For two gases P and Q with molecular weights ${\mathrm{M}}_{\mathrm{P}} \mathrm{and} {\mathrm{M}}_{\mathrm{Q}}$ it is observed that at a certain temperature T the mean velocity of P is equal to the root and mean square velocity of Q. Thus the root mean square velocity of P and Q will be equal if

Average velocity = $\sqrt{\frac{8\mathrm{RT}}{\mathrm{\pi M}}}$

Root mean square velocity = $\sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}}$

$\Rightarrow$Average velocity of P at temperature $\mathrm{T} , {\mathrm{u}}_{\mathrm{av}\left(\mathrm{P}\right)}=$$\sqrt{\frac{8\mathrm{RT}}{{\mathrm{\pi M}}_{\mathrm{P}}}}$

Root mean square velocity of Q at temperature T

                                                                        =$\sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{\mathrm{Q}}}}={\mathrm{u}}_{\mathrm{RMS}\left(\mathrm{Q}\right)}$

$\Rightarrow$$\sqrt{\frac{8\mathrm{RT}}{{\mathrm{\pi M}}_{\mathrm{P}}}}$= $\sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{\mathrm{Q}}}}$           $\Rightarrow$$\sqrt{\frac{8\times 7}{22{\mathrm{M}}_{\mathrm{P}}}}=\sqrt{\frac{3}{{\mathrm{M}}_{\mathrm{Q}}}}$

$\Rightarrow$$\frac{56}{22 {\mathrm{M}}_{\mathrm{P}}}=\frac{3}{{\mathrm{M}}_{\mathrm{Q}}} \Rightarrow {\mathrm{M}}_{\mathrm{P}} = \frac{56}{66} {\mathrm{M}}_{\mathrm{Q}}$

Let the temperature of Q be maintained at T and that of P be changed to $T_2$

$\Rightarrow$ Now RMS of both P and Q should be equal

$\Rightarrow \sqrt{\frac{3{\mathrm{RT}}_2}{{\mathrm{M}}_{\mathrm{P}}}}= \sqrt{\frac{3\mathrm{RT}}{{\mathrm{M}}_{\mathrm{Q}}}} \Rightarrow \frac{{\mathrm{T}}_2}{{\mathrm{M}}_{\mathrm{P}}}=\frac{\mathrm{T}}{{\mathrm{M}}_{\mathrm{Q}}}$

Since ${\mathrm{M}}_{\mathrm{P}}=\frac{56}{66} {\mathrm{M}}_{\mathrm{Q}}$

$\Rightarrow \frac{66{\mathrm{T}}_2}{56{\mathrm{M}}_{\mathrm{Q}}}=\frac{\mathrm{T}}{{\mathrm{M}}_{\mathrm{Q}}}\Rightarrow 66{\mathrm{T}}_2=56\mathrm{T}$

${\mathrm{T}}_2=\frac{56}{66}\mathrm{T}; {\mathrm{T}}_2<\mathrm{T}$