At what temperature is the R.M.S. velocity of Hydrogen molecule equal to that of an oxygen molecule at…
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 $=32$ )
$80 \mathrm{~K}$
$20 \mathrm{~K}$
$40 \mathrm{~K}$
$60 \mathrm{~K}$
Solution
The r.m.s. velocity of a molecule is given by
$\begin{aligned}
\mathrm{C} &=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}} \\
\therefore \frac{\mathrm{C}_{\mathrm{H}}}{\mathrm{C}_{\mathrm{o}}} &=\sqrt{\frac{\mathrm{T}_{\mathrm{H}}}{\mathrm{T}_{\mathrm{o}}} \cdot \frac{\mathrm{M}_{\mathrm{o}}}{\mathrm{M}_{\mathrm{H}}}} \\
\therefore \mathrm{C}_{\mathrm{H}} &=\mathrm{C}_{\mathrm{o}} \\
\frac{\mathrm{T}_{\mathrm{H}}}{\mathrm{T}_{\mathrm{o}}} \cdot & \frac{\mathrm{M}_{0}}{\mathrm{M}_{\mathrm{H}}}=1, \quad \mathrm{~T}_{\mathrm{H}}=47+273=320 \\
\therefore \mathrm{T}_{\mathrm{H}} &=\frac{\mathrm{M}_{\mathrm{H}}}{\mathrm{M}_{\mathrm{o}}} \cdot \mathrm{T}_{\mathrm{o}}=\frac{2}{32} \cdot 320=20 \mathrm{~K}
\end{aligned}$