Speed Distribution, Real Gases & Brownian Motion
Not every molecule has the average speed, not every gas is ideal, and pollen grains prove it all.
Lesson at a glanceShow the key points
- dN = 4πN(m/2πkT)^(3/2) v² e^(−mv²/2kT) dv (24.13); a long tail of fast molecules.
- Most probable speed v_p = √(2kT/m) (24.14); v_p < v̄ < v_rms.
- State: specified when its parameters are fixed; for an ideal gas, p and V fix it. Equation of state pV = nRT.
- Van der Waals: (p + a/V²)(V − b) = nRT (24.15).
- Brownian motion: random jiggling of tiny suspended particles from unbalanced molecular impacts; stronger at higher temperature and in less viscous liquids. Einstein's theory gave molecular sizes.
The rms speed is an average. Inside a gas, some molecules dawdle and a few race. Maxwell worked out the whole spread, and Brown saw its effects through a microscope.
Maxwell’s speed distribution
Oxygen at 300 K has an rms speed of about 480 m/s, but plenty of molecules are slower and plenty faster. If dN molecules have speeds between v and v + dv:
The peak of the curve is the most probable speed:
The long tail matters: a small fraction of molecules move many times faster than average. In nuclear fusion experiments, it is these fast particles that react.
Find the most probable, average and rms speeds of helium atoms (4.0 g/mol) at 400 K.
Thermodynamic state and equation of state
The relation between p, V and T is the equation of state: pV = nRT for an ideal gas. For a real gas the molecules’ size and attraction matter. Van der Waals proposed
Brownian motion
In 1827 the botanist Robert Brown watched pollen grains in water under a microscope. Though the water was still, the grains jiggled and wandered randomly. Water molecules, too small to see, hit each grain from all sides; at any instant the kicks don’t quite balance, so the grain lurches one way, then another.
- The particles must be light: a floating log is too massive to respond to molecular kicks.
- Brownian motion grows with temperature, and is livelier in liquids of low viscosity and density.
- Einstein’s 1905 theory of Brownian motion let the size of molecules be worked out, strong evidence that atoms are real.
- 1Maxwell's distribution gives how many molecules have each speed; most probable speed v_p = √(2kT/m).
- 2v_p < v̄ < v_rms; a long tail of fast molecules.
- 3State: fixed measurable parameters; equation of state relates p, V, T. Van der Waals: (p + a/V²)(V − b) = nRT.
- 4Brownian motion: random jiggling of small suspended particles from molecular impacts.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.