$5.[10$ Pts$]$ Consider an unrestrained expansion of air from $\backslash (1\backslash$mathrm$\{$~m$\}^\{3\}\backslash )$ to $\backslash (2\backslash$mathrm$\{$~m$\}^\{3\}\backslash )$ at $300$ K without any external influence. Treat air as ideal gas, and use $0$ K for reference point, if needed.

$($a$)$ What are the specific entropy change and lost work? $[5$ Pts$]$

$($b$)$ Draw $\backslash ($ P$-$v $\backslash )$ and $\backslash ($ T$-$s $\backslash )$ diagrams and indicate areas corresponding to lost work. $[3$ Pts$]$

$($c$)$ If the air expanded at $\backslash (0\mathrm{.}1\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$s$\}\backslash )$ during this process, what was the specific entropy generation rate at the beginning of the process? $($hint: chain rule of derivative$)[2$ Pts$]$