General gas process relation — identify the most general law For quasi-static expansion or compression of a gas between two equilibrium states, which relation represents the general polytropic form that encompasses special cases like isothermal and adiabatic?

Introduction / Context:
Many real compression/expansion paths can be approximated by the polytropic relation p * v^n = C. This flexible model reduces to familiar special cases for particular n values and is widely used in compressor and expander performance analysis.

Given Data / Assumptions:

• Simple compressible gas, quasi-static path between two states.
• Ideal-gas equation of state may be used for interpretation.
• Constant specific heats over the range (for adiabatic special case).

Concept / Approach:
The polytropic law p * v^n = C includes: n = 1 (isothermal, p * v = C), n = γ (reversible adiabatic, p * v^γ = C), and other n values modeling heat transfer during compression/expansion. It is more general than any one special case and fits measured data by choosing n appropriately.

Step-by-Step Solution:
Recognize that option B is the equation of state (not a process relation).Option A is the specific isothermal case (n = 1).Option D is the adiabatic special case and does not represent all real processes.Thus, the general process relation sought is p * v^n = C.

Verification / Alternative check:
Experimental compressor/expander data often fit a constant n over a pressure ratio range, validating the polytropic model for engineering prediction.

Why Other Options Are Wrong:
Equations for specific cases (A, D) lack generality; the ideal-gas equation (B) is a state equation; (E) is Charles’ law at constant pressure.

Common Pitfalls:
Using γ indiscriminately for non-adiabatic processes; confusing process path laws with equations of state.

Final Answer:
p * v^n = C