A MODIFIED UNIQUAC EQUATION FOR MIXTURES CONTAINING SELF-ASSOCIATING COMPOUNDS

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1 rala Joural of Chemcal Egeerg ISSN Prted ral Vol., No. 03, pp , July - September, 005 MOIFIE UNIQUC EQUTION FOR MIXTURES CONTINING SELF-SSOCITING COMPOUNS P.. Pessôa Flho *, R. S. Mohamed H ad G. Maurer 3 epartameto de Egehara Químca, Escola Poltécca, Uversdade de São Paulo, Phoe: +(55) () , Fax: +(55) () , C. P. l 6548, CEP , São Paulo - SP, ral. E-mal: pedro.pessoa@pol.usp.br epartameto de Termofludodâmca, Faculdade de Egehara Químca, Uversdade Estadual de Campas, Cx. P. 6066, CEP , Campas - SP, ral 3 Lehrstuhl für Techsche Thermodyamk, Uverstät Kaserslauter, Postfach 3049, , Kaserslauter, Germay. (Receved: July 6, 004 ; ccepted: March 5, 005) bstract - The UNIQUC model for the excess Gbbs eergy s modfed usg chemcal theory to accout for cha-lke assocato occurrg self-assocatg compouds such as alcohols. The equato cosders the alcohol to be a mxture of clusters chemcal equlbrum. The UNIQUC equato s used to model the behavor of the mxture of clusters, wth se ad surface parameters related to the umber of alcohol molecules volved ther formato. The values of assocato ethalpy ad etropy were obtaed through fttg vapor pressure data. The model s used to correlate phase behavor of alcohol-hydrocarbo mxtures at low pressures, presetg excellet results bubble pot calculatos. further exteso was made to allow for cross-assocato, the formato of a hydroge bod betwee the molecules of a alcohol ad a actve solute. Ths exteso was used to model alcohol-aromatc mxtures wth equally good results. Keywords: Model; Excess Gbbs eergy; Vapor-lqud equlbra; lcohol; Hydrocarbo. INTROUCTION The fact that mxtures cotag self-assocatg compouds show strog devatos from deal behavor has log bee recoged. The frst mportat attempt to accout for t thermodyamc modelg was made by Kretschmer ad Webe (954), who regarded a alcohol as a mxture of lear clusters chemcal equlbrum. The devato from deal behavor a mxture of these clusters ad a ert compoud was obtaed usg oly the combatoral part of the Flory-Huggs model. The ma assumpto of Kretschmer ad Webe s that the Gbbs eergy of the geeral reacto: + + () s depedet of whe occurrg betwee solated molecules. Furthermore, the volume of a olgomer s cosdered to be proportoal to the umber of moomer segmets. Ths assumpto, alog wth the Flory-Huggs equato for the Gbbs eergy of mxg, leads to the expresso: K + ϕ + = () ϕ ϕ e *To whom correspodece should be addressed H Ths paper s dedcated to the memory of Prof. r. R. S. Mohamed, who passed away after fshg ths work.

2 47 P.. Pessôa Flho, R. S. Mohamed H ad G. Maurer whch ϕ s the volume fracto ad K s the equlbrum costat. fter some rearragemets, oe ca relate the rato: c C + K+ = (3) c c to K through the expresso: K + = (4) ev C + K+ where V represets the volume of the moomer ad c s the cocetrato amout of substace per volume. The depedece of K + o, due to the dfereces olgomer volumes accouted for whe the stadard state s chaged to the pure lqud, C results K + ot depedg o. Whe aromatc hydrocarbos are preset, a crossassocato or solvato reacto (betwee a hydrocarbo molecule ad a molecule of ay cluster) was cosdered to occur by Kretschmer ad Webe (954). The chemcal equlbrum was the calculated usg some mathematcal smplfcato volvg the arbtrary defto of the solvato equlbrum costat as a rato of cocetratos smlar to K C. Reo ad Praust (967) obtaed almost the same expressos as Kretschmer ad Webe (954) through a more rgorous dervato; they, however, dd ot address the problem of cross-assocato. Nagata ad Kawamura (977) preseted a modfed UNIQUC equato, based o the same assumptos of Kretschmer ad Webe. I ther work, the rato of volume fractos: K ϕ ϕ + = (5) ϕ ϕ + was used to calculate the chemcal equlbrum amog clusters ad was cosdered to be depedet of cocetrato ad cluster se. The authors also preseted aother model based o the Kempter ad Mecke (940) equlbrum costat. Nagata (985) exteded that model order to tackle mxtures cotag ay umber of alcohols ad mxtures of a alcohol ad a actve compoud (. e, a compoud that ca udergo solvato). Nath ad eder (98a) proposed to use the ormal bolg pot temperature ad the ethalpy of vaporato to obta the value of the equlbrum costats. They recoged that the determato of equlbrum costats through expermetal phase equlbrum ad excess ethalpy data, as doe htherto, was coveet due to the ecessary troducto of mxture data the calculato of a property of a pure substace. The authors subsequetly exteded the proposed model to mxtures of oe alcohol ad ert compouds (Nath ad eder, 98b) ad to mxtures of ay umber of alcohols ad ert compouds (Nath ad eder, 983). It s worth otg that the authors used equato (5) for the equlbrum costat. rada (983) ad rada ad Evagelsta (984) publshed mportat papers o ths subect. The authors dd ot defe a equlbrum costat, as doe most prevous studes, but stead used the UNIQUC equato to obta ths equlbrum costat. I other words, they bega wth Flory s (94) referece state (pure substace whose molecules are oreted a crystalle arragemet) ad the, through a seres of steps, obtaed a expresso for the equlbrum costat, whose parameters (ethalpy ad etropy of assocato) were foud by fttg vapor pressure data; the gas phase was modeled usg a trucated vral equato of state. rada ad Evagelsta (984) replaced the crystalle state by the pure lqud as the referece state, order to mata cosstecy wth the UNIQUC model. Recetly, the Statstcal ssocated-flud Theory (SFT) has foud extesve applcato the developmet of models for mxtures cotag assocatg compouds (ether for excess Gbbs eergy or for volumetrc equatos of state): for stace, oe ca meto Fu et al. (995), Megarell et al. (999) ad Che et al. (004), amog others. Nevertheless, chemcal theory s stll a appealg theory, presetg ramfcatos such as the wdely used ERS model, coceved by Het et al. (986), ad the cotuous thermodyamc equato of state develped by rowark (004). theoretcal aspect, v. the fact that ofte the chemcal equlbrum s calculated depedetly of the excess Gbbs eergy model, ustfes further work ths feld. Strctly, the chemcal equlbrum costat s related to the rato of actvtes, whch are calculated usg a excess Gbbs eergy equato. Other smlar ratos, as the rato of cocetratos or volume fractos, are true equlbrum costats oly some specal cases,. e, whe certa excess Gbbs eergy models are appled. s Hofma (990) poted out, the thermodyamcs of assocato depeds o the thermodyamcs of the rala Joural of Chemcal Egeerg

3 Modfed UNIQUC Equato 473 multcompoet system, ad the expresso for the equlbrum costat s defed beforehad by the excess Gbbs eergy model. Ths kd of dscrepacy occurs mostly whe cross-assocato reactos are cosdered cf. Kretschmer ad Webe (954), Nath ad eder (983) ad rada ad Evagelsta (984) whch the solvato equlbrum s calculated by a rato smlar to that obtaed for the self-assocato equlbrum costat. The purpose of ths work s to preset a modfcato of the model by rada ad Evagelsta (984). It cossts of two parts: the model s altered by adoptg deas of Kretschmer ad Webe (954), ad some assumptos from a prevous developmet of a equato of state for self-assocato compouds (Pessoa Flho ad Mohamed, 999) are corporated. No expresso for the equlbrum costat s postulated ad hoc, opposto to the way t s usually doe lterature: the model here developed uses the same UNIQUC equato to calculate both the chemcal equlbrum amog the clusters ad phase equlbrum. The exteso of the model thus developed to cross-assocatg mxtures follows the deas preseted by spro et al. (003). The model developed results alteratve expressos that provde good correlato of phase equlbrum solutos volvg self-assocato ad solvato. THEORETICL EVELOPMENT Whe descrbg a mxture of a assocatg compoud ad some ert compouds, two dfferet procedures are dstgushed. Oe procedure eglects ay assocato / solvato. The mxture cossts of N compoets (e.g.,,...). Its propertes are desgated by superscrpt, ad the composto s charactered by stochometrc amout fractos x, e.g. x, x, etc., whose sum equals oe. The other procedure takes assocato / solvato to accout. The mxture the cossts of N > N speces (e.g.,, 3,..., +,...,,...), where,, 3 represet moomers, dmers ad trmers of compoet ad s a ert,.e. o-assocatg compoet). The propertes are desgated by superscrpt, but the composto s charactered by mcroscopc amout fractos, e.g. sum also equals oe. Self-ssocato Model,,,...,,...etc., whose 3 The developmet s based o a seres of hypotheses cocerg the occurrece of selfassocato ad ts relatoshp wth the thermodyamc model. t frst a mxture of a selfassocatg compoud () ad a ert oe () s cosdered. ased o stochometrc amout fractos, oe obtas the followg expressos for the chemcal potetal ad the actvty coeffcet (ormaled accordg to the Lews ad Radall rule) for compoet : µ (T,p,x ) =µ (T, p) + pure lqud + RT l ( x γ ) (6) where x s the stochometrc amout fracto of compoet, calculated through: x = + ad ñ ad (7) ñ beg the stochometrc amout of compoets ad, respectvely. alogous expressos hold for the ert compoet. Hypothess. The self-assocatg flud s a mxture of clusters chemcal equlbrum accordg to equato () for. The bary mxture of compoets ad s therefore cosdered to be a multcompoet mxture of speces,.e. assocates,, 3, 4,... ad the ert substace. ased o mcroscopc amout fractos oe gets for the chemcal potetal ad the actvty coeffcet (ormaled accordg to Lews ad Radall rule) for ay speces : µ (T,p, ) =µ (T,p) + pure lqud + RT l ( γ ) (8) where stads for all speces ad s the mcroscopc amout fracto of speces (.e.,,... or ): = (9) N = (0) = rala Joural of Chemcal Egeerg Vol., No. 03, pp , July - September, 005

4 474 P.. Pessôa Flho, R. S. Mohamed H ad G. Maurer The relatve values of the equlbrum costats wll be the subect of aother hypothess. The secod hypothess s cocered wth the use of the UNIQUC equato ad ts parameters. Hypothess. The o-deal behavor of the lqud phase s descrbed by the UNIQUC excess Gbbs eergy model. The actvty coeffcet of a substace (ether compoet or speces) a multcompoet mxture from the UNIQUC equato s: ϕ θ ϕ lγ = l + ql + l yl y ϕ y θk k ql θτ + q q wth θ τ τ k () The UNIQUC parameters related to teractos betwee stes of ay two olgomers ad are ull: a = a = 0 (8),, Expressos for the mxture terms preset the UNIQUC equato ca ow be developed. s assocato / solvato results a reducto of the al amout of substace, a parameter ξ s defed to descrbe that reducto. ξ s the rato of macroscopc amout of substace, ad mcroscopc amout of substace: ξ= whch leads to: ñ, the ñ, the (9) (0) x = ξ= + = l = (r q) (r ) () age se parameter r usg the mcroscopc amout fractos s defed: a τ = exp ( ) (3) T For the sake of smplcty, equato () s wrtte wthout superscrpts (ether or ) ad wth amout fracto y (stead of ether x or ). s the umber of earest eghbors the lattce ( ths work, as usual, = 0). The volume ad surface parameters of a speces (assocate) are gve by: r = r (4) q = q (5) where superscrpts ad have bee omtted for the sake of smplcty. The bary parameters of the UNIQUC model for teractos betwee teracto stes o a olgomer ad o a ert speces are assumed to be depedet of the olgomer se: a = a (6),, a = a (7),, = = + = () r r r r r as well as a age se parameter usg stochometrc amout fractos (.e. amout fractos x ad x of compoets ad ): r x r x r x r () = = + Combg equato (0) wth equatos () ad () results : r r ξ= (3) The mcroscopc volume fractos ϕ ad ϕ of a speces (olgomer) ad of the ert speces, respectvely, the mxture are: r ϕ = = ad: r r r (4) rala Joural of Chemcal Egeerg

5 Modfed UNIQUC Equato 475 ϕ = r (5) r The volume fractos of the compoets ad from the overall amout fractos x ad x ( ϕ ad ϕ ) are expressed smlarly: x ϕ = r r x ϕ = r r (6) (7) It ca be otced that the volume fracto of s depedet of the cosderato or eglect of assocato. Oe ca also express the volume fracto of compoet o stochometrc amout fracto scale ϕ through the volume fractos of the olgomers o mcroscopc amout fracto scale: ϕ = ϕ (8) = alogous expressos for aged surface parameters ad surface fracos ca be obtaed exactly the same way. pplyg equato () to descrbe the actvty coeffcet of speces (.e. o mcroscopc amout fracto bass) gves: ϕ θ lγ = l + ql + ϕ ϕ + l l θ τ,, θk τk, k ql θ τ + q q (9), s, from equatos (6) to (8), τ =τ ad oe gets:,,, τ =,, τ =τ, for a olgomer S = θ + θ τ (3), whch does ot deped o the umber of moomer uts, beg heceforth refered to as S. For the speces, a smlar procedure gves: S = The sum, θ τ + θ = S (3) be rearraged to: l l = + ( r q) r apeearg equato (9) ca (33) Isertg ths equato as well as equato () for l, ad troducg ξ from equato (3) ad ts aalogous for the surface parameters to equato (9) results the actvty coeffcet for the assocatg speces: r r lγ = + l + ξ r r ξ q r r q + q l + + (34) r q q r θ θ + q ls S S τ I a very smlar procedure the fal expresso for the actvty coeffcet γ of the ert speces s: r r lγ = + l + ξ r r The deomator of the last term o the rght had sde of ths equato s: S = θ τ = θ τ +θ τ (30) k k,,, k q r r q + q l + + rq q r θ θ + q ls τ S S (35) rala Joural of Chemcal Egeerg Vol., No. 03, pp , July - September, 005

6 476 P.. Pessôa Flho, R. S. Mohamed H ad G. Maurer The thrd hypothess s cocered wth the relatve value of the chemcal equlbrum costats. Hypothess 3. Followg Kretschmer ad Webe (954), t s assumed that the chemcal reacto () s accompaed by a Gbbs eergy chage that s depedet of whe ocurrg betwee solated molecules. ccordg to the defto of equlbrum costat: oe gets: + rξ + lk = l r Settg K exp( lk) = K + e rξ e = + results : r (4) (4) lk + + G + H + S = = + (36) RT RT R where K + s the equlbrum costat for the formato of speces + accordg to equato () ad the chages Gbbs eergy, ethalpy ad etropy are related to the reacto occurrg betwee pure lqud speces. I order to aalye the depedecy of K + o, oe must cosder that, whe the stadard state s chaged to the pure lqud speces, there s a adtoal etropy chage due to dffereces volumes per amout of substace. Sce the volume of the olgomer s cosdered to be proportoal to the umber of moomers, oe ca wrte after Kretschmer ad Webe (954): 0 + G G + = l (37) RT RT where G 0 does ot deped o. Ths expresso leads to the followg depedecy of the equlbrum costat: + lk+ = lk+ l (38) From equato (34), oe recoges that the actvty of a olgomer ca be wrtte as: ( ) lα = l γ = + r + l + g(x,x, ξ) r ξ (39) whch g s a fucto that does ot deped o. s the thermodyamc chemcal equlbrum costat s: α = (40) α α + K + Equato (4) s a recurso formula that allows for the calculato of the amout fractos of the speces from the moomer amout fracto. Thus, oly ad ξ rema fact to be determed. There are two depedet equatos relatg the two ukow varables: the defto of ξ, equato (9) ad the sum of all amout fractos, equato (0). etals o the mathematc soluto of the problem are preseted appedx ; the fal expressos are: ad: ξ = x x x = + x ξ x r Ke x r + x r (43) (44) The value of, whch s ecessary to calculate the actvty of the ert compoud, s obtaed from equato (0). The model expressed by the set of equatos (34), (35), (43) ad (44) s referred to hereafter as the -UNIQUC model. Cross-ssocato: Cocepts ad Equatos There are two ways to cosder a chemcal equlbrum betwee a actve compoud (e. g. a aromatc hydrocarbo) ad a assocatg compoud (e. g., a alcohol). Oe way s to cosder that the hydrocarbo molecule may bod to ay cluster regardless of ts se cf. Kretschmer ad Webe (954) ad Nagata (985). The other approach cosders that the hydrocarbo molecule bods oly to a sgle cluster of solvet molecules, as the treatmet preseted by spro et al. (003) ad, by aalogy, Yu et al. (993). The frst approach rala Joural of Chemcal Egeerg

7 Modfed UNIQUC Equato 477 troduces some mathematcal dffcultes whch stad the way of a rgorous soluto of chemcal equlbrum. The secod s mathematcally smpler ad provdes smlar results; cosequetly, t was adopted to represet cross-assocato the preset work. Oe must be aware that t s ot a exact represetato of the molecular pheomeo: crossassocato s accouted for oly through ts effect correctg the abormal departure from the deal behavor caused by self-assocato rather tha ts molecular mplcatos for t would be mpossble to take ay possble electro dog reacto to accout. Ths assumpto s a break-eve betwee mathematcal feasblty ad the correct descrpto of the macroscopc effects of the mcroscopc pheomeo. Thus, besdes the self-assocato reactos cosdered earler, a sgle cross-assocato reacto: + (45) s cosdered to occur. The compoud s regarded as a ert oe,. e, t wll ot udergo ay other chemcal reacto wth other compouds. The equlbrum costat for the reacto s K : K α γ = = α α γ γ (46) Ths reacto causes chages the mass balace. Therefore, a ew parameter, the dmesoless extet of cross-assocato χ s troduced. χ s defed as the amout of udergog cross-assocato dvded by the overall amout of substace the soluto: χ= = (47) The umerator of ths equato ca be substtuted usg the amout of the assocatg speces,. e, speces cotag oly segmets, resultg : χ= = (48) The values of χ ad K are related to each other, as wll be see later. The followg relatos hold for the composto: =ξχ (49) =ξ(x χ ) (50) The value of s to be obtaed through solvg the chemcal equlbrum. I order to adapt the model, the secod hypothess, preseted earler, s exteded: the pure compoud UNIQUC parameters for the ew speces are: q = q + q (5) r = r + r (5) lso the hypothess that the solvato does ot modfy the value of the teracto parameters for the ert compoud s made, leadg to a = a, a, = a ad,,,,,,, k, a = a = a = a = a = 0. These expressos are obvously a smplfcato of the actual problem. I prcple, t s possble to accout for the dffereces betwee the dstct speces usg the defto of a as the dfferece of teracto eerges; however, t would complcate the subsequet developmet, ad other smplfyg assumptos would be ecessary. The defto of aged values r, r, q ad q remas the same, although the expressos of r ad accout for the ew speces: = + + = q are chaged to r r r r (53) ad smlarly for q. s before, the values r ad r, ad q ad q are related through ξ, as gve by equato (3) for the se parameters, for stace. Se ad surface fractos are also calculated through the same expressos. The actvty coeffcet takes dfferet forms accordg to the compoud to whch t s related. I ay case, the sum S m s wrtte as: S m k k = θ τ = k,m ( ) = θ τ + θ +θ τ =, m,m (54) rala Joural of Chemcal Egeerg Vol., No. 03, pp , July - September, 005

8 478 P.. Pessôa Flho, R. S. Mohamed H ad G. Maurer S For a olgomer t becomes: ( ) = θ + θ +θ τ = S = (55) ad for ad : S ( ) (56) = S = θ τ + θ +θ = Substtutg these expressos equato (9), oe gets for the actvty coeffcet of the olgomers: θ r, r q r r q ( ) θ +θ τ = lγ = + l q + l + + q ls ξ r r ξ rq q r S S (57) ad for ad : θ r m r m qm r rm q θ +θ = lγ m = + l q + m l q + + m ls, r r τ ξ rq m qm r S S (58) where m stads for ether or. ga, the actvty of the olgomers ca be wrtte as: r lγ = + l + g (x,x, χξ, ) r ξ (59) where g s ot a fucto of. The use of the chemcal equlbrum relatoshp, equato (40), leads to the same recursve equato (4). However, the procedure to obta ad ξ s slghtly dfferet, as the mass balace has to be adapted. s show appedx, t results : ad: ξ = x (x χ) (x χ) = + x ξ (x χ)r Ke x r + x r Wth these expressos, the cocetrato (60) (6), ad, as well as the actvty coeffcet γ, γ ad γ ca be calculated whe χ s kow. Whle the hypotheses made so far facltated the mathematcal mapulato of the equatos, t s ot possble to elmate the eed of a tral-ad-error soluto for χ. Oe way to do so would be to substtute the expressos for the actvty coeffcets equato (46) ad solve t; however, as there are other equatos beg cosdered (the self-assocato oes), t s dffcult to tell advace whether the soluto to be foud s a actual stable soluto. The other way s to mme the Gbbs eergy of the system. Total Gbbs eergy s gve by: (6) = G = µ + µ + µ The codto of equlbrum betwee the olgomers ( µ =µ ) leads to: G = µ + µ + µ (63) = Itroducg the cocept of actvty gves: rala Joural of Chemcal Egeerg

9 Modfed UNIQUC Equato 479 G = µ + µ + µ =,pure lqud,pure lqud,pure lqud + RT lα + lα + l α = Usg the defto of the extet of cross-assocato extet, the expresso for G ca be rewrtte: ( ),,pure lqud,,pure lqud pure lqud G = µ + µ +χ µ µ µ + RT lα + lα + l α = (64) (65) The stadard chemcal potetals the above equato are related to the equlbrum costats through: ( ) ( ) RT lk ( )lk l() µ µ µ = + + (66) pure lqud The amout of the speces are related to χ through equatos (47) ad (48). The term,,pure lqud, µ + µ s costat for,pure lqud a gve stochometrc composto, ad has o fluece the mmato procedure. Therefore, χ s foud by mmg the followg expresso for a gve stochometrc composto (x ad x ) of a lqud mxture: G m χ (,,pure lqud,,pure lqud ) ( ) ( ) ( ) µ + µ RT = χ lk + ( )lk+ l() + x χ l α + + x χ lα +χlα s.t. 0 χ = (67) For the sake of smplcty, the modfed model takg both self ad cross-assocato to accout s called S-UNIQUC. Phase Equlbrum The prevous dscusso was restrcted to lqud phases. pplyg those results descrbg the vapor-lqud equlbrum requres some addtoal assumptos for the vapor phase. We assume that the vapor s a deal gas of moomers ad speces. Whe the pure lquds (. e, ad ) are chose as the referece state, the codtos for vapor lqud equlbrum result : sat γ P = y P (68) sat γ P = y P (69) sat P s the saturato pressure of a hypothetcal lqud cosstg of moomerc speces oly. sat s ot drectly accessble by expermet as t s ot sat the saturato pressure P above pure lqud compoet (whch s a mxture of moomers ad assocates ). sat P s elmated by cosderg the vapor lqud equlbrum of pure lqud compoet : (pure ) (pure ) sat sat γ P = P (70) where (pure ) ad (pure ) P γ are the amout fracto ad the actvty coeffcet of moomers pure lqud. Itroducg equato (70) to equato (68) gves: γ y P = (pure ) (pure ) sat γ P (7) rala Joural of Chemcal Egeerg Vol., No. 03, pp , July - September, 005

10 480 P.. Pessôa Flho, R. S. Mohamed H ad G. Maurer The composto of the vapor ad the pressure above a lqud mxture of a alcohol ad a ert compoet of gve stochometrc composto at temperature T s the calculated several steps: Calculato of (pure ) ad at temperature T. Calculato of amout fractos (pure ) equatos (43), (44) ad (0). Calculato of actvty coeffcets γ pure lqud from equatos (34) ad (35). Calculato of vapor phase fugactes ad from γ ad γ yp ad yp from equatos (69) ad (7). If cross-assocato has to be accouted for, the procedure s smlar, except for the fact that solvato equlbrum must be calculated pror to phase equlbrum, through solvg the mmato problem, equato (67). Obtag the Parameters The model requres three parameters to charactere a alcohol : se parameter r, surface parameter q ad the assocato equlbrum costat K, equato (38). We assume that K depeds o temperature through: 0 lk = 0 + (7) T 0 ad 0 are obtaed a procedure adopted from rada (983), usg expermetal data for sat the saturato pressure of the alcohol P ad the saturato pressure of a ert hydrocarbo havg (early) the same molecular mass ( P homomorph sat ). The procedure cossts of mmg the followg obectve fucto: k = 0 0 O.F.( S, H) = (73) = P (T ) (T ) γ (T )P (T ) sat k * k * k sat k alcohol homomorph For a bary mxture of a alcohol ad a ether assocatg or cross-assocatg compoet the model further requres two bary teracto parameters (a ad a ). These are obtaed as usual by fttg bary vapor-lqud equlbrum data. I the preset work the followg obectve fucto s used: O.F.(a,a ) = (74) = P (T,x,x ) P (T,x,x ) k = calc k k k exp k k k bubble bubble Whe cross-assocato betwee a ter speces ad a assocate has to be take to accout, the chemcal equlbrum costat K, equato (46) has also to be determed from a exteso of equato (74): O.F.(a,a,K ) = k = = P (T,x,x ) P (T,x,x ) calc k k k exp k k k bubble bubble (75) The depedecy of K o the temperature s wrtte the usual patter for equlbrum costats: lk = + (76) T The bary teracto parameters, regardless whch equato they refer to, were cosdered to be depedet of the temperature. Self-ssocato RESULTS The equatos prevously developed were appled for the determato of the parameter 0 ad 0 of equato (7) for fve alcohols; these values are gve Table. rada (983) also reported umercal values for 0 ad 0, whch are, for comparso purposes, also gve Table.The dfferece the values of 0 s small ad s maly due to the dfferet rage of temperature for whch vapor pressure data were used. The dfferece the value 0 s larger, due to the dstct stadard states chose. The model was subsequetly used to calculate the vapor-lqud equlbrum of sx bary systems of a alcohol ad a paraff at sevetee temperatures. The mea relatve errors the bubble pot pressure calculated wth -UNIQUC were compared to those obtaed wth the UNIQUC Gbbs excess eergy model (wthout takg assocato to accout) Table. The parameters of both equatos are gve Table C- of ppedx C. The correlato wth -UNIQUC always produced a better agreemet wth expermetal data tha rala Joural of Chemcal Egeerg

11 Modfed UNIQUC Equato 48 obtaed wth the UNIQUC. The relatve dfferece the bubble pot pressure was reduced by a mmum of almost 0% (ethaol / methylcyclohexae system) ad by a maxmum factor of eght (for the ethaol / heptae system). The composto of the vapor phase calculated usg the -UNIQUC equato s also closer to the expermetal oe (Table ). Whe assocato was eglected, the correlato predcted a lqud-lqud phase separato, e. g., for the systems ethaol / hexae, ethaol / octae ad -propaol / cyclohexae. Takg assocato to accout avoded ths false predcto ad resulted a better agreemet wth expermetal data. s both - UNIQUC ad UNIQUC requred two adustable parameters, the mprovemet correlato was a result of the explct cosderato of the selfassocato of the alcohol. Table : Parameters of the equlbrum costat of self-assocato. Comparso wth data from rada (983). Compoud T m / K T max / K 0 0 / K 0 # ethaol propaol propaol butaol petaol * Referece for vapor pressure data of alcohols: Smth ad Srvastava (986b) ad of homomorph: Smth ad Srvastava (986a) # ata from rada (983) 0 # / K Table : Comparso of expermetal data for the vapor pressure above bary mxtures of a alcohol ad a paraff wth calculatos for -UNIQUC ad UNIQUC equato. * P/ P y System T / K Ref. exp. data # -UNIQUC UNIQUC -UNIQUC UNIQUC ethaol / hexae ethaol / cyclohexae ethaol / heptae ethaol / methylcyclohexae ethaol / -octae propaol/ cyclohexae * calc exp P 00 P P = P N exp = P, calc exp y= N y, y,, = # Leged:. Gmehlg ad Oke (977),. Gmehlg et al. (98a). Solvato I lqud mxtures of a alcohol ad a aromatc hydrocarbo, such as beee or toluee, the aromatc rg acts as a electro door, ad hydroge bodg would occur betwee the aromatc rg ad the alcohol. lthough ths cross-assocato s weaker tha the selfassocato of a alcohol, t would also eed to be cosdered Gbbs excess eergy models. I the preset work solvato betwee a alcohol ad beee or toluee was take to accout as dscussed before. s the chemcal costat for the reacto could ot be estmated from pure compoet data, t had to be troduced as a thrd adustable parameter. alteratve way to take selfassocato to accout was gve by Kretschmer ad Webe (954) for the system ethaol / toluee. s cross-assocato reduces self-assocato, Kretschmer ad Webe used for the chemcal equlbrum costat for the self-assocato of ethaol toluee a much smaller value tha for ethaol a ert hydrocarbo. rala Joural of Chemcal Egeerg Vol., No. 03, pp , July - September, 005

12 48 P.. Pessôa Flho, R. S. Mohamed H ad G. Maurer The S-UNIQUC model was appled to ft VLE data of four mxtures of a alcohol (ethaol, - propaol, -butaol) wth beee or toluee at eleve temperatures. s beee ad toluee have oly a sgle electro door ste, oly crossassocato betwee oe aromatc rg ad oe alcohol molecule or olgomer was expected. The fluece of temperature o cross-assocato was descrbed by equato (76). Thus, whe temperature vares, the S-UNIQUC model has four adustable parameters, plus the age umber of the olgomer cross-assocated. The parameters were obtaed by fttg the bubble pot pressure data of the four bary systems. s the parameter ca assume oly teger values, t was vared depedetly; all cases studed, the optmum value was foud to be oe. These parameters are gve Table C- of appedx C. The relatve mea devato betwee calculated ad expermetal values for the bubble pot pressure ad the absolute mea devato the vapor phase composto are gve Table 3. The parameter of equato (76) for the crossassocato s always smaller tha the same parameter for the self-assocato of a alcohol, as ca be see Table 4. s s drectly related to the assocato ethalpy, ths fdg s agreemet wth the expectato that the absolute value for the self-assocato ethalpy should be larger tha the absolute value for the cross-assocato ethalpy. Table 3: Comparso of expermetal data for the vapor pressure above bary mxtures of a alcohol ad a cross-assocatg hydrocarbo wth calculatos for S-UNIQUC, -UNIQUC ad UNIQUC equato. System T / K P / P y S-UNIQUC -UNIQUC UNIQUC S-UNIQUC -UNIQUC UNIQUC ethaol / beee ethaol / toluee propaol / beee butaol / toluee # Leged:. Gmehlg ad Oke (977),. Gmehlg et al. (98a). 3. Gmehlg et al. (98b). Ref # Table 4: Values for ad. System / K ethaol / beee.55 - ethaol / toluee propaol / beee butaol / toluee Table 3 also gves a comparso of the results obtaed whe correlatg the bubble pot pressure data usg UNIQUC (. e, wthout ay assocato) to those obtaed usg -UNIQUC (. e, wth selfassocato oly) ad S-UNIQUC. s expected, S-UNIQUC gves the best agreemet wth the expermetal data. However, ths case the mprovemet s ot as remarkable as for alcohol / paraff mxtures, as the UNIQUC equato already results reasoable accuracy. Takg selfassocato to accout, but eglectg crossassocato (. e, usg -UNIQUC) results comparatvely large devatos betwee the calculated ad the expermetal data. Fgure shows the extet of the crossassocato reacto χ as a fucto of cocetrato for the -propaol / beee system at two dfferet temperature. I ths fgure, the value of χ /x (. e, the fracto of hydrocarbo molecules that udergo solvato) s plotted versus x (the stochometrc amout fracto of the alcohol). It ca be verfed that the extet of cross-assocato s ot large: whle eglectg t worses sgfcatly the correlato, ts low value may be the resposble for the fact that oly oe alcohol molecule s calculated to be assocated to each electro-door ste. The extet of cross-assocato creases wth creasg amout fracto of the alcohol, passes rala Joural of Chemcal Egeerg

13 Modfed UNIQUC Equato 483 through a maxmum ad decreases to a early cocetrato depedet umber that creases wth creasg temperature. Ths s the result of two couteractg effects: at hgher temperatures there s a lower extet of self-assocato (. e, a hgher amout fracto of alcohol moomers, Fgure ), whch leads to a hgher extet of solvato. Ths hgher extet overcompesates the otherwse reducg effect of temperature o solvato. ccoutg for cocurret reactos also expla the exstece of a maxmum for χ /x at lower alcohol amout fractos, as ca be verfed Fgure 3 the al fracto of self-assocato f, defed as the rato of the calculated umber of hydroge bods to the maxmum allowed, s preseted for the same system: both self- ad crossassocato reactos experece a steep crease for lower cocetratos of alcohol χ/ x beee K 38.5K K K x -propaol Fgure : Extet of cross-assocato reacto the lqud phase of the system -propaol / beee x -propaol Fgure : mout fracto of moomers of -propaol as a fucto of the stochometrc amout fracto of -propaol beee K 333.5K 0.6 f x -propaol Fgure 3: Total fracto of self-assocato the lqud phase of the system -propaol / beee. rala Joural of Chemcal Egeerg Vol., No. 03, pp , July - September, 005

14 484 P.. Pessôa Flho, R. S. Mohamed H ad G. Maurer CONCLUSIONS The effect of lear cha self-assocato ad solvato was corporated to the UNIQUC model for Gbbs eergy a straghtforward way, wthout postulatg ay expresso for the equlbrum costat, but usg stead the actvty of the olgomers as they are obtaed from the UNIQUC equato to calculate the chemcal equlbrum. The pure self-assocato model thus developed preseted good results for the correlato of VLE data of alcohol / alkae mxtures, beg able to correlate systems at varous temperatures wth expermetal ucertaty. For systems cotag a alcohol ad a aromatc hydrocarbo a crossassocato reacto was cosdered; t was foud that the correlato was also excelet ths case, wth low devatos from expermetal data, otwthstadg the fact that the extet of crossassocato was calculated to be small. Lat Letters NOMENCLTURE moomer of a self-assocatg compoet 0, parameters the equato of the equlbrum costat olgomer of moomors of, =,,... a, bary UNIQUC teracto parameter betwee stes of compoet ad. actve compoud 0, parameters the equato of the equlbrum costat c cocetrato of speces (amout of substace per volume) ert compoud e f dmesoless extet of self-assocato g fucto defed by equato (39) G Gbbs eergy [J.mol - ] G al Gbbs eergy of a sample [J] H ethalpy [J.mol - ] umber of moomers a multmer umber of moomers a multmer K equlbrum costat of solvato reacto K chemcal reacto equlbrum costat for the formato of ; =, 3, 4... l parameter the UNIQUC equato defed by equato () ñ al amout of compoets [mol] ñ overall amout of compoet [mol] ñ or mcroscopc amout of compouds [mol] ñ mcroscopc al amout of speces [mol] N umber of compoets N umber of speces P pressure [Pa] sat P saturato pressure of speces [Pa] q surface parameter of the UNIQUC equato q age surface parameter o stochometrc amout fracto bass q age surface parameter o mcroscopc amout fracto bass R uversal gas costat [8.34 J.mol -.K - ] r se parameter of the UNIQUC equato r age se parameter o stochometrc amout fracto bass r age se parameter o mcroscopc amout fracto bass S etropy [J.mol -.K - ] S parameter the UNIQUC equato defed by equato (30) T thermodyamc temperature [K] V volume per amout of substace [L.mol - ] x stochometrc amout fracto of compoet y amout fracto of compoet the vapor phase or mcroscopc amout fracto of speces umber of earest eghbors the lattce Greek Letters α actvty of compoet χ dmesoless extet of reacto, defed by equato (47) ξ = ñ / ñ γ actvty coeffcet of compoet surface fracto of compoet θ ϕ volume fracto of compoet µ chemcal potetal of compoet [J.mol - ] exp( a /T) τ Subscrpts age moomer of self-assocatg compoet rala Joural of Chemcal Egeerg

15 Modfed UNIQUC Equato 485 s Superscrpts olgomer speces cosstg of moomors of compoet ; =,..., ert compoet or ert speces solvato al c based o cocetrato calc calculated E excess exp expermetal L lqud sat saturato x stochometrc mcroscopc 0 stadard ϕ based o volume fracto * pure alcohol REFERENCES spro, N., Hasse, H. ad Maurer, G., Thermodyamc ad IR Spectroscopc Studes of Solutos wth Smultaeous ssocato ad Solvato, Flud Phase Equlbra, 08, 3 (003). rada, V., Cotuous Lear ssocato Model for etermg the Ethalpy of Hydroge-od Formato ad the Equlbrum ssocato Costat for Pure Hydroge-oded Lquds, Flud Phase Equlbra,, 87 (983). rada, V., ad Evagelsta, F., The UNIQUC ssocated-soluto Theory: Vapor-Lqud Equlbra of ary Systems Cotag oe ssocatg ad oe Iert or ctve Compoet, Flud Phase Equlbra, 7, 8 (984). Che, J., M, J. G., Yu, Y. M., ad Luo, G. S., alytcal Equato of State for Water ad lkaols, Chem. Eg. Sc., 59, 583 (004). rowark,., Phase-equlbrum Calculatos for - lkae plus lkaol Systems Usg Cotuous Thermodyamcs, Flud Phase Equlbra, 7, 5 (004). Flory, P. J., Thermodyamcs of Hgh Polymer Solutos, J. Chem. Phys., 0, 5 (94). Fu, Y. H., Sadler, S. I., ad Orbey, H., Modfed UNIQUC Model that Icludes Hydroge odg, Id. Eg. Chem. Res., 34, 435 (995). Gmehlg, J. ad Oke, U., Vapor-Lqud Equlbrum ata Collecto, Chemstry ata Seres, v., part a. echema, ortmud (977). Gmehlg, J., Oke, U. ad rlt, W., Vapor-Lqud Equlbrum ata Collecto, Chemstry ata Seres, v., part c. echema, ortmud, (98a). Gmehlg, J., Oke, U. ad Wedlch, U., Vapor- Lqud Equlbrum ata Collecto, Chemstry ata Seres, v., part d. echema, ortmud (98b). Het,., olch, E. ad Lchtethaler, M., New Expermetal VLE-ata for lkaol-lkae Mxtures ad ther escrpto by a Exteded Real ssocato (ERS) Model, Flud Phase Equlbra, 7, 6 (986). Hofma, T., Thermodyamcs of ssocato of Pure lcohols, Flud Phase Equlbra, 55, 39 (990). Kempter, H. ad Mecke, R., Spektroskopcsche estmmug vo ssoatos-glechgewtche, Z. Phys. Chem., 46, 9 (940). Kretschmer, C.. ad Webe, R., Thermodyamcs of lcohol-hydrocarbo Mxtures, J. Chem. Phys.,, 697 (954). Megarell,. C., rgole, E.. ad ott, S.., ctvty Coeffcets of ssocatg Mxtures by Group Cotrbuto, Flud Phase Equlbra, 63, 95 (999). Nagata, I., O The Thermodyamcs of lcohol Solutos. Phase Equlbra of ary d Terary Mxtures Cotag y Number of lcohols, Flud Phase Equlbra, 9, 53 (985). Nagata, I., Kawamura, Y., Excess Thermodyamc Fuctos of Solutos of lcohols wth Saturated-Hydrocarbos - pplcato of UNIQUC Equato to ssocated Soluto Theory, Z. Phys. Chem. Neue Folge 07, 4 (977). Nath,. ad eder, E., O the Thermodyamcs of ssocated Solutos. I. alytcal Method for etermg the Ethalpy ad Etropy of ssocato ad Equlbrum Costat for Pure Lqud Substaces, Flud Phase Equlbra, 7, 75 (98a). Nath,. ad eder, E., O the Thermodyamcs of ssocated Solutos. II. Vapor-Lqud Equlbra of ary Systems wth oe ssocatg Compoet, Flud Phase Equlbra, 7, 89 (98b). Nath,. ad eder, E., O the Thermodyamcs of ssocated Solutos. III. Vapor-Lqud Equlbra of ary ad Terary Systems wth y Number of ssocatg Compoets, Flud Phase Equlbra, 0, 43 (983). Pessôa Flho, P.. ad Mohamed, R. S., Chemcal Theory ased Equato of State for Self- ssocatg Compouds, Thermochmca cta 38, 65 (999). rala Joural of Chemcal Egeerg Vol., No. 03, pp , July - September, 005

16 486 P.. Pessôa Flho, R. S. Mohamed H ad G. Maurer Reo, H. ad Praust, J..M., O the Thermodyamcs of lcohol-hydrocarbo Solutos, Chem. Eg. Sc.,, 99 (967). Smth,.. ad Srvastava, R. Thermodyamc ata for Pure Compouds, Part : Halogeated Hydrocarbos ad lcohols. Elsever, msterdam (986b). Smth,.. ad Srvastava, R., Thermodyamc ata for Pure Compouds, Part : Hydrocarbos ad Ketoes. Elsever, msterdam (986a). Yu, M., Nshum, H. ad ros, J. S., Thermodyamcs of Phase Separato queous Solutos of Polymers, Flud Phase Equlbra, 83, 357 (993). PPENIX pplyg the recursve formula, equato (4), gves: ( x ) Kr e =ξ rξ (7) Kr e = r ξ ( ) from whch oe obtas: () = () Kr e rξ ad: (3) = Kr e r ξ From equato (0): =ξ ( ) =ξ x (4) ad: = = ξ x (5) Relatg equatos () to (5), ad (3) to (4), oe obtas a set of two equatos wth the ukow ad ξ: ( x ) Kr e = ξ rξ (6) vdg the square of equato (6) by equato (7), ad recallg that x = x, gves: ξ = x x (8) The expresso for ξ s obtaed by sertg equato (8) to equato (6). fter some rearragemet, oe gets: x x e Kr x r x = (9) whch s a quadratc equato that ca be solved by meas of usual algebra. The oly meagful soluto s obtaed by cosderg that for x = 0, x = ad ξ= : e / r + + 4Ke x r = + x ξ r K r (0) I order to avod umercal problems whe Ke 0 ad the self-assocato vashes, rearragemet s made resultg equato (44). Wth some varatos, ths form of equato s usually foud chemcal-theory based models. rala Joural of Chemcal Egeerg

17 Modfed UNIQUC Equato 487 PPENIX The procedure to be followed to obta ad ξ whe both self ad cross-assocato are cosdered s slghtly dfferet from that preseted appedx. Equatos () to (3) rema the same, but ther relato wth overall quattes s chaged. I ths case, from equatos (0), (49) ad (50): From equatos (3) ad (): Kr e rξ = (x χξ ) fter smlar rearragemet: (4) = ξ x () From equatos (9) ad (48): χal al al = = = (x χξ ) From equatos () ad (): Kr e rξ = ξx () (3) ξ = x (x χ) (5) I order to obta the value of ξ, equatos () ad (5) are substtuted equato (), leadg to: x (x χ) Kr x r (x χ) e = whose oly meagful soluto s equato (6). (6) PPENIX C Table C-: Parameters for -UNIQUC (pure self-assocato model) ad UNIQUC equatos. System T / K -UNIQUC UNIQUC a / K a / K a / K a / K ethaol / hexae ethaol / heptae ethaol / cyclohexae ethaol / methylcyclohexae ethaol / -octae propaol/ cyclohexae Table C-: Parameters for S-UNIQUC ad UNIQUC equatos. System T / K S-UNIQUC UNIQUC a / K a / K a / K a / K ethaol / beee ethaol / toluee propaol / beee butaol / toluee rala Joural of Chemcal Egeerg Vol., No. 03, pp , July - September, 005