Air $(1$mol$)$ at $1$ bar and $298\mathrm{.}15K$ is compressed to $3$ bar and $298\mathrm{.}15K$ by the closed$-$system mechanically reversible process of heating at constant volume followed by cooling at constant pressure The following heat capacities for air may be assumed independent of temperature: $cv=20\mathrm{.}785$ and $cp=29\mathrm{.}100J*mo{l}^{-1}*K^{-1}.$ Assume that air remains a gas for which PV$/$T is a constant, regardless of the changes it undergoes. At $298\mathrm{.}15K$ and $1$ bar, the volume of $1$mol of air is $0\mathrm{.}02479m^3.$ Please answer with the following unit: $P(\frac{?}{\text{b}\text{a}\text{r}}),T(K),V(m^3),Q(J),W(J),U(J),$ and $H(J).$

$($a$)$ What are the values of $P,T,V$ of each state? $(2$ points$)$

$($b$)$ What are the values of $Q,W,U,H$ of the first step? $(2$ points$)$

$($c$)$ What are the values of $Q,W,U,H$ of the second step? $(2$ points$)$

$($d$)$ What are the values of $Q,W,U,H$ of the overall process? $(2$ points$)$

$($e$)$ Plot the path of this process $($P$-$V graph$)$ with $V$ as $y-$axis and $P$ as $x-$axis $(1$ point$)$

$($f$)$ Plot the path of this process $($P$-$T graph$)$ with $T$ as $y-$axis and $P$ as $x-$axis $(1$ point$)$