Thermodynamics of Mixing in Solids

PA CANDELA

GEOLOGY 614

Consider the binary system
FeCO_{3}-MgCO_{3}. 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, w_{IE}, 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 N_{o} = 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, w_{IE}=0,
W_{ij}=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:

where

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, G^{ex}=H^{ex}, 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.