${\mathrm{K}}_{\mathrm{C}}$for 

${\mathrm{H}}_2+{\mathrm{I}}_2\rightleftharpoons 2\mathrm{HI}$is ' ${\mathrm{K}}^{\text{'}}$, then for $\mathrm{HI}\rightleftharpoons 1/2{\mathrm{H}}_2+1/2{\mathrm{I}}_2$, it is

${\mathrm{H}}_2+{\mathrm{I}}_2\rightleftharpoons 2\mathrm{HI}$

Equilibrium constant -

$K_c=\frac{[HI{]}^2}{\left[H_2\right]\left[I_2\right]}=k$

$k_c$for

H I & $\rightleftharpoons \frac{1}{2}{H}_2$+

$\frac{1}{2}{I}_2 K^{\text{'}}$=

$$\begin{gathered} \frac{{\left[I_2\right]}^{1/2}{\left[H_2\right]}^{1/2}}{\left[HI\right]} \\ {\left(k^{\text{'}}\right)}^2 \end{gathered}$$=

$$\begin{gathered} {\left(\frac{1}{\frac{\left[HI\right]}{{\left[I_2\right]}^{1/2}{\left[H_2\right]}^{1/2}}}\right)}^2 \\ {K}^{{\text{'}}^2} \end{gathered}$$ & =$\frac{\frac{1}{[HI{]}^2}}{\left[I_2\right]\left[H_2\right]}$=

$$\begin{gathered} \frac{1}{K} \\ \text{ So, }{K}^{\text{'}} \end{gathered}$$ =$\frac{1}{\sqrt{K}}$

Hence the correct answer is (C) $\frac{1}{\sqrt{\mathrm{K}}}$.