Computer simulation of vapor-liquid equilibria of linear dipolar fluids: Departures from the principle of corresponding states

Compu e simula ion o apo -liquid equilib ia o linea dipola luids: Depa u es om he p inciple o co esponding s a es Beni o Ga zo ´n, San iago Lago,a) and Ca los Vega Depa amen o de Quı ´mica Fı ´sica, Facul ad de Ciencias Quı ´micas, Uni e sidad Complu ense de Mad id. 28040 Mad id, Spain Luis F. Rull Depa amen o de Fı ´sica A o ´mica Molecula y Nuclea , Uni e sidad de Se illa, Ap do 1065, Se illa 41080, Spain ~Recei ed 14 No embe 1994; accep ed 25 Janua y 1995! Liquid- apo equilib ium o linea dipola luids has been de e mined by using he Gibbs ensemble simula ion echnique. Se e al elonga ions and alues o he dipole momen we e conside ed. Dipole momen inc eases he c i ical empe a u e and a ec s sligh ly he c i ical densi y and p essu e. Comp essibili y ac o a he c i ical poin dec eases as he dipole momen o he molecule inc eases. Dipole momen p o okes de ia ions om he p inciple o co esponding s a es. I is shown ha he empe a u e-densi y coexis ence cu e is b oadened and ha he slope o he apo p essu e cu e inc eases wi h inc easing dipole momen . We p opose a new way o educing he dipole momen so ha he inc ease o he c i ical empe a u e becomes almos independen on he molecula elonga ion. We ha e also ob ained he apo -liquid equilib ium o models ha ing bo h a dipole and a quad upole momen . The ob ained da a we e used o desc ibe he beha io o some ela i ely complex luids, namely, 1,1,1- i luo oe hane and 2,2,2- i luo oe hanol. Good ag eemen o coexis ence densi ies and p essu es was ob ained. The esul s p esen ed in his wo k o linea dipola luids along wi h p e ious wo k on linea quad upola luids p o ide a e y comp ehensi e iew o he e ec o pola o ces on he apo -liquid equilib ium o linea luids. © 1995 Ame ican Ins i u e o Physics. I. INTRODUCTION The p inciple o co esponding s a es is one o he mos use ul concep s in liquid s a e heo y. This p inciple i s enuncia ed by an de Waals in 1873 s a es ha he equa ion o s a e ~EOS!o a luid when educed by he c i ical p op- e ies is he same o all subs ances. This p inciple is qui e success ul o desc ibing he beha io o sphe ical o quasi- sphe ical molecules. A molecula de i a ion o his p inciple was ca ied ou by Pi ze 1and Guggenheim.2Howe e , i was soon clea ha his p inciple is only app oxima e and does no hold o all kind o subs ances. De ia ions om he p inciple o co esponding s a es3we e clea ly isible in sub- s ances ha ing sho - ange epulsi e aniso opic o ces ~non- sphe ical shape!and in luids p esen ing long- ange a ac- i e o ces p o oked by mul ipole momen s. F om an empi ical poin o iew, de ia ions om he p inciple o co esponding s a es a e usually desc ibed by he acen ic ac o , in oduced by Pi ze e al.4Howe e , i is clea ha a molecula unde s anding o he o igin o he de- ia ions om he p inciple o co esponding s a es should be p e e able. The ole o he molecula shape on apo -liquid equilib- ium ~VLE!o linea and simple nonlinea luids is now well unde s ood. Pe u ba ion heo ies o nonpola linea sys ems ha e been de eloped du ing he las decade o he wo- cen e Lenna d-Jones model5–7 and o he Kiha a model.8–10 These heo ies we e able o desc ibe he e ec o molecula aniso opy on VLE. Recen Gibbs ensemble Mon e Ca lo ~GEMC!simula ions o he VLE o Gay-Be ne,11,12 Kiha a,13 and wo-cen e Lenna d-Jones14 luids ha e comple ed he pic u e o how he molecula shape p o okes depa u es om he p inciple o co esponding s a es. The nex s ep is o y o unde s and he ole o pola o ces on VLE o molecula luids. Wi h ha pu pose we ha e ecen ly pe o med Gibbs ensemble simula ions o de- e mine he VLE o linea quad upola luids.15 F om his s udy we lea ned he e ec o a quad upole momen on he VLE o a linea luid. The nex na u al s ep is o analyze he e ec o a dipole momen on he coexis ence p ope ies o a linea luid. This is he pu pose o he p esen wo k. P e ious wo k conce ning he e ec o a dipole momen on VLE should be men ioned. The e ec o a dipole momen on he coexis ence p ope ies o a sphe ical model has been s udied by GEMC.16–19 Howe e , sphe ical molecules wi h a pe ma- nen dipole momen a e no commonly ound in na u e. Typi- cally, molecules ha ing a pe manen dipole momen p esen also a nonsphe ical shape. Lupkowsky and Monson ha e de- eloped a pe u ba ion heo y o he wo-cen e Lenna d- Jones model wi h an embedded dipole momen .20 Mo eo e , Dubey e al. ha e s udied by compu e simula ion he VLE o his model and ound good ag eemen wi h he heo e ical p edic ions.21 In his wo k we ollow his line o wo k and we s udy he coexis ence p ope ies o a linea luid wi h a pe manen dipole momen . We choose he Kiha a po en ial o desc ibe he molecula shape and a dipole–dipole in e ac ion e m is added. The choice o he Kiha a po en ial p esen s some ad an ages, in pa icula , ha he VLE o bo h he non- pola and quad upola Kiha a luids has been p e iously ob- ained om simula ions.13,15 In his way we a e able o dis- a!Au ho o whom co espondence should be add essed. 7204 J. Chem. Phys. 102 (18), 8 May 1995 0021-9606/95/102(18)/7204/12/$6.00 © 1995 Ame ican Ins i u e o Physics cuss he e ec o he dipole momen on he coexis ence p ope ies which is common wi h he wo k o Dubey e al.21 Mo eo e , we can also discuss di e ences and simila i ies be ween he e ec o a dipole momen on VLE wi h espec o he e ec o he quad upole momen o wi h espec o he ole o he molecula shape. The ob ained esul s may be use ul no only in p o iding an unde s anding o he ole o pola o ces on coexis ence p ope ies bu desc ibing he apo -liquid equilib ium o eal luids as well. In an a emp o asses he abili y o he dipola Kiha a model, we ha e applied ou simula ions o desc ibe he VLE o a luid o echnical in e es like he e ige an 1,1,1- i luo oe hane. Finally we shall conside he apo -liquid equilib ium o linea models ha ing bo h a dipole and a quad upole mo- men . The beha io o hese sys ems will be compa ed wi h ha o a pu ely dipola o a pu ely quad upola model. Tha allows o s udy he addi i i y o di e en mul ipole momen s on he apo -liquid coexis ence p ope ies o a gi en luid. Simula ion da a o a dipola model wi h quad upole we e used o desc ibe he apo -liquid coexis ence p ope ies o a complex luid like 2,2,2- i luo oe hanol. The scheme o he pape is as ollows. In Sec. II he molecula model and simula ion me hod a e desc ibed. In Sec. III he esul s o he simula ions, and he ob ained e- sul s o models a e desc ibed. The in luence o he dipole momen upon he coexis ence p ope ies, c i ical pa ame e s and depa u es om he p inciple o co esponding s a es is analyzed and a compa ison wi h expe imen al esul s o 1,1,1- i luo oe hane is also gi en. Sec ion IV p esen s he esul s o models ha ing bo h a dipole and a quad upole momen , and hei applica ion o he desc ip ion o VLE o 2,2,2- i luo oe hanol. Conclusions o his wo k a e p esen ed in Sec. V. II. SIMULATION METHOD Le us conside a dipola linea luid consis ing o ods o leng h Lwi h an embedded poin dipole m , in e ac ing h ough a po en ial gi en by u~ , 1, 2!5uK~ , 1, 2!1u mm ~ , 1, 2!,~1! whe e is he dis ance be ween he cen e s o mass o he molecules and i[ $ u i, i % s ands o he pola angles o molecule iwi h espec o a e e ence ame ha ing i s pola axis aligned along he cen e o mass sepa a ion ec o , .uK is he Kiha a po en ial,22 gi en by uK~ , 1, 2!54 e F S s ~ , 1, 2! D 12 2 S s ~ , 1, 2! D 6 G ~2! and u mm is he dipole–dipole po en ial,23 u mm ~ , 1, 2!5 m 1• m 2 323~ m 1• !~ m 2• ! 5.~3! In Eq. ~2!, ~ , 1, 2!is he sho es dis ance be ween he molecula co es ~see Fig. 1!, e is an ene ge ic pa ame e and s a size pa ame e . In Eq. ~3!, m iis he dipole ec o loca ed in he cen e o molecule i, aligned wi h he molecu- la axis. When L50 and m 50, he po en ial unc ion gi en by Eqs. ~1!–~3! educes o he well known Lenna d-Jones po en ial. When L50 and m Þ0 he luid unde conside a ion is he S ockmaye luid.23 Finally, i LÞ0 and m 50, we ha e a Kiha a luid. The Kiha a po en ial is a eliable model o desc ibing he modynamic beha io o luids. I has been used o de- sc ibe he gas,24 liquid,25–28 and solid phases29 o eal sub- s ances. Howe e , we should ecognize a his poin ha he Kiha a dipola model is somewha a i icial in one espec . Linea dipola luids a e usually made up by he e onuclea dia omic molecules ~ o ins ance, HCl!. The use o he Ki- ha a po en ial gi en by Eq. ~1!implies a sphe ocylinde -like co e ha is adequa e only when he wo a oms o g oups o ming he molecule ha e simila sizes. Howe e , he use o his model p esen s an impo an ad an age. Since he mo- lecula co e used is he same as in p e ious wo k on nonpola 13 and quad upola 15 models any di e ence in he beha io o he dipola model will be exclusi ely a ibu ed o he dipole momen . E alua ion o he Kiha a po en ial equi es he calcula- ion o he sho es dis ance be ween wo linea ods. This seems a e y ime consuming ask, bu e y e icien algo- i hms o i s de e mina ion a e a ailable,28,30–32 so ha he compu e ime expended in he e alua ion o he Kiha a po- en ial be ween wo linea ods is simila o he ime equi ed o e alua e he wo-cen e Lenna d-Jones in e ac ion be ween wo molecules. To simula e a dipola luid, one has o deal wi h he long- ange dipola in e ac ions. Two me hods ha e been de- signed o deal wi h long- ange e ec s in o simula ions o pola luids: he Ewald summa ions ~EW!me hod33 and he eac ion ield ~RF!app oach.34 In he EW me hod, he cen al simula ion box is su ounded by an in ini e numbe o epli- cas. To conside he long- ange dipola in e ac ions, la ice ec o sums a e aken o e sphe ical shells o an in ini e sphe ical la ice su ounded by a con inuum. The RF ap- FIG. 1. Sho es dis ance be ween wo linea ods o leng h L. 7205Ga zo ´n e al. : Vapo -liquid equilib ia in dipola luids J. Chem. Phys., Vol. 102, No. 18, 8 May 1995 p oach eplaces pa icles beyond a cu o dis ance by a di- elec ic con inuum. The e ec o his con inuum is aken in o accoun by including a new e m in o he dipola pai po en- ial. In p e ious wo k he VLE o dipola luids has been de e mined wi h he GEMC echnique in combina ion wi h he EW me hod. Howe e , he RF echnique can be imple- men ed in GEMC simula ions yielding compa ible esul s. We ha e ecen ly pe o med19 GEMC o he S ockmaye 23 model ~Lenna d-Jones1dipole!by using he RF echnique. Fo his po en ial model GEMC simula ions using he EW me hod a e also a ailable.16–18 Coexis ence densi ies and p essu es and c i ical magni udes ob ained om bo h me h- ods we e undis inguishable.19 Tha p o es ha bo h RF and EW can be implemen ed in GEMC simula ions yielding iden ical esul s. In his wo k we shall use he RF echnique o accoun o long- ange o ces since i is simple and less compu a ionally demanding han he EW echnique. This is an impo an ac o since he simula ions e en in he absence o dipola o ces a e al eady qui e demanding om a com- pu a ional poin o iew. Wi hin he RF geome y, he dipola pai in e ac ion po- en ial is35 uRF mm ~ , 1, 2!5 5 m 1• m 2 323~ m 1• !~ m 2• ! 522~ e RF21! 2 e RF11 3 m 1• m 2 c 3, , c 0, > c~4! whe e cis he cu o dis ance and e RF he dielec ic cons an o he con inuum. To de e mine he VLE o dipola linea Kiha a mol- ecules we use he Gibbs ensemble Mon e Ca lo simula ion echnique. This me hod, de eloped by Panagio opoulos,36 al- lows he di ec de e mina ion o he coexis ence cu e, simu- la ing simul aneously bo h phases. A mo e de ailed desc ip- ion o his me hod can be ound in he o iginal pape s.36,37 We ha e ob ained VLE o linea dipola Kiha a luids o educed leng h L*5L/ s 50.3, 0.6 and 0.8 using he Gibbs ensemble echnique. F om a p e ious wo k,13 we know he coexis ence cu e o hese sys ems o m 50. GEMC simula- ions o 512 molecules we e pe o med. A empe a u es close o he c i ical poin , he ini ial con igu a ion was aken om an a -N2la ice, wi h 256 molecules in each box. A lowe empe a u es, inal con igu a ions om p e ious uns we e used. The Kiha a in e ac ion was unca ed a 53 s and long- ange co ec ions we e applied by ha he luid was uni o m beyond he cu o .38 The long- ange dipola in e ac- ion was conside ed wi hin he RF geome y by including a RF e m in o he pai po en ial @see Eq. ~4!#, plus he addi ion o a RF sel - e m as long- ail co ec ion.39 The dipola in e - ac ion was unca ed a c53 s 1L, and he RF dielec ic cons an was se equal in bo h phases: e RF~liquid!5 e RF~ apo !5`. I has been p o ed ha his ap- p oach does no a ec o he coexis ence p ope ies.19 To ob ain a poin o he coexis ence cu e, we pe o med 3000– 6000 s eps o equilib a ion plus 4000–8000 s eps o a e - ages. A s ep consis s o one a emp o mo ing each pa icle in bo h phases, ollowed by one a emp o changing he olume and Nex a emp s o exchanging pa icles be ween he simula ion boxes. Accep ance a ios o pa icle mo es and olume exchanging we e kep in he ange 30–60 % and Nex was chosen o ge an exchange a io o 1–3 %. We ob- ained VLE o T>0.75Tc. To ob ain a poin o he coex- is ence cu e, we need abou 6ho CPU ime on a DEC 3000/600 wo ks a ion. The c i ical empe a u e, Tc *, densi y, nc *, and p essu e, Pc *we e es ima ed by i ing he simula ion da a o he ex- p essions n1 *1ng * 25a1bT*,~5! n1 *2ng *5c S 12T* Tc * D b ,~6! ln P*5d1e T*,~7! whe e n1 *and ng *a e he liquid and apo educed densi ies ~n*5n s 3, wi h nbeing he numbe densi y!,T*5kT/ e is he educed empe a u e, and P*5P s 3/ e is he educed a- po p essu e. Equa ion ~5!is he ec ilinea diame e s law.2In Eq. ~6!, we assumed a c i ical exponen b 51/3, close o he uni e sal alue gi en by he eno maliza ion g oup heo y.40 Equa ion ~7!is he Clausius–Clapey on equa ion o he a- po p essu e.41 We ha e also es ima ed he acen ic ac o , i s de ined by Pi ze e al.,4 52log S P Pc D T50.7Tc 21~8! which is a measu e widely used in Chemical Enginee ing o he depa u es om he p inciple o co esponding s a es. III. RESULTS AND DISCUSSION We ha e ob ained he coexis ence cu e o linea Kiha a luids o he men ioned elonga ions and o wo di e en alues o he educed dipole o each elonga ion. The e- duced dipole, m *2, is de ined as m *25 m 2 es 3.~9! We ha e s udied he ollowing sys ems: L*50.3 and m *251.5, 3; L*50.6 and m *252, 4; and L*50.8 and m *252.3, 4.6. The esul s o simula ions a e p esen ed in Tables I–III, including apo and liquid densi ies and p essu es on he coexis ence cu e o di e en empe a u es. The es ima ed e o s we e ob ained om he s anda d de ia ions o e blocks o 100 s eps. Fo some empe a u es, se e al addi- ional uns we e also pe o med. In Figs. 2, 3, and 4 we compa e he esul s o his wo k o L*50.3, 0.6, and 0.8, espec i ely, wi h he p e ious da a ob ained o he nonpola Kiha a model.13 Table IV shows he c i ical p ope ies ~ empe a u e, densi y, p essu e, pack- ing ac ion, and comp essibili y ac o !as es ima ed om he simula ion esul s making use o Eqs. ~5!–~7!, and an 7206 Ga zo ´n e al. : Vapo -liquid equilib ia in dipola luids J. Chem. Phys., Vol. 102, No. 18, 8 May 1995 es ima e o acen ic ac o s. Resul s o he Lenna d-Jones c i ical p ope ies om Re . 42 a e also shown. These esul s show ha he c i ical empe a u e inc eases signi ican ly as he educed dipole m *2inc eases. The e o e, he boiling empe a u e o a luid inc eases wi h he dipole momen , a ac widely quo ed in gene al chemis y ex books.43,44 Dipole momen does no a ec s ongly he c i i- cal densi y. Fo mode a e dipole momen s he c i ical densi y is almos iden ical wi h he co esponding nonpola model o he same elonga ion. Fo la ge dipole momen s he e is a sligh dec ease o he c i ical densi y. These indings ag ee wi h p edic ions o a an de Waals like heo y o pola luids ecen ly p oposed.45 Mo eo e , he packing ac ion a he c i ical poin h c, de ined as h c5ncVm,~10! emains almos cons an and independen on bo h he dipole momen and he molecula elonga ion. App oxima ely he 16% o he a ailable olume is occupied by molecules a he c i ical poin . In Eq. ~10!,Vmis he molecula olume, Vm5 p 6 s 3~113 2L*!.~11! Quad upole and dipole momen s seem o a ec c i ical densi ies in a di e en way. Quad upole momen inc eases c i ical densi y15 whe eas he dipole momen p o okes a sligh dec ease in he c i ical densi y. C i ical p essu e shows a simila beha io . Fo mode a e dipole momen s he c i ical p essu e is almos iden ical wi h ha o he nonpola model. Fo la ge dipole momen s he e is a sligh dec ease o he c i ical p essu e. In Table IV we p esen alues o he com- TABLE I. Resul s o phase coexis ence p ope ies o linea dipola Kiha a luids o L*5L/ s 50.3 and m *25 m 2/~ es 3!51.5 and 3. All he modynamic p ope ies a e gi en in educed uni s. The numbe s in pa en heses indica e he unce ain y in uni s o he las decimal digi i.e., 0.474~11!means 0.474 60.011. T*ng *Pg *nl *Pl * m *251.5 1.18 0.1102~76!0.0657~35!0.351~23!0.067~29! 1.16 0.0903~41!0.0571~25!0.362~19!0.048~19! 1.14 0.0674~54!0.0490~27!0.352~23!0.041~20! 1.10 0.0627~32!0.0427~14!0.4096~92!0.046~21! 1.075 0.0548~37!0.0384~18!0.425~12!0.034~17! 1.05 0.0434~32!0.0315~18!0.4333~89!0.016~20! 1.025 0.0359~13!0.026 98~86!0.4535~85!0.020~22! 1 0.0333~16!0.024 35~56!0.4690~95!0.016~21! 0.975 0.0273~17!0.0203~12!0.474~11!0.013~30! 0.95 0.021 51~96!0.016 31~61!0.4891~83!0.007~20! 0.925 0.015 83~71!0.012 56~53!0.4978~59!0.004~18! 0.9 0.012 67~74!0.010 30~63!0.5084~66!20.008~27! m *253 1.28 0.0653~15!0.0475~14!0.377~22!0.043~42! 1.26 0.0533~25!0.0410~16!0.386~19!0.021~35! 1.25 0.0447~12!0.036 64~98!0.373~30!0.005~50! 1.225 0.0425~20!0.0337~12!0.412~19!0.024~46! 1.2 0.0358~14!0.029 79~82!0.419~19!20.001~46! 1.175 0.0315~13!0.0261~11!0.444~11!0.016~29! 1.175 0.0327~16!0.026 70~93!0.4358~85!0.003~33! 1.15 0.0288~17!0.0235~13!0.4485~76!20.022~41! 1.125 0.022 57~94!0.019 11~79!0.4681~61!20.002~34! 1.1 0.018 82~87!0.016 18~83!0.4767~80!20.008~29! 1.04 0.011 57~50!0.010 87~54!0.4940~62!20.034~48! 0.975 0.008 65~61!0.007 24~87!0.5255~47!20.029~34! 0.95 0.006 05~22!0.005 47~31!0.5326~52!20.051~33! 0.925 0.004 62~15!0.004 56~36!0.5409~59!20.011~61! TABLE II. Resul s o phase coexis ence p ope ies o linea dipola Kiha a luids o L*5L/ s 50.6 and m *25 m 2/~ es 3!52 and 4. T*ng *Pg *nl *Pl * m *252 1.035 0.0630~24!0.0356~19!0.286~17!0.035~24! 1.025 0.0588~23!0.0343~10!0.299~13!0.041~29! 1.015 0.0531~19!0.0324~14!0.298~12!0.031~19! 1 0.0517~28!0.0306~14!0.311~14!0.032~30! 0.975 0.0383~22!0.0249~12!0.3209~87!0.024~22! 0.95 0.0281~17!0.019 33~82!0.332~10!0.012~23! 0.925 0.022 73~41!0.015 84~41!0.3476~64!0.012~23! 0.9 0.016 62~58!0.012 28~36!0.350~12!20.015~47! 0.875 0.016 33~80!0.011 41~37!0.3671~64!0.006~20! 0.825 0.009 57~30!0.007 10~34!0.3853~38!20.009~15! 0.8 0.007 03~15!0.005 80~30!0.3958~66!20.011~26! m *254 1.21 0.0978~72!0.0446~41!0.240~22!0.041~15! 1.2 0.0904~71!0.0421~40!0.246~37!0.035~25! 1.175 0.0602~83!0.0356~35!0.2705~96!0.024~10! 1.15 0.0466~35!0.300~11!0.292~15!0.025~21! 1.125 0.0382~12!0.025 89~86!0.3120~95!0.014~26! 1.1 0.0337~15!0.2297~71!0.3242~98!0.008~20! 1.075 0.023 11~91!0.017 21~53!0.3297~99!20.001~25! 1.05 0.020 3~11!0.015 16~69!0.3404~70!20.010~23! 1.025 0.018 6~10!0.013 58~60!0.3562~53!20.014~18! 1 0.014 61~55!0.010 94~38!0.3640~53!20.011~23! 0.975 0.010 31~48!0.008 09~31!0.3675~55!20.022~22! 0.95 0.009 25~61!0.007 21~43!0.3835~49!20.016~24! TABLE III. Resul s o phase coexis ence p ope ies o linea dipola Kiha a luids o L*5L/ s 50.8 and m *25 m 2/~ es 3!52.3 and 4.6. T*ng *Pg *nl *Pl * m *252.3 1 0.0575~18!0.0314~11!0.214~28!0.022~20! 0.99 0.0495~42!0.0289~14!0.210~30!0.028~15! 0.985 0.0489~38!0.0279~14!0.222~23!0.024~17! 0.975 0.0436~32!0.02608~93!0.244~11!0.027~16! 0.95 0.0422~22!0.02396~84!0.2706~72!0.026~20! 0.925 0.0308~25!0.0193~11!0.2825~75!0.021~18! 0.9 0.0221~18!0.01461~96!0.2909~92!0.015~23! 0.875 0.0180~11!0.012 04~67!0.3035~59!0.016~17! 0.85 0.016 43~66!0.010 87~47!0.3145~47!0.009~14! 0.825 0.011 36~48!0.007 89~28!0.3227~52!0.002~16! 0.8 0.008 91~89!0.006 27~48!0.3295~35!0.002~15! 0.775 0.007 15~22!0.005 10~19!0.3376~43!20.002~22! m *254.6 1.145 0.0545~25!0.0321~22!0.221~13!0.034~24! 1.135 0.0484~26!0.0298~13!0.233~15!0.042~30! 1.125 0.0548~30!0.0304~19!0.234~14!0.033~18! 1.1 0.0446~22!0.0259~13!0.256~17!0.035~29! 1.075 0.0325~20!0.020 78~84!0.264~11!0.018~12! 1.05 0.300~37!0.0192~19!0.2828~63!0.010~13! 1.025 0.208~24!0.0145~14!0.2872~64!0.008~16! 1 0.0158~11!0.011 38~64!0.2961~59!20.004~15! 0.95 0.011 57~41!0.008 59~41!0.3192~55!0.002~19! 0.925 0.009 10~98!0.007 05~60!0.3243~51!20.006~18! 0.9 0.007 24~58!0.006 09~39!0.3315~63!20.008~20! 7207Ga zo ´n e al. : Vapo -liquid equilib ia in dipola luids J. Chem. Phys., Vol. 102, No. 18, 8 May 1995 p essibili y ac o a he c i ical poin , Zc. Dipole momen educes signi ican ly he alue o Zc. This ag ees wi h ex- pe imen al da a3and esul s o Table IV p o ide an illus a- ion o his ac . Dipole momen does no s ongly a ec ei he c i ical densi y o p essu e. Howe e , i p o okes an impo an inc ease o he c i ical empe a u e. The e o e, he dec ease o Zcwi h he dipole is mos ly due o he inc ease o he c i ical empe a u e. All he esul s p esen ed he e o dipola linea luids ag ee wi h p e ious indings o sphe i- cal dipola models.16–19 The e ec o dipole momen on he apo p essu e is shown in Fig. 5 o L*50.3. Dipole momen dec eases he apo p essu e a a gi en empe a u e. The same e ec is obse ed o he dipola Lenna d-Jones luid.16–19 Acco ding o he Clausius equa ion, which becomes accu a e a low empe a u es, he slope o a ln P* s 1/T*plo is ela ed wi h he apo iza ion en halpy by dln P* d1/T*52 DH N e 52DH *.~12! In Fig. 5 he loga i hm o he educed apo p essu e is ep esen ed s in e se educed empe a u e a di e en e- duced dipole momen s. We can conclude ha since he slope o he lines is almos cons an ha implies, om Eq. ~12!, ha a low empe a u es DH *is ela i ely cons an . We see om Fig. 5 ha DH *inc eases as he dipole is inc eased. This con i ms he idea ~widely quo ed on ex books o chemis y! ha apo iza ion en halpy inc eases when mol- ecules ha e a dipole momen .43,44 In Fig. 6~a! he ela i e a ia ion o he c i ical empe a- u e, DTc/Tc 0 o se e al elonga ions and dipole momen s is shown. The magni ude DTcis de ined as DTc5Tc~ m *!2Tc 0~13! and Tc 0is he c i ical empe a u e o a nonpola model o he same elonga ion. Acco ding o he esul s p esen ed in Fig. 6~a! he same educed dipole momen p o okes la ge changes in c i ical empe a u e as he molecule becomes mo e sphe ical. In ou p e ious wo k on quad upola Kiha a luids,15 we al eady poin ed ou ha molecules wi h di e en elonga ions should be compa ed when hey p esen he same densi y o mul i- pole. In his way we we e able o show ha he inc ease o he c i ical empe a u e as a unc ion o he densi y o quad- upole p esen s uni e sal beha io o , in o he wo ds, is in- FIG. 2. Vapo -liquid coexis ence densi ies o dipola Kiha a luids o L*5L/ s 50.3. The esul s a e gi en in educed uni s T*5kT/ e and n*5n s 3. Da a co esponding o a educed dipole m *25 m 2/~ es 3!53 a e plo ed wi h iangles. Squa es co espond o m *251.5. Ci cles ep esen da a o he nonpola sys em o Re . 13. Lines a e i ings o simula ion da a o Eqs. ~5!and ~6!o main ex . FIG. 3. Same as in Fig. 2, bu o L*5L/ s 50.6. Da a co esponding o a educed dipole m *25 m 2/~ es 3!54 a e plo ed wi h iangles. Squa es co e- spond o m *252. Ci cles ep esen da a o nonpola sys em o Re . 13. FIG. 4. Same as in Fig. 2, bu o L*5L/ s 50.8. Da a co esponding o a educed dipole m *25 m 2/~ es 3!54.6 a e plo ed wi h iangles. Squa es co - espond o m *252.3. Ci cles ep esen da a o non-pola sys em o Re . 13. 7208 Ga zo ´n e al. : Vapo -liquid equilib ia in dipola luids J. Chem. Phys., Vol. 102, No. 18, 8 May 1995 dependen o he molecula elonga ion. He e we shall ollow he same app oach by de ining he educed densi y o dipole, m,as~see Re . 15 o de ails! m25 m 2 e Vm 3/3 .~14! Figu e 6~b!shows ha he magni ude o DTc/Tc 0 ollows now a uni e sal cu e as a unc ion o m2, ega dless o he alue o L*. The uni e sali y o he cu e in Fig. 6~b!sup- po s ou choice o m2 o compa ing luids wi h di e en L*. Consequences o he uni e sali y shown in Fig. 6~b!a e qui e in e es ing. Fo ins ance, i he c i ical empe a u e o a linea Kiha a luid wi h elonga ion L*and m *50 is known (Tc 0), Tcmay be p edic ed o any alue o m *2~m2!,by eading DTc/Tc 0in Fig. 6~b!and sol ing o Tc. The si ua- ion conce ning he e ec o pola o ces on he c i ical em- pe a u e o linea luids can be summa ized as ollows. Fo quad upola luids15 when DTc~ educed by e /k!is plo ed as a unc ion o he educed densi y o quad upole esul s co - esponding o molecules wi h di e en elonga ions also all on a single line. Fo dipola luids when DTc/Tc 0is plo ed as FIG. 5. Loga i hm o he educed apo p essu es, P*5P s 3/ e s he in- e se o educed empe a u e, 1/T*51/(kT/ e ), o L*50.3 and m *250 ~solid ci cles!, m *251.5 ~open iangles!, and m *253~solid iangles!. FIG. 6. ~a!Rela i e a ia ion o he educed c i ical empe a u e @see Eq. ~13!in he ex #as a unc ion o he educed dipole o L*50~ci cles! ~ esul s ob ained om Re s. 16–19!,L*50.3 ~squa es!,L*50.6 ~open i- angles!, and L*50.8 ~open diamonds!.~b!Rela i e a ia ion o he educed c i ical empe a u e as a unc ion o he educed densi y o dipole, m, de- ined in Eq. ~14!. Symbols a e as in ~a!. Lines a e plo ed as a guide o he eye. TABLE IV. C i ical p ope ies o di e en dipola linea luids. h cis he c i ical packing ac ion, de ined in Eq. ~10!,Zcis he comp essibili y ac o a he c i ical poin and is he acen ic ac o , de ined in Eq. ~8!. L* m *2m2Tc *nc *Pc * h cZc 0 0 0 1.310a0.314a0.126a0.164a0.306a20.03a 0.3 0 0 1.114~12!0.219 ~6!0.073~10!0.166 ~5!0.30~4!0.00~12! 1.5 2 1.210~40!0.220~17!0.075~17!0.167~15!0.28~9!0.04~13! 3 4 1.365~48!0.206~13!0.065~15!0.156 ~9!0.23~5!0.06~13! 0.6 0 0 1.000~12!0.161 ~5!0.051~10!0.160 ~5!0.32~6!0.15~16! 2 2 1.090~29!0.167~10!0.050~15!0.166~10!0.28~10!0.15~19! 4 4 1.225~20!0.161~16!0.047~13!0.160~16!0.24~7!0.23~16! 0.8 0 0 0.952~11!0.140 ~3!0.038 ~8!0.161 ~4!0.29~6!0.11~12! 2.3 2 1.026~24!0.143~15!0.037 ~7!0.165~17!0.25~6!0.15~11! 4.6 4 1.179~37!0.137~13!0.038~13!0.158~15!0.24~8!0.15~20! aResul s om Re . 42. 7209Ga zo ´n e al. : Vapo -liquid equilib ia in dipola luids J. Chem. Phys., Vol. 102, No. 18, 8 May 1995 a unc ion o he educed densi y o dipole esul s co e- sponding o molecules wi h di e en elonga ions all on a single line. The sea ch o an explana ion o hese uni e sali- ies should be a challenge o heo ies o molecula pola luids so a p esen ed. In ac we ha e ecen ly p oposed a simple heo y45 explaining some o hese indings bu u he wo k is s ill needed. In Fig. 7 we show he coexis ence cu e, when he em- pe a u e and densi y a e educed by hei co esponding c i i- cal pa ame e s, o wo alues o L*. Al hough he esul s shown in Fig. 7 a e qui e sensi i e o e o s in he de e mi- na ion o c i ical p ope ies, so ha cau ion is needed, we obse e ha end is a b oadening o he VLE coexis ence cu e due o he dipole momen . The e o e, dipola o ces p o oke a b oadening o he coexis ence cu e and his is in common wi h quad upola o ces.15 Depa u es om he p inciple o co esponding s a es in he apo p essu e due o he dipole a e illus a ed by plo s o ln (P/Pc) sT c /T. This is ep esen ed in Fig. 8 o L*50.3 and 0.8. We ha e also ep esen ed he apo p essu e o a Lenna d-Jones luid, aken om Re . 42. Thus, i can be obse ed he e ec ha bo h shape and dipole exe upon de ia ions om co esponding s a es o apo p essu e. The slope ~in absolu e alue!o he ln(P/Pc) cu e inc eases wi h he molecula aniso opy ~i.e., compa e L*50, m *250 wi h L*50.3, m *250!also wi h he dipole momen ~i.e., compa e L*50.3, m *250 wi h L*50.3, m *253!. Al hough hese esul s should be aken wi h ca e due o di icul ies in ob aining accu a ely c i ical magni udes, we belie e his FIG. 7. Reduced coexis ence densi ies, n/nc, as a unc ion o he educed empe a u e, T/Tc o se e al dipola sys ems. ~a!L*50.3 and m *250~solid line!, m *251.5 ~long-dashed line!and m *253~sho -dashed line!.~b! L*50.8 and m *250~solid line!, m *252.3 ~long-dashed line!and m *254.6 ~sho -dashed line!. FIG. 8. Reduced apo p essu es as a unc ion o he in e se o educed empe a u e o L*50, 0.3 and 0.8. ~a!Resul s o ~ om op o bo om! L*50 and m *250, and L*50.3 wi h m *250, 1.5, and 3. ~b!Resul s o ~ om he op o he bo om!L*50 and m *250, and L*50.8 wi h m *250, 2.3 and 4.6. 7210 Ga zo ´n e al. : Vapo -liquid equilib ia in dipola luids J. Chem. Phys., Vol. 102, No. 18, 8 May 1995 end o be co ec . We conclude ha he e ec o a dipole momen on co esponding s a es plo s is simila o ha ound p e iously o quad upola models. An impo an p ope y widely used in Chemical Enginee ing is he acen ic ac o . In he las column o Table IV compu ed alues o a e p esen ed. Al hough a p ecise de e mina ion o om GEMC is a qui e di icul ask ~see he la ge e o ba s o ! some endencies a e clea . Aniso opy and dipole momen inc ease he magni ude o he acen ic ac o . In p e ious wo k15 we ha e shown ha he Kiha a qua- d upola model is a good e ec i e po en ial o desc ibing apo -liquid equilib ium o eal luids. In ac he compu ed coexis ence p ope ies o he model we e in excellen ag ee- men wi h expe imen al esul s o ca bon dioxide. He e we shall illus a e how his is also he case o he Kiha a dipola model. As an example we ake 1,1,1- i luo oe hane ~CH3–CF3!. This choice is mo i a ed by he ac ha he size o F is simila o ha o H ~Re . 46!and ha makes ou choice o he molecula shape ~a sphe ocylinde ! easonable. Mo eo e , his molecule p esen s a dipole momen aligned wi h he C–C bond which is in common wi h ou model. C i ical empe a u e and densi y ob ained om simula ions o L*50.6 ~a simila alue o L*was used by Fische e al. o desc ibe he he modynamic p ope ies o e hane7!and m *254 we e i ed o he expe imen al c i ical empe a u e and densi y o CH3–CF3. In his way we ob ain he pa am- e e s s 53.73 Å and e /k5282.54 K. Expe imen al47 and simula ion coexis ence cu e and a- po p essu es o he e ige an 1,1,1- i luo oe hane a e shown in Fig. 9. The dipole momen ob ained wi h he GEMC da a, m GEMC52.8310218 esu cm, in easonable ag eemen wi h he expe imen al alue,48 m 52.32310218 esu cm. We ha e ob ained an excellen desc ip ion o coex- is ence p ope ies o his luid by using a simple model o i s in e ac ion ene gy. The e o e, we can conclude ha he in e - ac ion po en ial o Eqs. ~1!–~3!is a good e ec i e pai po- en ial o he he modynamic desc ip ion o a ela i ely complex dipola luid, as he e ige an 1,1,1- i luo oe hane. We ha e hus a p esen ed esul s o models p esen ing a dipole model. In ou p e ious wo k we ob ained esul s o models p esen ing only a quad upole momen . Howe e , i is o en ound in na u e ha molecules p esen ing dipole mo- men ha e also a signi ican quad upole momen ~ o in- s ance, wa e !.49 In Sec. IV we p esen esul s o models ha ing simul aneously bo h a dipole and a quad upole mo- men . IV. DIPOLAR MODELS WITH A QUADRUPOLE In his sec ion we p esen GEMC esul s o a Kiha a model wi h L*50.8 and m *252.3 and Q*251.5. The pai po en ial is gi en by u~ , 1, 2!5uK~ , 1, 2!1u mm ~ , 1, 2!1uQQ1u m Q. ~15! Exp essions o uKand u mm a e gi en by Eqs. ~1!and ~2!. Exp ession o uQQ was aken om Re . 49. The u m Q e m is gi en by49 u m Q53 m Q 2 4@~c12c2!~115c1c222e1•e2!#.~16! In Eq. ~16!,ci5cos u i~see Fig. 1!and eiis a uni ec o in he di ec ion o m i. In Table V he VLE o his model is p esen ed. Es ima ed c i ical pa ame e s a e Tc * 51.153(30), nc *50.139(15), Pc *50.037(8), and Zc50.23~5!. The acen ic ac o is 50.20~12!. Resul s o a pu ely dipola model, a pu ely quad upola and o a nonpo- la model o he same elonga ion a e also p esen ed in Fig. 10. As expec ed he c i ical empe a u e o he dipole 1quad upole model is highe han ha o he pu ely dipola o ha o he pu ely quad upola model. The alue o DTc *, de ined as [Tc( m *,Q*)2Tc 0]/( e /k), o he dipole 1quad upole model is DTc *50.201. I he alues o DTc * o he pu ely dipola o he pu ely quad upola model a e added hen one ob ains DTc *50.136. The e o e he inc ease o he c i ical empe a u e o he dipola model wi h quad upole is la ge han he summa ion o he inc ease un- de gone by he pu ely dipola and he pu ely quad upola FIG. 9. Coexis ence p ope ies o 1,1,1- i luo oe hane. ~a!Coexis ence den- si ies. ~b!Vapo p essu es. Symbols ep esen he expe imen al da a ~ aken om Re . 47!. Lines a e i ings o he GEMC da a o L*50.6 and m *254, wi h he pa ame e s ob ained as desc ibed in he ex . 7211Ga zo ´n e al. : Vapo -liquid equilib ia in dipola luids J. Chem. Phys., Vol. 102, No. 18, 8 May 1995 luid. The explana ion o ha is as ollows. The dipola model wi h quad upole as desc ibed by Eq. ~15!p esen s no only he uQQ and u mm e ms bu also he addi ional con ibu- ion a ising om he in e ac ion be ween he dipole and he quad upole gi en by he u m Q e m. As a gene al ule e ms in he Hamil onian o a sys em dec easing he ee ene gy a a gi en densi y and empe a u e aise he c i ical empe a u e. Con e sely, e ms in he Hamil onian inc easing he ee en- e gy o he sys em dec ease he c i ical empe a u e. The e ms uQQ,u mm , and u m Qdec ease he ee ene gy o he sys em and, he e o e, all o hem aise he c i ical empe a- u e. The dipola model wi h quad upole is mo e han he summa ion o con ibu ions due o he dipole and con ibu- ions due o he quad upole and his is so because o he p esence o he u m Q e m. Ou esul s ag ee wi h heo e ical p edic ions made by Bena ides e al.50 In Fig. 11 co esponding s a es plo s o he VLE o he dipola model wi h quad upole a e p esen ed. As an icipa ed, he dipola model wi h quad upole p esen s la ge de ia ions om he p inciple o co esponding s a es han he pu ely quad upola o he pu ely dipola model. In Fig. 11~a!i can also be obse ed ha he b oadening o he coexis ence cu e is la ge o he dipola model wi h quad upole han o he pu e dipola o quad upola model. Fig. 11~b!shows ha he slope o he ln P/Pc s Tc/Tplo is la ge o he dipola model wi h quad upole luid. Ou conclusions a e simila o he conclusions exposed by Dubey and O’Shea o Lenna d-Jones dipola plus qua- d upola luids.51 As i has been al eady said, dipola molecules in na u e o en also p esen a quad upole momen .49 To desc ibe he liquid- apo coexis ence p ope ies o a eal dipola luid wi h quad upole like he 2,2,2- i luo oe hanol ~TFE!, b oadly used as sol en o p o eins,52 we ha e es ima ed he molecula pa ame e s o a linea Kiha a dipola model wi h quad upole ha could gi e an accu a e desc ip ion o VLE o TFE. We ha e assumed he same shape o TFE han o 1,1,1- i luo oe hane, ha is, L*50.6, and he same e and s Kiha a pa ame e s. Then, we ha e es ima ed he dipole and quad upole momen s o eal TFE om he expe imen al di- pole momen s o 1,1,1- i luo oe hane and e hanol. By as- suming a disc e e cha ge model we ha e es ima ed ha he mul ipole momen s o eal TFE a e m TFE51.8310218 esu cm and QTFE55.2310226 esu cm2. The ial educed mul ipole momen s ob ained by ha way o TFE a e m *25 m TFE 2/~ es 3!>1.5 and Q*25QTFE 2/~ es 5!>1. We ha e pe o med GEMC simula ions o ha model ~see Table V!. The c i ical pa ame e s ob ained o his model ~L*50.6, m *251.5, and Q*251!a e Tc *51.143(14), nc * 50.160(6) and Pc *50.050(8). C i ical empe a u e and densi y we e i ed o he expe imen al c i ical empe a u e and densi y o eal TFE, ob aining he ollowing Kiha a pa ame e s: e /k5436.66 K and s 53.83 Å. Expe imen al53–56 and simula ion coexis ence cu e and apo p essu es o TFE a e shown in Fig. 12. The dipole and quad upole momen s ob ained wi h he simula ion da a a e m GEMC52.26310218 esu cm and QGEMC57.06310226 esu cm2, in good ag eemen wi h he es ima ed expe imen al mul ipoles. The desc ip ion made by simula ion o he dipo- la Kiha a model wi h quad upole o liquid- apo coexis - ence o TFE is easonably good. A ema kable ac obse ed in all ou simula ion desc ip- FIG. 10. Liquid- apo coexis ence cu es o pola Kiha a luids wi h L*50.8. Resul s o ~ om he op o he bo om!a dipola model wi h quad upole ~ m *252.3, Q*251,5! luid, a pu e dipola ~ m *252.3!, a pu e quad upola ~Q*251.5!, and he nonpola luid. Lines a e i ings o he GEMC da a. Da a o he pu e quad upola luid ob ained om Re . 15. Da a o he nonpola Kiha a luid ob ained om Re . 13. TABLE V. Coexis ence p ope ies o dipola luids wi h quad upole. See he main ex o an es ima e o c i ical p ope ies. T*ng *Pg *nl *Pl * L*50.8, m *252.3, Q*251.5 1.1 0.0406~22!0.0278~14!0.234~20!0.024~22! 1.075 0.0322~17!0.0233~11!0.267~13!0.017~24! 1.05 0.0279~14!0.0202~10!0.2808~95!0.017~31! 1.025 0.021 70~83!0.016 52~70!0.289~11!0.009~27! 1 0.016 35~66!0.012 68~47!0.294~15!0.008~54! 0.975 0.015 24~42!0.011 45~42!0.3173~78!0.005~37! 0.975 0.015 63~32!0.011 80~33!0.3147~12!0.000~48! 0.95 0.012 65~47!0.009 59~40!0.3263~93!0.011~33! 0.925 0.010 02~54!0.007 63~40!0.3378~70!20.001~33! 0.9 0.008 36~27!0.006 29~24!0.3457~66!0.001~33! 0.85 0.004 97~15!0.003 69~12!0.3569~59!20.020~36! L*50.6, m *251.5, Q*251 1.07 0.0466~17!0.0324~15!0.296~37!0.034~40! 1.06 0.0436~27!0.0302~14!0.299~25!0.021~40! 1.05 0.0391~11!0.0281~13!0.312~15!0.019~33! 1.025 0.0371~16!0.0256~10!0.333~12!0.017~25! 1 0.0257~13!0.019 27~94!0.338~14!0.016~31! 0.975 0.0220~10!0.016 43~73!0.352~13!0.008~38! 0.95 0.0207~10!0.015 18~61!0.3653~79!0.002~29! 0.95 0.0191~19!0.0143~12!0.3677~84!0.006~32! 0.925 0.0157~12!0.011 65~71!0.3759~64!0.002~31! 0.9 0.011 78~60!0.008 91~41!0.3872~74!20.011~33! 0.875 0.009 46~53!0.007 13~35!0.3946~62!20.006~31! 0.875 0.011 39~26!0.008 34~22!0.3951~65!20.006~36! 0.85 0.008 08~59!0.005 95~43!0.4060~59!20.003~29! 0.8 0.004 43~38!0.003 21~27!0.4220~50!0.000~29! 7212 Ga zo ´n e al. : Vapo -liquid equilib ia in dipola luids J. Chem. Phys., Vol. 102, No. 18, 8 May 1995