$(30$ points$)$ The P$-$V isotherms of $3$ moles $C{O}_2$ described by the van der Waals equation $($vdW eqn$)$ are shown in Figure $3.$ In the process of isothermal compression from gas volume $V^{**}$ to liquid volume $V^{in}$ at $P_{\text{s}\text{a}\text{t}}$ below $T_r.($a$)$ Briefly explain the physical process between $V^{in}$ and $V^s.$ Are the curves reasonable? What do you expect for a reasonable curve? Choose the proper equation from the given ones below to describe where the decreased energy of gas comes from and go to$?(5$ points$)$ Please $($b$)$ calculate the volume and critical volume of $C{O}_2$ molecules. $(5$ points$)($c$)$ Calculate the interactive pressure between $C{O}_2$ molecules at the critical point. Will the interactive pressure increase or decrease the actual gas pressure P$?(5$ points$)($d$)$ Calculate the work $W_{280}$ and $W_{300}$ done by the gas at $280$ and $300K.(5$ points$)($e$)$ Calculate the changed internal energy $U_{280}$ and $U_{300}$ at $280$ and $300K.(5$ points$)($f$)$ Calculate the changed enthalpy at $280$ and $300K$ and explain the observed energy and phase changes in $($d$),($e$),$ and $($f$).(5$ points$)$ Given $R=0\mathrm{.}082\frac{\text{a}\text{t}\text{m}*liter}{(K*\text{m}\text{o}\text{l}\text{e})}=8\mathrm{.}314\frac{J}{(K*\text{m}\text{o}\text{l}\text{e})},V_{280}^{\text{l}\text{i}\text{q}}=0\mathrm{.}085,V_{280}^g=0\mathrm{.}265,V_{300}^{\text{l}\text{i}\text{q}}=0\mathrm{.}105,V_{300}^g=0\mathrm{.}155$ liter for $1$ mole. All pressures in Figure $3$ are the saturation pressure $P_{\text{s}\text{a}\text{t}}$ at each temperature. The critical pressure and temperature $P_c=72\mathrm{.}9$ atm and $T_{cr}=304K.$ Given vdW eqn $(P+\frac{n^{2}a}{V^2})(V-nb)=nRT,H=U+PV,dU=TdS-PdV,$ and $(del\frac{S}{d}$elV${)}_P=(del\frac{P}{d}$elT${)}_V.$ NOTE: Be careful about mole number!