# Compression and Expansion of Gases

The relationship between pressure and density when compressing - or expanding - a gas depends on the nature of the process. The process can be

- isothermal,
- isentropic (adiabatic)
- polytropic

### Isothermal Compression/Expansion Processes

If compression or expansion of gas takes place under **constant temperature** conditions - the process is said to be **isothermal.** The isothermal process can be expressed with the Ideal Gas Law as

p / ρ = constant(1)

where

p= absolute pressure(Pa, N/m^{2})

ρ= density(kg/m^{3})

The isothermal process can also be expressed as

pV = constant (1a)

or

p_{1}V_{1}= p_{2}V_{2 }(1b)

where

V = gas volume (m^{3}, ft^{3}...)

suffix_{1}denotes initial conditions and suffix_{2}denotes final conditions

### Isentropic (or adiabatic) Compression/Expansion Processes

If compression or expansion of gas takes place with **no flow of heat energy either into or out of the gas** - the process is said to be **isentropic or adiabatic.** The isentropic (adiabatic) process can be expressed with the Ideal Gas Law as

p / ρ^{k}= constant(2)

where

k=c_{p}/ c_{v}- the ratio of specific heats - the ratio of specific heat at constant pressure -c_{p}- to the specific heat at constant volume -c_{v}

The isentropic or adiabatic process can also be expressed as

pV^{k }= constant (2a)

or

p_{1}V_{1}^{k}= p_{2}V_{2}^{k }(2b)

*Polytropic *Compression/Expansion Process

An ideal isothermal process must occur very slowly to keep the gas temperature constant. An ideal adiabatic process must occur very rapidly without any flow of energy in or out of the system. In practice most expansion and compression processes are somewhere in between, or said to be **polytropic**.

The polytropic process can be expressed as

pV^{n }= constant (3a)

or

p_{1}V_{1}^{n}= p_{2}V_{2}^{n }(3b)

where

n = polytropic index or exponent (ranging 1 to 1.4)

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