Lab 03 · Graduate investigation

What can a consistency test actually establish?

An experimental VLE data set gives a small integrated area. Is that enough to trust it? Establish a synthetic control, construct a counterexample, then defend a limited conclusion about measured data.

Complete Steps 1–5. Submit numerical comparisons, an assumption audit, and a decision that distinguishes consistency evidence from measurement accuracy.

2027 prototype
Review the student workflow

Before you start: which data will you use?

No experimental file is needed for this lab. Use the supplied cases to learn what each consistency test can detect. Start with case A, compare B and C, then inspect the published measurements in Step 4.

Component 1 is 2-propanol (isopropanol); component 2 is water. x₁ and y₁ are the mole fractions of 2-propanol in the liquid and vapor, respectively. T is in K and P is in kPa.

Cases A–C · Generated practice data

The generator uses the article's NRTL parameters and NIST Antoine correlations. These are simulated states, not laboratory measurements.

A: fix T = 350 K; calculate P and y₁ at 81 liquid compositions.
B: alter the generated activities deliberately, then recalculate P and y₁.
C: fix P = 60 kPa; solve T and y₁ at each liquid composition.

The generator supplies endpoint activity limits and H^E. Changing 41 / 81 / 161 points changes the sampling grid.

Case D and the 80 kPa option · Measured data

Barbieri, De Guido and Moioli (2024), Tables 4–5, report 20 measured T–P–x–y states at each pressure.

60 kPa: 18 usable activity pairs; two pure endpoints cannot determine the absent component's activity.
80 kPa: 17 usable pairs; two pure endpoints and one state outside the common Antoine range are excluded from activity inference.

All original rows remain visible. P^sat and γ are calculated, not measured here. Independent H^E is unavailable.

What your controls change: endpoint options add assumed curves; Step 4 measurement shifts create separate sensitivity scenarios. The original measured rows remain unchanged. After learning the tests here, use Lab 08 to enter your own VLE measurements.

Where do the model and property values come from?

The generating NRTL parameters are from Table 7 of the same article. Pure-component P^sat values use the supplied 2-propanol and water Antoine records from NIST, within their common range of 334–362.41 K. Synthetic H^E is derived from the generating NRTL model; it is not independent calorimetry.

See the Antoine equation, constants and activity-inference steps in Lab 02 · Download the supplied data and provenance.

One data set, three different questions

A conceptual sketch, not calculated data

On a narrow screen, scroll the drawing horizontally. The three explanations below remain readable without scrolling.

1. Do areas cancel?

A small net area can hide errors of opposite sign.

2. Does the local identity hold?

Derivatives can reveal a defect, but amplify noise and spacing effects.

3. Can this model reproduce the data?

A fit tests a chosen model, property assumptions and numerical procedure together.

Teal is the reference. Amber highlights a local departure or model prediction; sand marks integrated area. No single criterion certifies the accuracy of every measurement.

Loading the source measurements and calculation methods…

1. Start with a control whose origin is known

2-Propanol + water · binary mixture

Keep the starting case: synthetic, isothermal at 350 K, consistent reference, 81 points.
Read the assumptions below. Predict which methods can assign a result on this path.
Press Calculate diagnostics. In Step 2, run the modeling test separately and record the control before changing a setting.

Load an investigationEach button loads settings. Calculate next.

Evidence and experimental path

Current data source

Model-generated practice data for 2-propanol + water; no measurement file is required.

Controlled alteration to the log-activity ratio

Alteration amplitude a

no unit

Allowed teaching range: −0.5 to +0.5. Used only for an altered case.

Sampling resolution n

Equally spaced liquid compositions from 0 to 1 mol/mol.

Include the supplied synthetic excess-enthalpy term H^E

2. Calculate, then read the claim behind each result

Calculate the starting case in Step 1. Results are cleared whenever a scientific setting changes.

Not run for this calculated case.

The model fit uses five starting guesses. Wait for a result before interpreting its thresholds. A nonconverged fit supplies no verdict.

Evidence to inspect

Read the numerical observations and exclusions

Row	x₁ / mol mol⁻¹	y₁ / mol mol⁻¹	T / K	P / kPa	ln γ₁	ln γ₂	Status

Read local residuals and model-fit details

x₁	Raw residual	Heat term	Residual used

Modeling test not run.

Name this calculated comparison

Work through the equations before making a claim

Area: an integrated requirement

I = \int_{0}^{1} \ln (\frac{γ_{1}}{γ_{2}}) d x_{1}

D = 100 \frac{| A_{+} - A_{-} |}{A_{+} + A_{-}}

Integrate the log ratio over the full composition range.
Separate positive area A₊ and the magnitude of negative area A₋. Segments are split at zero crossings.
Compute D. If A₊ = 0.40 and A₋ = 0.39, then I = 0.01 and D = 100(0.01)/0.79 = 1.27%.
D < 5% meets this empirical isothermal criterion. It cannot rule out canceling local errors. If both areas vanish, D is undefined.

Differential: a local requirement

g = \frac{g^{E}}{R T} = x_{1} \ln γ_{1} + (1 - x_{1}) \ln γ_{2}

r = \frac{d g}{d x_{1}} - \ln (\frac{γ_{1}}{γ_{2}}) + \frac{H^{E}}{R T^{2}} \frac{d T}{d x_{1}}

At fixed T, the heat term is zero; the excess-volume pressure term is neglected. At fixed P, the displayed heat term is needed. Three-point derivatives use actual composition spacing. The teaching score is 100 mean |r|, with a threshold of 5. It is not a percentage uncertainty or the NIST/Kojima Padé point test.

Calculate one local residual from the current data

After calculating, follow these three neighboring points. The derivative uses a three-point Lagrange formula, including unequal spacing when needed.

Calculate a case first.

Herington: an empirical allowance

For an isobaric path, J = 150(T_max − T_min)/T_min, with temperatures in K. This lab evaluates D − J < 10 only when a full-range area is available. J is not a measured excess-enthalpy correction. This is one published criterion variant, not the complete NIST TDE decision tree.

Modeling: reproduction by a chosen model

The NRTL teaching fit holds α = 0.47. It uses two constant parameters on an isothermal path, or four temperature-dependent parameters on an isobaric path. Thresholds are mean absolute relative pressure error < 1% and mean absolute vapor-composition error < 0.01 mol/mol. Different model flexibility, property errors or numerical convergence can change the result. This is not the full NIST five-parameter Van Ness implementation.

3. Construct a counterexample and separate numerical error

Synthetic evidence only

Load case B in Step 1. Keep amplitude a = 0.25 and 81 points. Calculate, run the modeling test, then save a comparison.
Compare its area and local diagnostic with case A. Explain which criterion misses the deliberately imposed error.
Run the controlled comparisons below. Compare 41, 81 and 161 points before attributing a residual to physics.
Load case C. Calculate once with the supplied H^E term and once without it. Explain why a missing correction means “not assessed.”

Why the canceling case is a useful trap

Let f(x₁) = a cos(2πx₁). Add (1 − x₁)f to ln γ₁ and subtract x₁f from ln γ₂. The weighted sum g stays unchanged, while ln(γ₁/γ₂) increases by f.

The full-range integral of cos(2πx₁) is zero, so the imposed error cancels in I.
The local residual changes by −f, up to differentiation error. At x₁ = 0.5 and a = 0.25, f = −0.25, so the added local residual is +0.25.
Therefore a small area does not establish the local partial-molar relationship. Use the calculated numbers to test this reasoning.

Synthetic endpoint activity limits and H^E are supplied by the generator. They are not inferred experimental measurements.

Calculate a synthetic case first.

Same path and resolution, different imposed errors

Alteration	I	D / %	Area criterion	Mean local score	Maximum \|r\|	Local criterion

Inspect the localized defect. Can both aggregate criteria meet their thresholds while a narrow defect remains? Compare the mean score, maximum residual and local trace. No separate peak threshold is imposed here.

Same alteration, different numerical resolution

Points n	I	D / %	Mean local score	Change from preceding grid

Grid refinement probes numerical approximation for this smooth generator. It does not add independent experimental evidence or reduce measurement uncertainty. These comparisons do not rerun the modeling fit.

4. Ask what measured data cannot tell you

Article data · 60 or 80 kPa

Load case D, then calculate with Leave missing. Read the measured composition interval and excluded rows in Step 2.
Run the investigation below. Compare no extension, linear extension and a quadratic fit. Identify every conclusion that depends on unmeasured endpoints.
Enter assumed common measurement shifts with units. Compare the signed shifts with the nominal calculation and count rejected scenarios.
Run the modeling test in Step 2. Explain why its result cannot supply the missing H^E, endpoint evidence or measurement covariance.

These measurements are isobaric. No independent H^E is supplied, so the raw differential residual is descriptive and cannot assign a consistency verdict. Endpoint extrapolation does not fix this missing information.

Assumed systematic measurement shifts

Each scenario shifts one measured quantity by + or − its entered value for all nominally eligible observations. These are eight deterministic scenarios, not independent random errors or confidence intervals. Original measurements are retained.

Temperature half-width ±ΔT

Pressure half-width ±ΔP

kPa

Liquid composition half-width ±Δx₁

mol/mol

Vapor composition half-width ±Δy₁

mol/mol

Calculate an article data set first.

Endpoint treatment of the same original observations

Assumption	I	D / %	J	D − J	Herington claim

One common measurement shift at a time

Scenario	Included / nominal pairs	I on selected area curve	D / %	Raw local score	Interpretation / rejection

The nominal inclusion mask is retained. If any nominally included row becomes invalid, the entire scenario is rejected. No row is silently dropped, clipped or extrapolated. None of these scenarios provides the missing excess enthalpy, and their frequency has no probability interpretation.

Graduate questions. Which conclusion survives the tested endpoint assumptions? Which measurement shift changes the diagnostic most? Explain how correlated composition errors, shared pressure calibration and uncertain P^sat could alter that conclusion. Specify the covariance, coverage and independent property evidence needed for a statistical uncertainty statement.

Why can a pressure error be invisible to the area?

\ln (\frac{γ_{1}}{γ_{2}}) = \ln (\frac{y_{1}}{1 - y_{1}}) - \ln (\frac{x_{1}}{1 - x_{1}}) - \ln (\frac{P_{1}^{sat}}{P_{2}^{sat}})

The same measured P multiplies both inferred activity coefficients, so it cancels from their ratio under the adopted ideal-vapor and pressure-correction approximations. A common pressure calibration shift changes each γ, while leaving this log ratio unchanged. Compare the P-shift rows. An insensitive area does not establish an accurate pressure measurement. Vapor fugacity or liquid pressure corrections can break this simple invariance.

5. Defend a limited conclusion and retain your work

Use at least two numerical comparisons, one disagreement between methods, and one numerical-resolution comparison. A result that meets a criterion is evidence for its stated claim under its assumptions.

Assumptions and applicability

Uncertainty and numerical sensitivity

Evidence and conflicting diagnostic results

Supported decision and next measurement

Saved comparisons

Save the control and at least one changed case from Step 2.

Open a saved Lab 03 study (.json)

Restore this lab's saved settings, comparisons and notes. Start with the supplied cases above; you do not need to upload a VLE table.

Loading the starting case…

Calculations and notes stay in this open page. Download one study file to retain them. Reopening validates the file and recalculates derived results; it does not grade your reasoning.

Methods, source records and limits

NIST: area and Herington criteria · NIST: point test · NIST: modeling test · NIST: uncertainty propagation and covariance.

Measured data: Barbieri, De Guido and Moioli, J. Chem. Thermodynamics 198 (2024), 107342, Tables 4–5. The authors' published consistency result is separate from every teaching result calculated here. Lab 02 gives Antoine constants, units and original inference steps.

The modeled objective is the unweighted mean of squared 100ΔP/P and 100Δy. Five-start bounded Nelder–Mead estimates NRTL parameters with α = 0.47 fixed. Isobaric τ_ij = c_ij + d_ij(350/T − 1), with c ∈ [−3, 8], d ∈ [−40, 40]. Isothermal fits use two constant τ values in [−3, 8]. Pure endpoints are excluded; bounds and convergence are reported. A favorable score is neither parameter-identifiability evidence nor a confidence interval.

Reference calculations reuse the existing course engines. Inspect the method source · Download source data and property provenance · Open the original Lab 03.