Three phases.
One tangent.

Equality checks use every phase on the selected coexistence line, including a phase with zero selected amount at a boundary. Global support tests both the liquid and vapor Gibbs curves over the full composition interval. Small equality residuals alone are insufficient.

At a binary three-phase state, fixed T fixes the equilibrium P and all phase compositions. The phase rule concerns intensive variables; it does not supply a missing balance for the amounts. For a feed strictly inside the span of the phase compositions, the feasible fractions generally form a line segment. An additional independent constraint, such as total enthalpy or volume with suitable property models, is needed to choose a unique point.

The slider interpolates between the two feasible endpoints, ordered by vapor fraction. It is a freely chosen assignment, not a calculated heat input, flash path or time evolution. Every point has the same total Gibbs energy and satisfies the balances. At the endpoints, one phase can disappear. For a feed at an extreme phase composition, the feasible set can collapse to a single point.

If the vapor composition lies outside the two liquid compositions, some feeds require a nonzero minimum vapor fraction. This is why the feasible interval is calculated from all three phase compositions. At a homogeneous azeotrope, liquid and vapor have the same composition, so their fractions can also be indeterminate at saturation.

Here g is molar Gibbs energy divided by RT, after subtracting the same pure-liquid standard-state linear contribution from both phases. This vapor curve is not just ideal mixing entropy: the pressure-dependent standard-state offsets are essential. The model assumes ideal vapor fugacity coefficients and neglects liquid pressure corrections (Poynting factors).

For a globally supporting liquid common tangent, define aᵢ=xᵢᵅγᵢᵅ=xᵢᵝγᵢᵝ. Then:

For partially miscible liquids, P* is generally not P₁sat+P₂sat. The latter is the limiting result for completely immiscible liquids with unit activities in their respective pure-liquid phases. Above P*, the vapor lies above this liquid tangent. Below P*, a vapor state lies below it and destabilizes that LLE construction; the actual equilibrium is recalculated for the selected feed.

Margules conventions and liquid spinodal calculations are inherited from Lab 04. Equal A₁₂=A₂₁ recovers the symmetric model. NRTL uses dimensionless τ₁₂, τ₂₁ and symmetric α at the selected fixed temperature. All distinct globally supporting liquid gaps are retained; select a gap to explore its three-phase pressure.

At the chosen pressure, all liquid–vapor candidates solve P=xγ₁P₁sat+(1−x)γ₂P₂sat. Every spinodal and homogeneous azeotrope partitions this bubble function into monotone intervals. For Margules, the azeotrope condition is quadratic. For NRTL, the derivative of ln(γ₁/γ₂) has at most one internal zero, obtained analytically from its positive cubic denominators. These stationary points bracket every isolated azeotrope, including a tangent root. Each bracket is searched, then any candidate whose tangent fails global support is discarded. The liquid–liquid tangent is retained only if the vapor cannot undercut it.

For a line ℓ(w)=μ₂/RT+(μ₁−μ₂)w/RT, the liquid-distance minimum is evaluated at both endpoints and every root g′ᴸ(w)=ℓ′. The ideal-vapor minimum is analytical:

The lowest feasible Gibbs energy across single phases and supported coexistence lines determines the result. Support tolerance is 10⁻⁹ in g; pressures within 10⁻¹⁰ in ln(P/P*) are treated as numerically coincident with VLLE. The diagram samples these verified constructions; it does not determine phase compositions by grid interpolation. Near-degenerate liquid states that cannot be verified are reported as unresolved.

Pure feeds are handled with their own saturation condition. A full-mixture tangent/TPD is not evaluated there because the absent-component chemical potential diverges. The present component's fugacity equality still applies.

All curves, supported phase candidates, feasible fraction endpoints and fugacity checks are retained in the JSON export. Calculation source · Regenerate figures (Python).

Background: LearnChemE: immiscible-liquid phase diagrams; Howard DeVoe: Gibbs phase rule.

This is a synthetic, isothermal model study. No experimental vapor pressures, physical temperature sweep, energy balance, volume closure, kinetics or distillation design is claimed.