Polytropic Process Work of a Closed System

Polytropic process work equation for a closed system
\( \text{Polytropic process: } pV^n = \text{constant} \)
\( \displaystyle W = \begin{cases} p_1 V_1 \ln\left(\frac{V_2}{V_1}\right) & \text{if } n = 1.0 \\ \frac{p_2 V_2 - p_1 V_1}{1 - n} & \text{if } n \neq 1.0 \end{cases} \)
Symbols
  • \(W\) — work done by the gas
  • \(V_1\) — initial volume
  • \(V_2\) — final volume
  • \(p\) — absolute pressure
  • \(n\) — polytropic index
Note: \( p_1V_1 = p_2V_2 \) for \( n = 1.0 \)
Derivation steps
The work associated with the change in the volume of a closed system due to the movement of its boundary can be evaluated by:
\( \displaystyle W = \int_{V_1}^{V_2} p \, dV \)

For \( n = 1.0 \):
\( \displaystyle W = \int_{V_1}^{V_2} \frac{\text{constant}}{V} \, dV = \text{constant} \ln\left(\frac{V_2}{V_1}\right) = p_1V_1 \ln\left(\frac{V_2}{V_1}\right) \)

For \( n \neq 1.0 \):
\( \begin{aligned} W &= \int_{V_1}^{V_2} \frac{\text{constant}}{V^n} \, dV = \frac{\text{constant}(V_2^{1-n} - V_1^{1-n})}{1-n} = \frac{(p_2V_2^n)V_2^{1-n} - (p_1V_1^n)V_1^{1-n}}{1-n} \\ &= \frac{p_2 V_2 - p_1 V_1}{1 - n} \end{aligned} \)

Schematic

Piston-cylinder Assembly with Polytropic Process

Inputs & Outputs

Final pressure \(p_2\)
... kPa
Work \(W\)
... kJ

Plot

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