Modified associate formalism without entropy paradox: Part II. Comparison with similar models

0.33 MB·English

by  Dmitry N. Saulov

3
1
0 ModifiedAssociateFormalismwithoutEntropyParadox:
2
PartII. ComparisonwithSimilarModels
n
a
∗
J DmitryN. Saulov ,a, A. Y. Klimenkoa

5 aSchool of Engineering, The University ofQueensland, AUSTRALIA
1

]
r
e
hAbstract
t
oThemodifiedassociateformalismiscompared withsimilarmodels, suchastheclassicalassociatemodel, theassociatespeciesmodel andthe
.modified quasichemical model. Advantages of the modifiedassociate formalismaredemonstrated.
t
a
m
Key words: Thermodynamic modeling (D),Entropy (C)
-
d
n
o
c1. Introduction – dependenceofcoordinationnumbersofthesolutioncompo-
[ nentsonthe mole numbersofdifferentpairsthatallow one

1 In this paperwe compare the modified associate formalism tosetthecompositionofmaximumorderingsoastocomply
v(MAF) proposed in the first paper[1] with the classical asso- withtheexperimentaldataforeachbinarysystem individu-
5ciatemodel(CAM)describedbyPrigogineandDefay[2]and ally.
1theassociatespeciesmodel(ASM)describedbyBesmannand We refer to the second version of the modified quasichemical
2
Spear [3]. Where it is possible, we compare MAF with the modelasMQM(2000).
3
.classicalquasichemicalmodel(CQM)ofFowlerandGuggen-
1
heim [4]. We also compare MAF with two modifications of
0
2. ConfigurationalEntropyofMixing inBinarySolutions
3CQM. ThefirstmodificationsuggestedbyPeltonandBlander
1[5] include:
:– selection of the coordination numbers that allow one to set In this section, we compare the molar configurational en-
v
i thecompositionofmaximumorderingsoastocomplywith tropiesofmixinggivenbythesolutionmodelsmentionedabove
X
the experimentaldata; with theoretical boundaries for the configurational entropy of
r– anempiricalexpansionofthemolarGibbsenergyofthequa- mixing.Theseboundariesarediscussedimmediatelybelow.
a
sichemicalreactionaspolynomialsintermsofcoordination Theconfigurationalentropyofmixingisameasureofdisor-
equivalent mole fractions of solution components,which is der and has its maximum value for an ideal solution which is
usedforfittingavailableexperimentaldata. completely disordered.For any solid or liquid solution A−B,
The first version of the modified quasichemical model is re- the configurational entropy of mixing can not exceed that of
ferred to as MQM(1986)in the presentstudy. Furthermodifi- idealsolution.Ontheotherhand,whenvacanciesarenottaken
cationsofthequasichemicalmodelintroducedbyPeltonetal. into account (which is the case in all considered models), the
[6] are: configurationalentropyofmixingcannotbenegative.Theval-
– expansion of the Gibbs energy of the quasichemical reac- uesofmolarconfigurationalentropyofmixingwhichareout-
tion as polynomials in pair fractions (instead of equivalent sideofthese boundariesareparadoxical.
molefractionsofsolutioncomponents),whichprovidesad- For each model, we consider the following three limiting
vantagesin data-fitting; cases.
(i) ThecaseofidealsolutionofA-particlesandB-particles.
In this case, the Gibbs free energy changes on forming
associatesthatconsistofbothA-particlesandB-particles.
∗
Corresponding author: School ofEngineering, The University of Queens-
The configurationalentropy of mixing should reduce to
land,St.Lucia,QLD4072,Australia. Phone:[phone]Fax:(07)3365-
thatofthe idealsolutionmodel.
3670
Emailaddress:[e-mail] (Dmitry N. Saulov). (ii) ThecaseofimmisciblecomponentsAandB.Inthiscase,
ArticlepublishedinJournalofAlloysandCompounds473(12)(2009)157–162
the configurational entropy of mixing should equals to 0.9
zero.
0.8
(iii) The case of the highly ordered solution A−B. In this
case,theGibbsfreeenergychangeonformingoftheas- 0.7
sociates of a particular composition and of a particular R
0.6
spacial arrangement of particles approaches −¥ . If the /
quasichemecal model is considered, the Gibbs free en- onf 0.5
c
ergychangeon forming (A−B)-bondsapproaches−¥ . s
0.4
The configurational entropy of mixing should be equal
to zeroatthe compositionofmaximumordering. 0.3
0.2
2.1. ClassicalAssociateModel
0.1
Consider a solution of x moles of the component A and
A
0
xtiBon=al1e−ntxroApmyoolfesmoixfitnhgescompgoivneenntbBy.tTheheclmasosliacralcoanssfiogcuiarate- 0A 0.2 0.4 xB 0.6 0.8 B1
conf
model[2]is expressedas A B-associates only A2 B-associates and Theoretical
2 AB-associates range
2
(cid:229)
s =−R(n ln(x )+n ln(x )+ n ln(x )).
conf A1 A1 B1 B1 AiBj AiBj Fig.1.Molarconfigurationalentropyofmixinggivenbytheclassicalassociate
i,j≥1 model inthe limit ofideal solution ofA-particles and B-particles.
(1)
Here, n , n and n are the mole numbers (in one mole 0.7
A1 B1 AiBj
of the solution A−B), while x , x and x are the mo-
A1 B1 AiBj
larfractionsof A -monoparticles,B -monoparticlesand AB - 0.6
1 1 i j
associates, respectively; R is the universal gas constant. Note
that compositions of associates and sizes of associates (i+ j) R 0.5
are determined by a modeller. The equilibrium values of the /
mole numbers are determined by minimisation of the Gibbs onf 0.4
c
s
free energy of the solution subject to the mass balance and
nonnegativityconstraints(see,forexample,themonographby 0.3
PrigogineandDefay[2]formoredetails).
NowconsiderthelimitwhentheGibbsfreeenergychanges 0.2
on forming the associates equal to zero. As pointed out by
Lu¨cketal.[7],theclassicalassociatemodelgivesparadoxical 0.1
values of the molar configurational entropy of mixing, which
arelargerthanthosegivenbytheidealsolutionmodel.Fig. 1,
0
for example, shows the configurational entropy curves given 0A 0.2 0.4 xB 0.6 0.8 B1
by the classical associate model where only A B-associates
2 Theoretical
CAM
areconsidered(solidline)andwhere A2B-associatesandAB2- range
associatesareconsidered(dashedline).
Fig.2.Molarconfigurationalentropyofmixinggivenbytheclassicalassociate
As shown in the figure, the curves are outside of the the-
model inthe limit ofthe highly ordered solution.
oretically possible range. Note also that the entropy curve is
notsymmetricwithrespecttoverticallinewithx =1/2when equalstozeroatthecompositionofmaximumordering,since
B
onlyA2B-associatesareconsidered.Toobtainasymmetricen- differentspatial arrangementsof particles in an A2B-associate
tropy curve (which is expected in the limit of ideal solution) arenotconsideredintheclassical associatemodel.
oneshouldconsiderboth A B-associatesandAB -associates.
2 2
Consider the limit when the Gibbs free energy changes on
2.2. AssociateSpeciesModel
formingallassociatesapproach+¥ .Inthiscase,noassociates
formandtheconfigurationalentropyofmixingequalstothatof
Accordingtotheassociatespeciesmodel[3],theconfigura-
theidealsolutionmodel.Intheory,however,theconfigurational
tionalentropyof mixingof x molesof the componentA and
entropyofmixingshouldbe zeroforany x . A
B x =1−x molesof thecomponentBis givenby
Nowconsiderthelimitofhighlyorderedsolutionandassume B A
that the Gibbs free energy change of forming A2B-associates R (cid:229)
approaches −¥ . The entropy curve given by the classical as- sconf=−q nAa·qBb·qln(xAa·qBb·q) . (2)
sociatemodelinthiscaseisshowninFig.2.Asshowninthis a+b=1 !
figure,theentireentropycurveiswithinthetheoreticallypossi- Here, q is the size of species (the number of particles per an
blerange.Notealsothattheconfigurationalentropyofmixing associate specie); n are the mole numbers and x
Aa·qBb·q Aa·qBb·q
158
1 .4 0.7
1 .2 0.6
R 1 .0 R 0.5

/ /
onf 0.8 onf 0.4
c c
s s
0.6 0.3
0.4 0.2
0.2 0.1
0 0
0 0.2 0.4 x 0.6 0.8 1 0 0.2 0.4 x 0.6 0.8 1
A B B A B B
q= 1 q= 2 q= 3 Theoretical q= 1 q= 2 q= 3 Theoretical
range range
Fig.3.Molarconfigurationalentropyofmixinggivenbytheassociatespecies Fig.4.Molarconfigurationalentropyofmixinggivenbytheassociatespecies
modelwiththespeciesAq,Bq,A2q/3Bq/3 andAq/3B2q/3 inthelimitofideal model inthe limit ofimmiscible components.
solution.
0.7
0.6
are the molar fractions of the Aa·qBb·q-species. Similar to the
classicalassociatemodel,thesizeandcompositionsofspecies R 0.5
aredeterminedbya modeller.
/
Consider the limit of ideal solution, assuming that the so- onf 0.4
c
lution consists of the associate species A , B , A B and s
q q 2q/3 q/3
A B . The Fig. 3 shows the entropy curves given by the
q/3 2q/3 0.3
associatespeciesmodelforq=1,q=2andq=3.Incontrast
to the classical associate model, one can adjust the configura-
0.2
tionalentropyofmixing,usingtheadditionalparameterq.For
thisparticularsetofspecies,theselectionofq=2resultinthe
0.1
entropycurvewhichisclosed(thoughlocatedoutsidethetheo-
reticallypossiblerange)totheentropycurvegivenbytheideal
0
solutionmodel.AsshowninFig. 3,theselectionq=1results 0 0.2 0.4 x 0.6 0.8 1
A B B
in substantial overestimationof the configurationalentropyof
mixing,whiletheselectionq=3givesunderestimatedvalues. q= 1 q= 2 q= 3 Theoretical
range
Consider the limit of immiscible components, that is, the
Gibbs free energy changes on forming the species A2q/3Bq/3 Fig.5.Molarconfigurationalentropyofmixinggivenbytheassociatespecies
and A B approach +¥ . The entropycurvesgiven by the model inthe limit ofhighly ordered solution.
q/3 2q/3
associate species model in this case are shown in Fig. 4. As
2.3. ModifiedAssociateFormalism
shown in this figure, the associate species modeloverestimate
theconfigurationalentropyofmixinginthelimitofimmiscible
components.However,theentropycurvetendstotheoretically In the modified associate formalism the configurationalen-
expectedone(s ≡0),whenqincreases. tropyofmixingis givenby(seeEq.(44)in Ref.[1])
conf
Now consider the limit of highly ordered solution. Assume
alsothattheGibbsfreeenergychangeonformingtheassociate s =−R (cid:229) Jk1(cid:229),...,krn[j] ln x[kj1],...,kr , (3)
scpuervceiegsivAe2nq/b3yBqth/3eaaspsporcoiaactehespse−ci¥es.mFoigd.e5linshthoiwsscatshee.Aenstrsoepeny conf k1,...,kr∈Sm j=1 k1,...,kr dk[1j],...,kr
 
fromFig.5,theentropycurvewith q=3isentirelylocatedin where all the notationshave the same meaning as in Ref. [1].
the theoretically possible range. If q=2, however, small part In contrast to previous modifications of the associate model,
of the entropy curve (approximately, for 0 <x <0.075) is the compositions of associates that have to be considered are
B
located outside the theoretically possible range. The selection definedbythesize ofassociates.
q=1 results in paradoxical values of configurationalentropy Considerthe limit of ideal solution. In this limit, the Gibbs
approximatelyfor0<x <0.25and0.6<x <1. free energies of the reactions of forming different associates
B B
159
0.7 0.7
0.6 0.6
R 0.5 R 0.5

/ /
onf 0.4 onf 0.4
c c
s s
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 0.2 0.4 x 0.6 0.8 1 0 0.2 0.4 x 0.6 0.8 1
A B B A B B
m=3 m= 6 Theoretical m=3 m= 6 Theoretical
range range
Fig.6.Molarconfigurational entropy ofmixinggiven bythemodifiedasso- Fig.7.Molarconfigurational entropy ofmixinggivenbythemodifiedasso-
ciate formalism in the limit of immiscible components. ciate formalism in the limit ofhighly ordered solution.
equal to zero. As demonstrated in the previous paper [1], the
equivalent fractions y and y are defined as (see Eq. (6) in
A B
configurationalentropy of mixing correctly reduces to that of
Ref.[6])
theidealsolutionmodelforarbitrarysize ofassociates.
Now consider the limit of immiscible components, where yA =ZAnA/(ZAnA+ZBnB)
(6)
the Gibbs free energies of the associate that consist of both
y =Z n /(Z n +Z n ),
A-particles and B-particles approach +¥ . In this case, Eq. (3) B B B A A B B
reducesto whereZ andZ arethecoordinationnumbersofthesolution
A B
R components A and B, respectively. As demonstrated by Pel-
s =− (x ln(x )+n ln(x )). (4)
conf m A A B B tonandBlander[5]forconstantcoordinationnumbersandby
Pelton et al. [6] for variable coordination numbers, the modi-
wheremisthesizeofassociates.Theentropycurvesform=3
fied quasichemical model correctly reduces to the ideal solu-
and m=6 are shown in Fig. 6. As seen from this figure, the
tion model, when the Gibbs free energy change on forming
entropycurvesareentirelylocatedinthetheoreticallypossible (A−B)-pairsD g equalstozero.
range and tendsto the theoreticallyexpectedcurve(s ≡0) AB
conf IntheframeworkofMQM(1986)(themodifiedquasichemi-
with increasingthesize ofassociates.
calmodelafterPeltonandBlander[5]),thecoordinationnum-
Consideringthelimitofhighlyorderedsolution,weassume
bersareassumedtobeconstantandareselectedtosetthecom-
also thatthe Gibbsfreeenergyofassociate with the composi- positionofmaximumorderingx∗ andtosatisfytheconditions
tionA2Bandwithaparticularspatialarrangementofparticles s =0atx∗ whenD g →−¥ B.Thisresultsin(seealsoEqs.
approaches−¥ .Theentropycurvesform=3andm=6given conf B AB
(30)and(31)inRef. [5])
bythemodifiedassociateformalisminthiscaseispresentedin
Fig. 7. In this case, the curves are also entirely located in the
Z =−(x∗ln(x∗)+(1−x∗)ln(1−x∗))/(x∗ln(2))
theoreticallypossiblerange. B B B B B B (7)
Z =Z x∗/(1−x∗).
A B B B
2.4. ModifiedQuasichemicalModel(1986)
Similartotheexamplesdescribedabove,consideronemole
ofbinarysolutionA−Bandassumethatx∗ =1/3.Inthiscase,
B
In the modified quasichemical model, the molar configura- Z =2.75489 and Z =1.37744. In the limit of immiscible
B A
tionalentropyofmixingisgivenby(see Eq.(10)in Ref.[6]) components,D g →+¥ .Themolarconfigurationalentropyof
AB
mixinggivenbythecurvesgivenbyMQM(1986)inthiscaseis
sconf=−R(xAln(xA)+xBln(xB)) presentedinFig.8.Asshowninthisfigure,MQM(1986)gives
−R n ln xAA +n ln xBB +n ln xAB . paradoxical, negative values of the configurational entropy of
(cid:20) AA (cid:18) y2A (cid:19) BB (cid:18) y2B (cid:19) AB (cid:18)2yAyB(cid:19)(cid:21) mixingfor0<xB<1/3.
(5) Now consider the limit of highly ordered liquid, that
Here, x and x are the molar fractions of the components A is, D g → −¥ . Fig. 9 shows the entropy curve given by
A B AB
and B; n and x (i,j=A,B) are the mole number and the MQM(1986) in this case. As seen from Fig. 9, the curve is
ij ij
molar fraction of (i− j)-pairs, respecively. The coordination- entirelylocatedinthe theoreticallypossiblerange.
160
0.7 assuggestedbyPeltonetal. [6],namely:
0.6 ZA =ZB =ZB =6; ZA =3. (9)
AA BB BA AB
0.5 Fig. 10 demonstrates that MQM(2000) gives paradoxical val-
R uesoftheconfigurationalentropyofmixingfortheentirecom-
/ 0.4 positionalrangein thelimitofofimmisciblecomponents.
nf
o
c
s
0.3 1.0
0.2
0.5
0.1
R
0
/
nf 0
o
c
-0.1 s
0 0.2 0.4 x 0.6 0.8 1
A B B -0.5
Theoretical
MQM(1986)
range
Fig. 8. Molar configurational entropy of mixing given by the modified qua- -1.0
sichemical model (1986) in the limit ofimmiscible components.
0.7
-1.5
0 0.2 0.4 x 0.6 0.8 1
0.6 A B B
Theoretical
MQM(2000)
range
R 0.5
Fig. 10. Molar configurational entropy of mixing given by the modified
/
onf 0.4 quasichemical model (2000) inthe limit ofimmiscible components.
c
s
Now consider the limit of highly ordered solution, that is,
0.3 D g → −¥ . The entropy curve given by MQM(2000) in
AB
this case is presented in Fig. 11. As seen form this figure,
0.2
MQM(2000) gives negative values of the configurational en-
tropyofmixinginthevicinityofthecompositionofmaximum
0.1 ordering.
0 0.8
0 0.2 0.4 x 0.6 0.8 1
A B B
0.6
Theoretical
MQM(1986)
range
0.4
Fig. 9. Molar configurational entropy of mixing given by the modified qua- R
sichemical model (1986) in the limit ofhighly ordered solution. / 0.2
nf
o
c
2.5. ModifiedQuasichemicalModel(2000) s 0
-0.2
IntheframeworkofMQM(2000)(themodifiedquasichem-
icalmodelafterPelton etal.[6]),the configurationalnumbers
-0.4
areassumedtobecomposition-dependant.Thefollowingequa-
tions for the coordination numbers have been suggested (see
-0.6
Eqs.(19)and(20)inRef. [6]):
-0.8
1 1 2n 1 n
= AA + AB 0 0.2 0.4 x 0.6 0.8 1
ZA ZAAA2nAA+nAB ZAAB2nAA+nAB A B B
(8) Theoretical
1 1 2nBB 1 nAB MQM(2000) range
= + .
Z ZB 2n +n ZB 2n +n
B BB BB AB BA BB AB Fig. 11. Molar configurational entropy of mixing given by the modified
Considerthelimitofimmisciblecomponents(D g →+¥ ), quasichemical model (2000) inthe limit ofhighly ordered solution.
AB
assumingthatthe constantsZA , ZB , ZA andZB are chosen
AA BB AB BA
161
2.6. ClassicalQuasichemicalModel Now consider the solution discussed in in Ref. [1] in the
frameworkoftheassociatespeciesmodel.Thatis,weconsider
Itisimportanttonotethattheclassicalquasichemicalmodel the associate species Aq, Bq, A2q/3Bq/3 and Aq/3B2q/3 of the
ofFowlerandGuggenheim[4](withZA=ZB=1/2)correctly size q.Inthiscase,chemicalactivityaB isgivenby
reducestotheidealsolutionmodelwhenD gAB=0.Inthelimit a =x1/q , (10)
ofimmisciblecomponents,theconfigurationalentropyofmix- B Bq
ing given by this model correctly reduces to the theoretically where x is the molar fraction of the species B . Eq. (28) in
Bq q
expectedcurve(s ≡0).Inallothercases,theclassicalqua- Ref.[1] reads
conf
smicixhienmgicwahlimchodaerelgwivitehsinvatlhueesthoefothreeticcoanllfiygpuorastsiiobnlealreanntgreo.pTyhoef daB = 1x1/q−1 dxA −1 . (11)
dx q Bq dx
applicabilityoftheclassicalquasichemicalmodel,however,is A (cid:18) Bq(cid:19)
limitedtothosechemicalsolutions,inwhichmaximumorder- Using the technique described in Section 6 of the previous
ingoccursatthecompositionx =1/2. paper[1],oneverifiesthat
B
dx 4 1
A =−1− e t− e t2
2.7. Summary 2 1
dx 9 9
The classical associate model results in paradoxically large Bq +29(e 2+e t12t)+11+e +32e 22te+t13e 1t2 (12)
values of the configurational entropy in the limit of ideal so- (cid:0)3 2 3 1 (cid:1)
lution,whilegivingtheoreticallypossiblevaluesoftheconfig- Here,t≡ x /x 1/3,e ≡(cid:0)exp −D g (cid:1) /RT ande ≡
urationalentropy of mixing in othertwo limits. The associate Aq Bq 1 A2q/3Bq/3 2
species modelcan give paradoxicallylarge values in the limit exp −D g(cid:0)A B /(cid:1)RT , while D (cid:16)gA B and D gA(cid:17) B are
q/3 2q/3 2q/3 q/3 q/3 2q/3
ofidealsolutionandinthe limitofhighlyorderedsolution. them(cid:16)olarGibbsfreeen(cid:17)ergiesofformationofA2q/3Bq/3-species
The modified quasichemical model after Pelton and Blan- andA B -species,respecively.
q/3 2q/3
der [5] results in paradoxically small, negative values of the SubstitutingEq.(12)intoEq.(11)andtakingthelimitwhen
configurationalentropyinthelimitofimmisciblecomponents. xA→0,oneverifiesthat
ThemodifiedquasichemicalmodelafterPeltonetal.[6]gives 3
paradoxicallysmall,negativevaluesinthelimitofimmiscible lim(daB/dxA)=− . (13)
xA→0 q
componentsandin thelimitofhighlyorderedsolution.
Thisdemonstratesthat,foranyfinite valuesof D g and
Themodifiedassociateformalism[1]andtheclassicalqua- A2q/3Bq/3
D g ,chemicalactivitya givenbytheassociatespecies
sichemical model [4] give theoretically possible values of the Aq/3B2q/3 B
modelvariesas(1−x )onlywhenq=3.Inthelimitofhighly
configurational entropy in all considered limits. The classical A
orderedsolution(D g →−¥ ),a variesas(1− 3 x )in
quasichemicalmodelalsocorrectlyreducestothetheoretically A2q/3Bq/3 B 2q A
thevicinityofx =0.
expectedcurveinthelimitofimmisciblecomponents.Theap- A
It is also important to note that, in the framework of the
plicability of the classical quasichemical model, however, is
associate species model, one can consider the species A , B
limited. q q
andA B only,sincethesetofspeciestobeconsideredis
2q/3 q/3
determinedby a modeller. Formally setting D g =+¥ ,
A B
3. Dilute Solutions q/3 2q/3
Eq.(12)reducesto
Pelton et al. [6] consideredbinarysolution, in which maxi- dxA =−1−1e t2+2e 1t 1+13e 1t2 . (14)
mumorderingoccursatthe composition xB=1/3inthelimit dxBq 9 1 9 t(cid:0)2+23e 1t (cid:1)
ofhighlyorderedsolution(Gibbsfreeenergychangeonform-
SubstitutionofEq.(14)intoEq.(11)(cid:0)andtaking(cid:1)thelimitwhen
ing(A−B)-pairsapproaches−¥ ).Itwasdemonstratedthat,in
x →0 gives
A
theframeworkofthemodifiedquasichemicalmodel,chemical 3
activity a of the componentB in the varies in the vicinity of lim(daB/dxA)=− . (15)
B xA→0 2q
x =0as (1−x )eveninthislimitingcase.
A A In this case, a varies in the vicinity of x =0 as (1− 3 x )
Similar solution has also been considered in Ref. [1] in the B A 2q A
foranyvalueofD g .
framework of the modified associate formalism. As demon- A2q/3Bq/3
The examples presented above indicate that the set of as-
strated, chemical activity a varies as (1−x ) only for finite
B A
values of D g and D g . Here, D g and D g are the mo- sociate species and the size of species should be carefully se-
2,1 1,2 2,1 1,2
lected, when the associate species model is applied for dilute
lar Gibbs free energy changes on forming A B-associates and
2
solutions.
AB -associates, respectively.Inthelimitofhighlyorderedso-
2
lution (D g →−¥ ), a varies as (1−x /2). Note also that
2,1 B A
the interval in the vicinity of x =0, in which a can be ap- 4. Settingthe CompositionofMaximumOrdering
A B
proximated as (1−x ), decreases with the decrease in D g .
A 2,1
Thisshouldbetakenintoconsideration,whenthemodifiedas- Modelling of multicomponentsystems requires a provision
sociate formalismisusedformodellingdilute solutions. for setting the composition of maximum ordering so as to
162
comply with the experimental data for each sub-system (bi- Acknowledgment
nary,ternary,quaternaryandsoon)individually.Associate-type
modelsallow to dothis in a straightforwardway by including ThepresentworkhasbeensupportedbytheAustralianRe-
ineachsub-systemtheassociates,whichcompositioncoincides searchCouncil.
with that of maximum ordering in a particular sub-system. In
thecaseoftheclassicalassociatemodelortheassociatespecies
References
model,amodellercanarbitrarilyincludetherequiredassociates
or the associate species. In the case of the modified associate
[1] D. N. Saulov, I. G. Vladimirov, A. Y. Klimenko, Modi-
formalism,inwhichthecompositionsofassociatesaredefined
fied associate formalism without entropy paradox Part I:
bytheirsize,oneshouldselecttheadequatelylargesizeofas-
Model Description, Submitted to Journal of Alloys and
sociatestoincludetherequiredcompositions.
Compounds.
Incontrasttoassociate-typemodels,therearenoprovisions
[2] I.Prigogine,R.Defay,ChemicalThermodynamics,Long-
for setting the composition of maximum ordering for each
mans,GreenandCo., 1954.
ternary, quaternary and so on sub-systems individually nei-
[3] T. M. Besmann, K. E. Spear, Thermochemical modeling
therintheframeworkofMQM(1986)norintheframeworkof
ofoxideglasses,JournaloftheAmericanCeramicSociety
MQM(2000). Furthermore, the composition of maximum or-
85(12)(2002)2887–2894.
deringcannotbesetindependentlyevenforbinarysub-systems
[4] R. H. Fowler, E. A. Guggenheim, Statistical Thermody-
intheframeworkofMQM(1986).Toovercomethisdisadvan-
namics,CambridgeUniversityPress, 1939.
tage forbinary sub-systems, Pelton et al. [6] suggested to use
[5] A. D. Pelton, M. Blander,Thermodynamicanalysisof or-
variable coordination numbers given by Eqs. (8). This modi-
dered liquid solutions by a modified quasichemical ap-
fication,however,leadstomathematicalandthermodynamical
proach- application to silicate slags, MetallurgicalTrans-
inconsistencies of MQM(2000).These inconsistencies are de-
actionsB:ProcessMetallurgy17B (1986)805–815.
scribedimmediatelybelow.
[6] A. D. Pelton, S. A. Degterov, G. Eriksson, C. Robelin,
Consider the case, in which the coordination numbers are
givenbyEqs.(8)andtheconstantsZA ,ZB , ZA andZB are Y. Dessureault,The modifiedquasichemicalmodelI - Bi-
AA BB AB BA narysolution,MetallurgicalandMaterialsTransactionsB:
givenbyEqs.(9).Firstly,asdemonstratedinRef.[8],thetotal
ProcessMetallurgyandMaterialsProcessingScience31B
mole number of pairs n depends on the extent of the qua-
tp
(2000)651–659.
sichemical reaction (Eq. (1) in Ref. [6]). More precisely, n
tp
[7] R.Lu¨ck,U.Gerling,B.Predel,Anentropyparadoxofthe
decreasesby1/2molepereachmoleof(A−B)pairsformed.
association model, Zeitschrift fur Metallkunde 80 (1989)
Thisresultcontradictstotheequationofthequasichemicalre-
270–275.
action(seeEq.(1)inRef.[6]),whichunderliesthequasichemi-
[8] D.Saulov,Shortcomingsoftherecentmodificationsofthe
calmodel.AccordingtoEq.(1)inRef.[6],n doesnotchange
tp
quasichemical solution model, Calphad - Computer Cou-
inthequasichemicalreaction.Secondly,whenthecoordination
pling of Phase Diagramsand Thermochemistry31 (2007)
numbersZ andZ varywithn ,theexpressionfortheGibbs
A B AB
390–395.
free energy of the solution given by Eq. (9) in Ref. [6]) is in
error. Using Eqs. (1-3,19,20)in Ref. [6], one verifies that the
correctexpressionin thiscaseis
Z Z n
G= A n g◦+ B n g◦−TD Sconfig+ ABD g . (16)
ZA A A ZB B B 2 AB
AA BB
Here,allthenotationshavethe samemeaningasin Ref.[6].
5. Conclusions
The modified associate formalism has been compared with
similarmodels.Asdemonstrated,theformalismgivestheoret-
icallypossiblevaluesoftheconfigurationalentropyinallcon-
sidered limits. Being an associate-type model, the formalism
allows to set the composition of maximum ordering for each
sub-systemofamulticomponentsystemindividually.Thecor-
rectedGibbsfreeenergyequationofthequasichemicalmodel
isalso presented.
163