v_rms = √(3RT/M) — Root-Mean-Square Molecular Speed

Root-mean-square molecular speed from kinetic theory — with a worked nitrogen example.

v_{rms} = \sqrt{\frac{3RT}{M}}

RMS speed equals the square root of three R T over molar mass.

What it means

Temperature is molecular motion: the rms speed grows with the square root of absolute temperature. Light molecules move faster than heavy ones at the same temperature — which is why hydrogen escapes Earth's atmosphere but oxygen stays.

Variables

Symbol	Name	Unit	Meaning
$v_rms$	RMS speed	m/s	Typical molecular speed
$R$	Gas constant	J/(mol·K)	Ideal gas constant
$T$	Temperature	K	Absolute temperature
$M$	Molar mass	kg/mol	Molar mass in kg per mole

Assumptions

Ideal gas in thermal equilibrium
Maxwell–Boltzmann speed distribution
T in kelvin; M in kg/mol

Worked example

Problem

What is the rms speed of nitrogen molecules (M = 0.028 kg/mol) at 300 K?

Given:

T = 300 K
M = 0.028 kg/mol (N₂)
R = 8.314 J/(mol·K)

Write the formula: $v_{rms} = \sqrt{3RT/M}$
Substitute: $v_{rms} = \sqrt{3 \times 8.314 \times 300 / 0.028}$
Calculate: $\sqrt{7482.6/0.028} = \sqrt{267{,}236} \approx 517\ \mathrm{m/s}$

≈ 517 m/s

Air molecules at room temperature zip around at ~517 m/s — faster than a jetliner. They don't get far because they collide billions of times per second, randomizing direction.