$K_{1}, K_{2}$ and $K_{3}$ are the equilibrium constants of the following reactions (I), (II) and (III)…

$K_{1}, K_{2}$ and $K_{3}$ are the equilibrium constants of the following reactions (I), (II) and (III) respectively:
(I) $\mathrm{N}_{2}+2 \mathrm{O}_{2} ightleftharpoons 2 \mathrm{NO}_{2}$
(II) $2 \mathrm{NO}_{2} ightleftharpoons \mathrm{N}_{2}+2 \mathrm{O}_{2}$
(III) $\mathrm{NO}_{2} ightleftharpoons \frac{1}{2} \mathrm{~N}_{2}+\mathrm{O}_{2}$
The correct relation from the following is
  1. $K_{1}=\frac{1}{K_{2}}=\frac{1}{K_{3}}$
  2. $K_{1}=\frac{1}{K_{2}}=\frac{1}{\left(K_{3}ight)^{2}}$
  3. $K_{1}=\sqrt{K_{2}}=K_{3}$
  4. $K_{1}=\frac{1}{K_{2}}=K_{3}$

Solution

$\quad \mathrm{N}_{2}+2 \mathrm{O}_{2} \stackrel{\mathrm{K}_{1}}{ightleftharpoons} 2 \mathrm{NO}_{2}$
$K_{1}=\frac{\left[\mathrm{NO}_{2}ight]^{2}}{\left[\mathrm{~N}_{2}ight]\left[\mathrm{O}_{2}ight]^{2}}$...(i)
(II) $2 \mathrm{NO}_{2} \stackrel{\mathrm{K}_{2}}{ightleftharpoons} \mathrm{N}_{2}+2 \mathrm{O}_{2}$
$K_{2}=\frac{\left[\mathrm{N}_{2}ight]\left[\mathrm{O}_{2}ight]^{2}}{\left[\mathrm{NO}_{2}ight]^{2}}$ ...(ii)
(III) $\mathrm{NO}_{2} \stackrel{K_{3}}{ightleftharpoons} \frac{1}{2} \mathrm{~N}_{2}+\mathrm{O}_{2}$
$K_{3}=\frac{\left[\mathrm{N}_{2}ight]^{1 / 2}\left[\mathrm{O}_{2}ight]}{\left[\mathrm{NO}_{2}ight]}$
$\therefore\left(K_{3}ight)^{2}=\frac{\left[\mathrm{N}_{2}ight]\left[\mathrm{O}_{2}ight]^{2}}{\left[\mathrm{NO}_{2}ight]^{2}}$...(iii)
from equation (i), (ii) and (iii) $K_{1}=\frac{1}{K_{2}}=\frac{1}{\left(K_{3}ight)^{2}}$ *

Asked in: JEE-TOPICTESTS-CHEMISTRY

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