Chapter 3: Solutions and Thermodynamics of Multicomponent Systems

Transcription

1 W. M. Whte Gechemstry Chapter 3: Slutns Chapter 3: Slutns and Thermdynamcs f Multcmpnent Systems 3.1 Intrductn In the prevus chapter, we ntrduced thermdynamc tls that allw us t predct the equlbrum mneral assemblage under a gven set f cndtns. Fr example, havng specfed temperature, we were able t determne the pressure at whch the assemblage anrthte+frsterte s n equlbrum wth the assemblage dpsde+spnel+enstatte. In that reactn the mnerals had unque and nvarant cmpstns. In the Earth, thngs are nt qute s smple: these mnerals are present as sld slutns*, wth substtutns f Fe fr Mg, Na fr Ca, and Cr and Fe 3+ fr Al, amng thers. Indeed, mst natural substances are slutns; that s, ther cmpstns vary. Water, whch s certanly the mst nterestng substance at the surface f the Earth and perhaps the mst mprtant, nevtably has a varety f substances dsslved n t. These dsslved substances are themselves ften f prmary gechemcal nterest. Mre t the pnt, they affect the chemcal behavr f water. Fr example, the freezng temperature f an aqueus NaCl slutn s lwer than that f pure water. Yu may have taken advantage f ths phenmenn by spreadng salt t de-ce sdewalks and rads, r addng salt t ce t make ce cream. In a smlar way, the equlbrum temperature and pressure f the plagclase+lvne clnpyrxene+spnel+rthpyrxene reactn depends n the cmpstn f these mnerals. T deal wth ths cmpstnal dependence, we need t develp sme addtnal thermdynamc tls, whch s the bjectve f ths chapter. Ths may seem burdensme at frst: f t were nt fr the varable cmpstn f substances, we wuld already knw mst f the thermdynamcs we need. Hwever, as we wll see n Chapter 4, we can use ths cmpstnal dependence t advantage n recnstructng cndtns under whch a mneral assemblage r a hydrthermal flud frmed. A fnal dffculty s that the valance state f many elements may vary. Irn, fr example, may change frm ts Fe 2+ state t Fe 3+ when an gneus rck weathers. The tw frms f rn have very dfferent chemcal prpertes; fr example Fe 2+ s cnsderably mre sluble n water than s Fe 3+. Anther example f ths knd f reactn s phtsynthess, the prcess by whch CO 2 s cnverted t rganc carbn. These knds f reactns are called xdatn reductn, r redx reactns. The energy yur bran uses t prcess the nfrmatn yu are nw readng cmes frm xdatn f rganc carbn carbn rgnally reduced by phtsynthess n plants. T fully specfy the state f a system, we must specfy ts redx state. We treat redx reactns n the fnal sectn f ths chapter. Thugh Chapter 4 wll add a few mre tls t ur gechemcal tlbx, and treat a number f advanced tpcs n thermdynamcs, t s desgned t be ptnal. Wth cmpletn f ths chapter, yu wll have a suffcent thermdynamc backgrund t deal wth a wde range f phenmena n the Earth, and mst f the tpcs n the remander f ths bk. 3.2 Phase Equlbra Sme Defntns Phase Phases are real substances that are hmgeneus, physcally dstnct, and (n prncple) mechancally separable. Fr example, the phases n a rck are the mnerals present. Amrphus substances are als phases, s glass r pal wuld be phases. The sugar that wn't dsslve n yur ce tea s a *The naturally ccurng mnerals f varyng cmpstn are referred t as plagclase rather than anrthte, lvne rather than frsterte, clnpyrxene rather than dpsde, and rthpyrxene rather than enstatte. 62 September 26, 2001

2 W. M. Whte Gechemstry Chapter 3: Slutns dstnct phase frm the tea, but the dsslved sugar s nt. Phase s nt synnymus wth cmpund. Phases need nt be chemcally dstnct: a glass f ce water has tw dstnct phases: water and ce. Many sld cmpunds can exst as mre than ne phase. Nr need they be cmpstnally unque: plagclase, clnpyrxene, lvne, etc., are all phases even thugh ther cmpstn can vary. Thus a fssl n whch the aragnte (CaCO 3 ) s partally retrgraded nt calcte (als CaCO 3 ) cnssts f 2 phases. Systems, and reactns ccurrng wthn them, cnsstng f a sngle phase are referred t as hmgenus ; thse systems cnsstng f multple phases, and the reactns ccurrng wthn them, are referred t as hetergeneus Speces Speces s smewhat mre dffcult t defne than ether phase r cmpnent. A speces s a chemcal entty, generally an element r cmpund (whch may r may nt be nzed). The term s mst useful n the cntext f gases and lquds. A sngle lqud phase, such as an aqueus slutn, may cntan a number f speces. Fr example, H 2 O, H 2 CO 3, HCO 3, CO 2 + 3, H +, and OH are all speces cmmnly present n natural waters. The term speces s generally reserved fr an entty that actually exsts, such as a mlecule, n, r sld n a mcrscpc scale. Ths s nt necessarly the case wth cmpnents, as we shall see. The term speces s less useful fr slds, althugh t s smetmes appled t the pure end-members f sld slutns and t pure mnerals Cmpnent In cntrast t a speces, a cmpnent need nt be a real chemcal entty, rather t s smply an algebrac term n a chemcal reactn. The mnmum number f cmpnents* f a system s rgdly defned as the mnmum number f ndependently varable enttes necessary t descrbe the cmpstn f each and every phase f a system. Unlke speces and phases, cmpnents may be defned n any cnvenent manner: what the cmpnents f yur system are and hw many there are depend n yur nterest and n the level f cmplexty yu wll be dealng wth. Cnsder ur aragntecalcte fssl. If the nly reactn ccurrng n ur system (the fssl) s the transfrmatn f aragnte t calcte, ne cmpnent, CaCO 3, s adequate t descrbe the cmpstn f bth phases. If, hwever, we are als nterested the precptatn f calcum carbnate frm water, we mght have t cnsder CaCO 3 as cnsstng f 2 cmpnents: Ca 2+ and CO 2 3. There s a rule t determne the mnmum number f cmpnents n a system nce yu decde what yur nterest n the system s; the hard part s ften determnng yur nterest. The rule s: c = n - r 3.1 where n s the number f speces r phases, and r s the number f ndependent chemcal reactns pssble between these speces. Let's try t fr ur fssl. n s 2 (calcte and aragnte), r s 1 (the transfrmatn reactn), s c =1; the aragnte calcte fssl has nly ne cmpnent. Thus a system may have nly 1 cmpnent (here n =2 and r = 1), but several phases. Nw let s try the rule n the speces we lsted abve fr water. We have 6 speces: H 2 O, H 2 CO 3, HCO 3, CO 2 + 3, H +, and OH. We can wrte 3 reactns relatng them: H + + CO 3 2 HCO 3 H + + HCO 3 H 2 CO 3 H + + OH H 2 O S equatn 3.1 tells us we need 3 = 6 3 cmpnents t descrbe ths system: CO 3 2 +, H +, and OH. In gneus and metamrphc petrlgy, cmpnents are ften the majr xdes (thugh we may ften chse t cnsder nly a subset f these). On the ther hand, f we were cncerned wth the stpc equlbratn f mnerals wth a hydrthermal flud, 18 O wuld be cnsdered as a dfferent cmpnent than 16 O. * Cautn: sme bks use the term number f cmpnents as synnymus wth mnmum number f cmpnents. 63 September 26, 2001

3 W. M. Whte Gechemstry Chapter 3: Slutns Al 2 O 3 AlO(OH) Al(OH) 3 H 2 O Fgure 3.1. Graphcal Representatn f the System Al 2 O 3 -H 2 O. Perhaps the mst straghtfrward way f determnng the number f cmpnents s a graphcal apprach. If all phases can be represented n a ne-dmensnal dagram (that s, a straght lne representng cmpstn), we are dealng wth a tw cmpnent system. Fr example, cnsder the hydratn f Al 2 O 3 (crundum) t frm behmte (AlO(OH)) r gbbste Al(OH) 3. Such a system wuld cntan 4 phases (crundum, behmte, gbbste, water), but s nevertheless a tw cmpnent system because all phases may be represented n ne-dmensn f cmpstn space, as s shwn n Fgure 3.1. Because there are tw plymrphs f gbbste, ne f bhemte, and tw ther pssble phases f water, there are 9 phases pssble phases n ths tw-cmpnent system. Clearly, a system may have many mre phases than cmpnents. Smlarly, f a system may be represented n 2 dmensns, t s a three-cmpnent system. Fgure 3.2 s a ternary dagram llustratng the system Al 2 O 3 H 2 O SO 2. The graphcal representatn apprach reaches t practcal lmt n a fur cmpnent system because f the dffculty f representng mre than 3 dmensns n paper. A fur cmpnent system s a quaternary ne, and can be represented wth a three-dmensnal quaternary dagram. It s mprtant t understand that a cmpnent may r may nt have chemcal real- H 2 O w ty. Fr example n the exchange reactn: NaAlS 3 O 8 + K + = KAlS 3 O 8 + Na + d,b g,by,n p ka,ha,d,na c q Al 2 O 3 a,k,s SO 2 Fgure 3.2 Phase dagram fr the system Al 2 O 3 H 2 O SO 2. The lnes are called jns because they jn phases. In addtn t the end-members, r cmpnents, phases represented are g: gbbste, by: bayerte, n: nrstrandte (all plymrphs f Al(OH) 3 ), d: daspre, b: bhemte (plymrphs f AlO(OH)), a: andeluste, k: kyante, s: sllmante (all plymrphs f Al 2 SO 5 ), ka: kalnte, ha: hallyste, d: dckte, na: nacrte (all plymrphs f Al 2 S 2 O 5 (OH) 4 ), and p: pyrphyllte (Al 2 S 4 O 10 (OH) 2 ). There are als 6 plymrphs f quartz (ceste, stshvte, trdymte, crstbalte, a-quartz, and b-quartz). we culd alternatvely defne the exchange peratr KNa -1 (where Na -1 s -1 ml f Na n) and wrte the equatn as: NaAlS 3 O 8 + KNa 1 = KAlS 3 O 8 Here we have 4 speces and 1 reactn and thus a mnmum f 3 cmpnents. Yu can see that a cmpnent s merely an algebrac term. There s generally sme freedm n chsng cmpnents. Fr example, n the ternary (.e., 3 cmpnent) system SO 2 Mg 2 SO 4 MgCaS 2 O 6, we culd chse ur cmpnents t be quartz, dpsde, and frsterte, r we culd chse them t be SO 2, MgO, and CaO. Ether way, we are dealng wth a ternary system (whch cntans MgSO 3 as well as the three ther phases) Degrees f Freedm The number f degrees f freedm n a system s equal t the sum f the number f ndependent ntensve varables (generally T & P) and ndependent cncentratns (r actvtes r chemcal ptentals) f cmpnents n phases that must be fxed t defne unquely the state f the system. A system 64 September 26, 2001

4 W. M. Whte Gechemstry Chapter 3: Slutns that has n degrees f freedm (.e., s unquely fxed) s sad t be nvarant, ne that has ne degree f freedm s unvarant, etc. Thus n an unvarant system, fr example, we need specfy nly the value f ne varable, fr example, temperature r the cncentratn f ne cmpnent n ne phase, and the value f pressure and all ther cncentratns are then fxed,.e., they can be calculated (assumng the system s at equlbrum) The Gbbs Phase Rule The Gbbs Phase Rule s a rule fr determnng the degrees f freedm, r varance, f a system at equlbrum. The rule s: ƒ = c - f where ƒ s the degrees f freedm, c s the number f cmpnents, and f s the number f phases. The mathematcal analgy s that the degrees f freedm are equal t the number f varables less the number f equatns relatng thse varables. Fr example, n a system cnsstng f just H 2 O, f tw phases f cexst, fr example, water and steam, then the system n unvarant. Three phases cexst at the trple pnt f water, s the system s sad t be nvarant, and T and P are unquely fxed: there s nly ne temperature and ne pressure at whch the three phases f water can cexst ( K and bar). If nly ne phase s present, fr example just lqud water, then we need t specfy 2 varables t descrbe cmpletely the system. It desn t matter whch tw we pck. We culd specfy mlar vlume and temperature and frm that we culd deduce pressure. Alternatvely, we culd specfy pressure and temperature. There s nly 1 pssble value fr the mlar vlume f temperature and pressure are fxed. It s mprtant t remember ths apples t ntensve parameters. T knw vlume, an extensve parameter, we wuld have t fx ne addtnal extensve varable (such as mass r number f mles). And agan, we emphasze that all ths apples nly t systems at equlbrum. Nw cnsder the hydratn f crundum t frm gbbste. There are 3 phases, but there need be nly tw cmpnents. If these 3 phases (water, crundum, gbbste) are at equlbrum, we have nly 1 degree f freedm,.e., f we knw the temperature at whch the reactn s ccurrng, the pressure s als fxed. Rearrangng equatn 3.2, we als can determne the maxmum number f phases that can cexst at equlbrum n any system. The degrees f freedm cannt be less than zer, s fr an nvarant, ne cmpnent system, a maxmum f three phases can cexst at equlbrum. In a unvarant necmpnent system, nly 2 phases can cexst. Thus sllmante and kyante can cexst ver a range f temperatures, as can kyante and andaluste. But the three phases f Al 2 SO 5 cexst nly at ne unque temperature and pressure. Let's cnsder the example f the three cmpnent system Al 2 O 3 H 2 O SO 2 n Fgure 3.2. Althugh many phases are pssble n ths system, fr any gven cmpstn f the system nly three phases can cexst at equlbrum ver a range f temperature and pressure. Fur phases, e.g., a, k, s and q, can cexst nly at pnts f ether fxed pressure r fxed temperature. Such pnts are called unvarant pnts. Fve phases can cexst at nvarant pnts at whch bth temperature and pressure are unquely fxed. Turnng ths arund, f we fund a metamrphc rck whse cmpstn fell wthn the Al 2 O 3 H 2 O SO 2 system, and f the rck cntaned 5 phases, t wuld be pssble t determne unquely the temperature and pressure at whch the rck equlbrated. J. Wllard Gbbs ( ) s vewed by many as the father f thermdynamcs. He receved the frst dctrate n engneerng granted n the U. S., frm Yale n He was Prfessr f Mathematcal Physcs at Yale frm 1871 untl hs death. He als helped t fund statstcal mechancs. The mprtance f hs wrk was nt wdely recgnzed by hs Amercan clleagues, thugh t was n Eurpe, untl well after hs death. 65 September 26, 2001

5 W. M. Whte Gechemstry Chapter 3: Slutns The Clapeyrn Equatn A cmmn prblem n gechemstry s t knw hw a phase bundary vares n P-T space, e.g., hw a meltng temperature wll vary wth pressure. At a phase bundary, tw phases must be n equlbrum,.e., DG must be 0 fr the reactn Phase 1 Phase 2. The phase bundary therefre descrbes the cndtn: d(dg r ) = DV r dp - DS r dt = 0. Thus the slpe f a phase bundary n a temperature-pressure dagram s: dt Vr = D 3.3 dp DSr where DV r and DS r are the vlume and entrpy changes asscated wth the reactn. Equatn 3.3 s knwn as the Clausus-Clapeyrn Equatn, r smply the Clapeyrn Equatn. Because DV r and DS r are functns f temperature and pressure, ths, f curse, s nly an nstantaneus slpe. Fr many reactns, hwever, partcularly thse nvlvng nly slds, the temperature and pressure dependences f DV r and DS r wll be small and the Clapeyrn slpe wll be relatvely cnstant ver a large T and P range. Because DS = DH/T, the Clapeyrn equatn may be equvalently wrtten as: dt T Vr = D 3.4 dp DH r Slpes f phase bundares n P-T space are generally pstve, mplyng that the phases wth the largest vlumes als generally have the largest entrpes (fr reasns that becme clear frm a statstcal mechancal treatment). Ths s partcularly true f sld-lqud phase bundares, althugh there s ne very mprtant exceptn: water. Hw d we determne the pressure and temperature dependence f DV r and why s DV r relatvely T and P ndependent n slds? 3.3 Slutns Slutns are defned as hmgeneus phases prduced by dsslvng ne r mre substances n anther substance. In gechemstry we are ften cnfrnted by slutns: as gases, lquds, and slds. Free energy depends nt nly n T and P, but als n cmpstn. In thermdynamcs t s generally mst cnvenent t express cmpstns n terms f mle fractns, X, the number f mles f dvded by the ttal mles n the substance (mles are weght dvded by atmc r mlecular weght). The sum f all the X must, f curse, ttal t 1. Slutns are dstnct frm purely mechancal mxtures. Fr example, salad dressng (l and vnegar) s nt a slutn. Smlarly, we can grnd anrthte (CaAl 2 S 2 O 8 ) and albte (NaAlS 3 O 8 ) crystals nt a fne pwder and mxture them, but the result s nt a plagclase sld slutn. The Gbbs Free Energy f mechancal mxtures s smply the sum f the free energy f the cmpnents. If, hwever, we heated the anrthte-albte mxture t a suffcently hgh temperature that the knetc barrers were vercme, there wuld be a rerderng f atms and the creatn f a true slutn. Because ths rerderng s a spntaneus chemcal reactn, there must be a decrease n the Gbbs Free Energy asscated wth t. Ths slutn wuld be stable at 1 atm and 25 C. Thus we can cnclude that the slutn has a lwer Gbbs Free Energy than the mechancal mxture. On the ther hand, vnegar wll never dsslve n l at 1 atm and 25 C because the Gbbs Free Energy f that slutn s greater than that f the mechancal mxture. 66 September 26, 2001

6 W. M. Whte Gechemstry Chapter 3: Slutns Example 3.1: The Graphte-Damnd Transtn At 25 C the graphte-damnd transtn ccurs at 1600 MPa (megapascals, 1 MPa =10 b). Usng the standard state (298 K, 0.1 MPa) data belw, predct the pressure at whch the transfrmatn ccurs when temperature s 1000 C. Graphte Damnd a (K 1 ) b (MPa -1 ) S (J/K-ml) V (cm 3 /ml) Answer: We can use the Clapeyrn equatn t determne the slpe f the phase bundary. Then, assumng that DS and DV are ndependent f temperature, we can extraplate ths slpe t 1000 C t fnd the pressure f the phase transtn at that temperature. Frst, we calculate the vlumes f graphte and damnd at 1600 MPa as: V = V (1 bdp) 3.5 where DP s the dfference between the pressure f nterest (1600 MPa n ths case) and the reference pressure (0.1 MPa). Dng s, we fnd the mlar vlumes t be fr graphte and fr damnd, s DV r s cc/ml. The next step wll be t calculate DS at 1600 MPa. The pressure dependence f entrpy s gven by equatn 2.143: S/ P) T = av. Thus t determne the effect f pressure we ntegrate: S P =S + P 1 P S 1 dp =S + avdp 3.6 P Pref T (We use S p t ndcate the entrpy at the pressure f nterest and S the entrpy at the reference pressure.) We need t express V as a functn f pressure, s we substtute 3.5 nt 3.6: S P =S + P 1 Pref Pref av (1 bp)dp =S av DP b 2 (P 1 2 P 2 ref ) 3.7 The reference pressure, P ref, s neglgble cmpared t P 1 (0.1 MPa vs 1600 PMa), s that ths smplfes t: S P =S av DP b 2 P 1 2 Fr graphte, S p s 5.66 J/K-ml, fr damnd, t s 2.34 J/K-ml, s DS r at 1600 MPa s J-K -1 - ml -1. The Clapeyrn slpe s therefre: DS DV = = JK-1 cm -3 One dstnct advantage f the SI unts s that cm 3 = J/MPa, s the abve unts are equvalent t K/MPa. Frm ths, the pressure f the phase change at 1000 C can be calculated as: P 1000 = P DT DS DV = = 3584 MP The Clapeyrn slpe we calculated (sld lne) s cmpared wth the expermentally determned phase bundary n Fgure 3.3. Our calculated phase bundarys lnear whereas the expermental ne s nt. The curved nature f the bserved phase bundaryndcates DV and DS are pressure and temperature dependent Ths s ndeed the case, partcularly fr graphte. A mre accurate estmate f the vlume change requres b be expressed as a functn f pressure. Pressure, MPa Damnd Graphte Temperature, C Fgure 3.3. Cmparsn f the graphte-damndphase bundarycalculated frm thermdynamc data and the Clapeyrn slpe (sld lne) wth the expermentally bserved phase bundary (dashed lne). 67 September 26, 2001

7 W. M. Whte Gechemstry Chapter 3: Slutns Rault's Law Wrkng wth slutns f ethylene brmde and prpylene brmde, Rault ntced that the vapr pressures f the cmpnents n a slutn were prprtnal t the mle fractns f thse cmpnents: P = X P 3.8 where P s the vapr pressure f cmpnent abve the slutn, X s the mle fractn f n slutn, and P s the vapr pressure f pure under standard cndtns. Assumng the partal pressures are addtve and the sum f all the partal pressures s equal t the ttal gas pressure (SP = P ttal ): P = X P 3.9 ttal Thus partal pressures are prprtnal t ther mle fractns. Ths s the defntn f the partal pressure f the th gas n a mxture. Rault's Law hlds nly fr deal slutns,.e., substances where there are n ntermlecular frces. It als hlds t a gd apprxmatn where the frces between lke mlecules are the same as between dfferent mlecules. The tw cmpnents Rault was wrkng wth were very smlar chemcally, s that ths cndtn held and the slutn was nearly deal. As yu mght guess, nt all slutns are deal. Fg. 3.4 shws the varatns f partal pressures abve a mxture f water and dxane. Sgnfcant devatns frm Rault's Law are the rule except where X appraches P B 40 P (mm Hg) P A 10 (A) H 2 O (B) Fgure 3.4. Vapr pressure f water and dxane n a water-dxane mxture shwng devatns frm deal mxng. Shaded areas are areas where Rault's Law (dashed lnes). Henry's Law slpes are shwn as dt-dashed lnes. After Nrdstrm and Munz (1986). Francs Mare Rault ( ), French chemst. 68 September 26, 2001

8 W. M. Whte Gechemstry Chapter 3: Slutns Henry's Law Anther useful apprxmatn ccurs when X appraches 0. In ths case, the partal pressures are nt equal t the mle fractn tmes the vapr pressure f the pure substance, but they d vary lnearly wth X. Ths behavr fllws Henry s Law, whch s: P = hx fr X << where h s knwn as the Henry's Law cnstant. 3.4 Chemcal Ptental Partal Mlar Quanttes Free energy and ther thermdynamc prpertes are dependent n cmpstn. We need a way f expressng ths dependence. Fr any extensve prperty f the system, such as vlume, entrpy, energy, r free energy, we can defne a partal mlar value, whch expresses hw that prperty wll depend n changes n amunt f ne cmpnent. Fr example, we can defne partal mlar vlume f cmpnent n phase f as: Ê V v f = ˆ Á such that V = Â nv Ë n 3.11 T, P, n j, j π (we wll use small letters t dente partal mlar quanttes; the superscrpt refers t the phase and the subscrpt refers t the cmpnent). The Englsh nterpretatn f equatn 3.11 s that the partal mlar vlume f cmpnent n phase f tells us hw the vlume f phase f wll vary wth an nfntesmal addtn f cmpnent, f all ther varables are held cnstant. Fr example, the partal mlar vlume f Na n an aqueus slutn such as seawater wuld tell us hw the vlume f that slutn wuld change fr an nfntesmal addtn f Na. In ths case wuld refer t the Na cmpnent and f wuld refer t the aqueus slutn phase. In Table 2.2, we see that the mlar vlumes f the albte Pl and anrthte end-members f the plagclase sld slutn are dfferent. We culd defne v Ab as the partal mlar vlume f albte n plagclase, whch wuld tell us hw the vlume f plagclase wuld vary fr an nfntesmal addtn f albte. (In ths example, we have chsen ur cmpnent as albte rather than Na. Whle we culd have chsen Na, the chce f albte smplfes matters because the replacement f Na wth Ca s accmpaned by the replacement f S by Al.) Anther example mght be a slutn f water and ethanl. The varatn f the partal mlar vlumes f water and ethanl n a bnary slutn s llustrated n Fgure 3.5. Ths system llustrates very clearly why the qualfcatn fr an nfntesmal addtn s always added: the value f a partal mlar quantty f a cmpnent may vary wth the amunt f that cmpnent present. Equatn 3.11 can be generalzed t all partal mlar quanttes and als expresses an mprtant prperty f partal mlar quanttes: an extensve varable f a system r phase s the sum f t s partal mlar quanttes fr each cmpnent n the system. In ur example abve, ths means that the vlume f plagclase s the sum f the partal mlar vlume f the albte and anrthte cmpnents. Generally, we fnd t mre cnvenent t cnvert extensve prpertes t ntensve prpertes by dvdng by the ttal number f mles n the system, Sn. Dvdng bth sdes f equatn 3.11 by Sn we have: V = Â X v 3.12 Ths equatn says that the mlar vlume f a substance s the sum f the partal mlar vlumes f ts cmpnents tmes ther mle fractns. Fr a pure phase, the partal mlar vlume equals the mlar vlume snce X=1. named fr Englsh chemst Wllam Henry ( ), wh frmulated t. 69 September 26, 2001

9 W. M. Whte Gechemstry Chapter 3: Slutns Defntn f Chemcal Ptental and Relatnshp t Gbbs Free Energy We defne µ as the chemcal ptental, whch s smply the partal mlar Gbbs Free Energy: Ê m = G ˆ Á 3.13 Ë n PT,, n j, jπ The chemcal ptental thus tells us hw the Gbbs Free Energy wll vary wth the number f mles, n, f cmpnent hldng temperature, pressure, and the number f mles f all ther cmpnents cnstant. We sad that the Gbbs Free Energy f a system s a measure f the capacty f the system t d chemcal wrk. Thus the chemcal ptental f cmpnent s the amunt by whch ths capacty t d chemcal wrk s changed fr an nfntesmal addtn f cmpnent at cnstant temperature and pressure. In a NCd battery (cmmn rechargeable batteres) fr example, the chemcal ptental f N n the battery (ur system) s a measure f the capacty f the battery t prvde electrcal energy per mle f addtnal N fr an nfntesmal addtn. The ttal Gbbs Free Energy f a system wll depend upn cmpstn as well as n temperature and pressure. The equatns we ntrduced fr Gbbs Free Energy n Chapter 2 fully descrbe the Gbbs Free Energy nly fr sngle cmpnents systems r systems cntanng nly pure phases. The Gbbs Free Energy change f a phase f varable cmpstn s fully expressed as: dg = VdP - SdT + Â mdn Prpertes f the Chemcal Ptental We nw want t cnsder tw mprtant prpertes f the chemcal ptental. T llustrate these prpertes, cnsder a smple tw-phase system n whch an nfntesmal amunt f cmpnent s transferred frm phase b t phase a, under cndtns where T, P, and the amunt f ther cmpnents s held cnstant n each phase. One example f such a reactn wuld be the transfer f Pb frm a hydrthermal slutn t a sulfde mneral phase. The chemcal ptental expresses the change n Gbbs Free Energy under these cndtns: a b a a b b dg = dg + dg = m dn + m dn 3.15 snce we are hldng everythng else cnstant, what s ganed by a must be lst by b, s dn a = dn b and: a b dg = m - m dn 3.16 ( ) a At equlbrum, dg = 0, and therefre m = m b Equatn 3.17 reflects a very general and very mprtant relatnshp, namely: In a system at equlbrum, the chemcal ptental f every cmpnent n a phase s equal t the chemcal ptental f that cmpnent n every ther phase n whch that cmpnent s present V 52 (ml/ml) Ethanl Water X Ethanl Fgure 3.5. Varatn f the partal mlar vlumes f water and ethanl as a functn f the mle fractn f ethanl n a bnary slutn. Ths fgure als llustrates the behavr f a very nn-deal slutn September 26, 2001

10 W. M. Whte Gechemstry Chapter 3: Slutns Equlbrum s the state tward whch systems wll naturally transfrm. The Gbbs Free Energy s the chemcal energy avalable t fuel these transfrmatns. We can regard dfferences n chemcal ptentals as the frces drvng transfer f cmpnents between phases. In ths sense, the chemcal ptental s smlar t ther frms f ptental energy, such as gravtatnal r electrmagnetc. Physcal systems spntaneusly transfrm s as t mnmze ptental energy. Thus fr example, water n the surface f the Earth wll mve t a pnt where t s gravtatnal ptental energy s mnmzed,.e., dwnhll. Just as gravtatnal ptental energy drves ths mtn, the chemcal ptental drves chemcal reactns, and just as water wll cme t rest when gravtatnal energy s mnmzed, chemcal reactns wll cease when chemcal ptental s mnmzed. S n ur example abve, the spntaneus transfer f Pb between a hydrthermal slutn and a sulfde phase wll ccur untl the chemcal ptentals f Pb n the slutn and n the sulfde are equal. At ths pnt, there s n further energy avalable t drve the transfer. We defned the chemcal ptental n terms f the Gbbs Free Energy. Hwever, n hs rgnal wrk, Gbbs based the chemcal ptental n the nternal energy f the system. As t turns ut, hwever, the quanttes are the same: Ê m = G ˆ Ê Á = U ˆ Á 3.18 Ë n Ë n PT,, n j, jπ SV,, nj, jπ It can be further shwn (but we wn t) that: Ê m = G ˆ Ê Á = U ˆ Ê Á = H ˆ Ê Á = A ˆ Á Ë n Ë n Ë n Ë n The Gbbs-Duhem Relatn PT,, n π SV n π SPn j, j,, j, j,, j, jπ T, V, n j, jπ Snce µ s the partal mlar Gbbs Free Energy, the Gbbs Free Energy f a system s the sum f the chemcal ptentals f each cmpnent: Ê G ˆ G = ÂÁ = Ânm 3.19 Ë n PT,, n j, jπ The dfferental frm f ths equatn (whch we get smply by applyng the chan rule) s: dg = n dm + m dn   3.20 Equatng ths wth equatn 3.14, we btan: ndm + m dn = VdP - SdT + m dn    3.21 Rearrangng, we btan the Gbbs-Duhem Relatn: VdP -SdT -  n dm = The Gbbs-Duhem Equatn descrbes the relatnshp between smultaneus changes n pressure, temperature and cmpstn n a sngle-phase system. In a clsed system at equlbrum, net changes n chemcal ptental wll ccur nly as a result f changes n temperature r pressure. At cnstant temperature and pressure, there can be n net change n chemcal ptental at equlbrum:  nd m = Ths equatn further tells us that the chemcal ptentals d nt vary ndependently, but change n a related way. In a clsed system, nly ne chemcal ptental can vary ndependently. Fr example, cnsder a tw cmpnent system. Then we have n 1 dµ 1 + n 2 dµ 2 = 0 and dµ 2 = (n 1 /n 2 )dµ 1. If a gven varatn n cmpstn prduces a change n µ 1 then there s a cncmtant change n µ September 26, 2001

11 W. M. Whte Gechemstry Chapter 3: Slutns Fr mult-phase systems, we can wrte a versn f the Gbbs-Duhem relatn fr each phase n the system. Fr such systems, the Gbbs-Duhem relatn allws us t reduce the number f ndependently varable cmpnents n each phase by ne. We wll return t ths pnt later n the chapter. We can nw state an addtnal prperty f chemcal ptental: In spntaneus prcesses, cmpnents r speces are dstrbuted between phases s as t mnmze the chemcal ptental f all cmpnents. Ths allws us t make ne mre characterzatn f equlbrum: equlbrum s pnt where the chemcal ptental f all cmpnents s mnmzed Dervatn f the Phase Rule Anther sgnfcant aspect f the Gbbs-Duhem Equatn s that the phase rule can be derved frm t. We begn by recallng that the varance f a system (the number f varables that must be fxed r ndependently determned t determne the rest) s equal t the number f varables less the number f equatns relatng them. In a multcmpnent sngle phase system, cnsstng f c cmpnents, there are c +2 unknwns requred t descrbe the equlbrum state f the system: T, P, µ 1, µ 2,...µ c. But n a system f f phases at equlbrum, we can wrte f versns f equatn 3.23, whch reduces the ndependent varables by f. Thus the number f ndependent varables that must be specfed t descrbe a system f c cmpnents and f phases s: f = c + 2 -f whch s the Gbbs phase rule. Specfcatn f ƒ varables wll cmpletely descrbe the system, at least wth the qualfcatn that n thermdynamcs we are nrmally unnterested n the sze f the system, that s, n extensve prpertes such as mass, vlume, etc. (thugh we are nterested n ther ntensve equvalents) and utsde frces r felds such as gravty, electrc r magnetc felds, etc. Nevertheless, the sze f the system s descrbed as well, prvded nly that ne f the ƒ varables s extensve. 3.5 Ideal Slutns Havng placed anther tl, the chemcal ptental, n ur thermdynamc tlbx, we are ready t cntnue ur cnsderatn f slutns. We wll begn wth deal slutns, whch, lke deal gases, are fctns that avd sme f the cmplcatns f real substances. Fr an deal slutn, we make an assumptn smlar t ne f thse made fr an deal gas, namely that there were n frces between mlecules. In the case f deal slutns, whch may be gases, lquds, r slds, we can relax ths assumptn smewhat and requre nly that the nteractns between dfferent knds f mlecules n an deal slutn are the same as thse between the same knds f mlecules Chemcal Ptental n Ideal Slutns Hw des chemcal ptental vary n an deal slutn? Cnsder the vapr pressure f a gas. The dervatve f G wth respect t pressure at cnstant temperature s vlume: Ê G ˆ Á = V Ë P Wrtten n terms f partal mlar quanttes: If the gas s deal, then: T Ê m ˆ Á = Ë P and f we ntegrate frm P t P we btan: T, deal RT P Ê m ˆ Á = v Ë P T September 26, 2001

12 W. M. Whte Gechemstry Chapter 3: Slutns P P P m - m = RT ln 3.25 P where µ P s the chemcal ptental f the pure gas at the reference (standard state) pressure P. Ths s the standard-state chemcal ptental and s wrtten as µ. If we let P be the vapr pressure f pure and P be the vapr pressure f n an deal slutn, then we may substtute X fr P/P nt Rault's Law (Equatn 3.8) and t btan the fllwng: m, deal = m + RT ln X 3.26 Ths equatn descrbes the relatnshp between the chemcal ptental f cmpnent and ts mle fractn n an deal slutn Vlume, Enthalpy, Entrpy, and Free Energy Changes n Ideal Slutns We wll be able t generalze a frm f ths equatn t nn-deal cases a bt later. Let's frst cnsder sme ther prpertes f deal mxtures. Fr real slutns, any extensve thermdynamc prperty such as vlume can be cnsdered t be the sum f the vlume f the cmpnents plus a vlume change due t mxng: V =  XV + DVmxng 3.27 The frst term n the rght reflects the vlume resultng frm mechancal mxng f the varus cmpnents. The secnd term reflects vlume changes asscated wth slutn. Fr example, f we mxed 100 ml f ethanl and 100 ml f water (Fgure 3.5), the vlume f the resultng slutn wuld be 193 ml. Here, the value f the frst term n the rght wuld be 200 ml, the value f the secnd term wuld be -7 ml. We can wrte smlar equatns fr enthalpy, etc. But the vlume change and enthalpy change due t mxng are bth 0 n the deal case. Ths s true because bth vlume and enthalpy changes f mxng arse frm ntermlecular frces, and, by defntn, such ntermlecular frces are absent n the deal case. Thus: DV deal mxng = 0 therefre: V = X v = XV  deal  and DH deal mxng = 0 and therefre: H = Xh = X H  deal  Ths, hwever, s nt true f entrpy. Yu can magne why: f we mx tw substances n an atmc level, the number f pssble arrangements f ur system ncreases even f they are deal substances. The entrpy f deal mxng s (cmpare equatn 2.110): D S = -R X ln X 3.28 deal mxng S = X S - R X X deal slutn   ln 3.29 Because DG mxng = DH mxng TDS mxng and DH mxng = 0, t fllws that: D G = RT X ln X 3.30 deal mxng We stated abve that the ttal expressn fr an extensve prperty f a slutn s the sum f the partal mlar prpertes f the pure phases (tmes the mle fractns) plus the mxng term. The partal mlar Gbbs Free Energy s the chemcal ptental, s the full expressn fr the Gbbs Free Energy f an deal slutn s: 73 September 26, 2001

13 W. M. Whte Gechemstry Chapter 3: Slutns   D G = X m + RT X ln X 3.31 deal slutn Rearrangng terms, we can re-express equatn 3.31 as: D G = X m + RT ln X 3.32  deal slutn ( ) The term n parentheses s smply the chemcal ptental f cmpnent, µ, as expressed n equatn Substtutng equatn 3.26 nt 3.32, we have  D G = X m deal slutn Nte that µ s always less than r equal t µ because the term RTln X s always negatve (because the lg f a fractn s always negatve). Let's cnsder deal mxng n the smplest case, namely bnary mxng. Fr a tw cmpnent (bnary) system, X 1 = (1 X 2 ), s we can wrte equatn 3.30 fr the bnary case as: [ ] D Gdeal mxng = RT ( 1- X )ln( 1- X ) + X ln X Snce X 2 s less than 1, DG s negatve and becmes ncreasngly negatve wth temperature, as llustrated n Fgure 3.6. The curve s symmetrcal wth respect t X;.e., the mnmum ccurs at X 2 = 0.5. Nw let s see hw we can recver nfrmatn n µ frm plts such as Fgure 3.6, whch we wll call G-bar X plts. Substtutng X 1 = (1 X 2 ) nt equatn 3.33, t becmes: D G = m ( 1 - X ) + m X =m + ( m -m ) X 3.35 deal slutn Ths s the equatn f a straght lne n such a plt wth slpe f (µ 2 µ 1 ) and ntercept µ 1. Ths lne s llustrated n Fgure 3.7. The curved lne s descrbed by equatn The dashed lne s gven by equatn Bth equatn 3.31 and 3.35 gve the same value f G fr a gven value f X 2, such as X 2. Thus the straght lne and the curved ne n Fgure 3.7 much tuch at X 2. In fact, the straght lne s the tangent t the curved ne at X 2. The ntercept f the tangent at X 2 = 0 s µ 1 and the ntercept at X 2 = 1 s µ 2. The pnt s, n a plt f mlar free energy vs. mle fractn (a G-X dagram), we can determne the chemcal ptental f cmpnent n a tw cmpnent system by extraplatng a tangent f the free energy curve t X = 1. We see that n Fgure 3.7, as X 1 appraches 1 (X 2 appraches 0), the ntercept f the tangent appraches µ 1,.e., µ 1 appraches µ 1. Lkng at equatn 3.26, ths s exactly what we expect. Fgure 3.7 llustrates the case f an deal slutn, but the ntercept methd apples t nn-deal slutns as well, as we shall see. Fnally, nte that the sld lne, the lne cnnectng the µ s s the Gbbs Free Energy f a mechancal mxture f cmpnents 1 and 2, whch we may express as:  D G = X m mxture 3.36 Yu shuld satsfy yurself that the DG mxng s the dfference between ths lne and the free energy curve: G G G 15 = X 2 1 deal mx. deal sl. mxture G deal mxng kj/ml T = 0 K K 500 K 1000 K 1500 K 2000 K Fgure 3.6. Free energy f mxng as a functn f temperature n the deal case. 74 September 26, 2001

14 W. M. Whte Gechemstry Chapter 3: Slutns G (per mle) m 1 G mxture = X 1 m 1 +X 2 m 2 m' 1 G deal sl. = m 1 +(m 2 m 1 )X 2 Gdeal sl. = G m xture+ D G m xng 0 1 X 2 Fgure 3.7. Mlar free energy n an deal mxture and graphcal llustratn f equatn After Nrdstrm & Munz, ' X Real slutns We nw turn ur attentn t real slutns, whch are smewhat mre cmplex than deal nes, as yu mght magne. We wll need t ntrduce a few new tls t help us deal wth these cmplextes Chemcal Ptental n Real Slutns Let s cnsder the behavr f a real slutn n vew f the tw slutn mdels we have already ntrduced: Rault s Law and Henry s Law. Fgure 3.8 llustrates the varatn f chemcal ptental as a functn f cmpstn n a hypthetcal real slutn. We can dentfy 3 regns where the behavr f the chemcal ptental s dstnct: 1.) The frst s where the mle fractn f cmpnent X s clse t 1 and Rault's Law hlds. In ths case, the amunt f slute dsslves n s trvally small, s mlecular nteractns nvlvng slute mlecules d nt sgnfcantly affect the thermdynamc prpertes f the slutn, and the behavr f µ s clse t that n an deal slutn: m, deal = m + RT ln X (3.26) 2.) At the ppste end s the case where X s very small. Here nteractns between tw cmpnent mlecules are extremely rare, and the behavr f µ s essentally cntrlled by nteractns between and thse f the slvent. Whle the behavr f µ s nt deal, t s nnetheless a lnear functn f ln X. Ths s the regn where Henry s Law hlds. The cmpstnal dependence f the chemcal ptental n ths regn can be expressed as: m = m + RT ln hx 3.38 where h s the Henry s Law cnstant defned n equatn Ths equatn can be rewrtten as: m = m + RT ln X + RT ln h 3.39 DG mxng By defntn, h s ndependent f cmpstn at cnstant T and P and can be regarded as addng a fxed amunt t the standard state chemcal ptental (a fxed amunt t the ntercept n Fg. 3.8). By ndependent f cmpstn, we mean t s ndependent f X, the mle fractn f the cmpnent f nterest. h wll, f curse depend n the nature f the slutn. Fr example, f Na s ur cmpnent f nterest, h Na wll nt be the same fr an electrlyte slutn as fr a slcate melt. We can defne a new term, µ*, as: * m m + RT ln h 3.40 Substtutng 3.40 nt 3.39 we btan: m * m + RT ln X 3.41 m 2 m' 2 75 September 26, 2001

15 W. M. Whte Gechemstry Chapter 3: Slutns When pltted aganst ln X, the chemcal ptental f n the range f very dlute slutns s gven by a straght lne wth slpe RT and ntercept µ* (the ntercept s at X = 1 and hence ln X = 0 and µ = µ*). Thus µ* can be btaned by extraplatng the Henry's Law slpe t X = 1. We can thnk f µ* as the chemcal ptental n the hypthetcal standard state f Henry's Law behavr at X = 1. 3.) The thrd regn f the plt s that regn f real slutn behavr between the regns where Henry's Law and Rault's Law apply. In ths regn, µ s nt a lnear functn f ln X. We wll ntrduce a new parameter, actvty, t deal wth ths regn Fugactes The tls we have ntrduced t deal wth deal slutns and nfntely dlute nes are based n bservatns f the gaseus state: Rault s Law and Henry s Law. We wll cntnue t make reference t gases n dealng wth real slutns that fllw nether law. Whle ths apprach has a largely hstrcal bass, t s nevertheless a cnsstent ne. S m* m m 0 ln X - fllwng ths pattern, we wll frst ntrduce the cncept f fugacty, and derve frm t a mre general parameter, actvty. In the range f ntermedate cncentratns, the partal pressure f the vapr f cmpnent abve a slutn s generally nt lnearly related t the mle fractn f cmpnent n slutn. Thus chemcal ptental f cannt be determned frm equatns such as 3.26, whch we derved n the assumptn that the partal pressure was prprtnal t the mle fractn. T deal wth ths stuatn, chemsts nvented a fcttus partal pressure, fugacty. Fugacty may be thught f as the escapng tendency f a real gas frm a slutn. It was defned t have the same relatnshp t chemcal ptental as the partal pressure f an deal gas: f m = m + RT ln 3.42 f where ƒ s the standard-state fugacty, whch s analgus t standard-state partal pressure. We are free t chse the standard state, but the standard state fr ƒ and µ must be the same. ƒ s analgus t the standard state partal pressure, P, f an deal gas. If we chse ur standard state t be the pure substance, then ƒ s dentcal t P, but we may wsh t chse sme ther standard state where ths wll nt be the case. Snce the behavr f real gases appraches deal at lw pressures, the fugacty wll apprach the partal pressure under these crcumstances. Thus the secnd part f the defntn f fugacty s: f lm = P Æ0 P Fr an deal gas, fugacty s dentcal t partal pressure. Snce, as we stated abve, fugacty bears the same relatnshp t chemcal ptental (and ther state functns) f a nn-deal substance as pressure f a nn-deal gas, we substtute fugacty fr pressure n thermdynamc equatns. The relatnshp between pressure and fugacty can be expressed as: Rault's Law m = m * + RT ln X m = m + RT ln X Real Slutns Henry's Law Fgure 3.8. Schematc plt f the chemcal ptental f cmpnent n slutn as a functn f ln X. Here µ s the chemcal ptental f pure at the pressure and temperature f the dagram. After Nrdstrm and Munz (1986). 76 September 26, 2001

16 W. M. Whte Gechemstry Chapter 3: Slutns ƒ = fp 3.44 where f s the fugacty ceffcent, whch wll be a functn f temperature and pressure dffer fr each real gas. The fugacty ceffcent expresses the dfference n the pressure between a real gas and an deal gas under cmparable cndtns. Kerrch and Jacbs (1981) ftted the Redlch-Kwng equatn (equatn 2.15) t bservatns n the vlume, pressure and vlume f H 2 O and CO 2 t btan values fr the ceffcents a and b n equatn Frm these, they btaned fugacty ceffcents fr these gases at a seres f temperatures and pressures. These are gven n Table Actvtes and Actvty Ceffcents Fugactes are thermdynamc functns Frm Kerrck and Jacbs (1981). that are drectly related t chemcal ptental and can be calculated frm measured P-T-V prpertes f a gas, thugh we wll nt dscuss hw. Hwever, they have meanng fr slds and lquds as well as gases snce slds and lquds have fnte vapr pressures. Whenever a substance exerts a measurable vapr pressure, a fugacty can be calculated. Fugactes are relevant t the equlbra between speces and phase cmpnents, because f the vapr phases f the cmpnents f sme sld r lqud slutns are n equlbrum wth each ther, and wth ther respectve sld r lqud phases, then the speces r phases cmpnents n the sld r lqud must be n equlbrum. One mprtant feature f fugactes s that we can use them t defne anther thermdynamc parameter, the actvty, a: a f 3.45 f Table 3.1. H 2 O and CO 2 Fugacty Ceffcents H 2 O T C P, MPa CO 2 T C P, MPa ƒ s the standard state fugacty. Its value depends n the standard state yu chse. Yu are free t chse a standard state cnvenent fr whatever prblem yu are addressng. If we substtute equatn 3.45 nt equatn 3.42, we btan the mprtant relatnshp: m = m + RT ln a 3.46 The catch n selectng a standard state fr ƒ, and hence fr determnng a n equatn 3.46, s that ths state must be the same as the standard state fr µ. Thus we need t bear n mnd that standard states are mplct n the defntn f actvtes and that thse standard states are ted t the standardstate chemcal ptental. Untl the standard state s specfed, actvtes have n meanng. Cmparng equatn 3.46 wth3.26 leads t: a,deal = X 3.47 Thus n deal slutns, the actvty s equal t the mle fractn. Chemcal ptentals can be thught f as drvng frces that determne the dstrbutn f cmpnents between phases f varable cmpstn n a system. Actvtes can be thught f as the effectve cncentratn r the avalablty f cmpnents fr reactn. In real slutns, t wuld be cnvenent t relate all nn-deal thermdynamc parameters t the cmpstn f the slutn, because cmpstn s generally readly and accurately measured. T relate actvtes t mle fractns, we defne a new parameter, the ratnal actvty ceffcent, l. The relatnshp s: 77 September 26, 2001

17 W. M. Whte Gechemstry Chapter 3: Slutns Example 3.2. Usng Fugacty t Calculate Gbbs Free Energy The mnerals bructe (Mg(OH) 2 ) and perclase (MgO) are related by the reactn: Mg(OH) 2 MgO + H 2 O Whch sde f ths reactn represent the stable phase assemblage at 600 C and 200 MPa? Answer: We learned hw t slve ths srt f prblem n Chapter 2: the sde wth the lwest Gbbs Free Energy wll be the stable assemblage. Hence, we need nly t calculate DG r at 600 C and 200 MPa. T d s, we use equatn 2.131: Ú D G = D G - D S dt + D V dp TP ' ' T ' T ref r Ú T ' T ref r (2.131) Our earler examples dealt wth slds, whch are ncmpressble t a gd apprxmatn, and we culd smply treat DV r as beng ndependent f pressure. In that case, the slutn t the frst ntegral n the left was smply DV r (P - P ref ). The reactn n ths case, lke mst metamrphc reactns, nvlves H 2 O, whch s certanly nt ncmpressble: the vlume f H 2 O, as steam r a supercrtcal flud, s very much a functn f pressure. Let s slate the dffculty by dvdng DV r nt tw parts: the vlume change f reactn due t the slds, n ths case the dfference between mlar vlumes f perclase and bructe, and the vlume change due t H 2 O. We wll dente the frmer as DV S and assume that t s ndependent f pressure. The secnd ntegral n then becmes: Ú P' Pref Ú s D VdP = DV ( P' - P ) + V dp r ref Hw d we slve the pressure ntegral abve? One apprach s t assume that H 2 O s an deal gas. RT Fr an deal gas: V = P P' RT s that the pressure ntegral becmes: P dp RT P' Ú = ln Pref Steam s a very nn-deal gas, s ths apprach wuld nt yeld a very accurate answer. The cncept f fugacty prvdes us wth an alternatve slutn. Fr a nn-deal substance, fugacty bears the same relatnshp t vlume as the pressure f an deal gas. Hence we may substtute fugacty fr pressure s that the pressure ntegral n equatn becmes: f ' RT P df RT ' Ú = ln f ref ref where we take the reference fugacty t be 0.1 MPa. Equatn 3.47 thus becmes: P' f ' s s f ' Ú D VdP r = DV ( P' - Pref ) + Ú VH Odf = DV ( P' -Pref )ln 3.49 Pref f 2 ref fref We can then cmpute fugacty usng equatn 3.44 and the fugacty ceffcents n Table 3.1. Usng the data n Table 2.2 and slvng the temperature ntegral n n the usual way (equatn 2.139), we calculate the DG T,P s 3.29 kj. Snce t s pstve, the left sde f the reactn,.e., bructe, we predct that bructe s stable. The DS f ths reactn s pstve, hwever, mplyng that at sme temperature, perclase plus water wll eventually replace bructe. T calculate the actual temperature f the phase bundary requres a tral and errr apprach: fr a gven pressure, we must frst guess a temperature, then lk up a value f f n Table 2.1 (nterplatng as necessary), and calculate DG r. Dependng n ur answer, we make a revsed guess f T and repeat the prcess untl DG s 0. Usng a spreadsheet, hwever, ths ges farly quckly. Usng ths methd, we calculate that bructe breaks dwn at 660 C at 200 MPa, n excellent agreement wth expermental bservatns. P' Pref P ref H O 78 September 26, 2001

18 W. M. Whte Gechemstry Chapter 3: Slutns a = X l 3.50 The ratnal actvty ceffcent dffers slghtly n defntn frm the practcal actvty ceffcent, g, used n aqueus slutns. l s defned n terms f mle fractn, whereas g s defned n terms f mles f slute per mles f slvent. Cnsder fr example the actvty f Na n an aqueus sdum chlrde slutn. Fr l Na, X s cmputed as: nna X Na = nna + ncl + nh 2O Na whereas fr g Na, X Na s: n nh 2 O where n ndcates mles f substance. In very dlute slutn, the dfference s trval. g s als used fr ther cncentratns unts that we wll ntrduce n sectn Excess Functns The deal slutn mdel prvdes a useful reference fr slutn behavr. Cmparng real slutns wth deal nes leads t the cncept f excess functns, fr example: G excess = G real - G deal 3.51 whch can be reslved nt cntrbutns f excess enthalpy and entrpy: G excess = H excess - TS excess 3.52 The excess enthalpy s a measure f the heat released durng mxng the pure end-members t frm the slutn, and the excess entrpy s a measure f all the energetc effects resultng frm a nnrandm dstrbutn f speces n slutn. We can express excess enthalpy change n the same way as excess free energy,.e.: DH excess = DH real - DH deal 3.53 But snce DH deal mxng = 0, DH excess = DH real ; n ther wrds, the enthalpy change upn mxng s the excess enthalpy change. Smlar expressns may, f curse, be wrtten fr vlume and entrpy (bearng n mnd that unlke vlume and enthalpy, DS deal s nt zer). Cmbnng equatn 3.46 wth equatn 3.50 leads t the fllwng: m m + RT ln X l 3.54 We can rewrte ths as: m m + RT ln X + RT ln l 3.55 Equatn 3.55 shws hw actvty ceffcents relate t Henry's and Rault's Laws. Cmparng equatn 3.55 wth equatn 3.39, we see that n the regn where Henry's Law hlds, that s dlute slutns, the actvty ceffcent s equal t Henry's law cnstant. In the regn where Rault's Law hlds, the actvty ceffcent s 1 and equatn 3.55 reduces t equatn 3.26 snce RT ln l = 0. Ê G ˆ Snce we knw that Á =m =m + RT ln Xl Ë n T, P, n j, jπ cmparng equatns 3.51 and 3.55, we fnd that: Ê G ˆ excess Á = RT ln l Ë n T, P, n j, jπ whch s the same as: Gexcess, = RT ln l 3.56 S that the mlar excess free energy asscated wth cmpnent s smply RT tmes the lg f the actvty ceffcent. The ttal mlar excess free energy f the slutn s then: G = RT Â X ln l 3.57 excess 79 September 26, 2001

19 W. M. Whte Gechemstry Chapter 3: Slutns Depressn f the Meltng Pnt In nrthern clmates such as Ithaca, NY saltng rad and sdewalks t melt snw and ce s a cmmn practce n wnter. We have nw acqured the thermdynamcs tls t shw why salt melts ce and that ths effect des nt depend n any specal prpertes f salt r water. Depressn f the meltng pnt by addtn f a secnd cmpnent t a pure substance s a general phenmenn. Suppse that we have an aqueus slutn cntanng sdum chlrde cexstng wth pure ce. If the tw phases are at equlbrum, then the chemcal ptental f water n ce must equal that f water n the slutn,.e.: ce aq m = m 3.58 H 2O H 2O (we are usng subscrpts t dente the cmpnent, and superscrpts t dente the phase; aq dentes the lqud aqueus slutn). We defne ur standard state as that f the pure substance. Accrdng t equ. 3.48, the chemcal ptental f water n the slutn can be expressed as: aq aq m = m + RT ln a 3.59 H 2O H 2O H 2O µ H 2 O dentes the chemcal ptental f pure lqud water. Substtutng 3.56 nt 3.57 and rearrangng, we have: ce aq mh O - m H O = RT ln a 2 2 l H 2O 3.60 Ice wll ncrprate very lttle salt; f we assume t s a pure phase, we may wrte 3.60 as: aq mh O - m H O = RT ln a 2 s 2 l H 2O 3.60a aq r mh O - m H O = -RT ln a 2 l 2 s H 2O 3.61 (The rder s mprtant: equatn 3.60a descrbes the freezng prcess, 3.61 the meltng prcess. These prcesses wll have equal and ppste entrpes, enthalpes, and free energes). The left hand sde f 3.61 s the Gbbs Free Energy f meltng fr pure water, whch we dente as DG m (DG m s 0 at the meltng temperature f pure water, whch we dente T m, but nn-zer at any ther temperature). We may rewrte 3.61 as: aq D G = -RT ln a 3.62 m H 2O If we assume that DH and DS are ndependent f temperature (whch s nt unreasnable ver a lmted temperature range) and we assume pressure s cnstant as well, the left hand sde f the equatn may als be wrtten as: D G = DH -TDS 3.63 m Substtutng 3.59 nt 3.58: DH -TD S = -RT ln a m m m m aq H 2O 3.64 At the meltng temperature f pure water, D G m s zer, s that: D Hm = Tm DSm Substtutng ths nt 3.60 and rearrangng: aq DS T -T RT ln a 3.65 m ( m ) =- H 2O Further rearrangement yelds: Tm R aq - 1 = - ln a H 2O T DSm Fr a reasnably dlute slutn, the actvty f water wll apprxmately equal ts mle fractn, s that: Tm R aq - 1 = - ln X H O T DSm The entrpy f meltng s always pstve and snce X s always less than 1, the left hand sde f 3.66 must always be pstve. Thus the rat T m /T must always be greater than 1. S the temperature at whch an aqueus slutn wll freeze wll always be less than the meltng pnt f pure water. Saltng f rads s nt a questn f gechemcal nterest, but there are many examples f depressn f the freezng pnt f gelgcal nterest. Fr example, the freezng pnt f the cean s abut 2 C. And ths phenmenn s mprtant n gneus petrlgy, as we shall see n the next chapter. A related phenmenn f gelgcal nterest s elevatn f the blng pnt f a lqud: fr example hydrthermal slutns bl at temperatures sgnfcantly abve that f pure water. Can yu demnstrate that elevatn f the blng pnt f an deal slutn depends nly n the mle fractn f the slute? We wll see the usefulness f the cncept f excess free energy shrtly when we cnsder actvtes n electrlyte slutns. It wll als prve mprtant n ur treatment f nn-deal sld slutns and exslutn phenmena n the next chapter. 80 September 26, 2001