Partial Molar Quantities Activities Mixing Properties Composition X

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

Partial Molar Volume

Example - spinel solid solution

Spinel volumes

Activity Composition Relations:

The Entropy of Mixing in Solid Solutions Contributions 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 = -R[x. A ln x. A + x. B ln x. B] and ∆s¯(A) = -R ln x. A , ∆s¯(B) = -R ln x. B

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 [x. A ln x. A + x. Bln x. B] ∆µº (A) = RT ln x. A , ∆µº (B) = RT ln x. B a(A) = x. A , a(B) = x. B. 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 Ba. SO 4 -Ra. SO 4 solid solutions, Fe. Cr 2 O 4 -Ni. Cr 2 O 4 spinel solid solutions if one assumes Ni and Fe mix on tetrahedral sites only, and UO 2 -Pu. O 2 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 = x. Ax. B = WH x. Ax. B

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 x 1 or x 2 approach unity: where the coefficients r are called interaction parameters. Activity coefficients can be obtained by the partial differentiation of over the mole fraction x 1 or x 2:

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/2 R 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)

FREE ENERGY CURVES RESULTS • 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

Zn. O – Co. O solid solutions If the surface energy in wurtzite phase is smaller than in rocksalt, wurtzite will be favored at the nanoscale. Solid solubility of Zn. O in rocksalt will decrease while that of Co. O in wirtzite will increase Relevant to Chencheng Ma thesis work

Spinodal

Spinodal

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