Among the following, identify the gas which has the greatest average speed of its molecules?
Among the following, identify the gas which has the greatest average speed of its molecules?
$0.5 \mathrm{~mol}$ of $\mathrm{O}_2$ at $500 \mathrm{~K}$
$0.2 \mathrm{~mol}^2 \mathrm{CO}_2$ at $400 \mathrm{~K}$
$1.0 \mathrm{~mol}$ of $\mathrm{He}$ at $200 \mathrm{~K}$
$0.4 \mathrm{~mol}$ of $\mathrm{NH}_3$ at $300 \mathrm{~K}$
Solution
The average molecular speed is inversely proportional to the square root of molar mass.
$v=\sqrt{\frac{8 R T}{\pi M}}$ i.e. $v \propto \frac{1}{\sqrt{M}}$
Among $\mathrm{O}_2, \mathrm{CO}_2$, He and $\mathrm{NH}_3$, He has very low molar mass. Therefore, it has greatest average velocity.
Order of molar mass $(\mathrm{g} / \mathrm{mol})$ is
$
\begin{array}{ccc}
\mathrm{CO}_2> & \mathrm{O}_2 < \mathrm{NH}_3>\mathrm{He} \\
44.01 & 32.0 & 17.03
\end{array}
$
Order of average velocity is $\mathrm{He}>\mathrm{NH}_3>\mathrm{O}_2>\mathrm{CO}_2$. Hence, helium has the greatest average speed of its molecules