Phase equilibria Introduction General equilibrium conditions

Transcription

1 .5 hase equlbra.5. Introducton A gven amount of matter (usually called a system) can be characterzed by unform ntensve propertes n ts whole volume or only n some of ts parts; a porton of matter wth unform ntensve propertes s referred to as a phase. A heterogeneous system, therefore, conssts of more than one phase: crossng the boundary separatng two phases there s a sharp change n at least one ntensve property (densty, for eample). Consequently sngle phase system and homogeneous system are, n ths contet, synonyms. Equlbrum condtons are partcular condtons reached by a system fulfllng well-defned constrants (such as constant temperature and pressure) after some tme (theoretcally nfnte) from whch the system has no tendency to move. he state (defned as the ensemble of all the values of the thermodynamc varables of the system n a gven condton; such thermodynamc varables are called state varables) of stable equlbrum s ndependent of tme and prevous system hstory, and t can resst mnor fluctuatons of ts state varables. Such a defnton dstngushes stable equlbrum condtons not only from every other non-equlbrum condton, whether statonary or not, but also from metastable equlbrum condtons. Systems n a state of equlbrum are much easer to descrbe than systems where transformatons are beng carred out. hs a smplcty allows the equlbrum state to be descrbed wth a small number of state varables. hese partcular states (of equlbrum) are the feld of applcaton of the thermodynamc laws. hese basc laws cannot be mathematcally demonstrated; ther valdty reles on the absence of contrary epermental evdence. hs chapter deals wth equlbrum condtons n multphase mult-component systems wthout chemcal transformatons, meanng wthout any echange of the atoms present n a system wthn the molecules of the varous components. Snce the fundamental work by Josah Wllard Gbbs at the end of the neteenth century (Gbbs, 98) the theory whch defnes all the condtons for system equlbrum s well known. he man problem, whch has not yet been completely resolved, for arrvng at solutons n a numercal form sutable for practcal applcatons s the representaton of realty va models. In other words, the not yet satsfactory descrpton of molecular nteractons often does not allow the correct evaluaton of the thermodynamc functons needed to characterze the equlbrum condtons of a real system..5. General equlbrum condtons All systems, dependng on the constrants they must fulfl, spontaneously proceed n a well-defned drecton. It s common knowledge that the concept of entropy (a thermodynamc functon whch cannot be measured drectly) must be ntroduced n order to take nto account the drecton n whch real systems spontaneously proceed. ecause entropy provdes ths nformaton, t s also able to defne the equlbrum state as that partcular state from whch the system has no tendency to spontaneously evolve. Entropy s an etensve state functon defned n such a way that, followng a spontaneous transformaton n an solated system, t can only ncrease. As a consequence, equlbrum condtons n an solated system requre entropy to assume the mamum value possble provded that all the constrants mposed by the system tself are fulflled. Equlbrum states, as was mentoned earler, can be defned by a small number of varables. In partcular, t can be shown epermentally that the stable equlbrum etensve state (.e. the value of all the etensve varables) of an sotropc sngle-phase mult-component system can be completely defned once the values of ts nternal energy U, volume, and mole number n (n,n, ) of each speces are gven. hs s known as the frst law of thermodynamcs. Instead, the requrement of mamum entropy for equlbrum law of an solated system s known as the second postulate of thermodynamcs: there ests a functon of the etensve varables of a system (U, and n) called entropy, S, whch s defned for all the equlbrum states and mples that the values of U, and n, when no nternal constrants are enforced n an solated system, make the value of entropy mamum. As a consequence, when an solated system s n equlbrum condtons the frst dfferental of entropy must OUME / ISUMES 95

2 HYSICA AD CHEMICA EQUIIIA be equal to zero. Snce entropy s a state functon, t has an eact dfferental and the followng relaton holds true: [] S + n dn U,, nj A mult-phase system s consttuted by many sngle-phase systems, for each of whch all the aforementoned consderatons must be fulflled. Snce entropy s an etensve varable, the system entropy can be calculated smply as the sum of the entropy of each phase. hs can be put forward as the thrd law of thermodynamcs: entropy s an etensve (n other words, the entropy of a system made up of several subsystems s gven by the sum of the entropes of the sngle subsystems), contnuous and dfferentable varable, whch monotoncally ncreases wth the nternal energy. hs allows the followng defntons: [] [3] [4] S S ds du d U + + U S S U U S U S µ n n, n U, n, n, n S, n U, n S S,, n j U,, nj S U U U, n S, n U µ n, n S,, nj As wll be dscussed below, and are the measurable varables temperature and pressure, whle m s a chemcal potental. he prevous relaton therefore becomes: [5] ds du µ + d dn From ths relaton all the general equlbrum condtons among phases can easly be deduced. All thermodynamc problems, ncludng those of phase equlbra, can be descrbed by the general model of an solated system represented by a contaner wth rgd, adabatc, mpermeable walls dvded by a wall nto two subsystems, A and. When the nternal wall s mpervous to matter (.e. t does not allow mass echange between the two subsystems), rgd (.e. t does not allow work echange between the two subsystems), and non-adabatc (.e. t allows heat echange between the two subsystems), both volume and mole number of the two subsystems are constant and therefore dn A dn d A d. Moreover, as the whole system s solated the total nternal energy s also constant; t follows that the nternal energy ncrease of a subsystem must be equal to the nternal energy decrease of the other subsystem: du A du. Snce entropy s an etensve varable, usng these relatons t follows that: [6] whch means that under equlbrum condtons necessarly A. herefore, has the physcal meanng of temperature and relaton [6] requres the temperature of the systems connected by a non-adabatc boundary to be equal once equlbrum condtons are attaned. In other words, the equlbrum condton wth respect to heat transfer s that temperature s unform, as epermentally verfed. Analogously, t s possble to deduce the equlbrum condtons wth respect to work echange. In ths case the boundary separatng the two subsystems s assumed to be mpervous to matter, but nether rgd nor adabatc (ths means that t allows heat and work echange between the two subsystems). Along the same lnes prevously dscussed, t can be demonstrated that: dn A dn ; d A d ; du A du. he mamum entropy requrement thus becomes: A A [7] ds d S S ds ds du A ( + ) + + A hs relaton mples that (seeng that A ) A must be equal to. herefore, has the physcal meanng of pressure and relaton [7] requres the pressure of the systems connected by a non-rgd boundary to be equal once equlbrum condtons are attaned. In other words, the equlbrum condton wth respect to work echange s that pressure s unform, as epermentally verfed. Fnally, equlbrum condtons wth respect to mass transfer can be deduced. In ths case the boundary separatng the two subsystems s consdered to allow echange of matter, heat and work. It s worthwhle notng that n ths case the systems beng eamned can be thought of as two dstnct phases n an solated system. hese dfferent phases are subsystems that can echange mass and energy and so are separated by permeable, dathermal and moble walls. It follows that, analogously to what was prevously dscussed, dn A dn ; d A d ; du A du and the equlbrum relaton can be recast as the followng: [8] + + A ds d S + S A ds ds A du A + du du A A A d A du A A A du A A + A A ds d ( S + S ) ds + ds A A du A d A µ + dn A A A + A du d µ dn A du A + A A d µ µ d A A A A d dn A 96 ECYCOAEDIA OF HYDOCAOS

3 HASE EQUIIIA hs relaton s fulflled ( A and A beng n equlbrum condtons) when m A m. he varable m, whose physcal meanng s less ntutve than that of pressure and temperature, s usually referred to as the chemcal potental. he chemcal potental of the th speces can be thought of as the nternal energy change nduced by addng to the system one mole of speces whle keepng constant both the entropy and volume of the system, as well as the mole numbers of all the other speces. Equaton [8] requres, n equlbrum condtons, that the chemcal potentals of each speces n dfferent subsystems separated by boundares permeable to mass be equal. Snce the prevous equlbrum condtons wth respect to heat and work echange requre that also temperature and pressure be the same, the general equlbrum condton between two mult-component phases s: α α β β [9] µ (,, ) µ (,, ) where a and b represent two dfferent phases, beng the composton and where a sngle value of both temperature and pressure for both the phases has been eplctly accounted for. ote that the composton of the two phases can be dfferent. However, for sngle-component systems, meanng those contanng only one pure speces, the composton s constant and the prevous relaton becomes: α β [] µ (, ) µ (, ) elatons [9] and [] are, n practce, the frst two steps descrbed n the ntroducton for solvng a problem usng thermodynamc tools. he problem of phase equlbra has been epressed n the abstract terms of thermodynamcs and the soluton has been found, represented by the equatons whch mpose the equalty of chemcal potentals. ow the problem s to brng these results back to the real world, n other words, to relate chemcal potentals wth those measurable ntensve varables of nterest, such as pressure, temperature and the composton of the dfferent phases. It s useful to look at the problem frst n terms of determnng the number of varables that need to be defned to dentfy the ntensve state of a system,.e. the value of all ts ntensve varables. In partcular, the degree of freedom, or varance, of a system s the number of ntensve varables that can be arbtrarly assgned. et us consder a system wth F phases and speces: the number of ntensve varables requred to defne the ntensve state of each phase (temperature, pressure and composton) s equal to ( ). ote that when composton s gven as mole fractons, the number of mole fractons to be defned for each phase s equal to snce the th value can be calculated usng the normalzaton relaton:. he number of ntensve varables that must be gven to completely characterze the ntensve state of a system nvolvng F phases s therefore equal to F( ). he aforementoned equlbrum condtons requre the temperature and pressure of each phase to be equal; the chemcal potental of each speces n the dfferent phases must be equal as well. Snce chemcal potental, whch s an nternal energy dervatve, depends on the ntensve varables of the system (temperature, pressure and composton), the relatons lnkng these ntensve varables are the followng: [] F... F relatons F... F relatons µ... µ F... F relatons F µ... µ Generally there are (F ) (F ) (F ) ( )(F ) equatons nvolvng F( ) varables. In order for the system of equatons to have a sngle root, n other words the number of ndependent equatons must be equal to the number of varables, t s necessary to defne the values of F( ) ( )(F ) F varables, whch s equal to the degree of freedom,, of the system: [] F For nstance, when a system nvolves two phases and two speces t s necessary to gve the values of ntensve varables. When F, the prevous relaton gves whch s the well-known epermental fndng statng that the ntensve state of a sngle-component sngle-phase system s completely defned once the values of two ntensve varables are gven. Once the values of ntensve varables are gven, the values of all the other ntensve varables can be calculated through the assgned varables and the aforementoned equlbrum relatons. How to use these relatons, wth partcular reference to the chemcal potental equalty, to compute the values of the equlbrum ntensve varables values for a system nvolvng F phases and speces wll be eamned later. It s useful, however, to start wth a bref descrpton of the equlbrum condtons for closed systems subject to condtons other than mantanng volume and nternal energy constant. he frst and second law of thermodynamcs taken together can provde the followng general relaton (here means volume) whch s vald for spontaneous transformatons n closed systems: [3] ds du d For a system where volume and nternal energy are constant (.e. an solated system) du d and the prevous relaton leads to: [4] ds U, whch means that entropy can only ncrease and therefore, n equlbrum condtons, t must be mamum followng the second law of thermodynamcs. For a system where volume and entropy are constant, ds d and relaton [3] becomes: [5] du S, whch means that nternal energy can only decrease and, therefore, n equlbrum condtons, t must be mnmum. For a system where volume and temperature are constant, d d and relaton [3] becomes: [6] ds du d d ( S) du d ( S U) da, whch means that Helmholtz free energy, A, can only decrease and, therefore, n equlbrum condtons, t must be mnmum. Fnally, for a system where pressure and OUME / ISUMES 97

4 HYSICA AD CHEMICA EQUIIIA temperature are constant, d d and relaton [3] becomes: [7] ds du d d ( S ) du d ( ) d( S U ) dg whch means that Gbbs free energy, G, can only decrease and, therefore, n equlbrum condtons, t must be mnmum..5.3 Equlbrum between sngle-component phases A sold at a gven temperature and pressure when heated at constant pressure usually changes ts temperature and molar volume as shown n Fg.. At the begnnng the sold s represented by pont. When heated, ts temperature and molar volume ncrease untl pont, where t starts to melt: here a phase transton proceeds wth the presence of both a sold and a lqud phase n equlbrum condtons. he sold phase molar volume s equal to v S, whle the molar volume of the lqud phase whch s formed from meltng the sold s equal to v 3. Snce the two phases are n equlbrum condtons, ther temperatures must be equal and consequently the te lne connectng ponts and 3 n equlbrum condtons s horzontal (all the sngle-component phase transtons proceed at constant temperature and pressure). hs also mples that at a gven pressure value, meltng (or fuson) temperature of a pure speces (pont ) s equal to the freezng (or soldfcaton) temperature (pont 3): at a gven pressure value there ests only one temperature value where lqud and sold phases can coest, called meltng or freezng temperature. Obvously, at a gven temperature value, there ests only one pressure value where lqud and sold phases can coest, called meltng or freezng pressure. hs s evdent f one remembers that the degree of freedom of a sngle-component system s equal to 3 F; when two phases are nvolved the degree of freedom s equal to one: gven the value of an ntensve varable, for nstance pressure, the values of all the other ntensve varables, ncludng temperature, are fed. temperature b d m s sold lqud vs v 3 v 4 v 5 molar volume Fg.. rends of temperature and molar volume durng heatng of a pure speces at constant pressure , vapour When the partally melted speces s heated further, the amount of lqud ncreases, as does the molar volume of the two-phase mture (that s, the volume occuped by one mole of the speces, partally sold and partally lqud) because the lqud molar volume s usually larger that that of the sold. he molar volume of the two-phase mture s n between the values of the lqud and sold molar volume n equlbrum condtons (ponts and 3) and t s consequently represented by a pont on lne 3. hs value can be calculated from a weghted average (based on the mole numbers) of the molar volumes values of the sold and lqud: (n n S )v n v 3 n S v S. From ths relaton the molar volume of a two-phase mture can be epressed as a functon of the proporton of lqud n the mture, n (n n S ): S n [8] v n n v n n n v S v v S + + S 3 S hs relaton s fully general and permts the calculaton of the value of a generc molar varable of a two-phase mture, m, as m m ( )m S. he amount of heat requred to completely melt one mole of compound s called molar latent heat of meltng. Snce ths heat s echanged at constant pressure, t s also called molar meltng enthalpy. he molar latent heat of freezng (or molar freezng enthalpy) s equal to the meltng value dsregardng the sgn, snce n ths case heat s removed from, not added to, the system. When all the matter s melted (pont 3) the heat put nto the system ncreases the temperature and the molar volume untl pont 4, and the speces begns to bol. At ths pont lqud and vapour n equlbrum condtons are smultaneously present. he molar volume of the vapour phase s equal to v 5, whle that of the lqud phase s equal to v 4. Snce the two phases are n equlbrum they are at the same temperature and therefore the te lne connectng pont 4 to pont 5 s horzontal. As n the prevous case of lqud-sold equlbrum, at a gven pressure value the bolng temperature of a pure speces (pont 4) and ts dew pont (pont 5) are equal. At a gven pressure value there ests only one temperature value where lqud and vapour n equlbrum condtons can est, called bolng or dew temperature. Obvously, at a gven temperature value there ests only one pressure value where lqud and vapour n equlbrum condtons can est, called vapour pressure. As n the sold-lqud phase transton, contnung to heat the system leads to an ncrease n the amount of vapour, and consequently the molar volume of the two-phase mture ncreases snce the molar volume of the vapour s larger than the molar volume of the lqud. he molar volume value of the two-phase mture s n between the values of the lqud and vapour phases (ponts 4 and 5); t corresponds to one pont on the lne 4-5 and can be calculated wth a relaton analogous to the one seen earler as: v v 5 ( )v 4 where n (n n ) s the vapour qualty of the two-phase mture. hs relaton s also fully general and the value of any molar quantty of the mture, m, can be calculated as m m ( )m. he amount of heat requred to completely vaporze one mole of compound s called molar latent heat (or enthalpy) of vaporzaton, whch corresponds (dsregardng the sgn) to the molar latent heat (or enthalpy) of condensaton. Heatng the system further, ts temperature and molar volume ncrease to pont ECYCOAEDIA OF HYDOCAOS

5 HASE EQUIIIA he same nformaton can be summarzed n a - dagram, as shown n Fg.. he three lnes represent the boundares of the regons where dfferent phases est (namely: sold, lqud and vapour): crossng each of these boundares ndcates a phase transton. For nstance, the vaporzaton boundary dentfes the regon where only vapour ests at a gven value of temperature,, as the regon wth pressure values lower than ( ), whch s the vapour pressure at temperature equal to for the consdered speces. For pressure values larger than ( ) only the lqud phase can est. Only when the pressure value s equal to ( ) can vapour and lqud phases est smultaneously n equlbrum condtons. Smlar conclusons can be derved from the sublmaton lne (vapour-sold equlbrum) and from the meltng lne (lqud-sold equlbrum). Above the crtcal temperature no phase transton can est and the equlbrum lne vanshes. In Fg. the trple pont s also dentfed as the only pont where three phases can est smultaneously n equlbrum condtons. hs pont s unque (.e. the pressure and temperature are fed) n that when the number of phases s equal to three the degrees of freedom s equal to zero and no ntensve varables can be arbtrarly chosen. Gven that n equlbrum condtons two phases a and b must necessarly have the same temperature and pressure, the equlbrum condtons for mass transport between the phases means that: α β [9] µ (, ) µ (, ) where s the unque value of pressure at whch the two phases of the speces n queston can be present n equlbrum when the system temperature s equal to the gven value. It s known that the chemcal potental s drectly related to the Gbbs free energy, through the followng dfferental equaton: pressure ( ) d ( ) meltng lne for speces that reduce ther volume when meltng d sold sublmaton lne meltng lne for speces that ncrease ther volume when meltng crtcal pont lqud trple pont Fg.. hase dagram for a pure speces. d d temperature evaporaton lne vapour [] dg d Sd + µ dn showng that the chemcal potental s equal to the partal molar Gbbs free energy. artal molar propertes represent the change of a generc mture property M due to the addton of one mole of speces whle keepng temperature, pressure and mole number values of all the other speces constant: M ( M n ),,n j. An alternatve meanng of partal molar propertes s the molar value of the property when the speces s n a mture. Consequently, all the mture propertes can be calculated through the values of the partal molar propertes and the mture composton as M(,,n) M (,,)n or, n molar terms, as m(,,) M (,,). Snce for a pure speces the partal molar Gbbs free energy s equal to the molar Gbbs free energy, the chemcal potental and the Gbbs free energy are related to each other through the followng relatons: G [] (,, ) G (,, ) n µ,, nj µ (,) g(, ) ( purespeces) he general equlbrum relaton [9] for a sngle-component system becomes, consderng for the sake of eample the vapour-lqud equlbrum: [] g (, ) g (, ) A system nvolvng a lqud and a vapour phase n equlbrum condtons at temperature and pressure ( ) s represented by pont n Fg.. When the system temperature ncreases by a small amount d, the vapour pressure must also ncrease by a small amount d n order that the two phases contnue to est n equlbrum condtons. he system evolves towards pont n Fg.. oth for the condtons of pont and of pont the equlbrum relaton must be fulflled: [3] g, g, g + d, + d g + d, ( )+ d Snce for both the phases g[ d, ( ) d] g[, ( )] dg, t follows that dg dg. Usng the molar Gbbs free energy dfferental for a pure speces: [4] dg s d + v d dg s d + v d an equaton relatng the vapour pressure to the temperature through latent evaporaton heat ( h ev h h ) and the molar volume change n the phase transton ( v ev v v ), whch s known as the Clapeyron equaton, can be deduced: [5] d s s h h h ev d v v v v vev Smlar equatons can be deduced for all the other phase transtons, that s, lqud-sold and vapour-sold, usng the heat and molar volume changes correspondng to the consdered phase transton. Snce the heat of evaporaton, meltng and sublmaton are always postve, the slope (d/d ) of the equlbrum lnes n Fg. depends on the sgn of v. he molar volume OUME / ISUMES 99

6 HYSICA AD CHEMICA EQUIIIA change from lqud to vapour and from sold to vapour s always postve and consequently the slope of the evaporaton and sublmaton lnes s always postve. In other words, the vapour pressure of lquds and solds always ncreases wth temperature. For many speces v fus, that s, the lqud molar volume s larger than the correspondng sold molar volume n equlbrum. For these speces d/d s postve also for the lqud-sold equlbrum boundary and the meltng temperature ncreases wth pressure. However, for other speces such as water, whose lqud molar volume s smaller than the correspondng sold molar volume, the slope of the meltng equlbrum lne s negatve and the meltng temperature decreases as pressure ncreases. For an evaporaton curve when pressure values are nether too hgh nor too close to the crtcal pont, the prevous relaton can be appromated consderng the vapour molar volume to be much greater than the lqud molar volume and smlar to that of an deal gas, v ev v v v, whch provdes to the Clausus-Clapeyron equaton: [6] d h d h ev ln ev d d hs equaton can be used to estmate the latent evaporaton heat through two values of the vapour pressure, or to deduce the trend of the vapour pressure wth respect to temperature. eglectng the dependence of the evaporaton heat on the temperature, a reasonable appromaton n a small temperature range, the prevous relaton can be recast as: [7] ln ( ) h ev showng that the logarthm of the vapour pressure s lnearly related to the nverse of the temperature..5.4 Equlbrum between mult-component phases Havng establshed that temperature and pressure n all equlbrum phases must be equal also for mtures, to show mass transfer equlbrum condtons among the phases t s necessary to relate the chemcal potental of a speces n a mture to measurable propertes. Followng the approach proposed by Glbert ewton ews, fugacty, fˆ, s defned through the dfferental relaton: [8] Snce, as wll be dscussed below, for a mture of deal gases the chemcal potental s related to the partal pressure as: [9] dµ, dln f ˆ dµ, dln ln h ev and rememberng that all fluds at suffcently low pressure values behave lke deal gases, the prevous fugacty defnton s complemented by the followng: [3] From these relatons t s clear that the fugacty of an deal gas s equal to the partal pressure (or the total pressure for a pure speces). Consderng a system contanng two phases n equlbrum, defned as a and b, the prevous relaton can be ntegrated between the condtons of the two phases n equlbrum: [3] α α fˆ µ α α µ β β (,, ) (,, ) (,, ) ln f ˆ β (,, β ) Snce n equlbrum condtons the chemcal potentals of each speces n the two phases must be equal, ths relaton leads to the followng condton for phase equlbrum, absolutely equvalent to the equalty of chemcal potental: [3] f ˆ α (,, α ) f ˆ β (,, β ) In order for a system to be n equlbrum wth respect to mass transfer, the fugacty of each speces must be the same n all the phases. For a system nvolvng speces and two phases, the ntensve varables requred to fully characterze the system are temperature and pressure (two varables snce we know that temperature and pressure of both the phases must be equal) and the mole fractons n each phase ( varables for each phase, snce the th mole fracton can be calculated from the other mole fracton as a complement to one). he total number of ntensve varables s therefore equal to. he degrees of freedom for ths system s equal to, and consequently ntensve varables must be gven n order to compute the other ntensve varables. he equatons requred to compute the unknowns are relatons [3], one for each speces present n both the phases. hese equatons are complemented by the two stochometrc equatons to compute the th mole fracton n both the phases: [33] ˆf lm β β,,, dµ dln f ˆ α α,,, α β β,,, β α α,,, All the partcular cases that wll be dscussed below requre the soluton of the set of algebrac equatons [3] and [33]. Obvously, the frst problem to be faced refers to the correlaton of fugacty to measurable propertes, whch means that the soluton of the problem s n tryng to brng the abstract language of thermodynamcs down to the real world. Fugacty from equatons of state: drect methods Drect methods use as a reference an deal gas mture, that s, a mture wthout nteractons between molecules whose behavour s represented by the equaton of state v and by the Gbbs theorem (all the partal molar propertes, ecept for volume, of a speces n an deal gas mture are equal to the correspondng molar propertes of ECYCOAEDIA OF HYDOCAOS

7 HASE EQUIIIA the pure speces at the mture temperature but at a pressure equal to the partal pressure of the consdered speces n the mture): M * (,,) m * (, ), where M s a generc thermodynamc property. he Gbbs theorem allows the calculaton of all the partal molar propertes of an deal gas mture from the molar propertes of the pure speces, apart from the partal molar volume whch obvously s calculated * (,,) ( * n ),,n j v * (,). Usng the Gbbs theorem t can be easly demonstrated that there s a smple relaton [9] between chemcal potental and partal pressure for one component of a mture of deal gases: [34] dµ * (,, ) dln hs equaton for a pure speces becomes: [35] dµ * (, ) dln So called drect methods, snce they use as a reference an deal gas mture, ntroduce the fugacty coeffcent /ˆ, a varable that takes nto account the departure of the system behavour from that of an deal gas. hs coeffcent s defned as the rato of the fugacty of a speces n a mture to the fugacty of the same speces n an deal gas mture under the same condtons: fˆ,, ˆ,, [36] ˆ ( f φ (,, ) ) ˆ * f,, ( ) It s clear that the fugacty coeffcent of a speces n an deal gas mture s equal to, whle for a pure spece the fugacty coeffcent becomes: f(, f, [37] φ(, ) ) * f (, ) As a consequence, the problem of phase equlbrum characterzaton has been traced to the calculaton of chemcal potentals to the fugactes and now fnally to the fugacty coeffcents. Drect methods deal wth ths last problem by relatng the fugacty coeffcents to the Gbbs free energy. he calculaton of the Gbbs free energy usng drect methods s based on the calculaton of the departure of the thermodynamc functon of a real flud, G, from that of an deal gas, G *, called departure functon: G G G *. Snce the thermodynamc propertes of an deal gas can be calculated eactly, knowng ether the Gbbs free energy or the departure Gbbs free energy provdes the same nformaton. he dmensonless form of the departure molar Gbbs free energy, g, can be easly calculated from the followng dfferental relaton: [38] g g g v h d d d d * * * v h d v h d d d where the relatons dg vd sd and g h s have been used. As usual, n thermodynamcs the value of a state functon s calculated followng the easest thermodynamc pattern. In ths case, the value of g can be easly calculated through an sothermal path: d [39] he zero value of the lower ntegraton lmt arses from the fact that all the fluds behave lke an deal gas when the pressure approaches zero, and consequently when the pressure approaches zero all the departure functons are equal to zero. he prevous relaton, ntroducng the defnton of the compressblty factor Z v/, leads to the followng equaton: * g (, ) v v [4] d v d d ( Z ) he mportant mplcaton of ths equaton s that, based on a gven Equaton Of State (EOS), that s, the relaton Z(,,), deduced from epermental measurements -v--, the departure molar Gbbs free energy can be calculated, and thus the value of the Gbbs free energy. hs plays a specal role n the thermodynamcs of mtures, as t s possble to demonstrate that all the other varables n queston can be derved from t: Gbbs free energy can be consdered the foundaton functon of all thermodynamc propertes. hs means that by usng an EOS t s possble to calculate all thermodynamc functons, confrmng the great mportance of EOS n thermodynamcs. For a real flud the fugacty coeffcent, /ˆ, can be related to the departure partal molar Gbbs free energy and consequently calculated through an EOS. Integratng the fugacty from an deal gas state to a real flud state t follows: [4] fˆ * µ (,, ) µ (,, ) ln ln ˆ φ Snce the chemcal potental of a speces n a mture s equal to the partal molar Gbbs free energy and ntroducng the defnton of departure functon, the prevous relaton can be recast n the followng manner: * [4] G (,, ) G (,, ) G (,, ) ln ˆ φ (,, ) As a consequence, the fugacty coeffcent of a speces n a mture can be calculated from the departure Gbbs free energy as: [43],,,,,* g v d d constant g / dµ dln f ˆ,,,,,* ln ˆ G (,, ) φ (,, ) n ng (,, ) n n Z n d,, n j n,, j, n, j OUME / ISUMES

8 HYSICA AD CHEMICA EQUIIIA It should be noted that the dependence of Z on n derves from the mng rules used n the EOS. For a pure speces, the compressblty factor does not depend on composton; therefore, the prevous relaton becomes: [44] nz n ln φ(, ) d ( Z ) n,, j d If an EOS s known, the ntegrals nvolved n these relatons can be calculated analytcally (n the case of a mture, preceded by the dervatve) and then computng the fugacty coeffcent value. One of the smplest EOS s that of the correspondng states. hs EOS states that the compressblty factor of all the fluds depends only on the reduced temperature and the reduced pressure (whch are defned as the rato of temperature or pressure and the related crtcal value: C and C ); ths means that Z Z (, ). hs relaton n practce states that two fluds that are n two correspondng states (.e. characterzed by the same reduced temperature and reduced pressure) have the same compressblty coeffcent. nz n n,, j d A large amount of work has been undertaken devoted to obtanng general correlatons of the compressblty factor as a functon of the reduced varables that are able to correctly reproduce the epermental results. he complety of these correlatons makes t more useful to provde the nformaton n tables or graphs, such as the well-known tables by. I. ee e M. G. Kesler (975). Apart from the regon close to the saturaton boundary or to the crtcal pont, ths EOS foresees the compressblty factor of low-polar speces wthn 5%. On the other hand, for large-polar speces or quantum gases (hydrogen, helum and neon) large dfferences wth the epermental values of the compressblty factor can be found. o ncrease the agreement wth the epermental fndngs a thrd parameter s often ntroduced n the correspondng state equaton, the tzer acentrc factor w: Z [45] Z Z( ) Z ( ) +,, ω, ( ω ) ω ω Z (, ) + Z (, )ω In ths relaton (usually referred to as the generalzed correspondng state equaton) the value of Z (whch s also avalable n a graph or a table as a functon of the reduced varables) s usually a small correcton to the Z value, and consequently t can often be gnored n a frst analyss. Fg. 3. Generalzed dagram of fugacty coeffcent as a functon of reduced varables (Sandler, 989). fugacty coeffcent saturaton lne reduced pressure ECYCOAEDIA OF HYDOCAOS

9 HASE EQUIIIA Usng ths EOS also allows the fugacty coeffcent relaton for pure speces to be generalzed as: [46] d (,,ω) Z constant It should be noted that the fugacty coeffcent depends only on the reduced pressure and on the compressblty factor. Introducng the generalzed correspondng state equaton t follows: [47] lnφ Z d d d lnφ Z + ωz Z ( ) + d + ω Z ln( φ) + ωln( φ) constant oth the ntegrals depend only on and, as well as on Z (, ) and Z (, ). It follows that the two terms ln(/) and ln(/) depend only on the reduced varables and they can be gven n tabular or graphcal form n the same way as for the two functons Z (, ) and Z (, ). Analogously to the case of the compressblty factor, the functon ln(/) represents a correcton to the values of / and consequently t can often be gnored at frst. he values of / as a functon of the reduced temperature and reduced pressure, both for lqud phase and vapour phase, are summarzed n the graph n Fg. 3. Dagrams lke ths permt a quck evaluaton of the departure behavour of a gven flud compared to an deal gas at gven temperature and pressure values. An mportant dfference between pure fluds and mtures s, n the case of mtures, that Z does not depend only on the reduced varables but also on the composton. hs means that for mtures, but not for pure fluds, t s not possble to derve generalzed correlatons for computng the fugacty coeffcents vald for any mture as a functon of only the reduced varables. For pressure values that are not ecessvely hgh (lower than 5 bar) and for the gas phase, vral EOS truncated at the second coeffcent, Z, gves reasonable predctons of the compressblty factor. Usng ths EOS the followng relaton for the fugacty coeffcent can be deduced: d d [48] lnφ ( Z ) + ral EOS can represent only the gas phase behavour. When the lqud phase s nvolved, more comple EOS are requred, such as cubc equatons. he general equaton representng a cubc EOS for volume (or for the compressblty factor) s the followng: v v a v [49] v b ( v + εb )( v + σb ) where Ω α(, ω) a ( c) Ωb c [5] a b c c he value or meanng of the symbols n these equatons for some common cubc EOS s summarzed n able. he prevous relaton can be recast n terms of compressblty factor as: 3 [5] Z + βz + γz + δ he coeffcents of ths equaton for some common cubc EOS are also summarzed n able. Snce cubc EOS gve the compressblty factor not as an eplct functon of pressure, but as an eplct functon of volume, a change of varables s requred n the ntegral n equaton [44] n order to compute, nsertng relaton [49], the fugacty coeffcent whch takes the general form: [5] ln φ ln ( v b ) Z a ln v+ σb Z + v b ε σ v+ εb hrough the parameters e and s, gven n able for the varous cubc EOS, dfferent operatve relatons for computng the fugacty coeffcent values of both lquds and vapours can be deduced. able. alues or meanng of some symbols found n the general cubc EOS form for van der Waals (vdw), edlch-kwong (K), edlch-kwong-soave (KS) and eng-obnson (). For all EOS A a/() ecept K where A a/[() / ] and b/() vdw K KS a(,w).5 [ ( w.76w ) (.5 )] [ ( w.699w (.5 )] e.444 s.444 W a 7/ W b / b g A A A A 3 d A A A A 3 OUME / ISUMES 3

10 HYSICA AD CHEMICA EQUIIIA Gven temperature and pressure values, the parameters n equatons [5] and [5] can be calculated for a gven EOS. Solvng cubc equaton [5] three roots are always found, but the only roots wth a physcal meanng are the real and postve roots. When more than one real postve root s found, the larger value s retaned when the pressure s lower than the vapour pressure at that temperature and so only the vapour phase s present, whle the lower value s retaned when the pressure s larger than the vapour pressure at that temperature and so only the lqud phase s present. When the pressure s equal to the vapour pressure at the gven temperature both the phases est n equlbrum condtons and both the larger and the lower values must be retaned. he central value does not represent a stable equlbrum state and therefore t s never retaned. Once the value of the compressblty factor, Z, s calculated, the value of the molar volume can be compute too, v Z, as well as the value of the fugacty coeffcent though equaton [5]. In the case of mtures, the same EOS can be used provded that the energetc and geometrc parameter values used wth these EOS refer to the gven mtures. hese parameters can be calculated based on the parameter values for pure speces as well as on the mture composton usng sutable mng rules (whch gve the dependence on the composton) and combnaton rules (whch gve the dependence on the pure-speces parameters). Composton dependence of the compressblty factor, whch s requred to compute the dervatve n equaton [43], s therefore confned to the mng rules consdered. For the vral EOS truncated at the second term, the mng rule that gves the mture coeffcent: [53] j j j In ths relaton the parameters characterzed by the same double subscrpt,, are the second coeffcents of the vral of the pure speces and so account for two-body nteractons between lke molecules. he parameters characterzed by dfferent subscrpts, j, account for twobody nteractons between dfferent molecules. Snce these parameters account only for two-body nteractons, ther values can be derved from eperments nvolvng pure speces or, at most, mtures of two components. he behavour of mtures wth more than two components can be foreseen usng only these parameters. Introducng ths mng rule n general equaton [43], the followng equaton for computng the fugacty coeffcent of a speces n a mture can be derved: [54] ln φ ˆ (,, ) ( j j ) As prevously seen for pure speces, usng cubc EOS t s possble to obtan the followng general relaton for computng the fugacty coeffcent of a speces n a mture: [55] ln ˆ b φ (,, ) ln b Z v b Z ( ) + v a + b ( ε σ) + a b a b ln v+ σb v+ εb where the dependence on the composton s confned n the value of the dervatves wth respect to the mole number of the th speces of both the geometrc and energetc mture parameters, whch n turn depend only on the partcular mng rule consdered: ( na ) [56] a n Usually, for cubc EOS quadratc (or van der Waals) mng rules are used for any parameter p (whch can correspond both to a and b); these mng rules are equal to the aforementoned mng rule for the vral EOS: [57] p p p Also n ths case the parameters characterzed by the same double subscrpt account for two-body nteractons between lke molecules, whle the parameters characterzed by dfferent subscrpts account for two-body nteractons between dfferent molecules. Assumng as a combnaton rule a smple mean, p j (p p j ), the quadratc mng rule smplfes to a lnear mng rule: [58] p p p whch s normally used for the geometrc parameter b. When usng as a combnaton rule a geometrc mean, p j 3 p p j, the quadratc mng rule reduces to: [59] ( nb ) b n p p whch s normally used for the energetc parameter a. he choce of a geometrc mean for the energetc parameter arses from the analogy wth the rule used for computng the ntramolecular potental between dfferent molecules when ondon dsperson forces preval. As a consequence, ths mng rule s epected to perform correctly for mtures when the ondon dsperson forces preval. Usng the quadratc mng rule, eplct relatons for the dervatves nvolved n the general relaton for the fugacty coeffcents can be obtaned: np [6] p p+ p j j n n, j from whch the specal cases arsng from the two combnaton rules usually used for parameters a e b can be deduced: [6] b b j n, j n, j j j a a+ aa Usng the varous EOS summarzed n able several relatons for computng the fugacty coeffcent of a speces n a mture can be deduced. Quadratc mng rules cannot correctly reproduce mtures wth strong ntramolecular nteractons. Moreover, snce EOS arse from the deal gas equaton of state, ther predctons are more accurate for the gas phase than for the lqud phase. Among the many alternatve mng rules proposed to remedy ths problem, the Wong-Sandler rules are partcularly effectve. hese mng rules try to make the EOS reproduce relable results both n the regon at hgh densty (the lqud phase) and n the regon at low densty (the vapour phase). o acheve these results these mng rules are derved requrng on the one hand that the EOS be 4 ECYCOAEDIA OF HYDOCAOS

11 HASE EQUIIIA able to reproduce correctly the ecess Helmholtz free energy of the mture (ecess functons wll be dscussed etensvely later; here t s suffcent to note that ecess functons account for the departure of the behavour of a mture, usually a lqud, from that of an deal mture where the ntramolecular nteractons between lke and dfferent molecules are the same) at nfnte pressure, that s, n the regon at hgh densty. hs s carred out usng the epermental nformaton avalable n terms of ecess Gbbs free energy, g E, at medum-low pressure. On the other hand, the mng rules must also gve the correct quadratc dependence of the second vral coeffcent from the composton. hs means that the EOS wll perform correctly also where the vral EOS performs well, that s, n the low-pressure regon. he mng rules arsng from these constrants are the followng: [6] where C s a parameter whose value depends on the partcular EOS consdered. When usng KS (edlch-kwong-soave) EOS the C parameter s equal to.6935, whle when usng (eng-obnson) EOS t s equal to.633. From these mng rules, eplct relatons for the dervatves n the general relaton for the fugacty coeffcents can be easly obtaned: [63] b j a b a b a + a j b+ b k j j E a g + b C E g C a + a j b b k j + j j b E ak g + k b C k a b + lnγ C b a E k g k b + C k ( j ) a lnγ a b a b b C + b where g s the actvty coeffcent, a parameter (whch wll be fully dscussed below) related to the ecess Gbbs free energy. he bnary nteracton parameter k j can be estmated by comparson wth the same epermental data of the ecess Gbbs free energy used n the mng rules. Ecept for the mng rules and the EOS used, the procedure for computng the fugacty coeffcent of a speces n a mture s the followng: gven the values of temperature, pressure and mture composton, for a gven EOS and a gven mng rule the values of the parameters nvolved n the general equatons [5] and [55] can be calculated. y solvng cubc equaton [5] three roots are found: also n ths case the only roots wth a physcal meanng are the real, postve ones. When more than one real, postve root s found, the larger value s consdered when dealng wth a vapour phase, whle the lower value s k k ( j ) retaned when a lqud phase s nvolved. he central value s meanngless snce t does not represent a stable equlbrum condton. Once the value of the compressblty factor, Z, has been calculated, the molar volume of the mture can be also calculated as v Z, and fnally the fugacty coeffcent value s recovered from equaton [55]. Condensed phases fugacty: ndrect methods Indrect methods use as a reference an deal mture, that s, a mture where mng effects related to both volume and enthalpy are nl. hs means that the sobarc mng process proceeds wthout volume and temperature changes. hs appromaton s reasonable when the components of the mture are smlar and therefore the ntramolecular nteractons between lke and dfferent molecules are smlar. hs s the smplest model for lqud mtures snce for lqud mtures t s not possble to neglect ntramolecular nteractons as n the deal gas model because the attractons among molecules that are responsble for condensaton must est. hs behavour s verfed f the partal molar Gbbs free energy depends on the composton as does an deal gas mture: [64] G (,, ) g (, ) + ln hs relaton dffers from that of an deal gas mture snce the molar Gbbs free energy of the pure speces, g (,), does not refer to the deal gas state but to the real state of the speces at and, for eample the lqud phase. In ths way the dfference between the behavour of the deal gas and the condensed phase (whch n departure functons s evaluated va an approprate EOS) s calculated through the Gbbs free energy of the pure speces. hs s the man advantage of ths approach wth respect to the drect use of an EOS. Fugacty computaton of a speces n an deal mture s qute smple snce, ntegratng the fugacty defnton between the pure speces state at and and the state related to a speces n an deal mture at, and t follows that: [65],,,,,,, dµ dln f ˆ, µ (,, ) fˆ,, µ (, ) ln f (, ) hs relaton, rememberng that the chemcal potental s equal to the partal molar Gbbs free energy that, n turn, for pure speces s equal to the molar Gbbs free energy, can be recast nto: fˆ (,, ) [66] G g (,, ) (, ) ln f (, ) From equatons [66] and [64] t follows that: [67] fˆ (,, ) f, hs relaton, usually referred to as the ews-andall rule, allows the calculaton of the fugacty of a speces n a mture through the value of the pure speces fugacty n the same phase and at the same temperature and pressure as the mture, as well as of the mture composton. From the defnton of the fugacty coeffcent of a speces n a mture t follows that: OUME / ISUMES 5

12 HYSICA AD CHEMICA EQUIIIA [68] hs means that the fugacty coeffcent of a speces n an deal mture s equal to that of the pure speces at the same temperature and pressure as the mture. hs greatly smplfes matters, even for all those mtures n vapour phase that cannot be consdered deal gases, but whose behavour can be appromated as that of deal mtures. hs allows the calculaton of fugacty coeffcents for the components of the mture usng the smpler, pure-speces relatons. Indrect methods use an deal mture as a reference pont and so dong ntroduce a new parameter whch accounts for the departure from the behavour of an deal mture: the actvty coeffcent, g ; t s defned as the rato of the fugacty of the speces n the mture to the fugacty of the speces n an deal mture at the same condtons: [69] Clearly the actvty coeffcent n an deal mture s equal to one, as they are when the related mole fracton approaches one (that s, the mture approaches pure speces behavour). he problem of computng the equlbrum condtons has thus been recast as that of computng the chemcal potentals and the fugactes and, n turn, the actvty coeffcents. Indrect methods deal wth the calculaton of actvty coeffcents, relatng them to the Gbbs free energy of the mture. he computaton of the Gbbs free energy through ndrect methods s based on the calculaton of the departure of the thermodynamc functon of a real flud from ts value when the mture behaves deally, called ecess functons, G E G G. Snce thermodynamc propertes of an deal mture can be easly calculated, knowng the Gbbs free energy or the ecess Gbbs free energy provdes the same nformaton. Just as the fugacty coeffcent s related to the departure partal molar Gbbs free energy, the actvty coeffcent s also related to the ecess partal molar Gbbs free energy. y ntegratng the fugacty defnton from deal to non-deal mture condtons the followng relaton can be deduced: [7] ˆ,, φ γ fˆ,, (,, ) fˆ,,,,,, dµ dln f ˆ µ (,, ),,,,,, ˆ f,, f, f (, ) φ (, ) fˆ,, µ (,, ) ln fˆ,, lnγ ememberng that the chemcal potental of a component n a mture s equal to the partal molar Gbbs free energy and usng the ecess functon defnton, the prevous relaton can be recast as: E [7] G (,, ) G (,, ) G (,, ) fˆ,, f (, ) ln γ (,, ) from whch the relaton between actvty coeffcents and partal ecess molar Gbbs free energy s clear: [7] lnγ As a consequence E E G ng [73] n G E or, n molar terms: E g [74] lnγ As prevously dscussed, when the mole fracton of a speces approaches one ts actvty coeffcent also approaches one. Snce n these condtons the mole fractons of all the other speces approach zero, the followng relaton holds true: E g [75] f,..., o descrbe non-deal mture behavour, the actvty coeffcents or, equally vald, the ecess Gbbs free energy must be known. On the other hand, the knowledge of the fugacty of the pure speces at the same temperature and pressure as the mture s requred to descrbe the behavour of the deal mture. hs problem wll be dscussed frst; then some models to represent the ecess Gbbs free energy wll be presented. As dscussed prevously, cubc EOS permt the calculaton of pure-speces fugacty through drect methods. On the other hand, EOS usually predct the gas phase behavour more accurately than the lqud phase. An alternatve approach for computng the condensed-phase fugacty that does not requre the use of EOS for the condensed phase nvolves the use of further epermental nformaton,.e. vapour pressure. In lqud-vapour equlbrum condtons the fugacty values of a pure speces n the two phases must be the same. Moreover, at a gven temperature the two phases coest n equlbrum at only one pressure value, the vapour pressure at the gven temperature, where the followng relaton must be fulflled: [76] f, f, ( ) φ, ( ) As a consequence, the fugacty of the compound n the lqud phase at a pressure value equal to () can be calculated by multplyng the vapour pressure value tmes the fugacty coeffcent n the vapour phase, that s, wthout usng the EOS for the condensed phase. he fugacty value at a dfferent pressure value (but at the same temperature) can be calculated through the general relaton: [77] dµ dln f dg vd E G n lnγ y ntegratng between the vapour pressure value and a generc pressure value for a component n the lqud-phase, the followng relaton whch gves the fugacty dependence on pressure s obtaned:, v [78] f f d (, ), ep, ( ) he eponental term n ths relaton s usually called the oyntng factor. In condtons far from crtcal and for 6 ECYCOAEDIA OF HYDOCAOS