Last updated: 2026-06-15

RMS Speed

The molecules in the air around you are not still — they are racing about at hundreds of metres per second, colliding billions of times a second. Kinetic theory connects that microscopic chaos to a single number you can measure: temperature. This calculator returns the root-mean-square speed of gas molecules, along with their average and most probable speeds, from just the temperature and the molar mass.

Where the formula comes from

Kinetic theory treats a gas as countless tiny particles in random motion. Their average translational kinetic energy depends only on temperature: $\tfrac{1}{2}m\overline{v^2} = \tfrac{3}{2}kT$ per molecule. Rearranging for the average of the squared speed and taking the square root gives the root-mean-square speed. Written per mole, with the gas constant $R$ and molar mass $M$, it becomes $v_\text{rms} = \sqrt{3RT/M}$.

Formula

Here $R = 8.314\ \text{J/(mol·K)}$, $T$ is the absolute temperature and $M$ the molar mass in kg/mol. Enter the molar mass in the familiar g/mol; the calculator converts it.

Worked example

Air ($M = 28.97\ \text{g/mol}$) at room temperature, $T = 298.15\ \text{K}$:

Just over 500 m/s — faster than the cruising speed of an airliner, and that is only the average; individual molecules range from nearly stationary to well over a kilometre per second.

The three speeds

Because molecules share out a range of speeds, the Maxwell–Boltzmann distribution, three different averages are useful. The most probable speed $v_p$ marks the peak of the distribution, the average speed $v_\text{avg}$ is the plain mean, and the RMS speed $v_\text{rms}$ is the largest, since squaring gives extra weight to the fast molecules. Their ratio is always $v_p : v_\text{avg} : v_\text{rms} = 1 : 1.128 : 1.225$. The RMS speed is the one tied to kinetic energy, which is why it appears most often in physics.

Limitations

These results assume an ideal gas: point-like molecules with no forces between them except during collisions. Real gases deviate at high pressure or near condensation, where molecular size and attractions matter. The formula gives speeds, not velocities — the molecules move in all directions equally, so the average velocity is zero even though the average speed is large. It also describes only translational motion, not the rotation or vibration that also store energy in molecules with more than one atom.