Thermodynamic Properties of Real Fluids, Departures and Fugacity

Why this unit matters

Bubble, dew, activity coefficients and flash separation. The preceding prerequisite is Phase equilibrium; the destination is Other equilibria. Engineering students should be able to move from a physical scenario to a transparent mathematical model and back to an interpretation of what would be observed in equipment. Thermodynamic design becomes unreliable when an input is precise to many decimal places while the phase model, standard states or material identities remain unspecified.

Four connected lessons

1. Raoult-law bubble pressure

For an ideal liquid solution at modest pressure, vapor partial pressures scale with liquid mole fractions and pure-component saturation pressures.

Useful calculation: P_bubble=Σ x_i P_i^sat; y_i=x_i P_i^sat/P.

Physical interpretation: A richer light-component liquid typically has a higher bubble pressure at fixed temperature when its pure saturation pressure is greater.

Failure to watch: An ideal mixture approximation may fail for hydrogen-bonding or strongly dissimilar liquids; vapor nonideality matters at high pressure.

Decision: Estimate vaporization onset for binary solvent handling.

2. Dew-pressure and dew-temperature calculations

The first droplet formed from vapor must satisfy vapor and incipient-liquid compositions simultaneously.

Useful calculation: At ideal low pressure Σ y_i P/P_i^sat(T)=1 at dew temperature.

Physical interpretation: A condenser can be screened for its first-liquid appearance by solving a nonlinear scalar temperature equation.

Failure to watch: A dew calculation performed with feed liquid fractions instead of vapor fractions solves a different problem and produces false results.

Decision: Set condenser approach temperatures with uncertainty margins.

3. Nonideal liquid activities

Activity coefficients correct chemical potential in a liquid mixture when interactions depart from ideal mixing.

Useful calculation: Modified Raoult relation y_i P≈x_i γ_i P_i^sat under low-pressure vapor conditions.

Physical interpretation: A nonideal mixture may exhibit an azeotrope where vapor and liquid compositions become equal at a particular equilibrium state.

Failure to watch: Using γ_i=1 without evidence can conceal azeotropes and cause impossible separation specifications.

Decision: Use experimental VLE data to decide whether simple correlations are appropriate.

4. Flash vaporization and Rachford–Rice

A constant-T,P flash solves component mass balances and equilibrium partitioning together with unknown vapor fraction.

Useful calculation: F(β)=Σ z_i(K_i−1)/(1+β(K_i−1))=0, with 0≤β≤1 and denominator positive.

Physical interpretation: For known equilibrium ratios a feasible two-phase solution produces consistent liquid and vapor compositions and an overall component balance.

Failure to watch: Blindly solving the scalar equation without a phase-feasibility check may return a nonphysical vapor fraction.

Decision: Perform process-level flash checks and sensitivity analysis.

Unit assessment

• Problem 1A — Raoult-law bubble pressure. Given an engineering case of your choice, identify a defensible thermodynamic boundary and relevant equation, and justify its variables.
• Problem 1B — Data reliability. Describe a measurement or property-data uncertainty that affects raoult-law bubble pressure and a way to quantify its consequence.
• Problem 1C — Counterexample. Construct an operating case where the assumption “An ideal mixture approximation may fail for hydrogen-bonding or strongly dissimilar liquids; vapor nonideality matters at high pressure.” becomes important and would invalidate a simplistic answer.
• Problem 2A — Dew-pressure and dew-temperature calculations. Given an engineering case of your choice, identify a defensible thermodynamic boundary and relevant equation, and justify its variables.
• Problem 2B — Data reliability. Describe a measurement or property-data uncertainty that affects dew-pressure and dew-temperature calculations and a way to quantify its consequence.
• Problem 2C — Counterexample. Construct an operating case where the assumption “A dew calculation performed with feed liquid fractions instead of vapor fractions solves a different problem and produces false results.” becomes important and would invalidate a simplistic answer.
• Problem 3A — Nonideal liquid activities. Given an engineering case of your choice, identify a defensible thermodynamic boundary and relevant equation, and justify its variables.
• Problem 3B — Data reliability. Describe a measurement or property-data uncertainty that affects nonideal liquid activities and a way to quantify its consequence.
• Problem 3C — Counterexample. Construct an operating case where the assumption “Using γ_i=1 without evidence can conceal azeotropes and cause impossible separation specifications.” becomes important and would invalidate a simplistic answer.
• Problem 4A — Flash vaporization and Rachford–Rice. Given an engineering case of your choice, identify a defensible thermodynamic boundary and relevant equation, and justify its variables.
• Problem 4B — Data reliability. Describe a measurement or property-data uncertainty that affects flash vaporization and rachford–rice and a way to quantify its consequence.
• Problem 4C — Counterexample. Construct an operating case where the assumption “Blindly solving the scalar equation without a phase-feasibility check may return a nonphysical vapor fraction.” becomes important and would invalidate a simplistic answer.

Assessment evidence

Unit 5 · Lesson 1: Raoult-law bubble pressure

Learning objectives. Explain raoult-law bubble pressure in physical terms; identify the relevant state or balance variables; apply a quantitative relation only within its validity range; and critically review a proposed process-engineering decision.

The physical question

For an ideal liquid solution at modest pressure, vapor partial pressures scale with liquid mole fractions and pure-component saturation pressures.

Governing relation and interpretation

Key relation: P_bubble=Σ x_i P_i^sat; y_i=x_i P_i^sat/P.

Engineering example and model boundary

A richer light-component liquid typically has a higher bubble pressure at fixed temperature when its pure saturation pressure is greater.

Worked investigation

At 350 K take illustrative vapor pressures 110 kPa and 40 kPa and liquid x1=0.35. Ideal bubble pressure becomes 64.5 kPa.

The vapor is enriched in the more volatile first species, so its y1 exceeds the liquid x1 in the low-pressure ideal model.

Where the relation breaks down

An ideal mixture approximation may fail for hydrogen-bonding or strongly dissimilar liquids; vapor nonideality matters at high pressure.

Engineering decision and data checklist

Estimate vaporization onset for binary solvent handling.

Guided problems

1. Definition and boundary. Describe the relevant system and identify which properties must be uniform before using the governing relation for raoult-law bubble pressure.

2. Derivation. State which conservation principle, potential differential or equilibrium condition supports the key relation. Explain each approximation that is needed to reach the displayed form.

3. Units and sign. Determine the natural units of each term. Describe a sign error or reference-state error that would produce a misleading yet numerical-looking result.

4. Sensitivity. Change one physically adjustable input by approximately 10% and predict the direction of the response before performing an exact calculation.

5. Independent validation. Propose a laboratory measurement or alternate property relation that would help detect a model error.

6. Engineering judgment. Explain which additional information would be mandatory before the calculation could support purchase, construction or operation of real equipment.

Mini-case: two competing models

Suppose a project team has a quick spreadsheet estimate and a detailed simulation package that disagree. Model A is simpler and transparent; Model B includes correlations for effects missing from Model A. First, compare both results under the same physical conditions and reference convention. Then determine whether they are truly independent or share the same underlying property parameters. A difference between them is not automatically proof that Model B is better. Ask which result fits experimental observations, and whether the disagreement is material to the stated decision. Apply this comparison specifically to Raoult-law bubble pressure using the assumptions discussed in this lesson.

Lesson synthesis

Subject focus: The key distinguishing result of this lesson is the link between raoult-law bubble pressure and the application “Estimate vaporization onset for binary solvent handling.” The governing relation is P_bubble=Σ x_i P_i^sat; y_i=x_i P_i^sat/P.; the stated failure mode is An ideal mixture approximation may fail for hydrogen-bonding or strongly dissimilar liquids; vapor nonideality matters at high pressure.

Unit 5 · Lesson 2: Dew-pressure and dew-temperature calculations

Learning objectives. Explain dew-pressure and dew-temperature calculations in physical terms; identify the relevant state or balance variables; apply a quantitative relation only within its validity range; and critically review a proposed process-engineering decision.

The physical question

The first droplet formed from vapor must satisfy vapor and incipient-liquid compositions simultaneously.

Governing relation and interpretation

Key relation: At ideal low pressure Σ y_i P/P_i^sat(T)=1 at dew temperature.

Engineering example and model boundary

A condenser can be screened for its first-liquid appearance by solving a nonlinear scalar temperature equation.

Worked investigation

A vapor contains 65 mol% of the more volatile species; use illustrative Clausius vapor-pressure curves to find its first condensation temperature at fixed pressure.

Dew calculations begin with y rather than x; a robust root solver requires a temperature bracket with a change of sign.

Where the relation breaks down

A dew calculation performed with feed liquid fractions instead of vapor fractions solves a different problem and produces false results.

Engineering decision and data checklist

Set condenser approach temperatures with uncertainty margins.

Guided problems

1. Definition and boundary. Describe the relevant system and identify which properties must be uniform before using the governing relation for dew-pressure and dew-temperature calculations.

2. Derivation. State which conservation principle, potential differential or equilibrium condition supports the key relation. Explain each approximation that is needed to reach the displayed form.

3. Units and sign. Determine the natural units of each term. Describe a sign error or reference-state error that would produce a misleading yet numerical-looking result.

4. Sensitivity. Change one physically adjustable input by approximately 10% and predict the direction of the response before performing an exact calculation.

5. Independent validation. Propose a laboratory measurement or alternate property relation that would help detect a model error.

6. Engineering judgment. Explain which additional information would be mandatory before the calculation could support purchase, construction or operation of real equipment.

Mini-case: two competing models

Suppose a project team has a quick spreadsheet estimate and a detailed simulation package that disagree. Model A is simpler and transparent; Model B includes correlations for effects missing from Model A. First, compare both results under the same physical conditions and reference convention. Then determine whether they are truly independent or share the same underlying property parameters. A difference between them is not automatically proof that Model B is better. Ask which result fits experimental observations, and whether the disagreement is material to the stated decision. Apply this comparison specifically to Dew-pressure and dew-temperature calculations using the assumptions discussed in this lesson.

Lesson synthesis

Subject focus: The key distinguishing result of this lesson is the link between dew-pressure and dew-temperature calculations and the application “Set condenser approach temperatures with uncertainty margins.” The governing relation is At ideal low pressure Σ y_i P/P_i^sat(T)=1 at dew temperature.; the stated failure mode is A dew calculation performed with feed liquid fractions instead of vapor fractions solves a different problem and produces false results.

Unit 5 · Lesson 3: Nonideal liquid activities

Learning objectives. Explain nonideal liquid activities in physical terms; identify the relevant state or balance variables; apply a quantitative relation only within its validity range; and critically review a proposed process-engineering decision.

The physical question

Activity coefficients correct chemical potential in a liquid mixture when interactions depart from ideal mixing.

Governing relation and interpretation

Key relation: Modified Raoult relation y_i P≈x_i γ_i P_i^sat under low-pressure vapor conditions.

Engineering example and model boundary

A nonideal mixture may exhibit an azeotrope where vapor and liquid compositions become equal at a particular equilibrium state.

Worked investigation

A solvent–water mixture forms a nonideal azeotrope. Compare γ_i=1 with activity-coefficient predictions and assess whether distillation alone can reach the requested purity.

A strongly nonideal mixture can invalidate ordinary relative-volatility assumptions and motivate alternate separation technologies.

Where the relation breaks down

Using γ_i=1 without evidence can conceal azeotropes and cause impossible separation specifications.

Engineering decision and data checklist

Use experimental VLE data to decide whether simple correlations are appropriate.

Guided problems

1. Definition and boundary. Describe the relevant system and identify which properties must be uniform before using the governing relation for nonideal liquid activities.

2. Derivation. State which conservation principle, potential differential or equilibrium condition supports the key relation. Explain each approximation that is needed to reach the displayed form.

3. Units and sign. Determine the natural units of each term. Describe a sign error or reference-state error that would produce a misleading yet numerical-looking result.

4. Sensitivity. Change one physically adjustable input by approximately 10% and predict the direction of the response before performing an exact calculation.

5. Independent validation. Propose a laboratory measurement or alternate property relation that would help detect a model error.

6. Engineering judgment. Explain which additional information would be mandatory before the calculation could support purchase, construction or operation of real equipment.

Mini-case: two competing models

Suppose a project team has a quick spreadsheet estimate and a detailed simulation package that disagree. Model A is simpler and transparent; Model B includes correlations for effects missing from Model A. First, compare both results under the same physical conditions and reference convention. Then determine whether they are truly independent or share the same underlying property parameters. A difference between them is not automatically proof that Model B is better. Ask which result fits experimental observations, and whether the disagreement is material to the stated decision. Apply this comparison specifically to Nonideal liquid activities using the assumptions discussed in this lesson.

Lesson synthesis

Subject focus: The key distinguishing result of this lesson is the link between nonideal liquid activities and the application “Use experimental VLE data to decide whether simple correlations are appropriate.” The governing relation is Modified Raoult relation y_i P≈x_i γ_i P_i^sat under low-pressure vapor conditions.; the stated failure mode is Using γ_i=1 without evidence can conceal azeotropes and cause impossible separation specifications.

Unit 5 · Lesson 4: Flash vaporization and Rachford–Rice

Learning objectives. Explain flash vaporization and rachford–rice in physical terms; identify the relevant state or balance variables; apply a quantitative relation only within its validity range; and critically review a proposed process-engineering decision.

The physical question

A constant-T,P flash solves component mass balances and equilibrium partitioning together with unknown vapor fraction.

Governing relation and interpretation

Key relation: F(β)=Σ z_i(K_i−1)/(1+β(K_i−1))=0, with 0≤β≤1 and denominator positive.

Engineering example and model boundary

For known equilibrium ratios a feasible two-phase solution produces consistent liquid and vapor compositions and an overall component balance.

Worked investigation

For a binary feed z1=0.5, K1=2.2 and K2=0.4, solve Rachford–Rice only after checking F(0)>0 and F(1)<0.

A zero of F inside [0,1] provides a phase fraction; component balances must still be checked against reconstructed x_i and y_i.

Where the relation breaks down

Blindly solving the scalar equation without a phase-feasibility check may return a nonphysical vapor fraction.

Engineering decision and data checklist

Perform process-level flash checks and sensitivity analysis.

Guided problems

1. Definition and boundary. Describe the relevant system and identify which properties must be uniform before using the governing relation for flash vaporization and rachford–rice.

2. Derivation. State which conservation principle, potential differential or equilibrium condition supports the key relation. Explain each approximation that is needed to reach the displayed form.

3. Units and sign. Determine the natural units of each term. Describe a sign error or reference-state error that would produce a misleading yet numerical-looking result.

4. Sensitivity. Change one physically adjustable input by approximately 10% and predict the direction of the response before performing an exact calculation.

5. Independent validation. Propose a laboratory measurement or alternate property relation that would help detect a model error.

6. Engineering judgment. Explain which additional information would be mandatory before the calculation could support purchase, construction or operation of real equipment.

Mini-case: two competing models

Suppose a project team has a quick spreadsheet estimate and a detailed simulation package that disagree. Model A is simpler and transparent; Model B includes correlations for effects missing from Model A. First, compare both results under the same physical conditions and reference convention. Then determine whether they are truly independent or share the same underlying property parameters. A difference between them is not automatically proof that Model B is better. Ask which result fits experimental observations, and whether the disagreement is material to the stated decision. Apply this comparison specifically to Flash vaporization and Rachford–Rice using the assumptions discussed in this lesson.

Lesson synthesis

Subject focus: The key distinguishing result of this lesson is the link between flash vaporization and rachford–rice and the application “Perform process-level flash checks and sensitivity analysis.” The governing relation is F(β)=Σ z_i(K_i−1)/(1+β(K_i−1))=0, with 0≤β≤1 and denominator positive.; the stated failure mode is Blindly solving the scalar equation without a phase-feasibility check may return a nonphysical vapor fraction.

Laboratory 9: Two-phase steam quality

Model: h=hf+x hfg

Controllable variables: Mass vapor quality (fraction) and Latent enthalpy (kJ kg⁻¹).

Governing relation

h=hf+x hfg. This relation is used for an educational screening calculation, not certified physical-property or process-safety design. Every output must be checked against dimensional consistency, equilibrium assumptions and an independent reference when applied to real chemicals.

Stated limitations

Reference saturated-liquid h_f=600 kJ/kg; both enthalpies at one saturation pressure.

Step-by-step investigation

1. Record the default inputs and the readout with its units. Sketch the qualitative expected trend before inspecting the chart.
2. Decrease Mass vapor quality to its minimum while keeping Latent enthalpy fixed. Record the change and explain which term in the governing equation is responsible.
3. Increase Mass vapor quality to its maximum and compare the response with the first observation. Determine whether the dependence is linear, logarithmic, exponential or nonlinear for a reason you can explain.
4. Reset, then alter Latent enthalpy and repeat. Estimate a finite-difference sensitivity between two nearby parameter values, including its units.
5. Export the response curve to CSV. Explain how a plotted sensitivity helps select a range for a laboratory measurement or a process-control test.
6. Identify the first physical mechanism omitted by the model that would become important if the system were pressurized, heated, mixed or scaled up substantially.

Questions

Reasoning: Why is quality not a volume fraction?

Extension: Compare 50% quality with the midpoint of vapor and liquid specific volumes.

Uncertainty: How would uncertainty in latent enthalpy influence your interpretation of the response curve? Which single additional measurement would best distinguish model error from input uncertainty?

Evidence to submit

Export a parameter sweep from the interactive model and attach it with your explanation of h=hf+x hfg. Demonstrate the predicted change in Mass vapor quality at two different settings of Latent enthalpy. Conclude by explaining whether the laboratory's approximation (Reference saturated-liquid h_f=600 kJ/kg; both enthalpies at one saturation pressure.) would remain credible in a real chemical process.

Laboratory 10: Ideal-liquid bubble pressure

Model: P=xA PsatA+(1−xA)PsatB

Controllable variables: Liquid mole fraction A (fraction) and Temperature (K).

Governing relation

P=xA PsatA+(1−xA)PsatB. This relation is used for an educational screening calculation, not certified physical-property or process-safety design. Every output must be checked against dimensional consistency, equilibrium assumptions and an independent reference when applied to real chemicals.

Stated limitations

Low-pressure Raoult law; hypothetical saturation data anchored at 350 K.

Step-by-step investigation

1. Record the default inputs and the readout with its units. Sketch the qualitative expected trend before inspecting the chart.
2. Decrease Liquid mole fraction A to its minimum while keeping Temperature fixed. Record the change and explain which term in the governing equation is responsible.
3. Increase Liquid mole fraction A to its maximum and compare the response with the first observation. Determine whether the dependence is linear, logarithmic, exponential or nonlinear for a reason you can explain.
4. Reset, then alter Temperature and repeat. Estimate a finite-difference sensitivity between two nearby parameter values, including its units.
5. Export the response curve to CSV. Explain how a plotted sensitivity helps select a range for a laboratory measurement or a process-control test.
6. Identify the first physical mechanism omitted by the model that would become important if the system were pressurized, heated, mixed or scaled up substantially.

Questions

Reasoning: Why does vapor enrich the higher-saturation-pressure component?

Extension: Change temperature and compare with changing composition.

Uncertainty: How would uncertainty in temperature influence your interpretation of the response curve? Which single additional measurement would best distinguish model error from input uncertainty?

Evidence to submit

Export a parameter sweep from the interactive model and attach it with your explanation of P=xA PsatA+(1−xA)PsatB. Demonstrate the predicted change in Liquid mole fraction A at two different settings of Temperature. Conclude by explaining whether the laboratory's approximation (Low-pressure Raoult law; hypothetical saturation data anchored at 350 K.) would remain credible in a real chemical process.

Unit 5 Assessment — Vapor–liquid systems

Topic 1: Raoult-law bubble pressure

1. State the governing physical assumption for the relation P_bubble=Σ x_i P_i^sat; y_i=x_i P_i^sat/P., and describe a failure case.
2. Derive or interpret the connection between raoult-law bubble pressure and the following equipment context: A richer light-component liquid typically has a higher bubble pressure at fixed temperature when its pure saturation pressure is greater.
3. Create two credible numerical inputs using correct units, and explain the predicted response direction before computing it.
4. Assess this warning using a counterexample: An ideal mixture approximation may fail for hydrogen-bonding or strongly dissimilar liquids; vapor nonideality matters at high pressure.
5. Propose an independent reference or experimental measurement that would distinguish numerical accuracy from physical accuracy.

Answer guidance: A correct explanation must state: For an ideal liquid solution at modest pressure, vapor partial pressures scale with liquid mole fractions and pure-component saturation pressures. The central decision is: Estimate vaporization onset for binary solvent handling. Check unit conventions, the sign of the outcome and whether the selected thermodynamic state exists in the assumed phase region.

Topic 2: Dew-pressure and dew-temperature calculations

1. State the governing physical assumption for the relation At ideal low pressure Σ y_i P/P_i^sat(T)=1 at dew temperature., and describe a failure case.
2. Derive or interpret the connection between dew-pressure and dew-temperature calculations and the following equipment context: A condenser can be screened for its first-liquid appearance by solving a nonlinear scalar temperature equation.
3. Create two credible numerical inputs using correct units, and explain the predicted response direction before computing it.
4. Assess this warning using a counterexample: A dew calculation performed with feed liquid fractions instead of vapor fractions solves a different problem and produces false results.
5. Propose an independent reference or experimental measurement that would distinguish numerical accuracy from physical accuracy.

Answer guidance: A correct explanation must state: The first droplet formed from vapor must satisfy vapor and incipient-liquid compositions simultaneously. The central decision is: Set condenser approach temperatures with uncertainty margins. Check unit conventions, the sign of the outcome and whether the selected thermodynamic state exists in the assumed phase region.

Topic 3: Nonideal liquid activities

1. State the governing physical assumption for the relation Modified Raoult relation y_i P≈x_i γ_i P_i^sat under low-pressure vapor conditions., and describe a failure case.
2. Derive or interpret the connection between nonideal liquid activities and the following equipment context: A nonideal mixture may exhibit an azeotrope where vapor and liquid compositions become equal at a particular equilibrium state.
3. Create two credible numerical inputs using correct units, and explain the predicted response direction before computing it.
4. Assess this warning using a counterexample: Using γ_i=1 without evidence can conceal azeotropes and cause impossible separation specifications.
5. Propose an independent reference or experimental measurement that would distinguish numerical accuracy from physical accuracy.

Answer guidance: A correct explanation must state: Activity coefficients correct chemical potential in a liquid mixture when interactions depart from ideal mixing. The central decision is: Use experimental VLE data to decide whether simple correlations are appropriate. Check unit conventions, the sign of the outcome and whether the selected thermodynamic state exists in the assumed phase region.

Topic 4: Flash vaporization and Rachford–Rice

1. State the governing physical assumption for the relation F(β)=Σ z_i(K_i−1)/(1+β(K_i−1))=0, with 0≤β≤1 and denominator positive., and describe a failure case.
2. Derive or interpret the connection between flash vaporization and rachford–rice and the following equipment context: For known equilibrium ratios a feasible two-phase solution produces consistent liquid and vapor compositions and an overall component balance.
3. Create two credible numerical inputs using correct units, and explain the predicted response direction before computing it.
4. Assess this warning using a counterexample: Blindly solving the scalar equation without a phase-feasibility check may return a nonphysical vapor fraction.
5. Propose an independent reference or experimental measurement that would distinguish numerical accuracy from physical accuracy.

Answer guidance: A correct explanation must state: A constant-T,P flash solves component mass balances and equilibrium partitioning together with unknown vapor fraction. The central decision is: Perform process-level flash checks and sensitivity analysis. Check unit conventions, the sign of the outcome and whether the selected thermodynamic state exists in the assumed phase region.