At $550 \ K$,the $K_c$ for the following reaction is $10^4 \ mol^{-1} \ L$: $X_{(g)} + Y_{(g)} \rightleftharpoons Z_{(g)}$. At equilibrium,it was observed that $[X] = \frac{1}{2}[Y] = \frac{1}{2}[Z]$. What is the value of $[Z]$ (in $mol \ L^{-1}$) at equilibrium?

One mole of $N_2O_{4(g)}$ at $300 \ K$ is kept in a closed container under $1 \ atm$ pressure. It is heated to $600 \ K$ when $20 \%$ of $N_2O_{4(g)}$ decomposes to $NO_{2(g)}$. The resultant pressure at $600 \ K$ is $..... \ atm$.

$A$ solid $XY$ kept in an evacuated sealed container undergoes decomposition to form a mixture of gases $X$ and $Y$ at temperature $T$. The equilibrium pressure is $10 \, bar$ in the vessel. $K_p$ for this reaction is

One mole of $N_2O_4(g)$ is taken in a closed vessel at $300 \, K$ under $1 \, atm$ pressure. When it is heated to $600 \, K$,$20 \%$ of $N_2O_4(g)$ dissociates into $NO_2(g)$. What is the final pressure in $atm$?

For a reaction,$A \rightleftharpoons P$,the plots of $[A]$ and $[P]$ with time at temperatures $T_1$ and $T_2$ are given below. If $T_2 > T_1$,the correct statement$(s)$ is (are) (Assume $\Delta H^{\ominus}$ and $\Delta S^{\ominus}$ are independent of temperature and ratio of $\ln K$ at $T_1$ to $\ln K$ at $T_2$ is greater than $T_2 / T_1$. Here $H, S, G$ and $K$ are enthalpy,entropy,Gibbs energy and equilibrium constant,respectively.)
$(A)$ $\Delta H^{\ominus} < 0, \Delta S^{\ominus} < 0$
$(B)$ $\Delta G^{\ominus} < 0, \Delta H^{\ominus} > 0$
$(C)$ $\Delta G^{\ominus} < 0, \Delta S^{\ominus} < 0$
$(D)$ $\Delta G^{\ominus} < 0, \Delta S^{\ominus} > 0$

One mole each of $He$ and $A(g)$ are taken in a $10 \text{ L}$ closed flask and heated to $400 \text{ K}$ to establish the following equilibrium: $A(g) \rightleftharpoons B(g)$. $K_{c}$ for this reaction at $400 \text{ K}$ is $4.0$. The partial pressures (in $\text{atm}$) of $He$ and $B(g)$ are respectively (at equilibrium) (Assume $He$, $A(g)$ and $B(g)$ behave as ideal gases) (Given: $R = 0.082 \text{ L atm K}^{-1} \text{ mol}^{-1}$)