Consider a frictionless $($i$.$e$.,$ isobaric$)$ piston$/$cylinder assembly filled with air $($R $=0\mathrm{.}2870$ kJ$/$kg$-$K$)$ and housing a pure copper ball. The cross$-$sectional area and mass of the piston are $0\mathrm{.}1$ square meters and $51$ kg$,$ respectively. Gravity acts toward the bottom of the page at $9\mathrm{.}81$ m$/$s$2,$ such that the weight of the piston influences the pressure of the air inside the piston$/$cylinder$.$ The surrounding pressure is $100$ kPa. Take the air and copper ball as a combined system in thermodynamic equilibrium at the initial state and final state $($i$.$e$.,$ T$\_$air $=$ T$\_$copper at state $1$ and state $2,$ so T$\_2-$T$\_1$ for air $=$ T$\_2-$T$\_1$ for copper$).$

The mass of the copper ball is $0\mathrm{.}5$ kg$,$ and c $=397$ J$/$kg$-$K$.$ The initial temperature of the system is $300$K$.$ The initial volume of the air is $0\mathrm{.}4$m$3.$ The final temperature is $420$ K$.$ Determine the following:

The initial pressure of the air $($hint: force balance on the piston; it$$s larger than the surrounding pressure$).$

The boundary work

The heat added $($or removed$)$

Draw the process on a P$-$v diagram

Draw the process on a T$-$v diagram