Polytropic process

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Thermodynamic process involving heat and work transfer between the system and surroundings

title: "Polytropic process" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["thermodynamic-processes"] description: "Thermodynamic process involving heat and work transfer between the system and surroundings" topic_path: "general/thermodynamic-processes" source: "https://en.wikipedia.org/wiki/Polytropic_process" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Thermodynamic process involving heat and work transfer between the system and surroundings ::

A polytropic process is a thermodynamic process that obeys the relation: p V^{n} = C

where p is the pressure, V is volume, n is the polytropic index, and C is a constant. The polytropic process equation describes expansion and compression processes which include heat transfer.

Particular cases

Some specific values of n correspond to particular cases:

Where \gamma is the ratio of the heat capacity at constant pressure (C_P) to heat capacity at constant volume (C_V).

Equivalence between the polytropic coefficient and the ratio of energy transfers

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/ad/Polytropic.gif" caption="Polytropic processes behave differently with various polytropic indices. A polytropic process can generate other basic thermodynamic processes."] ::

For an ideal gas in a closed system undergoing a slow process with negligible changes in kinetic and potential energy the process is polytropic, such that p v^{(1- \gamma)K + \gamma} = C where C is a constant, K = \frac{\delta q}{\delta w}, \gamma = \frac{c_p}{c_v}, and with the polytropic coefficient

Relationship to ideal processes

For certain values of the polytropic index, the process will be synonymous with other common processes. Some examples of the effects of varying index values are given in the following table.

::data[format=table title="Variation of polytropic index ''n''"] | Polytropic index | Relation | Effects | |---|---|---| | n | — | Negative exponents reflect a process where work and heat flow simultaneously in or out of the system. In the absence of forces except pressure, such a spontaneous process is not allowed by the second law of thermodynamics ; however, negative exponents can be meaningful in some special cases not dominated by thermal interactions, such as in the processes of certain plasmas in astrophysics, or if there are other forms of energy (e.g. chemical energy) involved during the process (e.g. explosion). | | n = 0 | p=C | Equivalent to an isobaric process (constant pressure) | | n = 1 | pV=C | Equivalent to an isothermal process (constant temperature), under the assumption of ideal gas law, since then pV=nRT. | | 1 | — | Under the assumption of ideal gas law, heat and work flows go in opposite directions (K 0), such as in vapor compression refrigeration during compression, where the elevated vapour temperature resulting from the work done by the compressor on the vapour leads to some heat loss from the vapour to the cooler surroundings. | | n = γ | — | Equivalent to an isentropic process (adiabatic and reversible, no heat transfer), under the assumption of ideal gas law. | | γ | — | Under the assumption of ideal gas law, heat and work flows go in the same direction (K | | n = +∞ | V=C | Equivalent to an isochoric process (constant volume) | ::

When the index n is between any two of the former values (0, 1, γ, or ∞), it means that the polytropic curve will cut through (be bounded by) the curves of the two bounding indices.

For an ideal gas, 1 \gamma = \frac{c_p}{c_v} = \frac{c_v+R}{c_v} = 1+\frac{R}{c_v} = \frac{c_p}{c_p-R}.

Other

A solution to the Lane–Emden equation using a polytropic fluid is known as a polytrope.

The term "polytropic poison" has been used exclusively in publications from Russia regarding lead poisoning and chloroprene to indicate multisystemic toxic effects.

In entomology it has been used to denote insects visiting various flowers for nectar.

References

References

  1. Horedt, G. P.. (2004-08-10). "Polytropes: Applications in Astrophysics and Related Fields". Springer.
  2. (2021-09-01). "The Study of Airborne Particulate Matter in Dalnegorsk Town". International Journal of Environmental Research and Public Health.
  3. Sanotskii, I. V.. (1976). "Aspects of the Toxicology of Chloroprene: Immediate and Long-Term Effects". Environmental Health Perspectives.
  4. "Definition of POLYTROPIC".

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::

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