Modified Associate Formalism without Entropy Paradox: Part I. Model Description

0.13 MB·English

by  Dmitry N. Saulov

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0 ModifiedAssociateFormalismwithoutEntropyParadox:
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PartI.ModelDescription
n
a
J DmitryN.Saulov ∗,a , IgorG. Vladimirov a, A. Y. Klimenko a

5 aSchool of Engineering, The University ofQueensland, AUSTRALIA
1

]
r
e
hAbstract
t
oAModifiedAssociateFormalismisproposedforthermodynamicmodellingofsolutionphases.Theapproachisfreefromthe entropyparadox
.describedbyLu¨cketal.(Z.Metallkd.80(1989)pp.270–275).Themodelisconsideredinitsgeneralformforanarbitrarynumberofsolution
t
acomponents andanarbitrarysizeof associates. Asymptotic behaviour of chemical activitiesof solutioncomponents in binarydilutesolutions
m
isalsoinvestigated.
-
d
nKey words: Thermodynamic modeling (D),Entropy (C)
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c
[

1 1. Introduction The Gibbs free energy G of the associated solution of nA
molesof the solution componentA and n moles of B is then
v B
3 The associate model in its various modifications has been givenby
1successfully used for modelling solution phases of metallur-
G=n g +n g +n g −TS , (1)
2gical and chemical engineering interest [1–7]. Besmann and A1 A1 B1 B1 AB AB conf
3
.Spear[3],forexample,utilisedthemodifiedassociatedspecies wheretheconfigurationalentropyofmixing Sconf isexpressed
1model for glasses used in nuclear waste disposal. Recently, as
0
3Yazhenskikh et al. [5–7] have successfully applied the asso- Sconf=−R(nA1lnxA1+nB1lnxB1+nABlnxAB). (2)
1ciate species model to model melting behaviourof coal ashes Here, nA1, nB1, nAB are the mole numbers and gA1, gB1, gAB
:which is an important problem in coal gasification technolo- are the molar Gibbs free energies of the A-monoparticles, B-
v
igies.Goodagreementbetweenmodelpredictionsandavailable monoparticles and AB-associates, respectively; T is the abso-
Xexperimentaldatawas reported. lutetemperatureandRistheuniversalgasconstant.Themolar
r AccordingtotheclassicalassociatemodeldescribedbyPri- fractions x , x and x are defined in a usual way. For ex-
a A1 B1 AB
gogineandDefay [8], the stronginteractionsresultin the for- ample,x =n /(n +n +n ).Theothermolarfractions
A1 A1 A1 B1 AB
mation of stable configurations of mixing particles, the so- aredefinedsimilarly.
calledassociationcomplexesor,briefly,associates.Thosepar- Theequilibriumvaluesofthemolenumbersn ,n andn
A1 B1 AB
ticles that are not involved in the formation of associates are are determined by minimising the Gibbs free energy G given
called free particles or, interchangeably, monoparticles. The byEq.(1),subjectto themassbalanceconstraints
associated solution is then considered to be an ideal solution
of monoparticles and different associates. For example, a bi- n =n +n ,
A A1 AB (3)
nary associated solution of components A and B, in which
n =n +n .
only AB-associates are formed, is considered to be a ternary B B1 AB
idealsolutionoftheA-monoparticles,B-monoparticlesandAB- TheadjustableparameterofthemodelisthemolarGibbsfree
associates. energyD g of thereaction
AB
A+B⇄AB. (4)
Besmann and Spear [3] considered an ideal mixture of as-
∗ Corresponding author: School ofEngineering, The University of Queens-
sociate species (instead of monoparticlesand associates). The
land, St. Lucia, QLD4072, Australia.
stoichiometryoftheassociatespecieswasspecifiedso thatall
Phone: [phone] Fax: [phone]
Emailaddress:[email protected] (Dmitry N. Saulov ). thespeciescontaintwonon-oxygenatomsperformulaunit.In
ArticlepublishedinJournalofAlloysandCompounds473(12)(2009)167–175
8 canresultin eitherunderestimationof the non-configurational
Associated solution
entropy or overestimation of the enthalpy of mixing or both.
7 This, in tern, can undermine the predictive capabilities of the
model.
6
1] As pointed out by Lu¨ck et al. [9] and later by Pelton et
−
K 5 al.[10],theexpressionfortheconfigurationalentropyofmix-
1
−ol Ideal solution ing used in the quasichemical model does reduce to the ideal
J m 4 configurationalentropyof mixingwhen D gAB=0.In the next
[onf 3 section, we proposethe Modified Associate Formalism which
Sc isfreeoftheentropyparadox.Theparadoxisresolvedbydis-

2 tinguishingbetweenallpossiblespatialarrangementsofparti-
cles in an associate that have not been taken into account in
1
previousassociate models.
0
0 0.2 0.4 0.6 0.8 1
A xB B 3. Modelassumptions

Themodelproposedinthispaperis basedonthefollowing
Fig. 1. The configurational entropy of mixing for the associated solution,
in which only the AB-associates are formed, in the high temperature limit assumptions.
compared with the ideal entropy ofmixing. 1) Similarlytotheclassicalassociatemodel[8],weassume
that interactions between mixing particles result in the
this approach,contributionsof differentspecies to the config- formation of associates which are in a stable dynamic
urationalentropyofmixingareequallyweighted. equilibriumwitheachother.Theassociatesofthemodel
Lu¨cketal.[9]describedaremarkablefeatureoftheconfig- areunderstoodasatoolformodellingshort-rangeinter-
urationalentropyofmixinggivenbyEq.(2).Thisfeaturewas actions between mixing particles. The associates of the
referredtoasanentropyparadoxandisbrieflydescribedinthe model,however,mayrepresentrealassociatedcomplexes
nextsection. presentin solutionphases.
2) We assume that the associates do notinteract with each
otherandare uniformlydistributed(ideallymixed)over
2. Entropyparadox
alattice.Equivalently,theoccupanciesofthesitesofthe
associatelattice arestochasticallyindependentandhave
Lu¨cketal.consideredthehightemperaturelimitfortheas-
identicalprobabilitydistributions.
sociatedsolution.ThetemperatureT isassumedtobeso high
3) Incontrasttotheclassicalmodel,allpuresolutioncom-
that the entropy term plays a dominant role in the Gibbs free
ponents and the chemical solution of these components
energy of the solution and the enthalpy changes on forming
are treated in a unified way. More precisely,we assume
different associates can be neglected. The configurational en-
thatthesolutionandallitscomponentsconsistofnonin-
tropyofmixinggivenbyEq.(2)in thehightemperaturelimit
teractingassociatesofthesamesize,sothattheassociates
is higherthan that used in the regularsolution model(the en-
are composed of the same number of particles. In this
tropy of ideal mixing of the solution components).For exam-
approach,thecontributionsofdifferentassociatestothe
ple,theconfigurationalentropyofmixingoftheassociatedso-
configurationalentropyofmixingareequallyweighted.
lutioninwhichonlytheAB-associatesareformediscompared
4) Following the convention (see, for example, Ref. [11]
with the ideal entropy of mixing in Fig. 1. At the same time,
for more details), we also assume that particles of the
the formationofassociatesmodelsshortrangeorderingin the
sametypeareindistinguishable,whileparticlesofdiffer-
solution. As pointedout by Lu¨ck et al. [9], it is a paradoxical
enttypesandparticlesiteswithinanassociatearedistin-
result that the configurational entropy, which is a measure of
guishable.
disorder,appearsto behigherin a solutionwith orderingthan
in acompletelydisorderedsolution.
Anotherinterpretationofthisentropyparadoxwasdescribed 4. Modeldescription
by Pelton et al. [10]. Consider the binary associated solution
whereonlyAB-associatesareformedandassumethatthereare For simplicity of exposition, we exemplify the model by
no Gibbs free energychanges on forming AB-associates from considering a binary solution A−B where the associates are
monoparticles, so that D g =0. In this case, the configura- composedofthreeparticles.Letn andn bethemolenumbers
AB A B
tionalentropyofmixingofthesolutionshouldbeequaltothat of A and B particles in the solution. Since, by assumption 4)
ofanidealsolution,sincenointeractionsbetweenmixingpar- above,theparticlesiteswithinanassociatearedistinguishable,
ticlesareassumed.Eq.(2)however,leadstohighervaluesfor wealsoassumethattheyarenumbered.Ifthe1stand2ndsites
the configurationalentropyofmixingthatreducestothe ideal inanassociatearebothoccupiedbyA-particles,whilethe3rd
configurationalentropyofmixingonlywhen D g =+¥ ;see oneisoccupiedbyaB-particle,suchanassociateissaidtobe
AB
also Fig. 1. The overestimationof the configurationalentropy of type [AAB]. Othertypes of associates are defined similarly.
168
Thus,inthesolutionconsidered,thereare23=8differenttypes of the model, one can employ the Gibbs free energies of the
ofassociates,[AAA],[BAA],[ABA],[AAB],[BBA],[BAB],[ABB], reactions of forming mixed associates from pure [AAA] and
[BBB]. It is important to note that all these types should be [BBB]-associates.Forexample,insteadofg ,theGibbsfree
[AAB]
takenintoaccountin calculatingtheconfigurationalentropy. energyD g ofthefollowingreactioncan beused,
[AAB]
Let n be the mole numberof [ijk]-associates, where the
[ijk] 2 1
triplet of symbolic indices i,j,k =A,B specify the associate [AAA]+ [BBB]⇄[AAB], (11)
3 3
type.Thenthe massbalanceconstraintstaketheform
sothat
nA = 3n[AAA] D g =g −2g −1g . (12)
[AAB] [AAB] 3 [AAA] 3 [BBB]
+ 2(n +n +n )
[BAA] [ABA] [AAB] TheotherGibbsfreeenergies D g aredefinedsimilarly.
[ijk]
+ n +n +n , NotethatEqs.(5)definen andn aslinearfunctions
[BBA] [BAB] [ABB] (5) [AAA] [BBB]
ofthemolenumbersofthesolutioncomponentsn andn and
n = 3n A B
B [BBB] of the mole numbers n of the mixed associate types with
[ijk]
+ 2(n[ABB]+n[BAB]+n[BBA]) [ijk]6=[AAA],[BBB].Themolenumbersofthelatterassociates,
whichconsistofbothAandBparticles,aretheinternalvariables
+ n +n +n .
[AAB] [ABA] [BAA] of the model. These are determined by minimising the Gibbs
Byastandardcombinatorialargument,thenumberofavailable freeenergyofthesolutionatconstantn andn ,subjecttothe
A B
microstatesW is massbalanceconstrainsofEqs.(5).Theequilibriumvaluesof
n ,with [ijk]6=[AAA],[BBB],are foundfrom
[ijk]
N (cid:229) n !
◦i,j,k=A,B [ijk]! ¶ G
W = (cid:213) N◦n[ijk] ! , (6) (cid:18)¶ n[ijk](cid:19)nA,nB,n[i′j′k′] =0, (13)
i,j,k=A,B
(cid:0) (cid:1) wherethe derivativein n is calculated forfixed n , n and
where N is the Avogadro number. Under assumption 2) of [ijk] A B
idealmix◦ingofassociates,theconfigurationalentropyofmix- fixedfivevariablesn[i′j′k′],with[i′j′k′]6=[ijk],[AAA],[BBB].By
thechainrule,Eq.(13)reads
ingS is
conf
Sconf=kBlnW , (7) ¶ G ¶ n[AAA] ¶ G ¶ n[BBB] ¶ G
+ + =0. (14)
where kB is Boltzmann’s constant. Applying the Stirling for- ¶ n[AAA] ¶ n[ijk] ¶ n[BBB] ¶ n[ijk] ¶ n[ijk]
mulato Eqs.(6)and(7),
Therefore, combining the last equation with Eqs. (5) and (9)
(cid:229)
Sconf=−R n[ijk]lnx[ijk], (8) gives
i,j,k=A,B
n D g
[ijk] [AAB]
where =exp − . (15)
n[ijk] (n )a [ijk]/3(n )b [ijk]/3 RT
x = [AAA] [BBB] (cid:18) (cid:19)
[ijk] (cid:229) n
i′,j′,k′=A,B [i′j′k′] Here,a [ijk]andb [ijk]standforthenumbersofAandBparticles
isthemolarfractionofthe[ijk]-associates.Therefore,theGibbs inthe[ijk]-associate,respectively.Forexample,a [AAB]=2and
freeenergyofthe solutionis givenby b [AAB]=1.
Recall that Eqs. (5) define n as a linear function of n
(cid:229) [AAA] A
G= n g +RTlnx , (9)
[ijk] [ijk] [ijk] and n[ijk]’s, and n[BBB] as a linear function of nB and n[ijk]’s,
i,j,k=A,B
where [ijk]6=[AAA],[BBB]. The chemical potential µ of the
(cid:0) (cid:1) A
where g is the molar Gibbsfree energyof [ijk]-associates. solutioncomponentA iscalculatedas follows
[ijk]
InonemoleofpuresolutioncomponentAthereare1/3moles
¶ G
of[AAA]-associates.Hence,g[AAA]=3gA,wheregAisthemolar µA = ¶ n
gGibbs=fre3egen,ewrghyeroeftgheipsutrheesomluotliaornGciobmbspofrneeentenAe.rSgiymoilfarthlye, (cid:18)(cid:229) A¶ (cid:19)GnB¶ n[ijk]
[BBB] B B =
puresolutioncomponentB. [ijk]¶ n[ijk] ¶ nA
Theassociates,whichconsistofbothAandBparticles,will ¶ G ¶ n[AAA] (cid:229) ¶ n[AAA]¶ n[ijk]
bereferredtoasmixedassociates.Forthebinarysolutioncon- + +
¶ n ¶ n ¶ n ¶ n
sidered, there are 8−2=6 types of mixed associates. Their [AAA] A [ijk] [ijk] A !
(16)
molarGibbsfreeenergies g[ijk], with [ijk]6=[AAA],[BBB],are + ¶ G ¶ n[BBB]+(cid:229) ¶ n[BBB]¶ n[ijk]
adjustable parametersof the model.These parameterscan de- ¶ n ¶ n ¶ n ¶ n
[BBB] A [ijk] [ijk] A !
pendonthe temperatureT as ¶ G ¶ n[AAA] (cid:229) ¶ G ¶ n[AAA]
= +
g =h −Ts , (10) ¶ n ¶ n ¶ n ¶ n
[ijk] [ijk] [ijk] [AAA] A [ijk](cid:18) [AAA] [ijk]
whereh[ijk]ands[ijk]arethemolarenthalpyandentropyof[ijk]- + ¶ G ¶ n[BBB]+ ¶ G ¶ n[ijk].
associates,respectively.Alternatively,asadjustableparameters ¶ n ¶ n ¶ n ¶ n
[BBB] [ijk] [ijk](cid:19) A
169
Here, the sums are taken over i,j,k =A,B such that [ijk]6=
[AAA],[BBB].Using Eqs.(9)and(14),oneverifiesthat
¶ G ¶ n[AAA] 1
µ = =g + RTlnx . (17)
A ¶ n ¶ n A 3 [AAA]
[AAA] A
Thechemicalpotentialµ of thesolutioncomponentB is cal-
B
culatedina similarway.
Note that [AAB], [ABA] and [BAA] associates differ in the
spatial arrangement of the constituent particles, though they
havethesame“chemicalcomposition”A B.Thesamedistinc-
2
tion holds for [BBA], [BAB] and [ABB] associates of common
compositionAB . Letus assume, fora moment,thatonly one
2
spatialarrangementofassociatesisallowedforeachcomposi-
tion,whiletheotherarrangementsareprohibitedenergetically,
for example, g =g =+¥ and g =g =+¥ .
[BAA] [ABA] [BAB] [ABB]
In this case, the proposed formalism reduces to the associate
(a) Particlesitesarearrangedalonga(horizontal)line.
speciesmodel[3],inwhichassociatespeciesarecomposedof
Theassociates [BAA]and[AAB]aremirrorreflections
threeparticles.
of each other about axis a; they are not symmetric,
Unlike previous modifications of the associate model, we however, to [ABA].
takeintoaccountallpossiblespatialarrangementsofparticles
inanassociate.Iftwoormoreassociatesofdifferenttypesare
spatially symmetric to each other, then their molarGibbs free
energies are considered equal. For example, 2-particle asso-
ciates[AB]and[BA]aresymmetricandthereforeareendowed
with equalGibbsenergiesg =g .
[AB] [BA]
Inmorecomplexcases,however,Gibbsfreeenergylevelsare
ascribedtoassociatesdependingonthespatialarrangementof
particlesinthem,sothatbothenergysplitsandmultiplelevels
mayoccurforassociatesofcommonchemicalcomposition.For
example, if particle sites in 3-particle associates are arranged
linearlyasshowninFig.2(a),theng =g whileg
[AAB] [BAA] [ABA]
canbedifferent.Alternatively,iftheparticlesitesarearranged
as in Fig. 2(b), the three associate types are all symmetric to
eachother,andhence,g =g =g . (b) Particle sites are arranged at the vertices of an
[AAB] [ABA] [BAA]
Notethatnoassumptionsonspatialarrangementsofparticle equilateral triangle. All associates are symmetric to
each other by rotation about axis b, perpendicular to
sites within an associate have been made so far in the frame-
the plane ofthe triangle.
workoftheproposedmodel.Ingeneral,itisimpossibletode-
termineinadvancethenumberofenergylevelsandtheirmul-
Fig. 2. Schematic views ofdifferent arrangements ofparticle sites.
tiplicities for the associates of a particular composition. Fur-
indicesunderstoodappropriately.Moregeneralcaseisconsid-
thermore, such associates may have the same Gibbs free en-
eredinSection7.
ergy of formation, even if they are not spatially symmetric.
UsingEqs.(15)and(18),
Asareasonableinitialapproximation,allassociatesofagiven
composition can be endowed with the same Gibbs energy of
n ≡n ,
[AAA] 3,0
formation.Thisassumptioncanberefinedsubsequentlyinthe
processofthermodynamicmodeloptimisationforrealchemi- n[AAB]=n[ABA]=n[BAA]≡n2,1/3,
(19)
cal systems, if use of severalenergylevels appearsto provide
n =n =n ≡n /3,
[BBA] [BAB] [ABB] 1,2
a betterfittoexperimentaldata.
For the rest of this section and also in Sections 5 and 6, n[BBB]≡n0,3.
we assume, for simplicity, that the molar Gibbs free energy
Intermsofmolarfractions,Eqs.(19)read
of an associate type is completely specified by its chemical
composition, x[AAA]≡x3,0,
D g =D g =D g ≡D g , x[AAB]=x[ABA]=x[BAA]≡x2,1/3,
[AAB] [ABA] [BAA] 2,1 (20)
(18)
D g =D g =D g ≡D g . x[BBA]=x[BAB]=x[ABB]≡x1,2/3,
[BBA] [BAB] [ABB] 1,2
x ≡x .
Here, the subscript “2,1” signifies that the associate consists [BBB] 0,3
of two A-particles and one B-particle, with “1,2” and similar SubstitutionofEqs.(19)and(20)intoEq.(9)gives
170
G = n g +n g +n g +n g g =3g ,
3,0 3,0 0,3 0,3 2,1 2,1 1,2 1,2 3,0 A
+ RT(n lnx +n lnx ) (21) g =2g +g ,
3,0 3,0 0,3 0,3 2,1 A B
(26)
x x
2,1 1,2
+ RT n ln +n ln . g =g +2g ,
2,1 1,2 1,2 A B
3 3
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) g =3g .
ThemassbalanceconstraintsofEqs.(5)reduceto 0,3 B
Finally,substitutionofEqs.(25)and(26)intoEq.(21)yields
n = 3n +2n +n ,
A 3,0 2,1 1,2
(22) G=n g +n g +RT(n lnx +n lnx ). (27)
A A B B A A B B
n = 3n +2n +n .
B 0,3 1,2 2,1
Thus, the proposed model correctly reduces to the ideal solu-
The equilibriumvaluesof the associate mole numbersare de- tionmodelinthecasewheretherearenointeractionsbetween
terminedby mixingparticles.
n D g
2,1 2,1
=3exp − , 6. Dilute solutions
n2/3n1/3 RT
3,n0 0,3 (cid:18) D g (cid:19) (23)
1,2 1,2
=3exp − . Pelton et al. [10] pointed out another interesting feature of
n13/,03n20/,33 (cid:18) RT (cid:19) theassociatemodelthatoccursindilutesolutions.Theyconsid-
ered the associated solutionof monoparticlesA and B and as-
5. CaseD g =0andD g =0 sociatesA2B.ThehighlyorderedsolutionrichincomponentB
2,1 1,2
consistsprimarilyofB-monoparticlesandA B-associates.Ac-
2
cordingtothe authors,the chemicalactivity a of thecompo-
Recalling Eq. (18), consider the situation where there are B
nentBbehavesasymptoticallyas(1−x /2)ratherthan(1−x )
no Gibbs free energychanges of forming differentassociates, A A
for small x . Thus, lim(da /dx )=−1/2. Note, however,
that is, D g2,1 =0 and D g1,2 = 0. In this case, the A and B A xA→0 B A
particle species mix ideally. The composition of a randomly that this behaviourof aB is observedonly in the limiting case
selected triplet of particles follows the binomial distribution, D g2,1 =−¥ . For any finite value, lim(daB/dxA)=−1. We
well-knownin probabilitytheory;see, forexample,Ref. [12]. xA→0
present the proof of this result for the model proposed in this
More precisely, the probabilities of choosing an associate of
paper.Using thepresentedtechnique,similarresultcanbe es-
particularcompositions,or,equivalently,themolarfractionsof
tablishedfortheexampleconsideredbyPelton etal.[10].
appropriateassociates, are 1/3
AsobtainedinSection4,a =x .Therequiredderivative
B 0,3
isthencalculatedasfollows.First,
x =x3,
3,0 A
da 1 dx 1 dx −1
x2,1 =3x2AxB, (24) dxB = 3x−0,32/3 dx0,3 = 3x−0,32/3 dxA . (28)
x =3x x2, A A (cid:18) 0,3(cid:19)
1,2 A B Secondly,assumingD g andD g finiteanddenotingtheright
2,1 1,2
x0,3 =x3B. handsidesofEqs.(??)by
Here, xA and xB are the molar fractions of A and B particles. e ≡3exp −D g2,1 , e ≡3exp −D g1,2 , (29)
2,1 1,2
Sincethetotalmolenumberofassociatesn isequalto(n + RT RT
tot A (cid:18) (cid:19) (cid:18) (cid:19)
nB)/3,thenthemolenumbersofassociatesaregivenby weobtain
n = n3A , x2,1=e 2,1x23/,03x10/,33, (30)
3,0 3(nnA2+n nB)2 x1,2=e 1,2x13/,03x20/,33.
n2,1 = (n +A nB )2, Substitution of Eqs. (30) into the mass balance constraints of
A B (25) Eqs. (22), with the latter written in terms of molar fractions,
n n2
n = A B , gives
1,2 (n +n )2
A B
n0,3 = 3(n n+3Bn )2. 1=x3,0+x0,3+e 2,1x23/,03x10/,33+e 1,2x13/,03x20/,33, (31)
A B
1 2
1−x =x + e x2/3x1/3+ e x1/3x2/3. (32)
Bydirectinspection,themolenumbersgivenbyEqs.(25)sat- A 0,3 3 2,1 3,0 0,3 3 1,2 3,0 0,3
isfyEqs.(22)andEqs.(23)withD g2,1=0andD g1,2=0.Since Now,Eq.(31)implicitlydefinesx0,3asafunctionofx3,0,while
the Gibbsfree energy G givenbyEq. (21)is a strictly convex
Eq.(32)definesx asafunctionofx andx .Differentiating
A 3,0 0,3
functionof the mole numbersof associates, then Eq. (23) has
Eq.(31)gives
noothersolutionsatisfyingthemassbalance.
Furthermore,ifD g2,1=0andD g1,2=0,thenthemolarGibbs dx3,0 =− 1+23e 1,2t+31e 2,1t2 , (33)
freeenergiesofassociatesaregivenby dx0,3 1+31e 1,2t−1+23e 2,1t−2
171
where In one mole of the solution component A, there are 1/m
i
t≡ x3,0 1/3 . (34) moles of m-particle associates of composition s i. Thus, their
(cid:18)x0,3(cid:19) molarGibbsfreeenergy gs i is
DifferentiationofEq.(32)withrespecttox0,3 andsubstitution gs i =mgi, (39)
ofEq.(34)intothe resultantexpressionyields
where g is the molar Gibbs free energy of the solution com-
i
dxA =−1−4e t−1e t2 ponentAi.
dx 9 1,2 9 2,1 Ingeneral,thetotalnumberN ofdistinguishabletypes
0,3 k1,...,kr
+2 e 1,2t−2+e 2,1t−1 1+23e 1,2t+31e 2,1t2 of associates which have the same composition k1,...,kr is
9 1+1e t−2+2e t−1 describedbythemultinomialcoefficient
(cid:0) 3 1,2(cid:1)(cid:0) 3 2,1 (cid:1) (35)
4 1 m!
=−1−9e 1(cid:0),2t−9e 2,1t2 (cid:1) Nk1,...,kr = r . (40)
+2(e 1,2+e 2,1t) 1+32e 1,2t+13e 2,1t2 . i(cid:213)=1ki!
9 t2+(cid:0)13e 1,2+32e 2,1t (cid:1) We assume that all distinguishable types of the associates of
Note that x0,3→1, x(cid:0)3,0→0 and t→0(cid:1)as xA→0. Therefore, composition(k1,...,kr)areendowedwithJk1,...,kr differentval-
substitutingEq.(35)intoEq.(28)andtakingthelimitasx → uesofthemolarGibbsfreeenergy.Let d[j] bethenumber
A k1,...,kr
0, one verifies that lim(daB/dxA)=−1 for any finite values of the associate types at the jth energy level g[j] , that is,
ofD g andD g . xA→0 themultiplicityofthelevel,sothat k1,...,kr
2,1 1,2
DEgqN.1,(2o3wr5e)mcgoainivnsesidfienritthe.eRliemcaitlilninggcEaqses.w(2h9e)raenDdgta2,k1in→gt−he¥ l,imwihtiilne Jk1j(cid:229)=,...1,krdk[1j],...,kr =Nk1,...,kr. (41)
dx 2 1 1 [j]
e 2,1li→m+¥ dx0A,3 =−3−6e 1,2t−6t3. (36) eAnlelrtghyeodfk1f,.o..r,kmraatsisoonciDagte[j]typesahcacvoerdthinegstaomtehemroelaacrtiGoinbbsfree
Finally,substitutingEq.(36)intoEq.(28)andtakingthelimit k1,...,kr
k k
asxA→0,weobtainxlAi→m0(daB/dxA)=−1/2asD g2,1→−¥ . m1(A1)m+...+mr(Ar)m⇄((A1)k1...(Ar)kr)[j]. (42)
Moreprecisely,theGibbsfreeenergyofthereactionisdefined
7. Model equations for arbitrary number of solution by
componentsandarbitrarysize ofassociates D g[j] =g[j] −k1gs −...−krgs . (43)
k1,...,kr k1,...,kr m 1 m r
[j]
Consideranr-componentsolutionA −...−A andassume The molar Gibbs free energies g of mixed associates of
1 r k1,...,kr
that the components and the solution itself all consist of m- composition(k ,...,k )∈S◦,oralternatively,thecorrespond-
1 r m
particleassociates. Thereare rm distinguishabletypesofasso- ing formation energies D g[j] from Eq. (43) are adjustable
ciates. Some of these typeshavethe same compositions.Now k1,...,kr
parametersofthemodel.
consideran m-particle associate that consists of k1,...,kr par- Now let n[j] denote the mole number of the associates
ticles of types A1,...,Ar, respectively. Thus, the composition k1,...,kr
of composition (k ,...,k ) at the jth energy level. The mass
oftheassociateisspecifiedbyther-tupleofnonnegativeinte- 1 r
balanceconstraintsthenread
gers (k ,...,k ) satisfying k +...+k =m. Omitting the de-
1 r 1 r
pendenceonr,thesetofsuchtuplets,whichrepresentallpos- (cid:229) Jk1(cid:229),...,kr [j]
n = k n , i=1,...,r, (44)
siblecompositionsofm-particleassociatesinther-component i i k1,...,kr
solution,is denotedby S . Its cardinality,thatis, the number (k1,...,kr)∈Sm j=1
m
wheren isthemolenumberofA particlesinthesolution.One
N ofdifferentcompositionsis computedas i i
comp
verifiesthat
(m+r−1)!
Ncomp= m!(r−1)! . (37) (cid:229) r n =m (cid:229) Jk1(cid:229),...,krn[j] ≡mn . (45)
i k1,...,kr tot
ConsiderapuresolutioncomponentAi.Itsm-particleassociates i=1 (k1,...,kr)∈Sm j=1
allhavethecomposition Note that the totalnumberof associates n is independentof
tot
composition of the solution since they are assumed to be of
s i≡(0,...,0,m,0,...,0). (38)
equalsize. The Gibbsfree energyof the solution is computed
i−1 r−i as
tTyhpeescoamreporseiftieornresd s t1o,..a|.s,{szrp}uraen.|d{Tzchoe}rrecsopmonpdlienmgenatsasroyciasetet G = (cid:229) Jk1(cid:229),...,krn[j] g[j]
mSSatom◦mlre\eiass{dtcsifto1wfne,o.sre.tin.nt,uotstnperzad}ertrbioocyfleetnmhstoirpxiseeeecsd.ire-csto,umpislpewosrs(iikttti1eo,nn..sb.,r,ickeorfl)nytfaraionsmiSngS◦m.tmwTowhuiotshr, + R(kT1,..(.k,k1r,.).∈.,(cid:229)kSr)m∈Sjm=J1k1j(cid:229)=,...1k,k1r,.n..[k,kj1]r,...k,k1r,.l..n,kr dx[kk[j11j]],,......,,kkrr, (46)
 
172
r
where m!/(cid:213) k!. This assumption, which can be subsequently re-
[j] i
n i=1
x[j] = k1,...,kr (47) fined in thermodynamic model optimisation of real chemical
k1,...,kr ntot systems, allows substantially reduce the numberof adjustable
isthe molarfractionof anappropriateassociatetype.
parametersofthemodel.SincetheGibbsfreeenergiesofpure
Eqs.(44)definens i asalinearfunctionofthemolenumber associates are fixed, the number of adjustable parameters is
n ofthesolutioncomponentA andofthemolenumbersofthe
i i equalto (N −r), where N is given by Eq. (37). Then,
mixed associates nk[j1],...,kr, where (k1,...,kr)∈S◦m. The equi- omittingthecosmupperscript [j], Eqco.m(p46)reducesto
[j]
librium values of n , which are internal variables of the r
k1,...,kr x (cid:213) k!
mthoedseoll,uatrioendeattecrmoninsteadntbny1m,.i.n.i,mnris,isnugbtjhecetGtoibtbhsefmreaesesnbearglaynocef G = (cid:229) nk1,...,krgk1,...,kr+RTln k1,...,mkr!i=1 i 
constraintsofEqs.(44).Theminimumisfoundbysetting (k1,...,kr)∈Sm
  
  
¶ G   (53)
=0 (48) Similarly to Section 5, consider the case when all Gibbs free
¶ n[j]  energiesD g offormationofassociatesareequaltozero,
k1,...,kr n1,...,nr sothatpartikc1l,e..s.,korfthesolutioncomponentsaremixedideally.
 
forall(k1,...,kr)∈S◦m andforall[j]=1,...,Jk1,...,kr.Recall- In thiscase, the compositionof m randomlyselected particles
ing that ns i are functions of n[kj1],...,kr, where (k1,...,kr)∈S◦m, follows the multinomial distribution; see Ref. [12] for more
andusingthechainrule, details. The molarfraction of the associates with composition
¶ G (cid:229) r ¶ G ¶ ns (k1,...,kr)∈Sm isexpressedas
+ i =0. (49)
¶ nk[j1],...,kr i=1¶ ns i ¶ n[kj1],...,kr xk1,...,kr =m!(cid:213) r xkki!i (54)
SubstitutionofEqs. (44)and(46)intoEq.(49)gives i=1 i
andhence,theirmolenumberis
n[j] D g[j]
i(cid:213)=r1(kn1s,..i.),kkri/m =dk[1j],...,krexp− Rk1T,...,kr. (50) nk1,...,kr = ((cid:229)rmn−i1m)!−1(cid:213)i=r1nkiki!i. (55)
UsingEqs.(44),oneverifiesthatns i isafunctionofni andof (cid:18)i=1 (cid:19)
nth[kej1],.s..o,klru,twiohnecreom(kp1o,n..e.n,tkrA)1∈caSn◦mn.oTwhebechceamlciuclaaltepdotaesnftoiallloµw1so:f tHheerseo,lnuitiaonndcxoimapreontheentmAoi.leOnnuemvbereirfiaensdthtahtethmeomlaorlferancutimonbeorsf
givenbyEq.(55)describethesolutionofEq.(50)subjecttothe
¶ G
µ1 = ¶ n massbalanceconstraintsofEqs.(44)inthecaseD gk1,...,kr =0.
(cid:18) 1(cid:19)n2,...,nr Theuniquenessofthesolutionisensuredbythestrictconvexity
= (cid:229) Jk1(cid:229),...,kr ¶ G ¶ n[kj1],...,kr of the Gibbs free energy G given by Eq. (53) in the variables
(k1,...,kr)∈S◦m j=1 ¶ n[kj1],...,kr ¶ n1 nk1F,.r..o,kmr.Eqs.(39)and(43),themolarGibbsfreeenergyofthe
+ (cid:229) r ¶ G ¶ ns i + (cid:229) Jk1(cid:229),...,kr ¶ ns i ¶ n[kj1],...,kr associatesofthecomposition(k1,...,kr)isgivenby
i=1¶ ns i ¶ n1 (k1,...,kr)∈S◦m j=1 ¶ n[kj1],...,kr ¶ n1  g =(cid:229) r kg , (56)
¶ G ¶ ns (cid:229) Jk1(cid:229),...,kr ¶ G  k1,...,kr i=1 i i
= 1 +
¶ ns 1 ¶ n1 (k1,...,kr)∈S◦m j=1 ¶ n[kj1],...,kr wpohneerne,tgAi.isStuhbestmituotliaornGoibfbEsqfsr.ee(5e4n)e-(r5g6y)oifnttoheEsqo.lu(5ti3o)nacnodma-
+ (cid:229) r ¶ G ¶ ns i ¶ n[kj1],...,kr  straightfoirward,thoughlengthy,verificationshowsthatthepro-
i=¶ 1G¶ n¶ snis¶ nk[j1],...,kr ¶ n1 ptioosnedmomdoedleilnctohrirseccatlsye.reTdhuacteiss,to the r-componentideal solu-
= 1
¶ ns 1 ¶ n1 G=(cid:229) r n (g +RTlnx). (57)
(51) i i i
i=1
Here, Eq. (49) have been used. Using Eqs. (39) and (44),one
verifiesthat Fromthisreduction,itisimmediatelyfollowsthatthemodel
1 withassociatesofsizercanbereproducedbythemodelswith
µ1=g1+mRTlnxs 1. (52) associates of size 2r, 3r and so on. For example,considerthe
Thechemicalpotentialsoftheothercomponentsarecalculated modelwithassociatesofsize2r.Any2r-associateisacombi-
in asimilarway. nation of two r-associate. Now assume that there is no Gibbs
Now assume that all the associate types of composition free energy change on forming any 2r-associate from corre-
(k1,...,kr) have the same Gibbs free energy of formation spondingtwo r-associates. As demonstratedabove,the model
for any (k1,...,kr)∈Sm, so that Jk1,...,kr =1 and dk11,...,kr = with 2r-associate reduces to the ideal mixture of r-associates.
173
Thus,themodelwith2r-associatesismoregeneralandinclude Gibbsfreeenergiesofassociates,theoptimalvaluesofthe ef-
themodelwithr-associatesasa particularcase. fectiveadjustableparametersZ canbedeterminedbythe
k1,...,kr
trial-and-errorprocedurewhichisconventionallyusedin ther-
8. Effectiveadjustableparametersofthe model modynamicmodeloptimisationofrealchemicalsystems.
The adjustable parameters of the model related to asso-
9. Excess Gibbsenergyterms
ciatesofthecomposition(k ,...,k )areD g[j] ,where[j]=
1 r k1,...,kr
1,...,J .Oneshouldalsotakeintoaccountthemultiplic-
k1,...,kr Similarlytothemodifiedassociatespeciesmodel[4],regular
[j]
ity of the energylevels d . In general, the numberof ad- or,ingeneral,polynomial,excessGibbsfreeenergytermscan
k1,...,kr
justable parameters of the model increases exponentiallywith be included into the proposed formalism. These terms take
the increase in size of associates m. However, the number of into account interactions between associates. A probabilistic
adjustableparameterscanbesubstantiallyreducedwithoutloss interpretationofthepolynomialexcessGibbsfreeenergyterms
ofgeneralityofthe modelasdescribedbelow. ispresentedinRef. [13]. Infact,this interpretationprovidesa
Considerassociateswiththecomposition(k1,...,kr).Ingen- theoreticaljustification forsuch terms. Treatingthe associates
eral,Jk1,...,kr energylevelsarepossiblefortheseassociates.The asparticles,theresultsofRef.[13]areapplicabletothemodel
mole number nk1,...,kr of all associates with the composition presentedinthispaper.
(k1,...,kr) isgivenby Notethatitisdesirabletousetheexcesstermsonlyfor“fine
n =Jk1(cid:229),...,krn[j] . (58) ttuhneimngo”laorfGthibebmsofrdeeel,ewnehrigleietsheofmaasisnocaidajtuesst.aTbhleepfaorlalomweitnegrstawroe
k1,...,kr j=1 k1,...,kr conditionson the absolute value of the interaction parameters
should be satisfied; see, for example, the monograph by Pri-
ThemassbalanceconstraintsofEqs.(44)taketheform
gogineandDefay[8] formoredetails.
(cid:229)
ni= kink1,...,kr, i=1,...,r. (59) (i) Theabsolutevaluesoftheinteractionparametersshould
(k1,...,kr)∈Sm besmallcomparedwiththemolarGibbsfreeenergiesof
Summingupover[j] in Eq.(50),oneverifiesthattheequilib- theassociates.AspointedoutbyPrigogineandDefay[8],
riumvalueof n iscalculatedas iftheinteractionbetweenassociates,say“C1”and“C2”,
k1,...,kr
issufficientlystrongtoalterthevibrationalandrotational
n
r k1,...,kr =Zk1,...,kr (60) states of the associates, then the associate “C1C2” is
(cid:213) (ns )ki/m included into the set of associates by the definition of
i
i=1 the classical associate model. In the framework of the
where proposedformalism, one should considerthe associates
Z ≡Jk1(cid:229),...,krd[j] exp −D g[kj1],...,kr . (61) (ii) Tofhelavrgaeluressizseh.ouldalsobesmallincomparisonwithRT.
k1,...,kr j=1 k1,...,kr  RT  Otherwise,the assumptionof idealmixingof associates
  islessjustified.Again,largerassociatesshouldbetaken
ThefunctionZ definedbyEq.(61)playsa roleofthe
k1,...,kr intoaccount.
partitionfunctionwhichdescribesthedistributionofassociates
Note, however, that the second condition is sometimes re-
ofthe composition(k ,...,k )overenergylevels.Thatis,
1 r laxed in order to fit the experimental data available for real
n[j] x[j] d[j] D g[j] solutions.This is the case, forexample,forthe solutionswith
k1,...,kr = k1,...,kr = k1,...,kr exp − k1,...,kr (62) immiscibility,whichistheresultofrelativelyweak,compared
n x Z  RT 
k1,...,kr k1,...,kr k1,...,kr with the Gibbs free energies of associates, repulsive interac-
  tionsbetweentheassociates.
Infact,Z isasingleeffectiveadjustableparameterrelated
k1,...,kr
toassociatesofthecomposition(k ,...,k ).Theotherthermo-
1 r
dynamicparametersofthemodelcanbeexpressedintermsof
10. Discussion onapplicabilityofthe model
Z , where k ,...,k ∈S , and their derivatives. Indeed,
k1,...,kr 1 r m
usingEqs.(39),(43),(59)and(62),oneverifiesthattheGibbs
The suggested modified associate formalism belongs to
freeenergyofthe solutiongivenbyEq.(46)takesthe form
associate-type models. As a result, the range of applicability
G=(cid:229) r ng +RT (cid:229) n ln xk1,...,kr . (63) of the formalism is at least the same as that of the classical
i i k1,...,kr Z associate model or the associate species model. There are,
i=1 (k1,...,kr)∈Sm (cid:18) k1,...,kr(cid:19)
however, some distinctions in using the suggested formalism
According to Eq. (61), Z varies from zero to infinity.
k1,...,kr and the previous modifications of the associate model. These
Whentheassociatesofthecomposition(k ,...,k )areprohib-
1 r distinctions are discussed below. Since the present study is
ited energetically,that is (D g[kj1],...,kr →+¥ ) for all [j], Zk1,...,kr intended as a theoretical introduction to the modified asso-
approacheszero.Iftheassociatesofthecomposition(k ,...,k ) ciate formalism,the results on its applicationto real chemical
1 r
ishighlypreferable,Z approachesinfinity.Similartothe systemswillbereportedelsewhere.
k1,...,kr
174
In contrast to the previous modifications, where associates levels:g[1] =g +g +g withthemultiplicityd[1] =
A3B3 [AA] [BB] [AB] A3B3
ofarbitrarycompositionscanbe includedintothe setof asso- [2] [2]
12 and g =3g with the multiplicity d =8. A B -
ciates, the compositionsofassociatesare definedbytheirsize A3B3 [AB] A3B3 2 4
[1]
in the frameworkof the modified associate formalism. There- associate also have two energy levels: gA2B4 =g[AA]+2g[BB]
fore,amodellershouldpayspecialattentiontoselectionofthe withthemultiplicityd[1] =3 andg[2] =g +2g with
sizeofassociates.Ifexperimentalinformationaboutcomposi- A2B4 A2B4 [BB] [AB]
[2]
themultiplicityd =12,whileAB -associateshaveonlyone
tions of associates that present in the solution phase is avail- A2B4 5
energy level g =2g +g with the multiplicity equals
able,thisinformationshouldclearlybetakenintoaccount.The AB5 [BB] [AB]
to6.B -associatesalsohaveonlyoneenergylevelg =3g
size of associates shouldbe largeenoughto incorporatethose 6 B6 [BB]
withthemultiplicityequalsto1.ThebinarysystemsA−Cand
compositions.
B−Caretreatedsimilarly.
Whenthesuggestedformalismisappliedforthermodynamic
Previous modifications of the associate model allow more
descriptionofmulticomponentsolutionphases,the size of as-
flexibility in fitting experimentaldata comparedwith the sug-
sociateshouldbelargeenoughtoincorporatetheassociatesof
gested approach, since stoichiometry of associates can be ar-
the compositions, which coincide with those of maximum or-
bitrarily altered to improve the fit. This additional flexibility
deringinallbindingbinarysystems.Forexample,considerthe
canbehelpfulformodellingabinarysystemwithhighdegree
ternarysystemA−B−Candassumethatinthebinarysystem
of ordering, where the described paradox is of relatively low
B−C the compositionof maximumorderingis thatof theas-
practical importance. However, when such a binary system is
sociate B C, while in the systems A−B and A−C maximum
2
combinedwith otherbinariesto form multicomponentmodel,
ordering occurs at the compositions of the associates AB and
arbitrary selection of associate stoichiometry can result in the
AC, respectively.To incorporatethe requiredcomposition,the
negativeconsequencesofthe entropyparadox.
sizeofassociatesshouldbedivisibleby2andby3.Therefore,
Similar to the previousmodifications,a miscibility gap can
the size of associates forthe ternary system A−B−C should
notbereproducedwithoutexcessGibbsfreeenergytermsinthe
beatleast6.
proposedformalism.However,asdemonstratedbyBesmannet
Itisdesirabletouse6-particleassociatesforthermodynamic
el. [4] forthe associate species model,immiscibility,whichis
model optimisation for all the binary systems. There is, how-
the resultof repulsiveinteractionsbetween associate (or asso-
ever, no need to use 6-particle associates from the beginning.
ciate species), can be accurately represented by the associate-
Let us assume that the systems A-B and A-C are initially op-
type models with the polynomial excess Gibbs free energy.
timised with 2-particle associate, while the system B-C is op-
Treatingassociates(orassociatespecies)asparticlesandusing
timised with 3-particle associate. As discussed in the end of
theresultsofRef.[13],oneverifiesthatthecoefficientsofthe
Section 7, the model with associates of size r can be repro-
polynomialexcess Gibbs free energyexplicitly relate to ener-
duced by the models with associates of size 2r, 3r and so on.
gies of interactionsbetweenassociates. It is importantto note
Usingthispropertyoftheproposedformalism,thedescriptions
that,inthecaseofthemodifiedquasichemicalmodel[10],im-
of the binary systems with 2- and 3-particleassociates can be
miscibilityisdescribedbyempiricalpolynomialexpansionsof
replaced by the equivalent descriptions with 6-particle asso-
theGibbsfreeenergyofthequasichemicalreaction.Incontrast
ciates in a straightforward way. Then, the binary systems can
to associate-type models, physicalmeaningof the coefficients
becombinedintotheternaryone.
ofsuchexpansionsis notclear.
As a demonstrational example, consider the system A−B.
Anotherdistinctionoftheproposedformalismfromthepre-
When the system is optimised with 2-particle associates the
vious modifications is related to multicomponent associates,
Gibbsfreeenergiesoftheassociatesg ,g ,g andg
[AA] [BB] [AB] [BA] thatis,theassociateswhichconsistofparticlesofthreeormore
areknown.Duetospatialsymmetry,g =g .Intheequiv-
[AB] [BA] types.Inthepreviousmodifications,nomulticomponentasso-
alentdescriptionofthe system A−B,6-particleassociatesare
ciatesareinitiallyconsidered.Thisimpliesthatallmulticompo-
formedfromthree2-particleassociateswithnoGibbsfreeen-
nentassociatesareassumedtohaveinfinitepositiveGibbsfree
ergychangesonsuchformations.Theassociateofthecomposi-
energy.Multicomponentassociatesareusuallyincludedintothe
tionA isformedfromthree[AA]-associateswithg =3g .
6 A6 [AA] model,whensuchassociatesarerequiredtofitavailableexper-
TheassociateofthecompositionA Bisformedetherfromtwo
5
imentaldatainmulticomponentsystems.Incontrast,thecom-
[AA]-associatesand[AB]-associatesorfromtwo[AA]-associates
positions of multicomponent associates are prescribed by the
and [BA]-associates. Then, recalling that g = g , A B-
[AB] [BA] 5 selected size of associate in the proposedformalism.It would
associateshaveonlyoneenergylevelg =2g +g with
A5B [AA] [AB] bebeneficialtodevelopamethodforestimatingtheGibbsfree
multiplicity 6. The A B -associate can be formed from two
4 2
energies of the multicomponent associates from those of bi-
[AA]-associates and one [BB]-associate with the energy level
nary associates. The developmentof such a method,however,
[1] [1]
g =2g +g and the multiplicity d =3. Alterna-
A4B2 [AA] [BB] A4B2 isthetopicforseparatestudy.Asaninitialapproximation,the
tively,A B -associatecanbeformfromone[AA]-associateand
4 2 Gibbsfree energiesof formationof the multicomponentasso-
eithertwo [AB]-associates ortwo [BA]-associates orone [AB]-
ciates frompure associates can be set to zero.If experimental
associate andone[BA]-associate.Inthiscase,theenergylevel
data are available in multicomponentsystems, the Gibbs free
[2] [2]
isgA4B2 =g[AA]+2g[AB],anditsmultiplicityisdA4B2=12.Ina energiesofthemulticomponentassociatescanbeadjustedina
similarway,oneverifiesthatA3B3-associateshavetwoenergy conventionaltrial-and-errorprocedure.
175
A possibility to select compositions of multicomponentas- [6] E. Yazhenskikh,K. Hack, M. Muller,Critical thermody-
sociatesarbitrarily,whichexistsinthepreviousmodifications, namic evaluation of oxide systems relevantto fuel ashes
couldprovidemoreflexibilityindatafittingcomparedwiththe and slags. Part 2: Alkali oxide - alumina systems, Cal-
fixed set of compositions. It could be the case, that arbitrarily phad-ComputerCouplingofPhaseDiagramsandTher-
selected additional multicomponent associates provide better mochemistry30(2006)397–404.
fitting to particular experimental data. One should recognise, [7] E. Yazhenskikh,K. Hack, M. Muller,Critical thermody-
however,thatthepriceforthepossibilitytoselectthecomposi- namic evaluation of oxide systems relevantto fuel ashes
tionsofassociatesarbitrarilyistheentropyparadox,whichun- andslags. Part3:Silicaalumina system, Calphad- Com-
derminesthefundamentalpredictivecapabilitiesofthemodel. puterCouplingofPhaseDiagramsandThermochemistry
Inouropinion,thenecessityofadditionalmulticomponentas- Inpress.
sociatesforfittingtheavailableexperimentaldataindicatesthat [8] I.Prigogine,R.Defay,ChemicalThermodynamics,Long-
the size of associates should be increased to incorporate the mans,GreenandCo.,1954.
requiredcompositions. [9] R.Lu¨ck,U.Gerling,B.Predel,Anentropyparadoxofthe
associationmodel,ZeitschriftfurMetallkunde80(1989)
11. Conclusion 270–275.
[10] A. D. Pelton, S. A. Degterov, G. Eriksson, C. Robelin,
Y. Dessureault, The modified quasichemical model I -
In this paper, the Modified Associate Formalism has been
Binarysolution,MetallurgicalandMaterialsTransactions
proposedforthermodynamicmodellingof solutions. The pre-
B: Process Metallurgy and Materials Processing Science
sentedapproachisfreefromtheentropyparadoxandcorrectly
31B(2000)651–659.
reducesto the ideal solution model,where there are no Gibbs
[11] L. D. Landau, E. M. Lifshitz, Satistical Physics Part 1,
freeenergychangesonformingassociatesofdifferenttypes.
Vol.5ofCourseofTheoreticalPhysics,PergamonPress,
Asymptoticbehaviourofchemicalactivitiesofsolutioncom-
1980.
ponentsinbinarydilutesolutionshasbeeninvestigated.Ithas
[12] W. Feller, An Introductionto Probability Theory and Its
been demonstrated that the derivative of the chemical activity
Applications,Vol.1,JohnWiley& Sons,1971.
ofa solutioncomponentinitsmolarfractionatterminalcom-
[13] D. Saulov, On the multicomponent polynomial solution
positionhastheexpectedvalueforanyfinitevalueoftheGibbs
models,Calphad-ComputerCouplingofPhaseDiagrams
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andThermochemistry30(2006)405–414.
Acknowledgment
ThepresentworkhasbeensupportedbytheAustralianRe-
searchCouncil.
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