An Extended Regular Solution Model with Local Volume Fraction

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1 () An Etended Regular Slutn Mdel wth cal Vlume Fractn Shgetsh KOBUCHI, Ken ISHIGE (Department f Enrnmental Scence and Engneerng, Graduate Schl f Scence and Engneerng, Yamaguch Unersty) Setsuk YONEZAWA (Department f Chemcal Engneerng, Faculty f Engneerng, Kyushu Unersty) Ken FUKUCHI (Department f Chemcal and Blgcal Engneerng, Ube Natnal Cllege f Technlgy) Yasuhk ARAI (Prfessr Emertus f Kyushu Unersty) The regular slutn mdel (the Hldebrand-Scatchard equatn) cupled wth the Flry-Huggns equatn has been etended by usng the lcal lume fractn prpsed by Wlsn s that t can be appled t plar mtures. Frm the etended regular slutn mdel, actty ceffcent equatns can be dered. Par energy parameters requred t ealuate lcal lume fractns can be estmated by slublty parameters and mlar lumes predcted frm the mlecular structures f cnsttuent pure cmpnents by usng addte methds. The applcablty f the present regular slutn mdel s eamned by crrelatng apr-lqud equlbra f seeral ethanl hydrcarbn bnary systems, and t s fund that the prpsed mdel can be adpted fr plar mtures. Keywrds: Regular Slutn, cal Vlume Fractn, Actty Ceffcent, Vapr-qud Equlbrum, Ethanl, Hydrcarbn Intrductn The Hldebrand-Scatchard equatn based n the regular slutn mdel (RSM) seems t be a prmsng equatn t ge lqud phase actty ceffcents usng physcal prpertes (.e., slublty parameters and lqud mlar lumes) f cnsttuent pure cmpnents ). Hweer, the applcatn s lmted t nn-plar r slghtly plar mtures. Therefre, n a preus study ), the slutn mdel has been etended s that t can be appled t plar mtures by ntrducng an epnent-type mng rule and a cmpstn-dependent nteractn parameter between unlke mlecules. Ths mdel s termed here as RSM- (see Append). The etended regular slutn mdel can be successfully adpted t plar mtures such as ethanl hydrcarbn mtures usng the epnent parameter whch represents the nn-randmness n mlecular mng and ther tw nteractn parameters. As a cntnuatn, n ths study, an anther etended regular slutn mdel s prpsed by usng the lcal lume fractn gen by Wlsn 3) t epress the nn-randmness n mlecular mng. The applcablty f the prpsed regular slutn mdel s eamned by crrelatng the apr-lqud equlbra (VE) f seeral ethanl hydrcarbn bnary systems studed n the preus wrk ).. An Etended Regular Slutn Mdel The mlar ecess Gbbs energy based n the regular slutn mdel ) wth the lcal lume fractns s prpsed as g E RT ln A () where the Flry-Huggns equatn s added as a sze-effect term and A A and A A 0. Eq.() s termed here as RSM-. In the rgnal frm, the lcal lume fractn shuld be replaced by the usual lume fractn. When the frst term f rght hand sde f Eq.(), that s the energy f mng term, s mtted t becmes the well-knwn Wlsn equatn 3). In ths study, Wlsn equatn s cnsdered t be an ecess entrpy epressn 4). The ceffcent A s epressed n terms f slublty parameters ) as fllws: A ( δ δ ) lδ δ () Accrdng t Adach and Suge 5), the nteractn parameter l fr mtures s gen Mem Fac Eng Yamaguch Un

2 () by l ( ) m n (3) Based n the standard thermdynamc prcedure, the actty ceffcents f a bnary mture can be dered frm Eq.() as fllws: 4n ln ( A ) ( n ) ( W) A RT 4n ln {( ) δ δ / RT} ( ) ( A ) ( n ) ( W) A RT {( ) δ δ / RT} ( ) where the lcal lume fractn and the lume fractn are defned as, (6), (7) where A and the Wlsn parameters are gen by ( δ δ ) lδ, l m n( ) (8) A δ λ λ λ λ ep, ep RT RT. Par Energy Parameters and Slublty Parameters T apply Eqs.(4) and (5), the par energy parameters λ are requred t ealuate the Wlsn parameters and. The defntn (4) (5) (9) f slublty parameter s δ ( E/) 0.5 where E s the chese energy due t mlecular attractn frce. In ths study λ s apprmated as (/z)e, accrdng t Wng and Eckert 6), where usually E s gen V ) by ΔH RT and z s the c-rdnatn number. Then, the par energy parameters λ can be btaned as fllws: λ ( / z) δ, λ ( z) δ (0) / Further, fr the unlke par, the fllwng gemetrc mean s adpted. λ λ ( ε )( λ λ ) ( ε)( / z)( ) δδ () where ε s an nteractn parameter. The slublty parameters and the lqud mlar lumes at a gen temperature t needed n the abe calculatn can be btaned as fllws 7, 8) : δ t () and 5 δ 5 t ( ) t 5 β t 5, b 5 β (3) t 5 where b s the lqud mlar lume at the nrmal blng pnt t b [ ], whch can be estmated by usng the addte methd f e Bas 9) ecept small mlecules such as methanl and ethanl 7). The slublty parameters at 5 (δ 5 ) and the lqud mlar lumes at 5 ( 5 ) can be predcted by Fedrs methd 0) based n the grupcntrbutn methd. 3. VE Crrelatn and Dscussn Fr cmparsn f the present mdel (RSM-) wth the preus ne (RSM- ) ), VE f the same ethanl hydrcarbn bnary systems hae been crrelated. At suffcently lw pressures, VE (-y) f bnary mtures can be calculated by b Vl. 6 N. (00)

3 (3) 3 y γ p / p, y γ p / p (4) p γ p p γ (5) where and y dente mle fractns f lqud and apr phases, respectely. And p s the apr pressure gen by the Antne s equatn fr a pure cmpnent and p s the ttal pressure. Therefre VE can be calculated by usng the actty ceffcents γ and γ btaned frm Eqs.(4) and (5). The physcal prpertes f pure substances ( 5, δ 5, t b, b and the cnstants f Antne s equatn) are shwn n Table. The crrelatn perfrmances fr VE and the parameters requred are summarzed n Table, and typcal graphcal representatns are shwn n Fgures -3 fr C 6 hydrcarbns. As shwn n Table and Fgures and, the crrelatn perfrmance f RSM- s almst the same as the preus mdel RSM- fr cyclheane and armatcs. The c-rdnatn number z ptmzed s 0 as usually epected 6). The cntrbutn f nteractn parameter n, whch represents the cmpstn dependence f the nteractn parameter l, s fund t be ery small. Therefre, the parameter s let be n 0. On the ther hand, fr paraffn systems, the crrelatn perfrmance s pr and the c-rdnatn number shuld be z4 t btan better crrelatn results. Een thugh z4, the crrelatn perfrmance f RSM- s slghtly nferr t RSM- as shwn n Table. It s nted that the cntrbutn f (W), that s the ecess entrpy term, s pste fr paraffns thugh (W) fr cyclheane and armatcs s negate. The cntrbutns f ecess entrpy term t actty ceffcents, as shwn n Eqs. (4) and (5), are (W) gen by the Wlsn Table Physcal prpertes f pure substances 5 ) b ) Substance ) ) δ 5 t b Cnstants f Antne s equatn*, ) [cm 3 ml - ] [cm 3 ml - ] [(J cm -3 ) 0.5 ] [ ] A B C Heane Heptane Octane Cyclheane Benzene Tluene Ethanl * lg p [kpa] A B /( T [K] C) RSM (n 0) RSM b Hydrcarbn () a z m ε Δy * Δt** Δy * Δt** Heane Heptane Octane Cyclheane Benzene Tluene c Table Crrelatn perfrmances fr VE at 0.3 kpa f ethanl () hydrcarbn () bnary systems and nteractn parameters N N ( 00 / N ) [ y y / y ], ** Δ t [ ] ( / N ) t t, N number f data pnts * Δ y [%],calc,ep,ep calc ep a VE data surces are shwn n the preus paper ) ; b Three mdel parameters, m and n are adpted ) ; c Ttal pressure s 0 kpa Mem Fac Eng Yamaguch

4 4 (4) y [ ] y [ ] y [ ] [ ] Fg. Crrelatn f apr-lqud equlbra f ethanl () cyclheane () bnary system at 0.3 kpa. Epermental: ( ); Crrelatns : ( )RSM-; ( )RSM- ( 0.95, m 0.0, n 0.04) [ ] Fg. Crrelatn f apr-lqud equlbra f ethanl () benzene () bnary system at 0.3 kpa. Epermental: ( ); Crrelatns : ( )RSM-; ( )RSM- ( 0.96, m 0.056, n 0.08) [ ] Fg. 3 Crrelatn f apr-lqud equlbra f ethanl () heane () bnary system at 0.3 kpa. Epermental: ( ); Crrelatns : ( )RSM-; ( )RSM- ( 7, m 0.07, n 0.089) equatn. By fttng the present mdel (RSM-) t epermental VE data, t s fund that (W)>0 fr ethanl() paraffn() bnary systems whle (W)<0 fr the ther three bnary systems as mentned abe. Ths fact f (W)>0 means the mlar ecess entrpy s s E <0. On the ther hand, (W)<0 ndcates s E >0. If the Flry-Huggns(FH) equatn s adpted t ge the ecess entrpy, as studed n the preus wrk (RSM-), s E (FH)>0 and (FH)<0 fr any bnary system. The mlar ecess entrpy s E f ethanl() hydrcarbn() mtures s dscussed here usng the nfrmatns aalable n the lterature. Fr ethanl() heptane(), s E at 0, 40 and 60 are presented (Fg.4 n ref.) ). Ths result shws s E <0 thugh slghtly s E >0 near 0. And, fr ethanl() tluene(), s E at 00 t 35 are reprted (Fg.3 n ref.) 3). It s represented s E <0 whle slghtly s E >0 at 35 near 0. These results s E <0 supprt ln γ (W)>0. It s nted, hweer, that the temperatures (blng pnts at 0.3 kpa fr ethanl() heptane() and at 0 kpa fr ethanl() tluene(), see Table ) much dffer frm thse at whch s E reprted n the lterature cted because s E markedly depends n temperature. Therefre, the ealuatn f s E dscussed here may be bscure n rgrus sense. Cnclusns An etended regular slutn mdel (RSM-) has been deelped by adptng the lcal lume fractn f Wlsn 3). The actty ceffcent equatns dered frm RSM- can be adpted t crrelate apr-lqud equlbra f plar mtures such as ethanl() hydrcarbn(). Hweer, t s nted that the crrelatn perfrmance f RSM- s slghtly nferr t the preus mdel RSM- fr paraffns. The applcablty and physcal meanngs f nteractn parameters f RSM- shuld be further eamned and dscussed by adptng apr-lqud equlbrum data f ther plar mtures. Vl. 6 N. (00)

5 (5) 5 Nmenclature A nteractn term [J cm -3 ] g mlar Gbbs energy [J ml - ] l nteractn parameter between unlke mlecules m nteractn parameter between unlke mlecules n nteractn parameter between unlke mlecules p ttal pressure [Pa] p apr pressure f pure cmpnent [Pa] R gas cnstant [J ml - K - ] s mlar entrpy [J ml - K - ] T abslute temperature [K] t temperature [ ] lqud mlar lume [cm 3 ml - ] mle fractn f lqud phase y mle fractn f apr phase z c-rdnatn number γ lqud phase actty ceffcent δ slublty parameter [(J cm -3 ) 0.5 ] ε nteractn parameter between unlke mlecules λ nteractn energy due t attracte frce [J ml - ] lume fractn <Subscrpt> b nrmal blng pnt calc calculated alue ep epermental data cmpnent cmpnent cmpnent (ethanl) cmpnent (hydrcarbn) 5 standard temperature (5 ) <Superscrpt> E ecess prperty lcal quantty terature Cted ) Hldebrand, J. H., J. M. Prausntz and R.. Sctt, Regular and Related Slutns, Chap. 7, Van Nstrand Renhld C., New Yrk, U. S. A., 970 ) Kbuch, S., K. Ishzu, K. Hnda, Y. Shmyama, S. Ynezawa, K. Fukuch and Y. Ara, Crrelatn f Vapr-qud Equlbra fr Ethanl Hydrcarbn Bnary Systems Usng Regular Slutn Mdel wth Epnent Type Mng Rule, J. Chem. Eng. Japan, Vl. 4, pp , 009 3) Wlsn, G. M., Vapr-qud Equlbrum. XI. A New Epressn fr the Ecess Free Energy f Mng, J. Am. Chem. Sc., Vl. 86, pp. 7-30, 964 4) Wang, W. and K. -C. Cha, The Cmplete cal Cncentratn Mdel Actty Ceffcents, Chem. Eng. Sc., Vl. 38, pp , 983 5) Adach, Y. and H. Suge, A New Mng Rule Mdfed Cnentnal Mng Rule, Flud Phase Equl., Vl. 8, pp. 03 8, 986 6) Wng, K. F. and C. A. Eckert, Dlute Slutn Behar f Tw Cyclc Anhydrdes, Ind. Eng. Chem. Fundam., Vl. 0, pp. 0-3, 97 7) Ynezawa, S., S. Kbuch, K. Fukuch and Y. Ara, Predctn f qud Mlar Vlumes by Addte Methds, J. Chem. Eng. Japan, Vl. 38, pp , 005 8) Ynezawa, S., S. Kbuch, K. Fukuch, Y. Shmyama and Y. Ara, Predctn Methd f Slublty Parameter Based n Mlecular Structure (n Japanese), J. Sc. Mater. Eng. Resur. Japan, Vl. 9, pp. 5-7, 006 9) Plng, B. E., J. M. Prausntz and J. P. O Cnnell, The Prpertes f Gases and quds, 5th ed., McGraw-Hll, New Yrk, U. S. A. (00) 0) Fedrs, R. F., A Methd fr Estmatng Bth the Slublty Parameters and Mlar Vlumes f quds, Plym. Eng. Sc., Vl. 4, pp , 974 ) Bublk, T., V. Fred and E. Hála, The Vapur Pressures f Pure Substances, nd ed., Elseer, Amsterdam, The Netherlands (984) ) Van Ness, H. C., C. A. Sczek, and N. K. Kchar, Thermdynamc Ecess Prpertes fr Ethanl-n-Heptane, J. Chem. Eng. Data, Vl., , 967 3) Hwa, S. C. P. and W. T. Zegler, Temperature Dependence f Ecess Thermdynamcs Prpertes f Ethanl-Methylcyclheane and Ethanl-Tluene Systems, J. Phys. Chem., Vl. 70, pp , 966 4) Kbuch, S., S. Ynezawa, K. Fukuch and Y. Ara, Crrelatn f Vapr-qud Equlbra f Mtures Cntanng Plar Mlecules by Regular Slutn Mdel wth Epnent-Type Mng Rule (n Japanese), Mem. Faculty Eng., Yamaguch Un., Vl. 60, pp , 009 5) Prausntz, J. M., R. N. chtenthaler and E. G. de Azeed, Mlecular Thermdynamcs f Flud-Phase Equlbra, 3rd ed., p.354, Prentce Hall PRT, NJ, U. S. A., 999 Mem Fac Eng Yamaguch Un

6 6 (6) Append: RSM- where the epnent s treated as and the ther detals shuld be refered t the The mlar ecess Gbbs energy f RSM- preus paper ). The prpsed mdel s epressed by (RSM-) s fund t be useful t crrelate VE f ethanl() hydrcarbn() bnary systems and t s ntced that at nfnte E g A dlutn cndtn 4) : (A) RT ln The actty ceffcents dered fr a bnary system are gen as fllws. ( A ) ( n ) ( FH) < : ( 0) (A4) A : ( 0) ln RT (A5) > : ( 0) ln (A6) A RT 4n ( ) {( ) δ δ / RT} ln (A) ( A ) ( n ) ( FH) ln A ( ) RT (A3) 4n {( ) δ δ / RT} Fr ethanl() hydrcarbn() bnary systems, fr eample ethanl() heane(), 7 ( <) and Eq. (A) s fund t be effecte t represent the behar f cure (rapd ncrease) near 0 thugh Eq.(A) may be nadequate t be adpted at nfnte dlutn cndtn because f the fact f Eq. (A4). It s nted wth nterest, based n an asscated slutn mdel 5), that the nfnte dlutn actty ceffcent f a dmerzng cmpnent (all mlecules are dmerzed) n a nn dmerzng nert slent becmes t be nfnte ( ). (Receed Octber 7, 00) Vl. 6 N. (00)