Write an expression for kinetic energy of gas molecule.

The kinetic gas equation is given by $\mathrm{PV}=\frac{1}{3}{\mathrm{mnU}}_{\mathrm{rms}}^2$
for 1 mole gas n = N, mN = M
$\begin{array}{l}\mathrm{PV}=\frac{1}{3}{\mathrm{MV}}_{\mathrm{rms}}^2(\mathrm{M}=\text{ molar mass })\\ \mathrm{PV}=\frac{2}{3}\left[\frac{1}{2}{\mathrm{MV}}_{\mathrm{rms}}^2\right]\\ =\frac{2}{3}{\mathrm{E}}_{\mathrm{k}}\dots .\end{array}$
E_k= kinetic energy of one mole of a gas for 1 mole of a gas ideal gas equation is
PV = RT .... (2)
from 1 & 2 we have
$\frac{2}{3}{\mathrm{E}}_{\mathrm{k}}=\mathrm{RT};{\mathrm{E}}_{\mathrm{k}}=\frac{3}{2}\mathrm{RT}$
For n moles of a gas ${\mathrm{E}}_{\mathrm{k}}=\frac{3}{2}\mathrm{nRT}$