$K_1, K_2$ 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) $\mathrm{N}_2+2 \mathrm{O}_2 \rightleftharpoons 2 \mathrm{NO}_2$ (II) $2 \mathrm{NO}_2 \rightleftharpoons \mathrm{N}_2+2 \mathrm{O}_2$ (III) $\mathrm{NO}_2 \rightleftharpoons \frac{1}{2} \mathrm{~N}_2+\mathrm{O}_2$ The correct relation from the following is
  1. $K_1=\frac{1}{K_2}=\frac{1}{K_3}$
  2. $K_1=\frac{1}{K_2}=\frac{1}{\left(K_3\right)^2}$
  3. $K_1=\sqrt{K_2}=K_3$
  4. $K_1=\frac{1}{K_2}=K_3$

Solution

(I) $\mathrm{N}_2+2 \mathrm{O}_2 \stackrel{K_1}{\rightleftharpoons} 2 \mathrm{NO}_2$ $ K_1=\frac{\left[\mathrm{NO}_2\right]^2}{\left[\mathrm{~N}_2\right]\left[\mathrm{O}_2\right]^2} $ (II) $2 \mathrm{NO}_2 \underset{K_2}{\rightleftharpoons} \mathrm{N}_2+2 \mathrm{O}_2$ $ K_2=\frac{\left[\mathrm{N}_2\right]\left[\mathrm{O}_2\right]^2}{\left[\mathrm{NO}_2\right]^2} $ (III) $ \begin{aligned} & \mathrm{NO}_2 \underset{K_3}{\rightleftharpoons} \frac{1}{2} \mathrm{~N}_2+\mathrm{O}_2 \\ & K_3=\frac{\left[\mathrm{N}_2\right]^{1 / 2}\left[\mathrm{O}_2\right]}{\left[\mathrm{NO}_2\right]} \\ & \therefore\left(K_3\right)^2=\frac{\left[\mathrm{N}_2\right]\left[\mathrm{O}_2\right]^2}{\left[\mathrm{NO}_2\right]^2} \end{aligned} $ $\therefore$ from equation (i), (ii) and (iii) $ K_1=\frac{1}{K_2}=\frac{1}{\left(K_3\right)^2} $

Asked in: JEE Main 2012 (07 May Online)

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