Develop equations that may be solved to give the final temperature of the gas remaining in a tank after the tank has been bled from an initial pressure 
to a final pressure 
Known quantities are initial temperature, tank volume, heat capacity of the gas, total heat capacity of the containing tank, 
, and 
Assume the tank to be perfectly insulated.
What will be an ideal response?
We can re-use the sketch from the last problem, with slight modifications:
Our goal is to develop a relationship between the temperature in the tank and the pressure in the tank. We will write mole and energy balances on the tank. This time, however, the tank is insulated on the outside (so Q = 0) but the tank itself changes temperature with the gas. We will take as our control volume the tank and the gas inside the tank, up to a point just before the valve. Then the mole balance is
and the energy balance is
in the energy balance, Q = W = 0. 
is the total internal energy of the system, which includes both the tank and the gas inside. So, we can write 
where 
is the total internal energy of the tank and U is the molar internal energy of the gas. We can also write H = U + PV = U + RT.
Subsituting these in gives
or
where I have gone ahead and expanded the derivative on the left hand side using the product rule, and have used the mole balance to substitute for 
on the right-hand-side. Now, we can replace the internal energy terms with temperatures by noting that 
and 
where 
is the total heat capacity of the tank, and 
is the molar heat capacity of the gas. Canceling the term that appears on both sides of the
equation above and then substituting these gives
Rearranging this to put the T’s on one side and the n’s on the other, then integrating from the initial to final number of moles in the tank gives
This gives us a relationship between the number of moles in the tank and the temperature, but we want a relationship between the pressure in the tank and the temperature. We can substitute for the number of moles in terms of the
temperature and pressure using the ideal gas law:
This gives
or
This could be solved numerically for 
given all of the other information.
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