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A me ican M ne alogs , Voume63,pages 3-86, I 97

Theuseof thermodynamicxcessunctionsn theNernstdistributionawNonueNB. Holsr, Jn.Department f Geological ciences, niuersityof IllinoisChicago, linois 60680

AbstractThe Nernstdistribution aw,which s usedby manyauthors n geologyas he theoreticalbasisof geothermometrynd race-componentistribution,s derived singexcessunctions'It is demonstratedhat the ree-energyerm n theNernstequation, s t is applied n manyofthese ases,s an excessunctionand describeshe non-ideal ehavior fthe phasesnvolved.

It is then not necessaryo introduceany assumptionsoncerning ctivities r activitycoeffi-cients, nd onemayuse implemole ractionsnstead. nalysis f system ata n l ightof thisallows ne o makeuseful redictionsoncerninghe accuracyfthe dataand hebehavior fthe systemat elevated emperatures.nalysis s made of various data on the diop-side-enstatiteolvusas an example.

The Nernst distribution quations a veryusefultheoretical basis for approachinggeothermometryand t race-componentistribution. he derivation fthe Nernstequations dependent n the conventionone uses or dealingwith non-idealityn solutions.Thereare two commonconventions:ctivities, ndthermodynamicxcessunctions.There are two systemsof activities (Castellan,197): the rationalsystem singRaoult's aw as thelimiting case (activity coefficientapproaches asmole ractionapproaches); and hepractical ystemusingHenry's aw as he imitingcaseactivity oeffi-cent approaches as mole fraction approaches).Activities remorecommonly sed n geologicalit -erature,but excessunctionswould be more effectiveand eliminate onfusionn some nstances. morecomplete iscussion f thermodynamicxcessunc-tions can be found in Thompson 1967)or Swalin(1972).This paper will deal only with their appli-cation o the Nernstdistribution quation.Thermodynamicxcessunctions anbedefined sthe hermodynamicunctions f realsolutionsminusthe respectiveunctionsof ideal solutions.Thus theexcesshemical otential f component in phaseis the actual chemicalpotentialof componentminus hechemical otential newouldcalculatef Awerea perfectsolution.pt"" : p\ - tt^, ideal) (l)0003-004x/78l0I02-0083$02 0

Since hereare wo systems f activities,hereare worelationsbetween ctivityandexcesshemical oten-tial. In the rationalsystem:t t \ : t r \ + R T l n X t + R Z l n r t Q )

pt ideal= p! * RT ln X\ (3)p t * " : R? ' ln 7 { (4 )

where7 is the activitycoefficient, is the mole rac-tion andp! is thechemical otential f pure . In thepracticalsystem:r r \ : r r \ R Z l n X \ + RT n Kt * R I l n r t ( 5 )

pt*" : RI ln Kt * R?" n rt (6 )whereK is heHenry'saw constant nd7 is again nactivitycoefficient.The practicalsystem f activitiess usuallyused nworkingwith dilutesolutions nd he ational ystemwhen working with more concentrated olutions.However, y deriving he Nernstdistribution qua-tion with excessunctionsone may work with eitherdiluteor concentratedolutions ithoutactivities.ftwo phasesA andB) are n equilibrium,hechemicalpotential f any componentl) mustbe he sameneachphase.

p\ : t'tB, (7)Combiningeitherequations and 4 in the rational83

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HOLST: THERMODYNAMIC EXCESS FUNCTIONSsystem r 5 and 6 in the practical ystem nd sub-stituting nto equation yieldsdentical esults:p\- t RT ln xt + u]

u, * Rr ln xl * p"i" (8)RTln(X\/X!) : -(rr t ." - p1*")= -AG.". (9)From this it is apparent hat AG*" s the changenexcessreeenergyassociated ith the transferof onemoleof component from phase to A. SinceAG*"isanexcessunction,t describeshenon-ideal ehav-ior of the wo solutions nd t is not necessaryo dealwith activitiesn this orm of theNernstdistributionlaw.Pursuing his further et Ko : (X\/X\); then

l n K o : - I a G " " ( l o )R To nK, : _ l I a(d^c*/DI t , , IAT RL AT Io l n Ko t [ - aa* " ] , , - ,ar : -R t - -Tr - ) \ tz )

I 1' ' Al/*t ^-l nKo: -R J * aT (13)lnK" : - l ^ { " " *c . .r(J:- -r L. (14)For solid-stateeactions H andAS arealmost on -stantover fairly broad emperatureangesSwalin,1972). f they do not vary greatly with the molefractionof component in each hase r if thedegreeof solution n eachphase s small, hen AI1"" andAS*" can be considered onstant. t can then beshownby statistical rgumentsSwalin, 972,p.17l)that C : AS."/Rl so that:- RZln X" ' /X\: A11"" fA,S"". (15)

Thereforea plot of ln (XI/X!) versus /I shouldyielda straight ine with a slopeequal o -AH'"/Rand an intercept, t l /T: 0, equal o A,S*". p-proaches imilar o this areoften usedas he heoreti-cal basis or geothermometrye.g.Carmichael t al.,1974:Kern and Weisbrod, 967;Wood and Banno,1973)and tracecomponent istribution e.g.Broe-cker andOversby,97l). The approximations usu-ally made hat activityequalsmole raction.On ini-tial consideration his may seem to be a poorapproximation, ue o theverynon-ideal ehavior fthesesystems.n fact someauthorsare apologeticaboutusing t (Woodand Banno,1973). he abovederivationof the Nernst equation demonstrates,

however,ha t he hermodynamicunctions btainedby their plots or regressionsre n fact excessunc-tions and the non-ideal ehavior s accountedor .Correctionsor contaminantsn the systemmay bemadeby empirically orrecting G"" or its variationwith phase omposition.Anotheruseful lo t s to calculate G'" as-RIln(X\/ X"r) and plot this versus emperature. linearregression un on the points should produceastraight ine with a slopeapproximatingAS*' andan intercept pproximating H" at T : 0. Actuallythe data points n this plot as well as the plot dis-cussedn the previous aragraphwill not fall on astraight ine.This s because f1*"and AS*"do varywith temperaturend phase omposition. owever,theyshould ollow a smooth urvewith no nflectionsor singularpoints. nflections r singularpoints nAG curves nd the correspondingiscontinuitiesnthe first-orderderivativesA.F1 nd AS) indicateafirst-order ransformation r reaction.nflections rsingular oints n suchplotsare hengoodcauseosuspecthe accuracy f the data.A goodexamples hegeothemometeronstructedby Wood and Banno (1973) from the diop-side-enstatitehasediagramdetermined y Davisand Boyd (1966)at 30 kbar. There s a decidedn-flection n the diopside-richimb of the Davis andBoyd solvusat about 1475'Cand a correspondinginflection n the plot of ln(Xfip',*i,o"/X''?81r,,o")versul/7. The Wood and Bannoequation or the solvuswasobtained y inear egressionn thesolvus ata:ln xil[,",*"1xXJ,,,,o"-AH/Rr + A.S/R (-10202/ ) + s.3s.In l ight of new datadiscussedelow, he DavisandBoyddiagrams now known o beerroneousnd heWood and Bannoequation annotbe used n geo-thermometry.f theAG""versus plot is done or thesamesolvusdata (Fig. l), it is evident hat the in-flection n thisplot indicateshat the solvus etermi-nation is erroneous. he inflectionmust representeither an undetectedeactionor transformation,orsimplenaccuracyn the data.Since H*" andAS*"arevariable nd hedataplotis necessarily curve,whether t is ln (Xor/X})versusl/ T or AG'" versus , a linear regression annotrepresenthe datawith accuracy. muchmore accu-rate equationwould be affordedby a polynomialregressionn AG*" versus7. As an example he newdata on the diopside-enstatiteolvusat 30 kbar(Nehru and Wyllie, 1974;Mori and Green, 1975,1976; indsley ndDixon, 1976) re lottedaspoints

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HOLST: THERMODYNAMIC EXCESS FUNCTIONS 85in Figure l. The data are excellentlyrepresentedoverthe entire temperaturerange by a simple quadraticequation in Z:- R7 ln (fii,",,o,/ xiii,r',ou)=

AG*" : -4800 + 18.92 0.008297".An equation o be used n geothermometryhouldrepresenthe system s accurately spossible. his sof more interest han approximate aluesof AII*"and AS'". An equationobtained y polynomial e-gression hould henbe preferredo one obtainedbylinearregressionor this purpose. he difficulty nusingsuch an equation s that 7 appears n bothsides. t can be overcome y usingan equation e-rived by linear regression n ln (Xnf!l"t,6r/Xof,E}""o)versus/T to obtaina rough emperature.his canthenbe refinedn thequadratic quation y an terativetechniqueo obtain a more accurate olution. neither ase, owever, indsley ndDixon(1976) aveindicatedhat hesolvus ata n thissystem reno tofsufficient efinementor accurate eothermometry.It shouldalsobe noted hat hese ewdataplottedin Figure muchmoreclosely pproximatesmoothcurve than do the data of Davis and Boyd: theyfollow more closelyhe trend hat solvus ata mustfollow f thereareno phaseransformationsr reac-tions.There s variancen the points rom a smoothcurve,but this s to be expected henone considersthe remarks y all the aforementioneduthors on -cerning he difficultiesn obtainingequilibriumandaccuratenalysis y microproben thissystem. hereare also imitations n the accuracy f pressure ndtemperature eterminations it h the high-pressurepiston-cylinder pparatuswhich will contribute othisvariance.This type of plot has otherusefulaspects.f oneextrapolateshe curve o AG*" : 0,

-RI ln X+/n : 0r1."- P?'") AGxs 0; (6)ln X\/X\ : 0;

X \ : X \ .In the case f t race-componentistribution etweentwo immiscible hases, ne would expect quiparti-tioning of the componentat the temperature b-tainedby extrapolation.n the case f binarysolvusdata,since hereareonly two components,ompletesolidsolution hor,rldesult:

X t : l - X + a n d X E : l - X l ;xl : f r ;x t : t -n:xE.

2000

rooo t5 loEXCESSREE NERGYKCAL.Fig. l The molar change in excess ree energy of transferring

MgrSirO" from the orthopyroxene to the clinopyroxene phaseplotted against temperature. The solid line is calculated from thesolvusof Davis and Boyd (1966).Open circ lesare r om Nehru an dwyl l ie(1974); t r iangles rom Mori and Green 1975and 1976);andthe closedcircle from Lindsley and Dixon (1976).There are doubledatum points at 1673" and 1773"K.

Theseare of coursesubject o the condition that AG*"is not affectedappreciably by any increase n pressurewhich may be necessaryo raise the solidus to theindicated emperature.The data may also be used ocalculateactivit ies.To il lustrate his we may use th eDavis and Boyd solvus at a temperature 1100'C),where it closely agreeswith the later solvus deter-mined by Nehru and Wyll ie (1974).Since he ensta-tite phase contains very little diopside at all temper-atures,we make the approximation hat the chemicalpotential of MgzSirO. n it fol lows Raoult 's aw. As aresult of this:

R?'ln afi!i;,6" : AGo * RZln X'ff|,t,,o"The AG*" is about 5275 ca l the activity is 6.63, an dthe activity coefficient s 47.4.The enstatite-diopside solvus may be further com-plicated in that the high-temperatureMgrSirOu-richdiopsidephaseon the Davis and Boyd diagram s ofthe same composition as an "iron-free pigeonite"found by Kushiro (1969)at 20 kbar on the same oinbu t at somewhat ower temperatures'Kushiro's iden-tif ication was on the basisof X-ray diffraction.Davisand Boyd determined their solvus optically, and it

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