# Polytropic change of state

In thermodynamics , a change of state of a system in which the equation applies to pressure and specific volume is called polytropic . The exponent is called the polytropic exponent . In technical processes, the polytropic exponent can be viewed as constant. In the pv diagram, a polytrope takes the form of a power function with a negative slope. ${\ displaystyle p}$ ${\ displaystyle v}$ ${\ displaystyle pv ^ {n} = \ mathrm {const}}$ ${\ displaystyle n}$ Special cases of the polytropic change of state are:

• ${\ displaystyle n = 0}$ : isobaric
• ${\ displaystyle n = 1}$ : isothermal
• ${\ displaystyle n = \ infty}$ : isochoric
• ${\ displaystyle n = \ kappa = {\ frac {c _ {\ mathrm {p}}} {c _ {\ mathrm {v}}}}}$ : isentropic or adiabatic - reversible

The heat supplied to a gas during this change of state is given by:

${\ displaystyle Q_ {12} = m \ c _ {\ mathrm {v}} {\ frac {n- \ kappa} {n-1}} \ (T_ {2} -T_ {1})}$ Here referred to the mass , and start and end temperature of the process. The polytropy is characterized by a fixed heat capacity , which is from , and results. ${\ displaystyle m}$ ${\ displaystyle T_ {1}}$ ${\ displaystyle T_ {2}}$ ${\ displaystyle c _ {\ mathrm {p}}}$ ${\ displaystyle c _ {\ mathrm {v}}}$ ${\ displaystyle n}$ One also speaks of a polytropic equation of state :

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

For ideal gases with isentropic changes of 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, the following applies . With the isobaric heat capacity and the isochoric heat capacity . In the case of diatomic gases (for example air as a gas mixture ) and monatomic gases ( noble gases ) can be used. ${\ displaystyle n = c _ {\ mathrm {p}} / c _ {\ mathrm {v}}}$ ${\ displaystyle c _ {\ mathrm {p}}}$ ${\ displaystyle c _ {\ mathrm {v}}}$ ${\ displaystyle n = 1 {,} 403}$ ${\ displaystyle n = 1 {,} 66}$ ## Individual evidence

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