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
$(C)$ $\beta_{\text{equilibrium}}=0.7$
$(D)$ $K_c < 1$

• A
• B
• C
• D