Heat Capacities, Degrees of Freedom, and Mayer Relation
For a calorically perfect ideal gas with f active quadratic degrees of freedom, C_V = fR/2, C_P = C_V + R, and γ = C_P/C_V; vibrational modes are included only when thermally active as stated.
Why this shows up in the exam
Identifying monatomic or rigid-diatomic gases · Finding heat in constant-pressure or constant-volume heating · Deducing γ or molecular degrees of freedom
Learn the idea
Molecular degrees of freedom determine C_V, while ideal-gas expansion makes C_P exceed C_V by R. Heating at constant volume only raises microscopic energy; heating at constant pressure must also supply expansion work.
🧠 Memory hook: At constant pressure, add one R for expansion.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- C_V = fR/2; C_P = (f+2)R/2 — molar heat capacities for an ideal gas with f active quadratic degrees of freedom
- C_P - C_V = R — Mayer relation for one mole of an ideal gas
- γ = C_P/C_V = 1 + 2/f — heat-capacity ratio under the same active-mode assumption
How to approach it
- 1Determine active degrees of freedom
- 2Choose C_V or C_P from the constraint
- 3Keep molar, specific, and total heat capacities distinct
Common slip-ups that cost marks
- •Using Celsius temperature in absolute ratios
- •Always assigning five degrees of freedom to every diatomic gas
- •Confusing total heat capacity nC with molar heat capacity C
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A gas absorbs 500 J of heat and does 200 J of work. What is the change in its internal energy?
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