Polytropic processes — defining the polytropic index (n): For a polytropic process of a perfect gas described by p * v^n = constant between states 1 and 2, which expression correctly gives the polytropic index n?

Introduction / Context:
Many real compression and expansion processes do not follow purely isothermal (n = 1) or adiabatic (n = c_p/c_v = k) paths. The polytropic model p * v^n = constant fits intermediate behavior. Determining the polytropic index n from end states is a frequent task in cycle analysis and compressor/turbine testing.

Given Data / Assumptions:

• States 1 and 2 of a closed-system gas process are known.
• Polytropic relation holds: p * v^n = constant.
• Ideal-gas behavior (for interpretive clarity).

Concept / Approach:
From p1 * v1^n = p2 * v2^n, rearrange to isolate n. Taking natural logarithms yields n = ln(p2/p1) / ln(v1/v2). This relation uses only end-state pressure and volume ratios and is independent of path length, provided the process is truly polytropic.

Step-by-Step Solution:
Start: p1 * v1^n = p2 * v2^n.Divide: (p2/p1) = (v1/v2)^n.Take ln: ln(p2/p1) = n * ln(v1/v2).Solve for n: n = ln(p2/p1) / ln(v1/v2).

Verification / Alternative check:
Special cases: if n = 1, then p * v = constant (isothermal); if n = k = c_p/c_v, the path is adiabatic for an ideal gas—confirming the formula’s consistency when applied to appropriate p, v ratios.

Why Other Options Are Wrong:
Option B flips the volume ratio, giving a sign error. Option C mixes temperature and volume and is not the general definition. Option D equals k (adiabatic index), which applies only to reversible adiabatic processes, not general polytropes.

Common Pitfalls:
Using base-10 vs. natural logs inconsistently; any logarithm base works as long as the same base is used in numerator and denominator.

Final Answer:
n = ln(p2/p1) / ln(v1/v2)