$(30$ pts$)$ A natural gas stream at $310K$ and $1400$kPa is to be transmitted by underground pipeline. For this purpose, it is compressed adiabatically to $520K$ and $14500$kPa. For simplicity, let us consider the natural gas to be a pure species whose $\frac{C_P^{ig}}{R}=1\mathrm{.}702+9\mathrm{.}081{10}^{-3}T-2\mathrm{.}164{10}^{-6}{T}^2,T$ in $K$;$T_c=190\mathrm{.}6K$;$P_c=46$ bar; and $=0\mathrm{.}01.$ Determine the needed work from the steady$-$state energy balance: $+W_s=H,$ and $S$ of the natural gas using the Pitzer correlations of residual properties.