Equation for an adiabatic change
This pages considers the equation
for an adiabatic expansion or compression. The proof of the equation will be outside the
scope of some courses, so the result will be quoted first. The proof is given in a separate file
for interested students.
The pressure P and volume V of a gas undergoing an adiabatic
change are related by the formula:
Adiabatic change equation: PVγ = constant
where
g is a constant for the gas. This
constant is the ratio of the two principal specific heats of the
gas and has a value between 1.3 and 1.67.
Now when a gas expands or contracts
reversibly and adiabatically it still obeys the ideal gas equation (PV = nRT) and therefore we
have some alternative ways of expressing an adiabatic change, namely:
Equations for an adiabatic change
PVγ = constant
TVγ-1 = constant
P(1-γ)Tγ = constant
The equations may
also be written in the form:
P1V1γ = P2V2γ etc.,
where P
1V
1 and
P
2V
2 are the initial and final conditions of the gas
respectively.
Example problem
An ideal gas at 27 oC and a pressure of 760 mm of mercury is compressed isothermally until its volume is halved. It is then expanded reversibly and adiabatically to twice its original volume. If the value of γ for the gas is 1.4, calculate the final pressure and temperature of the gas.
For the isothermal change:
PV= P1V1 760xV = PxV/2 P= 1520 mm of mercury
For the adiabatic change:
PVγ = P2V2γ 1520 x (V/2)1.4 = P2 x (2V)1.4
P2 = 1520/6.97 = 218 mm of mercury
The gas obeys the ideal gas equation.
Therefore from: PV/T1 = P2V2/T2 we have 1520xV)/300x2 = 218x2V/T2
and so: T2 = 172 K = - 101 oC
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