If the absolute temperature of a gas is increased 5 times, the r.m.s. velocity of the gas molecules will be
If the absolute temperature of a gas is increased 5 times, the r.m.s. velocity of the gas molecules will be
5 times
10 times
25 times
$\sqrt{5}$ times
Solution
Roots mean square velocity, $\mathrm{v}_{\mathrm{rms}}=\sqrt{\frac{3 R T}{M}}$
Initial rms velocity, $\mathrm{v}_{\mathrm{rms}} \propto \sqrt{T}$
$\text { suppose, } \mathrm{v}_{r m s_1}=\mathrm{v}_{r m s} \text { and } \mathrm{T}_1=\mathrm{T}$
$\begin{aligned}
& \mathrm{v}_{r m s_2}=? \text { and } T_2=5 T \\
& \therefore \frac{\mathrm{v}_{r m s_2}}{r_{r m s_1}}=\sqrt{\frac{T_2}{T_1}}=\sqrt{\frac{5 T}{T}} \\
& \therefore \mathrm{v}_{r m s_2}=\sqrt{5} \mathrm{v}_{\mathrm{rms}}
\end{aligned}$