An adiabatic
free expansion of an ideal gas ie. where a greater volume suddenly becomes
available to the gas is an irreversible process which proceeds through
a chaotic non-equilibrium path. Nonetheless we can characterize the
beginning and end points and the net values of relevant changes in
energy. Since the gas expands against a vacuum it does no work and thus
combining this with our requirement that the process is adiabatic we have
If we are dealing with an ideal gas then the absence of a change in the
internal energy implies that the temperature is the same before and after the
expansion even though no temperature is defined during the irreversible process.
Using the equation of state for an ideal gas this implies that
Note the similarity to the result for an adiabatic reversible process
with the important difference of the power 
and the corresponding
change in temperature which occurs only in the adiabatic reversible
process.