Root Mean Square Speed (v_rms)

Definition. The root mean square speed, $v_{rms}$, is defined as the square root of the mean of the squared speeds of every molecule in the gas: $v_{rms}=\sqrt{\overline{v^2}}$. Rearranging equation (9.8), $kT=\tfrac13 m\overline{v^2}$, the mean square speed is

so the root mean square speed is

Writing this in terms of the universal gas constant $R$ and molar mass $M$ (using $N_Ak=R$ and $N_Am=M$), an equally common and more practical form is

What the formula tells you. $v_{rms}$ is directly proportional to $\sqrt{T}$ and inversely proportional to $\sqrt{M}$: (i) at a fixed temperature, lighter molecules move faster on average than heavier ones -- this is why hydrogen and helium have much higher $v_{rms}$ than oxygen or nitrogen at the same temperature; and (ii) raising the temperature of any gas always increases the rms speed of its molecules. Equation (9.6) can equivalently be written directly in terms of $v_{rms}$ as $P=\tfrac13 nmv_{rms}^2$, since $\overline{v^2}=v_{rms}^2$ by definition. It is also worth noting explicitly that $v_{rms}$ is not the same as the average speed -- the average speed of a gas turns out to be about 0.92 times the rms speed (see Section 9.2.6).

Worked illustration (Example 9.2). A room at $27^\circ$C ($T=300$ K) contains oxygen and hydrogen molecules in the ratio 3:1 ($M_{O_2}=32\times10^{-3}\ \text{kg mol}^{-1}$, $M_{H_2}=2\times10^{-3}\ \text{kg mol}^{-1}$, $R=8.32\ \text{J mol}^{-1}\text{K}^{-1}$).

(a) rms speeds: $v_{rms,O_2}=\sqrt{3\times8.32\times300/(32\times10^{-3})}\approx484\ \text{m s}^{-1}$, and $v_{rms,H_2}=\sqrt{3\times8.32\times300/(2\times10^{-3})}\approx1934\ \text{m s}^{-1}\approx1.93\ \text{km s}^{-1}$ -- since the molar mass of oxygen is 16 times that of hydrogen and $v_{rms}\propto1/\sqrt M$, hydrogen's rms speed is $\sqrt{16}=4$ times larger, matching $1934/484\approx4$. …