SMA5101 Thermodynamics of Materials Ceder 2001

Transcription

1 SMA5101 Thermdynamcs f Materals Ceder 2001 Chapter 4 Slutn Thery In the frst chapters we dealt prmarly wth clsed systems fr whch nly heat and wrk s transferred between the system and the envrnment. In the ths chapter, we study the thermdynamcs f systems that can als exchange matter wth ther systems r wth the envrnment, and n partcular, systems wth mre than ne cmpnent. Frst we fcus n hmgeneus systems called slutns. Next we cnsder hetergeneus systems wth emphass n the equlbrum between dfferent mult-cmpnent phases. 4.1 WHAT IS A SOLUTION? A slutn n thermdynamcs refers t a system wth mre than ne chemcal cmpnent that s mxed hmgeneusly at the mlecular level. A well-knwn example f a slutn s salt water: The Na +, Cl - and H2O ns are ntmately mxed at the atmc level. Many systems can be characterzed as a dspersn f ne phase wthn anther phase. Althugh such systems typcally cntan mre than ne chemcal cmpnent, they d nt frm a slutn. Slutns are nt lmted t lquds: fr example ar, a mxture f predmnantly N 2 and O 2, frms a vapr slutn. Sld slutns such as the sld phase n the S-Ge system are als cmmn. Fgure 4.1. schematcally llustrates a bnary sld slutn and a bnary lqud slutn at the atmc level. Fgure 4.1: (a) The (111) plane f the fcc lattce shwng a cut f a bnary A-B sld slutn whereby A atms (empty crcles) are unfrmly mxed wth B atms (flled crcles) n the atmc level. (b) A twdmensnal cut thrugh a bnary lqud slutn shwng a unfrm dstrbutn f A and B atms n the atmc level. -1-

2 SMA5101 Thermdynamcs f Materals Ceder 2001 T characterze a slutn, t s necessary t ntrduce varables specfyng the cmpstn f the dfferent chemcal cmpnents f the slutn. Several cmpstn varables are ften used, each havng partcular advantages n dfferent applcatns. The frst cmpstn varables f mprtance are the mle numbers. Fr a system wth N cmpnents, we wll refer t the number f mles f each cmpnent as n. When specfyng the cmpstn f a mult-cmpnent system n a phase dagram, mre practcal cmpstn varables are mle fractn and weght fractn. The mle fractn f cmpnent, dented by x, refers t the number f mle n f n the slutn dvded by the ttal number f mle n tt n the slutn. Smlarly, the weght fractn, w, f cmpnent s the rat f the weght f cmpnent, W, n slutn t the ttal weght f slutn, W tt. Weght fractns are ften used n practcal applcatns, where a mxture havng a partcular weght fractn can easly be prepared by weghng the pure cmpnents befre mxng them. Mle fractns are useful when vewng the slutn wthn a theretcal framewrk where detals f the slutn at the atmc level becme mprtant. Clsely related t mle fractn s the atmc percent f cmpnent whch s ften dented by (at%) and equals 100 tmes the mle fractn x. A furth mprtant cmpstn varable s the cncentratn C f cmpnent, defned as the number f mles f dvded by the vlume V f the slutn. Ths varable s ften mplemented n the study f rreversble prcesses, snce the cncentratn s a natural varable n Fck s dfferental equatns descrbng dffusn. As an vervew, the fur cncentratn varables are summarzed n table 1. Mle Fractn: x n wth n tt = n n tt ( ) Atmc Percent: at % 100 % x Weght Fractn: w W wth W tt = W W tt Cncentratn: C n = r smetmes C V W = V Table 1: Defntns f mprtant cmpstn varables n slutns. -2-

3 SMA5101 Thermdynamcs f Materals Ceder PARTIAL MOLAR QUANTITIES In prevus chapters, we saw that the state f a sngle phase, sngle cmpnent system that s subjected t nly P-V wrk s cmpletely specfed nce tw ndependent varables such as P and T are fxed. Thermdynamc varables such as the equlbrum vlume V r the Gbbs free energy G are then unquely determned nce partcular values f P and T have been mpsed n the system and we smetmes remnd urselves f ths functnal dependence by explctly wrtng the vlume as V(T,P) and the Gbbs free energy as G(T,P). In slutns, ths functnal dependence s n lnger cmplete. A multcmpnent system subjected t nly P-V wrk has added degrees f freedm due t the fact that we can change ts cmpstn. T fully characterze the equlbrum state f a mult-cmpnent system, t s therefre necessary t als specfy, n addtn t P and T, the number f mles n f the dfferent cmpnents n the slutn. Nw the equlbrum vlume f the slutn shuld be wrtten as V(P,T,n 1, n N ) and the Gbbs free energy as G(P,T,n 1, n N ), where n, =1, N, dentes the number f mles f each f the N cmpnents f the slutn. As a result f the added dependence n n, the perfect dfferental f thermdynamc quanttes lke V and G take the frms and V V T V dv = dt + dp + P dn (1) Pn, Tn n, PTn,, j G G G dg = dt + dp + T P dn. (2). Pn, Tn, n PTn,, j The subscrpts n n the partal dervatves ndcate that the number f mles f all cmpnents s kept cnstant. Expressns fr the partal dervatves wth respect t T and P fllw frm the results btaned fr clsed systems. Indeed, f we mpse the cnstrant that the mult-cmpnent system s clsed, t cannt exchange matter wth the envrnment and the dn are all zer. Then the dfferental expressns fr V and G reduce t thse fr clsed systems studed n prevus chapters and the partal dervatves wth respect t T and P becme by cmparsn V T V = Vα = V β P Pn, Tn, G T = S G P Pn, Tn, =V -3-

4 SMA5101 Thermdynamcs f Materals Ceder 2001 where α s the thermal expansn ceffcent and β s the sthermal ceffcent f cmpressblty. Of central mprtance n the study f slutns are the partal dervatves wth respect t the n. These partal dervatves measure the varatns f extensve thermdynamc prpertes as chemcal cmpnents are added r remved frm the system. We call these partal dervatves partal mlar quanttes. In general, the partal mlar quantty f the extensve varable Y wth respect t cmpnent, s dented by the symbl Y. The partal mlar vlume fr example s then wrtten as V V = (3) n TP,, n j Ths s the partal dervatve f V wth respect t n hldng cnstant T, P and the number f mles f all ther cmpnents j nt equal t. One exceptn t ths ntatn s the partal mlar quantty f thermdynamc ptentals such as the Gbbs free energy. These are referred t as chemcal ptentals and are dented by G µ = (4) n TP,, nj Puttng everythng tgether, the perfect dfferentals (1) and (2) becme dv = αvdt βvdp + V dn dg = SdT + VdP + µ dn. At cnstant T and P, a partal mlar quantty measures the amunt by whch the extensve thermdynamc quantty changes when an nfntesmal amunt f a partcular cmpnent s added r remved frm the slutn. Fr example, we may be nterested t knw hw much the vlume f ferrtc steel (prmarly rn n a bcc structure wth a dlute cmpstn f carbn resdng n the ntersttal stes) changes when the carbn cmpstn s ncreased. Under typcal crcumstances, where T and P are the cntrllng varables, the vlume f a blck f steel n equlbrum wll depend n T, P, n Fe and n C. Wrkng at cnstant T and P and keepng the amunt f rn fxed, the change f vlume f the steel when an amunt dn C f carbn s added s smply dv = V C dn C. Carbn s a small atm n cmparsn t rn, and furthermre t resdes n ntersttal stes whle rn frms the substtutnal framewrk f the bcc structure. Ths knwledge abut the atmc structure f the sld already gves us an mprtant clue that VC wll be smaller than V Fe snce addtn f rn extends the frame-wrk f the bcc crystal whle -4-

5 SMA5101 Thermdynamcs f Materals Ceder 2001 carbn smply flls sme f the numerus pen spaces already present wthn the structure. (fgure 4.2.) Whle n general, partal mlar vlumes are pstve, many examples exst where the partal mlar vlume s actually negatve. Addtn f a cmpnent wth a negatve partal mlar vlume causes the slutn t shrnk. Ths happens fr example n L 0.5 CO 2 where addtn f L results n a reductn f the vlume f the crystal. Fgure 4.2: In steel, Fe (large empty crcles) frms a bcc crystal structure and C (small flled crcle) resdes n the ntersttal stes. The partal mlar vlume f Fe s larger than that f C because the frmer extends the bcc crystal structure whle the latter flls empty ntersttal space. 4.3 PROPERTIES OF EXTENSIVE QUANTITIES Expermental bservatn has establshed that thermdynamc quanttes such as a system s vlume V, enthalpy H, and Gbbs free energy G are extensve prpertes, that s they are prprtnal t the sze f the system. Fr example, dublng the number f mles f each cmpnent results n the dublng f the equlbrum vlume f the system. The same hlds fr ther extensve quanttes lke H, S, G, etc. In general, f we ncrease the number f mles f each cmpnent by a factr λ at cnstant T and P the fllwng relatn can be wrtten fr any extensve quantty Y Y(λn 1, λn 2,..., λn N ) = λy( n 1, n 2,..., n N ) (5) (We have mtted the explct dependence f Y n T and P because we are assumng them fxed). The fact that extensve quanttes are state functns that satsfy equatn (5) has tw mprtant mplcatns. Frst t leads, as wll be shwn belw, t an mprtant relatn between an extensve quantty and ts partal mlar quanttes, namely that -5-

6 SMA5101 Thermdynamcs f Materals Ceder 2001 Y = ny (6) at cnstant T and P. Secndly t leads t the Gbbs-Duhem relatn whch s useful n surface thermdynamcs and n applcatns nvlvng nc dffusn n nnhmgenus slutns. T derve equatn (6), we start by settng v =λ n. Dfferentatng equatn (5) wth respect t λ yelds r N Y ( v, 1 v =1,... v ) v 2 N = λy ( n n,... n 1, 2 N ) v λ N Y( λ n 1, λ n 2,... λn N ) n = λy( n n 2 λn ),... n 1, N ) =1 ( Ntng that the peratn Y v s equvalent t Y n we btan after settng λ =1 N ( 1, n Yn n 2,... n ) N = Yn ( 1, n 2,... n N ) =1 n whch, when usng the defntn f partal mlar quanttes, reduces t equatn (6). Fr the vlume V f a mult-cmpnent slutn, equatn (6) becmes N V = n whle fr the Gbbs free energy, equatn (6) becmes =1 V (7) N G = n µ (8) =1 Equatn (6) (alng wth equatns (7) and (8)) shws that partal mlar quanttes are nt nly useful t ndcate hw extensve prpertes change wth varatns n cmpstn, but can als be used t descrbe the ttal value f the extensve prperty. An addtnal thermdynamc relatn, ften referred t as the Gbbs-Duhem equatn, can be btaned by takng the ttal dfferental f equatn (6) at cnstant T and P. Ths yelds Y =1 n TPnj,, N N N dn = n dy + Y dn =1 =1-6-

7 SMA5101 Thermdynamcs f Materals Ceder 2001 whch can be rewrtten as N N N Ydn = ndy + Y dn =1 =1 =1 Cancelng lke terms we btan the Gbbs-Duhem relatn N ndy = 0 (9). =1 The Gbbs-Duhem relatn fr the Gbbs free energy becmes N ndµ = 0 (10). =1 Ths equatn states that nt all chemcal ptentals can be vared ndependently. In a bnary slutn fr example, ndµ A A + n Bd µ B = 0. Hence a varatn f µ A by dµ A causes µ B t change by x A B A A n B x B dµ = n A dµ = dµ (11) Equatn (11) als shws that f the chemcal ptental f ne f the cmpnents n a bnary slutn s knwn as a functn f cmpstn t s pssble t btan the chemcal ptental f the ther cmpnent by ntegratn. The ntegral, thugh, s nt straghtfrward snce t extends frm µ A = when x A =0 and dverges as x B appraches zer (see fr example Chemcal Thermdynamcs f Materals by C. H. P. Lups fr mre detals) QUANTITIES OF MIXING Partal mlar quanttes tell us hw a thermdynamc prperty f a slutn changes when addng r remvng an nfntesmal amunt f a gven chemcal cmpnent. Often, thugh, we are als nterested n the change f thermdynamc quanttes when gng frm the unmxed state t the mxed state. Fr example, t may be mprtant t knw by hw much the vlume f a bnary slutn dffers frm the cmbned vlumes f the tw chemcal speces befre they are mxed. Ths dfference n -7-

8 SMA5101 Thermdynamcs f Materals Ceder 2001 vlume s called the vlume f mxng. We may als want t knw by hw much the enthalpy changes f a bnary slutn when gng frm the unmxed state t the mxed state. As was shwn n a prevus chapter, the change n enthalpy at cnstant T and P asscated wth an rreversble change f state (such as mxng) s equal t the heat exchanged between the system and the envrnment. Hence, the enthalpy f mxng at cnstant T and P tells us hw much heat wll be released r absrbed when cmbnng chemcal speces. Befre, further specfyng quanttes f mxng, t s useful t ntrduce the cncept f mlar quanttes. A mlar quantty refers t an extensve thermdynamc varable that has been dvded by the ttal number f mles n the system. The mlar quantty asscated wth the extensve varable Y s dented by Y and s gven by Y Y = (12) n tt where n tt s the ttal number f mles n the system. Fr bnary systems, mlar quanttes are ften pltted as a functn f the mle fractn x f ne f the tw cmpnents. Ths s llustrated n fgure 4.3 where the mlar vlume f a bnary mxture f T and Al s pltted as a functn f the Al mle fractn x Al. Fgure 4.3: Mlar vlume f a T-Al slutn pltted as a functn f the Al mle fractn. Plts such as the ne llustrated n fgure 4.3 are useful t graphcally dsplay quanttes f mxng. As an llustratn, cnsder the vlume change asscated wth mxng n Al mles f pure Al and n T mles f pure T. Befre mxng Al and T, the cmbned vlume f the tw pure cmpnents s smply nal V Al + n T V T -8-

9 SMA5101 Thermdynamcs f Materals Ceder 2001 where V Al and V T are the mlar vlumes f pure Al and T respectvely. Dvdng ths premxng vlume by n tt =n Al +n T yelds the mlar vlume befre mxng nal V Al + n T V T = xal V Al + x T V T n Al + n T Snce x Al +x T =1, the mlar vlume befre mxng becmes. xal V Al + (1 x Al )V T (13) Ths expressn fr the premxng mlar vlume represents the dashed lne f fgure 4.3 cnnectng V T at x Al =0 and V Al at x Al =1. When T and Al are mxed and frm a slutn, the mlar vlume changes frm that f the weghted average f the pure cmpnents gven by equatn (13) t the vlume gven by V( x Al ) (full curve n fgure 4.3). The change n vlume upn mxng s, therefre, the dfference between the dashed lne and the curve V( x Al ) as llustrated n fgure 3. Ths can be wrtten as ΔV mx = V ( Al ) (x Al V Al + (1 x Al )V T ) (14) Other mprtant quanttes f mxng are the mxng enthalpy, entrpy and Gbbs free energy. These can be wrtten as fllws fr a bnary A-B slutn. ΔH mx = H ( x B ) (x B H B + (1 x B )H A ) (15) ΔS mx = S ( x B ) (x B S B + (1 x B )S A ) (16) ΔG mx = G ( x B ) (x B G B + (1 x B )G A ) (17) As nted, the enthalpy f mxng ΔH mx s equal t the heat exchanged wth the envrnment upn mxng a ttal f ne mle f pure cmpnents A and B at cncentratn x B. When ΔH mx s negatve, heat s released and mxng s sad t be exthermc. When the enthalpy f mxng s pstve, heat s absrbed and mxng s sad t be endthermc. (See fgure4.4 a and b) -9-

10 SMA5101 Thermdynamcs f Materals Ceder 2001 Fgure 4.4: Enthalpes f mxng fr a hypthetcal bnary A-B slutn. A negatve enthalpy f mxng (a) means that heat s released upn mxng (exthermc) whle a pstve enthalpy f mxng (b) means that heat s absrbed upn mxng (endthermc). The value f ΔG mx s an mprtant quantty as ts sgn determnes whether mxng wll ccur r nt. A negatve Gbbs free energy f mxng means that there s a thermdynamc drvng frce fr mxng and the pure cmpnents when brught n cntact wll spntaneusly frm a slutn. A pstve Gbbs free energy f mxng means that the cmpnents are mmscble and wll nt frm a slutn when brught tgether, but rather a tw phase dspersn f a pure A phase mxed wth a pure B phase. (fgure 4.5 a and b). Fgure 4.5: Gbbs free energes f mxng fr a hypthetcal bnary A-B slutn. A negatve free energy f mxng (a) means that there s a thermdynamc drvng frce fr mxng t ccur (.e. t wll ccur spntaneusly), whle a pstve free energy f mxng (b) means that the pure cmpnents wll nt mx. -10-

11 SMA5101 Thermdynamcs f Materals Ceder RELATION BETWEEN MOLAR QUANTITIES AND PARTIAL MOLAR QUANTITIES (THE INTERCEPT RULE) Mlar quanttes are nrmalzed extensve varables. In general, t s mre practcal t wrk wth mlar quanttes than wth the actual extensve varables that pertan t a partcular system. Tabulatns f thermdynamc prpertes such as enthalpes, free energes etc. are ften expressed per mle. Furthermre, mlar quanttes f slutns are typcally pltted as a functn f the mle fractn f the cmpnents n the system. There exsts a cnvenent graphcal cnstructn t derve partal mlar quanttes frm plts f mlar quanttes versus mle fractns. Ths graphcal cnstructn s referred t as the ntercept rule. We wll llustrate the ntercept rule fr a bnary system. Fgure 4.6 schematcally llustrates the mlar free energy fr a bnary A-B slutn pltted as a functn f the mle fractn f B. Cnsder a slutn wth cmpstn x B. The mlar free energy f the slutn s Gx ( B '). The ntercept rule states that µ A s equal t the ntercept f the tangent t G at x B wth the axes at x B = 0 whle µ B s equal t the ntercept f the same tangent at x B = 1. Ths s llustrated n fgure 4.6. Ntce that the ntercepts change as the cmpstn f the slutn s changed; As wth G, µ A and µ B are all functns f x B. It s a straghtfrward matter t derve the ntercept rule. Startng frm the ttal dfferental f G dg = µ dx + µ dx and usng the fact that dx A =-dx B we btan Cmbnng equatn (18) wth A A B B dg dx B = µ B µ A. (18) G = x A µ + x A Bµ B (whch s smply equatn (8) dvded by the ttal number f mles n the system) t slve fr µ A and µ B yelds dg µ A = G x B (19a) dx B µ = G +( 1 x B ) dg (19b) B dx B -11-

12 SMA5101 Thermdynamcs f Materals Ceder 2001 Equatn (19a) and (19b) are the mathematcal expressn f the ntercept rule as llustrated n fgure 6. Fgure 4.6: Illustratn f the ntercept rule appled t the mlar Gbbs free energy (see text). It s mprtant t recgnze the dfference between equatn (19) and equatn (4). In equatn (19), the chemcal ptentals are btaned frm the mlar Gbbs free energy G whle n equatn (4) the chemcal ptentals are btaned frm the extensve Gbbs free energy G. It s a cmmn errr t set the chemcal ptental f a cmpnent equal t the dervatve f the mlar Gbbs free energy wth respect t the mle fractn f that cmpnent. It s als pssble t apply the ntercept rule t a plt f the mlar Gbbs free energy f mxng (equatn 17). Nw thugh, the ntercepts are n lnger the chemcal ptentals but rather µ A G A and µ B G B (see fgure 7). In ths cntext, G A and G B are ften dented by µ A and µ B respectvely. Ths ntatn stems frm the fact that µ A s the chemcal ptental f pure A (when x A = 1) and µ B s the chemcal ptental f pure B when (x B = 1). -12-

13 SMA5101 Thermdynamcs f Materals Ceder 2001 Fgure 4.7: Illustratn f the ntercept rule appled t the mlar Gbbs free energy f mxng (see text). In an analgus way as wth the mlar Gbbs free energy, the ntercept rule can be appled t any mlar quantty. The partal mlar vlumes fr example can be btaned frm the mlar vlume usng dv V A = V x B (20a) dx B dv V B = V + x A = V + (1 x B ) dv dx B dx B (20b). Fgure 4.8 llustrates the ntercept rule as appled t the mlar vlume f a bnary A-B slutn. It als graphcally llustrates the dfferent terms that appear n the expressns fr the mlar vlumes n equatns 20. Fgure 4.8: Illustratn f the ntercept rule appled t the mlar vlume f a bnary A-B slutn. The dfferent terms appearng n equatns 20 are als llustrated. -13-

14 SMA5101 Thermdynamcs f Materals Ceder PHASE EQUILIBRIUM BETWEEN SOLUTIONS The thermdynamcs f slutns, ntrduced n the prevus sectns, s a pwerful tl t study the equlbrum f a hetergeneus mxture f hmgeneus phases. Fr example, ne may be nterested n understandng the equlbrum f sld slcn wth trace amunts f rn n cntact wth a melt cntanng predmnantly alumnum. Ths equlbrum s f mprtance durng the fabrcatn f slar cells where an alumnum-rch layer s plated n the back surface f a slcn wafer. Durng heat treatment, the alumnum-rch layer serves as a getterng snk fr unwanted transtn metal ns such as rn n the slcn wafer. Ths example llustrates hw the equlbrum characterstcs between a lqud Al-S-Fe phase and a sld S-Fe-Al phase can be explted t engneer the cmpstn f the slcn wafer. In ths sectn we derve the thermdynamc cndtns f mult-phase equlbrum. Fr smplcty, we restrct urselves t bnary systems, yet t s a straghtfrward task t generalze the treatment f ths sectn t systems wth mre than tw cmpnents. Cnsder a bnary A-B system at cnstant T and P wth tw phases α and β as llustrated n fgure 4.9. The phases α and β culd each be n the sld, lqud r vapr frm and fr generalty, we assume that bth phases are slutns cntanng A and B atms. The number f mles f A and B n the α phase s dented by n α α A and n B and n β the β phase by n A and n β α B. Althugh n the analyss f equlbrum, we wll allw n A, α n B, n β β A and n B t vary, we assume that the ttal number f mles f each cmpnent n the system (defned as the cmbnatn f the α and β phases) s fxed. That s β n A = n α A + n A = cnstant β n B = n α B + n B = cnstant (21a) (21b) -14-

15 SMA5101 Thermdynamcs f Materals Ceder 2001 Fgure 4.9: Schematc llustratn f tw dfferent phases α and β n thermdynamc equlbrum. The Gbbs free energy s the characterstc ptental fr a system cnstraned t have cnstant T, P, and ttal number f mles f n A and n B. Equlbrum f a system under these externally mpsed cnstrants s characterzed by that state whch mnmzes the Gbbs free energy G(T, P, n A, n B ) f the system. Befre cntnung, let us elabrate n the meanng f the state that mnmzes the Gbbs free energy. Whle n ur tw phase system f fgure 4.9, the temperature, pressure and ttal number f A and B atms are fxed, there stll reman several nternal degrees f freedm. The cmpstns f A and B atms n each f the tw phases separately are nt cnstraned. All that s requred s that the ttal number f mle f each cmpnent remans cnstant (.e. equatns (21) a and b α are satsfed). Hence, f the number f mles f A n α s ncreased by an amunt dn A, α then the number f mles f A n β must be reduced by dn β = dn. Each set f values A A α α β β fr n A, n B, n A and n B that smultaneusly satsfy equatns (21) a and b are pssble states f the system under the cnstrant f cnstant T, P, n A and n B. And fr each ne f these states, a value fr the Gbbs free energy exsts. Indeed, snce the Gbbs free energy s an extensve quantty, t can be wrtten as the sum f the Gbbs free energes f the separate phases α and β accrdng t G tt ( T, Pn,,,,, n B ) = G ( T P n n B ) + G ( T, Pn,, α α α β β β A A A n B ) (22) where G tt s the free energy f the whle system, G α s the free energy f the α phase and G β s the free energy f the β phase. (In equatn 22, we are assumng that the α and β phases are large enugh such that surface effects can be neglected). In equlbrum, the α α β β system chses thse values f n A, n B, n A and n B that smultaneusly mnmze G tt and satsfy the cnstrants f equatns (21). T mnmze the Gbbs free energy f the tw phase system, we cnsder the ttal dfferental f G tt wth respect t ts nternal degrees f freedm dg tt = dg α + dg β (23) -15-

16 SMA5101 Thermdynamcs f Materals Ceder 2001 where, at cnstant T and P dg α = µ dn A + µ dn B α α α α A B dg β = µ dn A + µ dn B β β β β A B (24a) (24b). Cmbnng (23) and (24) alng wth the cnstrants that dn α A = dn and dn B = dn B (whch fllws frm (21) a and b) yelds β a B A dg tt = (µ µ )dn + (µ B µ )dn α β α α β α A A A B B (25). In equlbrum, G tt s mnmal and the dfferental f G tt wth respect t the nternal degrees f freedm, equatn (25), must equal zer. Ths must be true fr any α nfntesmal pstve r negatve perturbatn dn A r dn α B away frm equlbrum. Hence, equlbrum s characterzed by an equalty f chemcal ptentals f each speces n the tw phases α µ = µ β A A (26a) α µ = µ β B B (26b). Ths set f equatns cnsttutes the fundamental cndtns f equlbrum n a bnary tw-phase system. Equatn (25) nt nly sets the equlbrum crtera fr a bnary tw-phase system, t als ndcates hw the system evlves when the abve equlbrum cndtns, equatns (26a) and (26b), are nt satsfed. T cmply wth the secnd law f thermdynamcs, a system ut f equlbrum wll change ts state n a drectn that decreases the free energy. If fr example, the equlbrum cndtns (26) a and b are nt satsfed and the chemcal ptental f A n α s less than the chemcal ptental f A n β,.e. α µ < µ β A A then the system wll evlve n a drectn fr whch dg tt <0. Fr ths example, dg tt <0 when dn α A s pstve. In general, atms flw t the phase where there chemcal ptental s the lwest. Only when the chemcal ptentals fr each cmpnent n the dfferent phase are equal wll the system be n equlbrum and reman unchanged ver tme. A smlar analyss can be perfrmed fr a system wth mre than tw cmpnents and mre than tw phases. The general equlbrum crtera fr a system wth cmpnents 1,2,,N and phases α, β, γ,... s α β γ µ 1 = µ 1 = µ 1 =

17 SMA5101 Thermdynamcs f Materals Ceder 2001 α β γ µ 2 = µ 2 = µ 2 =.... (27).. α β γ µ N = µ N = µ N =... The cndtns f equlbrum, as encapsulated by equatn (26) r (27), can be represented graphcally by the cmmn tangent methd. Fgure 10 llustrates a plt cntanng the free energy curves f tw dfferent phases as a functn f mle fractn n an A-B system. Usng the ntercept rule, t s clear that n equlbrum the cncentratns n the tw phases must be such that the tw free energy curves have a cmmn tangent as llustrated n fgure Fgure 4.10: Schematc llustratn f the cmmn tangent methd fr a bnary system cntanng an α and β phase. 4.7 PHENOMENOLOGICAL EXPRESSIONS FOR CHEMICAL POTENTIALS. The Gbbs free energy f a slutn s a functn f T, P and the number f mle f each f the cmpnents n the system. Chemcal ptentals als depend n T and P. But snce chemcal ptentals, alng wth all partal mlar quanttes, are ntensve and, therefre, d nt depend n the sze f the system, t s suffcent t descrbe ther cmpstn dependence wth ntensve cmpstn varables such as mle fractns x. -17-

18 SMA5101 Thermdynamcs f Materals Ceder 2001 In prevus sectns, knwledge f the functnal dependence f chemcal ptentals n T, P and cncentratn varables was mplctly assumed. In ths sectn, we ntrduce and mtvate a cmmnly used explct expressn fr the chemcal ptental. A useful system t start wth t derve an expressn fr the chemcal ptental s a sngle cmpnent deal gas. Fr a ne cmpnent system, the chemcal ptental s equal t the mlar Gbbs free energy snce G µ = = ( ng) = G n n where n s the number f mle n the system and G s ndependent f n. Frm the prpertes f the Gbbs free energy f sngle cmpnent systems cvered n chapter 2 alng wth the equatn f state f an deal gas, we knw that r µ G RT = = V = P T P T P dp dµ = RT = RTd( ln P) (28). P Integratn f equatn (28) frm a reference state wth pressure P and temperature T, we btan µ( P, T ) µ( P,T) = RT ln P P (29). Often the reference state s defned as the gas at 1 atmspherc pressure,.e. P =1 atm. In the reference state, µ (P,T) s dented by µ ο (Τ). The chemcal ptental then becmes µ( P T ) = µ ( )+ RT ln ( ) (30), T P where P s measured n atm. S far, n apprxmatns have been made and equatn (30) s the exact expressn fr the chemcal ptental f a sngle cmpnent deal gas. Cnsder nw an deal gas cntanng several cmpnents. We are nterested n the chemcal ptental f each the cmpnents n ths deal gas mxture. Fr an deal gas mxture, we ntrduce partal pressures fr each cmpnent P defned as P = x P where x s the mle fractn f cmpnent and P s the ttal pressure f the gas mxture. These partal pressures shuld nt be cnfused wth partal mlar quanttes. They are fctve varables ntrduced fr cnvenence. Fr an deal-lke gas mxture, t turns ut t be a very gd apprxmatn t set the chemcal ptental f each cmpnent equal t -18-

19 SMA5101 Thermdynamcs f Materals Ceder 2001 µ (P, T, x ) = µ ( T )+ RT ln( P )= µ ( T)+ RT ln(x P ) (31). Equatn (31) has the same frm as equatn (30) fr a sngle cmpnent deal gas wth the exceptn that the partal pressure s used nstead f the ttal pressure. Fr an deal gas mxture at atmspherc pressure P=1 atm, the chemcal ptental becmes µ = µ + RT ln( x ) (32). The frm f the chemcal ptental n equatn (31) s retaned even fr nn-deal gas mxtures althugh then x s replaced wth a fugacty f whch tself s a functn f x. In general, gasses devate frm dealty nly under extremely hgh pressures and equatn (31) s a gd apprxmatn fr mst gas mxtures arund atmspherc pressure. µ n equatn (31) and (32) s the chemcal ptental f n the standard state. Fr gasses, the mst cmmn standard state s that f pure n the gas phase at ne atmspherc pressure. µ, therefre, nly has a temperature dependence. Equatn (32) s als a reasnable apprxmatn fr the chemcal ptentals n lqud and sld slutns. Often thugh, a mre accurate descrptn f the chemcal ptentals f the cmpnents n cndensed slutns s btaned after an actvty ceffcent γ s ntrduced nsde the lgarthm. The chemcal ptental s then wrtten as µ = µ + RT ln(γ x ) (33). The prduct γ x s typcally replaced by the varable a whch s called the actvty f cmpnent. The chemcal ptental s then wrtten as µ = µ + RT ln( a ) (34) µ represents the chemcal ptental f n the standard state. The standard state fr sld slutns s chsen as the pure cmpnent n the same phase as the slutn at the temperature and pressure f nterest. In ths standard state, µ s equal t the mlar free energy f pure. Nte that due t ths defntn f the standard state, µ depends bth n T and P but s ndependent f the cncentratn. Fgure 11 llustrates a schematc plt f an actvty n a bnary cndensed slutn. A slutn s sad t be deal f a = x fr all the cmpnents n the slutn (fgure 11a). The slutn then exhbts what s referred t as Raultan behavr. Usually, the actvty des nt vary lnearly wth mle fractn but devates frm t as llustrated n fgure 11b fr a bnary system. Althugh the cncentratn dependence f a may be cmplcated, there are tw features that are always satsfed n a bnary system: -19-

20 SMA5101 Thermdynamcs f Materals Ceder 2001 () When s the slvent, that s as x appraches 1, the actvty a appraches x. Ths s called the Raultan regme. () In the dlute lmt when x appraches 0, the actvty can be wrtten as a =. k x where k s a cnstant. Ths s called Henran behavr. Fgure 4.11: The actvty f cmpnent B n a bnary A-B slutn. (a) crrespnds t an deal slutn and (b) crrespnds t a mre realstc slutn. -20-