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Polytropic Process Work Calculator

Enter the start and end pressures, volumes, and the polytropic index to find the work.

How it is calculated

The tool uses W = (P1V1 − P2V2) / (n − 1). Pressures are entered in kPa and the tool multiplies them by 1000 to get pascals; volumes are in m³; n is the polytropic index. The work comes out in joules and also in kilojoules. A positive value means the gas does work on its surroundings (expansion), a negative value means work is done on the gas (compression). The tool does not check that your two states fit PVn = constant; it uses the four numbers as given. For n = 1 (isothermal) the formula divides by zero, so no result is produced.

Worked example

A gas expands from 500 kPa and 0.2 m³ to 203 kPa and 0.4 m³ with n = 1.3. At the start, 500,000 Pa × 0.2 m³ = 100,000 J. At the end, 203,000 Pa × 0.4 m³ = 81,200 J. The difference is 18,800 J. Dividing by n − 1 = 0.30 gives 62,666.67 J of work done by the gas, or 62.6667 kJ. The result is positive because the gas expanded.

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Common mistakes

  • Entering pressure in pascals or bar. The fields take kPa and are multiplied by 1000, so 5 bar must be typed as 500.
  • Entering volume in litres. The field is m³, so 200 L must be typed as 0.2.
  • Using gauge pressure. The formula needs absolute pressure, so add the atmospheric pressure to a gauge reading first.
  • Entering an end state that does not belong to the same process. The tool does not check that P2 matches P1, V1, V2 and n, so a mismatched pair gives a meaningless work value.
  • Trying n = 1 for an isothermal process. The tool returns nothing there; the isothermal work uses a logarithm, P1V1·ln(V2/V1), instead.

About this calculator

A polytropic process describes a gas that expands or compresses according to PV^n = constant, where n (the polytropic index) captures how much heat transfer occurs during the process — n=1 is isothermal, n equal to the specific heat ratio γ is adiabatic (no heat exchange), and everything in between represents partial heat transfer, which is what real compressors and expanders actually experience. The work done during such a process is W = (P1·V1 − P2·V2)/(n − 1), and this calculator evaluates it from the start and end pressures, volumes and the chosen index.

This formula is core material in engineering thermodynamics, used to size and analyze reciprocating and rotary compressors, gas turbines, and internal combustion engine cycles, where the polytropic index is often determined experimentally since real machines rarely behave exactly isothermally or adiabatically. Mechanical and chemical engineers use it during equipment selection to estimate the power a compressor motor must deliver, and it's a standard problem type in thermodynamics coursework for exactly that reason.

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