Binary Systems. Part 4: Binary Systems from Mn-Mo to Y-Zr: Phase Diagrams, Phase Transition Data, Integral and Partial Quantities of Alloys PDF
Preview Binary Systems. Part 4: Binary Systems from Mn-Mo to Y-Zr: Phase Diagrams, Phase Transition Data, Integral and Partial Quantities of Alloys
Introduction 1 Introduction Thefirst4volumesofthisseries,underthegeneralheadingThermodynamicPropertiesofInorganicMa- terials, presentsSGTE-compiledthermodynamicdataforpuresubstances, includingtheelementsintheir stablestates. Theseriesnowcontinueswithafurther4volumesofSGTEselectedandcompileddata–this timeforbinaryalloysystems. Forthermodynamiccalculationsinvolvingalloysolutionphases,Gibbsen- ergies of the pure elements in different stable and metastable states are required. Such data have been compiled on behalf of SGTE by Dinsdale [91Din] and have recently been updated [02Din]. The values havefoundwideuseinternationallyasthebasisforthermodynamicassessmentsofhigherordersystems. Aswiththepureelementvalues,thebinaryalloydescriptionscontainedinthepresent4-volumeseriesare notonlycompleteinthemselves,butalsoextendthebasisforthermodynamicassessmentsandcalculations relatingtomulticomponentalloys. MembersofSGTEhaveplayedaprincipleroleinpromotingtheconceptof“computationalthermochem- istry”asatimeandcost-savingbasisforguidingmaterialsdevelopmentandprocessinginmanydifferent areas of technology. At the same time, through organisation of workshops and participation in CODATA Task Groups, SGTE members have contributed significantly to the broader international effort to unify thermodynamicdataandassessmentmethods. TheSGTEdatacanbeobtainedviamembersandtheiragentsworld-wideforusewithcommerciallyavail- able software developed by some of the members, to enable users to undertake calculations of complex chemicalequilibriaefficientlyandreliably. TheSGTEMemberorganisationsare: Canada: –THERMFACTLTD.LTEE France: –InstitutNationalPolytechnique(LTPCM),Grenoble –AssociationTHERMODATA,Grenoble –IRSID,Maizie`res-le`s-Metz –Universite´ deParis-Sud(EA401) Germany: –Rheinisch-Westfa¨lischeTechnischeHochschule(MCh),Aachen –GTT-Technologies,Herzogenrath –MPIfu¨rMetallforschung(PML),Stuttgart Sweden: –RoyalInstituteofTechnology(MSE),Stockholm –Thermo-CalcSoftwareAB,Stockholm UnitedKingdom: –NationalPhysicalLaboratory(MATC),Teddington –AEATechnologyplc,Harwell USA: –TheSpencerGroup Assessmentandselectionprocedures Theassessmentsofthebinaryalloysystemspresentedinthis4-volumeserieshaveallbeenmadeusingthe so-called“CALPHADmethod”[98Sau]. Thismethodresultsinanoptimisedparametricdescriptionofthe Gibbs energies of the phases of the system when taking into account the crystallographic structure of the phases and all the experimental thermodynamic and phase boundary data available. The thermodynamic Landolt-Bo¨rnstein SGTE NewSeriesIV/19B 2 Introduction parametersprovideaconsistentanalyticaldescriptionofthephasediagram,chemicalpotentials,enthalpies ofmixing,heatcapacities,etc. Asanexample,therelationsbetweentheGibbsenergycurvesandthephasediagramfortheBi-Snsystem are demonstrated in Figs. 1 and 2, respectively. In Fig. 1 the Gibbs energy curves for the phases in the Bi-Sn system are given as a function of the mole fraction of Sn, x , at T = 450 K. At fixed pressure, Sn temperatureandcomposition,theequilibriumofthesystemisdeterminedbythestatewiththelowestGibbs energy. All equilibrium states are located on the convex hull of the set of G-curves which is constructed by applying double-tangents to the curves. The tangent points denote the boundaries between one- and two-phase regions. In Fig. 2, these points are marked on the selected isotherm of T = 450 K. If this constructionisrepeatedforothertemperaturesthecompletephasediagramofthesystemisobtained. Fig. 1. GibbsenergyfunctionsforthephasesintheBi-Sn Fig.2.PhasediagramforthesystemBi-Sn. systemat450K. Ifseveralpublishedassessmentsareavailableforaparticularsystem,selectionhasbeenmadefollowingan analysis of how well the available experimental data are reproduced by the description. Compatibility of themodellingusedwithrespecttoassemblyofadatasetforhigherordersystemshasalsobeentakeninto account. There are many different phases present in binary systems and, in order to combine their thermodynamic descriptionsinhigherordersystems,itisimportanttoknowtheircrystalstructuresaswellasthesolubilities ofalloyedelementsinthem. Inthesevolumes,thenamingofphaseshasbeencarriedoutasconsistentlyas possiblesoastofacilitateidentificationofthesamephaseappearingindifferentbinarysystems. Themaincharacteristicsofeachsystemarepresentedinindividualreportswhichgenerallyinclude – thecalculatedphasediagram – anabstractsummarisingthemainfeaturesofthesystem – a summary of the various stable and metastable phases defined in the system together with crys- tallographic information, the phase name used in the database and the thermodynamic model used, includingtheoccupationofthesublattices – atableoftheinvariantreactions – tablesanddiagramswithintegralquantities – tablesanddiagramswithpartialquantities – plotsofcalculatedthermodynamicfunctions SGTE Landolt-Bo¨rnstein NewSeriesIV/19B Introduction 3 Criteriaforselectionofbinaryalloyassessments Inordertoqualifyforselection,thefollowinginformationwasreviewed: – phasediagram – thermodynamicinformation – documentation – modelsusedforsolutionphases – modelsusedforstoichiometricphases – feasibilityofextrapolation – compatibilitywithSGTEunarydata ThermodynamicModelling Elements TheGibbsenergyofthepureelementi,◦Gφ(T),referredtotheenthalpyforitsstablestateφat298.15K, i ◦Hiφ(298.15K), is denoted by GHSERi. This quantity is described as a function of temperature by the followingequation: GHSERi = ◦Gφi(T) − ◦Hiφ(298.15K) = a+bT +cT ·lnT +dT2+eT3+f T−1+gT7+hT−9 (1) Anumberoftemperaturerangesmaybeused. Thefirstandsecondderivativesofthisquantitywithrespect totemperaturearerelatedtotheabsoluteentropyandheatcapacityofthecompoundatthesametemper- ature. Experimental values for heat capacities can thus be directly used in the optimisation and will be relatedtothecoefficientsc,d,e,f,gandh. Forelementswhichhaveamagneticordering,e.g. Co,Cr,Fe,NiandMn,thetermGHSERisreferredto apara-magneticstate. AnadditionaltermisthusaddedtothemolarGibbsenergyofthemagneticphase. Forelementsaswellasforsolutions,thistermisequalto: Gmag =RTln(β+1)f(τ) (2) where τ is T/T∗ , T∗ being the critical temperature for magnetic ordering (Curie temperature T for C ferromagnetic materials or the Ne´el temperature T for antiferromagnetic materials), and β the average N magneticmomentperatomofthealloyexpressedinBohrmagnetons. Thefunctionf(τ)isgivenas: τ <=1 : f(τ)=1−[79τ−1/140p+(474/497)(1/p−1)(τ3/6+τ9/135+τ15/600)]/A τ >1 : f(τ)=−[τ−5/10+τ−15/315+τ−25/1500]/A withA=518/1125+(11692/15975)(1/p−1). These equations were derived by Hillert et al. [78Hil] from an expression of the magnetic heat capacity Cmag describedbyInden[81Ind]. P Thevalueofpdependsonthecrystalstructure. Forexample,pisequalto0.28forfccandhcpmetalsand 0.40forbccmetals[81Ind]. Foranti-ferromagneticalloystheT∗ andβ aremodelledasnegativeandthey aredividedbyananti-ferromagneticfactorof-1forbccand-3forfccandhcpbeforethevaluesareused inequation(2). For each element, equation (1) is taken from the SGTE unary database. These data have been published previouslyastheSGTEdataforthepureelementsbyDinsdale[91Din,02Din]. Landolt-Bo¨rnstein SGTE NewSeriesIV/19B 4 Introduction ThefunctionGHSERi isalsooftenusedtoexpressthethermodynamicfunctionsofmetastablestructures ϕ,differentfromthestablestructureofthepureelement. Theexpression◦Gϕ(T) − ◦Hφ(298.15K)is i i equivalent to ◦Gϕi(T) − ◦Gφi(T)+GHSERi. The term ◦Gϕi(T) − ◦Gφi(T) is often called the lattice stabilityofelementiinphaseϕ. Binarycompounds TheGibbsenergyofthecompoundA B maybeexpressedas: a b G (T)−a◦Hφ(298.15K)−b◦Hφ(298.15K) = f(T) (3) AaBb A B whereaandbarestoichiometricnumbers.Theexpressionforf(T)isidenticaltothatgivenbyequation(1). Equation(3)canbetransformedbyapplyingequation(1)foreachcomponent f(T) = G (T)−a◦Gφ(T)−b◦Gφ(T)+aGHSER +bGHSER AaBb A B A B = ∆ G (T)+aGHSER +bGHSER (4) f AaBb A B Theterm∆ G (T)istheGibbsenergyofformationofthecompoundreferredtothestableelementsat f AaBb temperatureT. ItcanoftenbetakenasalinearfunctionofT. Gaseousspecies An expression identical to equation (1) may be used to describe the Gibbs energy of the gaseous species withtheadditionalRT ln(P/P )term,whereP isthetotalpressureandP thereferencepressure,usually 0 0 0.1MPa. Thespeciesinthegasphaseareassumedtoformanidealsolution. Thereferencestateforeach vapourspeciesistakentobethepurecomponentsat0.1MPapressure. Thethermodynamicpropertiesof the gas species are normally obtained fromvapour pressuremeasurements coupled to spectroscopic data. Dataforgaseoussubstancesarecoveredinmoredetailinsubvolume(A)forpuresubstances. Many species, i.e. molecules, may existin the gas phase and each has a Gibbs energy of formation. The equilibriumwithinagasforagivencompositionatagiventemperatureandpressureiscalculatedbymin- imisingtheGibbsenergyvaryingthefractionofthespecies. AstheGibbsenergyisusedasthemodelling functioninmostsolutiondatabasesitisnotpossibletocalculatethecriticalpointforgas/liquid.Themodels usedforthedifferentliquidsarealsonotcompatiblewiththeidealmodelforthegas. Condensedphases Thecondensedphasescanbedividedintothreegroups. 1: Substitutionalsolutions Forthesubstitutionalsolutionφ,themolarGibbsenergyisexpressedasfollows: Gφ =Gφ,srf +Gφ,id+Gφ,E (5) m m m m with (cid:1) Gφm,srf = xioGφ (6) i (cid:1) Gφm,id = RT xilnxi (7) i (cid:2) xi isthemolarfractionofcomponentiwith ixi = 1. ThetermGφm,srf istheGibbsenergyofthephase relativetothereferencestateforthecomponentsandGφ,idisthecontributionofidealmixingentropy. m SGTE Landolt-Bo¨rnstein NewSeriesIV/19B Introduction 5 TheRedlich-Kisterequation[48Red],apowerseriesexpansion,isusedtoexpresstheexcessGibbsenergy, Gφ,E,fortheinteractionbetweenthetwoelementsiandj asfollows: m (cid:1) Gφm,E = xixj νLφij (xi−xj)ν (8) ν=0 Themodelparameter νLφ canbetemperaturedependent. ij Ifexperimentalinformationforternarysolutionsisavailablethenanextratermcanbeaddedtoequation(8). ForaternarysystemA–B–C,thistermisequalto: x x x L (9) A B C ABC Theliquidisinmostcasestreatedasasubstitutionalsolution.Forliquidswithverystrongshortrangeorder theassociatemodel[78Som]ortheionicliquidmodel[85Hil]hassometimesbeenused. Formagneticalloys,thecompositiondependenceofT∗andβ areexpressedby: (cid:1) T∗(x) = xi◦Ti∗+T∗,E (10) (cid:1)i β(x) = xi◦βi+βE (11) i whereT∗,EandβEarebothrepresentedbyanexpressionsimilartoequation(8). 2: OrderedPhases Theuseofthesublatticemodel,developedbyHillertandStaffansson[70Hil]basedonTemkin’smodelfor ionicsolutions[45Tem]andextendedbySundmanandA˚gren[81Sun],allowsavarietyofsolutionphases tobetreated,forexampleinterstitialsolutions,intermediatephases,carbidesetc. Alloftheserepresentan orderingoftheconstituentsondifferentsublattices. As non-stoichiometric phases are formed by several sublattices, they can be schematically described as follows: (A,B,...)p(A,B,...)q... wheretheconstituentsA,B,... canbeatoms,vacancies,moleculesorionsonthedifferentsublattices....p, q,... arethenumberofsites. Ifp+q+...=1,thenthethermodynamicquantitiesarereferredtoonemole ofsites. Mostoftenpandqareselectedtobethesmallestsetofintegers. Foreachsublattices,thesitefractionofthespeciesi,ys,isequalto i ns ns (cid:1) (cid:1) ys = (cid:2) i = i with ys =1 and ns =n (12) i ns ns i j j i s where ns is the number of speciesiin sublattice s, ns the number of sitesin sublattice s, and n the total i numberofsites. ns isrelatedtonbyns = n·p/(p+q+...). Thenumberofsublatticesandthespecies occupyingthem,isgenerallyobtainedfromcrystallographicalinformation.Themolefractionofanelement isobtainedby (cid:2) nsys xi = (cid:2) nss(1−iys ) (13) s Va whereys isthefractionofvacantsitesonsublattices. Va Thismodelalsodescribesstoichiometricphases,inwhichcasethesublatticesareoccupiedonlybyasingle species,andsubstitutionalphaseswhichhaveasinglelattice. Landolt-Bo¨rnstein SGTE NewSeriesIV/19B 6 Introduction ThemolarGibbsenergyforaphaseφexpressedbythesublatticemodelisequalto Gφ =Gφ,srf +Gφ,id+Gφ,E (14) m m m m Asanexample,atwosublatticephasewithtwoelementsAandBineachofthesublatticesisconsidered. Denotingthesublatticeswithprimesatthesymbols,thesurfaceofreferencefortheGibbsenergyis Gsrf =y(cid:1) y(cid:1)(cid:1) oG +y(cid:1) y(cid:1)(cid:1) oG +y(cid:1) y(cid:1)(cid:1) oG +y(cid:1) y(cid:1)(cid:1) oG (15) A A A:A A B A:B B A B:A B B B:B Theterms oG and oG representtheGibbsenergiesofthephaseφfortheconstituentelementsA A:A B:B and B. The colon separates the different sublattices. The terms oG and oG represent the Gibbs A:B B:A energies of the stoichiometric compounds ApBq and BpAq, which may be stable or metastable. oGA:A, oG , oG and oG arenumericallygivenbyequations(3)and(1). B:B A:B B:A ThetermGidisrelatedtothemolarconfigurationalentropyandisequalto: m Gid =RT[p(y(cid:1) lny(cid:1) +y(cid:1) lny(cid:1) )+q(y(cid:1)(cid:1) lny(cid:1)(cid:1) +y(cid:1)(cid:1)lny(cid:1)(cid:1))] (16) m A A B B A A B B Finally,theexcessGibbsenergyGE isequalto m GEm =yA(cid:1) yB(cid:1) [yA(cid:1)(cid:1)LA,B:A+yB(cid:1)(cid:1)LA,B:B] +yA(cid:1)(cid:1)yB(cid:1)(cid:1)[yA(cid:1) LA:A,B+yB(cid:1) LB:A,B] +yA(cid:1) yB(cid:1) yA(cid:1)(cid:1)yB(cid:1)(cid:1)LA,B:A,B (17) The termsLi,j:i and Li:i,j represent the interaction parameters between the atoms on one sublattice fora givenoccupancyoftheother,andcanbedescribedbyaRedlich-Kisterpolynomial,asfollows: (cid:1) Li,j:i = (yi(cid:1) −yj(cid:1))ν νLi,j:i (18) ν=0 Theparameters νLi,j:icanbetemperaturedependent. ThetermLi,j:i,j isknownasthereciprocalparame- terwhichmayberelatedtotheexchangereactionofAandBbetweenthesublattices. Itisusuallyassumed tobecompositionindependentbutmaydependontemperature. Theaboveequationscaneasilybeextendedtoternaryandhigherordersystems. 3: Phaseswithorder-disordertransformation Phaseswithorder-disordertransformation,likeA2/B2andA1/L1 canalsobedescribedwiththesublattice 2 methodalthoughthisdisregardsanyexplicitshortrangeordercontributions.AsingleGibbsenergyfunction maybeusedtodescribethethermodynamicpropertiesofboththeorderedanddisorderedphasesasfollows: Gm =Gdmis(xi)+∆Gomrd(yis) (19) whereGdmis(xi)isthemolarGibbsenergyofthedisorderedphase,givenbyequation(5)and∆Gomrd(yis)is theorderingenergygivenby: ∆Gomrd =Gsmubl(yis)−Gsmubl(yis =xi) (20) where Gsubl(ys) is given by equation (14). This must be calculated twice, once with the original site m i fractionsysandoncewiththesesitefractionsreplacedbythemolefractions. Ifthephaseisdisorderedthe i sitefractionsandmolefractionsareequalandthus∆Gord equaltozero. m Toensurestabilityofthedisorderedphase,thefirstdifferentialofGsublwithrespecttoanyvariationinthe m siteoccupancymustbezeroatthedisorderedstate. Thisenforcessomerelationsbetweentheparameters inGsublasisdiscussedin[88Ans]. m SGTE Landolt-Bo¨rnstein NewSeriesIV/19B Introduction 7 DescriptionoftheTablesandDiagrams The diagrams and tables which are presented for the binary systems provide an overview of the major thermodynamic properties and the mixing behaviour of these systems. Depending on the nature of the respectivesystem,thenumberandthetypeofthepresenteddiagramsandtablesvaries. Forallsystems,a calculatedphasediagram,ashortabstractandatablelistingthecondensedphasesareprovided. Additional tablesanddiagramspresentdataforinvariantreactions,integralandpartialquantitiesoftheliquidandsolid phases,andstandardreactionquantitiesofintermetalliccompoundsinthesystem. Thefollowinglistgivesonoverviewofthequantitiesinthetablesanddiagramsandtheirdesignations. The definitionofthesequantitiesisprovidedinthefollowingparagraphs. Symbol Unit Quantity a thermodynamicactivityofthecomponentAinaliquidorsolidsolution A ∆ C◦ J mol−1K−1 change of the molar heat capacity at constant pressure upon formation of a f P compound ∆CP J mol−1K−1 change of the molar heat capacity at constant pressure upon formation of a liquidorsolidsolution ∆G J mol−1 integralGibbsenergyofaliquidorsolidsolution m GE J mol−1 integralexcessGibbsenergyofaliquidorsolidsolution m ∆G J mol−1 partialGibbsenergyofthecomponentAinaliquidorsolidsolution A GE J mol−1 partialexcessGibbsenergyofthecomponentAinaliquidorsolidsolution A ∆ G◦ J mol−1 standardGibbsenergyofformationofacompound f ∆H J mol−1 integralenthalpyofaliquidorsolidsolution m ∆H J mol−1 partialenthalpyofthecomponentAinaliquidorsolidsolution A ∆ H◦ J mol−1 standardenthalpyofformationofacompound f ∆ H J mol−1 enthalpyofreactionpermoleofatoms r p Pa partialpressureofspeciesi i ∆S J mol−1K−1 integralentropyofaliquidorsolidsolution m SE J mol−1K−1 integralexcessentropyofaliquidorsolidsolution m ∆S J mol−1K−1 partialentropyofthecomponentAinaliquidorsolidsolution A SE J mol−1K−1 partialexcessentropyofthecomponentAinaliquidorsolidsolution A ∆ S◦ J mol−1K−1 standardentropyofformationofacompound f T K thermodynamictemperature T K Curietemperature C x molefractionofcomponentAinanalloyorcompound A γ activitycoefficientofthecomponentAinaliquidorsolidsolution A The first diagram shows the phase diagram of the system. The single-phase fields and the compounds are marked with labels which are used in the tables to refer to the respective phases. All boundaries be- tween phases which transform into each other by first-order transformations are drawn with solid lines. Second-orderphasetransformationsandmagnetictransformationsaredenotedbydashedanddottedlines, respectively. Thetable“phases, structuresandmodels”, containscrystallographicdataandinformationonthethermo- dynamic model in the database. The designations of the phases according to Strukturbericht, prototype, Pearsonsymbolandthespacegrouphavebeencollectedfromvarioussources,includingtheoriginalpub- lication of the assessment and the reference books of Pearson [85Vil], Massalski [90Mas] and Smithells [92Bra]. The SGTE name is used by the accompanying software on the CD-ROM. The last column of thistabledenoteshowthesublatticesofthecrystalshavebeenmappedintoathermodynamicmodel. The species which dissolve in a common sublattice are enclosed in parentheses. The indices denote the stoi- Landolt-Bo¨rnstein SGTE NewSeriesIV/19B 8 Introduction chiometriccoefficientsoftherespectivesublattices. Ifasublatticeisoccupiedbyasinglespeciesonly,the parentheseshavebeenomitted. Vacanciesaredenotedbyabox((cid:1)). The table of “invariant reactions” provides detailed data for the invariant equilibria and special transition pointsshowninthephasediagram. Foreachofthesereactionsthetemperatureandthephasecompositions areprovided. Thecompositionsoftheparticipatingphasesarelistedinthesamesequenceasgivenbythe symbolicequation. Thelastcolumngivesthereactionenthalpyoncoolingforonemoleofatomsaccording totherespectivetransformation. Thethermodynamicquantitiesfortheliquidandsolidsolutionsareprovidedbyasetofthreetableswhich are denoted by a suffix a–c after the Roman number. The first of these tables lists the integral quantities as well as the change of the molar heat capacity. The other two tables give the partial quantities for the respectivetwocomponents. Theintegralandpartialquantitiescanoftenbeobtainedeasilyfromexperiments. Partialmolarquantities are used to describe the thermodynamic behaviour of the individual components. In a binary system, the partial molar Gibbs energy G of component A can be calculated from the molar Gibbs energy, G , at A m constanttemperatureandpressurebythewell-knownrelation: GA =Gm+(1−xA)(∂Gm/∂xA)P,T (21) G is also known as the chemical potential of component A and denoted by the symbol µ . Similar A A relationsholdforthepartialmolarenthalpy,H ,andthepartialmolarentropy,S . A A Partialquantitiesprovidethedifferencebetweenthevaluesofthermodynamicfunctionsofacomponentin asolutionandthecorrespondingvaluesforthepurecomponents. Thus,thepartialGibbsenergy∆G of A componentAiscalculatedfromG inthesolutionandG◦ inthepuresubstanceby: A A ∆G =G −G◦ (22) A A A Usually, the values of the pure components are given for their most stable modification at the respective temperature and pressure. But in order to avoid ambiguities the reference states for each component are givenatthetables. Thequantities∆H and∆S aredefinedaccordingly. A A Thethermodynamicactivitya ofacomponentAiscloselyrelatedtothepartialGibbsenergyby: A a =exp(∆G /RT) (23) A A Therefore,theactivityis1forpurecomponentsinthechosenreferencestate. The integral Gibbs energy, ∆G is equal to the difference between the Gibbs energy of one mole of a m solutionG andthesumofthemolarGibbsenergiesofthepurecomponentsG◦ atthesametemperature m i andpressure. ForabinarysystemtheintegralGibbsenergyis: ∆G =G −x G◦ −x G◦ (24) m m A A B B If the reference state of the components is the same phase as the mixture, ∆G is also called the Gibbs m energyofmixing. Ifthereferencestateofatleastonecomponentisdifferentfromthephaseofthemixture then∆G containsthedifferenceinGibbsenergiesforthepurecomponentsbetweentwophases. Inthese m cases ∆G is called the Gibbs energy of formation of the mixture. The quantities ∆H and ∆S are m m m definedaccordingly. The excess quantities describe the deviation of the mixture from the ideal mixing behaviour. The molar excessGibbsenergy,GE,isgivenbythedifferenceoftheintegralGibbsenergyandtheGibbsenergyof m mixingforanidealmixture: GE =∆G −Gid (25) m m m SGTE Landolt-Bo¨rnstein NewSeriesIV/19B Introduction 9 Incaseofaasimplesubstitutionalsolution,Gidisgivenbyequation(7)andforsolidsolutionswithseveral m sublatticesanexpressionsimilartoequation(16)applies. Thepartialexcessquantitiescanbederivedfromtheintegralexcessfunctionsbyrelationssimilartothose betweenpartialandintegralquantities. Thus, analogoustoequation(21), thepartialexcessGibbsenergy ofcomponentAisgivenby: GEA =GEm+(1−xA)(∂GEm/∂xA)P,T (26) Sincetheheatofmixingiszeroforanidealmixture,theexcessenthalpyisidenticaltotheheatofmixing and the partial excess enthalpy of a component is equal to its partial enthalpy. Therefore, the partial ex- cessentropycanbecalculatedfromthepartialexcessGibbsenergybyatemperaturederivativeorbythe differencefromthepartialenthalpy: SAE =−(∂GEA/∂T)P,xA =(∆HA−GEA)/T (27) The activity coefficient is related to the partial excess Gibbs energy by an expression analogous to equa- tion(23): γ =exp(GE/RT) (28) A A ForthecaseofsimplesubstitutionalsolutionstheactivityofacomponentAisrelatedtoitsmolefraction by: aA =γAxA. The preceding equations describe the thermodynamic behaviour of a single phase. In an unconstrained equilibriumbetweentwophaseseachcomponenthasthesamechemicalpotentialandthesameactivityin eachphaseandtheintegralquantitiesarelinearfunctionsofthecompositioninatwo-phaseregion. Inthe diagrams,thefunctionsaredrawnwithdashedlinesintheseregions. Special considerations apply to stoichiometric compounds. Here, the partial quantities cannot be defined bytheexpressiongiveninequation(21)becausethecompositioncannotbevaried. Instead,thechemical potentialsaredefinedbytheequilibriumwiththenextadjacentstablephase. The table of “standard reaction quantities” provides the Gibbs energy, the enthalpy, and the entropy of formation for the given compounds from the pure elements in their most stable state at 298.15 K and 0.1MPa. Phosphorusdeviatesfromthisrulesinceherethewhitemodificationisconventionallychosenas areferencestateinsteadofthemorestableredform. Allvaluesinthistablearegivenforthereactionofa totalamountof1moleofatoms. DescriptionoftheSoftware The software provided with the volumes can calculate the printed phase diagrams but it also has some additionalcapabilities. PhaseNames Thephasenamesarethesameasusedinthevolumes. Ifthephasehasamiscibilitygaporcouldappearas bothorderedanddisorderedinthesamesystem,a”COMPOSITIONSET”numberisappendedtothename afterahashsign. ForexampleLIQUIDandLIQUID#2mayappearasphasenamesifthereisamiscibility gap in the liquid phase. Normally the composition set 1 is not identified explicitly. As both phases are thermodynamicallyidenticaltheassignmentofaspecificcompositionsetnumberisarbitrary. Forordering intheAu-CusystemforexampletherearefourdifferentcompositionsetsfortheFCCphase. Landolt-Bo¨rnstein SGTE NewSeriesIV/19B 10 Introduction DiagramSelection ThetwobasicwindowsforSGTEbinareshowninFig. 3. Inthetextareaofthebasewindowreferences fordataandotherkeytextualinformationmayappear. Fortheselectionofasystempressanytwoofthe elements highlighted in bold print. The four buttons at the bottom of the window will become available. Fourbasictypesofdiagramscanbegeneratedbyuseofspecificbuttons. Theseare, – thephasediagram, – theGibbsenergycurvesforallphasesasafunctionofcompositionataspecifictemperature – theactivitycurvesofthetwoelementsasafunctionofcompositionataspecifictemperature – aplotofthephasefractionsasafunctionofthetemperatureforagivencomposition Fig.3.Basewindowandperiodicchartwindow. The basic diagrams are obtained by just selecting two elements and the specific button. From these four calculations an infinite number of modified diagrams can be generated. Some of these will be discussed below. Inadditiontoselectingthetwoelementsonecanalsoselectthesetofphases. Thefoldertagged”PHASE” gives the default selection of stable phases for the selected system. By changing this selection various metastablediagramscanbecalculated. PhaseDiagram Thisbuttonwillgenerateastandardtemperature-compositionphasediagramwiththeaxesinmolefrac- tions and degrees Celsius, see the example in Fig. 4a and 4b. Magnifications and phase labels can be obtainedusingspecificbuttonsinthegraphicalwindow. TheREDEFINEbuttonprovidesamenu, which will allow a change of the axes as shown in Fig. 5. Fig. 6 is equivalent to Fig. 4 but now plotted with activityandtemperatureinCelsiusasaxesvariables. SGTE Landolt-Bo¨rnstein NewSeriesIV/19B
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