Relation between pressure volume and temperature for reversible adiabatic process
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The mathematical equation for an ideal gas undergoing a reversible (i.e., no entropy generation) adiabatic process can be represented by the polytropic process equation[3]
PV^{n}={constant}
where P is pressure, V is volume, and for this case n = γ, where
gamma = CP/CV= (f+2)/f
CP being the specific heat for constant pressure, CV being the specific heat for constant volume, γ is the adiabatic index, and f is the number of degrees of freedom (3 for monatomic gas, 5 for diatomic gas and collinear molecules e.g. carbon dioxide).
For a monatomic ideal gas, γ = 5/3, and for a diatomic gas (such as nitrogen and oxygen, the main components of air) γ = 7/5.
Note that the above formula is only applicable to classical ideal gases and not Bose–Einstein or Fermi gases.
For reversible adiabatic processes, it is also true that
P^{1-gamma }T^{gamma }= constant
VT^{f/2}={constant}
where T is an absolute temperature. This can also be written as
TV^{gamma-1}= constant
PV^{n}={constant}
where P is pressure, V is volume, and for this case n = γ, where
gamma = CP/CV= (f+2)/f
CP being the specific heat for constant pressure, CV being the specific heat for constant volume, γ is the adiabatic index, and f is the number of degrees of freedom (3 for monatomic gas, 5 for diatomic gas and collinear molecules e.g. carbon dioxide).
For a monatomic ideal gas, γ = 5/3, and for a diatomic gas (such as nitrogen and oxygen, the main components of air) γ = 7/5.
Note that the above formula is only applicable to classical ideal gases and not Bose–Einstein or Fermi gases.
For reversible adiabatic processes, it is also true that
P^{1-gamma }T^{gamma }= constant
VT^{f/2}={constant}
where T is an absolute temperature. This can also be written as
TV^{gamma-1}= constant
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