HW#$6-4(15$ pt$)$ A piston$-$cylinder device contains $0\mathrm{.}5$ kg of air initially at $300$kPa and $27C.$ During a process, the air expands slowly until its volume doubles. Determine the temperature and pressure at the final state and the entropy change of air $(T_2,P_2,$ and $S)$ during the expansion process assuming the process to be

$($a$)$ polytropic process, $P{V}^k=$ constant,