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

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