Polytropic Process

Statement

A Polytropic Process is a reversible expansion or compression process described by the relation $PV^n = C$, where $n$ is the polytropic exponent and $C$ is a constant.

Mathematical Form

$\boxed{PV^n = C}$

Work Calculation (Closed System)

$W_{12} = \frac{P_2V_2 - P_1V_1}{1 - n} \quad (n \ne 1)$ $W_{12} = P_1V_1 \ln\left(\frac{V_2}{V_1}\right) \quad (n = 1)$

Conditions & Constraints

• Reversibility: Assumes a quasi-static process.
• Ideal Gas: Often assumed for specific property relations.

Special Cases

Process	Exponent ($n$)	Description
Isobaric	$0$	Constant Pressure ($P=C$)
Isothermal	$1$	Constant Temperature ($PV=C$ for ideal gas)
Adiabatic	$\gamma = c_p/c_v$	No Heat Transfer ($PV^\gamma=C$)
Isochoric	$\infty$	Constant Volume ($V=C$)

Relationships

• Related: Thermodynamic Work, First Law of Thermodynamics