Heat capacity ratio of air

The ratio of specific heats governs what happens when a gas is compressed too fast to exchange heat: pV^γ stays constant, and the temperature rises. It is why a bicycle pump gets hot, why diesel engines ignite without a spark, and why the discharge of an air compressor needs an aftercooler. It also sets the speed of sound, a = √(γRT), giving 343 m/s in air at 20 °C.

The value 1.400 is not arbitrary. Nitrogen and oxygen are diatomic, with three translational and two rotational degrees of freedom active at room temperature, so equipartition predicts γ = (5+2)/5 = 1.4 exactly — one of the cleanest confirmations of kinetic theory. It falls slowly as temperature rises and vibrational modes wake up: about 1.40 at 20 °C, 1.35 at 500 °C, 1.30 at 1000 °C.