Physical chemistry: equilibrium depends on the imposed constraint
“The equilibrium constant is the same” does not mean the conversion is the same. The vessel constraint changes partial pressures as a reaction proceeds. Derive those pressures before solving an equilibrium equation.
Course topic: Physical Chemistry
Study route and exercise: University Reading List editorial project. How we curate · Sources checked:
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Scope and responsibility
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.
Physical Chemistry
Peter Atkins; Julio de Paula; James Keeler · Edition: 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.
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.
Check your preparation
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.
A six-step study plan
Steps completed: 0 / 6
These are your own study marks, not an assessment. They are not sent to this site.
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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.
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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.
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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°.
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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.
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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°.
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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.
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.
- 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).
- Q=(pB/p°)²/(pA/p°)=4x²/(1−x²). Set Q=4: x²=1−x², hence the admissible root x=1/√2.
- 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.
Answer and interpretation
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
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.
- Now pA/p°=1−x and pB/p°=2x, so Q=4x²/(1−x). Pressure rises as molecules are produced.
- 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°.
- 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.
Answer and interpretation
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.
Before moving on
- Derive activities from a vessel constraint.
- Solve and check two distinct equilibrium conversions.
- Explain stability without claiming a reaction rate.
Sources and edition checks
Sources checked:
- DeVoe: thermodynamic potentials and chemical equilibrium
The hosted text separates potentials, mixtures and reaction equilibrium, supporting the constraint-first route.
- OUP: exact twelfth-edition paperback and contents
Exact twelfth-edition contents identify mixtures, chemical equilibrium and statistical thermodynamics as separate focuses.
- University Science 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.