$A$ flask contains argon and chlorine in the ratio of $2: 1$ by mass. The temperature of the mixture is $27\,^{\circ}C$. Obtain the ratio of
$(i)$ average kinetic energy per molecule,and
$(ii)$ root mean square speed $v_{rms}$ of the molecules of the two gases.
Atomic mass of argon $= 39.9\,u$; Molecular mass of chlorine $= 70.9\,u$.

(A) The important point to remember is that the average kinetic energy (per molecule) of any ideal gas (whether monatomic like argon,diatomic like chlorine,or polyatomic) is always equal to $\frac{3}{2}k_{B}T$. It depends only on temperature and is independent of the nature of the gas.
$(i)$ Since argon and chlorine both have the same temperature in the flask,the ratio of average kinetic energy per molecule of the two gases is $1:1$.
$(ii)$ The root mean square speed is given by $v_{rms} = \sqrt{\frac{3k_{B}T}{m}}$,where $m$ is the mass of a molecule. Therefore,the ratio of $v_{rms}$ for argon to chlorine is:
$\frac{(v_{rms})_{Ar}}{(v_{rms})_{Cl}} = \sqrt{\frac{m_{Cl}}{m_{Ar}}} = \sqrt{\frac{M_{Cl}}{M_{Ar}}} = \sqrt{\frac{70.9}{39.9}} \approx \sqrt{1.776} \approx 1.33$.
Thus,the ratio is $1.33:1$.