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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:      PVg = 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
PVg = constant
TVg-1 = constant
P(1-g)Tg = constant

The equations may also be written in the form:

P1V1g = P2V2g etc.,

where P1V1 and P2V2 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 g 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:
PVg = P2V2g         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
© Keith Gibbs 2010