Adiabatic Gas Constant

The relative amount of compression/expansion energy that goes into temperature versus pressure can be characterized by the heat capacity ratio

where is the specific heat (also called heat capacity) at constant pressure, while is the specific heat at constant volume. The specific heat, in turn, is the amount of heat required to raise the temperature of the gas by one degree. It is derived in statistical thermodynamics [138] that, for an ideal gas, we have , where is the ideal gas constant (introduced in Eq.(B.45)). Thus, for any ideal gas. The extra heat absorption that occurs when heating a gas at constant pressure is associated with the work (§B.2) performed on the volume boundary (fore times distance = pressure times area times distance) as it expands to keep pressure constant. Heating a gas at constant volume involves increasing the kinetic energy of the molecules, while heating a gas at constant pressure involves both that and pushing the boundary of the volume out. The reason not all gases have the same is that they have different internal degrees of freedom, such as those associated with spinning and vibrating internally. Each degree of freedom can store energy.

In terms of , we have

where for dry air at normal temperatures. Thus, if a volume of ideal gas is changed from to , the pressure change is given by

and the temperature change is

These equations both follow from Eq.(B.46) and the ideal gas law Eq.(B.45).

The value is typical for any diatomic gas.^B.31 Monatomic inert gases, on the other hand, such as Helium, Neon, and Argon, have . Carbon dioxide, which is triatomic, has a heat capacity ratio . We see that more complex molecules have lower values because they can store heat in more degrees of freedom.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.