In a piston cylinder system, initially there is $0\mathrm{.}15$ m$^3$ of nitrogen gas at a temperature of $350$ K$.$ The piston has a mass of $250$ kg and a cross$-$sectional area of $0\mathrm{.}1$ m$^2.$ The ambient pressure is $1$ bar and the acceleration due to gravity is $10$ m$/$s$^2.$ Assuming that the nitrogen gas follows the Redlich$-$Kwong equation of state, calculate the heat transferred to the gas for the following scenarios: a$)$ The piston is fixed at the initial level, and the final temperature and pressure are $365$ K and $1\mathrm{.}32$ bar, respectively.

The piston is released and the temperature reaches $361$ K$.$ Cv$\text{'}=20\mathrm{.}8$ J$/$mol$.$K Cp$\text{'}=29\mathrm{.}0$ J$/$mol$.$K