Questions: Polytropic Processes and the Polytropic Index

Question 1

A gas compressor operates with a polytropic index of n = 1.3. Which of the following best describes its thermodynamic behavior?

• A. It is isothermal — temperature stays constant throughout compression
• B. It is adiabatic — no heat is exchanged with the surroundings
• C. It exchanges some heat with surroundings, behaving between isothermal and adiabatic
• D. It is isochoric — volume stays constant during compression

Answer: (C)

n = 1 is isothermal and n = γ ≈ 1.4 is adiabatic. A value of n = 1.3, between these two extremes, describes a process that exchanges some (but not all) heat — neither perfectly insulated nor in complete thermal equilibrium. This is the typical regime for real compressors, which operate too fast for full heat exchange but are not perfectly insulated.

Question 2

Which polytropic index n corresponds to a constant-volume (isochoric) process?

• A. n = 0
• B. n = 1
• C. n = γ
• D. n = ∞

Answer: (D)

Writing the polytropic relation as P^(1/n) · V = constant and taking n → ∞ sends 1/n → 0, so P^0 · V = V = constant — a constant volume process. This is non-obvious; many students guess n = 0 (which is actually isobaric: PV^0 = P = constant) or n = γ (adiabatic). The ∞ limit captures isochoric because infinite 'stiffness' in the PV^n relation forces volume to be fixed.

Question 3

The isothermal and adiabatic processes are largely separate thermodynamic models with no unifying mathematical structure.

• T. True
• F. False

Answer: False

Both are special cases of the polytropic relation PV^n = constant — n = 1 for isothermal, n = γ for adiabatic. The polytropic framework is precisely the unifying structure that embeds all four fundamental process types (isobaric, isothermal, adiabatic, isochoric) as limiting or special values of a single parameter n.

Question 4

For a polytropic process with n between 1 and γ, the system exchanges some heat with its surroundings — it is neither fully insulated nor isothermal.

• T. True
• F. False

Answer: True

n = γ is fully adiabatic (no heat exchange) and n = 1 is isothermal (heat exchange maintains constant temperature). Any n strictly between 1 and γ represents intermediate behavior: partial heat exchange. This is why real engineering devices are often modeled polytropically — they fit into this intermediate regime.

Question 5

Why is the work formula W = (P₁V₁ − P₂V₂)/(n − 1) undefined when n = 1, and what formula is used instead?

Model answer: When n = 1, the denominator (n − 1) equals zero, making the formula undefined. This is not a mathematical accident — at n = 1 the process is isothermal, and integrating P = C/V (rather than C/V^n) gives W = nRT ln(V₂/V₁), which involves a logarithm rather than a power law. The singularity at n = 1 reflects a genuine change in the functional form of the work integral.

This question tests whether students understand that the polytropic work formula is derived by integrating PdV with P = C/V^n. For n ≠ 1, this integral produces a power-law result with (n−1) in the denominator. For n = 1, the integral of 1/V is ln(V), yielding the isothermal work formula. Knowing when a formula applies — and what replaces it at singular cases — is essential for correct application.