Partial Molar Quantities, Activities, Mixing Properties Composition (X) is a critical variable, as...
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Transcript of Partial Molar Quantities, Activities, Mixing Properties Composition (X) is a critical variable, as...
Partial Molar Quantities, Activities, Mixing PropertiesComposition (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
Partial Molar Volume
Example - spinel solid solution
Spinel volumes
Activity Composition Relations:
The Entropy of Mixing in Solid Solutions Contributions
vibrationalmagnetic and electronicconfigurational
For the random mixing of a total of one mole of species over a total of one mole of sites,
∆Smix = -R[xA ln xA + xB ln xB]
and ∆s¯(A) = -R ln xA , ∆s¯(B) = -R ln xB
The thermodynamic activity is defined as
Δµº(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) = ∆h¯ºT(A) - TΔs¯ºT(A) , ∆µºT(B) =
∆h¯ºT(B) - TΔs¯ºT(B)
Chemical Potential = Partial Molar Free Energy
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.
Raoult’s 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 Raoult’s 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.
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
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:
EmG
r
rr
Em xxxRTxfxfxRTG 21212211 lnln
...53ln 212122110221 xxxxxxxf
...53ln 121221210212 xxxxxxxf
Systematics in Mixing Propertieszz; (Davies and Navrotsky 1981)
Size Mismatch and Interaction Parameter
Henry’s Law Regions
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 (α and β) :simultaneous equalities of chemical potential or activities:
µ(A, phase α) = µ(A, phase β)
µ(B, phase α) = µ(B, phase β)
a(A, phase α) = a(A, phase β)
a(B, phase α) = a(B, phase β)
RESULTSFREE ENERGY CURVES
• Complete miscibility• Solvus- phases
derived from same structure and one free energy curve
• Immiscibility resulting from different structures
Phases with Different Structures Partial solid solution can exist among end members of different structure:A with structure “α” and B with structure “β”.
µ(A,α) = µº(A,α) + RT ln a(A,α)
µ(A,β) = µº(A,α) + ∆µ(A,α→β) + RT ln a(A,β)
µ(B,α) = µº(B,β) + ∆µ(B, β→α) + RT ln a(B,α)
µ(B,β) = µº(B,β) + RT ln a(B,β)
The limiting solubilities are given by equating chemical potentials: µ(A,α) = µ(A,β)
µ(B,α) = µ(B,β) The miscibility gap can not close and is not a solvus.
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
Spinodal
Spinodal
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.