One data set.
Different questions.

1. Redlich–Kister area criterion

For an isothermal path with negligible excess-volume pressure contribution, I should vanish. We report the empirical D < 5% criterion summarized by NIST. Piecewise-linear integration splits segments at each zero crossing. D is undefined if both area magnitudes vanish; this is not automatically a failure. Whole-range area cancellation is necessary under these assumptions, but cannot locate errors that cancel.

2. Herington empirical isobaric criterion

Temperatures are in kelvin. The boiling-range term J is an empirical allowance for temperature variation, not a measured excess-enthalpy correction. This is the D−J variant summarized by NIST, not its complete TDE decision tree with the alternative near-ideal |A*| criterion. Real-data results that extrapolate endpoints are explicitly conditional; no extrapolation means no full-area verdict.

3. Differential Gibbs–Duhem diagnostic

Three-point Lagrange derivatives use interior points and their actual composition spacing. The teaching threshold is score < 5. This illustrates the differential identity; it is not the NIST/Kojima point test using a fitted Padé slope. Local numerical differentiation is sensitive to resolution and noise. For isothermal data, the temperature term is zero and the excess-volume pressure term is neglected. For real isobaric data without Hᴱ, the raw residual is shown without a verdict.

4. Van Ness-type modeling capability

We fit a consistent NRTL model to P and y jointly, at observed T and x, with α fixed at 0.47. Isothermal fits use two constant τ values. Isobaric fits use four parameters: τᵢⱼ = cᵢⱼ + dᵢⱼ(350/T − 1). We minimize mean [(100 ΔP/P)² + (100 Δy)²] using five-start bounded Nelder–Mead, c ∈ [−3,8], d ∈ [−40,40]. Pure endpoints are excluded.

This teaching variant uses the NIST-reported Δp and Δy thresholds with a different parameterization from TDE's five-parameter model. Meeting both limits establishes reproduction by the chosen model. Missing either can reflect model flexibility, property correlations or measurement error; it does not identify the cause. Fits at parameter bounds are flagged. No measurement weighting or parameter confidence intervals are claimed.

For the synthetic data, add (1−x₁)f(x₁) to ln γ₁ and −x₁f(x₁) to ln γ₂. Their weighted sum, gᴱ/(RT), is unchanged, while ln(γ₁/γ₂) increases by f. This intentionally breaks the partial-molar relationship when f ≠ 0.

Uniform shift: f = amplitude. Canceling errors: f = amplitude cos(2πx₁), whose full-range integral is exactly zero. Local defect: f = amplitude exp[−((x₁−0.35)/0.035)²]. In the canceling case, the area test can meet its criterion while local residuals remain nonzero.

P and y are generated from these perturbed activities using the same ideal-vapor VLE equations. Isobaric T is solved at 60 kPa. Synthetic Hᴱ is derived from the baseline NRTL model; the weighted perturbations cancel, so that generating excess enthalpy is unchanged. This is supplied simulation information, not calorimetry inferred independently from VLE.

NIST: area and Herington tests · NIST: differential/point test · NIST: Van Ness modeling test.

Article: Barbieri, De Guido and Moioli, J. Chem. Thermodynamics 198 (2024), 107342, Tables 4–7. The authors' consistency result is separate from every result computed here. See Lab 02 for original observations, uncertainty, Antoine coefficients and parameter conventions.

Ideal vapor, negligible Poynting correction and a pressure-independent liquid excess Gibbs model are assumed. Joint vapor-pressure domain: 334–362.41 K. Original measurements outside it stay visible but are excluded from inferred activities and modeling. No original article values are altered. Endpoint assumptions never supply missing experimental Hᴱ.

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