Modeling Thermal Conductivity of Concentrated and Mixed-Solvent Electrolyte Systems

Transcription

1 5698 Id. Eg. Chem. Res. 28, 47, Modelg Thermal Coductvty of Cocetrated ad Mxed-Solvet Electrolyte Systems Pemg Wag* ad Adrzej Aderko OLI Systems Ic., 18 Amerca Road, Morrs Plas, New Jersey 795 A comprehesve model has bee developed for calculatg the thermal coductvty of aqueous, oaqueous, ad mxed-solvet electrolyte systems ragg from dlute solutos to fused salts or pure solutes. The model cossts of a correlato for calculatg the thermal coductvty of solvet mxtures ad a method for predctg the effect of electrolyte compoets. The thermal coductvty of multcompoet solvet mxtures ca be represeted usg surface area parameters ad thermal coductvtes of pure solvets cojucto wth a sgle bary parameter per solvet par. The effect of electrolytes s modeled by accoutg for a cotrbuto of dvdual os, whch s quatfed by the Redel coeffcets, ad a cotrbuto of specfc teractos betwee os or eutral speces. Formulatos have bee developed for the cotrbutos of dvdual os ad speces-speces teractos to represet the effect of multple solvets. I addto to solvet composto, the speces-speces teracto term s also a fucto of oc stregth. The model accurately reproduces expermetal thermal coductvty data over a wde rage of electrolyte cocetratos aqueous ad oaqueous systems. I partcular, the model has bee show to be accurate for aqueous acds ad bases (e.g., H 2 SO 4, HNO 3,H 3 PO 4, NaOH, ad KOH) up to the lmt of a pure acd or base, varous trates ragg from dlute solutos to fused salts, ad other salts water ad varous orgac solvets. The model has bee coupled wth thermodyamc equlbrum calculatos to reproduce the effects of complexato or other oc reactos o thermal coductvty. * To whom correspodece should be addressed. Tel.: (973) Fax: (973) E-mal: Pwag@olsystems.com. Itroducto The thermal coductvty of electrolyte solutos plays a sgfcat role the chemcal process dustry ad varous geologcal ad egeerg applcatos because of the mportace of heat trasfer a multtude of processes. The desg ad optmzato of varous processes ad devces such as those used refrgerato, geothermal power geerato, ad chemcal plats utlzg seawater as a coolg medum requre a detaled kowledge of thermal coductvty electrolyte solutos. 1 Icreasg atteto has bee focused o detaled studes of thermal coductvty of electrolyte solutos, as reflected by the large umber of expermetal results publshed recet years. Therefore, accurate models for represetg thermal coductvtes of electrolyte solutos are hghly desrable. However, aalyss of expermetal data has bee commoly performed oly o a case-by-case bass ad o attempt has bee made so far to develop a comprehesve thermal coductvty model for mxed-solvet electrolyte systems. The exstg models for the thermal coductvty of electrolyte solutos have bee desged maly for salt + solvet bary systems ad are applcable oly up to moderate cocetratos. A theoretcal equato for the cotrbuto of teroc forces to the thermal coductvty of dlute electrolyte solutos was derved by Bearma 2,3 based o the Debye-Hückel-Osager-Falkehage model. Ths equato predcts that the cotrbuto of log-rage electrostatc forces to thermal coductvty s a fucto of κ 3 D (where κ D s the verse Debye legth) or, equvaletly, c 3/2. However, Bearma 2 oted that eve the cocetrato rage where the Debye-Hückel model s vald ths o-o teracto cotrbuto does ot have a measurable effect o the overall thermal coductvty due to the fact that the cotrbutos of other effects are much greater ad vary as a fucto of κ D 2 (or c). Such behavor s qute dfferet from that foud for the vscosty ad, especally, electrcal coductvty of dlute electrolyte solutos. Thus, practce, the thermal coductvty of electrolyte solutos has bee reproduced by emprcal or sememprcal correlatos. The publshed correlato methods have bee revewed by Horvath 1 ad Cort et al. 4 The most wdely used expresso s that of Redel, 5 whch s a smple lear expaso terms of molar cocetratos: λ ) λ H2 O + R c (1) where λ H2O s the thermal coductvty of pure water, c s the molar cocetrato of o, ad R s the cotrbuto of o. Ths equato s a statemet of the addtvty of dvdual oc cotrbutos dlute solutos ad s aalogous to the Joes-Dole 6 equato for vscosty. Although the Redel equato s applcable to multcompoet systems ad ca be used for dlute ad moderately cocetrated electrolyte solutos wth good accuracy, t fals to represet expermetal data over exteded cocetrato rages such as those commoly ecoutered for cocetrated acds or alkale solutos of NaOH or KOH. Also, t s ot accurate for systems that show more complcated treds wth chagg cocetratos, such as the aqueous solutos of NaF where the thermal coductvty creases frst wth cocetrato ad the decreases. For the cases of NaOH ad KOH, Redel 7 exteded hs equato to clude a extra term φ(c) ad evaluated ths term at varous cocetratos of the bases. Alteratve approaches to modelg thermal coductvty clude the use of the cocept of apparet molar thermal coductvty aalogy to apparet molar thermodyamc quattes (e.g., volumes ad heat capactes). Ths quatty was related to c 1/2 through a lear equato. 8,9 Also, Vargaftk ad Os m 1 developed a method that relates the thermal 1.121/e71373c CCC: $ Amerca Chemcal Socety Publshed o Web 6/28/28

2 Id. Eg. Chem. Res., Vol. 47, No. 15, Fgure 1. Excess thermal coductvty of the water + ethylee glycol system at 5 C as a fucto of (a) mole fracto ad (b) weght fracto of ethylee glycol. The excess thermal coductvtes, λ x ex ad λ w ex, are defed as λ x ex ) λ m - k x kλ k ad λ w ex ) λ m - k w kλ k. Symbols are from expermetal data of Vaderkoo et al. 5 Table 1. r Coeffcets Equatos 18 for Selected Aqueous Ios catos R 1,H2 O R 2,H2 O aos R 1,H2 O R 2,H2 O H 3O Cl L NO Na SO K HSO Mg OH Ca F Ba HCO NH CO Fe PO Fe HPO N H 2PO Cu Z Cr coductvty to other propertes of the soluto ad of the solvet such as the heat capactes ad destes. These methods, however, are applcable oly to bary electrolyte solutos, ad ther accuracy deterorates wth rsg cocetrato. 1 More recetly, a geeralzed correspodg-states correlato has bee proposed by Quresh et al. 11 Usg two system-depedet parameters for each bary soluto ad 1 uversal parameters, ther model has bee show to reproduce the expermetal data for over 2 aqueous electrolyte systems wth 4% over wde rages of cocetrato, pressure, ad temperature. Although the model of Quresh et al. 11 s accurate for the solutos studed, t s applcable oly to aqueous bary systems. I mxed-solvet electrolyte solutos, thermal coductvty s determed ot oly by the cocetrato of electrolytes, but also by the composto of the solvet. The thermal coductvty of solvet mxtures aloe may chage sgfcatly wth composto. I addto, systems wth strog o assocato effect (e.g., fully mscble acds or bases), thermal coductvty s flueced by cocetratos of both os ad assocated o pars. Thus, a comprehesve treatmet of thermal coductvty of mxed-solvet electrolyte systems requres takg to accout ot oly the o-solvet ad o-o teractos that predomate aqueous solutos, but also the solvet-solvet ad o par-solvet teractos. The objectve of ths work s to develop a comprehesve, egeerg-oreted model for predctg thermal coductvty of mxed-solvet electrolyte solutos. I ths study, the term mxed-solvet electrolytes s used the broadest possble sese to clude (1) aqueous electrolyte solutos from the dlute rego to the molte salt lmt, (2) fully mscble acds or bases water, ad (3) electrolytes pure orgac or mxed orgac-water solvets. Further, the model s desged to accout for specato effects, such as complexato or o assocato, whe combed wth a specato-based thermodyamc model. The model developed ths study cossts of two parts: (1) computato of thermal coductvty of pure ad Table 2. Parameters of Equatos 1 ad 16 for Selected Solvet Pars solvet pars parameters () j k j (1) k j T ( C) relevat systems o. of pots AAD refereces methaol H 2O methaol + H 2O propaol + methaol + H 2O ethaol H 2O ethaol + H 2O , ethaol + ethylee glycol + H 2O ethylee glycol H 2O ethylee glycol+h 2O , 43, ethaol + ethylee glycol + H 2O ethylee glycol ethaol ethaol + ethylee glycol + H 2O , 47 dethylee glycol H 2O dethylee glycol + H 2O , 48, 54, 55 1-propaol H 2O propaol + H 2O , 43 2-propaol H 2O propaol + H 2O , 43, propaol + methaol + H 2O propaol methaol propaol + methaol propaol + methaol + ethylee glycol propaol + methaol + H 2O acetc acd H 2O acetc acd + H 2O acetoe H 2O acetoe + H 2O acetoe CCl acetoe + CCl , 57 toluee CCl toluee + CCl , 17, 58 25, 4 toluee + CCl 4 + cyclohexae cyclohexae CCl cyclohexae + CCl , 4 toluee + CCl 4 + cyclohexae , 57, 58 25, 4 bezee + CCl 4 + cyclohexae

3 57 Id. Eg. Chem. Res., Vol. 47, No. 15, 28 Thermal Coductvty of Solvet Mxtures Fgure 2. Thermal coductvty of the water + acetc acd system as a fucto of temperature at varous fxed compostos ( mass percet) of acetc acd. Expermetal data are from Bleazard et al., 56 ad the les are calculated from eqs 2, 8, 1, ad 16 usg parameters lsted Table 2. Fgure 3. Thermal coductvty of orgac + water mxtures as a fucto of the mole fracto of water at 2 C. The symbols are expermetal data from Rastorgu ad Gaev, 43 Le et al., 45 ad Bates et al., 42 ad the les are calculated from eqs 2, 8, 1, ad 16 usg the parameters lsted Table 2. Fgure 4. Percetage devatos for the predcto of thermal coductvtes as a fucto of the mole fracto of cyclohexae for terary systems cyclohexae + CCl 4 + bezee ad cyclohexae + CCl 4 + toluee. Expermetal data are from Rowley et al. 17 ad Rowley ad Gubler. 58 mxed solvets as a fucto of temperature ad solvet composto ad (2) computato of the depedece of thermal coductvty o electrolyte cocetrato. It has bee observed the lterature that thermal coductvtes of lqud mxtures are usually lower tha ether a mole or weght fracto average of pure-compoet coductvtes. 12 Varous models for represetg the thermal coductvty of lqud mxtures have bee descrbed the lterature. Some otable models that are applcable to multcompoet systems clude the power law method, 13 the harmoc mea method of L 14 ad ts modfcato, 15 ad models based o the local composto cocept such as those of Rowley, 16,17 Cao et al., 18 ad Huag. 19 Although the power law method has bee successfully used for a umber of lqud mxtures, t s lmted to oaqueous mxtures ad the rato of thermal coductvtes of ay two pure compoets ca ot exceed The harmoc method of L fals to predct the thermal coductvty behavor of azeotropc lqud mxtures. 14 Correlatos based o the correspodg-states prcple 2 23 have also bee proposed for calculatg the thermal coductvty of lquds ad lqud mxtures. The results obtaed from these methods deped o the selecto of referece fluds, whch may have a substatal fluece o the calculated values for lqud mxtures, especally whe the system goes beyod bary. Several hard-sphere theory-based models have bee developed for orgac mxtures However, these models are focused o mxtures cotag compoets wth smlar chemcal structures. A umber of other correlato methods have bee lmted oly to bary systems. 12 I ths secto, we develop a ew correlato that relates the thermal coductvty of solvet mxtures to those of pure compoets. Ths correlato s targeted prmarly at mxtures cotag dssmlar compoets such as water ad orgacs because of the preemece of such solvets electrolyte systems. The local composto cocept emboded the UNIQUAC model of Abrams ad Praustz 27 has bee used to derve the correlato. Ths approach reles o the use of local area fractos to represet the local compostos, whch appears to be a more approprate choce tha usg the mole fractos for modelg eergy trasport lqud mxtures. 14 The structural parameters used ths approach are readly avalable the lterature. 12 It has bee prevously oted 16 that the weght fracto average of thermal coductvty ( k w k λ k ), rather tha the mole fracto average ( k x k λ k ), leads to a more symmetrcal excess thermal coductvty, λ ex (defed as λ ex ) λ m - k y k λ k, wth λ m ad λ k beg the thermal coductvtes of the mxture ad of the pure compoet k, respectvely, ad y k s the weght or mole fracto of k). I addto, t ca be observed that the value of ths excess thermal coductvty s geerally much smaller whe weght fractos rather tha mole fractos are used, dcatg that a much smaller ad more symmetrcal correcto s eeded whe modelg the thermal coductvty of a mxture usg weght fractos. These observatos are demostrated Fgure 1 for the ethylee glycol + water system. Thus, the thermal coductvty of a -compoet mxture s assumed to be a modfed weght fracto (w ) average of the thermal coductvtes of the compoets usg local area fractos, θ j : λ m ) w θ j λ j (2) j where λ j (λ j ) λ j ) should be a approprately defed average of thermal coductvtes of pure compoets ad j ad t should also reflect teractos betwee the two soluto speces ad j. I the UNIQUAC model, 27 the local area fracto, θ j,

4 s the fracto of exteral stes aroud molecule that are occuped by molecule j. It ca be related to the excess free eergy of a lqud mxture through bary teracto parameters (a j ), whch ca be, prcple, determed from phase equlbrum data: θ j ) θ j τ j, θ k τ k k θ j ) 1 ( ) 1, 2,..., ) (3) j where τ j ) exp(- a j (4) RT) ad θ j s the average area fracto defed by θ j ) x j q j (5) x k q k k q j s the surface area parameter for molecule j ad x j s the overall mole fracto of j the mxture. Whe the parameters a j ad a j (a j * a j ) are determed, the local compostos, expressed terms of average local area fractos, ca be calculated from eqs 3 5. The scheme for evaluatg λ j s smlar to Rowley s 16 dervato usg the NRTL model. By substtutg eq 3 to eq 2, a expresso for the mxture thermal coductvty ca be obtaed: Id. Eg. Chem. Res., Vol. 47, No. 15, λ m ) w j q j x j τ j λ j (6) q k x k τ k k I the lmt of a pure compoet, t ca be easly determed from eq 6 that λ ) λ (7) I order to evaluate λ j for * j, oly a bary mxture of ad j eeds to be cosdered. We ow assume that the bary teracto parameter λ j s the thermal coductvty of the bary mxture whe the local area fractos θ j ad θ j are equal. Ths codto ca be satsfed oly at a sgle composto, whch ca be solved usg eqs 3 ad 5 ad expressed as weght fractos q w / j M τ j ), q j M τ j + q M j τ j w / / j ) 1 - w where M ad M j are the molecular weghts of ad j, respectvely. At the composto gve by eq 8 (.e., for θ j ) θ j ), whe the bary mxture thermal coductvty, λ m,sset equal to the teracto parameter λ j, eq 2 leads to a smple expresso for λ j : λ j ) w / λ + w / j λ j (9) whch has bee derved usg eq 7 ad the codto j θ j ) 1 for the bary system -j. (8) Table 3. Iteracto Parameters (Equatos 14, 16, ad 19) Used for Modelg Thermal Coductvtes of Selected Systems system ad codtos parameters HNO 3 + water a () β H3 O +,NO - 3 /H 2 O,H 2 O ) k H 2 O,HNO 3 ) (1) T ) -1 C β H3 O +,NO - 3 /H 2 O,H 2 O ) k H 2 O,HNO 3 ). x HNO3 ) -.93 β H3 O +,NO - 3 /H 2 O,H 2 O ) β H3 O +,NO - 3 /H 2 O,H 2 O ) KNO 3 + water a β K +,NO3 - /H 2 O,H 2 O ) β K +,NO3 - /H 2 O,H 2 O ) T ) C β K +,NO - 3 /H 2 O,H 2 O ) β K +,NO - 3 /H 2 O,H 2 O ). x KNO3 ) -1 NaOH + water a β Na +,OH - /H 2 O,H 2 O ) β Na +,OH - /H 2 O,H 2 O ). T ) C β Na +,OH - /H 2 O,H 2 O ) β Na +,OH - /H 2 O,H 2 O )-. x NaOH ) -.4 H 3PO 4 + water b () k H2 O,H 3 PO 4 ) k P2 O 5,H 3 PO 4 ) (1) T ) -15 C k H2 O,H 3 PO 4 )-.48 k P2 O 5,H 3 PO 4 ). 3 x P2 O 5 ) (wt % H 3PO 3 ) -115%) (3) FeCl 3 + water β FeCl 2+,Cl - /H 2 O,H 2 O ) β FeCl 2+,Cl - /H 2 O,H 2 O ) (31) T ) -1 C β FeCl 2+,Cl - /H 2 O,H 2 O ) β FeCl 2+,Cl - /H 2 O,H 2 O ). x FeCl3 ) -.1 β FeCl 2+,Cl - /H 2 O,H 2 O ) β FeCl 2+,Cl - /H 2 O,H 2 O ) β FeCl 2+,Cl - /H 2 O,H 2 O ) ZCl 2 + ethaol β Z 2+,Cl - /EtOH,EtOH ) β Z 2+,ZCl 2-4 /EtOH,EtOH ) T ) C β Z 2+,Cl - /EtOH,EtOH ). β Z 2+,ZCl 2-4 /EtOH,EtOH ). x ZCl2 ).-.19 β Z 2+,Cl - /EtOH,EtOH) β Z 2+,ZCl 2-4 /EtOH,EtOH ) β Z 2+,Cl - /EtOH,EtOH ) β Z 2+,ZCl 2-4 /EtOH,EtOH ) (3) (3) β Z 2+,Cl - /EtOH,EtOH ) β Z 2+,ZCl 2-4 /EtOH,EtOH ) (31) (31) β Z 2+,Cl - /EtOH,EtOH ) β Z 2+,ZCl 2-4 /EtOH,EtOH ) β Z 2+,Cl - /EtOH,EtOH ) β Z 2+,ZCl 2-4 /EtOH,EtOH ) SbCl 3 + acetoe a β Sb 3+,SbCl 3 /acetoe,acetoe ). β Cl -,SbCl 3 /acetoe,acetoe ). T ) 25-7 C β Sb 3+,SbCl3/acetoe,acetoe ). β Cl -,SbCl3/acetoe, acetoe ). x SbCl3 ).-.29 β Sb 3+,SbCl 3 /acetoe,acetoe ) β Cl -,SbCl 3 /acetoe, acetoe ) β Sb 3+,SbCl 3 /acetoe,acetoe ). β Cl -,SbCl 3 /acetoe,acetoe ). a β (3), β (31), ad β for the dcated speces pars are set equal to. b No teracto β parameters for ths system were troduced.

5 572 Id. Eg. Chem. Res., Vol. 47, No. 15, 28 Fgure 5. Thermal coductvtes of aqueous LCl solutos as a fucto of LCl molalty at varous temperatures. The expermetal data are from Aseyev, 32 Abdulagatov ad Magomedov, 59 ad Redel, 5 ad the les are calculated usg the model. The average percetage devato of the ft s.34. Fgure 6. Thermal coductvtes of aqueous NaF solutos as a fucto of NaF molalty at varous temperatures. The expermetal data are from Aseyev, 32 ad the les are calculated usg the model. The average percetage devato of the ft s.48. Fgure 8. Thermal coductvtes of the KOH + water system as a fucto of x KOH at varous temperatures. The expermetal data are from Vargaftk ad Os m, 1 Redel, 5,7 ad Losecky, 6 ad the les are calculated usg the model. The average percetage devato of the ft s.71. phase equlbrum data s mmal calculatg thermal coductvty usg eqs 6 9. Therefore, t s coveet to smplfy the proposed thermal coductvty model (eqs 2, 8, ad 9) by settg all of the teracto parameters, a j, ad a j equal to ad troducg a emprcal correcto factor to the parameter λ j eq 9. Thus λ j ) (w / λ + w / j λ j )(1 - k j ) where the values of w / ad w / j are determed from eq 8 by settg all τ j ad τ j equal to 1. The use of a sgle correcto factor k j s more effcet tha regressg two UNIQUAC teracto parameters o the bass of thermal coductvty data. After ths smplfcato, all of the bary terms ca be defed usg pure compoet thermal coductvtes, the surface area parameters, q ad q j, molecular weghts of the pure compoets, ad a sgle correcto factor, k j. The values of pure lqud thermal coductvtes, λ, are avalable from the complato of Daubert ad Daer 28 for orgac solvets, ad from Segers ad Watso 29 for water. The surface area parameters are well establshed 27 ad are avalable from Polg et al. 12 The bary parameter k j ca be determed from expermetal thermal coductvty data for the bary mxture of ad j. It ca be show usg algebrac mapulatos that whe the correcto factors k j are for all compoet pars, eq 2 reduces to a smple weght average of the thermal coductvtes of pure compoets: λ m ) w λ Fgure 7. Thermal coductvtes of the KNO 3 + water system as a fucto of x 1/2 KNO3 at varous temperatures. The expermetal data are from Abdullaev ad El darov, 37 Redel, 5 ad Gustafsso et al., 38 ad the les are calculated usg the model. The average percetage devato of the ft s.32. I a smlar approach, Rowley 16,17 oted that the thermal coductvtes predcted usg a correlato derved from the local-composto NRTL model were ot sestve to the choce of NRTL teracto parameters although the fal results agreed well wth expermetal data. At the same tme, a reverse procedure of calculatg vapor-lqud equlbra (VLE) from thermal coductvty data has faled. Also, our prelmary studes, t has bee determed that the practcal advatage of usg UNIQUAC eergetc teracto parameters derved from Depedece of Thermal Coductvty o Electrolyte Cocetrato I modelg the cocetrato depedece of several trasport propertes (e.g., vscosty, electrcal coductvty, ad selfdffusvty) electrolyte solutos, a log-rage electrostatc teracto term s geerally troduced to represet a lmtglaw slope dlute solutos. Ths cotrbuto s usually calculated usg a prmtve model of electrostatc teractos a delectrc cotuum. 4 However, ths cotrbuto s eglgble for thermal coductvty as derved by Bearma from the Debye-Hückel-Osager-Falkehage model. 2,3 Ideed, thermal coductvtes of very dlute electrolyte solutos (.e., -.2 molal) are ot much dfferet from those of pure solvets. Hece, a practcal thermal coductvty model does ot eed to explctly clude ths cotrbuto.

6 Id. Eg. Chem. Res., Vol. 47, No. 15, Fgure 9. Thermal coductvtes of the P 2O 5 + water system as a fucto of x P2 O 5 1/2 at varous temperatures. The expermetal data are from Aseyev, 32 Redel, 5 Luff ad Wakefeld, 41 Turbull, 61 ad Daubert ad Daer. 28 The les are calculated usg the model. The average percetage devato of the ft s Fgure 1. Thermal coductvtes of the H 2SO 4 + water system as a fucto of temperature at varous weght percets of H 2SO 4. The expermetal data are from Vargaftk ad Os m, 1 Redel, 5 ad Veart ad Prasad. 62 The les are calculated usg the model. The average percetage devato of the ft s Thus, a geeral model for the thermal coductvty of electrolyte solutos ca be postulated to clude the followg two cotrbutos: (1) a cotrbuto of dvdual os ( λ s ), whch s characterzed by o-specfc coeffcets ad ca be terpreted as the effect of o-solvet teractos; ths cotrbuto s a geeralzed verso of Redel s 5 addtvty rule (eq 1); (2) a cotrbuto of teractos betwee os or eutral speces ( λ s-s ). These two cotrbutos are aalogous to those used a prevously developed model for calculatg vscostes of electrolyte systems. 3 Accordgly, the dfferece betwee the thermal coductvty of a electrolyte soluto (λ) ad that of a solvet mxture (λ m ) ca be expressed as λ - λ m ) λ s + λ s-s (12) As wth the vscosty ad electrcal coductvty models developed prevously, 3,31 the effect of solvet composto o the cotrbutos of dvdual os ad o the teractos betwee speces must be take to accout the thermal coductvty model. I eq 12, λ m ca be evaluated usg eqs 2, 8, ad 1 as descrbed the prevous secto. A expresso based o the Redel equato 5 s used to represet the cotrbuto of dvdual os, λ s. However, mole fractos rather tha molar cocetratos have bee selected as more coveet composto varables. For aqueous electrolyte solutos, the use of ether molar cocetratos or mole fractos wth rescaled oc coeffcets R has bee foud to yeld smlar devatos betwee the calculated ad expermetal values. However, a mole fracto based coductvty model does ot ecesstate the use of a separate desty model order to calculate molar cocetratos, whch elmates the possblty of error propagato whe other composto varables are coverted to molar cocetratos. I fact, most thermal coductvty data the lterature are reported as a fucto of ether mass percet or molal cocetrato, whch ca be easly coverted to mole fractos. Thus, a mxedsolvet electrolyte soluto, the dvdual o cotrbuto ca be expressed as λ s ) j x j x R,j (13)

7 574 Id. Eg. Chem. Res., Vol. 47, No. 15, 28 Fgure 11. Thermal coductvtes of the SbCl 3 + acetoe system as a fucto of temperature at varous weght percets of SbCl 3. The expermetal data are from El darov, 63 ad the les are calculated usg the model. The average percetage devato of the ft s.54. Fgure 12. Thermal coductvtes of the ZCl 2 + ethaol system as a fucto of the mole fracto of ZCl 2 at varous temperatures. The expermetal data are from El darov, 63 ad the les are calculated usg the model. The average percetage devato of the ft s.24. where the subscrpt j deotes the solvet compoets, pertas to the solutes (os ad eutral speces), x s the mole fracto of the th speces, R,j s the R-coeffcet of the th speces a pure solvet j, ad x j s the mole fracto of solvet j o a salt-free bass. For a electrolyte soluto wth a sgle solvet, eq 13 reduces to λ s ) x R (13a) whch s a mole fracto based verso of the orgal Redel 5 term. For the λ s-s term, cotrbutos of teractos betwee all speces pars must be cluded. Also, mxed-solvet electrolyte solutos, the effects of dfferet solvets ad ther composto o the speces-speces teractos must be recogzed. To clude these effects, the λ s-s term s expressed as λ s-s ) j l x j x l f f k β k,jl (14) k where the frst ad secod sums (j ad l) are over all solvet compoets, the sums over ad k are over all solutes, x j ad x l are the mole fractos of solvets j ad l o a salt-free bass, Fgure 13. Predcted thermal coductvtes of the ZCl 2 + ethaol + water system as a fucto of temperature at varous mole fractos of ethaol (o a salt-free bass). All solutos cota 1% ZCl 2 (weght). The expermetal data (symbols) are from Abdulagatov ad Magomedov (at x EtOH ) ) 64 ad El darov (at x EtOH ) 1). 63 ad f ad f k are the solute-oly mole fractos of the th ad kth speces, respectvely, adjusted for the charges of speces,.e. f ) x /max(1, z ) x m /max(1, z m ) m (15) ad β k,jl s a bary parameter betwee the speces ad k a solvet mxture j-l. It should be oted that, whe j ) l, β k,ll becomes the -k teracto pure solvet l. The defto of the charge-adjusted fracto f has bee troduced followg a prevous study of mxg rules a vscosty model. 3 The sum eq 15 s over the solute speces ad the factor max(1, z ) esures that f reduces to the mole fracto for eutral speces. I cases whe there s oly oe solvet, such as aqueous electrolyte solutos, eq 14 reduces to λ s-s ) f f k β k k (14a) Parameter Evaluato Iteracto Parameter, k j, the Mxed Solvet λ m Model. The model for the thermal coductvty of mxed solvets (eqs 2, 8, 1) cludes oly a sgle adjustable parameter, k j, whch ca be determed usg expermetal data for bary mxtures. Whle the temperature depedece of the thermal coductvty of the solvet mxture, λ m, s prmarly determed by the varatos wth temperature of the thermal coductvtes of the pure compoets, λ, t has bee foud that the accuracy of the calculated thermal coductvty ca be mproved f a addtoal temperature depedece s troduced to the bary parameter k j : k j ) k () j + k (1) j T (16) The r Coeffcets. The R coeffcets as defed by Redel 5 for aqueous os have bee tradtoally based o molar cocetratos. These coeffcets must be rescaled to work wth the model proposed ths study, whch mole fractos are used as composto varables. The R coeffcets are also solvet-depedet. To determe the R coeffcets eq 13, expermetal thermal coductvty data of aqueous ad oaqueous electrolyte solutos at low or moderate electrolyte cocetratos have bee aalyzed. For aqueous solutos, data for bary systems from prmary lterature sources ad from the complato of Aseyev 32 were used to obta the R

8 Id. Eg. Chem. Res., Vol. 47, No. 15, Fgure 14. Predcted thermal coductvtes of the ZCl 2 + ethaol + water system as a fucto of the mole fracto of ethaol (o a salt-free bass) at 6 C. The expermetal data (symbols) are from Rastorgu ad Gaev (at ZCl 2 ) %), 43 Abdulagatov ad Magomedov (at x EtOH ) ), 64 ad El darov (at x EtOH ) 1). 63 Fgure 15. Thermal coductvtes of the NaCl + CaCl 2 + water system as a fucto of temperature at varous total weght percets of the salts. The mass rato of NaCl:CaCl 2 s 3:1 all solutos. The expermetal data are from Abdullaev et al., 65 ad the les are calculated usg the model. The average percetage devato of the ft s.31. Fgure 16. Thermal coductvtes of the NaCl + CaCl 2 + MgCl 2 + water system as a fucto of temperature at varous compostos ( weght percet) of the salts. The expermetal data are from Magomedov, 66,67 ad the les are calculated usg the model. The average percetage devato of the ft s.74. coeffcets accordg to the formula λ - λ H2 O ) x c R c + x a R a. As Redel s orgal work, 5 the R coeffcet for the Na + o has bee assged a value of ;.e., R Na + ). Wth ths assumpto, coeffcets for all other os ca be determed. I cases where there s a suffcet amout of expermetal thermal coductvty data at relatvely low cocetratos, or systems where specato effects are sgfcat, the R coeffcets have bee treated as adjustable parameters, together wth the teracto parameters β k,jl, ad have bee determed usg thermal coductvty data for bary electrolyte + solvet system. I Redel s orgal work, the thermal coductvty of electrolyte solutos was assumed to have the same temperature depedece as that of pure water. 5,33 At a gve temperature, λ was calculated usg the R coeffcets obtaed at 2 C, together wth the rato of thermal coductvtes of pure water at 2 C ad at the temperature of terest,.e. λ t t ) (λ H2 O/λ H2 O 2 2 )(λ H2 O + R 2 c ) (17) A smlar approxmato was also used by others 1 to estmate thermal coductvtes of aqueous solutos at elevated temperatures by usg λ data at 2 C. Although the temperature depedece expressed by eq 17 ca gve qute relable predctos at temperatures up to 1 C, 1 a explct expresso for the R coeffcets s expected to be more accurate over a wder temperature rage. A temperature-depedet fucto smlar to the oe used for the vscosty B coeffcets 34 has bee determed to be qute effectve for ths purpose ad has bee used ths work to calculate the R coeffcets as a fucto of temperature: R)R 1 +R 2 exp(-k(t - T )) (18) where T s the temperature K, T ) K, ad K has bee set equal to.23. It s of terest to ote that the value of the K coeffcet s the same for the vscosty B coeffcets ad for the thermal coductvty R coeffcets. Ths dcates a certa smlarty the shape of the temperature depedece for both propertes electrolyte solutos. The values of the R coeffcets are lsted Table 1 for selected os. Smlarly, the R coeffcets for os solvets other tha water ca also be determed. However, due to the fact that expermetal data are oly avalable for a lmted umber of oaqueous electrolyte systems, such parameters are much more dffcult to obta tha those for aqueous os. The β k, jl Parameters. For cocetrated solutos, t has bee foud that the parameter β k,jl eq 14 depeds o the oc stregth. A fucto of the form β k,jl ) β (1) k,jl + β k,jl (2) I 2 (3) x + β k,jl exp(β k,jl I x ) (19) has bee selected because t gves the best ft whe thermal coductvty data exted to hgher cocetratos ad whe the

9 576 Id. Eg. Chem. Res., Vol. 47, No. 15, 28 thermal coductvty shows a complex behavor. The quatty I x eq 19 s the exteded, mole fracto based oc stregth defed by eq 2 to clude the cocetratos of eutral o pars (as opposed to solvet molecules), whch typcally become predomat at hgh cocetratos because of specato equlbra. I x ) 1 z 2 2 x + x os o par The temperature depedece of each of the β (m) k,jl (m ) 1, 2, 3) parameters s gve by β (m) k,jl ) β (m) k,jl exp(β (m1) k,jl (T - T )) Results ad Dscusso Thermal Coductvtes of Solvet Mxtures. Expermetal data for a umber of bary ad terary solvet mxtures have bee used for valdatg the correlato descrbed the precedg secto. Table 2 lsts the parameters k j for selected systems, together wth the average percetage error, whch s defed by m AAD ) 1 λ [ exp - λ cal /λ k exp] /m (22) where m s the umber of expermetal data pots. Results for selected bary systems are show Fgures 2 ad 3. To valdate the model agast terary or hgher order systems, the parameters k j obtaed from bary data were used to predct the thermal coductvty of terary systems. The results are llustrated Fgure 4 the form of relatve devatos of the calculated thermal coductvtes from expermetal data for four terary mxtures cotag cyclohexae. The results show these fgures ad Table 2 dcate that the model (eqs 2, 8, ad 1) ca accurately reproduce expermetal data for solvet mxtures of ay composto. Thermal Coductvtes of Electrolyte Solutos. Valdato of the ew thermal coductvty model for the effect of electrolyte cocetrato has bee focused o two classes of systems: (1) aqueous electrolyte solutos (salts, acds, ad bases) ragg from the dlute rego to fused salts or pure acds or bases; (2) electrolytes pure orgac ad mxed solvets. Expermetal thermal coductvty data for aqueous electrolyte systems are avalable from extesve complatos 32,35 ad from other lterature sources. Compared to aqueous solutos, there s much less thermal coductvty data avalable for oaqueous electrolyte systems ad the expermetal coverage s eve sparser for mxed-solvet electrolyte solutos. Noetheless, the avalable lterature data provde a soud bass o whch the ew model ca be tested. For all of the systems for whch the thermal coductvty model has bee tested, thermodyamc model parameters 36 were frst developed to provde accurate specato put for thermal coductvty modelg. Table 3 lsts the adjustable parameters eqs ad, some cases, those eqs 2 ad 1 for selected aqueous ad oaqueous electrolyte systems. The performace of the model for bary aqueous systems s llustrated Fgures 5 1. I these fgures, lterature thermal coductvty data for the systems LCl + water, NaF + water, KNO 3 + water, KOH + water, H 3 PO 4 + water, ad H 2 SO 4 + water are compared wth calculated results at varous temperatures ad electrolyte cocetratos. The average percetage devatos, AAD, as defed by eq 22, are gve the captos to the fgures. The thermal coductvtes of most electrolyte solutos decrease as the cocetrato creases, as show Fgure 5 for aqueous LCl solutos. I cotrast, the thermal coductvtes of some other systems may exhbt a more complex behavor. For example, aqueous solutos of LOH, NaF, NaOH, Na 3 PO 4, ad Na 2 CO 3, the thermal coductvty creases wth rsg cocetrato ad may the decrease after a maxmum s reached, as show Fgure 6 for the NaF soluto. Such complex behavor of thermal coductvty varous bary aqueous electrolyte solutos ca be accurately reproduced by the model. Metal trates water ca be cotuously mscble from fte dluto to the fused salt lmt. Expermetal thermal coductvty data are avalable for such systems over a moderate cocetrato rage,.e. x trate ) (.5-8 mol kg -1 ) 37 ad the lmt of molte salts. 38,39 These data provde a good test case for evaluatg the performace of the model over the full cocetrato rage of electrolyte compoets. Fgure 7 shows the results for the system KNO 3 + water at temperatures ragg from 2 to 338 C ad cocetratos ragg from x w ) tox w ) 1.. Wth the cocetrato ad temperature rage where expermetal data are avalable, the agreemet betwee the calculated ad expermetal values s excellet. The model results betwee the upper ed of the cocetrato rage aqueous solutos (.e., 8 mol kg -1 ) ad the molte salt lmt ca be valdated whe ew expermetal data become avalable. However, the predcted tred appears to be reasoable. Fully mscble aqueous acds ad bases are aother mportat class of mxtures. Because of ther usually strog assocato effects, such systems provde ot oly good test cases, but also offer a excellet opportuty to exame the effect of specato o thermal coductvty. Whe modelg fully mscble acds, both water ad the udssocated acd molecules (e.g., H 2 SO 4, H 3 PO 4, HNO 3 ) have bee treated as solvet compoets. I these systems, specato ca chage dramatcally as acd cocetrato creases. I partcular, a sgfcat amout of eutral acd molecules may exst as the acd cocetrato approaches a mole fracto of uty ad the assocato s early complete a pure acd. 4 The teracto parameters that are used the model for ths type of systems clude the β parameters eq 14 betwee oc speces ad the k j parameters eq 1 betwee the solvet compoets (e.g., H 2 O ad HNO 3 ). For example, the best ft was obtaed for the HNO 3 + H 2 O system whe the parameters β H3 O +,NO 3 - /H 2 O,H 2 O ad k H2 O,HNO 3 were troduced. I the H 3 PO 4 + P 2 O 5 + H 2 O system, oly the k H2 O,H 3 PO 4 ad k P2 O 3,H 3 PO 4 parameters are used to reproduce the data from dlute to extremely cocetrated solutos that go beyod pure H 3 PO 4 (.e., the system H 3 PO 4 + P 2 O 5 ). Specato results for ths system dcated that oc speces are oly mportat at ftely dlute solutos where thermal coductvty of the soluto approaches that of pure water, whle udssocated acd molecules are the predomat speces elsewhere. These udssocated molecules have bee treated as solvets; therefore treatmet of ths system s smlar to those of the solvet mxtures, ad ts thermal coductvty ca be solely represeted by the solvet teracto parameters, k j,. Fgures 9 ad 1 show the results for the H 3 PO 4 + P 2 O 5 + H 2 O ad H 2 SO 4 + SO 3 + H 2 O systems, respectvely, at varous temperatures ad cocetratos. For both systems, data are avalable beyod the pure acd composto. I the moderately cocetrated phosphorc acd solutos (.e., x P2O 5 g.5), data from Aseyev 32 are cosstet wth those of Luff ad Wakefeld. 41 Aseyev s data are smoothed values whle the data of

10 Luff ad Wakefeld are cosstet wth the pure lqud H 3 PO 4 data of Daubert ad Daer, 28 whch were crtcally evaluated. Therefore, Aseyev s data at these cocetratos were excluded from the determato of model parameters, but were plotted Fgure 9 for comparso. Results for the KOH + water system are show Fgure 8. Excellet agreemet betwee expermetal data ad calculated results has bee obtaed for all of the vestgated acds ad bases over wde rages of temperature ad cocetrato. Results for modelg oaqueous electrolyte solutos ca be demostrated usg the SbCl 3 + acetoe ad ZCl 2 + ethaol systems as examples. The effect of specato o the model predctos ca also be aalyzed these cases. The prevalg complex, SbCl 3, acetoe solutos ecesstated the troducto of the β parameters for the {Cl -, SbCl 3 } ad {Sb 3+,SbCl 3 } teractos,.e., β Sb 3+,SbCl 3 /acetoe,acetoe ad β Cl -,SbCl 3 /acetoe,acetoe, to reproduce the expermetal results wth expermetal ucertaty. The results for ths system are show Fgure 11. Smlarly, thermal coductvtes for the system ZCl 2 + ethaol ca be accurately reproduced whe the teracto parameters β Z 2+,ZCl 2-4 /EtOH,EtOH ad β Z 2+,Cl - /EtOH,EtOH are used, as show Fgure 12. Due to the lack of expermetal data for electrolytes mxed orgac + water systems, oly predctos ca be made usg the parameters obtaed from the costtuet bary solutos. The predcted thermal coductvtes for the ZCl 2 + ethaol + water system are show Fgures 13 ad 14, where the results for ZCl 2 pure water ad pure ethaol are also plotted to llustrate the predcted treds wth chagg solvet composto. The model predcts that the solvet composto has the most sgfcat effect o thermal coductvty compared to the effects of electrolyte cocetrato ad temperature. It also predcts a crossover the λ vs x EtOH plot (Fgure 14) due to the opposte treds of λ wth respect to ZCl 2 cocetrato water ad ethaol. Whe expermetal data become avalable for the mxed system, the model results may be mproved, f ecessary, by troducg addtoal teracto parameters that are pertet to the mxed solvet, such as β Z 2+,Cl - /EtOH,H 2 O. The capablty of the model for predctg thermal coductvtes multcompoet electrolyte solutos has also bee tested ad s demostrated Fgures 15 ad 16 for the terary system NaCl + CaCl 2 + water ad the quaterary system NaCl + CaCl 2 + MgCl 2 + water. I each case, λ shows a temperature depedece that s smlar to that observed for pure water;.e., a maxmum value s reached at approxmately 14 C 29 at varous fxed electrolyte cocetratos. Also, λ decreases as the total electrolyte cocetrato creases these systems. Ths behavor has bee accurately reproduced. It should be oted that, for the multcompoet electrolyte solutos tested ths work, thermal coductvtes ca be geerally predcted wth 2.% usg oly parameters from bary fts, wth most pots beg wth 1%. A further mprovemet ca be obtaed whe lke-o teractos are troduced. Cocluso A geeral model has bee developed for calculatg the thermal coductvty of aqueous, oaqueous, ad mxed-solvet electrolyte solutos. The model cossts of two ma parts,.e., a correlato for computg the thermal coductvty of solvet mxtures as a fucto of temperature ad solvet composto, ad a expresso for the effect of electrolyte cocetrato. The correlato for the solvet mxtures has bee derved from the local composto cocept. It has bee subsequetly smplfed to use oly the surface area parameters ad thermal coductvtes for pure compoets as well as a sgle adjustable parameter for each bary par. It has bee show to be very effectve for represetg expermetal data for a varety of solvet mxtures. I partcular, the thermal coductvty of terary solvet mxtures ca be accurately predcted usg parameters determed from oly bary data. To represet the depedece of thermal coductvty o electrolyte cocetrato, the model cludes a cotrbuto of dvdual os ( λ s ), as quatfed by a Redel-type coeffcet, ad a cotrbuto of specfc teractos betwee os or eutral speces ( λ s-s ). Formulatos have bee developed for both the λ s ad λ s-s terms to accout for the effects of multple solvets. The thermal coductvty of multcompoet electrolyte solutos ca be predcted wth 2% by usg parameters derved from oly bary data. The predctos ca be further mproved by troducg lke-o teractos. The thermal coductvty model has bee coupled wth a thermodyamc equlbrum model 36 to provde specato, whch s ecessary for thermal coductvty calculatos for a large class of electrolyte systems. Ths makes t possble to reproduce the effects of complexato or other reactos the soluto. I all cases whch expermetal data are avalable, the ew model has bee show to be accurate for reproducg thermal coductvtes over wde rages of temperature ad cocetrato. It should be oted that although the thermal coductvty model descrbed ths paper does ot explctly gve the pressure depedece, t should be applcable to hgher pressures wth good accuracy, as log as the pressure effect the solvet thermal coductvtes are correctly accouted for. Ths ca be demostrated by comparsos made ths study for pure water ad for aqueous LCl solutos where expermetal data are avalable up to 1 MPa. 59 At 2 C, thermal coductvty of pure water creases from.663 W m -1 K -1 at saturated vapor pressure (1.56 MPa) to.733 W m -1 K -1 at 1 MPa; whle the crease the thermal coductvty of the.1 mass fracto LCl soluto uder the same codtos s from.637 to.7 W m -1 K The calculated λ ths soluto s.638 W m -1 K -1 at saturated vapor pressure ad.71 W m -1 K -1 at 2 MPa usg the ew model, wthout explctly cludg a pressure-depedet term. Ths dcates that the pressure effect o thermal coductvty of electrolyte solutos ca be adequately represeted by that of the solvet. A mproved accuracy calculatg the pressure effect o the thermal coductvty ca be obtaed by troducg a pressure depedece teracto parameters. Ackowledgmet The work reported here was supported by Alcoa, DuPot, Mtsubsh Chemcal, Nppo Chemcal, Rohm & Haas, ad Shell. Lterature Cted Id. Eg. Chem. Res., Vol. 47, No. 15, (1) Horvath, A. L. Hadbook of Aqueous Electrolyte Solutos. Physcal Propertes, Estmato ad Correlato Methods; Joh Wley & Sos: New York, (2) Bearma, R. J. Cotrbuto of Iteroc Forces to the Thermal Coductvty of Dlute Electrolyte Solutos. J. Chem. Phys. 1964, 41 (12), (3) Bearma, R. J.; Vadhyaatha, V. S. Theory of the Sgle-Io Heat of Trasport Nosothermal Electrolytc Solutos. J. Chem. Phys. 1963, 39 (12),

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