$$\begin{gathered} 2{\mathrm{NO}}_2 \rightleftharpoons 2\mathrm{NO} + {\mathrm{O}}_2, \mathrm{K} = 1.6 \times {10}^{-12}; \\ \mathrm{NO} + \frac{1}{2}{\mathrm{O}}_2 \leftrightharpoons {\mathrm{NO}}_2, \mathrm{K}\text{'}=? \end{gathered}$$

$$\begin{gathered} 2{\mathrm{NO}}_2 \rightleftharpoons 2\mathrm{NO} + {\mathrm{O}}_2 \left(\mathrm{i}\right) \\ \mathrm{K} = 1.6 \times {10}^{-12} \\ \mathrm{NO} + \frac{1}{2}{\mathrm{O}}_2 \leftrightharpoons {\mathrm{NO}}_2 \left(\mathrm{ii}\right) \end{gathered}$$

Reaction (ii) is half of reaction (i)

$$\begin{gathered} \mathrm{K}=\frac{{\left[\mathrm{NO}\right]}^2 \left[{\mathrm{O}}_2\right]}{{\left[{\mathrm{NO}}_2\right]}^2} ...\left(\mathrm{i}\right) \\ \mathrm{K}\text{'}=\frac{ \left[{\mathrm{NO}}_2\right]}{\left[\mathrm{NO}\right] {\left[{\mathrm{O}}_2\right]}^{1/2}} ...\left(\mathrm{ii}\right) \end{gathered}$$

On multiplying (i) and (ii)

$$\begin{gathered} \mathrm{K}\times \mathrm{K}\text{'}=\frac{{\left[\mathrm{NO}\right]}^2 \left[{\mathrm{O}}_2\right]}{{\left[{\mathrm{NO}}_2\right]}^2} \times \frac{ \left[{\mathrm{NO}}_2\right]}{\left[\mathrm{NO}\right] {\left[{\mathrm{O}}_2\right]}^{1/2}} \\ =\frac{ \left[{\mathrm{NO}}_2\right] {\left[{\mathrm{O}}_2\right]}^{1/2}}{\left[{\mathrm{NO}}_2\right] } =\frac{1}{\mathrm{K}\text{'}} \\ \mathrm{K}\times \mathrm{K}\text{'}= \frac{1}{\mathrm{K}\text{'}}; \mathrm{K}=\frac{1}{\mathrm{K}{\text{'}}^2}; \mathrm{K}\text{'}=\frac{1}{\sqrt{\mathrm{K}}} \end{gathered}$$