Result

Pressure from molecular motion

(T3.1.1) T3.1 · Molecular Model of Pressure →

P = \frac{1}{3} \frac{N m v ^{2}}{V}

Statement

Pressure expressed in terms of the number, mass, and mean-square speed of gas molecules.

Symbols and units

$P$	pressure	Pa
$N$	number of molecules	—
$m$	molecular mass	kg
$v^{2}$	mean-square speed	m²/s²

When it applies

An ideal gas: point-like molecules undergoing elastic collisions with the container walls.

Don't use when

Assumes an ideal gas of point-like molecules undergoing elastic collisions — not accurate where molecular volume or intermolecular forces matter.

Derivation

Each collision with a wall transfers momentum; averaging this transfer rate over all molecules and all directions of motion gives the pressure in terms of the mean-square speed:

P = \frac{1}{3} \frac{N m v ^{2}}{V}

Watch out for

The factor 1/3 comes from averaging the squared velocity component over three spatial dimensions — by symmetry, each direction carries one third of the total mean-square speed.