Root-Mean-Square Speed
Not every molecule in a gas moves at the same speed. Some are practically still, a rare few are moving extraordinarily fast, and most sit somewhere in a broad spread in between. This lesson is about how to sensibly average a spread like that, and why the fastest handful of molecules matter far more than their small numbers suggest.
Why you can't just average the speeds
The previous lesson's key result, $\tfrac12m\langle c^2\rangle = \tfrac32kT$, involves $\langle c^2\rangle$ — the mean of the squares of the molecular speeds — not the square of the mean speed. These are genuinely different numbers, and mixing them up is a common error in this topic.
Root-mean-square speed, $c_{rms} = \sqrt{\langle c^2\rangle}$, is defined precisely to match the physics in $pV=\tfrac13Nm\langle c^2\rangle$ and $\tfrac12m\langle c^2\rangle = \tfrac32kT$ — the "correct" average speed to use whenever kinetic energy (which depends on speed squared) is involved, rather than the everyday arithmetic mean.
"Root-mean-square" describes a specific sequence: square every value, then take the mean of those squares, then take the square root of that mean. Doing the steps in a different order — for instance, squaring the mean speed instead of taking the mean of the squared speeds — gives a different (and for this physics, wrong) answer.
The Maxwell-Boltzmann speed distribution
Rather than every molecule sharing one speed, real gas molecules span a continuous range, described by the Maxwell-Boltzmann distribution. Plotted as number (or fraction) of molecules against speed, this distribution has a characteristic shape:
- It starts at zero (no molecules have exactly zero speed) and rises to a peak — the most probable speed, shared by the largest fraction of molecules.
- It falls back towards zero at high speed, but with a long tail — a small but non-zero fraction of molecules always have unusually high speeds, however high a threshold you pick.
- It's not symmetric: skewed, with the tail on the high-speed side stretching further than the low-speed side ever can, since speed can't go below zero but has no upper limit.
Because the distribution is skewed, its three natural "average" measures — most probable speed, mean speed $\bar c$, and root-mean-square speed $c_{rms}$ — are all different, always in this order: most probable speed $<$ mean speed $<$ r.m.s. speed. The r.m.s. speed sits furthest into the high-speed tail because, as the worked example showed, squaring speeds before averaging always weights the faster molecules more heavily.
How temperature changes the distribution
Raising temperature changes the Maxwell-Boltzmann curve in two connected ways: the peak shifts to a higher speed (consistent with $\tfrac12m\langle c^2\rangle = \tfrac32kT$ — average kinetic energy, and hence typical speed, rises with $T$), and the curve broadens and flattens (a wider spread of speeds becomes populated, and since the total area under the curve — the fixed total number of molecules — must stay the same, a broader base means a lower peak).
Key takeaways
- Mean square speed $\langle c^2\rangle$ (mean of the squares) is not the same as the square of the mean speed — always square first, then average, then (for r.m.s.) take the square root.
- Root-mean-square speed $c_{rms} = \sqrt{\langle c^2\rangle}$ is the speed measure that correctly matches kinetic-energy-based results like $pV=\tfrac13Nm\langle c^2\rangle$ and $\tfrac12m\langle c^2\rangle=\tfrac32kT$.
- The Maxwell-Boltzmann speed distribution is skewed (not symmetric), starting at zero, rising to a peak at the most probable speed, and trailing off with a long high-speed tail; most probable speed < mean speed < r.m.s. speed.
- Raising temperature shifts the distribution's peak to higher speed and broadens/flattens the curve, while the total area (total number of molecules) stays constant.