# Polytropic process

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A polytropic process is a thermodynamic process that obeys the relation:

${\displaystyle pV^{\,n}=C}$

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

If the ideal gas law applies, a process is polytropic if and only if the ratio (K) of energy transfer as heat to energy transfer as work at each infinitesimal step of the process is kept constant:

${\displaystyle K={\frac {\delta Q}{\delta W}}={\text{constant}}}$

## Particular cases

Some specific values of n correspond to particular cases:

• ${\displaystyle n=0}$ is an isobaric process,
• ${\displaystyle n=+\infty }$ is an isochoric process.

In addition, when the ideal gas law applies:

• ${\displaystyle n=1}$ is an isothermic process,
• ${\displaystyle n=\gamma }$ is an adiabatic process.

## Equivalence between the polytropic coefficient and the ratio of energy transfers

Polytropic processes behave differently with various polytropic indices. Polytropic process can generate other basic thermodynamic processes.

Consider an ideal gas in a closed system undergoing a slow process with negligible changes in kinetic and potential energy. For an infinitesimal step of time, the first law of thermodynamics states that the energy added to a system as heat δq, minus the energy that leaves the system as work δw, is equal to the change in the internal energy du of the system:

${\displaystyle \delta q-\delta w=du}$        (Eq. 1)

Define the energy transfer ratio,

${\displaystyle K={\frac {\delta q}{\delta w}}}$ or ${\displaystyle \delta q=K\delta w}$.

Transfer of work to the environment can be expressed as ${\displaystyle {\delta w}=p{dv}}$ and internal energy change as ${\displaystyle du=nc_{\,v}\,dT}$

By substituting the above expressions for δw and δq into the first law:

${\displaystyle (K-1)p\,dv=nc_{\,v}\,dT}$        (Eq. 1)

Writing the ideal gas law in differential form

${\displaystyle p\,dv+v\,dp=nR\,dT\rightarrow {v\,dp \over p\,dv}+1={nR\,dT \over p\,dv}={nR\,(K-1)p\,dT \over p\,nc_{v}dT}={R\,(K-1) \over c_{v}}}$

By Mayer's relation, this becomes:

${\displaystyle {v\,dp \over p\,dv}={(c_{p}-c_{v})\,(K-1) \over c_{v}}-1=(K-1)\gamma -K\longrightarrow {dp \over p}+{((1-\gamma )K+\gamma )}{dv \over v}=0}$

where γ is the heat capacity ratio. Assuming K (and γ) remain constant during the transformation, as ${\displaystyle {df \over f}=d(log\,f)}$ this relation can be integrated as

${\displaystyle d\left(log\,p+{((1-\gamma )K+\gamma )}log\,v\right)=0\longrightarrow pv^{(1-\gamma )K+\gamma }=C}$

where C is a constant.

Thus, the process is polytropic, with the coefficient ${\displaystyle n={(1-\gamma )K+\gamma }}$.

This derivation can be expanded to include polytropic processes in open systems, including instances where the kinetic energy (i.e. Mach number) is significant. It can also be expanded to include irreversible polytropic processes.[1]

## 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 table.

Variation of polytropic index n
Polytropic
index
Relation Effects
n < 0 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[citation needed]; 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.[2]
n = 0 ${\displaystyle p=C}$ Equivalent to an isobaric process (constant pressure)
n = 1 ${\displaystyle pV=C}$ Equivalent to an isothermal process (constant temperature), under the assumption of ideal gas law, since then ${\displaystyle pV=nRT}$.
1 < n < γ 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 < n < ∞ Under the assumption of ideal gas law, heat and work flows go in the same direction (K < 0), such as in an internal combustion engine during the power stroke, where heat is lost from the hot combustion products, through the cylinder walls, to the cooler surroundings, at the same time as those hot combustion products push on the piston.
n = +∞ ${\displaystyle V=C}$ 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 < γ < 2, since by Mayer's relation

${\displaystyle \gamma ={\frac {c_{p}}{c_{v}}}={\frac {c_{v}+R}{c_{v}}}=1+{\frac {R}{c_{v}}}={\frac {c_{p}}{c_{p}-R}}}$.

## Other

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