Experimental VLE, consistency and fitting
Session 2: Independent study with calculations · 180 minutes
16 October 2026
A three-hour independent lesson
Your action: Read this schedule, then advance. All times are elapsed lesson time, including your calculations.
This lesson replaces the class on 16 October 2026.
| Activity | Minutes | Elapsed |
|---|---|---|
| Preparation and goals | 10 | 0–10 |
| Phase and material-balance checks | 25 | 10–35 |
| From Raoult’s law to liquid nonideality | 30 | 35–65 |
| Antoine conventions | 25 | 65–90 |
| Break | 10 | 90–100 |
| Lab 08 investigation | 45 | 100–145 |
| Review test, corrections, exit and feedback | 35 | 145–180 |
Times include reading and working through calculations. Pause before each solution.
Preparation · 0–10 min
Your action: Open worksheet pages 1–4 and read this slide. Do not start the exercises or Lab yet.
Download the 4-page worksheet. Keep this slide deck and the worksheet open side by side. A calculator and paper are enough until minute 100.
Last lesson: ideal-mixture VLE and phase calculations. Today: how we describe departures from the ideal-liquid prediction.
The 3-page reading is optional background. Skim it if time permits. This deck contains the instructions required for today.
By the end, you should be able to:
- Explain the new activity coefficient and calculate it from one VLE observation.
- Check an Antoine convention before using its constants.
- Compare two fitted models and state what the comparison supports.
1. Composition and phase amount · 10–15 min
Your action: Read this slide only. No worksheet answer is required yet. The next slide starts page 1, Section A.
| Symbol | Meaning |
|---|---|
| zᵢ | Overall mole fraction in the feed |
| xᵢ | Mole fraction in the liquid phase |
| yᵢ | Mole fraction in the vapor phase |
| β = V/F | Fraction of total moles in the vapor phase |
z_i=(1-\beta)x_i+\beta y_i
For each existing phase, its mole fractions sum to one.
Pause and calculate · 15–23 min
Your action: Open worksheet page 1. Complete A1–A3 using this slide and your calculator. No Lab is needed.
Ideal binary mixture at 350 K: z₁=0.5, P₁ˢᵃᵗ=150 kPa, P₂ˢᵃᵗ=60 kPa.
P_{bubble}=\sum_i z_iP_i^{sat},\qquad P_{dew}=\left(\sum_i\frac{z_i}{P_i^{sat}}\right)^{-1}
- Calculate both boundary pressures for this overall composition.
- Identify the phase state at 80, 95 and 110 kPa.
- At 110 kPa, what are β and x₁? Does a vapor composition exist?
Finish A1–A3 before advancing to the solution.
Phase check · 23–27 min
Your action: Compare this solution with page 1, A1–A3. Add corrections beside your original answers.
P_{bubble}=0.5(150)+0.5(60)=105\ \mathrm{kPa}
P_{dew}=[0.5/150+0.5/60]^{-1}=85.714\ \mathrm{kPa}
| Pressure | Equilibrium state |
|---|---|
| 80 kPa | Vapor, below the dew boundary |
| 95 kPa | Liquid + vapor, between the boundaries |
| 110 kPa | Liquid, above the bubble boundary |
At 110 kPa, β=0 and x₁=z₁=0.5. No vapor phase exists, so y is undefined.
Pause: material balance · 27–32 min
Your action: On worksheet page 1, complete A4: material balance and composition sums. Stop before the next solution.
At 95 kPa, use x₁=0.388889, y₁=0.614035 and β=0.493506.
Calculate:
z_1=(1-0.493506)(0.388889)+0.493506(0.614035)
Then find x₂ and y₂. Check each phase sum separately.
Use the numbers on this slide. No Lab is required for this calculation.
Material-balance check · 32–35 min
Your action: Check page 1, A4 against this slide. Keep the substituted balance and correct any mismatch.
z_1\approx0.196970+0.303030=0.500000
x₂=0.611111 and y₂=0.385965.
x_1+x_2=1,\qquad y_1+y_2=1
An incorrect phase sum can reveal a transcription error even when the plotted result looks plausible.
2. Starting from Raoult’s law · 35–39 min
Your action: Read this slide. On scratch paper, calculate 60/48 before advancing. No new worksheet entry is required.
For an ideal liquid mixture with ideal vapor:
y_iP=x_iP_i^{sat}(T)
The liquid mole fraction scales the pure-component vapor pressure.
At x₁=0.4 and P₁ˢᵃᵗ=120 kPa, this predicts a partial pressure of 48 kPa.
Suppose the observation instead gives y₁P=0.6(100)=60 kPa. The ideal-liquid prediction is too low.
Think: what factor would multiply 48 to give 60?
A new quantity: activity coefficient · 39–42 min
Your action: Read this definition of gamma. Compare 1.25 with your scratch calculation. No Lab or worksheet entry is required.
The factor is 60/48=1.25. We call it the activity coefficient, written γ (gamma).
y_iP=x_i\gamma_iP_i^{sat}(T)
- γᵢ=1 recovers Raoult’s law for the liquid.
- γᵢ>1 gives a higher partial pressure than the ideal-liquid prediction.
- γᵢ<1 gives a lower partial pressure under the same assumptions.
We still assume ideal vapor and negligible Poynting correction. Verify the observations and vapor-pressure inputs before attributing a mismatch to liquid nonideality.
Activity and its reference · 42–45 min
Your action: Read this slide and check 1.25 × 0.4 on your calculator. Keep worksheet B blank until minute 48.
a_i=\gamma_i x_i
Activity aᵢ represents liquid composition adjusted for nonideal behavior. Here the reference is the pure liquid. Both aᵢ and γᵢ are dimensionless.
For an ideal liquid solution, γᵢ=1 and aᵢ=xᵢ.
For x₁=0.4 and γ₁=1.25, a₁=0.50.
Calculating gamma from VLE · 45–48 min
Your action: Read the equation and assumptions here. You will use them in worksheet page 1, B1–B3 on the next slide.
Rearrange the modified Raoult equation to estimate gamma from an observation.
With ideal vapor and negligible Poynting correction:
y_iP=x_i\gamma_iP_i^{sat}(T)
\gamma_i=\frac{y_iP}{x_iP_i^{sat}(T)}
Use the same pressure units in P and Pˢᵃᵗ. The inferred γ depends on the vapor-pressure correlation and these assumptions.
Pause: one observation, two property inputs · 48–57 min
Your action: On worksheet page 1, complete B1–B3. Use only the observation on this slide. No Lab is needed.
Synthetic observation: P=100 kPa, x₁=0.4, y₁=0.6.
- With P₁ˢᵃᵗ=120 kPa, calculate γ₁ and a₁.
- Repeat using P₁ˢᵃᵗ=150 kPa.
- Which input changed? Did the observed mixture data change?
- Could fitting alone tell you which vapor pressure is correct?
Show substitution and unit cancellation in B1–B3.
A property error changes inferred activity · 57–65 min
Your action: Check page 1, B1–B3 with this slide. Correct your values and finish the explanation in B3.
| Assumed P₁ˢᵃᵗ | γ₁ = y₁P/(x₁P₁ˢᵃᵗ) | a₁ = γ₁x₁ |
|---|---|---|
| 120 kPa | 60/48 = 1.25 | 0.50 |
| 150 kPa | 60/60 = 1.00 | 0.40 |
The observations stay unchanged. Increasing the assumed vapor pressure lowers the inferred γ.
A fitted model can absorb a property error into its parameters. Check the property source independently.
Now correct your work and write one sentence explaining the difference.
3. Antoine conventions · 65–72 min
Your action: Read this slide. Open worksheet page 2, C, ready for the next exercise. Do not fill the Lab section D yet.
A common form is:
\log_{10}(P^{sat}/P_{unit})=A-\frac{B}{T_{source}+C}
Before entering constants, identify:
- The exact equation and log base, including the sign of C.
- The temperature and pressure units specified by the source.
- The valid temperature range and component identity.
Changing a menu setting does not automatically convert the constants.
Pause: Antoine calculation · 72–82 min
Your action: On worksheet page 2, answer C1–C4 using this synthetic correlation. Do not enter these constants in Lab 08.
A synthetic teaching correlation, valid from 20 to 80 °C:
\log_{10}(P^{sat}/\mathrm{kPa})=4-\frac{1000}{T_C+200}
- Calculate Pˢᵃᵗ at 50 °C.
- Explain why entering 323.15 directly for T_C is incorrect.
- If the left side were ln instead of log₁₀, could the same constants be used unchanged?
- Should this correlation be used at 100 °C?
Write the equation and substitutions, not only the answer.
Antoine check · 82–90 min
Your action: Check page 2, C1–C4 with this slide. Write corrections and their reasons before taking the break.
4-\frac{1000}{50+200}=0,\qquad P^{sat}=10^0\ \mathrm{kPa}=1\ \mathrm{kPa}
- The equation expects °C. Convert 323.15 K to 50 °C first.
- A change of log base requires corresponding constants. At this particular zero exponent, both bases happen to give 1, so this single point cannot detect the mistake.
- 100 °C lies outside the stated valid range.
Correct your calculation before moving on.
Break · 90–100 min
Your action: Pause for ten minutes. Keep your worksheet. At minute 100, open Lab 08 using the link below.
Take ten minutes away from the screen.
Keep your calculations. Next, you will use the same checks on the synthetic dataset in Lab 08.
4. Lab baseline · 100–110 min
Your action: Open Lab 08 and worksheet page 2, D1–D2. Follow the four steps below. Save the baseline JSON before continuing.
In Lab 08:
- Select Load synthetic example.
- Inspect the data units and both Antoine equation forms and constants.
- Select Validate & inspect data. Correct input errors. Keep endpoint handling at No extrapolation.
- Use Export complete study JSON to save your baseline.
Record the convention and valid range in D1–D2. Range fields use kelvin. A consistency result marked “not-assessed” or “not-applicable” is not an input error.
Pause: an independent Lab check · 110–120 min
Your action: Keep Lab 08 open. Complete page 2, D3 and page 3, E1 for component 1 at the same selected row.
Choose one interior observation from Imported observations.
Write its T, P, x₁ and y₁. The normalized table uses K and kPa.
- Convert T to the unit required by the correlation and calculate P₁ˢᵃᵗ.
- Convert P₁ˢᵃᵗ to kPa if needed.
- Calculate γ₁=y₁P/(x₁P₁ˢᵃᵗ). The table shows ln γ₁. Compare your result with exp(table ln γ₁), allowing for display rounding.
Complete D3 on page 2 and E1 on page 3.
If values disagree: check component order, units and log base first.
Model comparison · 120–135 min
Your action: In Lab 08, use Fit selected model for each model below. Complete worksheet page 3, E2 and save both model exports.
Fit Margules · 1 parameter, then NRTL · fixed α, using α=0.3.
Keep the observations and property inputs unchanged. Leave temperature-dependent parameters off for this comparison.
For each fit, select View: Pressure and vapor residuals. Record:
- Mean absolute pressure error (%) and vapor-composition error (percentage points).
- One residual pattern (e.g., sign or trend with x₁) and any parameter-bound warning.
Export each model’s complete study JSON and fitted residuals CSV before switching models. Give the files distinct names.
Interpreting the comparison · 135–145 min
Your action: Read this table, then complete worksheet page 3, F using your E2 results. No additional Lab or dataset is required.
| Evidence | What it establishes |
|---|---|
| Solver convergence | The optimizer met its stopping criterion |
| Small training residuals | The model reproduces the fitted observations |
| Consistency diagnostic | Agreement with a thermodynamic constraint under stated assumptions |
| Held-out observations | Evidence about prediction beyond the fitting data |
Complete F on worksheet page 3: state one finding, its numerical evidence and one limitation.
The example data were generated with NRTL. Its close fit is expected model recovery, not evidence of universal superiority. Consistency and prediction require separate checks.
Review test: purpose · 145–147 min
Your action: Open worksheet page 4, Review test. Read these rules before answering. The questions appear on the next two slides.
This is practice with attendance credit. Credit is for completing the activity, rather than a minimum accuracy score.
Work individually. You may consult the lesson and your calculations.
Write answers to the next six questions, then review the solutions. Preserve your original answers and corrections.
The attendance submission route will be provided separately. The anonymous exit/feedback form does not identify attendance.
Review test: phase and activity · 147–155 min
Your action: Write Q1–Q3 in the matching spaces on worksheet page 4. Show your calculations before advancing.
1. Ideal binary mixture: z₁=0.5, P₁ˢᵃᵗ=160 kPa, P₂ˢᵃᵗ=80 kPa. At P=130 kPa, identify the phase and its composition. Show a boundary calculation.
2. For a separate binary mixture in two-phase equilibrium, x₁=0.3, y₁=0.7 and z₁=0.5. Calculate β by material balance.
3. For P=90 kPa, x₁=0.3, y₁=0.5 and P₁ˢᵃᵗ=100 kPa, calculate γ₁ and a₁. Assume ideal vapor and negligible Poynting correction.
Keep your substitutions and units.
Review test: properties and evidence · 155–165 min
Your action: Write Q4–Q6 on worksheet page 4. For Q6, reuse your page 3, E2 results. Finish before viewing solutions.
4. A synthetic correlation is log₁₀(Pˢᵃᵗ/kPa)=5−1200/(T_C+200), valid at 20–80 °C. Find Pˢᵃᵗ at 40 °C. Can you insert 313.15 directly for T_C?
5. NRTL gives a lower training error than Margules. Does that establish thermodynamic consistency and better prediction? Explain.
6. From your Lab 08 results, report one comparison with its metric and units. State one limitation of your conclusion.
Finish your own answers before opening the next slide.
Review solutions · 165–169 min
Your action: Check worksheet page 4, Q1–Q4. Add corrections beside the original answers, using extra paper if needed.
1. P_bubble=0.5(160)+0.5(80)=120 kPa. At 130 kPa the mixture is liquid, x₁=0.5 and β=0.
2. β=(z₁−x₁)/(y₁−x₁)=(0.5−0.3)/(0.7−0.3)=0.5.
3. γ₁=0.5(90)/[0.3(100)]=1.5 and a₁=1.5(0.3)=0.45.
4. 5−1200/(40+200)=0, so Pˢᵃᵗ=1 kPa. Convert 313.15 K to 40 °C first.
Mark any correction beside your original answer.
Review of reasoning · 169–172 min
Your action: Check page 4, Q5–Q6. Explain each correction. Retain all four worksheet pages and the Lab exports for attendance submission.
5. Lower training error alone establishes neither claim. Consistency needs an applicable thermodynamic diagnostic. Prediction needs observations excluded from fitting.
6. A complete answer identifies the same dataset, reports both model results with the metric and units, and gives a defensible limitation.
For example: “Model ___ has lower mean absolute pressure error on this dataset. I have not tested it on excluded observations.” Fill the blank using your own result.
Exit ticket and feedback · 172–180 min
Your action: Open the Session 2 form using the QR code or link. Use page 3, F for reasoning, page 2, C/D for the property check, and E1 for numerical evidence.
5 min: explain and check
Does a small fitting residual establish thermodynamic consistency? Explain and name one independent check of the vapor-pressure inputs. Add one substituted γ or Pˢᵃᵗ calculation.
3 min: feedback on independent learning
Report pace and Lab access. Use the comment box to describe any difficulty with this format.
The form requests no name, ID or email. Wait for “Response received.” Keep attendance work separately.
