RMS speed of molecules

Kinetic theory begins with the pressure a swarm of molecules exerts on a container wall, derived from Newton's laws applied to elastic collisions:

where: $p$ is the gas pressure, $\rho$ is the gas density, and $\overline{c^2}$ is the mean of the squared molecular speeds (averaged over every molecule in the sample). Taking the square root of $\overline{c^2}$ defines the root-mean-square speed,

Two sub-points worth separating out before the main derivation:

• rms is not the plain average. Squaring every speed before averaging gives extra statistical weight to the fastest molecules, so $c_{rms}$ always comes out a little larger than the simple arithmetic-mean speed of the same sample. The ISC syllabus works only with $c_{rms}$; you are not asked to compute or compare the mean or most-probable speed.
• Two equivalent working formulas exist. $c_{rms}=\sqrt{3p/\rho}$ is the direct kinetic-theory result; combining it with the ideal gas law gives the far more usable $c_{rms}=\sqrt{3RT/M_0}$, worked out below.

The figure compares $c_{rms}$ for four common gases at the same room temperature, making the $1/\sqrt{M_0}$ dependence visible at a glance: hydrogen's molar mass is sixteen times smaller than oxygen's, so its rms speed is $\sqrt{16}=4$ times larger, exactly the pattern the bars show. Notice, too, that all four bars sit in the hundreds-to-thousands-of-metres-per-second range — comparable to rifle-bullet speeds — even though the gas as a whole (say, the air in a room) is not visibly rushing anywhere, because the molecules' motion is random in direction and constantly interrupted by collisions, not a steady drift.

Effect of temperature and pressure. $c_{rms}\propto\sqrt{T}$, so rms speed rises with absolute temperature — faster molecules carry more kinetic energy, matching the kinetic interpretation of temperature. At constant temperature, $c_{rms}$ is independent of pressure: compressing a gas raises $\rho$ in exact step with $p$, so $\sqrt{3p/\rho}$ is unchanged. And $c_{rms}\propto 1/\sqrt{M_0}$, which is why hydrogen and helium move far faster than oxygen or carbon dioxide at the same temperature, and why Earth retains its heavier atmospheric gases while light hydrogen slowly escapes over geological time.

Find the rms speed of oxygen molecules ($M_0 = 32\times10^{-3}\ \text{kg/mol}$) at $T=300\ \text{K}$.

Step 1. Write the formula: $c_{rms} = \sqrt{3RT/M_0}$.

Step 2. Substitute values:

Step 3. Evaluate: $c_{rms} \approx \mathbf{483.5\ m/s}$. Oxygen molecules at room temperature move at roughly 480 m/s on average — faster than the speed of sound in air, even though sound itself travels a little slower because it propagates via collisions between molecules, not via any single molecule's flight path. The mismatch is a useful sanity check to remember: a sound wave is a coordinated disturbance passed molecule to molecule, so it necessarily travels slower than the individual molecules' own random thermal speed.

Competitive stretch

Hydrogen gas at $27\,^\circ\text{C}$ (that is, $T_1=300\ \text{K}$) has rms speed $c_{rms,1}$. At what temperature does its rms speed double?

Step 1. Since $M_0$ is unchanged, $c_{rms}\propto\sqrt{T}$, so $c_{rms,2}/c_{rms,1}=\sqrt{T_2/T_1}$.

Step 2. Setting $c_{rms,2}=2c_{rms,1}$ gives $2=\sqrt{T_2/300}$, so $T_2=4\times300=\mathbf{1200\ K}$ ($927\,^\circ\text{C}$) — quadrupling the absolute temperature, not doubling it, because of the square-root dependence.

Cross-check. Compute both rms speeds directly with $M_0=2\times10^{-3}\ \text{kg/mol}$ instead of relying on the ratio: at $300\ \text{K}$, $c_{rms,1}=\sqrt{3(8.314)(300)/(2\times10^{-3})}\approx 1934\ \text{m/s}$; at $1200\ \text{K}$, $c_{rms,2}=\sqrt{3(8.314)(1200)/(2\times10^{-3})}\approx 3868\ \text{m/s}$, which is indeed double $c_{rms,1}$. This confirms the proportionality shortcut without needing to trust it blindly — a useful habit whenever a problem gives you a ratio rather than asking for an absolute number.

• Assuming rms speed doubles when temperature in degrees Celsius doubles — the $c_{rms}\propto\sqrt{T}$ relation only holds with $T$ in kelvin, never Celsius, since only absolute temperature has physical meaning as a measure of average molecular kinetic energy.
• Believing that compressing a gas at constant temperature speeds its molecules up — it does not, since $\rho$ rises in step with $p$, leaving $c_{rms}=\sqrt{3p/\rho}$ unchanged; only heating the gas, not squeezing it, raises molecular speed.
• Confusing rms speed with average speed or most-probable speed — these are three genuinely different statistical averages of the same underlying speed distribution; the ISC syllabus needs only $c_{rms}$, so resist the temptation to memorise formulas for the other two.
• Using molar mass in grams instead of kilograms in $c_{rms}=\sqrt{3RT/M_0}$ — since $R=8.314\ \text{J/mol}\,\cdot\,\text{K}$ is already in SI units, $M_0$ must be in kg/mol, or the computed speed comes out too large by a factor of about 31.6 (the square root of 1000).

Ask the tutor: want to walk through why pressure alone can't change rms speed, or practice comparing rms speeds across different gases at the same temperature?