Given that equilibrium constant for the reaction,

$2{\mathrm{SO}}_2\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right) \rightleftharpoons 2{\mathrm{SO}}_3\left(\mathrm{g}\right)$

has a value of 278 at a particular temperature. What is the value of the equilibrium constant for the following reaction at the same temperature?

${\mathrm{SO}}_3\left(\mathrm{g}\right) \rightleftharpoons {\mathrm{SO}}_2\left(\mathrm{g}\right)+\frac{1}{2}{\mathrm{O}}_2\left(\mathrm{g}\right)$

$2{\mathrm{SO}}_2\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)\rightleftharpoons 2{\mathrm{SO}}_3\left(\mathrm{g}\right)$

Equilibrium constant for this reaction,

$\mathrm{K}=\frac{{\left[{\mathrm{SO}}_3\right]}^2}{{\left[{\mathrm{SO}}_2\right]}^2\left[{\mathrm{O}}_2\right]}$-----(i)

${\mathrm{SO}}_3\left(\mathrm{g}\right)\rightleftharpoons {\mathrm{SO}}_2\left(\mathrm{g}\right)+\frac{1}{2}{\mathrm{O}}_2\left(\mathrm{g}\right)$

Equilibrium constant for this reaction is

${\mathrm{K}}^{\text{'}}=\frac{\left[{\mathrm{SO}}_2\right]{\left[{\mathrm{O}}_2\right]}^{1/2}}{\left[{\mathrm{SO}}_3\right]}$--------(ii)

On squaring both sides in Eq. (ii), we have

${\left({\mathrm{K}}^{\text{'}}\right)}^2=\frac{{\left[{\mathrm{SO}}_2\right]}^2\left[{\mathrm{O}}_2\right]}{{\left[{\mathrm{SO}}_3\right]}^2}=\frac{1}{\mathrm{K}}=\frac{1}{278}(\text{ as }\mathrm{K}=278)$

${\mathrm{K}}^{\text{'}}=\sqrt{\frac{1}{278}}=\sqrt{0.003597}$

$=5.99\times {10}^{-2}\text{ or }\approx 6.0\times {10}^{-2}$