Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 1.37.
As an example of adiabatic compression of an ideal gas, consider the compression of air in a diesel engine. Atmospheric air (at a temperature of, say, ) is quickly compressed to of its original volume. From the relation
where is the number of degrees of freedom of a gas molecule, we can estimate the temperature of the air after compression. As most air molecules are diatomic, we can take (3 translational + 2 rotational degrees of freedom; this assumes that vibrational modes are frozen out, although I’m not sure that’s true for higher temperatures), so the temperature after compression is
The autoignition temperature for diesel is so the fuel will automatically ignite when the air is compressed, which is why diesel engines don’t need spark plugs.