SKH Thermodynamics Labs

Lab 04 · Liquid stability and LLE

Is one homogeneous liquid the lowest Gibbs energy?

A 1.00 mol binary liquid feed at fixed temperature and pressure follows the symmetric Margules model with A = 3. The feed contains 0.50 mol of component 1.

Predict, calculate and read the Gibbs-energy construction, then explain when a liquid splits and how much of each liquid forms.

2027 prototype
Review the student workflow

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Core: predict, calculate, read the Gibbs construction and explain the result. Steps 1–5.

Graduate: the core steps plus local versus global evidence, parameter sensitivity and model-form comparisons in Step 4.

Graduate route. Complete Steps 1–3 to verify the calculation. In Step 4, compare local and global evidence, run bounded parameter scenarios and compare model forms matched at infinite dilution. Finish with evidence-backed claims. Go to the graduate investigation.

Choose a feedNamed cases keep A = 3 and change only z₁. Predict each state before you calculate.

Picture a liquid split

Composition and amount answer different questions
A homogeneous liquid feed can separate into two liquid layers One liquid feed enters a vessel at fixed temperature and pressure. When two liquids coexist, a component-1-poor liquid alpha and a component-1-rich liquid beta form separate layers. FeedLiquid βLiquid α
Feed · z₁

Of all feed moles, what fraction is component 1?

Liquid α · x₁α and fα

The component-1-poor liquid. fα is its share of the feed moles.

Liquid β · x₁β and fβ

The component-1-rich liquid. fβ is its share of the feed moles.

Which layer is on top depends on density, which this model does not calculate. For a 1 mol feed, fα + fβ = 1. Your calculation may instead find one liquid.

1. Set the model and feed

Basis · 1.00 mol feed · fixed T and P

The starting problem uses gE/RT = A x₁x₂ with A = 3.

Range 0–6. A two-liquid region exists only for A > 2.

mol/mol

Enter a fraction from 0 to 1. For this 1 mol feed, z₁ = 0.50 means 0.50 mol of component 1.

Predictions are optional and do not change the calculation. For symmetric Margules, g″(z₁) = 1/[z₁(1 − z₁)] − 2A gives the local clue.

Add a calculated result to the comparison before changing a setting.

Model assumptions

The liquid model gives the dimensionless mixing Gibbs energy g = ΔmixG/(nRT) at one fixed temperature and pressure. Pure liquids are the standard states. The model contains no vapor phase, interfacial energy, nucleation barrier or kinetics, so a metastable feed can persist in practice even though the split has lower Gibbs energy.

Parameters are hypothetical teaching values unless imported from a fit. No physical temperature is assigned to A.

Open the current Lab 04 for its original controls and the parameter map.

2. Identify the phases and amounts

Calculate the feed to see its phases, compositions, amounts and equilibrium checks.

4. Check your understanding

Write first, then compare

Use the calculations above to answer each question in your own words. Then open the explanation and compare. Your answers are saved with your study.

A. Why does z₁ = 0.12 split even though its curvature is positive?

Load the dilute feed, calculate, and read g″(z₁), the minimum TPD and the Gibbs plot.

Compare with the explanation

g″(0.12) = 1/(0.12 × 0.88) − 6 = 3.47 > 0, so the liquid resists small composition fluctuations. The minimum tangent-plane distance is −0.245, at a trial liquid near w₁ = 0.95: a liquid far from the feed lies below the tangent at the feed.

The equilibrium is two liquids (x₁ = 0.0707 and 0.9293). The homogeneous feed is metastable. Curvature is a local test; the tangent-plane distance is a global one.

B. At A = 3, which results change between z₁ = 0.12 and 0.50?

Calculate both named feeds and compare x₁α, x₁β and fβ.

Compare with the explanation

x₁α = 0.0707 and x₁β = 0.9293 stay the same; fβ rises from 0.057 to 0.50.

As in Lab 01, a binary with two coexisting phases at fixed T and P has no remaining freedom in the phase compositions. The feed only sets the amounts, through the lever rule fβ = (z₁ − x₁α)/(x₁β − x₁α).

C. Why can no liquid split occur when A ≤ 2?

Set A = 2 and then A = 1.5, calculate z₁ = 0.50, and look at the curvature plot.

Compare with the explanation

1/[x₁(1 − x₁)] has its minimum value, 4, at x₁ = 0.5. So g″ ≥ 4 − 2A, which is never negative when A ≤ 2. The Gibbs curve is convex everywhere and no common tangent can touch it twice.

A = 2 at x₁ = 0.5 is the critical point, where the two coexisting compositions merge.

4. Investigate what decides the split

Master's core work

Calculate a feed in Step 1 before running an investigation.

A. Separate local from global stability evidence

  1. Load the dilute feed z₁ = 0.12 and calculate it. Compute g″(z₁) = 1/[z₁(1 − z₁)] − 2A by hand and compare it with the lab.
  2. Run the audit. Compare the sign of the curvature with the minimum tangent-plane distance and the Gibbs energy decrease.
  3. Repeat for z₁ = 0.50 and 0.02. Explain why positive curvature alone cannot establish that a liquid is stable.

B. Test how sensitive the split is to the model parameters

  1. Enter an assumed half-width for each parameter and for z₁. Start with the illustrative values and explain why they are useful for this exercise.
  2. Run the one-at-a-time scenarios and the signed joint corners. Compare the change in x₁α with the change in fβ.
  3. For symmetric Margules, compare the finite change in x₁α with the local derivative dx₁α/dA from the binodal equation.
  4. Load the near-critical case A = 2.05 and rerun the same half-width. Explain why the phase count itself changes and why a local derivative cannot describe it.

C. Ask what the model form decides

  1. Use the current model to read its infinite-dilution limits ln γ₁∞ and ln γ₂∞.
  2. Run the comparison. Each candidate reproduces exactly the same two limits, so dilute activity data cannot tell them apart.
  3. Compare the coexistence compositions and the predicted phase count. Identify the data that would discriminate between the candidates.

D. Defend a limited, reproducible conclusion

Submission standard: report the model and parameter provenance; cite numerical results from A, B and C; identify retained assumptions and failed scenarios; and propose a discriminating measurement. The record check only confirms that the sections and calculations are present. Scientific reasoning still needs instructor review.

5. Compare and save

  1. Calculate and save the three named feeds at A = 3.
  2. Confirm that the coexistence compositions stay fixed while fβ changes.
  3. Answer questions A–C in Step 4 using the saved comparison.
  4. Download the complete study, reopen it, and reproduce a saved case.
  1. Calculate and save the three named feeds at A = 3. Confirm that the coexistence compositions stay fixed while fβ changes.
  2. Complete investigations A–C and write the graduate claims.
  3. Save the near-critical case and compare its phase count with the A = 3 cases.
  4. Download the complete study, reopen it, and reproduce a saved case and its investigation record.

Your calculated cases will appear here after you add them.

Starting case loaded. Press Calculate equilibrium.

Your inputs, comparisons, scenario settings and explanations are kept in this open page. Download one study file to keep the complete record and reopen it later. Results are recalculated on restore.