- 2073 Bhadra · 4 marks
Polytropic processes for different n; work
Short answer
A polytropic process obeys PVⁿ = constant. On a P-V diagram, higher n gives steeper curves: n=0 (constant pressure), n=1 (isothermal), n=k (adiabatic), n=∞ (constant volume). Work transfer is W = (P₁V₁ - P₂V₂)/(n-1) for n≠1, and W = P₁V₁ ln(V₂/V₁) for n=1.
Define polytropic process. Sketch polytropic processes on a common graph for different values of n. Also derive an expression for work transfer for a polytropic process.
Polytropic process
A polytropic process is one in which the pressure and volume of a gas are related by PVⁿ = constant, where n is the polytropic index. It is a general process which includes many particular ones.
Graphs for different n
P|*
|*\ n = inf (V = const, vertical)
|* \ \ n = k (adiabatic)
|* \ \ \ n = 1 (isothermal)
|*---\-\-\----- n = 0 (P = const)
+----------------- V
The standard cases are:
| n | Process | Relation |
|---|---|---|
| 0 | Constant pressure | P = C (horizontal line) |
| 1 | Isothermal (ideal gas) | PV = C |
| k = cp/cv | Reversible adiabatic | PVᵏ = C |
| ∞ | Constant volume | V = C (vertical line) |
Starting from the same point, the curve gets steeper as n increases: horizontal (n = 0), hyperbola (n = 1), steeper (n = k), vertical (n = ∞).
Work transfer
W₁₂ = ∫₁² P dV with P = C/Vⁿ:
W₁₂ = C (V₂¹⁻ⁿ − V₁¹⁻ⁿ)/(1 − n)
Since C = P₁V₁ⁿ = P₂V₂ⁿ:
W₁₂ = (P₁V₁ − P₂V₂)/(n − 1) = mR(T₁ − T₂)/(n − 1) (n ≠ 1)
For n = 1, W₁₂ = P₁V₁ ln(V₂/V₁).
✓ Answer checked by Claude Opus 5.5 (high effort)
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