Two gaseous equilibriums ${\mathrm{SO}}_{2\left(\mathrm{g}\right)}+\frac{1}{2}{\mathrm{O}}_{2\left(\mathrm{g}\right)}\to {\mathrm{SO}}_{3\left(\mathrm{g}\right)}\& 2{\mathrm{SO}}_{3\left(\mathrm{g}\right)}+ 2{\mathrm{SO}}_{2\left(\mathrm{g}\right)}+{\mathrm{O}}_{2\left(\mathrm{g}\right)}$have equilibrium constants ${\mathrm{K}}_1 \& {\mathrm{K}}_2$ respectively at 298 K. Which of the following relationship between  ${\mathrm{K}}_1 \& {\mathrm{K}}_2$ is correct 

The given reactions are as follows below. 

${\mathrm{SO}}_2+\frac{1}{2}{\mathrm{O}}_2\to {\mathrm{SO}}_2 ; 2{\mathrm{SO}}_2\to 2{\mathrm{SO}}_2+{\mathrm{O}}_2$

${\mathrm{K}}_1=\frac{\left[{\mathrm{SO}}_3\right]}{\left[{\mathrm{SO}}_2\right]{\left[{\mathrm{O}}_2\right]}^{1/2}} {\mathrm{K}}_2=\frac{\left[{\mathrm{SO}}_2\right]\left[{\mathrm{O}}_2\right]}{{\left[{\mathrm{SO}}_3\right]}^2}$

$$\begin{gathered} \frac{1}{{\mathrm{K}}_1}=\frac{\left[{\mathrm{SO}}_2\right]{\left[{\mathrm{O}}_2\right]}^{1/2}}{\left[{\mathrm{SO}}_3\right]} and then {\left[\frac{1}{K_1}\right]}^2=\frac{{\left[{\mathrm{SO}}_2\right]}^2\left[{\mathrm{O}}_2\right]}{{\left[{\mathrm{SO}}_3\right]}^2}=K_2 \\ \therefore {\mathrm{K}}_2=\frac{1}{{\mathrm{K}}_1^2} \end{gathered}$$

Therefore, the correct answer is option “C”.