p-value was deter-

mined using S-Plus software and the p-values differ-ence of the two successive models sorted by X2 werecompared with that sorted by G2 (Mann-Whitney Utest was used).

The concept of the null hypothesis in goodness-of-fit test is: “The model is adequate”; therefore; thelarge value of calculated p-value is an evidence foraccepting the model [37]. Thus, it can be inferred thatif the p-values difference of two successive sortedmodels regarding the first statistic (G2) be more thanthat of the second (X2), the first would be the moresensitive statistic to finding the best model.

The sorted isotherm models by X2 and G2 statis-tics from smallest to largest and the corresponding p-value differences are shown in Tables 6 and 7,respectively. For instance, as shown in the first col-umn of Table 6, the difference between the p-valuesfor Langmuir and Temkin models is 3.48E�03 andfor Temkin and Freundlich is 2.62E�02. The corre-sponding p-values for G2 statistic regarding the

Table 3Isotherm parameters, X2 and G2 statistics of two-parameter models

Model Parameter AB113 AB1 AB80 AR114 AY117

Jovanivic G2 17.427 135.870 30.752 22.067 55.201

X2 6.409 48.711 7.533 6.650 16.047

qmax 271.3 433.6 152.4 92.2 167.0

kj 0.081 0.492 0.120 0.114 0.166

Temkin G2 1.991 18.945 3.227 1.038 2.419

X2 1.986 19.251 3.113 1.016 2.380

kTe 6.892 126.8 2.029 1.993 3.507

bT 57.368 49.306 75.710 125.404 73.543

Harkins–Jura G2 1550.043 1423.123 914.523 634.621 951.459

X2 266.718 292.791 317.472 205.820 336.762

AH 177559.4 544231.1 27557.5 9856.8 31221.4

BH 6.970 7.196 5.477 5.399 5.144

Dubinin–Radushkevich G2 13.125 83.195 213.330 101.241 348.305

X2 9.839 57.883 635.807 184.519 9378.343

Qs 269.4 417.4 145.3 88.51 158.3

BD 0.028 0.001 0.006 0.007 0.003

Freundlich G2 1.920 20.522 8.672 2.627 5.340

X2 3.406 32.132 21.937 7.963 20.831

kf 124.1 253.4 46.90 28.06 56.48

n 5.740 7.570 3.475 3.445 3.534

Halsey G2 1.902 20.419 8.592 2.585 5.247

X2 3.406 32.137 21.947 7.966 20.843

kH 0.000 0.000 0.000 0.000 0.000

n �5.741 �7.572 �3.476 -3.446 �3.536

Langmuir G2 1.904 26.370 1.070 3.984 12.544

X2 0.836 8.907 1.116 1.329 2.476

qm 298.4 452.8 171.4 103.7 185.8

b 0.124 0.788 0.150 0.144 0.216

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Table 4Isotherm parameters, X2 and G2 statistics of three-parameter models

Model Parameter AB113 AB1 AB80 AR114 AY117

Toth G2 0.696 6.495 1.993 0.739 0.426

X2 0.597 3.790 1.251 1.167 0.984

mT 0.777 0.707 0.887 0.668 0.718

qmT 314.0 472.9 177.2 119.3 205.5

KT 0.271 1.312 0.203 0.359 0.426

Jossens G2 1.011 11.819 2.376 1.247 0.415

X2 0.639 5.402 1.157 0.912 0.857

j 0.177 1.113 0.193 0.312 0.388

Kj 44.2 440.8 28.3 20.6 51.9

b 0.965 0.968 0.965 0.899 0.921

Redlich–Peterson G2 0.797 11.863 2.313 1.220 0.406

X2 0.639 5.401 1.156 0.912 0.857

BR 0.177 1.113 0.193 0.312 0.388

AR 44.2 440.9 28.3 20.6 51.9

b 0.965 0.968 0.965 0.899 0.921

Langmuir–Freundlich G2 0.678 5.980 1.521 0.567 0.790

X2 0.588 3.481 1.300 1.322 1.113

qmLF 312.2 470.4 174.9 113.8 198.9

KLF 0.123 0.776 0.142 0.110 0.178

mLF 0.819 0.761 0.944 0.797 0.828

Koble–Corrigan G2 0.658 5.931 1.548 0.551 0.776

X2 0.588 3.480 1.298 1.319 1.112

AC 56.3 387.7 27.7 19.6 47.7

BC 0.180 0.824 0.159 0.173 0.240

b 0.819 0.761 0.944 0.797 0.828

Khan G2 0.911 12.653 2.364 1.248 1.024

X2 0.648 5.667 1.112 0.813 0.853

qmk 262.1 391.3 149.3 68.6 135.4

bk 0.162 1.079 0.186 0.280 0.362

b 0.960 0.967 0.953 0.875 0.904

UNILAN G2 0.649 4.331 1.893 0.647 0.257

X2 0.575 3.138 1.271 1.222 1.020

s 1.644 2.037 0.967 1.989 1.709

qmu 306.8 463.4 174.9 114.2 197.3

Ku 0.130 0.811 0.142 0.109 0.182

Vieth–Sladek G2 1.172 19.302 1.922 1.476 2.713

X2 0.703 7.227 1.048 0.584 0.907

Smax 287.4 437.1 162.1 86.8 163.9

kL 0.138 0.872 0.167 0.206 0.279

kd 0.082 0.189 0.104 0.195 0.328

Fritz–Schlunder G2 0.879 11.781 2.365 1.272 0.444

X2 0.639 5.399 1.156 0.912 0.857

qmFS 250.7 396.1 146.8 66.2 134.1

KFS 0.177 1.113 0.193 0.312 0.388

mFS 0.965 0.968 0.965 0.899 0.921

Radke I G2 0.890 12.6 2.305 1.287 0.991

X2 0.648 5.665 1.112 0.813 0.853

qmRP 262.1 391.2 149.3 68.6 135.3

(Continued)

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isotherm models are shown in Table 7. The differencebetween the p-values for Halsay and Langmuir mod-els is 1.11E�05 and for Langmuir and Freundlich is1.04E�04. The p-values difference values within eachcolumn of Table 6 compared with corresponding val-ues in Table 7.

The differences of X2 p-values (difference betweenp-values of two best successive three parameter mod-els) for sorted three-parameter models (Table 6) wascompared with corresponding values in Table 7 (dif-ference between G2 p-values of two best successivethree parameters models). The Mann–Whitney U testwas used for the comparison, and it was found at0.05 level the two distributions are significantly dif-ferent (p-value = 0.00323). The results of Mann–Whit-ney U test are shown in Table 8. Moreover, thevalue of the mean rank for X2 p-value difference dis-tribution was less than that for G2 (46.6 < 64.4). Theseresults show the sensitivity of G2 statistic is morethan that for X2 in finding the best three-parametermodel regarding goodness-of-fit. Therefore, the G2

statistic is found to be more appropriate than X2 forcomparing the models with three parameters. Inother words, the sensitivity of G2 statistic may beincreased by increasing the number of model param-eters in comparison with X2.

The differences of X2 p-values (difference betweenp-values of two best successive two-parameter mod-els) for sorted two-parameter models were also shownin Table 6. These values were compared with corre-sponding values in Table 7 (differences of G2 p-valuesof two best successive two-parameter models).

It was found at 0.05 level the two distributions arenot significantly different (p-value = 0.79012) (Table 8).Therefore, the G2 and X2 may have similar sensitivityfor the comparison of two-parameter models.

However, because the mean rank for X2 statistic ismore than that for G2 (31.1 > 29.8) (Table 8), the X2

statistic may be better to be used instead of the G2

statistic when the aim is finding the best amongtwo-parameter models.

Based on our findings, the best two- and three-parameter isotherm models to describe the sorption ofstudied dyes were found as follows:

AB113: Langmuir (two-parameter), UNILAN (Three-parameters)AB1: Langmuir (two-parameter), UNILAN (Three-parameters)AB80: Langmuir (two-parameter), Langmuir-Freund-lich (Three-parameters)AR114: Temkin (two-parameter), Langmuir-Freundlich(Three-parameters)AY117: Temkin (two-parameter), UNILAN (three-parameters)

The experimental isotherm data and best fittedmodels for the sorption of AB1 and AB113 dyes ontoGPAC and AB80, AR114 and AY117 dyes onto GACF400 are shown by Figs. 2 and 3 separately.

The two-parameter Langmuir and three-parameterUNILAN isotherm models were found statistically tobe the best models to describe AB1 and AB113 dyessorption onto GPAC. By comparison of their G2 andX2 statistics, it will be revealed that the UNILANmodel describes adequately experimental sorptiondata. This is usually attributed to the complexity ofthe GPAC which is not as homogeneous as isassumed in Langmuir model. Thus to account for sur-face heterogeneity, an energy distribution can beintroduced. In the UNILAN equation, a patch-wisesurface is assumed. Each patch is assumed ideal such

Table 4 (continued)

Model Parameter AB113 AB1 AB80 AR114 AY117

kRP 0.162 1.080 0.186 0.280 0.362

mRP 0.960 0.967 0.953 0.875 0.904

Radke II G2 0.887 11.7 2.314 1.271 0.445

X2 0.639 5.398 1.156 0.912 0.857

qmRP 250.7 396.1 146.8 66.2 134.1

kRP 0.177 1.114 0.193 0.312 0.388

mRP 0.965 0.968 0.965 0.899 0.921

Radke III G2 0.802 11.8 2.314 1.245 0.403

X2 0.639 5.404 1.157 0.914 0.857

qmRP 44.2 440.6 28.3 20.7 52.0

kRP 5.667 0.899 5.182 3.197 2.579

mRP 0.035 0.032 0.035 0.102 0.079

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Tab

le5

Previousresearch

esofthelinearan

dnonlinearisotherm

studies

Adsorben

tAdsorbate

Isotherm

models

Statistics

Model

types

Referen

ces

Activated

carbon

Congored

Freundlich

,Lan

gmuir,Red

lich

–Peterson

R2

Nonlinear

[38]

Ricehusk

ash

Brilliantgreen

dye

Freundlich

,Lan

gmuir,Red

lich

–Peterson,Tem

kin,

Dubnin–R

adush

kev

ich

SSE,SAE,ARE,

HYBRID

,MPSD

Linear&

Nonlinear

[39]

Activated

carbon

Methyleneblue

Freundlich

,Lan

gmuir,Red

lich

–Peterson

R2,

ERRSQ,ARE,

HYBRID

,MPSD,EABS

Nonlinear

[40]

Ricehusk

Methyleneblue

Lan

gmuir,Freundlich

,Tem

kin

Red

lich

-Peterson,

Lan

gmuir-Freundlich

R2

Nonlinear

[41]

Chitosan

Congored

Lan

gmuir,Freundlich

,Tem

kin,Koble–C

orrigan

,Toth

SSE,R2

Nonlinear

[42]

Shad

dock

peel

Methyleneblue

Freundlich

,Lan

gmuir,

R2

Nonlinear

[43]

Laterite

Phosp

horus

Lan

gmuir,Freundlich

,Red

lich

–Peterson

R2

Linear&

Nonlinear

[44]

Coal

flyash

Methylorange

Freundlich

,Lan

gmuir,Red

lich

–Peterson(R–P

),Dubnin–R

adush

kev

ich

SSE,SAE,ARE,

HYBRID

,MPSD

Linear&

Nonlinear

[45]

Pineconederived

activated

carbon

Methylorange

Lan

gmuir,Dubinin-Rad

ush

kev

ic,Tem

kin,

Freundlich

,Halsey,Harkins-Jura

RMSE,X2

Nonlinear

[46]

Pineconederived

activated

carbon&

Activated

carbonf400

Aciddyes

Lan

gmuir,Freundlich

,Dubinin–R

adush

kev

ich,

Tem

kin,Halsey,Jovan

ovic

,Harkins–Jura

G2,X2,RMSE,HYBRD,

MPSD,ARE,APE,CP,

AIC

c,

Nonlinear

[33]

Compost

Reactivedye

Lan

gmuir,Freundlich

,HarkinsJura

R2,X2,RMSE

Nonlinear

[47]

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that the local Langmuir equation is applicable on eachpatch, and the distribution of energy is assumed uni-form. The parameter s characterizes the heterogeneityof the surface, the larger the value of this parameteris; the more heterogeneous is the system. The otherparameters, the pre-exponential constant, Ku and the

monolayer capacity (qmu), are similar to the adsorptionparameters of the Langmuir equation. If s= 0 the UNI-LAN equation reduces to the Langmuir isotherm. Theparameter s for AB1 and AB113 dyes takes values of2.037 and 1.644 respectively. These show the heteroge-neity of the system.

Table 6Models sorted by X2 statistic from smallest to largest

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The two-parameter Temkin and three-parameterUNILAN isotherm models were found statistically tobe the acceptable models to describe AY117 dyesorption onto GAC F400. With regard to the numberof UNILAN parameters and by comparing its statis-tics with those of other models, UNILAN may be thebest fitted model to describe the AY117 adsorption.The UNILAN’s s parameter for AY117 dye is 1.709.This shows that the surface of GAC F400 for adsorp-

tion of free AY117 may be heterogeneous with a dif-ferent energy distribution.

The equilibrium sorption data of AR114 dye fittedwell to Koble–Corrigan equation. This model is validonly when b> 1 [23]. The corresponding Koble–Corrigan parameters are given in Table 4. Theconstant b for the sorption of all the studied dyes isless than unity, meaning that the model is unable todescribe the experimental data. Thus for the case of

Table 7Models sorted by G2 statistic from smallest to largest

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AR114 dye, based on G2, the second best model, thatis Langmuir–Freundlich, was chosen as the bestdescriptive model.

The Langmuir–Freundlich equation has the com-bined form of Langmuir and Freundlich equations. Atlow adsorbate concentrations, it reduces to Freundlichisotherm; while at high concentrations, it predicts amonolayer adsorption capacity characteristic of theLangmuir isotherm [24].

The parameter KLF in the Langmuir–Freundlichmodel is equilibrium constant for a heterogeneoussolid, and mLF is the heterogeneity parameter, liesbetween 0 and 1. The larger is this parameter, thehigher is the degree of homogeneity. However, thisinformation does not point to what is the source ofhomogeneity, whether it can be the GAC F400 struc-tural or energetic properties or the dye property. Theparameter mLF for AB80 sorption is 0.994 which isnear to unity, suggesting some degree of homogeneityof this dye/activated carbon system. The Langmuir–Freundlich isotherm exhibits the best fit for AR114 adsorption process. The mLF for AR114 adsorption

onto GAC F400 is 0.797 and unlike to AB80/GACF400 system, indicate that the adsorption process isheterogeneous. The adsorption isotherm with theLangmuir–Freundlich model is presented in Fig. 3. Asshown in Table 4 Langmuir–Freundlich modeldescribes the adsorption of AB80 and AR114 accept-ably. The values of maximum adsorption capacitydetermined using Langmuir–Freundlich model forAB80 and AR114 are 174.87 and 113.86mgg�1, respec-tively. These values are comparable to the experimen-tal adsorption isotherm plateau, which are acceptable.

4. Conclusion

The adsorption equilibrium isotherms of five dyes(Acid blue 113, Acid Black 1, Acid Blue 80, Acid Red114, and Acid Yellow 117), from aqueous solutiononto activated carbon were studied and modeled.A trial-and-error nonlinear method of seven

Fig. 2. Experimental points and the best model curves forthe adsorption of AB1 and AB113 dyes onto GPAC.

Fig. 3. Experimental and the best model curves for theadsorption of AY117, AR114, and AB80 dyes onto GACF400.

Table 8Mann–Whitney U test results

Distribution N Mean Rank Sum Rank U Z Prob >|U|⁄

Three parameters

X2 P-value diff. 55 46.6 2563 1023 �2.94514 0.00323

G2 P-value diff. 55 64.4 3542

Two parameters

X2 P-value diff. 30 31.1 933.5 468.5 0.26616 0.79012

G2 P-value diff. 30 29.8 896.5

⁄ Null Hypothesis: X2 P-value diff. = G2 P-value diff., Alternative hypothesis: X2 P-value difference <> G2 P-value difference.

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two-parameter and 12 three-parameter isotherm mod-els was used to determine the best-fit isotherm. Theabilities of two well-known G2 and X2 statistics tofinding the best isotherm model were compared. Thesensitivity of G2 statistic was more than that for X2 tofind the best three-parameter model regarding good-ness-of-fit. On the other hand, the X2 statistic may bemore appropriate than G2 for comparing the modelswith two parameters. Based on G2 statistic adsorptionisotherm data of AB1 and AB113 dyes onto GPACindicated a good-fit to the UNILAN three-parameterisotherm model and the sorption processes were het-erogeneous. The UNILAN model also found as thebest fitted model to describe the AY117 adsorptiononto GAC F400. Langmuir–Freundlich modeldescribed the adsorption of AR114 and AB80 ontoGAC F400 acceptably. The adsorption process forAR114 and AB80 was found heterogeneous andhomogeneous, respectively.

Acknowledgments

The authors thank the Hamadan University ofMedical Sciences for financial support. The corre-sponding author would like to thank Prof. GordonMcKay for the provision of some experimental data.Furthermore, thanks to Mr. Siavash Mirzaee Ramhor-mozi, M.Sc. student of biostatistics from HamadanUniversity of Medical Science for some suggestions toimprove the manuscript.

Nomenclature

Ce –– the equilibrium liquid-phase concentration of theadsorbate (mgL�1)

qe –– the equilibrium adsorbate loading onto theadsorbent (mgg�1)

C0 –– initial dye concentration (mgL�1)

m –– sorbent mass (g)

V –– The solution volume (L)

AR –– the Redlich–Peterson and isotherm constant(L g�1)

BR –– the Redlich–Peterson constant having unit of(Lmg�1)

AC –– the K-C constant (Lbmg1�b g�1)

BC –– the K-C constant having unit of (Lmg�1)b

qmLF –– the Langmuir–Freundlich maximum adsorptioncapacity (mgg�1)

KLF –– the Langmuir–Freundlich equilibrium constantfor a heterogeneous solid

mLF –– the Langmuir–Freundlich heterogeneityparameter, lies between 0 and 1

qmFS –– the Fritz–Schlunder maximum adsorptioncapacity (mgg�1)

(Continued)

KFS –– the Fritz–Schlunder equilibrium constant (Lmg�1)

mFS –– the Fritz–Schlunder model exponent

qmRP –– the Radke–Prausnitz maximum adsorptioncapacities (mgg�1)

KRP –– the Radke–Prausnitz equilibrium constants

mRP –– the Radke–Prausnitz model exponents

qmT –– the Toth maximum adsorption capacity (mgg�1)

KT –– the Toth equilibrium constant

mT –– the Toth model exponent

qmk –– maximum adsorption capacity in the Khanmodel (mgg�1)

bk –– the Khan constant (Lmg�1)

b –– the Jossens, Koble-Corrigan, Khan, Redlich-Peterson models exponent

Ku –– the UNILAN constant (Lmg�1)

qmu –– maximum adsorption capacity in UNILANmodel (mgg�1)

s –– the UNILAN model constant

kj –– the Jossens constant (L g�1)

j –– the Jossens constant having unit of (Lmg�1)

Smax –– maximum adsorption capacity in the Vieth–Sladek model (mgg�1)

kL –– the Vieth–Sladek constant (Lmg�1)

kd –– the parameters of the Vieth–Sladek model

qm –– maximum adsorption capacity in Langmuirmodel (mgg�1)

b –– the Langmuir constant related to the energy ofadsorption (Lmg�1)

n –– the Freundlich and Halsey equation exponents

Kf –– the Freundlich constant (mg1�1/nL1/n g�1)

R –– universal gas constant (kJ mol�1 K�1)

T –– temperature (K)

bT –– the Temkin constant related to heat of sorption(kJmol�1)

KTe –– the Temkin equilibrium isotherm constant (L g�1)

KJ –– the Jovanovic isotherm constant (L g�1)

qmax –– maximum adsorption capacity in Jovanovicmodel (mgg�1)

Qs –– monolayer saturation capacity in Dubinin–Radushkevich model (mgg�1)

BD –– Dubinin–Radushkevich model constant(mol2 kJ�2)

e –– Polanyi potential

AH –– the Harkins–Jura isotherm parameter

BH –– the Harkins–Jura isotherm constant

KH –– the Halsey isotherm constant

N –– number of experimental points

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