# Thermodynamics of solid solutions

For an introduction to the basics of thermodynamics, look at the Phase Diagrams and Solidification TLP.

### Entropy

The entropy of the two endmembers, A and B, of a solid solution are S_{A} and S_{B}, and are mainly vibrational
in origin (i.e. related to the structural disorder caused by thermal vibrations of the atoms at finite temperature). The
entropy of the solid solution will always be greater than the entropy of the mechanical mixture.

The entropy of the mechanical mixture is given by:

S = x_{A}S_{A} + x_{B}S_{B}

The excess entropy is called the *entropy of mixing* (ΔS_{mix}), and
is mainly configurational in origin (i.e. it is associated with the large number of energetically-equivalent ways of arranging
atoms/ions on the available lattice sites).

The configurational entropy is defined as:

S = k lnW

where k is Boltzmann's constant (1.38 x 10^{-23} JK^{-1}), and W is the number of possible
configurations.

If it is assumed that the entropy of mixing is equal to the configurational entropy,

ΔS_{mix} = k lnW

If we consider mixing N_{A} A atoms and N_{B} B atoms on N lattice sites at
random, then the number of different configurations of A and B cations is given by:

\[w = \frac{{N!}}{{{N_A}!{N_B}!}} = \frac{{N!}}{{({x_A}N)!({x_B}N)!}}\]

where x_{A} and x_{B} are the mole fractions of A and B respectively.

Hence

\[\Delta {S_{{\rm{mix}}}} = k\ln \frac{{N!}}{{({x_A}N)!({x_B}N)!}} = k[\ln N! - \ln ({x_A}N)! - \ln ({x_B}N)!]\]

Stirling's approximation states that, for large N:

lnN! ≈ N lnN − N

Hence

ΔS_{mix} = k(N lnN
- N) - k(x_{A}N lnx_{A}N
- x_{A}N) - k(x_{B}N lnx_{B}N
- x_{B}N)

ΔS_{mix} = -*Nk*(*-*lnN
+ 1* + **x*_{A}ln*x*_{A} + *x*_{A}lnN -
*x*_{A} + x_{B}lnx_{B} + x_{B}lnN -
x_{B})

ΔS_{mix} = -*Nk*(x_{A}lnx_{A}
+ x_{B}lnx_{B} + (x_{A} + x_{B})lnN
- (x_{A} + x_{B}) - lnN
*+ *1)

Since x_{A} + x_{B} = 1,

ΔS_{mix} = -*Nk*(x_{A}lnx_{A}
+ x_{B}lnx_{B})

If N is taken to be equal to Avogadro's number, then per mole of sites:

ΔS_{mix} = -*R*(x_{A}lnx_{A}
+ x_{B}lnx_{B})

### Enthalpy

The enthalpy of the two endmembers of the solid solution, A and B, are equal to H_{A} and H_{B}
respectively. For a mechanical mixture of these two endmembers, the enthalpy is given by:

H = x_{A}H_{A} + x_{B}H_{B}

The excess enthalpy relative to the mechanical mixture is known as the *enthalpy of mixing*, ΔH_{mix}.
This can either be positive or negative, or zero.

If ΔH_{mix} = 0, the solution is said
to be ideal, and for ΔH_{mix} ≠ 0,
the solid solution is said to be non-ideal.

A simple expression for the enthalpy of mixing can be derived by assuming that the energy of the solid solution arises only from the interaction between nearest-neighbour pairs.

Let *z* be the coordination number of the lattice sites on which mixing occurs. If the total number of sites is
N, then the total number of nearest-neighbour bonds is 0.5*Nz.* (The factor of 0.5 arises since there are
two atoms/ions per bond).

Let the energy associated with A-A, B-B and A-B nearest-neighbour pairs be W_{AA}, W_{BB}
and W_{AB} respectively. If the cations are mixed randomly, then the probability of A-A, B-B and A-B neighbours
is x_{A}^{2}, x_{B}^{2} and 2x_{A}x_{B}
respectively.

Hence the total enthalpy of the solid solution is given by:

H = 0.5 *Nz*(x_{A}^{2}W_{AA}
+ x_{B}^{2}W_{BB} + 2x_{A}x_{B}W_{AB})

This can be rearranged to:

H = 0.5 *Nz*(x_{A}W_{AA} +
x_{B}W_{BB}) + 0.5 *Nz*x_{A}x_{B}(2W_{AB}*
*-W_{AA} -W_{BB})

The first term in this expression is equal to the enthalpy of the mechanical mixture. Hence:

ΔH = 0.5 *Nz*x_{A}x_{B}(2W_{AB}*
*-W_{AA} -W_{BB}) =
0.5 *Nzx*_{A}x_{B}W

where W (= 2W_{AB}* *-W_{AA}
-W_{BB}) is known as the *regular solution interaction parameter*,
and its sign determines the sign of ΔH_{mix}.

A positive value of W indicates that it is energetically more favourable to have A-A and B-B neighbours, rather
than A-B neighbours. In order to maximise the number of A-A and B-B neighbours, the solid solution unmixes into A-rich and
B-rich regions. This process is called *exsolution*.

A negative value of W indicates that it is energetically more favourable to have A-B neighbours, rather than A-A or B-B neighbours. To maximise the number of A-B neighbours, the solid solution forms an ordered compound.

View a simple representation of exsolution and cation ordering