Polytropic process

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

PVⁿ = C,

where P is the pressure, V is volume, n is any real number (the polytropic index), and C is a constant. This equation can be used to accurately characterize processes of certain systems, notably the compression or expansion of a gas, but in some cases, possibly liquids and solids.

Under standard conditions, most gases can be accurately characterized by the ideal gas law. This construct allows for the pressure-volume relationship to be defined for essentially all ideal thermodynamic cycles, such as the well-known Carnot cycle. (Note however that there may be instances where a polytropic process occurs in a non-ideal gas.)

For certain indices n, the process will be synonymous with other processes:

• if n = 0, then PV⁰ = P = const and it is an isobaric process (constant pressure)
• if n = 1, then for an ideal gas PV = NkT = const and it is an isothermal process (constant temperature)
• if n = γ = c_p/c_V, then for an ideal gas it is an adiabatic process (no heat transferred)

Note that , since (see: adiabatic index)

• if n = , then it is an isochoric process (constant volume)

When the index n occurs between any two of the former values (0, 1, gamma, or infinity), it means that the polytropic curve will lie between the curves of the two corresponding indices.

The equation is a valid characterization of a thermodynamic process assuming that:

• The process is quasistatic
• The values of the heat capacities,c_p and c_V, are almost constant when n is not zero or infinity. (In reality, c_p and c_V are a function of temperature, but are nearly linear within small changes of temperature).

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 = Kρ^{(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 = Kρ^γ

where in this case, γ = (1 + 1 / n) (Note that γ need not be the adiabatic index (the ratio of specific heats), and in fact often it is not. This is sometimes a cause for confusion.)

Gamma

In the case of an isentropic ideal gas, γ is the ratio of specific heats, known as the adiabatic index.

An isothermal ideal gas is also a polytropic gas. Here, the polytropic index is equal to one, and differs from the adiabatic index γ.

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

Other

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

See also

• Adiabatic process
• Isentropic process
• Isobaric process
• Isochoric process
• Isothermal process
• Quasistatic equilibrium
• Thermodynamics