For the reaction $N_{2}\left ( g \right )+O_{2}\left ( g \right )\rightleftharpoons 2NO\left ( g \right )$, the equilibrium constant is $K_{1}$. The equilibrium constant is $K_{2}$ for the reaction$2NO\left ( g \right )+O_{2}\rightleftharpoons 2NO_{2}\left ( g \right )$What is K for the reaction$NO_{2}(g)\rightleftharpoons \frac{1}{2}N_{2}(g)+O_{2}(g)?$ Option 1) $1/\left ( 2K_{1}K_{2} \right )$ Option 2) $1/\left ( 4K_{1}K_{2} \right )$ Option 3) $[1/K_{1}K_{2}]^{\frac{1}{2}}$ Option 4) $1/\left ( K_1 K_2 \right )$

As we discussed in concept

Law of Chemical equilibrium -

At a given temperature, the product of concentration of the reaction products raised to the respective stoichiometric coefficient in the balanced chemical equation divided by the product of concentration of the reactants raised to their individual stoichiometric coefficients has a constant value.

- wherein

$aA+bB\rightleftharpoons cC+dD$

$K_{c}=\frac{[C]^{c\:[D]^{d}}}{[A]^{a}\:[B]^{b}}$

$[A],\:[B],\:[C]\:[D]$

are equilibrium concentration

$N_{2}+O_{2}\:\rightleftharpoons 2NO\:\:\:K_{1}$

$2NO+O_{2}\:\rightleftharpoons 2NO_{2}\:\:\:K_{2}$

$NO_{2}\:\rightleftharpoons \frac{1}{2}N_{2}+O_{2}\:\:\:K$

$K_{1}=\frac{[NO]^{2}}{[N_{2}][O_{2}]}\:,\:K_{2}=\frac{[NO_{2}]^{2}}{[NO]^{2}[O_{2}]}$

$K=\frac{[N_{2}]^{\frac{1}{2}}.[O_{2}]}{[NO_{2}]}=\sqrt{\frac{[N_{2}][O_{2}]\times [NO]^{2}[O_{2}]}{[NO]^{2}\times [NO_{2}]^{2}}}$

$K=\sqrt{\frac{1}{K_{1}K_{2}}}$

Option 1)

$1/\left ( 2K_{1}K_{2} \right )$

This option is incorrect.

Option 2)

$1/\left ( 4K_{1}K_{2} \right )$

This option is incorrect.

Option 3)

$[1/K_{1}K_{2}]^{\frac{1}{2}}$

This option is correct.

Option 4)

$1/\left ( K_1 K_2 \right )$

This option is incorrect.

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