Where Pressure Comes From
Where Pressure Comes From
- A tyre pushes back because molecules hammer its walls. Turning that sentence into is the most-derived result in Paper 4.
The three-mark story
- The qualitative chain, each link a mark (9702/42/O/N/24 Q4(b)): molecules collide with the wall and rebound, so their momentum changes; the wall causes that change, so by Newton's third law the molecules exert a force on the wall; billions of collisions spread over the wall's area make a steady pressure = force / area.
The derivation
- Set-up: one molecule of mass moving at speed (not the speed of light!) straight between opposite walls of a cube of side (9702/41/O/N/23 Q3(a)(ii)).
- Each bounce off wall A is elastic, so the velocity reverses: momentum change = .
- Between hits on wall A the molecule crosses the box and returns: distance , so time .
- Average force on the wall = rate of change of momentum = .
- Pressure = force / area of the wall = .
- Real gas: molecules with different speeds — replace by the average . Only a third of the motion is towards this pair of walls (three perpendicular directions), giving .
Symbols
- = number of molecules (—)
- = mass of one molecule (kg)
- = mean of the squares of the molecular speeds (m² s⁻²)
Common mistake
is the average of the squares, not the square of the average. Square every speed first, then average. In the derivation, the mark most often lost is the justification of the : say explicitly that molecular motion divides equally between three perpendicular directions.
Your turn— tap to reveal the worked answer (coursebook Q4, Ch22)
An oxygen molecule moves at 400 m s−¹ inside a spherical container of diameter 0.10 m. Estimate how many times per second it hits the walls.
Between hits it crosses about 0.10 m... twice per round trip of 0.20 m, so hits ≈ times per second. The enormous collision rate is why pressure feels perfectly steady.