Paragraph: Thermal decomposition of gaseous $\mathrm{X}_{2}$ to gaseous $\mathrm{X}$ at $298 \mathrm{~K}$ takes place according to the following equation: $\mathrm{X}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{X}(\mathrm{g})$ 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 $\mathrm{X}_{2}$ and no $\mathrm{X}$. As the reaction proceeds, the number of moles of $\mathrm{X}$ formed is given by $\beta$. Thus, $\beta_{\text {equilibrium }}$ is the number of moles of $\mathrm{X}$ formed at equilibrium. The reaction is carried out at a constant total pressure of $2$ bar. Consider the gases to behave ideally. (Given : $R=0.083 \mathrm{~L}$ bar $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$ )
Question: The equilibrium constant $K_{P}$ for this reaction at $298 \mathrm{~K},$ in terms of $\beta_{equilibrium}$ is

• A

  $\frac{8{\mathrm{\beta }}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{m}}^2}{2-{\mathrm{\beta }}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{m}}}$
• Correct answer
  B

  $\frac{8{\mathrm{\beta }}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{m}}^2}{4-{\mathrm{\beta }}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{m}}^2}$
• C

  $\frac{4{\mathrm{\beta }}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{m}}^2}{2-{\mathrm{\beta }}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{m}}}$
• D

  $\frac{4{\mathrm{\beta }}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{m}}^2}{4-{\mathrm{\beta }}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{u}\mathrm{m}}^2}$

Hence, the correct answer is option: 'B'.