### Adiabatic Gas Constant

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

*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

*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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Heat Capacity of Ideal Gases

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Isothermal versus Isentropic