Chemical Reaction Equilibria and Multi-Phase Reaction Systems

Why this unit matters

Thermodynamic workflows, cycles, separations and uncertainty management. The preceding prerequisite is Reaction equilibrium; the destination is capstone process decisions. 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. Compression and expansion efficiency

Adiabatic machines are benchmarked against reversible isentropic endpoints; real performance is defined by enthalpy changes.

Useful calculation: For an ideal gas with constant k, T2s/T1=(P2/P1)^((k−1)/k); compressor η_is=(h2s−h1)/(h2−h1).

Physical interpretation: A compressor with reduced efficiency delivers hotter discharge and consumes more shaft power for the same inlet and outlet pressures.

Failure to watch: Do not use ideal-gas k at supercritical conditions or near condensation; multi-stage compression may require intercooling and a real-fluid EOS.

Decision: Calculate heat rejection needs from efficiency trends.

2. Refrigeration and heat pumps

A vapor-compression cycle transfers heat from low to high temperature by consuming work and exploiting phase changes.

Useful calculation: COP_refrigerator=Q_c/W; COP_heatpump=Q_h/W; energy conservation gives Q_h=Q_c+W.

Physical interpretation: Refrigerant approach temperatures and compressor inefficiency strongly influence practical electricity consumption.

Failure to watch: An ideal Carnot COP is an upper bound and not a forecast of real equipment performance; refrigerant environmental and safety constraints also matter.

Decision: Analyze seasonal sensitivity without inventing vendor performance data.

3. Thermodynamics of separation work

Mixing creates entropy, and unmixing requires energy or available work even before real equipment losses are included.

Useful calculation: For ideal mixing ΔG_mix=nRT Σ x_i ln x_i; minimum reversible separation work is −ΔG_mix for an appropriate specified process.

Physical interpretation: Very dilute solutes may demand substantial work per mole recovered because an extreme purity specification is thermodynamically demanding.

Failure to watch: The minimum reversible work is not the steam or electric use of a real column, which includes finite heat transfer and imperfect separation.

Decision: Explain why recovery and purity targets increase cost.

4. Validation, uncertainty and process decisions

A defensible thermodynamic prediction has explicit references, assumptions, validity limits, sensitivity tests and property-data provenance.

Useful calculation: Propagated first-order variance: σ_y²≈Σ(∂f/∂x_i)²σ_i² when inputs are independent.

Physical interpretation: If vapor-pressure data and temperature measurements each have uncertainty, report a range of flash fractions rather than a false exact operating point.

Failure to watch: Numerically smooth graphs do not certify a physical model; uncertainty in phase behavior often exceeds floating-point solver error.

Decision: Build a review checklist and reject unqualified designs.

Unit assessment

• Problem 1A — Compression and expansion efficiency. 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 compression and expansion efficiency and a way to quantify its consequence.
• Problem 1C — Counterexample. Construct an operating case where the assumption “Do not use ideal-gas k at supercritical conditions or near condensation; multi-stage compression may require intercooling and a real-fluid EOS.” becomes important and would invalidate a simplistic answer.
• Problem 2A — Refrigeration and heat pumps. 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 refrigeration and heat pumps and a way to quantify its consequence.
• Problem 2C — Counterexample. Construct an operating case where the assumption “An ideal Carnot COP is an upper bound and not a forecast of real equipment performance; refrigerant environmental and safety constraints also matter.” becomes important and would invalidate a simplistic answer.
• Problem 3A — Thermodynamics of separation work. 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 thermodynamics of separation work and a way to quantify its consequence.
• Problem 3C — Counterexample. Construct an operating case where the assumption “The minimum reversible work is not the steam or electric use of a real column, which includes finite heat transfer and imperfect separation.” becomes important and would invalidate a simplistic answer.
• Problem 4A — Validation, uncertainty and process decisions. 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 validation, uncertainty and process decisions and a way to quantify its consequence.
• Problem 4C — Counterexample. Construct an operating case where the assumption “Numerically smooth graphs do not certify a physical model; uncertainty in phase behavior often exceeds floating-point solver error.” becomes important and would invalidate a simplistic answer.

Assessment evidence

Unit 8 · Lesson 1: Compression and expansion efficiency

Learning objectives. Explain compression and expansion efficiency 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

Adiabatic machines are benchmarked against reversible isentropic endpoints; real performance is defined by enthalpy changes.

Governing relation and interpretation

Key relation: For an ideal gas with constant k, T2s/T1=(P2/P1)^((k−1)/k); compressor η_is=(h2s−h1)/(h2−h1).

Engineering example and model boundary

A compressor with reduced efficiency delivers hotter discharge and consumes more shaft power for the same inlet and outlet pressures.

Worked investigation

Air enters an ideal-gas compressor at 300 K and experiences P2/P1=4 with k=1.4. Estimate isentropic discharge temperature and compare η_is=0.78.

Actual compressor discharge temperature exceeds the isentropic estimate when η_is is below one; the model ignores variable heat capacities.

Where the relation breaks down

Do not use ideal-gas k at supercritical conditions or near condensation; multi-stage compression may require intercooling and a real-fluid EOS.

Engineering decision and data checklist

Calculate heat rejection needs from efficiency trends.

Guided problems

1. Definition and boundary. Describe the relevant system and identify which properties must be uniform before using the governing relation for compression and expansion efficiency.

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 Compression and expansion efficiency using the assumptions discussed in this lesson.

Lesson synthesis

Subject focus: The key distinguishing result of this lesson is the link between compression and expansion efficiency and the application “Calculate heat rejection needs from efficiency trends.” The governing relation is For an ideal gas with constant k, T2s/T1=(P2/P1)^((k−1)/k); compressor η_is=(h2s−h1)/(h2−h1).; the stated failure mode is Do not use ideal-gas k at supercritical conditions or near condensation; multi-stage compression may require intercooling and a real-fluid EOS.

Unit 8 · Lesson 2: Refrigeration and heat pumps

Learning objectives. Explain refrigeration and heat pumps 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 vapor-compression cycle transfers heat from low to high temperature by consuming work and exploiting phase changes.

Governing relation and interpretation

Key relation: COP_refrigerator=Q_c/W; COP_heatpump=Q_h/W; energy conservation gives Q_h=Q_c+W.

Engineering example and model boundary

Refrigerant approach temperatures and compressor inefficiency strongly influence practical electricity consumption.

Worked investigation

A refrigerator removes 3.5 kW from cold storage while consuming 1.0 kW electrical-equivalent shaft input. Its COP is 3.5 and it rejects 4.5 kW.

The electric meter and compressor shaft can differ due to drive losses; COP boundaries must be aligned before comparing systems.

Where the relation breaks down

An ideal Carnot COP is an upper bound and not a forecast of real equipment performance; refrigerant environmental and safety constraints also matter.

Engineering decision and data checklist

Analyze seasonal sensitivity without inventing vendor performance data.

Guided problems

1. Definition and boundary. Describe the relevant system and identify which properties must be uniform before using the governing relation for refrigeration and heat pumps.

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 Refrigeration and heat pumps using the assumptions discussed in this lesson.

Lesson synthesis

Subject focus: The key distinguishing result of this lesson is the link between refrigeration and heat pumps and the application “Analyze seasonal sensitivity without inventing vendor performance data.” The governing relation is COP_refrigerator=Q_c/W; COP_heatpump=Q_h/W; energy conservation gives Q_h=Q_c+W.; the stated failure mode is An ideal Carnot COP is an upper bound and not a forecast of real equipment performance; refrigerant environmental and safety constraints also matter.

Unit 8 · Lesson 3: Thermodynamics of separation work

Learning objectives. Explain thermodynamics of separation work 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

Mixing creates entropy, and unmixing requires energy or available work even before real equipment losses are included.

Governing relation and interpretation

Key relation: For ideal mixing ΔG_mix=nRT Σ x_i ln x_i; minimum reversible separation work is −ΔG_mix for an appropriate specified process.

Engineering example and model boundary

Very dilute solutes may demand substantial work per mole recovered because an extreme purity specification is thermodynamically demanding.

Worked investigation

At 300 K, separate an equimolar ideal liquid mixture containing 100 mol total into pure components. The ideal mixing Gibbs energy has magnitude nRT ln2.

The minimum reversible separation work is positive because spontaneous mixing lowers Gibbs energy; real separation heat demand is larger.

Where the relation breaks down

The minimum reversible work is not the steam or electric use of a real column, which includes finite heat transfer and imperfect separation.

Engineering decision and data checklist

Explain why recovery and purity targets increase cost.

Guided problems

1. Definition and boundary. Describe the relevant system and identify which properties must be uniform before using the governing relation for thermodynamics of separation work.

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 Thermodynamics of separation work using the assumptions discussed in this lesson.

Lesson synthesis

Subject focus: The key distinguishing result of this lesson is the link between thermodynamics of separation work and the application “Explain why recovery and purity targets increase cost.” The governing relation is For ideal mixing ΔG_mix=nRT Σ x_i ln x_i; minimum reversible separation work is −ΔG_mix for an appropriate specified process.; the stated failure mode is The minimum reversible work is not the steam or electric use of a real column, which includes finite heat transfer and imperfect separation.

Unit 8 · Lesson 4: Validation, uncertainty and process decisions

Learning objectives. Explain validation, uncertainty and process decisions 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 defensible thermodynamic prediction has explicit references, assumptions, validity limits, sensitivity tests and property-data provenance.

Governing relation and interpretation

Key relation: Propagated first-order variance: σ_y²≈Σ(∂f/∂x_i)²σ_i² when inputs are independent.

Engineering example and model boundary

If vapor-pressure data and temperature measurements each have uncertainty, report a range of flash fractions rather than a false exact operating point.

Worked investigation

A modeled dew temperature is 342 K with ±2 K sensor error and uncertain vapor-pressure coefficients. Estimate the output sensitivity from finite differences.

A solver residual near zero is not evidence that the physical answer is accurate; property-parameter uncertainties can dominate.

Where the relation breaks down

Numerically smooth graphs do not certify a physical model; uncertainty in phase behavior often exceeds floating-point solver error.

Engineering decision and data checklist

Build a review checklist and reject unqualified designs.

Guided problems

1. Definition and boundary. Describe the relevant system and identify which properties must be uniform before using the governing relation for validation, uncertainty and process decisions.

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 Validation, uncertainty and process decisions using the assumptions discussed in this lesson.

Lesson synthesis

Subject focus: The key distinguishing result of this lesson is the link between validation, uncertainty and process decisions and the application “Build a review checklist and reject unqualified designs.” The governing relation is Propagated first-order variance: σ_y²≈Σ(∂f/∂x_i)²σ_i² when inputs are independent.; the stated failure mode is Numerically smooth graphs do not certify a physical model; uncertainty in phase behavior often exceeds floating-point solver error.

Laboratory 15: Henry-law dissolved-gas screening

Model: c=kH p

Controllable variables: Gas partial pressure (kPa) and Concentration-form kH (mol m⁻³ kPa⁻¹).

Governing relation

c=kH p. 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

Dilute dissolved gas, fixed temperature and chosen coefficient convention.

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 Gas partial pressure to its minimum while keeping Concentration-form kH fixed. Record the change and explain which term in the governing equation is responsible.
3. Increase Gas partial pressure 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 Concentration-form kH 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: What error arises if a reference defines the reciprocal form of Henry law?

Extension: Double gas partial pressure and compare predicted equilibrium concentration.

Uncertainty: How would uncertainty in concentration-form kh 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 c=kH p. Demonstrate the predicted change in Gas partial pressure at two different settings of Concentration-form kH. Conclude by explaining whether the laboratory's approximation (Dilute dissolved gas, fixed temperature and chosen coefficient convention.) would remain credible in a real chemical process.

Laboratory 16: Equilibrium liquid–liquid extraction

Model: Recovered fraction=K Vorg/(Vaq+K Vorg)

Controllable variables: Partition ratio K (dimensionless) and Organic solvent volume (L).

Governing relation

Recovered fraction=K Vorg/(Vaq+K Vorg). 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

Constant dilute partition ratio; aqueous volume fixed at 8 L.

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 Partition ratio K to its minimum while keeping Organic solvent volume fixed. Record the change and explain which term in the governing equation is responsible.
3. Increase Partition ratio K 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 Organic solvent volume 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 more solvent provide diminishing gains in recovered fraction?

Extension: Find a solvent volume that yields at least 75% extraction.

Uncertainty: How would uncertainty in organic solvent volume 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 Recovered fraction=K Vorg/(Vaq+K Vorg). Demonstrate the predicted change in Partition ratio K at two different settings of Organic solvent volume. Conclude by explaining whether the laboratory's approximation (Constant dilute partition ratio; aqueous volume fixed at 8 L.) would remain credible in a real chemical process.

Unit 8 Assessment — Engineering applications

Topic 1: Compression and expansion efficiency

1. State the governing physical assumption for the relation For an ideal gas with constant k, T2s/T1=(P2/P1)^((k−1)/k); compressor η_is=(h2s−h1)/(h2−h1)., and describe a failure case.
2. Derive or interpret the connection between compression and expansion efficiency and the following equipment context: A compressor with reduced efficiency delivers hotter discharge and consumes more shaft power for the same inlet and outlet pressures.
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: Do not use ideal-gas k at supercritical conditions or near condensation; multi-stage compression may require intercooling and a real-fluid EOS.
5. Propose an independent reference or experimental measurement that would distinguish numerical accuracy from physical accuracy.

Answer guidance: A correct explanation must state: Adiabatic machines are benchmarked against reversible isentropic endpoints; real performance is defined by enthalpy changes. The central decision is: Calculate heat rejection needs from efficiency trends. Check unit conventions, the sign of the outcome and whether the selected thermodynamic state exists in the assumed phase region.

Topic 2: Refrigeration and heat pumps

1. State the governing physical assumption for the relation COP_refrigerator=Q_c/W; COP_heatpump=Q_h/W; energy conservation gives Q_h=Q_c+W., and describe a failure case.
2. Derive or interpret the connection between refrigeration and heat pumps and the following equipment context: Refrigerant approach temperatures and compressor inefficiency strongly influence practical electricity consumption.
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 Carnot COP is an upper bound and not a forecast of real equipment performance; refrigerant environmental and safety constraints also matter.
5. Propose an independent reference or experimental measurement that would distinguish numerical accuracy from physical accuracy.

Answer guidance: A correct explanation must state: A vapor-compression cycle transfers heat from low to high temperature by consuming work and exploiting phase changes. The central decision is: Analyze seasonal sensitivity without inventing vendor performance data. Check unit conventions, the sign of the outcome and whether the selected thermodynamic state exists in the assumed phase region.

Topic 3: Thermodynamics of separation work

1. State the governing physical assumption for the relation For ideal mixing ΔG_mix=nRT Σ x_i ln x_i; minimum reversible separation work is −ΔG_mix for an appropriate specified process., and describe a failure case.
2. Derive or interpret the connection between thermodynamics of separation work and the following equipment context: Very dilute solutes may demand substantial work per mole recovered because an extreme purity specification is thermodynamically demanding.
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: The minimum reversible work is not the steam or electric use of a real column, which includes finite heat transfer and imperfect separation.
5. Propose an independent reference or experimental measurement that would distinguish numerical accuracy from physical accuracy.

Answer guidance: A correct explanation must state: Mixing creates entropy, and unmixing requires energy or available work even before real equipment losses are included. The central decision is: Explain why recovery and purity targets increase cost. Check unit conventions, the sign of the outcome and whether the selected thermodynamic state exists in the assumed phase region.

Topic 4: Validation, uncertainty and process decisions

1. State the governing physical assumption for the relation Propagated first-order variance: σ_y²≈Σ(∂f/∂x_i)²σ_i² when inputs are independent., and describe a failure case.
2. Derive or interpret the connection between validation, uncertainty and process decisions and the following equipment context: If vapor-pressure data and temperature measurements each have uncertainty, report a range of flash fractions rather than a false exact operating point.
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: Numerically smooth graphs do not certify a physical model; uncertainty in phase behavior often exceeds floating-point solver error.
5. Propose an independent reference or experimental measurement that would distinguish numerical accuracy from physical accuracy.

Answer guidance: A correct explanation must state: A defensible thermodynamic prediction has explicit references, assumptions, validity limits, sensitivity tests and property-data provenance. The central decision is: Build a review checklist and reject unqualified designs. Check unit conventions, the sign of the outcome and whether the selected thermodynamic state exists in the assumed phase region.