No can't say that I have. Now Dyno Dave may be up on this. He's just a hop further.
Isothermal and adiabatic expansion
Suppose that the temperature of an ideal gas is held constant by keeping the gas in thermal contact with a heat reservoir. If the gas is allowed to expand quasi-statically under these so called isothermal conditions then the ideal equation of state tells us that
This is usually called the isothermal gas law.
Suppose, now, that the gas is thermally isolated from its surroundings. If the gas is allowed to expand quasi-statically under these so called adiabatic conditions then it does work on its environment, and, hence, its internal energy is reduced, and its temperature changes. Let us work out the relationship between the pressure and volume of the gas during adiabatic expansion.
According to the first law of thermodynamics,
in an adiabatic process (in which no heat is absorbed). The ideal gas equation of state can be differentiated, yielding
The temperature increment can be eliminated between the above two expressions to give
which reduces to
Dividing through by yields
where
It turns out that is a very slowly varying function of temperature in most gases. So, it is always a fairly good approximation to treat the ratio of specific heats as a constant, at least over a limited temperature range. If is constant then we can integrate Eq. (6.57) to give
or
This is the famous adiabatic gas law. It is very easy to obtain similar relationships between and and and during adiabatic expansion or contraction. Since , the above formula also implies that
and
Equations (6.60)-(6.62) are all completely equivalent.