# Physical chemistry: equilibrium depends on the imposed constraint — Study worksheet

[Study guides](https://unireadinglist.com/guides/physical-chemistry/)

[Course topic: Physical Chemistry](https://unireadinglist.com/subjects/chemistry/#course-physical-chemistry)

Sources checked: 2026-11-04

Study route and exercise: University Reading List editorial project.

Chemical thermodynamics, ideal-gas activities and equilibrium stability for an invented reaction. DeVoe is the free macroscopic spine; Atkins adds the wider physical-chemistry course; McQuarrie and Simon connect molecular states to thermodynamics. The model gives no real compound data or reaction-rate prediction.

## Choose your starting point

- [ ] I need free thermodynamic foundations
  Choose DeVoe for systems, potentials, mixtures and chemical equilibrium.

- [ ] I need the wider physical-chemistry course
  Choose Atkins when thermodynamics must connect with quantum theory and kinetics.

- [ ] I want a molecular explanation
  Choose McQuarrie and Simon after calculus and basic chemical thermodynamics.

## Compare the three books

### Thermodynamics and Chemistry

Howard DeVoe
Free macroscopic spine

Choose it when: You can use derivatives, logarithms and chemical amount balances.
Use another route when: You need a whole quantum-chemistry or spectroscopy course from this book alone.
Selected reading: Thermodynamic potentials, mixtures, reactions and multicomponent equilibrium.
Limits and trade-offs: Careful definitions make constraints visible, but the formalism requires more than inserting numbers into K.

#### Try before choosing

Write dimensionless activities for two ideal gases with partial pressures pA and pB.
Check your work: They are pA/p° and pB/p°; K and Q must not acquire pressure units.

Your notes:

________________________________

[Official free reading](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry)

[Book record and access options](https://unireadinglist.com/subjects/chemistry/thermodynamics-and-chemistry-devoe/)

### Physical Chemistry

Peter Atkins; Julio de Paula; James Keeler · 12 · 2022
Broad course text

Choose it when: Your syllabus spans thermodynamics, molecular theory and kinetics.
Use another route when: You only need a focused low-cost equilibrium reference.
Selected reading: Twelfth edition: Focus 5 Simple Mixtures, Focus 6 Chemical Equilibrium; use Focus 13 for statistical thermodynamics later.
Limits and trade-offs: Breadth helps connect topics but can dilute a first pass; selected digital resources require the relevant package.

#### Try before choosing

At fixed total pressure p°, express pB/p° using conversion x for A⇌2B.
Check your work: It is 2x/(1+x), not 2x; the total mole amount changes.

Your notes:

________________________________

ISBN: 9780198847816

[Book record and access options](https://unireadinglist.com/subjects/chemistry/physical-chemistry/)

### Molecular Thermodynamics

Donald A. McQuarrie and John D. Simon · 1999
Molecular thermodynamics route

Choose it when: You want energy levels and molecular counting to explain macroscopic quantities.
Use another route when: You need the whole larger Physical Chemistry book, including its nonthermodynamic topics.
Selected reading: Molecular energy-level foundations followed by thermodynamic functions and chemical equilibrium.
Limits and trade-offs: A molecular viewpoint adds explanation but requires mathematical preparation; this selected book is thermodynamics-focused.

#### Try before choosing

For a fixed vessel V=n0RT/p°, express the two partial pressures from amounts.
Check your work: pA/p°=1−x and pB/p°=2x; total pressure is no longer fixed.

Your notes:

________________________________

ISBN: 978-1-891-38905-4

[Book record and access options](https://unireadinglist.com/subjects/chemistry/molecular-thermodynamics/)

## Check your preparation

### Can I separate extent from conversion?

Your notes:

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### Can I derive partial pressure rather than remember a formula?

Your notes:

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### Which potential is minimized?

Your notes:

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## A six-step study plan

- [ ] Specify the model

Selected reading: Systems and state variables.

What to do: Make a table of fixed quantities, unknown x and allowed range.

Check your work: 0\<x\<1; the two vessels impose different constraints.

Your notes:

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- [ ] Balance amounts

Selected reading: Mixtures and reactions.

What to do: Derive nA,nB and ntotal with units.

Check your work: ntotal=n0(1+x), not n0.

Your notes:

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- [ ] Construct activities

Selected reading: Ideal-gas mixtures.

What to do: Derive Q separately for the two vessels.

Check your work: Every pressure appears divided by p°.

Your notes:

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- [ ] Solve fixed pressure

Selected reading: Chemical equilibrium.

What to do: Solve Q=4 and reject inadmissible roots.

Check your work: x=1/√2 lies strictly between zero and one.

Your notes:

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- [ ] Change the constraint

Selected reading: Potentials and the follow-up.

What to do: Solve the fixed-volume case and calculate final total pressure.

Check your work: x=(√5−1)/2 and P=(1+x)p°.

Your notes:

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- [ ] Check stability and limits

Selected reading: Chemical potentials.

What to do: Show Q increases through K and state the relevant minimum criterion.

Check your work: The sign of RT ln(Q/K) changes from negative to positive, not a kinetic-rate conclusion.

Your notes:

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## Worked exercise: An equilibrium conversion at fixed total pressure

For fictitious ideal gases A⇌2B at fixed T, let Kp=4 be dimensionless and start with n0=1 mol of pure A. Hold total pressure P=p°. Find the equilibrium fractional conversion x.

Assumptions: Ideal mixture, only this reaction, fixed T, no phase change, equilibrium attainable, 0\<x\<1. Standard pressure p° is positive. x=ξ/n0 is dimensionless; Kp is given, not fitted to real data.

Your notes:

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## Change an assumption: A rigid vessel gives a different equilibrium

Keep initial amounts, T and Kp, but replace pressure control by a fixed volume V=n0RT/p°. Find equilibrium conversion and explain which minimization criterion applies.

Your notes:

________________________________

## Before moving on

- Derive activities from a vessel constraint.
- Solve and check two distinct equilibrium conversions.
- Explain stability without claiming a reaction rate.

## Answer and interpretation

### Can I separate extent from conversion?

Use dimensionless x=ξ/n0 with n0=1 mol. Amounts are n0(1−x), 2n0x; adding 1 to a dimensional extent would be invalid.

### Can I derive partial pressure rather than remember a formula?

Use p\_i=n\_iRT/V, or y\_i P when total pressure is imposed. The two constraints need different expressions.

### Which potential is minimized?

At fixed T,P use Gibbs energy; at fixed T,V use Helmholtz energy. Equilibrium chemical potentials give the same K at the same T.

### An equilibrium conversion at fixed total pressure

1. Amounts are nA=n0(1−x), nB=2n0x and n=n0(1+x). Thus pA/p°=(1−x)/(1+x), pB/p°=2x/(1+x).
2. Q=(pB/p°)²/(pA/p°)=4x²/(1−x²). Set Q=4: x²=1−x², hence the admissible root x=1/√2.
3. Q increases strictly from 0 to infinity over 0\<x\<1. Since ΔrG=RT ln(Q/Kp), Gibbs energy decreases before the root and increases after it under fixed T,P.

x=1/√2≈0.7071; nA≈0.2929 mol and nB≈1.4142 mol.

Test the result: At the root pA/p°≈0.1716 and pB/p°≈0.8284; their sum is 1 and their activity quotient is 4. Monotonicity establishes the unique global equilibrium minimum within this ideal model.

### Change an assumption: A rigid vessel gives a different equilibrium

1. Now pA/p°=1−x and pB/p°=2x, so Q=4x²/(1−x). Pressure rises as molecules are produced.
2. Q=4 gives x²=1−x. The physical root is (√5−1)/2≈0.6180; the other root is negative. Final total pressure is (1+x)p°≈1.6180p°.
3. Q increases from 0 to infinity. At fixed T,V the derivative of Helmholtz energy with respect to extent is the same reaction chemical-potential sum RT ln(Q/Kp), giving the unique minimum there. Gibbs minimization at fixed T,P cannot simply be carried into this constraint.

The rigid vessel gives x≈0.6180 rather than 0.7071. The equilibrium constant is unchanged, while the partial-pressure path and relevant constrained potential change.

Test the result: x²=1−x checks Q=4 exactly. Neither result predicts how quickly equilibrium is approached; no reaction kinetics were supplied.

## Common mistakes

### A dimensional equilibrium constant

Use dimensionless activities and a named standard state.

### Keeping total amount fixed

A⇌2B changes the total number of moles.

### Using one potential for every constraint

A minimum statement requires the variables held fixed to be stated.

## Sources and edition checks

- [DeVoe: thermodynamic potentials and chemical equilibrium](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry)
  The hosted text separates potentials, mixtures and reaction equilibrium, supporting the constraint-first route.

- [OUP: exact twelfth-edition paperback and contents](https://www.oup.com.au/books/higher-education/chemistry/9780198847816)
  Exact twelfth-edition contents identify mixtures, chemical equilibrium and statistical thermodynamics as separate focuses.

- [University Science Books: molecular thermodynamics](https://uscibooks.aip.org/books/molecular-thermodynamics/)
  The selected work focuses on thermodynamics with a molecular foundation, rather than every topic of the larger physical-chemistry text.

These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.

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