a) b - École Polytechnique de Montréal

Equation 2-1.
G = H − TS
where H is enthalpy, T is absolute temperature, and S is entropy. The necessary condition for
two substances A and B to be miscible is that the Gibbs free energy of the mixture must be lower
than the sum of the Gibbs free energies of the separate constituents, or in other words,
Δ mix
G < 0 . Therefore, the necessary condition can be extended to:
Equation 2-2. G = ΔH
− TΔS
< 0
Δ mix
mix
mix
where ΔH is the enthalpy of mixing and mix
Δ S is the entropy of mixing. Enthalpy and entropy
mix
of mixing have been modeled extensively. The combinatorial entropy is the most important
factor leading to miscibility in low molecular weight materials. For higher molecular weight
components, the TΔ S term is small and the enthalpy of mixing can dominate the miscibility.
mix
However, the Equation 2-2 is not a sufficient requirement, as the following expression must also
be satisfied:
2 ⎛ ∂ ΔG
⎞
Equation 2-3. ⎜ m ⎟
⎜
> 0
2 ⎟
⎝ ∂φi
⎠
Negative value of this equation can yield a region of the phase diagram where the mixture
separates into a phase rich in component A and another phase rich in component B(Denbigh,
1971). The entropy of mixing for dissimilar components can be determined(Clausius, 1864) with
the Boltzmann relationship as:
T , P
Equation 2-4. Δ = k ln Ω
S m
k is the Boltzmann constant and Ω represents the number of configurations or the summation of
combinations of arranging N1 and N2 molecules into a regular lattice of N cells:
Equation 2-5.
Ω
=
( N + N )
1
N ! N
1
2
2!
!
10