Polytropic processes – identify the hyperbolic (isothermal) case For a general polytropic process defined by p * v^n = C, the process is called hyperbolic when n equals which value?

Introduction / Context:
Polytropic processes unify many familiar paths (isothermal, isentropic, isobaric, isochoric) under the relation p * v^n = constant. Recognizing special values of n helps quickly identify process type and apply the right formulas for work and heat.

Given Data / Assumptions:

• Closed system with a simple compressible substance.
• Quasi-static process enabling state-variable relations.
• Ideal gas interpretation for the isothermal identification.

Concept / Approach:

When n = 1, the relation becomes p * v = constant, which plots as a rectangular hyperbola on the P–v plane. For an ideal gas, p * v = R * T, so n = 1 corresponds to constant temperature (isothermal). Other notable cases: n = 0 → isobaric; n → ∞ → isochoric; n = γ → isentropic (ideal gas). The term “hyperbolic” specifically refers to the n = 1 curve shape on P–v axes.

Step-by-Step Solution:

Verification / Alternative check:

P–v plots of isothermal ideal-gas compression/expansion show hyperbolic curves, matching the n = 1 polytropic case.

Why Other Options Are Wrong:

n = 0: isobaric; n = γ: isentropic; n → ∞: isochoric; n = −1 does not give the standard hyperbola shape used in this nomenclature.

Common Pitfalls:

Assuming “hyperbolic” means any curved line; here it specifically denotes p * v = constant.

Final Answer:

1