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1 Chemcal Engneerng Journal Contents lsts avalable at ScenceDrect Chemcal Engneerng Journal journal homepage: Revew Insghts nto the modelng of adsorpton sotherm systems K.Y. Foo, B.H. Hameed School of Chemcal Engneerng, Engneerng Campus, Unverst Sans Malaysa, 4300 Nbong Tebal, Penang, Malaysa artcle nfo abstract Artcle hstory: Receved 2 March 2009 Receved n revsed form 6 September 2009 Accepted 7 September 2009 Keywords: Adsorpton Isotherm Lnear Nonlnear Error functon Concern about envronmental protecton has ncreased over the years from a global vewpont. To date, the prevalence of adsorpton separaton n the envronmental chemstry remans an aesthetc attenton and consderaton abroad the natons, ownng to ts low ntal cost, smplcty of desgn, ease of operaton, nsenstvty to toxc substances and complete removal of pollutants even from dlute solutons. Wth the renassance of sotherms modelng, there has been a steadly growng nterest n ths research feld. Confrmng the asserton, ths paper presents a state of art revew of adsorpton sotherms modelng, ts fundamental characterstcs and mathematcal dervatons. Moreover, the key advance of the error functons, ts utlzaton prncples together wth the comparsons of lnearzed and non-lnearzed sotherm models have been hghlghted and dscussed. Conclusvely, the expandng of the nonlnear sotherms represents a potentally vable and powerful tool, leadng to the superor mprovement n the area of adsorpton scence Elsever B.V. All rghts reserved.. Introducton Over the past several decades, the exponental populaton and socal cvlzaton expanson, change affluent lfestyles and resources use, and contnung progress of the ndustral and technologes has been accompaned by a sharp modernzaton and metropoltan growth []. Wth the rsng awareness of the occurrences of ndustral actvtes whch has ntensfed numerous deteroratons on several ecosystems and serously threatens the human health and envronment, the enforcement of strngent rules and regulatons concernng the emsson of contamnants from ndustral waste streams by varous regulatory agences has been promulgated [2]. Smultaneously, a developng research by the nventon of a wde range of treatment technologes precptaton, coagulaton flocculaton, sedmentaton, flotaton, fltraton, membrane processes, electrochemcal technques, bologcal process, chemcal reactons, adsorpton and on exchange wth varyng levels of successes has accelerated a dramatc progress n the scentfc communty [3 3]. Of major nterest, adsorpton process, a surface phenomenon by whch a mult-component flud gas or lqud mxture s attracted to the surface of a sold adsorbent and forms attachments va physcal or chemcal bonds, s recognzed as the most effcent, promsng and wdely used fundamental approach n wastewater treatment processes [4], manly hnges on ts smplc- Correspondng author. Tel.: ; fax: E-mal address: chbassm@eng.usm.my B.H. Hameed. ty, economcally vable, techncally feasble and socally acceptable [5]. A notable trend n the development of actvated carbon AC, an adsorbent wth ts large porous surface area, controllable pore structure, thermo-stablty and low acd/base reactvty has been wtnesses [6], n terms ts versatlty for removal of a broad type of organc and norganc pollutants dssolved n aqueous meda, even from gaseous envronment [7]. Despte ts prolfc use n adsorpton processes, the bggest barrer of ts applcaton by the ndustres s the cost-prohbtve adsorbent and dffcultes assocated wth regeneraton [8]. Realzng the complcaton, a growng explotaton to evaluate the feasblty and sutablty of natural, renewable and low-cost materals bamboo dust, peat, chtosan, lgnte, fung, moss, bark husk, chtn, cor pth, maze cob, pnewood sawdust, rce husk, sugar cane bagasse, tea leaves, and sago waste as alternatve adsorbents n water polluton control, remedaton and decontamnaton processes has been exerted [9,20]. In the endeavor to explore novel adsorbents n accessng an deal adsorpton system, t s essental to establsh the most approprate adsorpton equlbrum correlaton [2], whch s ndspensable for relable predcton of adsorpton parameters and quanttatve comparson of adsorbent behavor for dfferent adsorbent systems or for vared expermental condtons [22,23]. In the perspectve, equlbrum relatonshps, generally known as adsorpton sotherms, descrbe how pollutants nteract wth the adsorbent materals, and thus are crtcal for optmzaton of the adsorpton mechansm pathways, expresson of the surface propertes and capactes of adsorbents, and effectve desgn of the adsorpton systems [24,25] /$ see front matter 2009 Elsever B.V. All rghts reserved. do:0.06/j.cej

2 K.Y. Foo, B.H. Hameed / Chemcal Engneerng Journal Nomenclature a K Khan sotherm model exponent a R sotherm constant /mg a RP Radke Prausntz sotherm model constant a S Sps sotherm model constant L/mg a T Toth sotherm constant L/mg A Koble Corrgan sotherm constant L n mg n /g A T Tempkn sotherm equlbrum bndng constant L/g b Langmur sotherm constant dm 3 /mg b K Khan sotherm model constant b T Tempkn sotherm constant B Koble Corrgan sotherm constant L/mg n B DR Dubnn Radushkevch sotherm constant C e equlbrum concentraton mg/l C o adsorbate ntal concentraton mg/l C s adsorbate monolayer saturaton concentraton mg/l C BET BET adsorpton sotherm relatng to the energy of surface nteracton L/mg d Interlayer spacng m ε Dubnn Radushkevch sotherm constant E mean free energy kj/mol g Redlch Peterson sotherm exponent G Gbbs energy change kj/mol k MacMllan Teller MET sotherm constant K ad Dubnn Radushkevch sotherm constant mol 2 /kj 2 K D Hll constant K F Freundlch sotherm constant mg/g dm 3 /g n related to adsorpton capacty K FH Flory Huggns sotherm equlbrum constant L/g K L Langmur sotherm constant L/mg K R sotherm constant L/g K s Sps sotherm model constant L/g K T Toth sotherm constant mg/g n adsorpton ntensty n FH Flory Huggns sotherm model exponent n H Hll cooperatvty coeffcent of the bndng nteracton p number of parameter q e amount of adsorbate n the adsorbent at equlbrum mg/g q e,calc calculated adsorbate concentraton at equlbrum mg/g q e,meas measured adsorbate concentraton at equlbrum mg/g q s theoretcal sotherm saturaton capacty mg/g q sh Hll sotherm maxmum uptake saturaton mg/l Q o maxmum monolayer coverage capactes mg/g r nverse power of dstance from the surface r R Radke Prausntz sotherm model constant R unversal gas constant 8.34 J/mol K R 2 correlaton coeffcent R L separaton factor t Toth sotherm constant T temperature K degree of surface coverage Frenkel Halsey Hll sotherm constant J m r /mole wth r s the sgn of nverse power of dstance from the surface ˇR Radke Prausntz sotherm model exponent ˇS Sps sotherm model exponent Meanwhle, lnear least-squares method s a tradtonal lnearly transformed approach wdely adopted to determne the sotherm parameters or the most ftted model, prmarly subjected to ts goodness ft to the expermental data, wth the magntude regresson correlaton coeffcents that close to unty [26]. Nevertheless, a substantal constrcton related to the lnearzed sotherm expressons has recently been ponted out, whch produce a vast amount of dfferent outcomes, mplctly alter the error structure, volate the error varance and normalty assumptons of standard least squares, leadng to the bas of the adsorpton data [27,28]. Dependng on the way the adsorptve equaton s lnearzed, the error dstrbuton changes worse. Ths has attested the utlzaton of nonlnearzed models n conjuncton wth a number of error analyss technques [29 3]. Wth the aforementoned, ths bblographc revew attempts to postulate a platform n descrbng the dstnct propertes, development and potental applcatons of adsorpton sotherm systems. The present work s amed at evaluatng ther accuracy and consstency n parameters predcton or estmaton. The extent of the error functons together wth ts comprehensve lterature comparsons has been hghlghted and outlned, to famlarze the knowledge defcences regardng non-lnearzed adsorpton sotherms. 2. Adsorpton sotherms models In general, an adsorpton sotherm s an nvaluable curve descrbng the phenomenon governng the retenton or release or moblty of a substance from the aqueous porous meda or aquatc envronments to a sold-phase at a constant temperature and ph [32,33]. Adsorpton equlbrum the rato between the adsorbed amount wth the remanng n the soluton s establshed when an adsorbate contanng phase has been contacted wth the adsorbent for suffcent tme, wth ts adsorbate concentraton n the bulk soluton s n a dynamc balance wth the nterface concentraton [30,34]. Typcally, the mathematcal correlaton, whch consttutes an mportant role towards the modelng analyss, operatonal desgn and applcable practce of the adsorpton systems, s usually depcted by graphcally expressng the sold-phase aganst ts resdual concentraton [35]. Its physcochemcal parameters together wth the underlyng thermodynamc assumptons provde an nsght nto the adsorpton mechansm, surface propertes as well as the degree of affnty of the adsorbents [36]. Over the years, a wde varety of equlbrum sotherm models Langmur, Freundlch, Brunauer Emmett Teller, Redlch Peterson, Dubnn Radushkevch, Temkn, Toth, Koble Corrgan, Sps, Khan, Hll, Flory Huggns and Radke Prausntz sotherm, have been formulated n terms of three fundamental approaches [37]. Knetc consderaton s the frst approach to be referred. Hereby, adsorpton equlbrum s defned beng a state of dynamc equlbrum, wth both adsorpton and desorpton rates are equal [38]. Whereas, thermodynamcs, beng a base of the second approach, can provde a framework of dervng numerous forms of adsorpton sotherm models [39,40], and potental theory, as the thrd approach, usually conveys the man dea n the generaton of characterstc curve [4]. However, an nterestng trend n the sotherm modelng s the dervaton n more than one approach, thus drectng to the dfference n the physcal nterpretaton of the model parameters [42]. 2.. Two parameter sotherms 2... Langmur sotherm model Langmur adsorpton sotherm, orgnally developed to descrbe gas sold-phase adsorpton onto actvated carbon, has tradtonally

3 4 K.Y. Foo, B.H. Hameed / Chemcal Engneerng Journal been used to quantfy and contrast the performance of dfferent bo-sorbents [38]. In ts formulaton, ths emprcal model assumes monolayer adsorpton the adsorbed layer s one molecule n thckness, wth adsorpton can only occur at a fnte fxed number of defnte localzed stes, that are dentcal and equvalent, wth no lateral nteracton and sterc hndrance between the adsorbed molecules, even on adjacent stes [43]. In ts dervaton, Langmur sotherm refers to homogeneous adsorpton, whch each molecule possess constant enthalpes and sorpton actvaton energy all stes possess equal affnty for the adsorbate [44], wth no transmgraton of the adsorbate n the plane of the surface [45]. Graphcally, t s characterzed by a plateau, an equlbrum saturaton pont where once a molecule occupes a ste, no further adsorpton can take place [33,46]. Moreover, Langmur theory has related rapd decrease of the ntermolecular attractve forces to the rse of dstance. The mathematcal expresson of Langmur sotherm models are llustrated n Table. Hereby, a dmensonless constant, commonly known as separaton factor R L defned by Webber and Chakkravort [47] can be represented as: R L = + K L C o where K L L/mg refers to the Langmur constant and C o s denoted to the adsorbate ntal concentraton mg/l. In ths context, lower R L value reflects that adsorpton s more favourable. In a deeper explanaton, R L value ndcates the adsorpton nature to be ether unfavourable R L >, lnear R L =, favourable 0 < R L < or rreversble R L = Freundlch sotherm model Freundlch sotherm [48] s the earlest known relatonshp descrbng the non-deal and reversble adsorpton, not restrcted to the formaton of monolayer. Ths emprcal model can be appled to multlayer adsorpton, wth non-unform dstrbuton of adsorpton heat and affntes over the heterogeneous surface [49]. Hstorcally, t s developed for the adsorpton of anmal charcoal, demonstratng that the rato of the adsorbate onto a gven mass of adsorbent to the solute was not a constant at dfferent soluton concentratons [9]. In ths perspectve, the amount adsorbed s the summaton of adsorpton on all stes each havng bond energy, wth the stronger bndng stes are occuped frst, untl adsorpton energy are exponentally decreased upon the completon of adsorpton process [50]. At present, Freundlch sotherm s wdely appled n heterogeneous systems especally for organc compounds or hghly nteractve speces on actvated carbon and molecular seves. The slope ranges between 0 and s a measure of adsorpton ntensty or surface heterogenety, becomng more heterogeneous as ts value gets closer to zero. Whereas, a value below unty mples chemsorptons process where /n above one s an ndcatve of cooperatve adsorpton [5]. Its lnearzed and non-lnearzed equatons are lsted n Table. Recently, Freundlch sotherm s crtczed for ts lmtaton of lackng a fundamental thermodynamc bass, not approachng the Henry s law at vanshng concentratons [23] Dubnn Radushkevch sotherm model Dubnn Radushkevch sotherm [52], s an emprcal model ntally conceved for the adsorpton of subcrtcal vapors onto mcropore solds followng a pore fllng mechansm. It s generally appled to express the adsorpton mechansm [53] wth a Gaussan energy dstrbuton onto a heterogeneous surface [54]. The model has often successfully ftted hgh solute actvtes and the ntermedate range of concentratons data well, but has unsatsfactory asymptotc propertes and does not predct the Henry s law at low pressure [55]. The approach was usually appled to dstngush the physcal and chemcal adsorpton of metal ons [4], wth ts mean free energy, E per molecule of adsorbate for removng a molecule from ts locaton n the sorpton space to the nfnty can be computed by the relatonshp [56]: [ ] E = 2B DR 2 where B DR s denoted as the sotherm constant. Meanwhle, the parameter ε can be correlated as: ] ε = RT ln [ + 3 where R, T and C e represent the gas constant 8.34 J/mol K, absolute temperature K and adsorbate equlbrum concentraton mg/l, respectvely. One of the unque features of the Dubnn Radushkevch sotherm model les on the fact that t s temperature-dependent, whch when adsorpton data at dfferent temperatures are plotted as a functon of logarthm of amount adsorbed vs the square of potental energy, all sutable data wll le on the same curve, named as the characterstc curve Temkn sotherm model Temkn sotherm s the early model descrbng the adsorpton of hydrogen onto platnum electrodes wthn the acdc solutons. The sotherm [57] contans a factor that explctly takng nto the account of adsorbent adsorbate nteractons. By gnorng the extremely low and large value of concentratons, the model assumes that heat of adsorpton functon of temperature of all molecules n the layer would decrease lnearly rather than logarthmc wth coverage [58]. As mpled n the equaton, ts dervaton s characterzed by a unform dstrbuton of bndng energes up to some maxmum bndng energy. Temkn equaton s excellent for predctng the gas phase equlbrum when organzaton n a tghtly packed structure wth dentcal orentaton s not necessary, conversely complex adsorpton systems ncludng the lqud-phase adsorpton sotherms are usually not approprate to be represented [59] Flory Huggns sotherm model Flory Huggns sotherm model [60], whch occasonally dervng the degree of surface coverage characterstcs of adsorbate onto adsorbent, can express the feasblty and spontaneous nature of an adsorpton process. In ths respect, s the degree of surface coverage, where K FH and n FH are the ndcaton of ts equlbrum constant and model exponent. Its equlbrum constant, K FH that used for the calculaton of spontanety free Gbbs energy, s related to the equaton [43]: G = RT ln K FH Hll sotherm model Hll equaton [6], that orgnated from the NICA [62] model, was postulated to descrbe the bndng of dfferent speces onto homogeneous substrates. The model assumes that adsorpton s a cooperatve phenomenon, wth the lgand bndng ablty at one ste on the macromolecule, may nfluence dfferent bndng stes on the same macromolecule [63] Three parameter sotherms sotherm model sotherm [64] s a hybrd sotherm featurng both Langmur and Freundlch sotherms, whch ncorporate three parameters nto an emprcal equaton [65]. The model has a lnear dependence on concentraton n the numerator and an exponental functon n the denomnator [66] to represent adsorpton equlbra over a wde concentraton range, that can be appled ether

4 K.Y. Foo, B.H. Hameed / Chemcal Engneerng Journal Table Lsts of adsorpton sotherms models. Isotherm Nonlnear form Lnear form Plot Reference Langmur q e = Qob +b = + bqo Qo vs C e [38] = Qo + bqo vs q e = Q o b q e vs b = bqo b vs q e Freundlch q e = K F C /n e log q e = log K F + log log vs log [48] n Dubnn Radushkevch q e =q s exp k ad ε 2 lnq e=lnq s k ad ε 2 Inq evsε 2 [52] Tempkn Flory Huggns Hll q e = RT b T ln A T C e q e = RT b T ln A T + Co = KFH n FH log q e = qs H Cn H e K D +C n H e q e = K R +a R C g e Sps q e = KsCˇS e +a S CˇS e Toth q e = K T log ln a T + /t ln Co RT ln C b e q e vs ln C e [57] T = log K FH + n FH log log qs H Co = n H log C e log K D log vs log [60] qs H K R = g lnc e + lna R ln K R ˇS lnc e = ln Ks K T + lna S ln Ks vs logc e [6] vs lnc e [64] vs lnc e [68] = lnc e t lnat + ln K T vs lnc e [69] Koble Corrgan q e = ACn e +BC n e = AC e n + B [70] A Khan q e = qsb K +b K a K [7] Radke Prausntz q e = a RP r R CˇR e a RP +r R CˇR e [43] BET q e = FHH ln qsc BET Cs [+C BET /Cs] Cs = RT Cs = + C BET qsc BET qsc BET Cs vs Cs Cs qs d r [74] /3 k MET q e = q s lncs/ [75] [73] n homogeneous or heterogeneous systems due to ts versatlty [22]. Typcally, a mnmzaton procedure s adopted n solvng the equatons by maxmzng the correlaton coeffcent between the expermental data ponts and theoretcal model predctons wth solver add-n functon of the Mcrosoft excel [26]. In the lmt, t approaches Freundlch sotherm model at hgh concentraton as the exponent ˇ tends to zero and s n accordance wth the low concentraton lmt of the deal Langmur condton as the ˇ values are all close to one [67] Sps sotherm model Sps sotherm [68] s a combned form of Langmur and Freundlch expressons deduced for predctng the heterogeneous adsorpton systems [53] and crcumventng the lmtaton of the rsng adsorbate concentraton assocated wth Freundlch sotherm model. At low adsorbate concentratons, t reduces to Freundlch sotherm; whle at hgh concentratons, t predcts a monolayer adsorpton capacty characterstc of the Langmur sotherm. As a general rule, the equaton parameters are governed manly by the operatng condtons such as the alteraton of ph, temperature and concentraton [45] Toth sotherm model Toth sotherm model [69], s another emprcal equaton developed to mprove Langmur sotherm fttngs expermental data, and useful n descrbng heterogeneous adsorpton systems, whch satsfyng both low and hgh-end boundary of the concentraton [43]. Its correlaton presupposes an asymmetrcal quas-gaussan energy dstrbuton, wth most of ts stes has an adsorpton energy lower than the peak maxmum or mean value [23] Koble Corrgan sotherm model Smlar to the Sps sotherm model, Koble Corrgan sotherm [70] s a three-parameter equaton, whch ncorporated both Langmur and Freundlch sotherm models for representng the equlbrum adsorpton data. The sotherm constants, A, B and n are evaluated from the lnear plot usng a tral and error optmzaton Khan sotherm model Khan sotherm [7] s a generalzed model suggested for the pure solutons, wth b K and a K are devoted to the model constant and model exponent. At relatvely hgh correlaton coeffcents and mnmum ERRSQ or ch-square values, ts maxmum uptake values can be well determned [72] Radke Prausntz sotherm model The correlaton of Radke Prausntz sotherm s usually predcted well by the hgh RMSE and ch-square values. Its model exponent s represented by ˇR, where a R and r R are referred to the model constants [43] Multlayer physsorpton sotherms Brunauer Emmett Teller BET [73] sotherm s a theoretcal equaton, most wdely appled n the gas sold equlbrum systems. It was developed to derve multlayer adsorpton systems

5 6 K.Y. Foo, B.H. Hameed / Chemcal Engneerng Journal Table 2 Lsts of error functons. Error functon Abbrevaton Defnton/expresson Reference Sum squares errors ERRSQ/SSE q e,calc q e,meas 2 [80] Hybrd fractonal error functon HYBRID 00 n p [,meas q e,calc,meas ] [66] Average relatve error ARE 00 n,meas q e,calc,meas [82] Sum of absolute error EABS q e,meas q e,calc Marquardt s percent standard devaton MPSD 00 n p,meas,calc 2,meas [83] [84] The coeffcent of determnaton R 2 r 2 =,meas q e,calc 2,meas q e,calc 2 +,meas q e,calc 2 [78] n Spearman s correlaton coeffcent r s 6,meas q e,calc 2 [78] nn 2 Standard devaton of relatve errors s RE n Nonlnear ch-square test 2 2 [,meas q e,calc ARE] [78] n q e,calc,meas 2 Coeffcent of non-determnaton K 2 [80],meas [78] Sum of normalzed errors SNE [88] wth relatve pressure ranges from 0.05 to 0.30 correspondng to a monolayer coverage lyng between 0.50 and.50. Its extncton model related to lqud sold nterface s exhbted as: q s C BET C e q e = 5 C s C e [ + C BET C e /C s ] where C BET, C s, q s and q e are the BET adsorpton sotherm L/mg, adsorbate monolayer saturaton concentraton mg/l, theoretcal sotherm saturaton capacty mg/g and equlbrum adsorpton capacty mg/g, respectvely. As C BET and C BET C e /C s are much greater than, the equaton s smplfed as: q s q e = 6 C e /C s Meanwhle, Frenkel Halsey Hll FHH sotherm [74], another multlayer adsorpton dervaton from the potental theory may be wrtten as: ln = qs r 7 C s RT q e d where d, and r are the sgn of the nterlayer spacng m, sotherm constant J m r /mole and nverse power of dstance from the surface about 3, respectvely. Smlarly, MacMllan Teller MET sotherm [75], an adsorpton model nterpreted from the ncluson of surface tenson effects n the BET sotherm s termed as: /3 k q e = q s 8 lnc s /C e where k s an sotherm constant. When C s /C e s approachng unty, the logarthmc term can be approxmated as: k /3 q e = q s 9 C s C e As a note, the emprcal sotherm s reasonable ft to Frenkel Halsey Hll FHH or MacMllan Teller MET sotherms for relatve pressures hgher than 0.8 and approxmately Brunauer Emmett Teller BET sotherm for relatve pressures lower than Error functons Wthn recent decades, lnear regresson has been one of the most vable tool defnng the best-fttng relatonshp [76] quantfyng the dstrbuton of adsorbates, mathematcally analyzng the adsorpton systems [77] and verfyng the consstency and theoretcal assumptons of an sotherm model [78]. Due to the nherent bas resultng from the transformaton whch rdng towards a dverse form of parameters estmaton errors and fts dstorton, several mathematcally rgorous error functons sum square error, Hybrd fractonal error functon, sum of absolute errors, average relatve error, Marquardt s percent standard devaton, coeffcent of determnaton, Spearman s correlaton coeffcent, standard devaton of relatve errors, nonlnear ch-square test, coeffcent of nondetermnaton and sum of normalzed errors Table 2 have lately drastcally been addressed and confronted [79]. Concomtant wth the development of computer technology n the 980s, the progresson of the nonlnear sotherm modelng has extensvely been facltated and motvated [78]. Contrary to the lnearzaton models, nonlnear regresson usually nvolves the mnmzaton or maxmzaton of error dstrbuton between the expermental data and the predcted sotherm based on ts convergence crtera [79]. 3.. Sum square error ERRSQ Despte ERRSQ s the most wdely used error functon [80], at hgher end of the lqud-phase concentraton ranges, the magntude

6 K.Y. Foo, B.H. Hameed / Chemcal Engneerng Journal and squares of the errors tend to ncrease, llustratng a better ft for the sotherm parameters dervaton [8] Hybrd fractonal error functon HYBRID The error functon was developed to mprove ERRSQ ft at low concentratons. Hereby, each ERRSQ value s dvded by the expermental sold-phase concentraton wth a dvsor ncluded n the system as a term for the number of degrees of freedom the number of data ponts mnus the number of parameters wthn the sotherm equaton [66] Average relatve error ARE ARE model [82] whch ndcates a tendency to under or overestmate the expermental data, attempts to mnmze the fractonal error dstrbuton across the entre studed concentraton range Sum of absolute errors EABS The approach s smlar to the ERRSQ functon, wth an ncrease n the errors wll provde a better ft, leadng to the bas towards the hgh concentraton data [83] Marquardt s percent standard devaton MPSD Marquardt s percent standard devaton MPSD error functon [84] has prevously practced by a number of researchers n the sotherm studes [26,83,85,86]. Accordng to the number of degrees of freedom n the system, t s smlar to some respects of a modfed geometrc mean error dstrbuton [87] Coeffcent of determnaton R 2, Spearman s correlaton coeffcent r s and standard devaton of relatve errors s RE Coeffcent of determnaton, whch represents the percentage of varablty n the dependent varable the varance about the mean s employed to analyze the fttng degree of sotherm and knetc models wth the expermental data [88]. Its value may vary from 0 to [89] where Spearman s correlaton coeffcent and standard devaton of relatve errors are ndvdually determned to evaluate the global correlaton and the dsperson of ts relatve errors [78] Nonlnear ch-square test 2 Nonlnear ch-square test s a statstcal tool necessary for the best ft of an adsorpton system, obtaned by judgng the sum squares dfferences between the expermental and the calculated data, wth each squared dfference s dvded by ts correspondng value calculated from the models. Small 2 value ndcates ts smlartes whle a larger number represents the varaton of the expermental data [78] Coeffcent of non-determnaton K 2 Another statstcal term, coeffcent of non-determnaton, s much useful n descrbng the extent relatonshp between the transformed expermental data and the predcted sotherms, and mnmzaton of the error dstrbuton [90] Sum of normalzed errors SNE Consequence of dfferent error crtera s lkely to produce dfferent sets of sotherm parameters, a standard procedure normalzng and combnng varous errors for better and meanngful comparson between the parameter sets for the sngle sotherm model s adopted [44,78,83,9]. The calculaton orentaton s revealed as follows: a Selecton of an sotherm model and error functon, and determnaton of the adjustable parameters whch mnmze the error functon. b Determnaton of all other error functons by referrng to the parameter set. c Computaton of other parameter sets assocated wth ther error functon values ntaton of the procedure by mnmzng the error functon. d Normalzaton and selecton of the maxmum parameter sets wth respect to the largest error measurement. e Summaton of each parameter set whch generates the mnmum normalzaton error. 4. Lterature revew on applcatons of lnear and nonlnear forms of sotherm models An accuracy of an sotherm model s generally a functon of the number of ndependent parameters, whle ts popularty n relaton to the process applcaton s an ndcatve of ts mathematcal smplcty [37]. Undoubtedly, lnear regresson analyss has frequently been employed n accessng the qualty of fts and adsorpton performance [44], prmarly owng to ts wde usefulness n a varety of adsorpton data [9] and partly reflectng the appealng smplcty of ts equatons [92]. However, durng the last few years, a development nterest n the utlzaton of nonlnear optmzaton modelng has been noted [65]. A number of researches have been advocated to nvestgate the applcablty of lnear or nonlnear sotherm models n descrbng the adsorpton of dyes, heavy metals and organc pollutants onto actvated carbons, zeoltes, chtosans, bentontes, montmorllontes, kaolntes and a lst of low-cost adsorbents Table 3. In 984, Harter [03] had frstly examned Langmur sotherm model n an ons adsorpton system adsorpton of on phosphate, znc, and copper by sol. Wthout suffcent ranges of adsorbate concentraton, he emphaszed that the estmaton of maxmum adsorpton capacty could be qute msleadng n error by 50% or more, reducng the varablty of ts lnearty. In 988, Persoff and Thomas [04] had proposed the use of nonlnear least-squares NLLS curve-fttng method for determnaton of the Mchaels Menten and Langmur adsorpton sotherm constants from the expermental data. From the applcaton, they concluded that weghted NLLS yelded a more precse and accurate estmaton. More recently, smlar observatons have been reported by several researchers [24,63,78,79,96,0]. The authors suggested that the lnearzed equatons apparently generate real problems and errors arsng from the complextes and complcatons for smultaneous transformaton of data, leadng to the volaton of theores behnd the sotherms. In certan cases, t has been llustrated that a dfferent axs settng dfferent lnearzed models would alter the regresson results, nfluencng ts consstency and accuracy [97]. Such tendency more statstcal functons are vald for nonlnear than lnear analyss could be drectly proportonal to the dstorton of the expermental errors, creatng an nherent errors estmaton problem whch lmts the valdty of the studed tools [35]. Moreover, lnear analyss method assumes that the scatter vertcal ponts around the lne follows a Gaussan dstrbuton, and the error dstrbuton s unform at every value of the lqud-phase resdual concentraton X-axs [89]. Nonetheless, such behavor s practcally mpossble wth the equlbrum relatonshps snce sotherm models had nonlnear shape,

7 8 K.Y. Foo, B.H. Hameed / Chemcal Engneerng Journal Table 3 Prevous researches of the lnear and nonlnear sotherm studes. Adsorbent Adsorbate Isotherm models Determnaton Preference types Reference Wheat bran Cadmum ons Freundlch, Langmur R 2 Nonlnear [5] Babasse fly ash Orange-G dyemethyl Volet dye Freundlch, Langmur, R 2 Nonlnear [9], Temkn, Dubnn Radushkevch, Elovch Peat Dvalent metal ons Freundlch, Langmur, R 2, ERRSQ, ARE, HYBRID, Both [23], Temkn, Dubnn Radushkevch, Toth MPSD, SAE, SNE Water hyacnth Methylene blue dye Freundlch, Langmur R 2 Nonlnear [24] Rce husk Safrann Freundlch, Langmur, R 2 Nonlnear [30] Fbrous bomass Drect Solophenyl Brown dye Langmur,, Temkn, ARED, EABS, MPSD, Nonlnear [35] Dubnn Radushkevch, Elovch HYBRID, R 2, RESID Iron oxde-coated cement Arsenc Freundlch, Langmur, SAE, ARE, HYBRID, MPSD, Both [44], Temkn EABS Montmorllonte, kaolnte Heavy metals Freundlch, Langmur R 2 Both [55] Yeast bomass Ochratoxn A Freundlch, Langmur, BET, R 2, EABS, HYBRID, ARE, Nonlnear [63] SAE, MPSD Actvated carbon Malachte green dye Freundlch, Langmur, R 2 Nonlnear [76] Actvated carbon Tetrahydrothophene Langmur R 2, r s, EABS, RSS, ARE, s RE, Nonlnear [78] HYBRID Actvated carbon Methylene blue Freundlch, Langmur, R 2, ERRSQ, ARE, HYBRID, Nonlnear [79] MPSD, EABS Actvated carbon Malachte green dye Freundlch, Langmur, R 2 Nonlnear [80] Rce husk ash Brllant green dye Freundlch, Langmur,, Temkn, Dubnn Radushkevch Chtosan Lead Freundlch, Langmur, SSE, SAE, ARE, HYBRID, MPSD Both [8] ERRSQ, ARE, HYBRID, Nonlnear [83] MPSD, EABS Zeolte Ammonum Freundlch, Langmur R 2 Nonlnear [88] Actvated carbon Basc dyes Freundlch, Langmur, R 2 Nonlnear [89] Actvated carbon Basc red 9 Freundlch, Langmur, R 2, EABS HYBRID, ARE, Nonlnear [90] SAE, MPSD Alumna cement granules Fluorde Freundlch, Langmur, R 2, SNE, EABS HYBRID, Both [92] Dubnn Radushkevch ARE, SAE, MPSD Eucalyptus bark Cadmum ons Freundlch, Langmur R 2 Nonlnear [93] Zeolte Methylene blue Freundlch, Koble Corrgan, Langmur, SAE, ARE, ARS, EABS Both [94], Temkn Bagasse fly ash Brllant green dye Freundlch, Langmur,, Temkn, Dubnn Radushkevch SSE, SAE, ARE, HYBRID, MPSD Both [95] Actvated carbon Basc blue 9 dye Freundlch, Langmur, R 2, ERRSQ, 2 Nonlnear [96] Sugarcane dust Basc dyes Freundlch, Langmur, R 2 Nonlnear [97] 2 Fly ash Dyes Freundlch, Langmur R 2 Nonlnear [98] Rce husk Bsmarck brown dye Freundlch, Langmur, R 2 Nonlnear [99] Bentonte Strontum Freundlch, Langmur RMSE Nonlnear [00] Ca-algnate beads Znc II ons Freundlch, Langmur R 2 Nonlnear [0] Mansona wood sawdust Methyl volet Freundlch, Langmur, R 2 Both [02] as the error dstrbuton tends to get altered after transformng nto a lnearzed order [26]. In another study, lnearzaton sotherms models Langmur and Freundlch sotherm models have been demonstrated napproprate n predctng the goodness of ft for a partcular set of condtons [95], and unable for provdng a fundamental understandng of the ons adsorpton systems, resultng n an mproper concluson. On the contrary, the nonlnear sotherm models are conducted on the same abscssa and ordnate, thus avodng such drawbacks of lnearzaton [92]. Nevertheless, a few researchers [23,44,94,95,02] also ndcated the smlartes and consstency of both lnear and nonlnear sotherms, lyng nto the same error dstrbutons and structures. Under such condtons, t would be more ratonal and relable to nterpret adsorpton data through a process of lnear and nonlnear regresson [92]. Irrespectve of ts technque ether the lnear or the nonlnear method, the avalablty and usefulness of the equlbrum data should be suffcent enough to effectvely represent an effcent and complete sotherm model [76]. 5. Concluson The past 0 years has seen a developng nterest n the preparaton of low-cost adsorbents as alternatves to actvated carbons n water and wastewater treatment processes. To date, lmted success of adsorbents n the feld applcatons has rased apprehensons over the use of adsorpton capacty generated from equlbrum data as a measure of ther effectveness n drnkng water treatment. Over the past few decades, lnear regresson has been developed as a major opton n desgnng the adsorpton systems. However, recent nvestgatons have ndcated the growng dscrepancy between the predctons and expermental data and dsablty of the model, propagatng towards a dfferent out-

8 K.Y. Foo, B.H. Hameed / Chemcal Engneerng Journal come. Despte ths obvous nherent bas of the model, lnearzaton remans a confdent opton n the lterature, appled n over 95% of the lqud-phase adsorpton systems. Hence, the next real challenge n the adsorpton feld s the dentfcaton and clarfcaton of both sotherm models n varous adsorpton systems. Further exploratons on developng n ths area are recommended. Acknowledgement The authors acknowledge the research grant provded by the Unverst Sans Malaysa under the Research Unversty RU Scheme Project No. 00/PJKIMIA/ References [] K.Y. Foo, B.H. Hameed, Utlzaton of bodesel waste as a renewable resource for actvated carbon: applcaton to envronmental problems, Renew. Sust. Energy Rev [2] K.Y. Foo, B.H. Hameed, Value-added utlzaton of ol palm ash: a superor recyclng of the ndustral agrcultural waste, J. Hazard. Mater. 2009, do:0.06/j.jhazmat [3] N. Mart, A. Bouzas, A. Seco, J. Ferrer, Struvte precptaton assessment n anaerobc dgeston processes, Chem. 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