A$)$ Estimate the equilibrium composition of a mixture of $1\mathrm{.}5mol{H}_2$ and $0\mathrm{.}5$mol of $N_2,0\mathrm{.}22$mol of $Ar,$ and $0\mathrm{.}22mo{l}^{2}C{H}_4$ at $150$ bar pressure at temperature of $673\mathrm{.}15K,$ taking into account the nonideal gas behavior of the reactants and products with assumption of an ideal solution of nonideal gases. In real ammonia plants the reactor feed always contains some argon and some $C{H}_4,$ which are practically inert. For this, use the below relationship taken by the Beattie$-$Bridgeman EOS. $($The equilibrium constant $K$:$0\mathrm{.}0137$ at $673\mathrm{.}15K)$

$log(\frac{1}{K_{}})=(\frac{0\mathrm{.}1191849}{T}+\frac{91\mathrm{.}87212}{T^2}+\frac{25122730}{T^4})*P$

$K_{}=\frac{{}_{N{H}_3}}{{}_{N_2}^{0\mathrm{.}5}{}_{H_2}^{1\mathrm{.}5}}$

for $T$ in $K$ and $P$ in atm.

B$)$ Discuss the role of inert gases $(Ar,C{H}_4)$ in terms of the conversion.