Chapter 9: Other Topics in Phase Equilibria

Transcription

1 Chapter 9: Other opcs n Phase qulbra hs chapter deals wth relatons that derve n cases of equlbru between cobnatons of two co-exstng phases other than vapour and lqud,.e., lqud-lqud, sold-lqud, and soldvapour. ach of these phase equlbra ay be eployed to overcoe dffcultes encountered n purfcaton processes that explot the dfference n the volatltes of the coponents of a xture,.e., by vapour-lqud equlbra. As wth the case of vapour-lqud equlbra, the objectve s to derve relatons that connect the copostons of the two co-exstng phases as functons of teperature and pressure. 9. qud-lqud qulbra () 9.. Phase Dagras Unlke gases whch are scble n all proportons at low pressures, lqud solutons (bnary or hgher order) often dsplay partal scblty at least over certan range of teperature, and coposton. If one attepts to for a soluton wthn that certan coposton range the syste splts spontaneously nto two lqud phases each coprsng a soluton of dfferent coposton. hus, n such stuatons the equlbru state of the syste s two phases of a fxed coposton correspondng to a teperature. he copostons of two such phases, however, change wth teperature. hs typcal phase behavor of such bnary lqud-lqud systes s depcted n fg. 9.a. he closed curve represents the regon where the syste exsts Fg. 9. Phase dagras for a bnary lqud syste showng partal scblty

2 n two phases, whle outsde t the state s a hoogenous sngle lqud phase. ake for exaple, the pont P (or Q). hs pont defnes a state where a hoogeneous lqud soluton exsts at a teperature P (or Q) wth a coposton x P (or x Q). However, f one tres to for a soluton of coposton x Y (at a teperature Y ) the syste autoatcally splts nto I II two lqud phases (I and II) wth copostons gven by x and x. he straght lne MYN represents a te-lne, one that connects the copostons of the two lqud phases that co-exst n equlbru. Any attept to for a soluton at Y wth a coposton correspondng to any I II pont wthn MYN always results n two phases of fxed copostons gven by x and x. he total ass (or oles) of the orgnal soluton s dstrbuted between the two phases accordngly. Note, however, that the copostons at the end te lnes (horzontal and parallel to MYN) change as the teperature changes. Indeed as one approaches ether pont U or, the te lne reduces to a pont, and beyond ether pont the syste exsts as a sngle hoogeneous soluton. he teperatures correspondng to ponts U and,.e., and are called the Upper Consolute oluton eperature (UC) and ower UC C Consolute oluton eperature (C) respectvely. hey defne the lt of scblty of the coponents of the bnary soluton. Not all lqud-lqud systes, however, depct the behavour descrbed by fg. 9.a. Other varants of phase behavour are shown n fgs. 9.b 9.c. In each case there s at least one consolute teperature. 9.. Phase tablty Crtera he passage of a lqud soluton fro a state of a sngle, hoogeneous phase to a bphasc state occurs when a sngle phase s no longer therodynacally stable. It s nstructve to derve the condtons under whch ths ay take place. et us consder a bnary lqud xture for exaple for whch the olar Gbbs free energy of xng ( G x ) s plotted as functon of ole fracton ( x ) of coponent for two teperatures and (fg. 9.). At teperature the value of Gx s always less than zero and also goes through a nu. hs sgnfes that the xture fors a sngle lqud phase at all copostons at the gven teperature. he atheatcal descrpton of ths condton s gven by the two followng equatons: G x x P, = 0..(9.)

3 And: G x x P, > 0..(9.) Alternately: G x x P, = 0..(9.3) And: G x x P, > 0..(9.4) However, at certan other teperature the value of G x, whle beng always less than zero, passes through a regon where t s concave to the coposton axs. Consder a soluton wth a coposton gven by the pont A. At ths coposton the Gbbs free energy of the xture s gven by: G = xg+ xg+ G..(9.5) A A A A x x Fg. 9. Molar Gbbs free energy xng vs. ole fracton x for a bnary soluton et the ponts B and C denote the ponts of at whch a tangent BC s drawn on the Gx curve for the sother, and D the pont vertcally below A on the straght lne BC (whch s a tangent to the Gx curve at ponts B and C). If the soluton at A splts nto two

4 B C phases wth copostons characterzed by x and x then ts olar Gbbs free energy s gven by: G = xg+ xg+ G..(9.6) B A A D x x Clearly, snce G D < G A,( or G D < G A ) a phase splttng s therodynacally favoured x x x x A over a sngle phase soluton of coposton x. Pont D represents the lowest Gbbs free A energy that the xture wth an overall coposton x can have at the teperature. In A other words, at the teperature a soluton wth an overall coposton x exsts n two B C phases characterzed by copostons x and x. If nb and ncrepresent the quanttes of A B C soluton n each phase then the followng equaton connects x to x and x : ( B C ) ( B C) x = nx + nx n + n..(9.7) A B C he above consderatons allow one to forulate a phase separaton crtera for lqud solutons. ssentally, for phase splttng to occur, the curve ust n part be concave to the coposton axs. hus n general atheatcal ters a hoogeneous lqud soluton becoes unstable f: ( x ) G x < 0; (where, x x )..(9.8) P, Gx It follows that the alternate crteron for nstablty of a sngle phase lqud xture s: ( Gx x ) < 0..(9.9) P, It ay be evdent that at the above condtons hold over the coposton range x < x < x Conversely, a hoogeneous lqud phase obtans for copostons over the B C. B C ranges 0 < x < x and x < x <. In these ranges, therefore, the followng atheatcal condton apples: ( Gx x ) > 0..(9.0) Or: P, ( Gx x ) > 0..(9.) P, Referrng agan to fg. 9. t ay be evdent that at ponts X and Y the followng atheatcal condton holds: ( Gx x ) = 0..(9.) P,

5 A seres of Gx curves at other teperatures (say between and ) ay be drawn, each showng dfferent ranges of unstable copostons. All such curves ay be ore concsely C expressed by the xplot n fg. 9.. here s an absolute teperature above whch the xture s stable at all copostons snce the condton descrbed by eqn. 9. apples at all copostons. he bnodal curve n fg. 9. represents the boundary between the sngle phase regon and the two phase regons. Wthn the two phase regon the spnodal curve represents the locus of ponts at whch ( Gx x ) = 0. As ay be evdent fro the xplot n fg. 9., t s the boundary between the unstable ( Gx x ) ( Gx x ) > 0 P, regons. P, < 0 P, and etastable he above atheatcal condtons ay be used to explan the phase behavour depcted n fg. 9.. nce the bnary xture dsplays both UC and C, t follows that at > UC and C teperatures the < the xture fors sngle phase at all copostons. hus for such > 0 for all Gx curve s descrbed by eqn. 9.,.e., ( Gx x ) P, copostons. Conversely for C < < UC the Gx curve dsplays a for correspondng to that for teperature n fg. 9.. hat s, for C < < UC there s partal scblty of the xture. he atheatcal condtons for the exstence of the consolute teperatures ay then be sursed by the followng relatons. For the exstence of UC: G = 0 for soe value of x at = x and G > 0 for all values of x at > x For the exstence of UC: UC G = 0 for soe value of x at = x and G > 0 for all values of x at < x C UC C he foregong atheatcal relatons are ore convenently expressed n ters of the excess olar Gbbs free energy functon as follows:

6 d G Gx Gx Or: =..(9.3) ( ln ln ) G = Gx x G+ xg + R x x+ x x..(9.4) hus: ( ln ln ) Gx = G + x G+ xg + R x x+ x x..(9.5) On applyng the crteron for nstablty gven by eqn. 9.8, an equvalent crteron obtans as follows: G + R + < 0 x x P, x G x P, R + < 0 xx..(9.6) For an deal soluton G = 0; hence the condton n eqn. 9.6 can never hold. hus, and deal soluton can never dsplay phase-splttng behavour. Consder now the case of a bnary xture descrbed by the excess Gbbs free energy functon: G = α xx. hus: G x P, = α On applyng the condton gven by eqn. 9.6 one obtans the followng relaton:..(9.7) R α + < 0..(9.8) xx Or: R α >..(9.9) xx he axu value of the ter xx = 0.5; hence the values of α that satsfes the nequalty 9.9 s gven by: α R..(9.0) An equvalent crteron of nstablty ay be derved usng actvty coeffcent as a paraeter n place of the excess Gbbs free energy functon. For splcty we consder a bnary soluton. hus we have: G = xln γ+ xln γ..(9.) R

7 On dfferentatng eqn. 9. one obtans: ( / ) ln γ ln γ d G R d d = ln γ ln γ + x + x..(9.) dx dx dx By Gbbs-Duhe relaton: dln γ dln γ x + x = 0..(9.3) dx dx Hence: ( / R ) d G dx ( ) = ln γ ln γ..(9.4) / ln ln =..(9.5) d G R d γ d γ dx dx dx Or, ultplyng both sdes by x : ( ) d G / R dln γ dln γ ( ) dx dx dx x = x x..(9.6) Applyng the Gbbs-Duhe relaton agan and upon splfcaton eqn. 9.6 reduces to: ( ) d G / R d ln γ =..(9.7) dx x dx For stablty of a soluton usng eqn. 9.6 we have: G x P, R + > 0 xx Upon substtutng eqn. 9.7 n 9.8 one obtans after due algebrac splfcaton:..(9.8) dln( xγ ) > 0..(9.8) dx he above relaton provdes a convenent startng pont for deternng f a certan actvty coeffcent odel ay predct phase nstablty. As an llustraton we apply t to Wlson eqn. for a bnary (see table 6.) wheren: Λ Λ ln γ = ln( x + xλ ) + x x +Λ x Λ x + x On usng eqn. 9.9 n 9.8 and after algebrac splfcaton one obtans: dln( xγ ) x Λ Λ dx x x x x x = + ( + Λ ) ( + Λ)..(9.9)..(9.30)

8 It ay be seen that the RH of the above equaton s always > 0, hence eqn. 9.8 s satsfed. Hence t follows that the Wlson equatons cannot be used to predct nstablty of a lqud soluton olvng the lqud-lqud phase equlbra proble We now present the equatons needed to obtan the phase copostons for a bphasc lqudlqud equlbra condton. Consder two lqud phases I and II, n equlbru wth each other and each contanng N speces. One starts wth the basc phase equlbra crteron: ˆ I ˆ II f = f ; ( =,,... N)..(9.3) In practce the ajorty of lqud-lqud extracton operatons are carred out at low to oderate pressures. hus the fugactes ay be expressed n ters of actvty coeffcents. On expandng each ter n the above equaton we get: ˆ I I f = ( xγ f )..(9.3) ˆ II II f = ( xγ f )..(9.33) Where, x = ole fracton of th speces n a phase nce both the phases are at the sae teperature and pressure at equlbru, the pure coponent fugactes are equal: f I = f II hus t follows that: I ( xγ ) = ( xγ ) II I I II II x γ = x γ ; ( =,,... N)..(9.34) Further by ole balance: I x =..(9.35) II x =..(9.36) quaton 9.34 represents the generalzed crteron applcable to systes wth any nuber of coponents dstrbuted between two co-exstng partally scble lqud phases. Addtonally, a sutable actvty coeffcent odel needs to be assued for solvng the proble. he phase copostons are derved by solvng a set of N equatons of the type 9.34 (one for each coponent), along wth two constranng equatons of the type 9.35 and xaple 9.

9 Use the van aar actvty coeffcent expresson to predct the copostons of co-exstng lqud phases (I and II) coprsed of two partally scble lquds () and () at 50 o C and 4 bar. At these condtons the van aar equatons are gven by: A A ln γ = ;ln γ = ; A =.5; A = 3.5 Ax Ax [ + ] [ + ] A x A x (Clck for soluton) ernary lqud-lqud equlbra As ay be evdent fro fg. 9.a, under two-phase condtons, one of the lqud phases usually contans one of the coponents n a ore concentrated for. hs feature of lqudlqud equlbra s exploted n the process ndustry, typcally n the for of ternary lqudlqud extracton. In any nstances of practcal nterest a bnary (or hgher) lqud xture cannot be adequately concentrated n one of the coponents by vapour-lqud separaton processes; ths, ether because of nadequate dfference n volatlty of the coponents, or because there s an azeotrope foraton. In such nstances an alternate approach to purfyng a speces to the requred extent fro other coponents could be through the addton of a thrd lqud (solvent) that s partally scble wth the orgnal soluton. In such a case a second lqud phase results, where one of the coponents n the orgnal xture ay becoe preferentally concentrated due to ts hgher affnty for the added solvent. As wth bnary systes one can solve ths ternary proble by applcaton of eqns uch phase equlbra data (ether experental or coputed) are typcally plotted as a trangular dagra. A very sple exaple of such a dagra s shown n fg More coplex versons of such dagras ay be found elsewhere (see.i. andler, Checal, Bochecal and ngneerng herodynacs, 4th dton, Wley Inda, 006). Fg. 9.3 rangular dagra for ternary lqud-lqud phase equlbra

10 In ths fgure A and B coprse the orgnal bnary xture to whch a thrd, partally scble solvent C s added to preferentally extract A. ach apex of the trangle represents 00% ole (or ass) fracton of the speces ndcated at the apex. ach sde of the trangle represents a xture of two speces that are ndcated at the two ends of the sde. he concentraton of each speces decreases lnearly wth dstance along the perpendcular lne that ay be drawn fro the apex to the opposte sde of the trangle. he ass or ole fractons of each speces correspondng to a pont wthn the trangle s found by frst drawng a lne through the pont parallel to the sde opposte to the apex for that speces. he pont of ntersecton of the lne wth the approprate sde of the trangle provdes the concentraton fracton of the speces. For exaple, for the pont J the consttuent copostons are shown as xaj, xbj and xcj respectvely on the three sdes of the trangle. he doe shaped regon DP corresponds to the two phase regon n the trangular dagra, whle any pont outsde t represents ether a bnary or a ternary hoogeneous xture. he lnes FG, HI, JK are typcal te lnes across the two-phase regon. he two end ponts of the lne correspond to the two phases that co-exst n equlbru. For exaple, the end pont on the left sde (here, F) of the doe DP corresponds to the solvent-rch extract phase where A s preferentally concentrated, whle the pont G represents the raffnate phase that s rch n B. hus any pont wthn the doe represents a bphasc syste wth the end ponts of the te lne through the pont correspondng to the extract and raffnate phases. As ay also be observed, the te lnes becoe shorter n length as one approaches the top part of the doe, untl t reduces to a pont at P. hs pont s tered the plat pont and t sgnfes the condton at whch the copostons of the two lqud phases n equlbru becoe dentcal, and ndeed transfor to a sngle phase. 9. old-qud qulbra he predcton of solublty of a sold n a lqud s portant for desgn of separaton processes that utlze ether preferental dssoluton or crystallzaton as a route to purfcaton of a speces. he presence of a dssolved sold n a lqud also changes the freezng (as well as bolng) pont of the latter, and ts estaton s portant for desgn of heat exchange equpents such as crystallzers (and evaporators). A wde varety of phase behavour has been observed n systes whch coprse xtures of solds, xture of a sold and a lqud and xtures coprsed of a lqud and ore than a sngle sold (see R.. DeHoff,

11 herodynacs n Materals cence, chaps. 9 and 0, McGraw-Hll, New York, 993). hs secton s devoted to consderaton of therodynac relatons that allow coputaton of the solublty of solds n lquds as a functon of teperature. uch relatons ay also be extended to estate changes n freezng or bolng pont of a lqud n presence of dssolved solds. he phase equlbra relatons developed here are lted to the splest case of systes coprsng a sold () and a lqud () (solvent). As wth all other phase equlbra, one starts wth the bass of equalty of the speces fugacty n the sold (s) and lqud (l) phases respectvely. fˆ ˆ = f..(9.37) xγ f = zγ f..(9.38) Where, x and z are, respectvely, the ole fractons of speces n the lqud and sold solutons. quvalently, xγ = zγ ς..(9.39) Where, ς f (, P) / f (, P)..(9.40) he rght sde of ths equaton, defnes ψ as the rato of fugactes at the teperature and pressure P of the syste. Note that for each pure speces at ts (say, noral) eltng pont, we have: f (,, P) = f (,, P)..(9.4) However at, ; f (, P) f (, P) hus, n general one needs to derve a sutable expresson for ς defned by eqn hs ay be done by through estaton of the enthalpy and entropy of on at the syste teperature, wth values of the enthalpy and entropy at the eltng pont at the syste pressure as datu. However, snce the propertes of solds are not dependent on pressure n a sgnfcant way, for convenence of coputaton one can assue the reference pont as the noral eltng pont., Accordngly, we ay wrte for ether the sold or the solvent speces: P P..(9.4) H = C d + H + C d

12 Or: = + ( P P) H H C C d..(9.43) In the sae anner one ay wrte: ( CP - CP) = d +..(9.44) At teperature, = 0, thus: G H = 0..(9.45) Or: H =..(9.46) hus, usng eqn n 9.44 we have: ( CP CP) H = d +..(9.47) Further, at teperature, we have: G = H..(9.48) herefore, substtutng eqns and 9.47 n 9.48 and re-arrangng we have: P P ( CP CP) ( )..(9.49) H G = H + C C d d If we further ake the splfyng assupton that ( CP CP) s constant over the teperature range to one obtans: G = H ( )( ) ( ) ln + CP CP CP CP..(9.50) Now, f f dg = R d ln f Or: f G = R ln..(9.5) f hus, coparng eqn and 9.5: f fu s Rln = H ( )( ) ( ) ln + CP CP CP CP f..(9.5)

13 herefore usng eqn. 9.5 n eqn. 9.40, we have: fu s Rlnς = H ( )( ) ( ) ln + CP CP CP CP Upon sutable rearrangeent: H ( C ) P C P lnς = + ln R R Now by eqn. 9.39:..(9.53)..(9.54) xγ = zγ ς..(9.39) Wthς gven by the general expresson as n eqn. 9.54, a sutable actvty coeffcent odel ay be chosen to represent the actvty coeffcents γ and γ n the sold and lqud phases, respectvely and then eqn ay be solved to yeld the phase copostons. We llustrate the typcal ethodology of such calculatons for a bnary syste. et us assue a sold () s to be solublzed n a lqud (). he startng pont s eqn However, n applyng t we ake the splfyng assupton that the lqud does not dssolve n the sold phase, and so the latter phase s a pure coponent sold. hs s a reasonable approxaton for any syste of practcal nterest n the checal ndustry. he lqud phase, however, wll be a bnary wth the sold transforng to a lqud phase and then xng wth the lqud solvent. hus, one ay rewrte the phase equlbru as: (, ) ˆ f P = f (, P, x )..(9.55) ˆ (,, ) = (,, )..(9.56) Now f Px xγ Px f hus fro the last two equatons, the equlbru solublty of the sold ( x ) n the lqud s gven by: f x =..(9.57) ( )( ) f γ( Px,, ) nce actvty coeffcents are functons of teperature and coposton only: f x = f γ(, x)..(9.58) It follows that f the teperature at whch the solublty s desred s the eltng pont of the sold, then f ( ) = f ( ). herefore, we have:

14 x = γ (, x )..(9.59) However, for any teperature (other than ) eqn needs to be eployed. hus, we have: f ln x = ln γ ln f Or usng eqns and 9.5 (or 9.53) we have: H ( C ) P C P ln x = ln γ + ln R R..(9.60)..(9.6) One ay solve eqn. 9.6 (teratvely) for the solublty x, by assung a sutable actvty coeffcent odel for the lqud phase (for ex: van aar, Regular oluton, NR, etc). In the specal case where the teperature of nterest s close to the eltng pont so that one can ake the approxaton eqn. 9.6 reduces to: H ln x = ln γ R..(9.6) xaple 9. o state solublty of a sold A n a lqud B at 300 K, usng () deal soluton assupton, () regular soluton odel for lqud phase. he followng data are avalable: ( ) ( ) 3 3 / / A = B = A = B = V 00 c / ol; V 5 c / ol; δ 9.5 cal / cc ; δ 7.5 cal / cc. on for A: 7.5 kj/ol. Meltng pont for A = 350 o K. (Clck for soluton) For the purpose of further llustraton of the results that typcally obtan fro applcaton of the foregong phase equlbra relatons, we consder two ltng or specal cases for a bnary syste assung that eqn. 9.6 apples. It ay be noted, however, that ths assupton s not essental to the consderatons below. Usng eqn we wrte: Heat of H lnς = R Or, for any of the speces:..(9.63)

15 ς H, = exp R,..(9.64) he ltng cases are:. Both phases for deal solutons,.e., γ = and γ = for all teperature and copostons.. he lqud phase behaves as an deal soluton ( γ = ), and all speces n the sold state are copletely scble,.e., ( z γ = ) Case Wrtng eqn for both speces, we have: x = zς..(9.65) x = z ς Further: x z x..(9.66) + =..(9.67) z + =..(9.68) Usng eqns , one obtans: x z ( ς ) ς = ς ς ς = ς ς If one uses the splfed for forς, then by eqn. 9.64:..(9.69)..(9.70) ς H, = exp R,..(9.7) ς H, = exp R,..(9.7) On puttng x = z = t follows that ς = ; thus, by eqn. 9.7, =., In the sae anner f we put x = z = thenς = ; and =., On choosng a seres of teperature between, and, and for each case solvng for x and zone obtans a x z plot that s shown scheatcally n fg he upper curve n fg. 9.4 s the freezng curve and the

16 lower curve s the eltng curve. Any xpont wthn the lens-shaped regon between the two curves corresponds to a two phase stuaton, whle ponts outsde the regon depct a sngle phase. If a pont s above the upper curve the state of the syste s a hoogeneous bnary lqud xture, whereas f the pont les beneath the eltng curve, the state of the syste s bnary sold xture. Any straght lne parallel to the coposton axs depcts a te lne (for exaple XY), the copostons at the two ends sgnfyng the lqud and sold phase copostons. hs behavour s analogous to that of Raoult s law x y plot shown n fg.7.7. Fgure 9.4 Case I, deal lqud and sold solutons An exaple of a syste exhbtng a phase dagra of ths type s a bnary xture of ntrogen/carbon onoxde at low teperatures. However, as ay be evdent, ths phase behavour s necessarly an dealzed one. Its utlty les n that t can provde a reference syste for nterpretng the behvour of ore coplex, real systes. Case l For ths case the lqud phase fors deal soluton ( γ = ), and there s coplete s scblty for all speces n the sold state ( z γ = ). hus the governng phase equlbra relatons are: x x = ς..(9.73) = ς..(9.74)

17 nce, ς and ςare functons of teperature alone (by eqns.9.7 and 9.7), t follows that x and x lkewse are functons of teperature only. hus eqns and 9.74 ay be solved ndependently upon whch one obtans two dstnct plots on the x dagra as shown n fg However, f eqns and 9.65 apply sultaneously, then we have: ς+ ς = and hence x+ x =. hus, t follows that under ths condton the followng relaton holds: Fg x-z plot for Case II H H,, exp + exp = R, R, (9.75) On solvng the above equaton one obtans the utectc teperature e shown n fg hs partcular phase dagra s to be nterpreted as follows. When eqn apples one has: H x exp R =..(9.76) hs equaton s vald over the teperature range =, and the correspondng coposton e range x < x<, where x = xe the eutectc coposton. hus n ths regon (I) of the phase dagra a lqud soluton can be n equlbru wth only pure speces as a sold phase. hs s represented by regon I on fg. 9.5 where lqud solutons wth copostons x gven by lne Y are n equlbru wth pure sold. Consder the syste to be ntally at a state

18 defned by pont M. If one cools the syste t eventually reaches the pont N on the x plot, at whch condton only the coponent just begns to freeze. hs happens tll one reaches the eutectc teperature e at whch pont coponent begns to co-precptate. In the sae anner regon II obtans by solvng eqn. 9.74,.e., x H sl = exp x R..(9.77) hs equaton s vald only fro =, where, x = 0, to = e, where x = x e, the eutectc coposton. quaton 9.77 therefore apples where a lqud soluton s n equlbru wth pure speces as a sold phase. hs s represented by regon II on fg. 9.5 where lqud solutons wth copostons x gven by lne X are n equlbru wth pure sold. Consder an ntal syste state defned by pont P whch s a bnary soluton. On coolng, the syste eventually reaches the pont Q where only pure sold begns to crystallze. On contnued coolng, the eutectc teperature s reached, whereupon coponent also precptates sultaneously. Fnally, f one coences coolng fro the pont R wth a soluton coposton gven by x, = x e no precptaton of any sold occurs tll the teperature e (or pont ) s reached; at whch state, as entoned above, eqn holds. hs s a state where two solds are n equlbru wth the soluton. Further coolng below e results n co-precptaton of both speces, and the syste enters a state where two scble solds are present xaple 9.3 Copute the eutectc coposton and teperature for a xture of two substances A and B usng the followng data: Property A B Noral ( o K) 80 8 H (J/ol) (Clck for soluton) old-vapour qulbru

19 hs secton deals wth the solublty of solds n gases. he phase equlbra consderaton here are relatvely straghtforward as the solublty of gases n solds s usually neglgble. herefore, the sold phase ay be consdered as pure; all non-dealty n the syste therefore, derves fro that of the gas phase. In the followng analyss we, thus, consder that a pure sold phase () co-exsts n equlbru wth a vapour-phase xture of the gas (or solvent, ) and the solute (.e., ). he phase equlbra relaton for the coponent ay be wrtten as: ˆ V f = f..(9.78) A relaton of the for of eqn. 6.9 used for pure lqud phase fugacty calculaton ay be eployed to copute the fugacty of a pure sold at the teperature and pressure P of the syste. l sat sat sat V ( P P ) f(, P) = φ P exp[ ]..(6.9) R For a pure sold the correspondng equaton s: sat ( ) V P P sat sat f = P φ exp R Here V s the olar volue of the sold, and sat P..(9.79) s the vapour-phase saturaton pressure for the sold solute at the gven teperature. Now for the vapour phase fugacty of the solute on has: ˆ V f = y ˆ φ P..(9.80) hus usng eqns one obtans: sat ( ) V P P sat sat P ˆ φ exp = yφp R he above equaton ay be wrtten n a copressed for as follows:..(9.8) sat P y = P Where, sat ( ) sat φ V P P = exp ˆ φ R..(9.8)..(9.83) sat he ter s called the enhanceent factor. he vapour pressure P of solds s typcally very low (~ sat bar), hence φ.0. Further f the syste pressure s also low or as P sat P, ˆ φ. hus, on nspectng eqn. 9.8, t s evdent that under such a condton

20 =. he correspondng solublty y s tered the deal solublty. At hgh pressures (wth P>> P, the exponental Poyntng factor tends to be greater than one, and the values of ˆ φ sat ) are lower than unty. As a result, the factor typcally s hgher than one. he enhanceent factor, therefore, provdes a easure of the degree to whch pressure augents the solublty of the sold n the gas phase xaple A certan sold A has a vapour pressure of 0.0 bar at 300 K. Copute ts solublty at the sae teperature n a gas B at a pressure of.0bar. he olar volue of the sold s 5cc/ol. (Clck for soluton) Assgnent- Chapter 9