# What is a polytropic gas

## linkCatch

In thermodynamics, a change of state of a system in which for pressure \ ({\ displaystyle p} \) and specific volume \ ({\ displaystyle v} \) the equation \ ({\ displaystyle pv ^ {n} = \ mathrm { const}} \) is considered to be **polytropic** designated. The exponent \ ({\ displaystyle n} \) becomes **Polytropic exponent** called. In technical processes, the polytropic exponent can be viewed as constant.^{[1]} In the p-v diagram, a polytrope takes the form of a power function with a negative slope.

Special cases of the polytropic change of state are:

- \ ({\ displaystyle n = 0} \): isobaric
- \ ({\ displaystyle n = 1} \): isothermal
- \ ({\ displaystyle n = \ infty} \): isochor
- \ ({\ displaystyle n = \ kappa = {\ frac {c _ {\ mathrm {p}}} {c _ {\ mathrm {v}}}}} \): isentropic or also adiabatically reversible

The heat supplied to a gas during this change of state is given by:^{[2]}

- \ ({\ displaystyle Q_ {12} = m \ c _ {\ mathrm {v}} {\ frac {n- \ kappa} {n-1}} \ (T_ {2} -T_ {1})} \)

\ ({\ Displaystyle m} \) denotes the mass, \ ({\ displaystyle T_ {1}} \) and \ ({\ displaystyle T_ {2}} \) start and end temperature of the process. The polytropy is characterized by a fixed heat capacity, which is made up of \ ({\ displaystyle c _ {\ mathrm {p}}} \)*,* \ ({\ displaystyle c _ {\ mathrm {v}}} \) and \ ({\ displaystyle n} \) results.

One also speaks of a polytropic equation of state:

- \ ({\ displaystyle p = K \ cdot \ rho ^ {\ gamma}} \)

with the pressure p, the density \ ({\ displaystyle \ rho} \), the polytropic constant K and the polytropic index m in \ ({\ displaystyle \ gamma = 1 + {\ frac {1} {m}}} \). It is used, for example, in astrophysics (Lane-Emden equation).

### Table of Contents

### Ideal gases

For ideal gases, with isentropic changes in state, the following relationships also apply:

- \ ({\ displaystyle {\ frac {T_ {2}} {T_ {1}}} = \ left ({\ frac {p_ {2}} {p_ {1}}} \ right) ^ {\ frac {n -1} {n}} = \ left ({\ frac {V_ {1}} {V_ {2}}} \ right) ^ {n-1}} \) or
- \ ({\ displaystyle {\ frac {p_ {2}} {p_ {1}}} = \ left ({\ frac {T_ {2}} {T_ {1}}} \ right) ^ {\ frac {n } {n-1}} = \ left ({\ frac {V_ {1}} {V_ {2}}} \ right) ^ {n}} \)

With

- \ ({\ displaystyle T} \): absolute temperature
- \ ({\ displaystyle p} \): Pressure
- \ ({\ displaystyle V} \): Volume.

For the isentropic change of state of an ideal gas, \ ({\ displaystyle n = c _ {\ mathrm {p}} / c _ {\ mathrm {v}}} \) applies. With the isobaric heat capacity \ ({\ displaystyle c _ {\ mathrm {p}}} \) and the isochoric heat capacity \ ({\ displaystyle c _ {\ mathrm {v}}} \). For diatomic gases \ ({\ displaystyle n = 1 {,} 403} \) (e.g. air as a gas mixture) and for monatomic gases (noble gases) \ ({\ displaystyle n = 1 {,} 66} \) can be used.

### literature

### Individual evidence

- ^ Fran Bosniakovic, "Technical Thermodynamics", 7th edition, Steinkopf-Verlag Darmstadt; Chapter 4.5 "Polytropic state change"
- ↑ Peter Stephan et al .:
*Thermodynamics. Fundamentals and technical applications, Vol. 1: One-component systems*. 18th edition Springer, Berlin 2013, p. 115, ISBN 3-642-30097-9.

### See also

**Categories:**Thermodynamic process

**Status of information:**02/25/2021 4:46:00 AM CET

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