Partial Molar Quantities, Activities, Mixing Properties

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Partial Molar Quantities, Activities, Mixing
Properties

• Composition (X) is a critical variable, as well
  at temperature (T) and pressure (P)
• Variation of a thermodynamic parameter with
  number of moles of one component, all other
  compositional variables, T, P held constant

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Partial Molar Volume
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Example - spinel solid solution
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Spinel volumes
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Activity Composition Relations
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The Entropy of Mixing in Solid Solutions   Contrib
utions vibrational magnetic and
electronic configurational   For the random
mixing of a total of one mole of species over a
total of one mole of sites,   ?Smix -RxA
ln xA xB ln xB   and   ?s(A) -R ln
xA , ?s(B) -R ln xB  
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The thermodynamic activity is defined as   ?µº(A)
RT ln a(A) and ?µº(B) RT ln a(B)
The changes in chemical potential on
mixing can be related to partial molar enthalpies
and entropies of mixing   ?µºT(A) ?hºT(A) -
T?sºT(A) , ?µºT(B) ?hºT(B) - T?sºT(B)
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Chemical Potential Partial Molar Free Energy
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If the enthalpy of mixing is zero, then the
simple ideal solution results,  ?Gºmix RT xA
ln xA xBln xB   ?µº (A) RT ln xA ,
?µº (B) RT ln xB   a(A) xA , a(B)
xB. Raoults Law holds only when one mole
total of species being mixed randomly, so always
introduces a microscopic meaning you can not get
away from. Thus Raoults Law applies directly to
BaSO4-RaSO4 solid solutions, FeCr2O4-NiCr2O4
spinel solid solutions if one assumes Ni and Fe
mix on tetrahedral sites only, and UO2-PuO2 solid
solutions if there is no variation in oxygen
content.
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Regular, Subregular and Generalized Mixing
Models   Starting point useful but inherently
contradictory assumption that, though the heats
of mixing are not zero, the configurational
entropies of mixing are those of random solid
solution.   ?Gºmix, ex ?Hºmix-T?Sºmix, ex
  For a two-component system, the
simplest formulation is  ?Gexcess ?Hmix
?xAxB WH xAxB
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Generalization

• For a binary system, the Guggenheim or
  Redlich-Kister based on a power-series expression
  for the excess molar Gibbs energy of mixing
  which reduces to zero when either x1 or x2
  approach unity
• where the coefficients ?r are called
  interaction parameters. Activity coefficients can
  be obtained by the partial differentiation of
  over the mole fraction x1 or x2

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Systematics in Mixing Propertieszz (Davies and
Navrotsky 1981)
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Size Mismatch and Interaction Parameter
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Henrys Law Regions
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IMMISCIBILITY Immiscibility (phase separation)
occurs when positive WG terms outweigh the
configurational entropy contribution. For the
strictly regular solution, the miscibility gap
closes at a critical point or consolute
temperature. T WH/2R Conditions for
equilibrium between two phases (a and ß)
simultaneous equalities of chemical potential
or activities   µ(A, phase a) µ(A, phase
ß) µ(B, phase a) µ(B, phase
ß)   a(A, phase a) a(A, phase
ß) a(B, phase a) a(B, phase ß)
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RESULTS

• FREE ENERGY CURVES

• Complete miscibility
• Solvus- phases derived from same structure and
  one free energy curve
• Immiscibility resulting from different structures

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Phases with Different Structures   Partial
solid solution can exist among end members of
different structureA with structure a and B
with structure ß.   µ(A,a) µº(A,a) RT ln
a(A,a)   µ(A,ß) µº(A,a) ?µ(A,a?ß) RT
ln a(A,ß) µ(B,a) µº(B,ß) ?µ(B,
ß?a) RT ln a(B,a) µ(B,ß) µº(B,ß) RT
ln a(B,ß) The limiting solubilities are
given by equating chemical potentials   µ(A,a)
µ(A,ß) µ(B,a) µ(B,ß) The
miscibility gap can not close and is not a
solvus.
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ZnO CoO solid solutions
If the surface energy in wurtzite phase is
smaller than in rocksalt, wurtzite will be
favored at the nanoscale. Solid solubility of ZnO
in rocksalt will decrease while that of CoO in
wirtzite will increase
Relevant to Chencheng Ma thesis work
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Spinodal
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Spinodal
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References

• Guggenheim, E.A., Thermodynamics An advanced
  treatment for chemists and physicists. 5th edn.
  Amsterdam North-Holland 390,1967
• Thompson, J.B., Thermodynamic properties of
  simple solutions. In Researches in geochemistry.
  Edited by Abelson PH. New York John Wiley and
  Sons 1967 340-361.
• Thompson, J.B., Chemical reactions in crystals.
  Amer. Mineral., 54 (1969) 341-375.
• Eriksson, G., Rosen, E., Thermodynamic studies of
  high temperature equilibria. VIII General
  equations for the calculation of equilibria in
  multiphase systems. Chemica Scripta, 4(4) (1973)
  193-194.
• Pelton, A. D., Bale, C. W., Computational
  techniques for the treatment of thermodinamic
  data in multicomponent systems and the
  calculation of phase equilibria. Calphad, 1(3)
  (1977) 253-273.
• Wood, B.J., Nicholls, J., The thermodynamic
  properties of reciprocal solid solutions.
  Contributions to Mineralogy and Petrology 66
  (1978) 389-400.
• Nordstrom, D.K., Munoz, J.L., Geochemical
  thermodynamics. 2nd edn. Boston Blackwell
  Scientific Publications (1994) 483.
• Ott, J.B., Boerio-Goates, J., Chemical
  thermodynamics Advanced applications. San Diego
  CA Academic Press, 438 (2000).
•  Ott, J.B., Boerio-Goates J., Chemical
  thermodynamics Principles and applications. San
  Diego CA Academic Press, 664 (2000).
• Ganguly, J., Thermodynamic modelling of solid
  solutions. In EMU Notes in Mineralogy Solid
  solutions in silicate and oxide systems of
  geological importance. Edited by Geiger CA.
  Budapest Eotvos University Press, 3 (2001)
  37-69.
• Geiger, C.A., Solid solutions in silicate and
  oxide systems of geological importance. In
  European mineralogical union notes in mineralogy.
  Edited by Papp G, Weiszburg TG. Budapest Eotvos
  University Press, 3 (2001) 458.