Actvty Coeffcent Calculaton for Bnary Systems Usng UNIQUAC roject Work n the Course Advanced Thermodynamcs: Wth Applcaton to hase and Reacton qulbra K8108 Department of Chemcal ngneerng, NTNU Ugochukwu. Aronu July, 009
Table of Contents 1.0 Introducton 3.0 Thermodynamc Framework 4.1 Thermodynac qulbrum Condton 4. artal Molar Gbbs Free nergy 5.3 Gbbs-Duhem quaton 6.4 qulbrum n Heterogenous Closed System 7.5 Chemcal otental 8.6 Fugacty and Fugacty Coeffcent 9.7 Actvty and Actvty Coeffcent 10.7.1 Normalzaton of Actvty Coeffcent 11.8 xcess Functons 1 3.0 Vapour Lqud qulbrum Calculaton 13 3.1 Chemcal qulbra 14 3.1.1 Chemcal qulbrum Constant 15 3. hase qulbra 15 3.3 G Model for Actvty Coeffcent 17 3.3.1 Local Composton Models 18 4.0 UNIQUAC 18 4.1 UNIQUAC Implementaton 0 4.1.1 MA-H0 System 1 4.1. Acetc Acd-H0 System 1 4.1.3 Other Systems 1 5.0 Results 5 Summary 5 Future Work 5 References 5 Dervaton of Actvty Coeffcent xpresson Based on UNIQUAC 7
1.0 Introducton Desgn of ndustral chemcal process separaton equpment such as absorpton and dstllaton columns as well as smulaton of chemcal plants requre reasonably accurate correlaton and predcton of phase equlbra because t s an ntegral part of vapour lqud equlbrum (VL) modelng. It s thus necessary to develope thermodynamc models sutable for practcal phase equlbrum calculatons. In general, VL models are based on fundamental equatons for phase and chemcal equlbra. The basc quanttes requred for VL calculatons are; chemcal reacton equlbrum constants, Henry s law constant, fugacty coeffcents and actvty coeffcents (oplstenova, 004), phase equlbra calculaton requres Henry s law constant, fugacty coeffcents and actvty coeffcents. The dfferences n VL models are manly n the way the phase non-dealtes are treated. In the vapor phase non-dealty s ether neglected or represented by a fugacty coeffcent calculated from one of the well-establshed equatons of state such as Soave-Redlch-Kwong equaton of state, the models wll then dffer on the type of equaton of state that were appled. These dfferences, however are of mnor mportance snce the calculaton of vapor phase fugactes are not crucal for the model performance. The treatment of the lqud phase, on the other hand, s very mportant. Accurate predcton of the lqud phase composton plays a key role n the VL modelng. Snce the evaluaton of equlbrum constants and fugacty coeffcents are reasonably well establshed, the actvty coeffcents were dentfed as the most mportant varables of the VL model. Thus proper representaton of actvty coeffcents s desrable (oplstenova, 004). The system of nterest n my research group s manly CO -H 0-Alkanolamne. An actvty coeffcent model for such system must be able to represent the non-dealty of an electrolyte soluton. Some actvty coeffcent models for non-electrolyte systems nclude, Wlson, NRTL, UNIFAC, UNIQUAC. Some have been modfed for electrolyte systems. Revew of electrolyte actvty coeffcent models were present by Maurer, 1983; Renon, 1986 and Anderko et al., 00. For the purpose of ths course the unversal quas-chemcal (UNIQUAC) equatons proposed by Abrams and rausntz, 1975 s used n calculatng actvty coeffcent for non-electrolyte systems. Further work wll nvolve extendng the model for calculaton actvty coeffcent for electrolyte solutons. 3
.0 Thermodynamc Frame Work Vapour lqud equlbra calculaton requres smultaneous soluton of phase and chemcal equlbra for reactve systems. Ths secton shows brefly the mathematcal framework for the VL calculatons whch s bult around basc concepts of thermodynamc. Most of the dscussons here were taken from llot and Lra, 1999; rausntz et al., 1999; Km, 009; Hartono, 009 and oplstenova, 004..1 Thermodynamc qulbrum Condton Homogenous Closed system A homogeneous s a system wth a unform propertes e. propertes such as densty s same from pont to pont, n a macroscopc sense example s a phase. A closed system do not exchange matter wth the surroundng eventhough t may exchange energy (rausntz et al., 1999). An equlbrum state s one wth no tendency to depart spontenously (havng n mnd certan permssble changes or process such as heat transfer, work of volume dsplacement and for open systems mass transfer across phase boundry). It s propertes are ndependent of tme. A change n equlbrum state of a system s called a process and a reversble process s one that mantans a state of vrtual equlbrum throughout the process, t s often referred to as one connectng a seres of equlbrum states. For a reversble process, the general condton for thermodynamc equlbrum s derved by combnaton of frst and second law of thermodynamcs as: du TdS dv 1 Ths condton could be expressed n terms of all extensve thermodynamc functons; nternal energy (U), enthalpy (H), Helmholtz energy (A) and Gbbs free energy (G): The condton for equlbrum s usually mantaned by keepng two of the thermodynamc varables constant. Condtons for equlbrum s often expressed n terms of Gbbs free energy because the two constant varables are often the temperature and pressure. Usng the fundamental thermodynamc relaton for Gbbs free energy G U TS V 4
q. 1 could be wrtten as dg SdT Vd 3 Thus at constant temperature and pressure dg 0 4 whch mples that for thermodynamc equlbrum at constant T and, the Gbbs free energy of a closed system reaches ts mnmum.. artal Molar Gbbs Free nergy Homogenous open System For a closed system, U s consdered a functon of S and V only; that s U = U(S, V) but for open system, there are addtonal ndepedent varables. For these varables we use the mole numbers of the varous compnents present, thus we consder U as the functon (,,,..., ), where m s the number of components present. Total dfferental s U U S V n1 n n m then U U U du ds dv dn S V S Vn, Sn, SVn,, j 5 where subsrpt n refers to all mole numbers and n j to all mole numbers other than th (the term under dfferencaton). Wrtng eq. 5 n the form du TdS dv + dn 6 where U n SV,, nj q. 6 s the fundamental equaton for an open system correspondng to eq. 1 for a closed system. The functon μ s an ntensve property whch depends on temperature, pressure and composton from ts poston n the equaton as the coeffcent of dn t can be refered to as mass or chemcal potental just as T (the coeffcent of ds) s a thermal potental and (the 5
coeffcent of dv) s a mechancal potental. Smlar expresson can be derved n terms of Gbbs free energy G (and other extentve thermodynamc propertes). dg SdT Vd + dn 7 where U H A G n n n n SV,, n j Sn,, j TV,, n j T,, nj 8 The quantty s the partal molar Gbbs free energy, but not partal molar nternal energy, enthalpy, or Helmholtz energy, because the ndependent varables T and chosen for defnng of partal molar quanttes are also fundamental ndependent varables for the Gbbs free energy G (rausntz et al., 1999). The quantty, G n T,, nj, the partal molar Gbbs free energy s also called the chemcal potental..3 Gbbs-Duhem quaton At constant temperature and pressure eq. 7 reduces to dg = dn 9, T For equlbrum, at constant and T, the Gbbs free energy s mnmzed (.e dg = 0) (llot and Lra, 1999). Also for a closed system dn = 0 thus equaton 7 equal to zero at equlbrum. From equaton 9, equlbrum condton n terms of chemcal potental could then be deduced dg dn = 0 10, T Ths s called Gbbs-Duhem quaton. It s a thermodynamc consstency relaton for a heterogenous system that s used for expermental data evaluton and theory development. It expresses the fact that among + varables consstng of temperature, pressure and 6
chemcal potentals of each component present n the system. Only +1 are ndependent varables and the last varable s a dependent varable calculated n such a way that Gbbs- Duhem equaton s satsfed. In terms of actvty coeffcent Gbbs-Duhem equaton s expressed as xd ln 0 11 quaton 11 s a dfferental relaton between the actvty coeffecents of all the components n soluton (rausntz et al., 1999). For a bnary soluton, the Gbbs-Duhem equaton may be wrtten as: dln dln x x 1 1 1 dx1 dx.4 qulbrum n a Heterogeneous Closed System A heterogeneous, closed system s made up of two or more phases wth each phase consdered an open system wthn the overall closed system. If the system s n nternal equlbrum wth respect to the three processes of heat transfer, boundry dsplacement and mass transfer, neglectng specal effects such as surface forces; sempermeable membranes; and electrc, magnetc or gravtatonal forces (rausntz et al., 1999), the general result for a closed system consstng of phases (where two of the phases are lqud (L) and vapour (V)) can be wrtten as v L T T... T v L... v L... 13 Ths s the basc crteron for phase equlbrum, whch states that at equlbrum, the temperature, pressure and chemcal potentals of all speces are unform over the whole system. 7
.5 Chemcal otental The task of phase-equlbrum thermodynamcs s to descrbe quanttatvely the dstrbuton at equlbrum of every component among all the phases present. Gbbs obtaned the thermodynamc soluton to the phase-equlbrum problem by ntroducng the abstract concept of chemcal potental. The task s then to relate the abstract chemcal potental of a substance to physcally measurable quantty such as temperature, pressure and composton. For a pure substance, the chemcal potental s related to the temperature and pressure by the dfferental equaton d sdt vd 14 where s s the molar entropy and v the molar volume. Integratng and solvng for at some temperature T and pressure, we have r r ( T, ) ( T, ) sdt vd r T T 15 r where superscrpt r s an arbtrary reference state. The ntegrals can be solved from thermal r r and volumetrc data over temperature range T to T and pressure range to but the r r chemcal potental ( T, ) s unknown. Chemcal potental at T and can thus only be r r evaluated relatve to an arbtrary reference states gve as T and these refence states are often known as standard states. Chemcal potental does not have an mmedate equvalent n the physcal world and t s desrable to express the chemcal potental n terms of some auxlaary functon that mght be more easly dentfed wth physcal realty. The term fugacty (f) was ntroduced by G.N. Lews n tryng to smplfy the equaton of chemcal equlbrum by frst consderng the chemcal potental for a pure, deal gas and then generalzed the result to all systems (rausntz et al., 1999). Auxlary thermodynamc functons such as fugactes and actvtes are often used n thermodynamc treatment of phase equlbra. 8
.6 Fugacty and Fugacty Coeffcent The fugacty of component n a mxture s defned as (llot and Lra, 1999) RTd In f d at constant T 16 where f s the fugacty of component n a mxture and s the chemcal potental of the component. For a pure deal gas, the fugacty s equal to the pressure, and for a component n a mxture of deal gases, t s equal to ts partal pressure y. The defnton of fugacty s completed by the lmt: f 1 as y 0 17 By ntegratng eq. 16 at constant T for any component n any system, sold, lqud or gas or pure mxed, deal or non-deal. For vapour phase we have o f RT ln 18 o f Whle ether o and f o s arbtrary, both may not be chosen ndependently; when one s chosen, the other s fxed. Wrtng an analogous expresson for the lqud and vapour phase and equatng the chemcal potentals usng equaton 13 obtan: v v L f RT ln 0 19 L f Ths transformaton consequently leads to addtonal crtera for equlbrum called sofugacty: v L f f... f 0 Ths tell us that the equlbrum condton n terms of chemcal potental can be replaced wthout loss of generalty by equaton n terms of fugacty. 9
Fugacty coeffcent, s the rato of fugacty to real gas pressure. It s a measure of nondealty. f y 1 It s a way of characterzng the Gbbs excess functon at fxed T,. For a mxture of deal gases = 1..7 Actvty and Actvty Coeffcent Actvty concept s an alternatve approach to express the chemcal potental n a real soluton. The actvty of component at gven temperature, pressure and composton s defned as the rato of the fugacty of at these condtons to the fugacty at standard state. Actvty of a substance gves an ndcaton of how actve a substance s relatve to ts standard state, t s expressed as: a f f 0 Substtutng equaton nto 18 gves relatonshp between chemcal potental and actvty. RTln a 3 0 A general expresson for the chemcal potental n an deal soluton n terms of deal mxng could be wrtten as RTln x 4 d 0 Actvty coeffcent gves a measure of non-dealty of soluton, t s the rato of actvty of component to ts concentraton, usually the mole fracton a 5 x 10
In an deal soluton the actvty s equal to the mole fracton and the actvty coeffcent s equal to unty. Introducng equaton (5) nto (3) 0 RT ln x RT ln 6 Actvty coeffcent relates chemcal potental n an deal soluton to the chemcal potental n a real soluton, thus representng a measure of non-dealty as llustrated by combnng equatons (6) and (4) d ln RT 7.7.1 Normalzaton of Actvty Coeffcent It s convenent to defne actvty n such a way that for an deal soluton actvty s qual to the mole fracton or equvalently, that the actvty coeffcent s equal to unty. Because we have two types of dealty (one leadng to Raoult s law and the other leadng to Henry s law), t mples there wll be two ways of normalzng actvty coeffcent. Symmetrc Conventon Ths conventon apples when all the components both solutes and solvent at the system temperature and pressure are lquds n ther pure state (reference state). The actvty coeffcent of each component then approaches unty as ts mole fracton approaches unty. Ths conventon leads to an deal soluton n the Raoult s law sense. It follows that: 1 as x 1 8 Unsymmetrc Conventon Ths conventon apples when pure component cannot be used as a reference state for nstance when some component are sold or gaseous at the system temperature and pressure. In ths case t s convenent to defne the reference state as the nfnte dlute state of the component at system temperature and reference pressure. Ths conventon therefore leads to an deal dlute soluton n the sense of Henry s law. 11
for solvent 1 as x 1 9 s for onc and molecular solutes 1 as x 0 30 Subscrpts s and refer to solvent and solute respectvely whle astersk(*) shows that the actvty coeffcent of the solute approaches unty as mole fracton approaches zero. Ths conventon s sad to be unsymmetrc because solvent and solute are not normalzed n same way. s.8 xcess Functons xcess functons are thermodynamc propertes of a soluton that are n excess of an deal (or deal dlute) soluton at the same condtons of temperature, pressure and composton. For an deal soluton all excess propertes are zero (rausntz, 1999). A general excess functon s defned as real deal e e e 31 One partcularly mportant excess functon s the excess Gbbs energy ( G ) defned by G G( actual soluton att, and x) G( deal solutonat the samet, and x) 3 Smlar defntons hold for excess volume V, excess entropy S, excess enthalpy H, excess nternal energy U, and excess Helmholtz energy A. Relatons between these excess functons are exactly the same as those between the total functons, for example: G H TS 33 Also, partal dervatve of extensve excess functon are analogous to those of the total functons. For example: G T x, S 34 1
artal molar excess functons are defned n a manner smlar to partal molar thermodynamc propertes. If M s an extensve thermodynamc property, then component, s defned by m, the partal molar M of m M n T,, n j smlarly m M n T,, nj 35 From uler s theorem, we have that M n m smlary M n m 36 From excess functon defnton, t can be seen from eq. 7 that RT ln s equal to excess chemcal potental. Snce chemcal potental at constant T and s equal to the partal molar Gbbs free energy as shown n secton., we obtan a very mportant relatonshp between the actvty coeffcent and the partal molar excess Gbbs free energy: g RTln 37 Usng equaton (37) n (36) gves an equally mportant relaton g RT xln 38 q. 38 forms the bass for calculaton of actvty coeffcents from descrbed n secton 3.3. G models as wll 3.0 Vapour Lqud qulbrum Calculaton Vapour lqud equlbrum calculaton requres smultanoeaus soluton of chemcal and phase equlbra, actvty and fugacty coeffcents s requred n chemcal equlbra calculatons whle fugacty coffcent s requred n phase equlbra. Both are used to express chemcal potentals n lqud and vapour phase respectvely 13
3.1 Chemcal qulbra Molecular electrolytes dssocates or react n the lqud phase to produce onc speces to an extent governed by the chemcal equlbrum. Chemcal reacton n the lqud phase enhances the mass transfer rate and the solublty of CO and thus affects the phase equlbrum and vce versa, the dstrbuton of speces between the two affects the chemcal equlbrum n the lqud phase. A system s n equlbrum when there s no drvng force for a change of ntensve varables wthn the system. Chemcal reacton moves towards a dynamc equlbrum n whch both reactants and products are present but have no further tendency to undergo net change. The generalzed chemcal reacton: A A... A A... 39 1 1 m m m 1 m 1 could be wrtten as (Ott and Goates, 000): A 0 40 where the coeffcents are postve for the products of the reacton and negatve for the reactants. The condton for equlbrum n a chemcal reacton s gven by: 0 41 The Gbbs free energy change n the chemcal reacton s gven by: o v G G RTln a 4 r r where o G r s the Gbbs free energy change wth reactants n ther standard states. 14
3.1.1 Chemcal qulbrum Constant The chemcal equlbrum s tradtonally defned by a chemcal equlbrum constant. At equlbrum, G r = 0 so that equaton eq. 4 becomes, (Ott and Goates, 000): o G RTln K 43 r where K s the equlbrum constant and s gven by: v K a 44 The actvtes n eq. (44) are now the equlbrum actvtes. An equlbrum constant K expressed n terms of actvtes (or fugactes) s called a thermodynamc equlbrum constant. qulbrum constants are requred for each of the reactons occurng n solutons. They are related to the actvtes of each speces as: bb cc dd e 45 K a a 46 a a d D b B e c C Here the lower case letters are the stochometrc coeffcents,, and the captal letters are labels for the chemcal speces. 3. hase qulbra The solublty of gas n a lqud s often proportonal to ts partal pressure n the gas phase, provded that the partal pressure s not large. The equaton that descrbes ths s known as Henry s law: y Hx 47 where Henry s constant (H) s the constant of proportonalty for any gven solute and solvent, dependng only on temperature (rausntz et al., 1999). At hgh partal pressures, Henry s constant must be multpled by actvty coeffcent and pressure by fugacty coffcent. 15
A measure of how chemcal speces dstrbutes tself between lqud and vapour phase, s the rato (erry, 1997): y f K 48 x The condton for phase equlbrum n a closed heterogeneous system at constant temperature and pressure s gven by eq. 0. v L f f 49 Where f and f v L are fugactes of component n the vapor and lqud phase respectrvely. From eq. 48 or by nsertng eqs. 1,, and 5 nto eq. 49 we obtan: y x f 50 ol where and are fugacty and actvty coeffcent respectvely. f ol s reference state fugacty coeffcent defned ether by symmertc conventons Raoult s law or unsymmetrc conventon Henry s law. A correcton term used to relate fugactes at the dfferent pressures s called oyntng factor, wrtten as and at constant temperature and pressure change from 1 to can further be shown to be: p f( T, ) vd exp f1( T, 1) RT 1 51 The complete phase equlbra for vapour lqud equlbrum of solute and solvent at systems temperature and pressure can thus be respectvely wrtten as (Austgen 1989): o v ( H O) For solute: y xh exp 5 RT 16
For solvent: o o o o v( s ) sy s sx s ss exp RT 53 where H and v represent Henry s law constant, partal molar volume of molecular solute at nfnte dluton n pure water at the system temperature and at saturaton water vapour pressure, whle v s s the molar volume of pure solvent at system temperature and o HO saturaton pressure. The exponental correcton (oynng factor) here takes nto account the o fact that lqud s at a pressure dfferent from the saturaton pressure. For molecular solutes such as carbon doxde, Henry s constant represents reference-state fugactes. To solve the phase equlbrum equaton, we need to evaluate the fugacty and actvty coeffcents. Gas phase fugacty coeffcent can be calculated from equaton of state such as Soave-Redlch-Kwong equaton of state. The real modelng task les n the calculaton of actvty coeffcent. In ths work actvty coeffcent calculaton usng, UNIQUAC s presented. 3.3 G Model for Actvty Coeffcent Non-dealty n lqud phase s represented by actvty coeffcent as descrbed n secton.8. They are usually obtaned from excess Gbbs free energy models usng eq.38. An approprate excess Gbbs energy functon most take nto consderaton the molecular nteractons between all speces n the system. For electrolyte solutons, dverse speces are usually present and nteractons among them must be represented. At hgh concentratons, nteractons between neutral molecules or between ons and neutral molecules are very short-range n character and domnates whle at low concentratons t s nteractons between ons whch are very strong long-range electrostatc nteractons that domnates. The usually practce s to assume that the contrbutons of the varous types of nteractons are ndependent and addtve (oplstenova, 004). The excess Gbbs energy s then calculated as the sum of short-range and long-range contrbutons: G GSR GLR 54 17
Most modelng applcatons combne the Debye-Hückel electrostatc theory for the long-range term wth modfcatons of well known non-electrolytes models for the short-range term. In ths work the local composton model for non-electrolytes, UNIQUAC wll be presented n more detal snce they were appled n ths work. 3.3.1 Local Composton Models Regular soluton theory assumes that the mxture of nteractons were ndependent of each other such that quadratc mxng rules provde reasonable approxmatons. However n some cases, the mxture nteracton can be strongly coupled to mxture composton. One way of treatng ths s to recognze the possblty that the local compostons n the mxture mght devate strongly from the bulk composton (llot and Lra, 1999). Some of the well-known local composton models for non-electrolytes are Wlson, NRTL, UNIFAC and UNIQUAC for these models to be used n electrolyte solutons, several assumptons have to be made regardng the local composton n the presence of ons. 4.0 UNIQUAC For ths work the UNIQUAC equaton descrbed by Abrams and rausntz, 1975 was derved, mplemented and used for calculaton of actvty coeffcents for non-electrolyte systems. Man advantages of UNIQUAC s that t uses only two adjustable parameters per bnary to obtan relable estmates for both vapor-lqud and lqud-lqud equlbra for a large varety of multcomponent systems usng the same equaton for the excess Gbbs energy (Abrams and rausntz, 1975). xpermental data for typcal bnary mxtures are usually not suffcently plentful or precse to yeld three meanngful bnary parameters, varous attempts were made (Abrams and rausntz, 1975; Maurer and rausntz, 1978; Anderson, 1978; Kemeny and Rasmussen, 1981) to derve a two-parameter equaton for g that retans at least some advantages of Wlson equaton wthout restrcton to completely mscble mxtures. Abrams derved an equaton that n a sense, extends the quaschemcal theory of Guggenhem for nonrandommxtures to solutons contanng molecules of dfferent sze. Ths s called unversal quas-chemcal (UNIQUAC) theory. 18
UNIQUAC equaton for g conssts of two parts combnatoral part that descrbe the domnant entropc contrbuton, and resdual part that accounts for ntermolecular forces whch are responsble for the enthalpy of mxng. The combnatoral part s determned only by composton and by the szes and shapes of the molecules, t requres only pure-component data. The resdual part depends on ntermolecular forces, thus the two adjustable bnary parameters appear only n the resdual part. The UNIQUAC equaton s g g g RT RT RT combnatoral resdual 55 For a bnary mxture g 1 z 1 x1ln xln xq 1 1ln xqln RT x1 x 1 combnatoral 56 g RT resdual xq ln( ) xq ln( ) 1 1 1 1 1 1 57 where coordnaton number z s set equal to 10. Segment fracton,, and the area fractons,, are gven by 1 xr 11 x r x r 11 xr x r x r 11 58 1 xq 1 1 x q x q 1 1 xq x q x q 1 1 59 The parameters r and q are the two pure-component structural parameter per component representng volume and area respectvely. They are dmensonless and are evaluated from bond angles and bond dstances. For a bnary mxture, there are two adjustable parameters, 1 and 1. They are gven n terms of charaterstc energes u1 u1 u ; u u u. 1 1 11 u1 1 exp RT u1 1 exp RT 60 19
For many cases eq. 60 gves the prmary effect of temperature on 1 and 1. u1 and u1 are often weakly dependent on temperature. Wrtng eq. 38 n terms of mole number n : ng T RT n ln 61 where n T s the total number of moles. The fnal expresson for actvty coeffcent s obtaned by takng the partal dervatve of excess Gbbs energy g wth respect to mole number. ng T RT ln n Tn,, j ( jì) 6 For a multcomponent system, actvty coeffcent expresson based on UNIQUAC s then derved to be: k k z k k ln k ln 1 q k ln 1 xk xk k k qk 1 ln j jk 63 k j j j j Detaled dervaton of ths UNIQUAC expresson for actvty coeffcent s shown n Appendx 4.1 UNIQUAC Implementaton The expresson for actvty coeffcent was added to the thermodynamc functon lbrary developed by Tore Haug-Warberg and Bjørn Tore Løvfall at Department of Chemcal ngneerng, NTNU. The model was coded n an n-house language called RGrad supportng automatc gradent calculatons, and from there exported to C-code whch s compled nto a set of DLL s accessble from Matlab, Octave and Ruby (Haug-Warberg, 008). 0
It has a folder <MODL>/src/mex that contans two makefles whch comples and set up the model called <model>. The model s then smply run as <model>_mexmake or <model>_octmake dependng on whether you want a matlab/m nterface or an Octave nterface. 4.1.1 MA-H O System Monoethanolamne(MA) s an mportant solvent for CO absorpton, so for ths work, UNIQUAC equaton was appled n calculaton of actvty coeffcent n bnary system of MA and water. In the model, the parameter adjustable energy nteracton parameters the UNIQUAC enthalpy term are assumed to be temperature dependent and are ftted to the followng temperature functon ( u j ) of u u u ( T 98.15) 64 0 T j j j o T The r and q parameters for MA and water as well as and were taken from Faramarz o T et al. 009. Other parameter values were set to large values whle parameters were set to u j zero. Result from the calculaton was compared wth lterauture data of Tochg, K et al., 1999, Belabbac, A et al. 009 and Km, I et al., 008 n fgure 1. Molar xcess Gbbs energy functon was also calculated and presented n fgure 5. u j u j u j 4.1. Acetc Acd-H0 System The r and q parameters as well as energy nteracton parameter a j for acetc acd-water system were taken from rausntz, et al. 1999. The actvty coeffcent plot for acetc acdwater system s shown n fgure. The molar excess Gbbs energy functon was also calculated and presented n fgure 6. 4.1.3 Other Systems The predcted model for formc acd acetc acd sysems as well as acetone chloroform syetems are presented n fgure 3 and 4 respectvely. All parameters were taken from rautntz et al. 1999. It has not been easy to get expermental data to use to compare the model result from ths work, however the shapes of the plots agrees very well wth plots n rausntz, et al. 1999. 1
5.0 Results 1 0.9 0.8 0.7 0.6 0.5 0.4 Tochg K. et al. 1999 Belabbac A. et al. 009 Km I. et al. 008 H0 Ths work MA Ths work 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xma Fgure 1: Actvty coeffcent plot for MA-H0 system 0.6 0.5 0.4 0.3 lls & Bahar 1956 Sebastan & Lacquant 1967 Hansen et al. 1955 Arch & Taglavn 1958 Marek & Standart 1954 H0 Ths work ACTIC Ths work log 0. 0.1 0-0.1 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xh0
Fgure : Actvty coeffcent plot for Acetc Acd H0 system. 1.35 1.3 ACTIC FORMIC 1.5 1. 1.15 1.1 1.05 1 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xformic Fgure 3: Actvty Coeffcent plot for Acetc Acd Formc Acd System 1 0.9 0.8 CHLOROFORM ACTON 0.7 0.6 0.5 0.4 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xacton Fgure 4: Actvty Coeffcent plot for Acetone -Chloroform System 3
0-100 xcess Gbbs nergy (g) -00-300 -400-500 -600-700 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xma Fgure 5: xcess molar Gbbs free energy plot for MA-H0 System 50 00 xcess Gbbs nergy g 150 100 50 0 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xactic Fgure 6: xcess molar Gbbs free energy plot for Acetc Acd -H0 System 4
Summary UNIQUAC actvty coeffcent model was derved and successfully mplemented for calculatng actvty coeffcent for dfferent non-electrolyte systems. Lterature actvty coeffcent data for MA-H0 system appear to vary among the dfferent authors presented, the model appear to ft much better to the data from Tochg K et al 1999. Actvty coeffcent data for acetc acd H0 system however s more consstent. Model predcton agrees well wth the lterature values. Future Work Further work wll nvolve the calculaton of actvty coeffcent for electrolyte systems by extendng the UNIQUAC model for electrolyte system through addton of the long-range term. References Abrams, Dens S.; rausntz, John M., 1975; Statstcal thermodynamcs of lqud mxtures. New expresson for the excess Gbbs energy of partly or completely mscble systems. AICh Journal, 1(1), 116-8. Anderko, Andrzej; Wang, emng; Rafal, Marshall., 00; lectrolyte solutons : from thermodynamc and transport property models to the smulaton of ndustral processes. Flud hase qulbra, 194-197 13-14. Anderson, T. F.; rausntz, J. M., 1978; Applcaton of the UNIQUAC equaton to calculaton of multcomponent phase equlbrums.. Lqud-lqud equlbrums. Industral & ngneerng Chemstry rocess Desgn and Development, 17(4), 561-7. Arch, Gudo; Taglavn, Guseppe., 1958; Lqud-vapor equlbrum sotherms for the water-acetc acd system. Rcerca sc., 8 493-500. Belabbac, Aoucha; Razzouk, Antono; Mokbel, Ilham; Jose, Jacques; Negad, Latfa., 009; Isothermal Vapor - Lqud qulbra of ( Monoethanolamne + Water ) and (4- Methylmorpholne + Water ) Bnary Systems at Several Temperatures. Journal of Chemcal & ngneerng Data ACS ASA. llot, J.R.; Lra, C.T., 1999; Introductory Chemcal ngneerng Thermodynamcs; rentce Hall TR: New Jersey. lls, S. R. M.; Bahar,.., 1956 Vapor-lqud equlbrum at low concentratons; acetc acd-water; ntrc acd-water. Brtsh Chemcal ngneerng, 1 10-11. 5
Faramarz, Lela; Kontogeorgs, Georgos M.; Thomsen, Kaj; Stenby, rlng H., 009; xtended UNIQUAC model for thermodynamc modelng of CO absorpton n aqueous alkanolamne solutons. Flud hase qulbra (009), 8(), 11-13. Hansen, Robert S.; Mller, Frederck A.; Chrstan, Sherrl D., 1955; Actvty coeffcents of components n the systems water-acetc acd, water-proponc acd, and water-butyrc acd at 5 Journal of hyscal Chemstry, 59 391-5. Hartono, Ard; 009, Characterzaton of dethylenetramne(dta) as absorbent for carbon doxde. h.d thess., Chemcal ngneerng Department, Norwegan Unversty of Scence and Technology, Trondhem, Norway. Kemeny, Sandor; Rasmussen, eter., 1981; A dervaton of local composton expressons from partton functons. Flud hase qulbra, 7(), 197-03. Km, Inna; 009, Heat of reacton and VL of post combuston CO absorbents. h.d thess., Chemcal ngneerng Department, Norwegan Unversty of Scence and Technology, Trondhem, Norway. Marek, Jan; Standart, George., 1954; ffect of assocaton on lqud-vapor equlbra. I. qulbrum relatons for systems nvolvng an assocatng component. Collecton of Czechoslovak Chemcal Communcatons, 19 1074-84. Maurer, G., 1983; lectrolyte solutons. Flud hase qulbra (1983), 13 69-96. Maurer, G.; rausntz, J. M., 1978; On the dervaton and extenson of the UNIQUAC equaton. Flud hase qulbra, (), 91-9. Ott, J. B., Boero-Goates J., 000: Chemcal Thermodynamcs: Advanced Applcatons. Academc ress erry, R.H., Green, D.W.,1997; erry s Chemcal ngneerng Handbook (7th edton), McGraw-Hll. oplstenova, Jana; 004, Absorpton of carbon doxde modellng and expermental characterzaton. h.d thess., Chemcal ngneerng Department, Norwegan Unversty of Scence and Technology, Trondhem, Norway. rausntz, J. M., Lchtenthaler, R.N., and de Azevedo,. G., 1999; Molecular Thermodynamcs of Flud-hase qulbra (3rd ed.), rentce Hall TR; New Jersey. Renon, Henr., 1986; lectrolyte solutons. Flud hase qulbra, 30 181-95. Sebastan, nzo; Lacquant, L. 1967; Acetc acd - water system thermodynamc correlaton of vapor-lqud equlbrum data. Chemcal ngneerng Scence, (9), 1155-6. Thomsen, Kaj; Rasmussen, eter., 1999; Modelng of vapor-lqud-sold equlbrum n gasaqueous electrolyte systems. Chemcal ngneerng Scence, 54(1), 1787-180. Tochg, Katsum; Akmoto, Kentarou; Och, Kenj; Lu, Fangyh; Kawase, Yasuhto., 1999; Isothermal Vapor - Lqud qulbra for Water + - Amnoethanol + Dmethyl Sulfoxde and Its Consttuent Three Bnary Systems. Journal of Chemcal and ngneerng Data, 44(3), 588-590. Tore Haug-Warberg, 008; K8108 Advanced Thermodynamcs: Wth applcatons to hase and Reacton qulbra, Lecture Handout, Department of Chemcal ngneerng, NTNU. 6
Dervaton of Actvty Coecent xpresson Based on UNIQUAC Ugochukwu Aronu July 13, 009 UNIQUAC xpresson for xcess Gbbs nergy g RT = Combnatoral z } { Resdual z } { x ln x + Z where j = exp h u j u RT q x ln q x ln( j j j ) (1) = q x j q = q n () jx j j q jn j = r x j r = r n (3) jx j j r jn j wrtng eq.1 n terms of the number of moles 'n' and factorng out 'n'. A B z } { ng z } { C RT = n Z z } { n ln + q n ln q n ln( n j! g @n( RT ln k = ) @A @B @C = + + T;n @n 6=k k T;n @n 6=k k T;n 6=k T;n 6=k j j ) (4) (5) 1
Smplfy q. 4A and derentate wth respect to mole number. n A = n ln n = n (ln n ln n ) = n (ln + ln n ln n ) A1 z } { A z } { = n (ln ln n ) + n ln n @A = @A 1 + @A (6) Usng kronecker-delta to derencate @A 1 = @A 1 = @ [ n (ln ln n )] (ln ln n ) k + n ( 1 @ 1 k ) n k @A 1 = ln n k + n @ 1 (7) @A @(n ln n) = = ln n @n + n @ ln n = ln n + 1 n n @A Substtutng eq. 7 and eq. 8 back n eq. 6 @A k = ln + ln n + n k = ln n + 1 (8) @A k = ln + x k n @ n @ (9)
From 3, = r n j r jn j ; thus @ = j r jn j r k (r n ) j r j jk j r jn j @ = r k j r jn j r n r k j r jn j (10) ut eq. 10 nto eq. 9 @A k = ln + x k = ln k x k + n n r j r jn j = ln k n k r + k x k k j r jn j 6 4 r k j r jn j r n r k ( j r jn j ) r n r k ( j r jn j ) 3 r n r k 7 5 j r jn j Factorng out k and and smplfyng = ln k + 1 n r k x k j r jn j nr k = ln k x k + 1 j r jn j Multplyng numerator and denomnator by n k and factor out k = ln k x k + 1 nr k n k j r : jn j n k @A T;n 6=k = ln k x k + 1 k x k (11) 3
Smplfy q. 4B and derentate wth respect to mole number. @B B = z = z q n ln = z q n (ln ln ) q (ln ln ) k + n ( 1 @ 1 @ ) @B = A z } { z q k ln k + B z } { z n @ n @ q (1) From 10 @ = r k j r jn j r n r k ( j r jn j ) ; therefore n @ = n r k j r jn j r n r k ( j r jn j ) = k n r k ( j r jn j ) (13) Smlarly; @ ( j q jn = j )q k q n q k ( j q jn j ) n @ = = q k j q jn j n q k j q jn j q n q k ( j q jn j ) n q n q k ( j q jn j ) = k n q k ( j q jn j ) (14) Substtute eq. 13 and eq. 14 nto eq. 1B z "! q n q k k j q jn j = z = z q n r k j r jn j q n r k j r jn j!# n r k k j r jn j n q k j q jn j!! q n q k j q jn j 4
Multply numerator and denomnator of by n k then factor out k and. Factor out k and. = z = z = q k z q n r k j r jn j n k n k q n k n k q n k q k n k Factorng out k and settng = 1. Therefore @B @B! q n q k j q jn j = z k q k 1 k q k!! = z q k ln k k + z q k( k k 1) T;n 6=k = z q k ln k k + k k 1 (15) Derencate q. 4C wth respect to mole number. @C = = C = q 4 ln 0 @ j q n ln( j j 1 A @n A z 0 } { q 6 k ln@ 1 j jk 4 j j j j ) 3 1 @ + n j j j 5 j j j B z } { n A j j @ j + q j j j 3 7 5 (16) But; j = clearty. q jn j l q ln l. Note that subscrpt 'l' s used n place of subscrpt '' for @ j = ( l q ln l )q j jk q j n j q k ( l q ln l ) 5
@ j = Substtutng eq. 17 n eq. 16B ) q j jk l q ln l q n j j q j n j q k ( l q ln l ) (17) h qj jk q jn j q k l q ln l ( l q ln l ) j j j q n l q ln l h q n k q k j j j j q jn j q k j ( l q ln l ) Factor out then substtute and smplfy further; ) q k k j j j q k j j j j j j Smplfy further and substtute back nto q.16 @C = 4 qk ln( j j jk ) + q k k j j j q k 3 5 But = 1 @C T;n 6=k = q k 4 1 ln( j j jk ) 3 k 5 (18) j j j Substtutng eq. 11, eq. 15 and eq. 18 nto eq. 5 and obtan the nal expresson for actvty coecent based on UNIQUAC ln k = ln k + 1 k x k x k + z q k + q k 4 1 ln( ln k k + k k 1 j j jk ) k j j j 3 5 6