Adiabatic Gas Constant

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

$$\gamma \isdefs \frac{C_p}{C_v}$$

where $C_p$ is the specific heat (also called heat capacity) at constant pressure, while $C_v$ 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 [139] that, for an ideal gas, we have $C_p=C_v+R$ , where $R$ is the ideal gas constant (introduced in Eq.(B.45)). Thus, $\gamma>1$ 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 (force 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 $\gamma$ 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 $\gamma$ , we have

$$P_1V_1^\gamma \eqsp P_2V_2^\gamma, \protect$$ (B.46)

where $\gamma\approx 1.4$ for dry air at normal temperatures. Thus, if a volume of ideal gas is changed from $V_1$ to $V_2$ , the pressure change is given by

$$P_2 \eqsp P_1 \left(\frac{V_1}{V_2}\right)^{\gamma}$$ (B.47)

and, converting pressure to temperature via the ideal gas law $P_iV_i=nRT_i$ , the temperature change is

$$T_2 \eqsp T_1 \left(\frac{V_1}{V_2}\right)^{\gamma-1}.$$ (B.48)

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

The value $\gamma=1.4$ is typical for any diatomic gas.^B.31 Monatomic inert gases, on the other hand, such as Helium, Neon, and Argon, have $\gamma\approx 1.6$ . Carbon dioxide, which is triatomic, has a heat capacity ratio $\gamma=1.28$ . We see that more complex molecules have lower $\gamma$ values because they can store heat in more degrees of freedom.