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Where Pressure Comes From: The Kinetic Model

The gas laws were found by experiment — but why should they be true? Here we derive pressure from scratch, molecule by molecule, and watch PV = nRT emerge from nothing but Newton's laws and a swarm of bouncing balls.

The rules of the game

To derive pressure we spell out the ideal-gas model precisely. (1) A gas is a huge number N of identical molecules, each of mass m. (2) They move in random directions with a spread of speeds. (3) They are so small that their own volume is negligible next to the container. (4) They exert no forces on each other except during collisions, which are perfectly elastic (no kinetic energy is lost). (5) Collisions with the walls are elastic too — a molecule bounces off with its speed unchanged.

The momentum kick of a single collision

Start with one molecule in a cubic box of side L, and focus on the wall facing the x-direction. When the molecule strikes that wall its x-velocity reverses from +v_x to -v_x, so its momentum changes by 2mv_x. The wall delivered that change; by Newton's third law the molecule delivers an equal and opposite kick to the wall.

\Delta p_x = 2 m v_x

The momentum handed to the wall in one bounce. Only the x-component matters for the x-wall; the y and z motions just slide the molecule along the wall.

How often does this molecule return to the same wall? It must cross to the far wall and back, a round trip of $2L$, taking a time \Delta t = 2L/v_x. Force is momentum delivered per unit time, so the average force this one molecule exerts on the wall is the kick divided by the time between kicks.

F_1 = \dfrac{\Delta p_x}{\Delta t} = \dfrac{2 m v_x}{2L/v_x} = \dfrac{m v_x^2}{L}

The average force from a single molecule on one wall. Notice it depends on the square of the velocity — a hint of the kinetic energy to come.

Adding up the whole swarm

One molecule gives a feeble, jittery force. But N of them, hammering the wall billions of times a second, add up to the smooth, steady push we call pressure. Follow the bookkeeping:

  1. Total force on the x-wall is the sum over all molecules: F = (m/L) Σ v_x², where the sum runs over every molecule's own v_x².
  2. Replace the sum by N times the average: Σ v_x² = N · ⟨v_x²⟩, where ⟨v_x²⟩ is the mean squared x-velocity.
  3. Motion is random, so no direction is special: ⟨v_x²⟩ = ⟨v_y²⟩ = ⟨v_z²⟩. Since v² = v_x² + v_y² + v_z², each one is a third of the whole: ⟨v_x²⟩ = ⟨v²⟩/3.
  4. Divide the total force by the wall area A = L² to get pressure, and use volume V = L³: P = F/L² = (m/L³)·N·⟨v_x²⟩ = (N m /V)·⟨v²⟩/3.

Pressure is a statistical average

Each molecular impact transfers a tiny packet of momentum 2mv_x to the wall. Countless impacts per second blur into the steady, uniform pressure a gauge reads. Pressure is not a force from any one molecule — it is the collective drumming of the whole swarm.

PV = \tfrac{1}{3} N m \, \overline{v^2}

The central result of kinetic theory: the macroscopic product PV equals one-third of N times the molecular mass times the mean-square speed. We derived it from Newton's laws alone.

Pause on what just happened. We started with balls bouncing under Newton's laws and arrived at a relation between the macroscopic quantities P and V. The mean-square speed \overline{v^2} carries the whole crowd's motion; its square root is the root-mean-square speed, the effective molecular speed we will meet next. Compare this equation with the experimental PV = Nk_BT and something wonderful is about to fall out — the true meaning of temperature.