$.(20$ marks$)$ A horizontal cylinder is separated into two compartments by an adiabatic frictionless piston. One side contains $0\mathrm{.}2$ m$3$ of nitrogen and the other side contains $0\mathrm{.}1$ kg of helium, both initially at $20$C and $95$ kPa. The sides of the cylinder and the helium end are insulated. Now heat is added to the nitrogen side from a reservoir at $500$C until the pressure of the helium rises to $120$ kPa. The properties of nitrogen near room temperature are R $=0\mathrm{.}2968$ kJ $/$kg$.$K$,$ cp $=1\mathrm{.}039$ kJ$/$kg$.$K$,$ cv $=0\mathrm{.}743$ kJ$/$kg$.$K$,$ k $=$ cp$/$ cv $=1\mathrm{.}4.$ The properties for helium near room temperature are R $=2\mathrm{.}0769$ kJ $/$kg$.$K$,$ cp $=5\mathrm{.}1926$ kJ$/$kg$.$K$,$ cv $=3\mathrm{.}1156$ kJ$/$kg$.$K$,$ k $=1\mathrm{.}67.$ Determine $($a$)$ the final temperature of the helium $($b$)$ the final volume of the nitrogen $($c$)$ the heat transferred to the nitrogen $($d$)$ the entropy generation during this proces