Process identification from p * v^n = C If a compressible system follows the law p * v^n = constant and the polytropic index n tends to infinity (n → ∞), then the process is effectively which type?

Introduction / Context:The generalized polytropic law p * v^n = constant unifies several familiar processes. Recognizing special values and limits of n helps identify the corresponding physical process (isothermal, adiabatic, isobaric, or isochoric).

Given Data / Assumptions:

• Quasi-equilibrium compression/expansion so that state variables are well defined.
• Polytropic index n is constant during the process.
• Ideal-gas intuition may be used for interpretation, though the identification via limits is general.

Concept / Approach:Important polytropic special cases: n = 1 → isothermal (for ideal gases), n = γ → reversible adiabatic, n = 0 → isobaric. The limit n → ∞ forces the volume to remain constant to keep p * v^n finite, representing an isochoric (constant volume) process.

Step-by-Step Solution:Start with p * v^n = C.Let n grow very large. Any small change in v would make v^n blow up or vanish unless v is fixed.Thus, to satisfy the relation with finite C, v must remain constant, i.e., an isochoric process.Therefore, the correct identification is constant volume process.

Verification / Alternative check:Graphically on a p–v diagram, as n increases, polytropes get steeper, approaching a vertical line (v = constant) in the limit n → ∞.

Why Other Options Are Wrong:

• Adiabatic corresponds to n = γ, not n → ∞.
• Isothermal corresponds to n = 1 (ideal gas).
• Isobaric corresponds to n = 0.
• Free expansion has no defined p–v path satisfying a polytropic law.

Common Pitfalls:Confusing the adiabatic value γ (about 1.3–1.67) with very large n; they have very different geometric interpretations.

Final Answer:constant volume process