Polytropic and General Quasistatic Paths
For a quasistatic ideal-gas path described by P(V) or PV^k = constant, work is ∫P(V)dV and heat follows Q = ΔU + W; for constant k ≠ 1, W_by = (P_fV_f − P_iV_i)/(1−k).
Why this shows up in the exam
Power-law pressure-volume paths · Temperature-dependent volume paths · Finding effective molar heat capacity
Learn the idea
A specified relation between pressure, volume, and temperature defines the path and its effective heat capacity. When the path is neither a standard named process nor a simple straight line, substitute its equation before integrating work.
🧠 Memory hook: Path equation first, integral second, first law last.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- W_by = ∫_(V_i)^(V_f) P(V) dV — signed quasistatic work for any specified pressure-volume path
- PV^k = constant => W_by = (P_fV_f-P_iV_i)/(1-k) — polytropic work for constant k not equal to one
- C_path = C_V + R/(1-k) — molar path heat capacity for an ideal-gas polytrope under constant heat capacities
How to approach it
- 1Rewrite the path in an integrable variable
- 2Find endpoint temperatures from PV = nRT
- 3Compute work, then ΔU and Q with signs
Common slip-ups that cost marks
- •Using endpoint-average pressure for a curved path
- •Applying the k ≠ 1 formula to an isotherm k = 1
- •Calling path heat capacity a universal material constant
🌟 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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