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$K_1, K_2$ 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
Options:
(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
Solution:
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Verified Answer
The correct answer is:
$K_1=\frac{1}{K_2}=\frac{1}{\left(K_3\right)^2}$
$K_1=\frac{1}{K_2}=\frac{1}{\left(K_3\right)^2}$
(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)
$$
K_1=\frac{1}{K_2}=\frac{1}{\left(K_3\right)^2}
$$
$$
K_1=\frac{\left[\mathrm{NO}_2\right]^2}{\left[\mathrm{~N}_2\right]\left[\mathrm{O}_2\right]^2}
$$
(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)
$$
K_1=\frac{1}{K_2}=\frac{1}{\left(K_3\right)^2}
$$
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