SOLID SOLUTIONS

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TOPIC 6

• SOLID SOLUTIONS

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REVIEW OF IDEAL MIXING

• Recall that for an ideal mixture, we have found
  the following relationships

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ANOTHER APPROACH TO THE ENTROPY OF MIXING

• Recall the expression from Boltzmann
• S k ln W
• where W the number of microstates corresponding
  to a single macrostate.
• We can calculate W for the mixing of N objects,
  some of which are indistinguishable
• where Xi is the fraction of objects of type i.

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EXAMPLE KAlSi3O8

• We have 4 tetrahedral sites. How do we arrange 3
  Si4 and 1 Al3 ion over 4 sites with the same
  energy?

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• Combining the Boltmann expression with the
  formula for calculating W we get
• Now, we introduce a mathematical approximation
  called Stirlings approximation.
• Note that, the following equality holds

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• Now, if N is sufficiently large, we can replace
  the summation with an integral, so we have
• Now, we need to integrate by parts
• so
• and

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• For very large N, we can neglect the 1 to get
• and now the entropy becomes

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• If we take N nN0, we get (n is the number of
  distinguishable sites)
• This gives the ideal, configurational entropy,
  assuming all sites are energetically equivalent.
• KAlSi3O8(xl, ordered) ? KAlSi3O8(xl, disordered)
• microcline ? sanidine

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• Measurement shows that the actual ?S 3.5 cal
  K-1 mol-1 which tells us that sanidine is not
  completely disordered.
• Aluminum avoidance principle No two adjacent
  formula units have aluminum in adjacent sites
  (Kerrick and Darken, 1975, GCA 39 1431-1442).
• Using the avoidance principle, the maximum
  configuration entropy should be 2.9 cal K-1
  mol-1.

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IDEAL IONIC MIXING IN OLIVINE

• For ideal mixing we have ?Hmix 0 ?Vmix 0
  ?Smix ?Sconfigurational, max
• Olivine solid solution
• XMg2SiO4 (1-X)Fe2SiO4 ? Mg2X Fe2(1-X)SiO4
• The olivine structure can be represented as
• M2 M1 TO4
• and 1 mole of olivine contains two moles of
  octahedral metal sites.

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• Winitial WforsteriteWfayalite
• Using Stirlings approximation we get

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(No Transcript)
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ACTIVITY OF FORSTERITE IN OLIVINE
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But recall that
So
Or alternatively
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Ternary diagram showing compositions of Fe-Ti
oxides
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ACTIVITY OF Fe-Ti OXIDES

• xFe3O4 (1-x)Fe2TiO4 ?? (Fe3)2x(Fe2)(2-x)Ti(1-x
  )O4
• Total number of sites 3N.

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where (nmag 2nulv) number of Fe2, nulv
number of Ti4, and 2nmag number of Fe3.
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xFe3O4 (1-x)Fe2TiO4 ?? (Fe3)2x(Fe2)(2-x)Ti(1-x
)O4
Note the 4 is a normalization factor to insure
that aulv for pure ulvospinel is unity.
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PYROXENES

• General structural formula for pyroxene
• M2 M1 T2 O6
• M2 Large cation octahedral site. Takes Na,
  Ca2, Mg2.
• M1 Small cation octahedral site. Takes Fe2,
  Fe3, Mg2, Ti4, Al3.
• T Tetrahedral site. Takes Si4 and Al3.
• Commonly used end member pyroxene components
• diopside CaMgSi2O6, hedenbergite CaFeSi2O6,
  Ca-tschermakite CaAl(AlSi)O6, jadeite
  NaAlSi2O6, and acmite NaFe3Si2O6.

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ACTIVITIES OF PYROXENES IN THE IDEAL IONIC MIXING
MODEL
If perfectly ordered.
If perfectly disordered.
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A REAL-LIFE PROBLEM

• Consider the following reaction relevant to a
  quartz eclogite
• NaAlSi2O6 SiO2 ? NaAlSi3O8
• We may wish to plot this reaction boundary in P-T
  space. Quartz is usually pure, but jadeite will
  be a component of a pyroxene solid solution, and
  albite will be a component of plagioclase solid
  solution. We therefore cannot assume that their
  activities are unity we must calculate them.
• For plagioclase, we can assume ideal molecular
  mixing, i.e., aalb ? Xalb.
• For pyroxene we use an ideal ionic mixing model.

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• Suppose we are given the following analysis of
  the pyroxene, determined by electron microprobe,
  in terms of atoms per 6 oxygens

We must determine which cations go into which
sites. This is done in a stepwise fashion. 1)
First apportion all Si on to the T sites. 2)
Next, fill the rest of the T sites with Al, and
put the rest of the Al into M1. 3) Put Ca and Na
into M2.
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• 4) Put sufficient Mg into M2 to fill it up.
• 5) Put Ti, Fe, Mn and the remaining Mg into M1.
• This results in the following
• T - 1.788 Si 0.212 Al (2.00)
• M1 - 0.034 Al 0.088 Ti 0.240 Fe Mn 0.005
  0.633 Mg (1.00)
• M2 - 0.873 Ca 0.050 Na 0.077 Mg (1.00)

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MOLECULAR VS. IONIC MIXING
M1
Mg
Al
Consider the simplified hypothetical cpx shown to
the right
M2
Ca
Na
Di
Jd
0.7

• Molecular mixing - assumes short-range order and
  that molecules of diopside and jadeite actually
  exist in the solid solution.
• Ionic mixing - Assumes no short-range order.
  Considers the system to be a random mixture of Mg
  and Al on M1, and Na and Ca on M2.

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• Unless there is evidence for short-range order,
  it is best to use mixing on sites (ionic model).
• For example, charge-balance considerations may
  force short-range ordering. This occurs in the
  plagioclase feldspars
• NaAl Si3O8 CaAl2Si2O8
• aalb Xalb
• aan Xan
• Thus, we need to know the structure of minerals
  before we can apply activity models.

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IONIC MIXING ACTIVITY MODEL FOR MICAS

• General structural formula for micas
• A M1 (M2)2 T4 O10 V2
• A alkali (interlayer) site - takes Na, K,
  Rb, Ca2, Ba2.
• M1 octahedral site - takes Li, Fe2, Mg2 and
  vacancies ().
• M2 octahedral site - takes Fe2, Fe3, Cr3,
  Mg2, Al3, Ti4, Mn2.
• T tetrahedral site - takes Si4, Al3.
• V volatile site - takes OH-, F-, Cl-, O2-.

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COMMONLY RECOGNIZED MICA END MEMBERS

• Dioctahedral Micas
• Muscovite - KAl2(AlSi3)O10(OH)2
• Paragonite - NaAl2(AlSi3)O10(OH)2
• Margarite - CaAl2(Al2Si2)O10(OH)2
• Mg-Al Celadonite - KMgAlSi4O10(OH)2
• Fe-Al Celadonite - KFe3AlSi4O10(OH)2
• Fuchsite - KCr2(AlSi3)O10(OH)2
• Ferrimuscovite - KFe32(AlSi3)O10(OH)2
• Ferriceladonite - KFe2Fe3Si4O10(OH)2

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• Trioctahedral Micas
• Phlogopite - KMg3(AlSi3)O10(OH)2
• Annite - KFe3(AlSi3)O10(OH)2 - all Fe2
• Fluorophlogopite - KMg3(AlSi3)O10F2
• Oxyannite - KFe2Fe32AlSi3O12
• Zinnwaldite - KLiFeAl(AlSi3)O10(OH)2
• Eastonite - KMg2Al(Al2Si2)O10(OH)2
• Wonesite (Na-phlogopite) - NaMg2Al(Al2Si2)O10(OH)2
  
• Lepidolite - K2Li3Al4Si7O21(OH)3

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• Suppose microprobe analysis of a mica yielded the
  following Na0.03K0.97Mg1.65Fe0.8Al1.5Si2.9O10(OH)
  1.7F0.3
• We need to first assign the cations to various
  sites.
• 1) All Si goes onto the T sites, leaving 4 - 2.9
  1.1 vacancies to be filled by Al.
• 2) Put the remainder of Al (0.4) onto the M4
  site.
• 3) Put all the Na and K onto the A site this
  just fills the A site with none left over.
• 4) Put all OH and F on volatile site.
• 5) The Fe and Mg go onto the M1 and the 2 M2
  sites. The total cations on all M sites is 0.4 Al
  1.65 Mg 0.8 Fe 2.85. This is 0.15 short
  suggesting vacancies on M1.

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• So now we have XKA 0.97 XNaA 0.03 XM1
  0.15 XAlM2 0.4/2 0.2 XAlT 1.1/4 0.275
  XSiT 2.9/4 0.725 XOHV 1.7/2 0.85 and
  XFV 0.3/2 0.15.
• However, we need to make some assumption about
  how Fe and Mg partition among M1 and M2!
• If we assume that Fe and Mg do not show any
  preference between M1 and M2, then we can write
• we also have

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• And
• Solving these three equations results in XFeM1
  0.277 XFeM2 0.261 XMgM1 0.572 XMgM2
  0.539.
• Now suppose we need to calculate the ideal
  activity of muscovite in this mica. We would
  write
• where C is a normalization factor to insure that
  amuscmica 1 for pure end member muscovite. A
  normalization factor different from unity arises
  when you have mixing on the same site, e.g.,
  here, Al-Si mixing on the T site, in the end
  member.

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• So, to calculate the normalization factor, assume
  we have pure muscovite. Then
• amuscmica 1 C(1)(1)(1)2(1/4)(3/4)3(1)2
• 1 C(27/256)
• C 9.48
• Now, the activity of muscovite in the actual mica
  for which we have an analysis is
• amuscmica 9.48(0.97)(0.15)(0.2)2(0.275)(0.725)3(
  0.85)2
• amuscmica 0.00418
• To calculate the activity of annite component in
  the mica, we write

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• To calculate the normalization factor, again
  assume we have pure annite. Then
• aannmica 1 C(1)(1)(1)2(1/4)(3/4)3(1)2
• 1 C(27/256)
• C 9.48
• For the activity of annite in the actual mica we
  write
• aannmica 9.48(0.97)(0.277)(0.261)2(0.275)(0.725)
  3(0.85)2
• aannmica 0.0131

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IONIC MIXING ACTIVITY MODEL FOR AMPHIBOLES

• General structural formula for hornblende
• A (M4)2 (M3) (M2)2 (M1)2 (T1)4 (T2)4 O22 V2
• 1) Al can only go on the T1, but not the T2,
  tetrahedral sites.
• 2) M2 is the smallest octahedral site and is
  preferred by trivalent cations.
• 3) M4 is the largest octahedral site and is
  preferred by alkali and alkaline earth ions.
• 4) A contains Na, K, .

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COMMONLY RECOGNIZED AMPHIBOLE END MEMBERS

• Tremolite - Ca2Mg5Si8O22(OH)2
• Tschermakite - Ca2(Al2Mg3)(Al2Si6O22)(OH)2
• Gedrite - Mg2(Al2Mg3)(Al2Si6O22)(OH)2
• Actinolite - Ca2Fe5Si8O22(OH)2
• Cummingtonite - Mg2Mg5Si8O22(OH)2
• Edenite - NaCa2Mg5(AlSi7)O22(OH)2
• Pargasite - NaCa2(AlMg4)(Al2Si6)O22(OH)2
• Glaucophane - Na2(Al2Mg3)Si8O22(OH)2
• Richterite - Na(NaCa)Mg5Si8O22(OH)2

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IDEAL ACTIVITY OF PARGASITE

• We write
• assuming pure end member pargasite we write
• apargamph 1
• C(1)(1)2(1)2(1/2)(1/2)(1)(1/2)2(1/2)2(1)4(1)2
• apargamph C(1/64)
• C 64

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REVIEW OF IDEAL MIXING

• Recall that for an ideal mixture, we have found
  the following relationships

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Gibbs free energy of dissolution for ideal binary
solutions as a function of temperature from 0 to
2000 K. ?Gideal dissoln and Gideal soln
decrease markedly at higher T, making solutions
more stable. From Anderson and Crerar (1993).
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REGULAR SOLUTIONS

• Consider two ions substituting for one another
  that have significantly different binding
  energies, but have the same charges and ionic
  radii. For such a regular solution we can write

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TRULY NON-IDEAL SOLUTIONS
In general, ?i ? 1.
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Energy-composition diagrams for two-component
mixtures or solutions at 1000 K. (a) Ideal
solution. (b) Similar except enthalpy is
non-ideal. From Anderson Crerar (1993).
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Energy-composition diagrams for two-component
mixtures or solutions. (c) Similar to (b) but at
higher temperature. (d) Similar to (b) except
excess enthalpy is smaller. From Anderson
Crerar (1993).
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A free-energy curve illustrating the range of
compositions for which a mechanical mixture is
more stable than a solution, and vice-versa.
The solvus is also called the binodal curve
T-X diagram corresponding to the above G-X curve.
The shaded area represents the region where a
single solution is stable. Two solutions exist
inside the solvus curve. From Anderson Crerar
(1993).
Tc critical temperature or consolute point
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(a) Schematic isothermal, isobaric G-X plot for a
real solution, showing contributions from
mechanical mixing, ideal mixing and excess
mixing. (b) Schematic G-X plot for the real
solution of part (a). (c) Solvus and spinodal for
immiscible solutions. Here Tc and Xc are the
critical temperature and critical composition,
respectively, of the solvus. From Nordstrom and
Munoz.
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Diagram to illustrate change in ?G with growth of
compositional fluctuations. A crystal of
composition E would become more stable if it
decomposed to e and e at constant T a crystal F
would become less stable if it decomposed to f
and f. Only by nucleating to f can it minimize
its free energy. From Carmichael et al.
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EXCESS THERMODYNAMIC FUNCTIONS
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GIBBS FREE ENERGY OF A REAL SOLUTION
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RELATIOSHIP BETWEEN EXCESS PROPERTIES AND THE
ACTIVITY COEFFICIENT

• Starting with
• We can write for a real solution
• and for an ideal solution
• Now we define

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• This leads to the relationship
• The excess partial molar entropy, enthalpy and
  volume can now be written

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MARGULES EQUATIONSSYMMETRICAL, REGULAR SOLUTIONS

• The main equation for a binary solution is
• GEX WGXAXB WGXA(1-XA) WG(XA - XA2)
• This Margules equation gives the total excess
  free energy per mole as a parabolic function of
  concentration of the two components. It is often
  referred to as the one-parameter Margules
  equation. This equation is symmetrical about the
  11 composition, i.e., XA XB 0.5. This yields
  a symmetrical solvus on a T-X diagram.
• The parameter WG has units of energy it is
  independent of X but dependent on T and P.

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• The Margules parameter WG can be thought of as
  the energy required to interchange a mole of A
  with one of B in the mixture, without changing
  composition.
• If WG gt 0, then molecules A and B prefer to be
  with molecules of the same type if WG lt 0, then
  they prefer to associate with each other.
• Other excess functions
• VEX WVXAXB
• HEX WHXAXB
• SEX WSXAXB

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THE ACTIVITY COEFFICIENT IN TERMS OF WG

• We can write
• But
• Now, inserting XB (1 - XA) and G ?A we
  write
• Now we know that for non-ideal solutions

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• Combining the three previous relations we can
  finally write
• Note that these equations have the form of a
  truncated virial equation. They have the same
  parabolic form as the original expression
• GEX WGXAXB
• The above equation has a very simple and
  convenient form, but relatively few solid
  solutions can be expected to behave so simply.

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RELATIONSHIP OF WG TO THE HENRYS LAW CONSTANT

• Recall that Henrys Law states that
• i Kh,iXi
• ai i/i ?iXi
• i ?iXii
• i Xiie(WG/RT)
• so
• Kh,i ie(WG/RT)

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ASYMMETRIC SOLUTIONS

• Most real, non-ideal solutions will be
  assymetric. One possible solution would be to
  extend the polynomial in a form such as
• RT ln ?i aX2 bX3 cX4
• This approach is called the Redlich-Kister
  expansion and works quite well for many non-ideal
  solutions.
• Another approach is the two-parameter Margules
  equation
• GEX X1(WG2X1X2) X2(WG1X1X2)
• or
• GEX WG2X2 (WG1 - 2WG2)X22 (WG2 - WG1)X23
• The latter has the form of a truncated virial
  equation once again.

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ACTIVITY COEFFICIENTS IN THE TWO-PARAMETER
MARGULES MODEL
Note that, when X1 1, we get
and that, when X2 1, we get
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MARGULES EQUATIONS FOR TERNARY AND HIGHER ORDER
SYSTEMS

• For a ternary symmetric solution we write for the
  excess Gibbs free energy
• GEX WG12X1X2 WG23X2X3 WG13X1X3
• and for the activity coefficients

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• For a quaternary symmetrical solution we write
• GEX WG12X1X2 WG13X1X3 WG14X1X4 WG23X2X3
  WG24X2X4 WG34X3X4
• and
• For a ternary, asymmetric system we have
• GEX (WG23X22X3 WG32X32X2)
• (WG13X12X3 WG31X32X1)
• (WG12X12X2 WG21X22X1)

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SOME PRACTICAL SIMPLIFICATIONS

• It is found experimentally that, in many
  minerals, WG for Fe mixing is small.
• Thus, when we have Ca-Mg-Fe mixing we can make
  the following simplifications
• WFe-Mg ? 0
• WCa-Mg ? WCa-Fe

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THE MOLECULAR PICTURE FOR A BINARY SYMMETRIC
SOLUTION

• Assume that molecules are located on a lattice of
  Na Nb sites that have a coordination number Z.
  When a and b are in separate phases we have
  1/2ZNa nearest-neighbor interactions for a and
  1/2ZNb nearest-neighbor interactions for b.
  Random mixing yields a probability Xa that any
  site contains a molecule of a and Xb that any
  site contains a molecule of b. Let ?aa, ?bb, and
  ?ab represent the energy of a-a, b-b and a-b
  interactions, respectively.

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• We can then write
• Esolution 1/2Z(Na Nb)-1(?aaNa2 ?bbNb2
  ?abNaNb)
• Emech mix 1/2Z(?aaNa ?bbNb)
• WG 1/2ZN0(2?ab -?aa - ?bb)

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Free energy of mixing (Gideal GEX) for binary
solutions of muscovite-paragonite (ms-pg) micas,
calculated for several isotherms at PT 2.07
kbars. At this pressure WGms 13,408 0.71 T
and WGpg 18,210 1.653 T.
From Eugster et al. (1972).
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Calculated miscibility gap (solid line) and
spinodal (dashed line) for muscovite-paragonite
solid solutions. Dots are the experimentally
obtained compositions of coexisting muscovite and
paragonite solid solutions at 2.07 kbars (Eugster
et al., 1972).
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Activities in the muscovite-paragonite system at
2.07 kbars for temperatures of 1000 K and 1200 K.
The temperature 1200 K is above the critical
point of the solvus 1000 K is below the critical
point.
Thus, the dashed lines on the 1000 K isotherm
give the activities of components in the
metastable homogeneous solid solution. Stable
immiscible compositions at 1000 K are Xams 0.18
and Xbms 0.62. Note that at these compositions,
aams abms and aapg abpg.
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Mixing properties of high-temperature alkali
feldspars at 800C and 10 kb. Note the asymmetry
of the ?Gmixsoln curve, despite the apparent
symmetry of the other thermodynamic properties of
the mixture. The reference state is a mechanical
mixture of pure end members at 800C and 10 kb.
After Waldbaum and Thompson (1969).
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Calculated solvus and spinodal for a symmetrical
regular solution together with the corresponding
curves (Waldbaum and Thompson, 1969) for the
high-temperature alkali feldspars at 1 bar.
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Activities of NaAlSi3O8 and KAlSi3O8 at 1000 bars
in high-temperature alkali feldspars plotted as a
function of mole fractions of the two components.
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a) Schematic plot of Greal soln vs. composition
(X2) in a binary compound showing a sequence of
Greal soln at different temperatures. b) A
schematic T-X plot showing the solvus (binodal),
spinodal, the critical point, the critical
temperature and the critical composition.
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ESTIMATING MARGULES PARAMETERS FROM SYMMETRIC
SOLVI

• Consider two phases of composition b and c,
  coexisting within the solvus at 1000 K. At
  equilibrium it is true that
• ?A ?A and ?B ?B
• and also
• ?B ?B RT ln XB WGXA2
• combining these two equations we get
• RT ln XB WG(XA)2 RT ln XB WG(XA)2
• Rearranging and using XA 1 - XB we get
• WG (RT ln (XB/XB)/((1 - XB)2 - (1 - XB)2

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Symmetric solvus in a binary T-X diagram. A and B
refer to the two components and and refer to
the two different phases.
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• Because it is a function of temperature, we have
  to obtain WG at a number of temperatures on the
  solvus. We would normally express WG as a
  function of temperature using linear regression.
• Next we can apply the usual thermodynamic
  relations
• To get WH and WS.

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ESTIMATING MARGULES PARAMETERS FROM ASYMMETRIC
SOLVI

• In this case we have two Margules parameters to
  estimate WG1 and WG2. Consider two phases of
  composition b and c, coexisting within the solvus
  at 1000 K. At equilibrium it is once again true
  that
• ?1 ?1 and ?2 ?2
• but now
• ?1 ?1 RT ln X1
• (2WG2 - WG1)X22 2(WG1 - WG2)X23
• and
• ?2 ?2 RT ln X2
• (2WG2 - WG1)X12 2(WG1 - WG2)X13

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• We now have two equations relating WG1 and WG2
• RT ln X1 (2WG2 - WG1)(X2)2 2(WG1 -
  WG2)(X2)3
• RT ln X1 (2WG2 - WG1)(X2)2 2(WG1 -
  WG2)(X2)3
• RT ln X2 (2WG1 - WG2)(X1)2 2(WG2 -
  WG1)(X1)3
• RT ln X2 (2WG1 - WG2)(X1)2 2(WG2 -
  WG1)(X1)3
• These equations are not nearly as convenient to
  solve as in the symmetrical case!