For the reaction, 

${\mathrm{N}}_{2\left(g\right)}+3{\mathrm{H}}_{2\left(g\right)}\rightleftharpoons 2{\mathrm{NH}}_{3\left(g\right)}\text{ at }400\mathrm{K},{\mathrm{K}}_p=41$

Find the value of ${\mathrm{K}}_{\mathrm{p}}$ for the following reaction 

$\frac{1}{2}{\mathrm{N}}_{2\left(\mathrm{g}\right)}+\frac{3}{2}{\mathrm{H}}_{2\left(\mathrm{g}\right)}\rightleftharpoons {\mathrm{NH}}_{3\left(\mathrm{g}\right)}$

${\mathrm{K}}_{\mathrm{p}}=\frac{{\left[{\mathrm{p}}_{{\mathrm{NH}}_3}\right]}^2}{\left[{\mathrm{P}}_{{\mathrm{N}}_2}\right]{\left[{\mathrm{p}}_{{\mathrm{H}}_2}\right]}^3}=41$

$\Rightarrow {\mathrm{p}}_{{\mathrm{NH}}_3}=\sqrt{{\mathrm{K}}_{\mathrm{p}}\left[{\mathrm{P}}_{{\mathrm{N}}_2}\right]{\left[{\mathrm{p}}_{{\mathrm{H}}_2}\right]}^3}$                      ….(i)

${\mathrm{K}}_{\mathrm{p}}^{\mathrm{\prime }}=\frac{\left[{\mathrm{p}}_{{\mathrm{NH}}_3}\right]}{{\left[{\mathrm{p}}_{{\mathrm{N}}_2}\right]}^{1/2}{\left[{\mathrm{p}}_{{\mathrm{H}}_2}\right]}^{3/2}}$                            …(ii)

Putting (i) in {ii), we get

${\mathrm{K}}_{\mathrm{p}}^{\mathrm{\prime }}=\frac{\sqrt{{\mathrm{K}}_{\mathrm{p}}\left[{\mathrm{P}}_{{\mathrm{N}}_2}\right]{\left[{\mathrm{p}}_{{\mathrm{H}}_2}\right]}^3}}{{\left[{\mathrm{p}}_{{\mathrm{N}}_2}\right]}^{1/2}{\left[{\mathrm{p}}_{{\mathrm{H}}_2}\right]}^{3/2}}=\sqrt{41}=6.4$