For the reaction,

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

if ${\mathrm{K}}_{\mathrm{P}}={\mathrm{K}}_{\mathrm{C}} {\left(\mathrm{RT}\right)}^{\mathrm{x}}$, where the symbols have usual meanings, then the value of x is (assuming ideality)

By using this formula, equilibrium constant,

${\mathrm{K}}_{\mathrm{P}}={\mathrm{K}}_{\mathrm{C}}{\left(\mathrm{RT}\right)}^{∆{\mathrm{n}}_{ }}$

where $∆\mathrm{n}$ = (no. of moles of products) - (no. of moles of reactants)

For the given reaction,

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

$∆{\mathrm{n}}_{\mathrm{g}}$ = $1-\left[\frac{1}{2}+1\right]=1-\left[\frac{3}{2}\right]=-\frac{1}{2}$