Polytropic process

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A polytropic process is a thermodynamic process that a system undergoes that obeys the relation

p V n = K

where p is pressure, V is volume, n is any real number, and K is a constant. Since this is a definition, it is absolutely general. The rest refers exclusively to ideal gases, where the processes are usually categorized more specifically depending on the index n:

if n = 0 then pV⁰=p=const and it is an isobaric process (constant pressure)
if n = 1 then pV=NkT=const and it is an isothermal process (constant temperature)
if n = $γ$ = c_p/c_V then it is an adiabatic process (no heat transferred)

Note that $1 \le \gamma \le 2$ , since $\gamma=\frac{c_p}{c_V}=\frac{c_V+R}{c_V}=1+\frac{R}{c_V}$

if n = $\infty$ then it is an isochoric process (constant volume)

Any number between two of this values means that the polytropic curve will lay between the corresponding curves.

The equation is valid assuming that

the process is quasistatic, and
the values of the heat capacities (c_p and c_V) are almost constant (Actually, they depend on the temperature, but are linear within small changes).

[edit] Polytropic fluids

Polytropic fluids are idealized fluid models that are used often in astrophysics. A polytropic fluid is a type of barotropic fluid for which the equation of state is written as

$P = K ρ (1 + 1 / n)$

where $P$ is the pressure, $K$ is a constant, $ρ$ is the density, and $n$ is a quantity called the polytropic index.

This is also commonly written in the form

$P = K ρ γ$

Note that $γ$ need not be the adiabatic index (or ratio of specific heats), and in fact often it is not. This is sometimes a cause for confusion.

In the case of an isentropic ideal gas, $γ$ is the ratio of specific heats, or adiabatic index:

$\gamma = \frac{c_P}{ c_V}$

An isothermal ideal gas is also a polytropic gas. Here, the polytropic $γ$ differs from the adiabatic index $γ$ , and in particular it is equal to one: $γ = 1$ .

In order to discriminate between the two gammas, the polytropic gamma is sometimes capitalized, $Γ$ .

To confuse matters further, some authors refer to $Γ$ as the polytropic index, rather than $n$ . Note that

$n = \frac{1}{\Gamma - 1}.$