Polytropic index for adiabatic (isentropic) behavior For a perfect gas undergoing a reversible adiabatic process, the polytropic relation p * v^n = C holds with n equal to which parameter?

Introduction / Context:
Polytropic processes unify many paths with a single exponent n in p * v^n = constant. Recognizing special n values helps you quickly select formulas for work, heat, and temperature changes, especially for adiabatic compression or expansion of gases.

Given Data / Assumptions:

• Perfect (ideal) gas.
• Reversible adiabatic (isentropic) process: no heat transfer and no internal dissipation.
• Constant specific heats for simplicity so that γ = cp/cv is constant.

Concept / Approach:

For a reversible adiabatic process of an ideal gas, p * v^γ = constant, T * v^(γ−1) = constant, and p^(1−γ) * T^γ = constant all hold. These are derived from the First law with Q = 0 and the ideal-gas equation, leading to the identification that the polytropic exponent n coincides with γ for isentropic behavior.

Step-by-Step Solution:

Verification / Alternative check:

On a log–log P–v plot, the slope of an isentropic line is −γ. Empirical compressor maps approximate this behavior at high efficiencies.

Why Other Options Are Wrong:

n = 0: isobaric; n = 1: isothermal for ideal gases; n → ∞: isochoric; n = −1: not the isentropic case.

Common Pitfalls:

Confusing any adiabatic with isentropic; irreversibilities change the effective exponent to a value different from γ in polytropic modeling.

Final Answer:

γ