I'm having a hell of a time understanding this. Can anyone give me a hand? How is it derived? etc?
You mean, how is it defined? It's a term with a definition. An adiabatic process (in thermodynamics, anyway) is one in which no heat is transferred between the various parts of the system undergoing the process.
Adiabatic process means no heat is transferred and therefore if volume increases temperature of the gas will decrease or; if it expands the gas gets cooler and if it compresses it gets hotter.
This also means, compared to a temperature constant process- isothermal process- if the gas expands adiabatically it will drop in pressure faster per change in volume, and it will increase in pressure faster when volume decrease.
To figure these use adiabatic gas law pressure volume to the power of heat capacity. Heat capacity is a fraction comprising (the energy to increase temperature of gas given constant pressure) / (energy to increase temperature given constant volume). This constant changes with the gas.
At Wikipedia under Ideal Gas Law There is a chart of different forms of adiabatic gas laws to get pressure 2 volume 2 or temperature 2 given original data. Isentropic is same as adiabatic
Since your talking about adiabatic processes Ive got a question. If you compress a gas adiabatically then preform a heat exchange process to bring back to original temperature can you use same heat capacity to figure an additional compression?