$(28$ points$)$ A vertical, frictionless, free$-$floating piston in

cylinder contains $3\mathrm{.}0$ kg of $N_2($molecular weight $\{$:$=28\frac{kg}{k}$mol$)$

that behaves as an ideal gas. The local atmospheric pressure is

$98$ kPa $.$ The piston diameter is $10$ cm $.$ The piston has a mass of

$70$ kg $.$ The initial temperature of the gas in the cylinder is

$25C.$ Assume the universal gas constant is $8\mathrm{.}3134k\frac{J}{k}$mol$-K$

and acceleration of gravity is $9\mathrm{.}807\frac{m}{s^2},$ find

a$.$ The pressure in the cylinder $($in units of kPa $)$

b$.$ The volume in the cylinder $($in units of $m^3)$

c$.$ Heat is added until the volume of the gas in the cylinder

doubles. What is the new temperature $($in $C)?$