The following concentrations were observed at 500K for the...

The equilibrium constant of concentration (denoted by ${K}_{c}$) of a chemical reaction at equilibrium can be defined as the ratio of the concentration of products to the concentration of the reactants, each raised to their respective stoichiometric coefficients.

The equilibrium reaction is 

${N}_{2\left(g\right)} + 3{H}_{2\left(g\right)}\mathop{⇋}_{}^{}2{NH}_{3\left(g\right)}$

${K}_{c} = \dfrac{{\left[{NH}_{3}\right]}^{2}}{{\left[{H}_{2}\right]}^{3}\left[{N}_{2}\right]}$

$= \dfrac{{\left(\dfrac{3}{2} \times {10}^{- 2}\right)}^{2}}{{\left(3 \times {10}^{- 2}\right)}^{3} \times 2 \times {10}^{- 2}}$

$= \dfrac{{3}^{2} \times {10}^{- 4}}{{2}^{2} \times {3}^{3} \times {10}^{- 6} \times 2 \times {10}^{- 2}}$

$= \dfrac{1}{4 \times 2 \times 3} \times {10}^{4} = \dfrac{10000}{24} = 416 . 6 \approx 417$