At what temperature is the R.M.S. velocity of Hydrogen molecule equal to that of an oxygen molecule at \(47^{\circ} \mathrm{C}\) ? (Molecular weight of hydrogen \(=2\), Molecular weight of oxygen \(=32\) )

The r.m.s. velocity of a molecule is given by
\(
\begin{aligned}
\mathrm{C} &=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}} \\
\therefore \frac{\mathrm{C}_{\mathrm{H}}}{\mathrm{C}_{\mathrm{o}}} &=\sqrt{\frac{\mathrm{T}_{\mathrm{H}}}{\mathrm{T}_{\mathrm{o}}} \cdot \frac{\mathrm{M}_{\mathrm{o}}}{\mathrm{M}_{\mathrm{H}}}} \\
\therefore \mathrm{C}_{\mathrm{H}} &=\mathrm{C}_{\mathrm{o}} \\
\frac{\mathrm{T}_{\mathrm{H}}}{\mathrm{T}_{\mathrm{o}}} \cdot & \frac{\mathrm{M}_{0}}{\mathrm{M}_{\mathrm{H}}}=1, \quad \mathrm{~T}_{\mathrm{H}}=47+273=320 \\
\therefore \mathrm{T}_{\mathrm{H}} &=\frac{\mathrm{M}_{\mathrm{H}}}{\mathrm{M}_{\mathrm{o}}} \cdot \mathrm{T}_{\mathrm{o}}=\frac{2}{32} \cdot 320=20 \mathrm{~K}
\end{aligned}
\)