2.1 The molecular picture and pressure

A gas is a swarm of molecules in ceaseless motion. Its macroscopic state variables — pressure, temperature, density — are statistical summaries of that motion, and kinetic theory is the bridge that recovers each of them from the mechanics of the molecules. This first lesson recovers pressure as the rate at which molecules deliver momentum to a wall.

Pressure as momentum flux

Take a gas of $N$ identical point molecules of mass $m$ in a box of volume $V$ , each with a velocity $\mathbf v$ drawn from some isotropic distribution. A molecule that strikes a wall and rebounds delivers an impulse to it; the steady stream of such impulses, averaged over the enormous number of molecules and over time, is a constant force, and that force per unit area is the pressure. Pressure is the time-averaged rate at which the wall receives momentum per unit area — the impulse–momentum theorem applied collision by collision.

▶ p = ⅓ n m ⟨v²⟩ Derivation

Consider the wall at $x = L$ of area $A$ , with box volume $V = AL$ , and write $n = N/V$ for the number density. A molecule hitting this wall with $x$ -component $v_x > 0$ rebounds elastically to $-v_x$ , delivering an impulse $2 m v_x$ .

Let $f(v_x)\,dv_x$ be the fraction of molecules with $x$ -velocity in $[v_x, v_x+dv_x]$ . Of those moving toward the wall ( $v_x>0$ ), the ones within a distance $v_x$ per unit time reach it, so the number striking area $A$ per unit time is $\tfrac12\, n\, A\, |v_x|\, f(v_x)\,dv_x$ — the factor $\tfrac12$ keeping only the half moving toward the wall. Pressure is the total momentum delivered per unit time per unit area,

p \;=\; \int_{-\infty}^{\infty} (2 m v_x)\,\tfrac12\, n\, |v_x|\, f(v_x)\, dv_x \;=\; n m \langle v_x^2\rangle.

For an isotropic distribution the three components share the speed equally, $\langle v_x^2\rangle = \langle v_y^2\rangle = \langle v_z^2\rangle = \tfrac13\langle v^2\rangle$ , so

p \;=\; \tfrac13\, n\, m\, \langle v^2\rangle. \;✓

The factor $\tfrac13$ is the three Cartesian directions sharing the molecular motion equally; that isotropy is the kinetic statement of equilibrium.

Number of particles N = 50 Characteristic speed v₀ = 2.00 ≈ √(k_BT/m)

Each particle hitting the right wall delivers an impulse 2m|v_x| (elastic bounce). Summing these over time and dividing by wall length and elapsed time gives the measured pressure. The kinetic theory predicts p = nm⟨v_x²⟩ — the two numbers agree to within statistical noise.

The simulation accumulates the impulse delivered to the right wall and computes the pressure; it tracks the theoretical $n m\langle v_x^2\rangle$ to within statistical noise. Speed the molecules up and the impulse rate rises; double the number and the pressure doubles.

Pressure already contains the temperature

Rewrite the result using the mean molecular kinetic energy $\langle\varepsilon\rangle = \tfrac12 m\langle v^2\rangle$ :

p V \;=\; \tfrac23\, N\, \langle\varepsilon\rangle.

Set this beside the ideal-gas law $pV = N k_B T$ , an experimental fact about dilute gases, and the two match only if

\boxed{\;\langle\varepsilon\rangle \;=\; \tfrac32\, k_B T.\;}

So the mean translational kinetic energy of a molecule is fixed by the temperature alone — the same $\tfrac32 k_B T$ for a helium atom and a nitrogen molecule at the same $T$ . The next lesson takes this as the entry to temperature and equipartition.

⏳ The history — From Bernoulli's bouncing balls to Boltzmann's H-theorem

Daniel Bernoulli, in Hydrodynamica (1738), gave the first kinetic derivation of pressure: he modelled a gas as a swarm of point particles bouncing off the walls of a container and recovered $p \propto v^2$ from rate-of-momentum arguments alone. The result was a century ahead of its time; chemistry was still pre-Daltonian and the reality of atoms was philosophically suspect.

Modern kinetic theory dates to the 1850s and 1860s. Rudolf Clausius (1857) gave the first rigorous derivation of $p = \tfrac13 n m\langle v^2\rangle$ and introduced the mean free path. James Clerk Maxwell (1860) wrote down the equilibrium velocity distribution. Ludwig Boltzmann (1872) gave a dynamical derivation through his H-theorem, showing that an arbitrary initial distribution evolves toward the Maxwell form under collisions.

The molecular reality of gases was disputed for a further generation. Einstein’s 1905 theory of Brownian motion and Perrin’s confirming measurements (1908) settled it: atoms are real, kinetic theory is exact in its classical limit, and the macroscopic gas laws are its statistical consequence.

Read the original: Introductio in analysin infinitorum (Leonhard Euler, 1748) — Latin