$3\mathrm{.}11.$ A well$-$insulated tank contains $1$ mole of air at $2$MPa and $673K.$ It is

connected via a closed valve to an insulated piston$/$cylinder device that is

initially empty. The piston may be assumed to be frictionless. The volumes

of the piping and valve are negligible. The weight of the piston and

atmospheric pressure are such that the total downward force can be

balanced with gas pressure in the cylinder of $0\mathrm{.}7$MPa. The valve between

the tank and piston$/$cylinder is cracked open until the pressure is uniform

throughout. The temperature in the tank is found to be $499\mathrm{.}6K.$ Air can be

assumed to be an ideal gas with a temperature$-$independent heat capacity $C_P$

$=29\mathrm{.}3\frac{J}{m}ol-K.$

a$.$ What is the number of moles left in the tank at the end of

the process?

b$.$ Write and simplify the energy balance for the process.

Determine the final temperature of the piston$/$cylinder gas.