IB Physics SLTopic B.3 — The Behaviour of GasesPaper 1 & 2Kinetic Theory Derivation · P = ⅓ρv²~9 min read
Deriving the Kinetic Theory Equation
Gas pressure feels like a single, steady number — but it’s really the statistical average of billions of individual molecular collisions. Follow just one molecule bouncing around a box, and that single collision builds all the way up to a formula linking pressure to density and speed.
📘 What you need to know
Gas pressure arises because molecules colliding with the container walls each exert a tiny force, and the sum of countless collisions produces one uniform pressure.
The derivation tracks a single molecule bouncing between two opposite faces of a cube, then scales the result up to N molecules moving in three dimensions.
The final result is the kinetic theory of gases equation: P = ⅓ρv².
You won’t be asked to reproduce this derivation from memory in an exam, but you do need to follow the physics reasoning behind each step and be able to use the final equation.
Setting Up the Model
Picture a single molecule of mass m, travelling at speed v, trapped inside a cube-shaped box with sides of length l. To keep the physics simple, imagine it moving parallel to one edge of the cube — bouncing directly back and forth between two opposite faces, colliding elastically with each one in turn.
A single molecule bounces elastically between two opposite faces of the cube, travelling a distance l each way — a round trip of 2l per collision with the same wall.
The Derivation, Step by Step
The full derivation builds up in six stages — each one uses a single piece of physics you already know.
🧮 From one collision to a full equation
Change in momentum per collision. The collision is elastic, so the molecule rebounds with the same speed but reversed direction — its momentum flips from +mv to −mv.
Δp = −mv − (mv) = −2mv
By Newton’s third law, the force the molecule exerts on the wall is equal and opposite to the force the wall exerts on it.
Time between collisions. To hit the same wall twice, the molecule has to travel to the opposite face and back — a round trip of distance 2l at speed v.
Δt = 2l ÷ v
Average force from one molecule. Force is the rate of change of momentum, so dividing the momentum change by the time between collisions gives the average force this one molecule exerts on the wall.
F = 2mv ÷ (2l ÷ v) = mv² ÷ l
Pressure from N molecules. Dividing by the area of one face, l², gives the pressure from a single molecule; multiplying by N molecules scales it up.
P = Nmv² ÷ l³
Accounting for three dimensions. Real molecules move in every direction, not just along one axis. Splitting speed into components (v² = vx² + vy² + vz²) and noting no direction is special (vx² = vy² = vz²) gives:
vx² = ⅓v²
Introducing density. Since l³ is the volume V of the cube, and the density ρ = Nm ÷ V (total mass of all N molecules over volume), substituting both gives the final result.
P = ⅓ρv²
The elastic collision reverses the molecule’s velocity from v to −v. By Newton’s third law, the force the molecule exerts on the wall is equal and opposite to the force the wall exerts on it.
Kinetic theory of gases equationP = ⅓ρv²
where P is the pressure of the gas (Pa), ρ is the density of the gas (kg m⁻³), and v² is the mean square speed of the molecules (m² s⁻²) — the average of the square of each molecule’s speed, since not every molecule moves at exactly the same rate.
Quick recap: one molecule’s momentum change, divided by the time between its collisions, gives a force; scaling up to N molecules moving in three dimensions and introducing density turns that into P = ⅓ρv².
WE 1
An ideal gas has a density of 3.8 kg m⁻³. The molecules have a mean square speed corresponding to a speed of 650 m s⁻¹. Calculate the pressure of the gas.
An ideal gas has a pressure of 8.5 × 10⁵ Pa and a density of 5.2 kg m⁻³. Determine the mean square speed of the gas molecules.
Step 1 — Rearrange for speedP = ⅓ρv² → v = √(3P ÷ ρ)
Step 2 — Substitutev = √[(3 × 8.5 × 10⁵) ÷ 5.2]v ≈ 700 m s⁻¹ (3 s.f.)This is a mean square speed, not the speed of any one molecule — real molecules move at a whole range of speeds around this average.
💡 Top tips
You need to understand the physics behind each step of this derivation, but you won’t be asked to reproduce it word-for-word — focus your revision on being able to use the final equation confidently.
The “mean square speed” in P = ⅓ρv² is an average — it’s shorthand for treating a gas as if every molecule moved at one representative speed.
Remember ρ is density, not the Greek letter for anything else in this equation — double-check you haven’t confused it with resistivity or another ρ from a different topic.
⚠ Common mistakes
Forgetting to square the speed before substituting — it’s v², not v, in the final equation.
Mixing up ⅓ and 3 when rearranging for speed — always double-check the rearrangement algebraically rather than guessing.
Treating the derivation’s intermediate step, P = Nmv² ÷ l³, as the final answer — that version still assumes every molecule moves along a single axis, before the 3D correction is applied.
Up next: Average Kinetic Energy of a Molecule — where we connect this equation to temperature itself, and see exactly why hotter gases have faster-moving molecules.
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Book a free session and we’ll work through the kinetic theory equation and its applications until they’re second nature.
