Polytropic Relation Identification – Special Case n = 1 For a perfect gas undergoing a polytropic process described by p * v^n = C, determine whether the case n = 1 corresponds to an isothermal process (constant temperature).

Introduction / Context:
Recognizing special cases of the polytropic relation p * v^n = C is a standard skill in thermodynamics and compressible flow. The correct identification helps select the right equations for work, heat transfer, and state changes during compression or expansion of gases.

Given Data / Assumptions:

• Working fluid behaves as an ideal (perfect) gas in the process.
• Polytropic relation: p * v^n = constant, with exponent n specifying the path.
• Quasi-static process so that p and v are well-defined state variables along the path.

Concept / Approach:

For an ideal gas, the equation of state is p * v = R * T (on a per-unit-mass basis). If the polytropic exponent n equals 1, then p * v^1 = constant implies p * v = constant. Combining with p * v = R * T gives R * T = constant, hence T = constant. Therefore, the polytropic case n = 1 coincides with an isothermal process for an ideal gas. Other important special cases include n = 0 (isobaric), n → ∞ (isochoric), and n = gamma (isentropic for a perfect gas with constant specific heats).

Step-by-Step Solution:

Verification / Alternative check:

On a T–s diagram, an isothermal path for an ideal gas has constant temperature; consistency is also seen on a p–v plot where n = 1 polytrope is a rectangular hyperbola, matching the classic isotherm shape for ideal gases.

Why Other Options Are Wrong:

Conditions like 'only at low pressure' or 'only if no heat transfer' are unnecessary; isothermal processes do require heat exchange to keep T constant during compression/expansion, but the identification n = 1 remains valid for ideal gases without those extra caveats.

Common Pitfalls:

Confusing isothermal with isentropic; for an ideal gas, isentropic requires n = gamma, not 1. Also, mixing up the behavior of real gases at high pressures where EOS deviations occur (outside the scope here).

Final Answer:

True