Gibbs free energy of mixing analyser

H. K. D. H. Bhadeshia and H. Yan, Queen Mary University of London

Interactive thermodynamic model of a binary regular solution displaying deep demixing energy wells, common tangent phase separation, and the entropic logarithmic singularity that forces the curve to meet both pure-component axes with an infinite vertical slope. The case where unlike atoms attract (tendency for ordering) can be studied by setting the enthalpy of mixing to a negative number. An accurate study might would not assume that the entropy of mixing (for solutions with a finite enthalpy of mixing) is calculated as if the atoms are distributed randomly.

The "well depth enhancer" is an empirical parameter to make the minima in the free energy of mixing curve more prominent for visual observation.

Gibbs free energy of mixing curve with deep demixing wells
ΔGmix (crimson red) ΔHmix (ocean blue) −TΔSmix (solid black)
Thermodynamic boundary slope: d(ΔGmix)/dx Asymptotes: ±∞

Thermodynamics of deep demixing minima and pure component boundary slopes

In a binary regular solution, the molar Gibbs free energy of mixing balances positive enthalpic repulsion (Ω > 0) with negative configurational entropy:

ΔGmix(x) = ΔHmix + (−TΔSmix) = Ω·x(1 − x) + R·T [ x ln x + (1 − x) ln(1 − x) ]

1. Why the two minima are deep and pronounced

When temperature drops below the critical threshold (T < Tc = Ω / (2R)), the curve undergoes a pitchfork bifurcation into a double-well potential. In the optimum demixing regime (T / Tc ≈ 0.65 – 0.85), the configurational entropy contribution remains substantial enough to drag the two equilibrium minima well below the pure-component baseline (ΔGmix < 0), while the repulsive enthalpy maintains a high crest at x = 0.5.

ΔGwell depth = ΔGmix(0.5) − ΔGmix(xmin)

Any overall mixture composition between xα and xβ lowers its total free energy by splitting into two coexisting solutions linked along the dashed horizontal common tangent tie-line.

2. Why the boundary slopes must diverge to infinity

Regardless of how deep the two minima dip or how tall the central demixing hump becomes, the curve never leaves the pure axes at a positive or zero slope. Differentiating the free energy of mixing gives:

d(ΔGmix)/dx = Ω(1 − 2x) + R·T [ ln(x) − ln(1 − x) ]

As composition approaches pure solvent A (x → 0+), the enthalpy derivative approaches a strictly finite constant (Ω), whereas the logarithmic entropy term diverges without bound:

limx → 0+ [ d(ΔGmix)/dx ] = −∞    limx → 1− [ d(ΔGmix)/dx ] = +∞

Because the entropic singularity unconditionally overwhelms any finite enthalpy term at dilute limits, the free energy curve must plunge vertically downwards before curving back upwards into the demixing hump.