For the reaction,$2 SO_{2(g)} + O_{2(g)} \rightleftharpoons 2 SO_{3(g)}$ at $300 \ K$,the value of $\Delta G^{\circ}$ is $-690.9 R$. The equilibrium constant value for the reaction at that temperature is ($R$ is gas constant).

$A_{(g)} + 3B_{(g)} \rightleftharpoons 4C_{(g)}$. The starting concentrations of $A$ and $B$ are equal. At equilibrium,the concentrations of $A$ and $C$ are the same. The value of $K_c$ is:

Match List-$I$ (Hypothetical reaction) with List-$II$ (Ratio of $K_p/K_c$ for the given reaction) and select the correct answer using the options given below.

$(1)$ $A_{2(g)} + 3B_{2(g)} \rightleftharpoons 2AB_{3(g)}$	$(i)$ $(RT)^{-2}$
$(2)$ $A_{2(g)} + B_{2(g)} \rightleftharpoons 2AB_{(g)}$	$(ii)$ $(RT)^0$
$(3)$ $A_{(s)} + 1.5B_{2(g)} \rightleftharpoons AB_{3(g)}$	$(iii)$ $(RT)^{1/2}$
$(4)$ $AB_{2(g)} \rightleftharpoons AB_{(g)} + 0.5B_{2(g)}$	$(iv)$ $(RT)^{-1/2}$

For the reaction $CO_{(g)} + 2H_{2(g)} \rightleftharpoons CH_3OH_{(g)}$,the true condition is:

$(i)$ $\frac{1}{2}N_{2(g)} + \frac{3}{2}H_{2(g)} \rightleftharpoons NH_{3(g)}$ at $298 \ K$ has $\Delta G^{\Theta} = -16.5 \ kJ \ mol^{-1}$. Find $K_p$.
$(ii)$ At $298 \ K$,for $N_{2(g)} + 3H_{2(g)} \rightleftharpoons 2NH_{3(g)}$,calculate $K_p$ and $\Delta G^{\Theta}$.

For the reaction $NH_4HS_{(s)} \rightleftharpoons NH_{3(g)} + H_2S_{(g)}$,the observed total pressure of the reaction mixture at equilibrium is $1.12 \ atm$ at $106 \ ^\circ C$. The value of $K_P$ for the reaction is: (in $atm^2$)