In Giancoli section 18-1, it is derived that if the average kinetic energy of a gas molecule is denote by $\bar{K}$ ,
$\begin{align}\bar{K} = \frac{1}{2} m v_{rms}^2 = \frac{3}{2}k T\end{align}$
This implies that

$\setcounter{equation}{1}\begin{align}v_{rms} = \sqrt{\frac{3 k T}{m}}\end{align}$
Therefore,
$\setcounter{equation}{2}\begin{align}\frac{v_{rms}\left(N_2 \right)}{v_{rms}\left(O_2 \right)} = \frac{\displaystyle\sqrt{\fr...$
The temperatures for the two gases are assumed to be the same, because they are both in the same mixture. Therefore,
$\setcounter{equation}{3}\begin{align}\frac{v_{rms}\left(N_2 \right)}{v_{rms}\left(O_2 \right)} = \sqrt{\frac{m\left(O_2 \righ...$

Solution to 2001 Problem 48