H. A. J. Oonk University of Utrecht The Netherlands
Some Aspects of the Graphical Calculation of Phase Diagrams
A s rnrly ns the 187th Gihhsl stressed the impnrtnnrc, of grnphicnl rrprcsentntions in studying hetcrogencous rquilihris. Such rrprcsentntions were usrrl, nmongothrrs, hy 13nkhuis Roosehoon12 in 1899 for t h r clrrivntion of t h r t!/prs of phnse clii~grnmapossihle for the isohnric binnry nluilihrium hrt\vrrn solid nnd liquid .solutions. Rnkhuis lloosehoom npplied qualilativr rhnrnctrristics of Gihhs' zetn function, now known ns the Gihhs function. For qrmnlitali~rcnlculntions of phnsr dingrnms, in gcnrrnl, one nlso hna to employ grnphicnl or rrlnted methods. A rrcent rxnmplr of .wrh rnlrulntio~~s is that g i w n hy Rnnd nnd Iiuhnchews k i v o r t h r soli(l-liquid quilihrium of thcsystrm urnnium-zirconium. T h r ohjrct of this pnper is to draw nttrntion to some nsprcts of the graphical cnlculntion of phnse dingrnms whirh mny h r of vnlur in t h r ducntionnl field. K c confinr ourselvn to isohnric hinnry rquilihrin; t h r pnrtiripnting stntrs will he cnllrd nolid nnd l i p i d . T o f:wilitntr rrndiug, complicntions surh ns hent-cnpncity influrnrrs and volumr ehnngn on mixing at constnnt prrswrr nrr nvoidrd. Theoretical
T h r compositions of the solutions nrr indicntd hy the mole fraction S of t h r swonrl romlmnent. Smnll cnse itnlirs nrr usrd for molnr qunntitin; t h r sperific molnr qunntitirs nre indientrri with dots. The drpen(lencc of t h r Gihhs r n r r g i n of the solid rind liquid solutions on nhwlutr trmpernture T nnd mole fraction cnn h r exprcssnl hy
The suprrscripts n nncl I rrfrr to t h r solid nnd liquid solutions, rrspretivrly. h st:~nds for rnthsl1)y itnd s for rntropy. whrrrns R denotrs t h r gas ronstnnt. T h r drvintion from idrnlity is expressed with the hrlp of exens fi~nctionsindieatnl hy the superscript 1.:. T h r rurvrs for !I' i ~ n d(1' can h r cnlrulntnl nt nny trmprrnturr. Figure 1 gives two such curvrs for n rrrtnin temprrnturr. The romlmsitions of t h r phnsw thnt are in thrrrnodyn:~mic nluilihrium, S.' nncl S.', nrr givrn hy the p i n t s of contnrt of the common tnn-
grnt. T h n r points of contnct nlxvnys lie 011 rilhrr aidr of the point of intersection of the two curvrs. This grnernl fact cnn he used ns follo~vs. The point of intersrction is givcn hy p.(T,S)
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fll(T,S)
(3)
At the intersection X' = St, w thnt in cqn. (3) t h r superscripts of S cnn he tlropprcl. The solutions of q n . (3) fit n curve in the T S plnnr. for which the name equal-1; n i n r hns hrcn introducrd.' T h r cqunl-C: curve stauds in n simplr rcl:~tionto t h 6 r s c n s functions. n.; follows from s u h s t i t u t i o ~of~ q n s . (1) nncl ( 2 ) in cqn. (3). This suhstih~tiong i v e . with Ah, = TI As, nnd Ahi = T2 A%, for t h r tempemture TFGC of t h r q u 8I-C'8 curvr
where TI and T2 denote the melting points of the cornponrnts. A denotrs the tliflerencr h e t ~ v w nthe qunntity on which it oprrntrs in the liquid nntl in the solid stnte nt the snmr molr frnctions; for instnnce Ah, = h - ' . On t h r other hnnd the rqunl-G curve stnnds in n simple relntion to the T S dingrnm, espwinlly h r n ~ u s c mnximn, minimn, nnd 1mints of horizo~~tnl inflexion in t h r rrlunl-G rurvr rnincidr with n~nximn,miuimn, nnd points of horisontnl iuflexion in the ph:~.;edingrnm.' 111 the celcul:~tionof phnsr dingrnms n good use cnn nlao h r mndr of t h r eirrunwtnncr. :IS mny h r kno\\.n, that t h r condition for thermoclynnmic cquilihriun~is insensitivr to r q r d lirtrnr m~rlril~uliorrs to (1' rind to B'. 111 ot hrr \vor(ls, t h r nhscisme of the points of contnct of t h r common tnngrnt., 1:igurr 1, rrmnin unehnngrcl if n contrihution AS is nddwl to hoth q'nntl !/'. T l ~condition r for thrrmotlynnmic rquilihrium, rsprrssml hy t h r rommon tnngent, cnn h r g i \ w with t h r hrlp o f t he thcrmoclyrinmic potentinlnpl and p2. Thrcondition rends a' n' nnd p,' = pd (.5)
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where
sec Fipurc 1. We now introduce two new pfunctions g', = p + A S and 9'' gt + A S (7)
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Volume 47, Number 3, Morch 1970
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Thnt the condition (T,) is not nffecterl by these linrnr contrihutio~~s follows from s ~ ~ h s t i t ~ ~oft ithe o r ~~ I I I I C ~ ~ O I I R ( i )in the rrlatior~s(6) which givrs r" =r ,a,"
=
, I , '
rr" r,"
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=
ri
+A +A
, I , '
the cxnmplc given below it is srcn thnt the nddition of proper linear contrihr~tionxmnkes pns-ihle n more a c c ~ ~ r ncnlrulatio~~ tr of thecorxisti~~g phnsrs. 111
Example A s :HI rxnmplr of the cnlculntio~~ of n phuw dingrnm 1vr chmsr :r hypthrtirnl orgnnic systrnr in which liqrid solutions rlrr i ~ l m l(I!"' : ~ r dn" nrr zero for :dl v : ~ l u nof S)n ~ i dnolid s o l ~ ~ t i o:we r ~ r rqplnr (nP' i* zrro for s l l v:dun of S).$ I.hr hB' we tnkr :I somr\vbnt 1111r n m m o n ~ ~ ~ ~ ~ nrrmrly c t i o ~ irR'(S) ~, = IIS(1-S) (1-L'S). T h r pnrnmrtws in w p . (4) nrc set at t h e fc~llo\vit~gvnltlrs:TI = :1!H)"Ii, T? = 41O01
