The reversible elementary gas$-$phase reaction $2Alongleftrightarro{w}^{k}B$ is carried out in a batch stirred tank reactor $($STR$)$ isothermally $($no pressure drop$)$ with specific rate $k$ and

equilibrium reaction constant $K_C.$ Only pure A is fed into the reactor. Choose the correct equation for the design equation for STR for A$.$

$\frac{dx}{dt}=k[C_{Ao}\frac{(1-x{)}^2}{(1-0\mathrm{.}5x)}-\frac{x}{2K_C}]$

$\frac{dx}{dt}=k{C}_{Ao}[C_{Ao}(1-x{)}^2-\frac{x}{2K_C}]$

$\frac{dx}{dt}=k{C}_{Ao}[1-x(1+\frac{1}{K_C})]$

$\frac{dx}{dt}=k{C}_{Ao}[C_{Ao}\frac{(1-x{)}^2}{(1-0\mathrm{.}5x)}-\frac{x}{2K_C}]$