Fit the data.
Test the phases.

The headline result has the lowest objective among converged starts. No stability-based filtering is used to hide a poor physical result. The composition error is calculated afterward from globally supporting binodals. A missing error means no verified branch was available.

Bounded Nelder–Mead fits τ₁₂ and τ₂₁ with fixed α. The simplex must shrink below 10⁻⁸ in parameter distance within 500 iterations. This stopping rule does not prove the global parameter optimum. Bounds and convergence status are reported, along with the full iteration history.

This is an unweighted chemical-potential residual fit, not a maximum-likelihood fit of composition measurements. Composition noise produces unequal, correlated residual errors after the logarithms. Experimental expanded uncertainties are displayed, not used as weights. The separately reported composition RMSE is in mole-fraction units.

NRTL convention: Gᵢⱼ=exp(−ατᵢⱼ), with α₁₂=α₂₁. Parameters apply only at the selected temperature. This lab neither reproduces the article’s temperature-dependent regression nor extrapolates it. See Lab 04 for the binary equations and global tangent solver.

One binary coexistence pair gives two compositions. With fixed α, two τ values can often reproduce it almost exactly. This is calibration, not independent validation. Repeated synthetic measurements reduce noise but do not add new equilibrium states. Changing α can produce different parameter values with similarly small composition errors.

The residual-Jacobian singular-value ratio describes local sensitivity with α fixed. A large ratio suggests a weakly constrained parameter direction. It is not a confidence interval, and the seven optimizer starts are not bootstrap samples. No uncertainty band is fabricated from them.

TPD at a noisy observed composition may be negative under the fitted model. Compare it with TPD at the model’s predicted coexisting liquid. Both predicted phases must lie on a globally supporting common tangent; chemical-potential equality by itself is insufficient. A nonnegative result is numerical evidence within the 10⁻¹⁰ TPD tolerance.

Experimental source: Atik and Kerboub, J. Chem. Eng. Data (2008), 53, 1669–1671. DOI: 10.1021/je8001688. Numerical values: NIST ThermoML, datasets 6 and 7. Nine reported temperature pairs from 278.15 to 316.15 K are selectable. This page uses cyclohexane as component 1, with α the cyclohexane-poor liquid. ThermoML uncertainties are compiler-evaluated expanded uncertainties at 95% confidence.

Data provenance, original phase labels and source-file SHA-256 are retained in the teaching data file. ThermoML numerical data are provided under the NIST license. The complete article is not redistributed.

The synthetic reference uses τ₁₂=3, τ₂₁=2, α=0.3. It is a hypothetical liquid, not cyclohexane–methanol. Data and parameter bounds are intentionally restricted to this binary teaching problem. No vapor, VLLE, temperature law, ternary fit or experimental data-consistency verdict is calculated.

Regression source · Regenerate figures (Python). Export retains observations, noise settings, all starts, selected parameters, stability diagnostics, curves and your explanation. All calculations run locally in your browser.