Appendix: Experimental VLE, consistency and fitting
Session 2 · optional reference
30 September 2026
Optional reference material
Return to the 12-slide classroom deck.
Use these details when a question from your investigation needs them. They are outside the required 45–60 minute practice.
Choose one extension: a controlled consistency defect in Lab 03, a fixed held-out comparison, bootstrap fit spread, or your own sourced VLE data. Exact synthetic bootstrap spread does not quantify real measurement uncertainty.
The measurement-to-model chain
Measured T, P, x and y → pure-component vapor pressures → inferred activity coefficients → consistency diagnostics → regression → prediction within a stated domain.
Each arrow adds assumptions. A fitted curve does not remove errors introduced earlier.
Antoine inputs are part of the model
A common form is \log_{10}P^{sat}=A-\frac{B}{T+C}
A, B and C belong to a specific equation, logarithm base, T unit and pressure unit. Lab 08 supports several forms; changing the selection does not convert constants.
Check Pˢᵃᵗ at one measured temperature and verify the source range.
Gibbs–Duhem constrains activities
At constant T and P for a binary liquid, x_1\,d\ln\gamma_1+x_2\,d\ln\gamma_2=0
A VLE path may change P or T. Pressure/excess-volume and temperature/excess-enthalpy terms need appropriate treatment or an explicit approximation.
Area cancellation is not a local test
Isobaric data need temperature information
At approximately constant P, \sum_i x_i\,d\ln\gamma_i=-\frac{h^E}{RT^2}\,dT
Independent excess-enthalpy data can support the temperature correction. An empirical range correction does not replace measured hᴱ.
Missing endpoints require an explicit extrapolation assumption.
Fitting objective and parameter convention
Lab 08 minimizes the mean of \left(100\frac{P_{pred}-P_{obs}}{P_{obs}}\right)^2+[100(y_{pred}-y_{obs})]^2
Compare one-parameter Margules, two-parameter Margules and NRTL on the same rows. NRTL α is fixed. This objective has no measurement-uncertainty weighting.
Residual structure matters
A lower scalar error may accompany biased residuals near an endpoint or a narrow temperature interval.
Compare residuals against composition and temperature. Report parameter bounds and temperature convention.
Lab 08 compares three models on one fixed held-out split. Its pairs bootstrap reports conditional fit spread, not a new-measurement prediction interval.
A fitted model has a domain
A single-temperature fit supplies composition dependence at that temperature. It does not establish dγ/dT.
A VLE fit need not identify LLE accurately. Similar VLE residuals can conceal different liquid Gibbs-energy curvature.
Session 3 adds stability and liquid coexistence evidence.
Sources and further reading
Module reference deck · Lab assumptions and sources
The worked examples use synthetic inputs. Each lab states its supported models and validity limits.