°F, well above the ignition point of many fuels. This is why a high-compression engine requires fuels specially formulated to not self-ignite (which would cause engine knocking when operated under these conditions of temperature and pressure), or that a supercharger with an intercooler to provide a pressure boost but with a lower temperature rise would be advantageous. A diesel engine operates under even more extreme conditions, with compression ratios of 16:1 or more being typical, in order to provide a very high gas temperature, which ensures immediate ignition of the injected fuel. Adiabatic free expansion of a gas For an adiabatic free expansion of an ideal gas, the gas is contained in an insulated container and then allowed to expand in a vacuum. Because there is no external pressure for the gas to expand against, the work done by or on the system is zero. Since this process does not involve any heat transfer or work, the first law of thermodynamics then implies that the net internal energy change of the system is zero. For an ideal gas, the temperature remains constant because the internal energy only depends on temperature in that case. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible. Derivation of P–V relation for adiabatic heating and cooling The definition of an adiabatic process is that heat transfer to the system is zero, . Then, according to the first law of thermodynamics, where is the change in the internal energy of the system and is work done by the system. Any work () done must be done at the expense of internal energy , since no heat is being supplied from the surroundings. Pressure–volume work done by the system is defined as However, does not remain constant during an adiabatic process but instead changes along with . It is desired to know how the values of and relate to each other as the adiabatic process proceeds. For an ideal gas (recall ideal gas law ) the internal energy is given by where is the number of degrees of freedom divided by 2, is the universal gas constant and is the number of moles in the system (a constant). Differentiating equation (a3) yields Equation (a4) is often expressed as because . Now substitute equations (a2) and (a4) into equation (a1) to obtain factorize : and