which ones will generally be moving faster?

The Root Mean Square Velocity

The root mean squared velocity of a molecule of a gas is defined as the square root of the average velocity square of every molecule.

If $u_1, u_2, u_3, ..., u_n$ be the velocity of $n$ molecules then the rms velocity is given as

$C=\sqrt{\frac{u_1^2+u_2^2+u_3^2+....+u_n^2}{n}}$

The pressure exerted by the random movement of molecules of a gas whose density is $\rho$ is

$p=\frac{1}{3}\rho C^2$

Let us consider that one mole of a gas occupies $V$ volume, therefore for one mole the pressure exerted by the gas is

$p=\frac{1}{3}\rho C^2=\frac{1}{3}\frac{M}{V}{C}^2$

$M$ the molecular weight of the gas. We get

$pV=\frac{1}{3}M{C}^2$

Then by ideal gas law equation for one mole

$$\begin{gathered} pV=\frac{1}{3}M{C}^2=RT \\ \Rightarrow C=\sqrt{\frac{3RT}{M}} \end{gathered}$$

$R$ the universal gas constant and $T$ is the temperature of the gas.

Let us consider the temperature of the container be $T$ and one mole each of $N_2$ and $O_2$ occupies the container. 

Given, the molecular weight of $N_2=M_N=28$ and

  the molecular weight of $O_2=M_O=32$

Therefore rms velocity of $N_2$

$C_N=\sqrt{\frac{3RT}{M_N}}=\sqrt{\frac{3RT}{28}}$

and rms velocity of $O_2$

$C_O=\sqrt{\frac{3RT}{M_O}}=\sqrt{\frac{3RT}{32}}$

Taking the ratio

$\frac{C_N}{C_O}=\frac{\sqrt{{\displaystyle \frac{3RT}{28}}}}{\sqrt{{\displaystyle \frac{3RT}{32}}}}=\sqrt{\frac{32}{28}}=1.07$

This gives

$C_N=1.07C_O$

This shows that rms velocity of nitrogen is greater and 1.07 times that of oxygen.