Pressure from Molecules

Add up the momentum delivered by every molecule bouncing off a wall.

10 minUpdated 22 October 2026pressurerms speedtranslational kinetic energy

Lesson at a glanceShow the key points

A molecule hitting a wall transfers 2mvₓ and returns every 2L/vₓ: average force mvₓ²/L.
Isotropy: v̄ₓ² = v̄²/3. So p = ⅓ρv̄² (24.1), pV = ⅓Mv̄² = ⅓Nmv̄² (24.2–24.3).
rms speed v_rms = √(v̄²) = √(3p/ρ) = √(3pV/M).
Total translational KE K = ½Mv²_rms = (3/2)pV (24.4).

Count the knocks of molecules on a wall, add up their momentum, and out comes the pressure of a gas.

One molecule, one wall

A gas fills a cubical box of side L. Take one molecule with velocity components vₓ, v_y, v_z. When it strikes the face A₁ perpendicular to x, vₓ reverses and the other components are unchanged (elastic collision). Its momentum changes by −2mvₓ, so the wall receives

\Delta p = 2mv_x

It then crosses to the opposite face and back, a distance 2L in x, before hitting A₁ again. So it strikes A₁ every 2L/vₓ seconds, that is vₓ/2L times per second. The average force it exerts on A₁ is

F = \frac{v_x}{2L}\cdot2mv_x = \frac{m}{L}v_x^2

All the molecules

Adding up, F = (m/L)Σvₓ². Since no direction is special, Σvₓ² = Σv_y² = Σv_z² = ⅓Σv². With N molecules, total mass M = mN and density ρ = M/L³, the pressure F/L² is

Pressure of an ideal gas(24.1–24.3)

p = \frac13\rho\overline{v^2}, \qquad pV = \frac13M\overline{v^2} = \frac13Nm\overline{v^2}

v̄² = Σv²/N is the mean square speed

Interactive

Pressure is molecules hitting the walls

Each molecule bounces elastically off the walls; every hit on the right wall delivers an impulse 2mvₓ. Averaged over time, the hits add up to a steady push: the pressure. Heat the gas (faster molecules) or squeeze it (more frequent hits) and the pressure rises. In a mixture at one temperature, heavy molecules move slower but carry the same average kinetic energy. (A 2D model; collisions between molecules are left out, as the theory allows.)

measured push on the right wallmeasuring…

kinetic theory predicts24.0

Temperature

T

300K

Width of the box (volume)1.00

The rms speed

The square root of the mean square speed is the root-mean-square (rms) speed:

rms speed

v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3p}{\rho}} = \sqrt{\frac{3pV}{M}}

Worked example

Oxygen at STP

Oxygen at 1 atm (1.0 × 10⁵ Pa) and 0°C has density 1.43 kg/m³. Find the rms speed of its molecules.

Translational kinetic energy

The total translational kinetic energy of the molecules is

(24.4)

K = \sum\tfrac12mv^2 = \tfrac12Mv_{rms}^2, \qquad K = \tfrac32pV

from pV = ⅓Mv²_rms; the average per molecule is ½mv²_rms

Worked example

Energy in a room's air

A room holds 60 m³ of air at 1.0 × 10⁵ Pa. What is the total translational kinetic energy of its molecules?

What to remember

1A molecule hitting a wall delivers 2mvₓ; it returns every 2L/vₓ.
2p = ⅓ρv̄²; pV = ⅓Mv̄² = ⅓Nmv̄².
3v_rms = √(3p/ρ).
4Translational KE K = ½Mv²_rms = (3/2)pV.

Check yourself

If the rms speed of the molecules doubles at fixed density, the pressure…

A gas at 2.0 × 10⁵ Pa fills 3.0 L. Its translational KE is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchMolecular kinetic theory: a simple derivationvt.physics PlayGas PropertiesPhET · CU Boulder ReadPressure, temperature and rms speedOpenStax University Physics Vol. 2 §2.2 ReadKinetic temperatureHyperPhysics

PreviousThe Kinetic Model Next Temperature & Molecular Speed

Pressure from Molecules

Add up the momentum delivered by every molecule bouncing off a wall.

10 minUpdated 22 October 2026pressurerms speedtranslational kinetic energy

Lesson at a glanceShow the key points

A molecule hitting a wall transfers 2mvₓ and returns every 2L/vₓ: average force mvₓ²/L.
Isotropy: v̄ₓ² = v̄²/3. So p = ⅓ρv̄² (24.1), pV = ⅓Mv̄² = ⅓Nmv̄² (24.2–24.3).
rms speed v_rms = √(v̄²) = √(3p/ρ) = √(3pV/M).
Total translational KE K = ½Mv²_rms = (3/2)pV (24.4).

Count the knocks of molecules on a wall, add up their momentum, and out comes the pressure of a gas.

One molecule, one wall

\Delta p = 2mv_x

F = \frac{v_x}{2L}\cdot2mv_x = \frac{m}{L}v_x^2

All the molecules

Adding up, F = (m/L)Σvₓ². Since no direction is special, Σvₓ² = Σv_y² = Σv_z² = ⅓Σv². With N molecules, total mass M = mN and density ρ = M/L³, the pressure F/L² is

Pressure of an ideal gas(24.1–24.3)

p = \frac13\rho\overline{v^2}, \qquad pV = \frac13M\overline{v^2} = \frac13Nm\overline{v^2}

v̄² = Σv²/N is the mean square speed

Interactive

Pressure is molecules hitting the walls

measured push on the right wallmeasuring…

kinetic theory predicts24.0

Temperature

T

300K

Width of the box (volume)1.00

The rms speed

The square root of the mean square speed is the root-mean-square (rms) speed:

rms speed

v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3p}{\rho}} = \sqrt{\frac{3pV}{M}}

Worked example

Oxygen at STP

Oxygen at 1 atm (1.0 × 10⁵ Pa) and 0°C has density 1.43 kg/m³. Find the rms speed of its molecules.

Translational kinetic energy

The total translational kinetic energy of the molecules is

(24.4)

K = \sum\tfrac12mv^2 = \tfrac12Mv_{rms}^2, \qquad K = \tfrac32pV

from pV = ⅓Mv²_rms; the average per molecule is ½mv²_rms

Worked example

Energy in a room's air

A room holds 60 m³ of air at 1.0 × 10⁵ Pa. What is the total translational kinetic energy of its molecules?

What to remember

1A molecule hitting a wall delivers 2mvₓ; it returns every 2L/vₓ.
2p = ⅓ρv̄²; pV = ⅓Mv̄² = ⅓Nmv̄².
3v_rms = √(3p/ρ).
4Translational KE K = ½Mv²_rms = (3/2)pV.

Check yourself

If the rms speed of the molecules doubles at fixed density, the pressure…

A gas at 2.0 × 10⁵ Pa fills 3.0 L. Its translational KE is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousThe Kinetic Model Next Temperature & Molecular Speed

One molecule, one wall#

All the molecules#

The rms speed#

Translational kinetic energy

Check yourself

Keep exploring

One molecule, one wall#

All the molecules#

The rms speed#

Translational kinetic energy

Check yourself

Keep exploring

One molecule, one wall

All the molecules

The rms speed

One molecule, one wall

All the molecules

The rms speed