Thermodynamics of Mixing in Solids




Consider the binary system FeCO3-MgCO3. The mixing of these two components in a carbonate solid solution is nearly ideal. Therefore, to a first approximation, Fe and Mg distribute themselves randomly over the available octahedral sites (assuming all sites are equally likely sites for Fe or Mg regardless of nearest neighbors). Further, in an ideal solid solution, no volume distortion occurs (i.e., the molar volume of all solution compositions is a LINEAR combination of the molar volume of the end members), and the vibrational (potential and kinetic) energy of an Fe (or Mg) atom in a 6-coordinated oxygen site surrounded by six OTHER Fe (or Mg) is a linear combination of the energies of the 6-coordinated oxygen site surrounded by six Mg (or Fe) atoms in the end members. If we represent these energies by a lower case w, then we can define, for the ideal mixing case:



For a solution that deviates from ideality for energetic reasons, equation 1 IS AN INEQUALITY, and an interaction energy, wIE, defined by the expression



is non-zero.

For one mole of a crystalline solution, we define an interaction parameter



where z = # of metal cation sites around any given cation, and No = Avagadro's number.

 For what is called a "Regular Binary Solution",




where we define two interaction parameters because, usually, the internal energy corrections are different for solid solutions rich in 1 vs. 2. A regular Solution is "symmetrical", when W1=W2 (In this case, a simpler formulation, Hex = WX1X2, is used).


For an ideal solution, wIE=0, Wij=0, and




In a regular solution,



General equations for non-ideal solutions: free energy and chemical potential

In terms of chemical potential and free energy:



Note that the free energy of mixing comprises the last two terms on the right-hand side, which can also be written as:






The first term on the right-hand side is the ideal entropy of mixing {which is all configurational (mixing of white and brown eggs in an egg carton)}.Note also that






The excess free energy is given by:



Note that:






For a regular solution, Gex=Hex, so we can write,





remember that





S ex is usually vibrational in character (the distribution of atoms/molecules in energy space is NOT a linear function of the distributions in the pure end members when excess entropy is non-zero), although, strictly it can also contain non-ideal, configurational (ordering) contributions to the entropy (these are always negative, because the ideal configurational entropy of mixing is the MAXIMUM configurational entropy of mixing). The excess vibrational (or in liquids and gases - rotational or translational) entropies may be positive or negative. Highly ordered liquid solutions may be "athermal solutions" wherein most of the contribution to the excess free energy is from the excess entropy rather than the excess enthalpy.