At $T$ $(K)$,$K_c$ for the dissociation of $PCl_5$ is $2 \times 10^{-2} \ mol \ L^{-1}$. The number of moles of $PCl_5$ that must be taken in a $1.0 \ L$ flask at the same temperature to get $0.2 \ mol$ of chlorine at equilibrium is:

Thermal decomposition of gaseous $X_2$ to gaseous $X$ at $298 \ K$ takes place according to the following equation :
$X_{2(g)} \rightleftharpoons 2 X_{(g)}$
The standard reaction Gibbs energy,$\Delta_r G^{\circ}$,of this reaction is positive. At the start of the reaction,there is one mole of $X_2$ and no $X$. As the reaction proceeds,the number of moles of $X$ formed is given by $\beta$. Thus,$\beta_{\text{equilibrium}}$ is the number of moles of $X$ formed at equilibrium. The reaction is carried out at a constant total pressure of $2 \ bar$. Consider the gases to behave ideally. (Given : $R=0.083 \ L \ bar \ K^{-1} \ mol^{-1}$)
$(1)$ The equilibrium constant $K_P$ for this reaction at $298 \ K$,in terms of $\beta_{\text{equilibrium}}$,is
$(A)$ $\frac{8 \beta_{\text{equilibrium}}^2}{2-\beta_{\text{equilibrium}}}$ $(B)$ $\frac{8 \beta_{\text{equilibrium}}^2}{4-\beta_{\text{equilibrium}}^2}$ $(C)$ $\frac{4 \beta_{\text{equilibrium}}^2}{2-\beta_{\text{equilibrium}}}$ $(D)$ $\frac{4 \beta_{\text{equilibrium}}^2}{4-\beta_{\text{equilibrium}}^2}$
$(2)$ The $INCORRECT$ statement among the following,for this reaction,is
$(A)$ Decrease in the total pressure will result in formation of more moles of gaseous $X$
$(B)$ At the start of the reaction,dissociation of gaseous $X_2$ takes place spontaneously
$(C)$ $\beta_{\text{equilibrium}}=0.7$
$(D)$ $K_c < 1$

For the following reactions,which oxide is more stable?
$X_2 + O_2 \rightleftharpoons 2XO, K_1 = 5$
$X_2 + 2O_2 \rightleftharpoons 2XO_2, K_2 = 10$

Which of the following statements is false?

For the reaction $2NO_{2(g)} \rightleftharpoons N_2O_{4(g)}$ at $300 \ K$,the value of $K_p$ is $2 \ atm^{-1}$. The total pressure at equilibrium is $10 \ atm$. If the volume of the container becomes two times its original volume,what will be its equilibrium pressure at $300 \ K$ (in $atm$)?

At $1 \ bar$ pressure and $310 \ K$ temperature,$25\%$ of $N_2O_4$ decomposes. The reaction is: $N_2O_{4(g)} \rightleftharpoons 2NO_{2(g)}$.
$(i)$ Find $K_p$.
$(ii)$ At $0.1 \ bar$ pressure and $310 \ K$,what is the percentage of $N_2O_4$ decomposed?