1.1 Air at rest

Stand in a quiet room. The air around you is not still, not really — every molecule is moving, fast, in some direction, colliding with its neighbours millions of times per second. What is still is the macroscopic state of that air: averaged over millimetres and microseconds, the pressure, density, and temperature are uniform and unchanging. This averaged state is equilibrium, and it is what we mean when we say “air at rest.”

Three macroscopic numbers describe equilibrium air:

• Pressure $p_0$ — the average force per unit area the molecules exert on any surface. At sea level, atmospheric pressure is about $101\,325$ Pa. That is roughly $10\,000$ kilograms of force on every square metre of you, balanced by an equal $10\,000$ kg pushing the other way from the air on your other side.
• Density $\rho_0$ — the mass per unit volume. Dry air at $20\,$°C has $\rho_0 \approx 1.204$ kg/m³. A cubic metre of air weighs about as much as a small bag of sugar.
• Temperature $T_0$ — a measure of the average kinetic energy of the molecules; for air at room temperature, $T_0 \approx 293\,$K (20°C).

These three are not independent. For an ideal gas they are tied by the ideal-gas law

where $n = N/V$ is the molecular number density, $k_B \approx 1.38 \times 10^{-23}\,$J/K is Boltzmann’s constant, $R = N_A k_B$ is the universal gas constant, and $M$ is the molar mass. For air, treating it as a single gas, $M \approx 29\,$g/mol.

Why “ideal”?

The ideal-gas law assumes the molecules occupy negligible volume and interact only by elastic collisions. At sea level and room temperature this is an excellent approximation: the molecules of air spend almost all of their time in flight between collisions, and the mean free path (about $7 \times 10^{-8}\,$m) is much larger than the molecular diameter (about $3 \times 10^{-10}\,$m). The gas is, on the molecular scale, mostly empty space.

What equilibrium does not mean

It does not mean the molecules are stationary. They are moving at thermal speeds of around $500\,$m/s — comparable to the speed of sound, which is no coincidence. We will see in the next lesson exactly what those molecular motions look like.

It does not mean the pressure is exactly $101\,325$ Pa everywhere all the time. There are tiny statistical fluctuations: in any small volume the molecular count, and therefore the pressure, jitters by a small amount on a fast timescale. We will encounter these fluctuations again in lesson 1.3 (Brownian motion) and again in chapter 10 (relaxation absorption).

What equilibrium does mean is that any coherent deviation from these averages — coherent across many molecules, sustained over many collision times — is not present. Sound is precisely such a coherent deviation. To make a sound, something must briefly displace the equilibrium and then let it relax.

We turn to the molecular picture next.