Q.For a gaseous homogeneous reaction at equilibrium, the number of moles of products is greater than the number of moles of reactants. Is $K_C$ larger or smaller than $K_P$?

$K_P$ and $K_C$ describe the same equilibrium, so a fixed relationship connects them. Starting from the ideal gas equation $P=(n/V)RT$ and noting that active mass equals molar concentration $n/V$, each species' partial pressure can be written as (concentration)$\times RT$; substituting this into the $K_P$ expression and comparing with $K_C$ gives

where $\Delta n_g$ is the difference between the total moles of gaseous products and the total moles of gaseous reactants in the balanced equation. Three regimes follow: $\Delta n_g=0$ gives $K_P=K_C$ exactly (e.g. $H_2+I_2\rightleftharpoons 2HI$, or ammonia synthesis's own $K_C/K_P=(RT)^2$ inverse case where $\Delta n_g=-2$ gives $K_P=K_C(RT)^{-2}$, so $K_C/K_P=(RT)^2$); $\Delta n_g>0$ gives $K_P>K_C$ (e.g. $PCl_5\rightleftharpoons PCl_3+Cl_2$, where $\Delta n_g=+1$, so $K_C/K_P=1/(RT)$); and $\Delta n_g<0$ gives $K_P<K_C$ (e.g. $2SO_2+O_2\rightleftharpoons 2SO_3$). …