$[40$ points$]$ A piston$-$cylinder device contains $0\mathrm{.}25$ kg of air initially at a temperature of $27C$ and an absolute pressure of $1$ bar $($State $1).$ The air undergoes a compression process, where $P{V}^{1\mathrm{.}3}=$ constant, until the volume is $20\%$ of the initial volume and the absolute pressure is $8\mathrm{.}1$ bar $($State $2).$ During the compression process, $44\mathrm{.}5$ kJ of work is done on the air. The cylinder is fitted with a cooling water jacket all around its outer wall. The cooling water jacket contains $1\mathrm{.}75$ kg of liquid water. The water is initially at a temperature of $25C$ and an absolute pressure of $1$ bar $($State $3)$ at the start of the air compression process. Heat transfer occurs only between air in the cylinder and water inside the cooling jacket since the water jacket is perfectly insulated on its outside.