Lab 02 · Experimental VLE and model evidence

Defend a model using thermodynamic evidence.

Graduate investigation: establish a 60 kPa NRTL fit as a control, check it at 80 kPa, then test how measurement bounds and fixed modeling assumptions affect your conclusion.

Complete the guided baseline, then G1–G3. Submit an assumption audit, two numerical sensitivity comparisons and a defended decision about model use.

2027 prototype
Review the student workflow

Separate fitting from checking

Starting case · fit 60 kPa, check 80 kPa

1. Fit · 60 kPa

Use 18 activity-coefficient pairs to choose the two parameters.

2. Retain the model

Keep the fitted parameters unchanged when you change the checking pressure.

3. Check · 80 kPa

Compare the model with 17 pairs that were not used in this fit.

Teal dots represent measurements. The amber line represents a model prediction. This sketch explains the workflow; the calculated data appear in Step 4.

1. Read one measured equilibrium state

Component 1 · 2-propanol (isopropanol)
Component 2 · water

At the same temperature and pressure, the experiment measures a liquid composition x₁ and a vapor composition y₁. Both are mole fractions of component 1.

Measured pressure / kPaMeasurement to inspect

Measured temperature T: — K
Measured pressure P: — kPa
Liquid composition x₁: —
Vapor composition y₁: —

Inspect all measured observations and exclusions

All original measurements remain visible. Only rows that supply both finite ln γ values within the property range enter the fit.

Row	P / kPa	T / K	Liquid x₁	Vapor y₁	Activity-pair status

Measured data from Barbieri, De Guido and Moioli (2024), Tables 4–5. Read the article.

2. Calculate activity coefficients from that measurement

Ideal vapor · negligible pressure correction

Find each pure-component saturation pressure at the measured temperature. Then use the measured x₁, y₁ and P to infer γ₁ and γ₂. An activity coefficient describes departure from ideal-liquid behavior; γ = 1 is the ideal-liquid limit.

γ_{1} = \frac{y_{1} P}{x_{1} P_{1}^{sat}}

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

Use the same pressure unit for P and both P^sat values. Activity coefficients γ are dimensionless.

Get P^sat from the Antoine equation

\log_{10} (\frac{P^{sat}}{bar}) = A - \frac{B}{T + C}

Use T in K. The equation gives P^sat in bar. Multiply by 100 to convert bar to kPa. The pressure is the equilibrium pressure of each pure liquid with its own saturated vapor.

2-Propanol · component 1

A = 4.8610
B = 1357.427
C = −75.814

Valid from 329.92 to 362.41 K.

NIST property source

Water · component 2

A = 5.0768
B = 1659.793
C = −45.854

Valid from 334 to 363 K.

NIST property source

Both correlations apply together from 334 to 362.41 K. The calculation uses this common range and does not extrapolate.

Try the calculation yourself

Optional practice. Enter your answers and check them, or continue to Step 3. Checks allow 1% rounding difference.

Your P₁^sat

kPa

Your P₂^sat

kPa

Your γ₁

no unit

Your γ₂

no unit

Worked calculation for this measurement

P₁^sat(T): — kPa
P₂^sat(T): — kPa
Activity coefficient γ₁: —
Activity coefficient γ₂: —

3. Fit, then check another pressure

Data used to fit the two parameters

Only these selected observations determine the fitted parameters. The fit matches inferred ln γ₁ and ln γ₂ at the measured T and x₁. The T–x–y curve is a subsequent model prediction.

Fixed NRTL nonrandomness α

no unit

The starting case uses α = 0.47 from the article. Keep α fixed while fitting τ₁₂ and τ₂₁. Teaching control range 0.10–0.60. This control applies to your teaching fit; the article reference keeps its reported α.

Compared with the fitting error, what checking error do you expect?

You can skip this prediction. It does not change the fit.

Pressure used to check the retained model / kPa

Changing this pressure keeps the fitted parameters. Do not refit before reading the checking error.

Compare with the article's NRTL reference

Model to inspect

The article reference uses τ = a + b/T and α = 0.47. Its parameters were fitted using a wider data collection. The 80 kPa measurements are not held out from that published model.

Read the fitting and checking errors

Fit the starting data to see the parameter values and compare the two errors.

Name this model comparison

4. Read the model against measurements

Plot to inspect

Use the checking-pressure selector in Step 3 to change the experiment shown here while keeping the same fitted model.

Inspect the numerical plot data

Graduate analysis · defend the model

Core graduate work · complete after the baseline

Use Steps 1–4 to establish your control calculation. Then complete G1–G3 below. A graduate conclusion must identify the assumptions, quantify their consequences and state what the available evidence cannot establish.

G1. Audit activity inference and test measurement sensitivity

Identify measured T, P, x₁ and y₁, and derived P^sat and γ. Explain which uncertainties can affect more than one derived value.
State what evidence would justify ideal vapor, negligible pressure correction and the selected Antoine range. Name the fugacity or liquid-volume information needed to quantify an omitted correction.
Choose assumed bounds below, then run the scenarios. Compare one-factor changes and the joint corner range. All original measurements and property constants are retained.

These are illustrative independent perturbation choices for sensitivity analysis. They are not the article's pointwise standard uncertainties. No distribution, covariance or probability is assigned. Eight one-factor scenarios and sixteen joint corners are sampled. Zero bounds repeat states.

Assumed ±ΔT

Assumed ±ΔP

kPa

Assumed ±Δx₁

mol/mol

Assumed ±Δy₁

mol/mol

Choose a valid interior observation and run the scenario analysis.

Quantity	Baseline	Minimum across accepted corners	Maximum across accepted corners

Rank one-factor effects

Compare the largest absolute change in each ln γ when one input moves to either assumed bound. Rows are ordered by the larger component effect. Rank each component separately. This finite scenario change is not a derivative or an uncertainty contribution.

Input	Accepted / attempted	Largest \|Δ ln γ₁\|	Largest \|Δ ln γ₂\|

Inspect all 24 scenarios and their rejections

Scenario	T / K	P / kPa	x₁	y₁	ln γ₁	ln γ₂	Status / reason

The displayed range covers accepted sampled corners only. It is not a confidence interval or a proven bound on all states inside the box. Rejected corners remain listed. Do not extend a conclusion into a rejected region or beyond a property correlation range.

Analysis question. Does γ₁ > 1 survive every accepted default corner? Quote the range, then explain why this is not a probability statement. Derive at least one first-order sensitivity from the inference equation and compare its local prediction with the finite one-factor change.

\frac{\partial \ln γ_{1}}{\partial x_{1}} = - \frac{1}{x_{1}} at fixed T, P and y

Explain why both inferred log activities share measurements and can be correlated even when errors in the raw measurements are independent. Near endpoints or outside a property domain, explain why a local linear approximation can become misleading.

Further study: NIST coverage factors · Sensitivity coefficients and covariance.

G2. Test sensitivity to a fixed modeling assumption

Fit your chosen training set in Step 3 and select the checking pressure.
Run three fixed-α fits. Each uses exactly the same training observations. Checking observations are used only for scoring unless already included in the selected training set.
Compare both τ parameters and both errors for at least two α values. Inspect convergence, bounds and curvature before recommending model use.

Establish the baseline model, then compare three conditional fits.

Fixed α	τ₁₂	τ₂₁	Training RMSE	Checking RMSE	Converged	At bound	Minimum sampled curvature	Status / interpretation

These are three conditional two-parameter fits to fixed observations. They do not jointly estimate α, provide parameter confidence intervals, bootstrap measurement uncertainty, or prove non-identifiability. Your retained model and source measurements have not been replaced.

If you choose α by inspecting the checking errors, that pressure has become model-selection data. It was excluded from each two-parameter fit, but it is no longer an untouched final test of the chosen α. Obtain another independent test set or use appropriately nested validation before claiming final predictive performance.

Open Lab 04 to investigate any fit with nonpositive sampled liquid curvature.

Analysis question. Can similar predictive errors accompany different parameter values? State what this supports, then propose the additional data or parameter-profile study needed before claiming parameter identifiability or a statistical confidence interval.

G3. Defend a model-use decision

Complete all four statements. Use at least two numerical comparisons from G1/G2 and one training/checking comparison. Distinguish measurement sensitivity, parameter sensitivity, model discrepancy and numerical error.

Assumptions and evidence needed Measurement sensitivity and missing covariance Conditional parameter sensitivity and checking evidence Decision, application range and next measurement

Name a model, use and supported T/P/composition range.
Use two numerical comparisons, including the checking-data role.
Name missing uncertainty/covariance information and a model limitation.
Propose a measurement that could challenge the conclusion.

Finish in Step 5 and download the existing study file. It includes these assumptions, bounds, results and conclusions. Reopening recomputes the investigations from their saved scientific context.

5. Compare and submit your defended analysis

Fit only 60 kPa and check 80 kPa. Save this comparison before changing the fitting data.
Reverse the experiment. Fit 80 kPa and check 60 kPa. Compare which model transfers better.
Fit both pressures. Explain why neither pressure is now held out from the fit.
Download your study, reopen it, and reproduce a saved comparison.

Saved model comparisons will appear here.

Optional additional note

Put the graduate claims in G3 above. Use this field for baseline observations or additional remarks. The downloaded study retains both.

Open a saved study

Loading the measured data…

Your calculations, graduate investigations and conclusions stay in this open page. Download one study file to retain them and reopen them in this prototype.

Methods and limits of this case study

Activity inference assumes ideal vapor, negligible pressure corrections and pure-liquid standard states. The teaching fit holds τ₁₂ and τ₂₁ constant across the selected temperature range. It minimizes an unweighted mean squared ln γ residual and does not estimate parameter confidence intervals.

The article reference uses τ = a + b/T. This equation requires b in K, although Table 7 prints 1/K. The numerical values are retained with the article's equation. This exercise uses different property assumptions and only the 40 newly reported observations, so it does not reproduce the article's wider Aspen regression.

The experimental path is isobaric and temperature varies with composition. An independent consistency verdict would need appropriate endpoint treatment and the temperature correction involving excess enthalpy. Fitting errors and a numerical Gibbs–Duhem check of NRTL do not supply that verdict.

Open the full case study for the controlled consistency experiment · Download the source measurements and property record

Defend a model using thermodynamic evidence.

Separate fitting from checking

1. Read one measured equilibrium state

2. Calculate activity coefficients from that measurement

2-Propanol · component 1

Water · component 2

3. Fit, then check another pressure

Read the fitting and checking errors

How is the error calculated?

4. Read the model against measurements

Graduate analysis · defend the model

G1. Audit activity inference and test measurement sensitivity

Rank one-factor effects

G2. Test sensitivity to a fixed modeling assumption

G3. Defend a model-use decision

5. Compare and submit your defended analysis